text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Pigments can be separated by chromatography. Pigments which appear to be of a uniform colour can in reality be composed of several different-coloured substances. A further segregation of the prepared leaf pigment and a felt-tipped pen ink are shown in this experiment. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,587 |
{"url":"https:\/\/tex.meta.stackexchange.com\/questions\/4579\/how-to-write-unicode-letter-in-tex-stackexchange-com","text":"# How to write unicode letter in tex.stackexchange.com?\n\nSorry for a very layman question. After trying for more than hour, I didn't succeed in writing\\unicode{f817}, which is a x with dot above and dot below. How we may write HERE in question ?\n\nYou can use HTML entities:\n\nThe above was typed as \n\nHowever, glyphs in the Private Use Area (PUA), that is from U+E000 to U+F8FF don't have a predefined meaning, so the output will be unpredictable. In my case, I see a \u201ct with bar above\u201d, but your browser could show something else (or nothing at all).\n\nThe entity method is useful for characters outside the PUA, when you just know their code point and have no practical input method available.\n\nIn order to get an \u201cx with dots above and below\u201d you can use a combining character:\n\n\u1e8b\u0323\n\nI typed the above by the combination U+1E8B U+0323, that is\n\nLATIN SMALL LETTER X WITH DOT ABOVE and COMBINING DOT BELOW\n\nYou can even use entities; the following was typed ẋ̣:\n\n\u1e8b\u0323\n\n\u2022 According to a quick search, this char is in the PUA so could display as anything! (I get the 'unknown char' symbol looking at your input here). \u2013\u00a0Joseph Wright Aug 11 '14 at 10:41\n\u2022 I see it as square box. IS something wrong with my browser or what ? \u2013\u00a0kaka Aug 11 '14 at 10:41\n\u2022 @kaka Like I said, font coverage is likely to be the issue here. I'm not sure this is even a standard Unicode char (it seems to be in the 'Private Use Area'), so 'all bets are off'. \u2013\u00a0Joseph Wright Aug 11 '14 at 10:42\n\u2022 @kaka No, there's nothing wrong. The particular code point is in the Private Use Area, so its output is not predictable. \u2013\u00a0egreg Aug 11 '14 at 10:42\n\u2022 so, how to put a dot below a letter? I know \\dot puts the dot above the letter , but how to put below? \u2013\u00a0kaka Aug 11 '14 at 10:46\n\u2022 @kaka \\dot puts a dot above a letter in TeX: it's got nothing to do with showing the accent in a question on the site (where you need something a web browser can render). As I say, you're likely to need an image here. \u2013\u00a0Joseph Wright Aug 11 '14 at 10:48\n\u2022 @kaka This website does not turn the LaTeX rendering on because we want to discuss about LaTeX commands and would like to see the raw input. So things like \\dot{} etc. won't work. However, Stackexchange network supports unicode input formatting as much as it is implemented. Some of the mappings are left open for customized input so they can show different characters depending on who put what on the so-called PUA. \u2013\u00a0percusse Aug 11 '14 at 10:49\n\u2022 @percusse This has nothing to do with LaTeX rendering. \u2013\u00a0egreg Aug 11 '14 at 10:51\n\u2022 See @kaka's comment above. \u2013\u00a0percusse Aug 11 '14 at 10:52\n\u2022 @percusse ;egreg ;Joseph Wright thanks for your time and making the clarification. \u2013\u00a0kaka Aug 11 '14 at 10:55\n\u2022 @egreg May I know that how did you combine and where did you type U+1E8B U+0323 ? \u2013\u00a0kaka Aug 11 '14 at 11:26\n\u2022 @kaka Here it is: \u1e8b\u0323. On my keyboard it's quite simple: Alt-w x Shift-Alt-x; Alt-w is for the dot above, Shift-Alt-x is for the combining dot below. Of course it depends on your keyboard and OS. \u2013\u00a0egreg Aug 11 '14 at 16:23\n\u2022 Layman way: 1) Open the site cs.tut.fi\/~jkorpela\/fui.html8 2) Type 1E8B and Press Enter 3) then Type 0323 and Press Enter Done \u1e8b\u0323 \u263a \u2013\u00a0kaka Aug 11 '14 at 23:14\n\nYou can view the characters in\n\nWindows: by opening the application Character Map,\n\nOS X: by opening the application Character Palette, or\n\nX11: by opening applications like KCharSelect if using KDE or GNOME Character Map if using GNOME.","date":"2020-09-21 02:03:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.6199301481246948, \"perplexity\": 2922.5372140118197}, \"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-2020-40\/segments\/1600400198887.3\/warc\/CC-MAIN-20200921014923-20200921044923-00460.warc.gz\"}"} | null | null |
We are pleased to release our full website with our first speakers and mentors. Registration will open in the coming days.
In the meantime, link with us on social media! Like us on Facebook and join our event, follow us on Twitter and on Instagram. Follow the latest news and join the discussions with our hashtag #INNOspaceWeekend. And if you haven't already, be sure to check out our teaser trailer!
Join us for an extended weekend in Cologne where you will where you will create a startup in just 60 hours. Whether you are a student, graduate, or young professional or want a change of career direction, INNOspace Weekend is for you! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,937 |
Q: How can give my app to client for testing hi i am developing an iphone app for a client who lives outside my country.
The client wants to check out the app every time i complete a functionality in order to note my progress and to suggest necessary changes.
How can give my app to client for testing before the project is completely done .
A: You can give them ADHOC or testing versions of your app.
It can be relatively easy to do on your own (and the instructions are easily found on Apple's "iOS Provisioning Portal", or there are also some services available from third parties online that make the process even more straighforward. I've heard good things about TestFlight, for example.
A: I suggest you to create an ad-hoc certificate. Then use xcode to create your archive.
ps: You need the UDIDs of your client iPhones to add the devices to the portal.
A: You can do it in this way-
1) Add the UDIDs of your client and use the particular profile and certificates to build your app.
2) Create a signed .ipa and .plist files.(Distribute for enterprise option).
3) Simpler way to distribute it as you can put it on the dropbox and attach it in an html file.
4) Send your client the public link of that html page you've put on dropbox.
Hope it helps. :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 47 |
{"url":"http:\/\/lt-jds.jinr.ru\/record\/67669?ln=en","text":"\/ hep-ex CERN-PH-EP-2014-264\n\nTwo-particle Bose-Einstein correlations in $pp$ collisions at $\\mathbf {\\sqrt{s} =}$ 0.9 and 7 TeV measured with the ATLAS detector\n\nPages: 26\n\nAbstract: The paper presents studies of Bose-Einstein Correlations (BEC) for pairs of like-sign charged particles measured in the kinematic range $p_{\\rm T}>$ 100 MeV and $|\\eta|<$ 2.5 in proton--proton collisions at centre-of-mass energies of 0.9 and 7 TeV with the ATLAS detector at the CERN Large Hadron Collider. The integrated luminosities are approximately 7 $\\mu$b$^{-1}$, 190 $\\mu$b$^{-1}$ and 12.4 nb$^{-1}$ for 0.9 TeV, 7 TeV minimum-bias and 7 TeV high-multiplicity data samples, respectively. The multiplicity dependence of the BEC parameters characterizing the correlation strength and the correlation source size are investigated for charged-particle multiplicities of up to 240. A saturation effect in the multiplicity dependence of the correlation source size is observed using the high-multiplicity 7 TeV data sample. The dependence of the BEC parameters on the average transverse momentum of the particle pair is also investigated.\n\nNote: Comments: 13 pages plus author list + cover pages (26 pages total), 6 figures, 2 tables, submitted to Eur. Phys. J. C. All figures including auxiliary figures are available at http:\/\/atlas.web.cern.ch\/Atlas\/GROUPS\/PHYSICS\/PAPERS\/STDM-2013-01\/\n\nTotal numbers of views: 1623\nNumbers of unique views: 673","date":"2019-02-16 18:00:22","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\": 0.8774254322052002, \"perplexity\": 2962.7807086367693}, \"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-2019-09\/segments\/1550247480905.29\/warc\/CC-MAIN-20190216170210-20190216192210-00361.warc.gz\"}"} | null | null |
Complete Kicking
The Ultimate Guide to Kicks for Martial Arts Self-defense & Combat Sports
Sang H. Kim, Ph.D.
COMPLETE KICKING
A Turtle Press Book / 2009
Copyright © 2009 Sang H. Kim. All Rights Reserved.
Printed in the United States of America. No part of this book may be reproduced without written permission except in the case of brief quotations embodied in articles or reviews. For information, address Turtle Press, PO Box 34010, Santa Fe, NM 87594-0410
Photographer: Marc Regis, Cynthia A. Kim
Photo models: Sang H. Kim, Kyu-hyung Lee, Carlos Sanchez, Marco Corea
ISBN 9781934903742
Warning-Disclaimer
This book is designed to provide information on specific skills used in martial arts and fitness training. It is not the purpose of this book to reprint all the information that is otherwise available to the author, publisher, printer or distributors, but instead to complement, amplify and supplement other texts. You are urged to read all available material, learn as much as you wish about the subjects covered in this book and tailor the information to your individual needs. Anyone practicing the skills presented in this book should be physically capable to do so and have the permission of a licensed physician before participating in this activity or any physical activity.
Every effort has been made to make this book as complete and accurate as possible. However, there may be mistakes, both typographical and in content. Therefore, this text should be used only as a general guide and not the ultimate source of information on the subjects presented here in this book on any skill or subject. The purpose of this book is to provide information and entertain. The author, publisher, printer and distributors shall neither have liability nor responsibility to any person or entity with respect to loss or damages caused, or alleged to have been caused, directly or indirectly, by the information contained in this book.
### Table of Contents
1 | Introduction
---|---
2 | Understanding the Body
3 | Fundamental Kicks
4 | Spinning and Turning Kicks
5 | Hopping Kicks
6 | Jumping Kicks
7 | Jumping Spinning Kicks
8 | Multiple Kicks
9 | Combination Kicks
INTRODUCTION
Kicking is for every fighter. It's powerful, fast, and deadly. Regardless of martial art style, the fundamentals are much the same: kick fast, kick hard, kick precisely. To meet these goals, you need to train your body, strengthen your muscles, and condition your mind.
You'll be surprised to find initially how natural kicking is, just like walking and running. However, to excel at kicking, you have to go beyond what is natural. You must learn to control your body at all times in order to acquire precision, speed and power. Then your body will become a deadly weapon for self-protection and an invincible tool for sport competition.
After you perfect the basic skills, you are only a few steps away from mastering challenging advanced kicks. As you progress and learn increasingly difficult kicks, you will find that practicing kicking enhances your mental clarity and concentration, as well as your physical fitness.
Preparation
If you're reading this book, you obviously have a serious interest in improving your kicking skills. You might already be a martial arts student or you might be thinking of taking up a martial art. Either way, here are some key things to know before you get started:
find a place
Kicking can be practiced almost anywhere that has a flat surface clear of obstacles. You can practice on your own, but if you are novice, professional instruction at a martial art school or program is recommended. To choose a class, visit the school(s) you are interested in and see for yourself: Is the instructor professional and knowledgeable? Do you feel like learning from him or her? Do the students have the right attitude? Check out the condition of the facility - is it clean and safe? Ask students what they like and don't like about the class. Once you're settled, either in a class or at home, develop a regular training schedule to achieve your goals.
proper outfit
Kicking requires free range of movement, so your training outfit should be loose fitting, especially in the hips and knees. Loose fitting shorts, sweat pants or a martial art uniform are safe, comfortable options.
equipment
In the beginning stage of learning a kick, you may need something to assist with your balance and posture, such as a stretching bar, chair, or even a wall. Once you've mastered the basic method of kicking, you'll use a target to improve your precision. Commonly used targets are a speed-bag hung from the ceiling, a free standing reflex bag, a double end ball or a hand target held by a partner. For power and speed practice, you'll use a heavy bag.
safety
Before practicing a new kick, learn the movement and visualize it. Know your physical limits and build from where you are. Gradually increase the height, speed, power and complexity of your movements. It is also very important to warm up for 10 to 15 minutes prior to practice and to cool down for about 5 minutes afterward to prevent injuries.
A First Lesson
fighting stance
Your stance is the position of your feet and hands. To make a basic fighting stance:
1. Place your feet about shoulder width apart with your weak side foot in front and your strong side foot in the rear. (If you're right handed, your right foot is your strong side.) For power, widen your stance. For speed and mobility, narrow your stance.
2. Turn your front foot inward 15° to 30° and turn your rear foot outward 60° to 90°. Keep your weight slightly more on the balls of your feet.
3. Bend your knees slightly.
4. Keep your torso upright and angled away from your opponent to protect your vital targets. Tuck your chin down slightly toward your chest and look at the target.
5. Bend your elbows and bring your hands in front of your chest to protect your torso and face.
6. Relax your shoulders and move rhythmically.
Foot position determines your kicking capacity and controls your center.
balance
Your knee and head positions are critical for maintaining your balance. Keep your head level and look at the target. Keep your knees slightly flexed to lower your center of gravity. When you kick, your standing knee should remain slightly flexed to maintain your balance in motion.
control
Kicking requires you to have two contradictory attributes: control and freedom of movement. You need to control your body to execute a kick precisely. However, constant controlling, and the consciousness of it, makes your body stiff. Through practice, you will develop "muscle memory" so that eventually your body will perform the kick without conscious thought. Muscle memory allows you to focus on suppleness without controlling the "moving parts" of each kick.
practice
Don't be afraid to experiment with changes that might improve your performance. Vary your practice routine so it is mentally and physically engaging, not routine or boring. When you become good at in-place kicking, add footwork and drills or vary the speed, power and height.
feedback
When you run into trouble with a kick, listen to your body and go back to basics. Everyone's body is shaped and functions differently, so you might need to adapt the instructions in this book to fit your own body type. Problems are often the foundation for enhanced performance, so don't get discouraged. Stay positive and experiment.
revision
There are many levels of mastery and your perception of perfection fluctuates because your kicking changes according to your condition, opponent, and circumstances. In training, you'll find that you have to constantly revisit a time when a kick felt right for you and try to recapture that feeling and build on it.
Developing Form
Form is the relationship of the key parts of the body illustrated at right. No matter how you alter your stance, the relationship of the four circled areas should remain constant. If you turn your hips 45°, then your chest and knees should move accordingly.
From the basic fighting stance, for example, your stance for each kick will be slightly different. Front kick begins from a conventional stance, but for side kick, your stance should be turned more to side to create a coiling force. As you bring your hip forward and knee up (photo 1), the body pivots (photo 2) and shoots the rear foot toward the target (photo 3). When you establish the correct form for your kick, your leg should naturally shoot along the most efficient path to the target.
arm position
To find the most natural position of your arms and hands, raise your arms higher than your usual fighting stance (photo 1) and slowly lower them until you find the right height (photo 2). Keep your elbows bent and swing your arms around your torso a few times, then stop where you feel most comfortable. Make they cover the vital targets of the torso.
knee control
The knee travels straight forward in front kick, in a circular motion in roundhouse kick, backward in back kick. How you manipulate your knee determines the angle of the kick. You can practice knee control in most of the twelve directions of the clock.
3 kicking essentials
Base
Your two feet form the base for your actions. The ideal base is about shoulder width or a bit wider. Positioning your feet too far apart or too close together creates an unstable base. Next, align your feet according to what type of kick you are planning to perform. For front kick, for instance, turn your feet forward. For side kick, align your feet sideways and parallel to each other. For roundhouse kick, turn your feet about 45°, between the positions of front kick and side kick.
pivot point
Your standing leg becomes a pivot point when you kick. If you do a roundhouse kick with your rear leg, your front leg is the pivot point. If you throw a front leg roundhouse kick, your rear leg is the pivot point. The pivot point plays an important role in kicking. The smoother you pivot, the better. The quicker you pivot, the more powerful your kick will be. The more stable your pivot is, the more precise your kick will be.
impact
Impact results from a high velocity strike, or a low velocity strike with sufficient weight behind it. You can strike a target with your foot, knee, or shin. For effectiveness, your strike should be precise, fast, and penetrating. To further enhance the impact, strive for maximum range of motion of your kicks through stretching and greater focus of your physical and mental energy through proper breathing and yelling (kihap, kiai). Impact is the ultimate goal of kicking.
5 Kicking Stages
ready
Before kicking, confirm that you have a strong base and align your feet according to what type of kick you are planning to use. Keep your arms close to your body in a guarding position and look at your target. Stay light on your feet and ready to move.
shift
Before turning your body, shift your weight to your front leg for a rear leg kick or to your rear leg for a front leg kick. Do not change the height of your head or your facial expression when you shift your weight, because this will reveal your intentions to your opponent. Be calm and move subtly as you initiate your kick.
pivot
Your standing leg becomes a pivot point when you kick. Always pivot on the ball of your foot, raising your heel slightly to facilitate the pivot. As you pivot, chamber your kick by raising your knee into the appropriate position.
kick
Impact results from a high velocity strike, so kick as quickly as you can while maintaining accuracy. Penetrate two to six inches into the target.
repose
Retract your foot and leg swiftly and prepare for the next action, which might be another kick as part of a combination or a defensive reaction to your opponent's counterattack.
3 power generators
knee
Bending your knee prior to kicking is called chambering. The way you chamber your knee will determine the direction, range of motion and power of your kick. In general, where you point your knee is the direction of the force of your kick. The sharper you bend your knee, the more power you can generate due to a larger range of motion. If your chamber is lazy, your kick will lack power.
hip
The hip is the primary mover of the body and it functions as a central powerhouse for kicking and punching. The muscles and bones of the hip are large and powerful and are the site of your center of gravity. By rotating your hip quickly and powerfully, you propel knee and foot into the target with maximum power. For many kicks, "snapping" your hip also adds to the impact power of your kick.
Your hip is the powerhouse for kicking. It brings all of the elements of your body's power together and directs your energy to one single point upon impact.
head
The head is a stabilizer. By keeping your head upright, you anchor the alignment of your spine and maintain the pivoting power of the lower body around the central axis of the spine. If you drop your head while kicking, the force of your kick is dispersed, much like an uncoiled spring.
Head up stabilizes your posture and integrates the body's force.
Kicking Physics
In the science of kicking, there are three axes of the body: torso, thigh, and shin. Each transmits force and the further a point is from the origin of the force the more potential force it carries.
The initial point of action is called the Action Point.
The multiplying point of force is called the Pivot Point, where mechanical force is magnified.
The effect of applying the action point is called the Reaction Point.
The striking spot is called the Impact Point.
The primary pivot point in the body is the lower abdomen which facilitates force magnification between the action point and reaction point.
* The best exercise for strengthening the pivot point (abdomen) is sit-ups.
For right leg roundhouse kick, the origin of force is the left shoulder (action point). The force passes through the lower abdomen (pivot point) and is magnified at the knee (reaction point) which propels the the foot toward the target (impact point).
UNDERSTANDING THE BODY
To advance in kicking practice, it is important to have some basic knowledge of how the body works, particularly the bones, joints and muscles. They are the three primary mechanical devices that maximize the impact of your kicks. When you understand how they work, you can fully take advantage of what you have available. Not only will your training be more effective, but you'll be less susceptible to injury. If you don't understand how your body works, you might work hard but risk poor results and even injury. So take a few moments and review this chapter before you get started.
the body structure
the torso
When your kicking form is correct, your torso stabilizes your body. If your form is incorrect, your torso can unbalance your body and reduce the power, height and speed of your kick. Because your torso is the heaviest and most cumbersome part of your body, it is important to keep it at the center of your movement and allow your arms and legs to rotate around the torso. Imagine that your torso is the eye of the hurricane of your movements.
the spine
The spine is a complex network of muscles, ligaments, bones, joints, cartilage and nerves that work together to support and mobilize the body: support to stand, walk and lift; mobility for movements such as turning, twisting, bending, jumping and kicking. Because the spine is the primary support and movement structure of the torso, it is very important to stretch the back properly before and after kicking for optimal flexibility.
the lower abdomen
Kicking power originates from the lower abdomen, also called the danjun or dantien. A sharp exhalation during the execution of each kick tenses the abdomen, increasing the power of the kick. Many martial artists combine this exhalation with a shout (called kihap or kiai).
the arms
For kicking, your arms aid in balance, acceleration and delivery of force. Because your arms can move freely on the horizontal, vertical or diagonal planes, they are excellent tools for counter-balancing the movement of your legs, particularly when learning a new kick. For example, when you learn the axe kick, sharply dropping your arms at your sides can help you raise your kicking leg higher.
the legs
The legs are the primary impact delivery tools for kicking, however they must work in concert with the rest of the body for optimum power, speed and flexibility. Kicking only with your legs will produce poor results and limit your potential for improvement. As you saw in Chapter 1, your legs are the transmitters of the power that originates in your hips and torso.
The legs also support and disperse the body's weight. The knees in particular function to generate force in moving, lifting, and striking and to lessen the impact when landing or falling.
INTEGRATED MOVEMENT
Practicing kicking alone is not sufficient to improve your kicking skills. Be sure to include strength, timing, accuracy, balance, power and flexibility exercises in your workout sessions to develop your overall physical condition. By taking a whole body approach, your physical movements will become integrated and your confidence will increase.
Some sample exercises: 1. Plyometric jumping drills, 2. Resistance band training (upper body), 3. Core strengthening bodyweight exercises, 4. Resistance band training (lower body), 5. Anti-gravity exercises, 6. Dynamic isometric drills.*
*For more details, see Ultimate Fitness Through Martial Arts and Ultimate Flexibility
BALANCE: It's about relationships
BALANCE IN KICKING requires an understanding of where the various parts of your body are located in relation to each other; where they are moving from and to; and how to orchestrate them to maintain your form throughout a kick.
THE KEY to this seemingly complex process is maintaining symmetry in your body, if not visually, then at least in terms of weight distribution.
FOR EXAMPLE, knee kick requires you to slightly hunch your upper body forward to balance the upraised knee. The same is true of axe kick. Because front kick travels more forward than upward, your upper body needs to decline slightly backward to balance the forward motion of your leg. Roundhouse kick and side kick require a similar decline of the upper body to balance the outstretched leg.
Every kick has a different balance angle:
Knee kick: forward hunch balances upraised knee
Axe kick: similar to knee kick balance
Front kick: slight upper body decline
Roundhouse kick: upper body declines in relation to upraised kicking leg
Side kick: upper body declines in relation to thrusting height of kicking leg
BALANCE IS ABOUT CENTERING
In any balancing activity, the most important thing is to center yourself. Relax your shoulders and arms to promote natural rhythm in your motions. Lower your center of gravity so that you feel comfortable and secure. Do not overextend your body.
EXERCISE: When you practice spinning kicks like those shown below, tense your lower abdomen to focus your center and relax your terminals (upper body and kicking leg). When you feel confident with a spinning kick, try practicing 2, 3, 4 or 5 in quick succession with the same leg to see how this affects your balance. If your kick is technically, you should have no problem.
After kicking, let go until the force becomes zero, then let your leg return to its natural original position.
Keep your torso balanced between both legs for high kicks.
the bones
key point:
Kicking requires coordination of complex movements of the legs, hips, spine, shoulders, arms and head. The bones of your skeleton facilitate these movements.
overview:
The skeletal structure of the body protects the organs and provides levers to facilitate movement. The placement of each bone is deliberate, exact, and in many cases, symmetrical. These characteristics are important to kicking because your kicking needs to be deliberate, precise and balanced. The symmetrical structure of the skeletal system enables us to balance our body by stabilizing it against gravity.
Additionally understanding the anti-gravitational nature of the spine, pelvic girdle and the lower limbs is critical since we must stand and kick in opposition gravity and often we must struggle enormously against an opponent. For standing kicks, for instance, you need to stabilize your standing leg and spine to free the kicking leg from the constraint of gravity (photo a). For jumping kicks, extend both legs when the spine reaches the highest point of the jump where gravity is weakest (photo b).
Another key point to understand about the skeleton is that it made up of bones of varying lengths and functions. For example, the femur, the most powerful bone in the human body, allows us to control the speed and power of a kick.
The tiny phalanges, or toe bones, play almost no role in creating powerful kicks but they are critical in maintaining your balance while standing on one leg. Without toes, you would no more be able to deliver a proper roundhouse kick than if you lacked a femur bone. Each bone is designed to play a particular role in movement.
The skeleton by itself, however, is useless. It can only function with the help of the muscles and nervous system.
major bones
skull
The skull is made up of 29 bones forming the brain case and face. For kicking, it is important to know the the jawbone is the only moving bone in the skull. Tuck your chin and close your mouth when kicking.
spine
The spine is attached to the pectoral girdle at the shoulder, the pelvic girdle at the hip, and the rib cage (12 ribs on each side) at the torso, giving shape and height to the body for support. The vertebrae of the spine are made up of alternating layers of bone and compressible cartilage, which reduces shock, bears the weight of the body and prevents the vertebrae from grinding against each other.
pectoral girdle
The pectoral girdle is the set of bones that connect the arms with the shoulders. It is composed of the clavicle and scapular.
arm
There are 30 bones in the arm, 27 of them in the hands and wrist. The arms are balance equalizers, initiators and accelerators in kicking.
pelvic girdle
The pelvic girdle is located at the bottom of the spine. It consists of the sacrum, coccyx, and a pair of hip bones (ilium, ischium, and pubis). The pelvic girdle is the source of power in kicking.
leg
There are 30 bones in the leg, 22 of them in the ankle and foot. The femur, the most powerful and largest bone in human body, can bear a tremendous amount of pressure and weight. It is engineered for maximum strength and power. As a whole, the leg bones function to resist gravity by bearing the weight of the body during motion and at rest.
Spinal Curve
The spine is the backbone of the body: not only carrying the weight but also facilitating movement through mechanical balancing and counterbalancing. The 4 alternating curves in the spine aid in this job.
the joints
A joint is a point where two or more bones are connected. Although each joint works differently according to how the bones are joined, they are all built for movement and function as mechanical fulcrums. Therefore they play invaluable roles in kicking: transmission of force, facilitation of coiling/uncoiling the body and control of speed.
transmission of force
The joints are responsible for chain reactions in kicking: force originates from one source (for example, the left shoulder for right roundhouse kick), activating the motion of the pelvic girdle, travels through the limb (right leg for right roundhouse kick) to the extremity (right foot). To visualize this, imagine cracking a whip. You initiate the force in your hand and the force travels through the whip to the opposite end, which strikes a target. The whip is a powerful weapon because it is an efficient transmitter of force. (See page 22-23 for a detailed explanation of this concept.)
coiling/uncoiling
For circular techniques like roundhouse kick, turn kick and spinning whip kick, when a joint pivots in preparation for kicking, a coiling of force occurs. In the execution of the kick, an uncoiling action delivers the force of the kick to the target. For example, in steps 4-7 on page 35, the hip and knee joints create the coiling action and your body has the greatest amount of potential energy at this point in the kick. In steps 8-9, the knee and hip joints have uncoiled to deliver the force of the kick to the target.
control of speed
When force passes through a joint, according to the pivoting angle and intensity, the joint magnifies the force which results in an acceleration of speed. However, if your joints are stiff, your speed will decrease as force passes through them. To understand why, think back to the example of the whip. When you crack the whip, your action has one point of initiation—your hand—and one point of final force transmission—the opposite end of the whip. Imagine if you tried to control the whip not only at the handle end, but at two or three points in the middle. The force would be chaotic and randomly dispersed. The same is true of your kick when your joints lack the fluidity to transmit the force smoothly from origin to target.
sequence of kicking
Action #1: The contraction of the abdomen muscles pulls the lumbar curve forward (1).
Action #2: Action #1 triggers the sacral curve to snap froward (2).
Action #3: Action #2 causes the sternum and neck (3) to counterbalance the forward motion of the hip. Actions #1-3 take place almost simultaneously, in a fraction of a second.
Action #4: The left shoulder begins to turn to the left.
Action #5: The right hip joint projects forward.
Action #6: As the right hip begins to rotate to the forward left, the elbows (6) move to the right to counterbalance.
Action #7: As the elbows swing to the right, the right knee moves to the left in a circular motion.
Action #8: As the right knee reaches the high point, both hands snap further to the right, accelerating the force of the leg.
Action #9: The right foot impacts the target at maximum speed and power.
WHAT IS BALANCE?
Balance is the ability to maintain your body position both in movement and at rest. There are two types of balance: static and dynamic. Static balance is a stationary object at equilibrium, like when you stand upright. Dynamic balance, a.k.a. balance in motion, is a body moving at constant linear and angular velocities.
WHERE DOES IT COME FROM?
1. Balance at rest and in motion both originate with good posture. Align your feet, hips, spine and head when kicking to maintain a stable upright posture.
2. Keep your eyes fixed on one spot, preferably your target. Vision provides your body with a ready means of finding your position relative to the ground through the presence or absence of visual cues.
3. Tactile cues, such as the network of sensors in the pads of your feet, relay important information to your brain, allowing it to adjust your weight distribution and foot position as necessary.
4. The vestibular apparatus in the inner ear controls the righting reflex which helps you remain upright in motion.
KEY POINT: If your lose your balance, restore it quickly by first positioning your head upright and fixing your line of sight on one object, then allow your body to follow.
the muscles
primary function
The primary function of the muscles is to create movement of the skeleton (bones) through contraction and relaxation. Strengthening your major kicking muscles will enhance the power and flexibility of your kicks and help prevent common injuries.
major kicking muscles
For fast, powerful, well-controlled kicks, you need to develop three major muscle groups in the body:
• abdominal muscles: rectus and transversus abdominis
muscles, oblique muscles
• frontal thigh muscles: quadriceps femoris
• rear leg muscles: gluteus muscles in the buttock, hamstring
The abdominal and thigh muscles produce power for front kick, roundhouse kick, axe kick and side kick. The rear leg muscles generate force for back kick, whip kick and spin whip kick as well as jumping kicks.
developing abdominal muscles
The best exercises to develop strong abdominal muscles are sit-ups, including side and twisting sit-ups, crunches, V-ups, knee raises, leg raises, side leg raises and hanging leg raises.* Use a wide variety of exercises to be sure you are working the lower, mid, upper and oblique ab muscles. And don't forget to include back strengthening exercises in your workout to develop a well-balanced torso musculature.
developing leg muscles
The best exercises to develop strong thigh and buttocks muscles are standing jumps, squats (with or without weight), lunges, sprinting, bounding, stair running and hill running.* Be sure to stretch at every workout so that your leg muscles do not become overly tight from strength training.
* For more information on the above exercises, see Ultimate Fitness Through Martial Arts, Sang H. Kim, Turtle Press.
sample kick strengthening workout*
Lunges, 10 reps, 1-3 sets
Squats, 10 reps, 1-3 sets
Crunches, 10 reps, 1-3 sets
Side sit-ups, 10 reps, 1-3 sets
V-ups, 5 reps, 1-3 sets
Alternating knee raise, 5 reps, 1-3 sets
Alternating leg raise, 5 reps, 1-3 sets
Double leg raise, 5 reps, 1-3 sets
Bench leg raise, 5 reps, 1-3 sets
50 yard sprint, 3 reps
50 yard uphill run, 3 reps
25 yard bounding, 3 reps
Stair run, 1 flight, 3 reps
Before Kicking...
1. warm up
Warm up before engaging in full-speed kicking to raise your body temperature and increase circulation. A good warm up enhances the ability of the muscles to contract and warm muscles stretch better. Begin with easy gross motor activities, gradually increasing in intensity from mild to vigorous. Control your movements while loading calculated stress on the muscles. Warm up for 15 minutes.
2. stretch
Before practice, always stretch your muscles to prepare them for the demands of kicking. Stretching increases the elasticity of the muscles, which enables you to have a greater range of motion and strength, and thus increases your capacity to generate more power. Stretching before kicking practice should include simple, dynamic exercises (see examples on page 41). This is different from the deep stretching that you do after practice to increase your flexibility. Stretch for 5-10 minutes.
3. breathe deeply
Deep breathing relaxes the muscles by providing ample oxygen to the cells. Deep belly breathing also enhances posture, circulation, relaxation and inner power. Consequently the mind becomes clear and the body works more efficiently. Deep breathing before practice eliminates the distractions of the body and mind and promotes mind-body unity. Breathe deeply for 3-5 minutes.
4. visualize
Visualization is active mental imagery to create familiarity with a specific target movement. For example, close your eyes and think of the beginning, middle and end of a side kick. Look closely at critical points such as pivoting your hip before impact. Replay it over and over until you get it right in your mind. Finally complete the entire sequence with a smooth and perfectly timed performance. Visualize one technique for 3-5 minutes.
warm-up options
Jumping jacks x 100
Push-ups x 50
Sit-ups x 100
Jump rope x 3 mins.
Running x 15 mins.
Dynamic leg raises
Hold the bar and raise your knee 10 times on each side, then swing your leg up, smoothly with control, 10 times on each side. This exercise develops kicking posture, precision and strength while warming up the kicking muscles.
leg raise & hold
Hold the bar and raise your leg in a controlled kicking motion (left: whip kick, right: back kick) as high as you can and hold it for as long as you can. This will stretch and strengthen your leg muscles and develop balance and posture simultaneously.
STRETCHING
There are two types of stretching: stretching as part of your warm-up, which is brief and dynamic, and stretching to develop flexibility (below), which is done after practice and can take up to 30 minutes.
Knockout Points
A knockout kick renders your opponent unconscious by either stopping the supply of oxygen and blood to the brain, causing insufferable pain or causing a shock to the brain.
MAJOR KNOCKOUT POINTS
1. Head: Kicks to the head can shock the brain and shut down the function of the nervous system. A kick to the jaw for instance causes a rebounding effect in the brain, momentarily nullifying the function of the nervous system, which renders the opponent unconscious.
2. Neck (Carotid Artery): Striking the carotid sinus can lead to a sudden drop in blood pressure and heart rate which can cause a loss of consciousness and, in some cases, death. Because striking the carotid sinus relies simply on triggering a physiological reaction, even a light strike can result in a knockout. Striking the neck is illegal in most combat sports.
3. Groin and 4. Pelvic Plexus: These targets are also generally not legal in combat sports, but are effective knockout targets for self-defense because they can be hit with a powerful low section kick.
5. Solar Plexus and 8. Cardiac Plexus: These two targets have to be struck with great force with a powerful kick like a side kick, to achieve a knockout. Kicking this area can also "knock the wind out of" an opponent.
6. Liver and 7. Kidneys (both sides): These three targets can be easily struck with roundhouse kick. Even if you don't achieve a knockout, kicks to the liver or kidneys will weaken an opponent's stamina and will.
9. Brachial Plexus: This target is vulnerable to an axe kick. If the plexus strike does not cause a knockout, it can result in serious damage to the clavicle bone.
FUNDAMENTAL KICKS
Kicking is one of the most fun and natural activities in the martial arts. This chapter has 12 fundamental kicks and for each kick you'll find the purpose of the kick, key points to pay attention to, common targets, how-to, variations, applications specific to self-defense, traditional martial arts practice and combat sports, common mistakes and practice drills.
FRONT KICK
purpose
Front kick is used to strike a target that is directly in front of you. If your opponent is on your side or rear, simply turn your body in that direction and kick. Front kick is the most direct way to knock down an opponent by kicking the groin, face or neck. You can also use it to deter an advancing opponent by attacking the kneecap, shin or thigh.
key points
Bend your kicking knee acutely and snap the ball or instep of your foot quickly and powerfully to the target. If you are wearing shoes, kick with the toe or instep of your shoe. In a self-defense situation, aim for the middle of the body or leg to increase your chances of striking the target.
Striking area
Ball, instep, toes, bottom of the foot
targets
1. groin
2. lower abdomen
3. solar plexus
4. chest
5. neck
6. mouth
7. frontal thigh
8. kneecap
9. shin
how to
1. From fighting stance, shift your weight to the front leg.
2. Bring your knee up, pointing it at the target.
3. As your hip snaps forward, extend your knee and kick. Keep your standing knee bent, using the thigh muscles to firmly stabilize your stance. Tuck your chin down, lean your torso slightly backward to add extra reach (length). Exception: If you are moving forward, bring your torso forward instead of leaning back.
4. Withdraw the foot quickly after the kick and repose.
FRONT VIEW (above) of front kick: notice the position of the left hand to protect the groin and the right hand to maintain balance. You can also keep your hands up in a guard position. The head, hip and standing foot are aligned vertically for maximum stability.
applications
Front kick is the most direct and easiest kick to execute, so you can instinctively throw it in almost any situation. For self-defense, it is useful for striking the groin, stomach or solar plexus directly, or in combination with a hold or lock. For sport competition, you can stop an opponent by kicking his lower abdomen or knock him down by kicking his neck vertically with the blade of your foot.
SPORT APPLICATION: Front kick to the neck using the blade of the foot. (Kicking the neck is not allowed in some combat sports.)
SELF-DEFENSE APPLICATION: Front kick to the chest in combination with a twisting wrist lock
SPORT APPLICATION: If an opponent rushes toward you, throw a front kick to push him away. As the opponent backs up, throw another front kick or a roundhouse kick or axe kick.
SELF-DEFENSE APPLICATION: A front kick to the pelvic plexus can knock an opponent down. Push deeply into the pelvis with the ball or bottom of the foot.
SPORT APPLICATION: Blade front kick to the carotid artery in the neck can result in a knock down or knockout. Recommended for advanced students only.
SELF-DEFENSE APPLICATION: For front kick to the groin, use the full instep of the foot to drive upward.
common mistakes
MISTAKE 1: The most serious mistake people make in executing a front kick is tilting the head, which disrupts your balance.
SOLUTION: Tuck your chin down toward your chest.
MISTAKE 2: Arm position: too wide, too narrow or too low.
SOLUTION: To practice keeping your arms close to your body in fighting stance, hold the collars of your uniform and practice.
MISTAKE 3: Excessively raising the heel of the standing foot.
SOLUTION: If you feel that you have to raise up onto the toes of your standing foot, pivot your standing foot slightly more when you kick or lower the height of your kick.
MISTAKE 4: Leaning too far back/away from the target.
SOLUTION: To kick, you should decline your torso slightly but not so much that you lose your balance. If you are leaning back too much, lower the height of your kick until you improve your flexibility or raise your knee higher when you chamber your kicking leg.
• remember
1. Keep your head straight.
2. Tuck your chin down.
3. Position both elbows at equal height.
4. Aim your knee at the target and kick directly to the target.
• avoid
1. Tilting your head
2. Leaning your torso too far backward
3. Opening your arms too wide
4. Dropping your hands
5. Holding your arms too tight in front of your chest
6. Lifting your standing heel excessively
ROUNDHOUSE KICK
purpose
Roundhouse kick is used to strike lateral targets on the trunk, head and legs of an opponent who is facing you. If your opponent stands sideways in front of you, kick the front of his torso. Due to its effectiveness and simplicity, roundhouse kick is the most commonly used kick in competition. It is a versatile weapon for attacking and counterattacking against any type of opponent.
key points
Bend your kicking knee forward acutely, rotate your hip and snap the ball or instep of your foot quickly and powerfully to the target. When attacking, bring your body forward for more power. For counterattacking, turn your body sideways or lean slightly backward to avoid being hit.
striking area
Ball, instep, shin, toes
targets
1. rib cage
2. thigh (inside, outside)
3. knee (inside, outside)
4. ankle (inside, outside)
5. neck
6. head
7. chest
8. solar plexus
9. lower abdomen
how to
1. From fighting stance, shift your weight to the front leg.
2. Bring your knee up, pointing at the target as in front kick.
3. Pivot your standing foot and rotate your knee perpendicular to the target, while your arms move in the opposite direction to maintain your balance.
4. Rotate your hip into the kick until it is parallel to the target surface. Snap your foot out at the target. At impact, your entire body should be on the same plane. Keep your head up and look at the target.
5. Withdraw the foot quickly along the same path after the kick and repose.
Roundhouse kick with the ball of the foot
FRONT VIEW: roundhouse kick with the instep
applications
Roundhouse kick is the most versatile kick for both competition and self-defense. For competition, it is safe, effective and powerful for striking low, middle, and high targets. For self-defense, it is useful for striking the knee, groin, and stomach. It is most effective when used in combination with punches, elbow strikes and knee kicks.
SPORT APPLICATIONS
When your opponent steps in to attack, use an intercepting roundhouse kick to stop him and score.
For high roundhouse kick, raise your knee high and shoot it quickly at the opponent's head.
When your opponent throws a high section roundhouse kick, counter with a middle section roundhouse kick.
When your opponent is retreating, step in and chase him with a roundhouse kick.
If your opponent hesitates while attacking, throw a middle section intercepting roundhouse kick.
Kick as your opponent steps in to attack.
MARTIAL ART APPLICATION: Roundhouse kick in one step sparring
SELF-DEFENSE APPLICATIONS
Kick over the opponent's strike, using your longer range kick to your advantage.
From the ground, use the ball of your foot to strike the opponent's chest. Place your hands on the ground to support your body and add power.
A low kick to the rear of the knee will cause an assailant to fall backward.
On the ground, you can use a downward roundhouse kick to a fallen opponent.
Maintain distance from your opponent on the ground with a roundhouse kick to the head.
common mistakes
MISTAKE 1: The most common mistake when executing roundhouse kick is incomplete rotation of the hip.
SOLUTION: For maximum power, rotate your body fully to align your hip, head and feet, and to put your bodyweight into your kick.
MISTAKE 2: Over-rotation of the body during and after kicking.
SOLUTION: It can be easy to over-rotate your body in an effort to make your kick more powerful, but power comes from speed and the snap of your hips, not from turning your upper body, so focus on these instead.
MISTAKE 3: Kicking in a wide arc.
SOLUTION: Bring your knee up straight at the target and then pivot tightly. Avoid letting your leg swing wide; instead snap it at the target.
BOTTOM LINE: Keep your opponent in sight at all times; kick fast and forcefully; repose as quickly as you can.
• remember
1. Keep your head straight.
2. Tuck your chin down.
3. Rotate your hip fully.
4. Move your arms naturally around your body for balance and power.
• avoid
1. Uncommitted kicking
2. Premature kicking (bad timing)
3. Tilting your head
4. Over-rotating your body
5. Kicking in a wide arc
6. Kicking at the wrong distance (too close to opponent)
roundhouse kick form drill
To improve your kicking, practice roundhouse kick holding a support. Start with your back to the wall and your kicking foot in the rear. Pivot so your hip faces the wall and your chest is parallel to the wall. Standing close to the wall will prevent your from over-rotating your hips or knees and from hunching your upper body forward. Finally, extend your kicking leg and hold briefly. Your body (chest, hips, kicking leg) should be parallel to the wall when your kick is fully extended.
Roundhouse kick power drill
The heavy bag is an excellent tool for developing power. Here are examples of heavy bag kicking drills: 1) 3 sets of 15 kicks with each leg at normal power and speed. 2) Be sure you're fully warmed up, then kick as high as you can 20 times on each side. 3) Kick as fast and as powerfully as you can to a middle level target, alternating legs for 50 kicks on each side.
Roundhouse kick precision drill
Ideally, you should develop precision from the start of the learning process. A good way to do this is to always have a specific target for each kick. You can use a hand target held by a partner, a ball suspended on a string or a heavy bag. When you practice with the heavy bag, mark it with tape as shown here so you do not fall into the habit of randomly kicking the bag. Whether you are training for precision with slow controlled kicks or you are working on power and speed, stay disciplined and try to hit your target every time.
SIDE KICK
purpose
Side kick is a powerful kick that transmits the weight of your entire body into the target. It is an effective weapon to stop an incoming opponent by pushing his knee joint or kicking his stomach. It is also useful as an initiative attack to push an opponent backward and then throw a powerful follow-up kick.
key points
Pivot your body and chamber your knee so that your kicking leg is perpendicular to the vertical target line before kicking. Pivoting and alignment of your hips, legs and shoulders are paramount.
striking area
Blade, bottom of the heel, bottom of the foot
targets
1. knee
2. thigh
3. pelvic crease
4. shin
5. ankle
6. ribcage
7. stomach
8. armpit
9. neck
how to
1. From fighting stance, shift your weight to the front leg.
2. Bring your knee up, pointing at the target.
3. Pivot your standing foot so your body is oriented laterally to the target. Be sure to fully pivot your hip so that your kicking foot, buttock, shoulder and head are aligned.
4. Snap your knee and shoot the blade of your foot into the target. Align your toes lower than your heel for maximum power. Keep your head up and your head, shoulder, hip, knee and foot on one plane.
5. Withdraw the foot quickly after the kick and repose.
FRONT VIEW of side kick: notice the kicking foot, hip, standing foot, shoulder and head are aligned and the arms cover the chest and lower torso.
variations
You can use side kick to hit any target: low, middle or high, standing or on the ground. The key is to kick quickly, but to align your body correctly no matter what the situation and penetrate the target deeply with your kicking foot. In a self-defense situation, it's acceptable and even helpful to place one or both hands on the ground for support and added power.
High side kick with the right arm stretched out along with the kicking leg for balance
High side kick has 3 key elements: 1. a strong support leg; 2. a flexible kicking leg; 3. strong ab muscles to stabilize the torso. To kick at your maximum height, for demonstration purposes or to test yourself, it helps to drop your torso (as shown here) while keeping your eye on the target.
applications
The side kick is popular for self-defense and board breaking demonstrations for its dynamic and aesthetic impression. But far more impressive than the look of a crisp high side kick is its destructive capacity in self-defense and fighting. With good timing and distance, one perfectly executed side kick can knock an opponent out of the ring or to his knees.
SPORT APPLICATIONS
A high side kick is a good counterattack when an opponent steps in to attack.
A middle section side kick is a good way to push an opponent backward to create additional distance in sparring.
Using the heel of your foot to kick to the face of an unguarded opponent can result in a knockout.
SELF-DEFENSE APPLICATIONS
When an opponent lunges toward you, kick the inside of his knee to stop his advance.
A side kick to the rear of the opponent's lead leg is a quick and easy way to knock him to his knees.
When you catch or trap an opponent's kicking leg, throw a low side kick to the upper calf to take him down.
Use the long reach of the side kick to ward off an opponent with a weapon.
If you've fallen or been thrown to the ground, quickly throw a side kick into the rear of the assailant's knee to knock him down and away from you.
Side kick can also be used when both fighters are on the ground.
SIDE KICK: TOES UP OR DOWN?
For maximum power, your toes should be lower than your heel when you perform a side kick. This creates a slight downward turn to your kicking foot and drives your heel into the target. It also aligns your body weight directly behind your kick and aids in balance. If your foot is not turned slightly downward on impact, you might be unbalanced backward by a stronger opponent or an opponent who resists by throwing his weight forward into you. It is easier to turn your foot downward when you completely rotate your hips.
common mistakes
MISTAKE 1: Curved body or hip sticking out to the rear.
SOLUTION: The key to power in the side kick lies in precise body alignment. Don't let your hip leave the force line of the kick. Stretch your body fully so that your torso, hip, knee and foot are aligned on the same plane.
MISTAKE 2: Dropping the head.
SOLUTION: Look at the target over your kicking side shoulder.
MISTAKE 3: Kicking too high.
SOLUTION: For practical applications, side kick should be powerful and a kick that's too high can lack power because your body is no longer behind the kick. Choose a realistic target height.
• remember
1. Align your hip, knee, and foot, and kick along the shortest path to the target.
2. Rotate your hip fully for power.
3. Strike the target with the blade or bottom of the foot.
4. Lower the toes slightly below the heel to enhance power and balance.
• avoid
1. Dropping your head or taking your eye off the target
2. Kicking too high for the situation
3. Curved body alignment, hip sticking out to the rear
one thought one action drill
Although every technique has multiple stages of execution, you should do each kick as a single action without separation in time and motion. In the beginning, this will be very hard; it might even seem impossible. But there are many exercises and drills you can use to develop muscle memory and reach the single action stage: visualization, kicking at the stretching bar, practicing with a handheld or suspended target, doing timed target kicking drills (how long does 5 kicks take? 10?), shadow sparring and incorporating kicks into sparring, self-defense or forms practice routines. Finally, ultimately, try to skip the thought process of bringing your knee up, rotating the body to the side, looking at the target, etc. Just kick. Fast.
Hold the stretching bar and visualize the motion of side kick before you kick.
Kick, internalize the process, and adjust your posture. Repeat at least 50 times a day on each side to develop muscle memory for every new kick you learn.
FOR A GREAT KICK:
1. Learn, understand and internalize the mechanics of the technique.
2. Aim at a target of a reasonable height and distance.
3. Kick fast. A fast kick provides you with less time to think and analyze, so your body naturally adjusts and controls itself.
Penetrating force
Kicking power comes from the combination of your body weight, the speed of the kick, and the distance covered. Since your body weight is constant, it is important to increase your kicking speed and to release your kick into the target to the fullest extent possible. Think of kicking several inches behind the target, into the center of the heavy bag or behind your opponent's body. Strive for penetrating force, not pushing force, by releasing the full weight of your body into the target at impact.
Aim for a target that is several inches behind the one you want to strike and stretch your kick to the fullest.
Explosive penetrating force causes the target to collapse rather than be pushed away. Imagine the effect on a human torso.
TIPS FOR HIGH SIDE KICK:
You should be practical with side kick; generally, you do not need to kick higher than stomach level. However, if you want to improve the height of your side kick, here are some exercises you can do:
1) Do low horse riding stance for 60 seconds (photo 2). Rest for 30 seconds.
2) Do a deeper horse riding stance for 60 seconds.
3) Squat down and hold your ankles. Breathe deeply for 60 seconds.
4) From step #3, put both hands on the floor in front of you. Slide your right foot all the way to the right and stretch your right leg for 60 seconds. Repeat with the left leg.
5) Stand up and walk around while breathing deeply and shaking your legs and ankles to loosen up the muscles and joints.
6) Hold a bar, chair or table for support, and raise your right leg for a slow side kick. Hold in the air as long as you can. Repeat on the left side. Do a minimum of 3 sets of 15 seconds on both sides.
7) If your have a partner, let him hold your leg and raise it slowly in side position to a little past the point where you are comfortable. If you don't have a partner, you can use a wall (photo 3) to support your kick. Repeat 3 times on each side, holding for 15 seconds each. Eventually you might be able to use a door frame for this exercise (photo 1)
Repeat steps #1-7 three times a week for twelve weeks. Monitor your progress weekly by keeping a journal so you can see your progress over the three month period. It is important to know your limits and progress from there. As long as you are making positive progress, where you began from isn't very important.
KNEE KICK
purpose
Knee kick is used to strike an opponent in close quarters or ground fighting. The main goal of knee kicking is to systematically drain the opponent's energy and will by inflicting pain on exposed targets like the thigh, rib cage, stomach and groin. For advanced practitioners, a jumping knee kick to the head can be a deadly weapon.
key points
Secure the opponent by holding his neck, head or torso before kicking. Precisely attack vital points such as the groin, outer thigh, and rib cage.
striking area
Upper knee, frontal knee, inner knee
targets
1. groin
2. lower abdomen
3. solar plexus
4. sternum
5. head
6. rib cage
7. thigh (outside, inside)
8. knee
how to
1. From fighting stance, shift your weight to the front leg.
2. Raise your knee toward the target.
3. Snap the pelvis and thrust the knee upward while pulling down with your hands and sharply exhaling as you tighten your abdomen muscles.
4. For repetitive knee strikes, set your foot down toward the rear of your stance. For combination strikes, set your foot down in front, close to the opponent.
FRONT VIEW: Begin practicing with a mid-height knee kick. Hold this position for 15 seconds at a time to develop pelvic control.
JUMPING KNEE KICK
A jumping knee kick is one of the most advanced fighting techniques. When you deliver a jumping knee kick, keep your head up, focus your attention on your lower abdomen, and try to form a triangular shape with your body (your striking knee is the tip of the triangle). Notice how compact the body is at the moment of impact.
variations
Knee kicks vary from style to style and performer to performer. You can kick vertically, horizontally or diagonally, upward, downward, inward or outward. Or you can kick in some combination of the above. The knee kick is the most versatile kick, particularly in combat sports.
BOTTOMLINE: Strive to develop absolute pelvic control to manipulate the direction and force of the knee kick.
Straight upward knee kick
Roundhouse knee kick to the middle section
Jumping roundhouse knee kick
Jumping horizontal knee kick
THE KNEE: the king of kicks
The knees are the most powerful weapon of your body. In most traditional martial arts competition, knee kicks are prohibited due to their potentially dangerous effects. In modern combat arts, knee kicks are known for their brutal destructive force in both standing and ground fighting.
Visualize your Kick
There are two types of visualization that you can use to improve your kicking. The more basic method is technique-specific. Begin with this and then move on to the mind-specific method as you advance.
1. Technique-specific: See or visualize yourself doing a technique step-by-step in great detail, such as the five stages of each kick: Ready, Shift, Pivot, Kick, and Repose. This is a good way to improve the accuracy of a technique.
2. Mind-specific: Mentally focus on the technique as a whole. In your mind, see yourself giving a perfect performance, like a movie star. This is a good way to improve the speed and efficiency of your techniques.
applications
Knee kick is the most powerful kick in a close quarters fight. A solid knee kick to the outer thigh or groin can easily send an opponent to the floor. A kick to the rib cage or solar plexus can stop the opponent's breathing. A kick to the chin or the nose can result in an instant knockout. On the ground, a knee kick is useful for keeping an opponent on defense or weakening his resolve.
SPORT APPLICATIONS
Grasp the side or rear of the opponent's head to hold him in place for an upward knee kick.
Grab behind the opponent's neck to drive his body downward into your roundhouse knee kick to the lower abdomen.
SELF-DEFENSE APPLICATIONS
A drop knee kick is a powerful way to finish a downed assailant.
Subdue a struggling assailant with knee kicks to the ribs.
common mistakes
MISTAKE 1: The most common error in knee kicking is angle control - either because the knee is not folded tightly or because the hip is opened too wide.
SOLUTION: Keep your knee folded as sharply as possible and thrust your pointed knee using the snapping force of your pelvis. By keeping your knee folded tightly and using your pelvis to control the direction of the kick, you'll find it easier to control the angle of your strike.
MISTAKE 2: Another common problem is distance control, often due to hesitation.
SOLUTION: Don't wait for perfect time to strike. In practice, focus on seizing the moment and throwing your knee instinctively to openings on the thigh, stomach, kidney, chest, neck or head.
• remember
1. Point your knee.
2. Use your pelvis for power and control.
3. Practice short, middle, and long knee kicks.
4. Synchronize the upward motion of your knee with the downward movement of your hands.
5. Exhale when kicking.
• avoid
1. Poor angle of kick, not folding the knee tightly
2. Wide arc of kick, opening your hip too wide
3. Tardy execution
4. Muscle tension
5. Striking with the edge of the kneecap
6. Self-injury
RAISING KICK
purpose
Raising kick is used to improve leg flexibility. However, if you drop your foot with force after raising it, it becomes an axe kick. If you kick an opponent who is behind you, it becomes a rear kick. It can also scare an opponent into backing up so you can follow up with a primary kick.
key points
Relax your body and keep your torso erect to gain kicking height. Keep your arms near the kicking leg for balance and control.
striking area
Ball, toes, bottom of the foot, heel
targets
1. head
2. sternum
how to
1. From fighting stance, shift your weight to the front leg.
2. Bring your straight leg up toward your chest and let your foot go over your shoulder. Do not put any power in this movement; simply focus on raising your leg as high as possible.
3. Return your leg along the same path in a relaxed but controlled way.
*Breathe out as you raise your leg and breathe in as you lower your leg.
variation
1. Pull your toes downward for an extra stretch in your hamstring muscles.
2. Point your toes to get extra length in your kick.
applications
Although raising kick is primarily used for improving flexibility and the height of your kicks, it can be practical in unusual ways, perhaps due to the element of surprise. In just the right situation, raising kick can be a practical way to shock an opponent with an unorthodox counter-movement.
SPORT APPLICATION
Use a quick, light raising kick to block an opponent's high front kick or inside crescent kick.
SELF-DEFENSE APPLICATION
Raising kick is a good way to improve your axe kick by dynamically stretching the hamstring muscles.
In this unorthodox self-defense application, you can throw a raising kick against an assailant who attacks from behind, striking with the the ball of the foot.
common mistakes
MISTAKE 1: Hunching the upper body.
SOLUTION: The most common cause of poor posture, especially in the torso, is excessive muscular tension, either because you are trying to kick too high or too hard. To remedy this, keep your upper body erect and relaxed while you raise your leg smoothly and without power.
MISTAKE 2: Raising up on the toes. This is also usually caused by kicking too high or too hard for your skill level and will often lead to another common mistake: lowering your upper body backwards.
SOLUTION: Relax and kick at an attainable height without power. Reduce muscle tension by having realistic expectations. Bend your standing leg (knee) slightly to eliminate the strain in your hamstrings that leads to raising up on your toes.
• remember
1. Keep your posture upright.
2. Relax your shoulders and neck.
3. Advance progressively.
4. Raise your leg smoothly, without power.
5. Exhale as you raise your leg.
Bending your standing knee slightly will allow you to kick higher while maintaining good posture. Keep your arms around your kicking leg for balance.
• avoid
1. Struggling to kick higher than your present skill level allows
2. Trying to progress too quickly
3. Hunching your upper body
4. Lifting your standing heel excessively
5. Leaning your body backward
timing & breathing
Breathing is key to converting internal energy to power. When your breathing and technique are in sync, your kick carries maximum force.
Breathe naturally as you move and your body will tell you when to breathe in and out. Here are some guidelines to help you get started:
1. For direct kicks, inhale before kicking and exhale when you kick. At the moment of impact, exhale sharply while tightening your lower belly.
2. For stepping kicks, inhale when you raise your leg, step forward, turn around, or jump. Exhale when you kick.
BOTTOMLINE: Power comes with exhalation at the moment of impact.
HOW TO KICK WELLwith your weaker leg:
1. Stretch the weaker leg muscles well and always warm up to prevent injuries.
2. Strengthen your weaker leg by doing very slow controlled kicks wearing a light (2 to 5 pound) ankle weight. Start with the lightest weight and work your way up. Start with 3 sets of 10 repetitions of front kick or roundhouse kick.
3. Remove the weight and do 10 repetitions slowly and then 10 more with speed.
4. Practice 3 sets of 15 repetitions on a heavy bag or handheld target with power.
5. Practice steps 1-4 consistently, 3 times per week. You should begin to see results in a few weeks. When you do, increase the number of sets, speed and power or try applying the exercises to more advanced kicks.
6. Track your progress in your training diary or notebook.
7. Re-evaluate your progress and revise your goals every two weeks.
OUTSIDE CRESCENT KICK
purpose
Outside crescent kick is used to strike the opponent's face at an in-to-outward angle. It is similar to whip kick; the difference is that, for whip kick, you should turn your body fully to the side with a larger arc whereas for outside crescent kick, your body faces more to the front than the side.
key points
From fighting stance, first rotate your upper body and coil your hip in the opposite direction of the target, then arc your foot in front of you, diagonally across the target.
striking area
Blade, bottom or ball of the foot
targets
1. face
2. neck
how to
1. From fighting stance, shift your weight to the front leg and begin to rotate your front leg and torso diagonally away from the target.
2. Bring your foot above your head, arcing outward.
3. Snap your foot outward in an arc toward the target, striking the target at the top of the downturn.
4. After kicking, control your foot as you complete the outward arc and return to fighting stance. Maintain your center on your standing leg throughout the kick so your kicking leg is easier to control.
FOR STRONG FLEXIBLE KICKS
Kicks like crescent kick and axe kick derive their power from a large range of motion, so they require flexibility as much as muscular strength.
To develop power in these kicks, try using a bike tube or resistance band, with one end attached to a fixed object and the other to your ankle. Perform your kicks so that they are traveling away from the fixed end of the band, creating resistance for each repetition. Control your kicks to avoid injuring your joints during resistance training.
As you get stronger, add more tubing or heavier tubing to increase the resistance.
applications
Outside crescent kick is useful for kicking at or knocking away anything entering the vertical centerline of your body. You can kick an opponent's face or you can use your foot to deflect a punch, grab or kick. Outside crescent kick can strike a target with the bottom of the foot, the blade, the ankle, or even the shin.
* Begin with a low kick and kick higher as you get used to the motion.
SELF-DEFENSE APPLICATIONS
Just after an opponent finishes an attack, counter with an outside crescent kick to the face using the bottom of the foot (timing and distance are the key factors).
Strike the assailant's knife wielding hand with the ankle of your outside crescent kick.
Use the long range of the outside crescent kick to knock the opponent's punching hand away (attack while defending).
* Outside crescent kick should be used with follow-up techniques such as:
Outside crescent kick + rear hand punch
Outside crescent kick + rear elbow strike
Outside crescent kick + rear leg front kick
Outside crescent kick + rear leg roundhouse kick
Outside crescent kick + rear leg knee kick
common mistakes
MISTAKE 1: Raising the leg straight upward.
SOLUTION: Balance and the correct arcing motion are the most difficult to achieve with this kick. If you have trouble making the in-to-outward arc, start at waist level where it is easier to control your leg.
MISTAKE 2: Bending your torso or leaning backward.
SOLUTION: When your upper body is tilted and the muscles are tight, the arc of your kick becomes smaller and you lose your balance. Focus on keeping your head upright and your shoulders relaxed. Emphasize flexibility rather than power.
• remember
1. Keep your head upright.
2. Relax your shoulders.
3. After kicking, let your foot go and repose your body naturally.
• avoid
1. Bending the torso
2. Raising the leg straight upward
3. Leaning backward
INSIDE CRESCENT KICK
purpose
Inside crescent kick is used to strike a close-range target located in front of you. The out-to-inward trajectory makes it a good surprise attack. The target is usually the face or side of the head. It is also useful for chopping downward on the clavicle area.
key points
Slap the target with your foot from the outside. Relax your shoulders, erect your spine and raise your foot lightly before releasing it into the target. Use the element of surprise to catch your opponent off guard.
striking area
Bottom of the foot, heel
targets
1. face
2. chest
3. clavicle
how to
1. From fighting stance, shift your weight to the front leg.
2. Bring your rear leg up about 15° outside of the target line. Snap the inner bottom of the foot inward in an arc to the target (usually the face or head). You can impact the target horizontally or at a diagonally downward angle.
3. To finish, continue through the target and return to fighting stance.
Your head should be at the vertical center at all times while your hands travel in the opposite direction of the kick for balance and power.
*Kick only flexible targets with inside or outside crescent kick. Kicking an immovable target can damage your knee or hip.
applications
Inside crescent kick has a surprise element that can be used to intimidate an eager opponent. It is useful when the opponent attacks your middle or low section, when he lands short after an attack, or when he has a weapon in his hand. The swinging of your foot should be well timed to hit a moving target. Strive for a light, fast and precise kick.
SPORT APPLICATIONS
Counter your opponent's punch to the stomach with an inside crescent kick to the temple.
Immediately after the opponent finishes a backfist strike, throw an inside crescent kick to the face.
common mistakes
MISTAKE 1: The most serious mistake is incorrect distribution of weight.
SOLUTION: If you throw your foot forward too heavily, you expose your face to the opponent. If you lean back while kicking to move away from the incoming opponent, you fall backward. To avoid these errors, make sure that you use short agile footwork before lifting your foot and stay centered over your kicking leg.
• remember
1. Center yourself for balance before kicking.
2. Move your feet first to create optimal distance from the opponent and then kick.
• avoid
1. Tilting the torso
2. Over-rotation of the body
3. Wide arc of the kick
4. Throwing your weight forward
AXE KICK
purpose
Axe kick is primarily used to strike the head and a successful kick often results in a knockout. It is also effective as a counterattack against a roundhouse kick or in stopping a reckless opponent from rushing in on you.
key points
Have a firm base (standing leg) for stability and shift your weight completely onto your standing leg to fully relax the muscles in your kicking leg. Raise your leg as quickly as possible to gain maximum height.
striking area
Bottom of the foot, heel
targets
1. face
2. clavicle
3. sternum
how to
1. From fighting stance, shift your weight fully to your front leg.
2. Bring your leg straight up on the target line with the knee slightly bent. Bring your knee to your chest and, when your foot reaches maximum height, snap it downward.
* You may raise your leg with the knee straight and chop downward the foot, keeping the knee straight. Axe kick may be executed with a snapping force or a chopping force, depending on the application.
As the leg raises vertically, the arms spread horizontally for balance. The entire body should move in unison to generate maximum force in the foot at impact. Tuck your chin down to direct the force of your body forward and prevent leaning back during kicking.
The higher the foot goes, the more power your kick produces. The power of axe kick originates in a large range of motion.
applications
The ultimate goal of axe kick is to strike a stationary or moving target at a perpendicular angle, utilizing the advantage of the length of the leg. Although it requires greater distance to be effective, it is especially useful against a shorter weapon. Thus it is popular in martial art competition to stop punchers and to create distance between you and the opponent.
SPORT APPLICATIONS
Use axe kick to the head as a close quarters counterattack against punching.
Axe kick is a good way to make the opponent move away from you, to create distance or stop an aggressive opponent.
SELF-DEFENSE APPLICATION
An axe kick is a safe and powerful technique against a crouching opponent.
axe kick combinations:
• Axe kick + roundhouse kick
• Axe kick + side kick
• Axe kick + punch + roundhouse kick
• Axe kick + hopping front kick
ACTIVE MUSCLE RELAXATION
When you first begin practicing kicks like axe kick and raising kick, you might find it quite hard to relax your leg muscles, which will cause your kicking height to suffer. Because the leg is so heavy, it can seem nearly impossible to lift it with the lightness necessary to kick above your head while maintaining your posture and balance. To help you relax your muscles while practicing, try using a support, like a stretching bar, to assist with your balance. The stretching bar will help you anchor your upper body position and take some of the stress off your standing leg, allowing you to relax your kicking leg fully.
1. Hold the bar with your rear side hand and anchor your posture with your rear hand and front leg.
2. Swing your rear leg up and down a few times, beginning small and increasing in height progressively.
3. As you feel your kicking leg relax and lighten, raise it once to maximum height, then return to step 2. Repeat 10 times on each side.
height drills
Using a visual, physical target is a good way to increase your kicking height and to keep track of your improvement. If you are having difficulty improving, take a measured progressive approach. For example, set a goal of increasing your target height one-quarter inch every other day. If you can achieve these small increases regularly, you'll have increased your kicking height by 12 inches in about 3 months!
common mistakes
MISTAKE 1: The most serious mistake with axe kick is overworking your muscles. When you try too hard to raise your leg high, the muscles get tight leading to pain and frustration.
SOLUTION: The easiest solution is to lower your target height and build up progressively. If you have prolonged pain in your muscles, rest and heal first, then begin at a height you feel comfortable with and build up gradually as described on pages 126-127.
MISTAKE 2: Pulling the toes of the kicking foot.
SOLUTION: If you lift your leg with your foot, you'll find that you have to pull your toes back to create enough lifting force. Instead, lift your leg from your hip and thigh, keeping your foot and ankle relaxed.
• remember
1. Create good posture to leave the kicking leg free of tension.
2. Plant your standing foot firmly for stability.
3. Build the height of your kick progressively.
4. Raise your leg straight upward quickly and drop your foot on the target.
• avoid
1. Pulling the toes, which causes the hamstring muscle to tighten
2. Lifting the heel, decreasing stability
3. Over-stretching, causing a loss of balance and opening yourself to counterattacks
WHIP KICK
purpose
Whip kick is used to strike the opponent across the head or face. In general, use a back leg whip kick for attacking and a front leg whip kick for counterattacking. Setting up the proper distance and height is the key to success in using the whip kick as a counterattack. The best option: just as the opponent finishes his kicking attack, throw a whip kick counter to his face.
key points
Whip kick is a good way to strike your opponent with a surprise counterattack. Wait until your opponent exposes his face after an attack and then strike.
striking area
Bottom of the foot, toes
targets
1. face
2. temple
3. neck
how to
1. From fighting stance, shift your weight to the front leg.
2. Bring your knee up pointing toward the target.
3. Pivot your standing foot and align your body perpendicular to the target, with your shin parallel to the floor and knee tightly folded.
4. Raise your knee as you unfold it and hook your foot across the target, pulling your foot toward your buttocks as you open your hip. Kick through the target and return to fighting stance.
Kick horizontally across the target with maximum range of motion.
The horizontal alignment of the hip shown here enables you to kick higher and wider.
applications
A front leg whip kick is faster and easier to do than a rear leg whip kick. The most effective targets for whip kick are the jaw, cheek, temple, and neck. The key to throwing a successful whip kick is timing; you must react as soon as you see an open target. If you delay even a fraction of a second, your opponent will be too close or too far away to strike. When you practice whip kick, focus on a quick initiation of the kick and a direct path of movement to the target.
SPORT APPLICATIONS
As the opponent steps in, throw a front leg whip kick to the jaw.
When the opponent lands after an attack, throw a front leg whip kick to the ear.
As the opponent raises his leg for axe kick, launch a whip kick counterattack.
SELF-DEFENSE APPLICATION: Use a whip kick to the neck of an opponent preparing to attack.
MARTIAL ARTS APPLICATION: Right: In a pre-arranged step sparring technique, grab the opponent as you kick over his punching arm.
range of motion
Range of motion, or ROM, is the measurable distance between the flexed position and the extended position of a joint. The greater the ROM of your kick is, the more power it generates. ROM can be increased by improving flexibility of the muscles in the legs and torso, and of the joints in the hip, spine and knee.
Whip kick practice to a hand-held target is an excellent way to improve ROM in a safe way. Avoid kicking immovable targets with whip kick.
For advanced practitioners, additional follow through after a whip kick is a good way to increase power.
The greater the range of motion in your kick, the more power your kick generates; the more flexible you are, the greater range of motion you have. The exercise shown here will increase the range of motion and flexibility of your hips for whip kick.
Holding the bar, point your toes and lift your right leg to the left, across your body. This stretches your hip and torso muscles.
Begin to rotate your upper body to the right and open your hips as you raise your leg. This strengthens the muscles of the buttocks while opening the hips.
Continue raising your leg, with your toes pointed. The final position of this stretch is the same as the moment of impact for whip kick. This stretches the hip joint and the muscles in the front of the leg.
common mistakes
MISTAKE 1: Exposing your torso to the opponent during or after kicking.
SOLUTION: This is generally caused by poor posture or kicking too slowly. Be sure to keep your hands and arms close to your body in a guarding position, whip your kick quickly across the target and avoid spinning your torso toward the target after kicking.
MISTAKE 2: Kicking too short or with limited range of motion.
SOLUTION: The main causes are lack of flexibility and coordination. Whip kick is a complex movement that works against the natural tendencies of the body because it forces the spine to bend backward. Improving the flexibility and strength of your back and ab muscles in addition to stretching your legs and hips will correct this problem.
• remember
1. Keep your head straight.
2. Firmly plant the standing foot.
3. Pivot your foot for a quick and smooth rotation of the body.
4. Keep your kicking leg free of tension.
Extend your foot about 6-12 inches in front of the intended target. More than this and your kick will be too wide. Less and you will not have enough space to generate a powerful slapping motion prior to contacting the target.
• avoid
1. Kicking too high
2. Exposing your torso to the opponent after kicking
3. Kicking too slowly and exposing vital targets to the opponent
TWIST KICK
purpose
Twist kick is an irregular type of kick which travels in the opposite direction of conventional kicks. The power and effectiveness of twist kick come from coiling and uncoiling the body from an unusual angle, however this also makes it vulnerable. The danger is that you may expose vital targets on your torso after kicking, so you need to protect your body during and after kicking.
key points
Keep your head and standing foot mostly fixed while rotating your chest and hips sequentially in an outward U shape.
striking area
Ball, blade, instep
targets
1. face
2. temple
3. neck
4. rib cage
5. stomach
6. inner thigh
7. inner knee
how to
1. From fighting stance, shift your weight to the front leg.
2. Bring your knee up and twist your body slightly inward, beginning with your chest followed immediately by your hips. As you do so, raise your kicking leg with the foot and knee tilting about 30 to 45 degrees upward.
3. Pivot your standing foot, reverse the rotation of your hip and torso outward, unfold your knee outward, and kick outward-upward with the outside, ball or blade of your foot. All of these steps happen sequentially and almost simultaneously, like coiling and releasing a spring.
variations
Twist kick is useful for kicking soft or vulnerable targets such as the inner thigh, groin, lower abdomen, solar plexus, neck, chin, nose, ears, and mandibular joint. For kicking to the lower abdomen, solar plexus or neck, striking with the ball of the foot (photo 1) is most effective due to its penetrating power. For most of the other targets, the outer side of the foot (photo 2) is effective due to its wide striking surface area.
When striking with the ball of the foot, rotate your body fully so that the ball contacts the target area as sharply as possible.
When striking with the outer side of the foot, put your body weight behind the kick for additional power
applications
This is a kick that is rarely used, so the element of surprise is one of its primary advantages. Use it to shock or confuse your opponent then follow up with a powerful combination such as an elbow strike or roundhouse kick.
SELF-DEFENSE APPLICATIONS
A long twist kick to the groin can stop an advancing assailant.
A good time for a twist kick is after the opponent misses with a forward attack.
hip joint exercises
Because of its unusual path of striking, twist kick can result in injuries to the muscles and joints of the hip or groin. You can improve your flexibility by doing these simple exercises:
Vertical foot lifting: Alternately lift your right and left foot to hip level. Inhale while resting and exhale when you lift your foot. You can do this in-place or while moving around the room, interspersing the knee lifts with footwork. Repeat 3 sets of 20 repetitions, alternating sides.
* After the exercises, shake your legs to release tension in your hips. You can stand and shake one leg at a time or sit on the floor and shake both legs simultaneously.
Outward knee lifting: Lift your knee outward, alternating legs. Raise your foot only to knee level. As you raise your knee and open your hip, your body will twist naturally, creating a coiling motion.
common mistakes
The most common mistake is putting too much power in the kick, resulting in a breakdown in posture and inefficient use of your energy as well as potential joint injuries. To correct this, hold a stretching bar and practice slowly without force until you master the path of the kicking motion and can generate force from range of motion rather than muscle power.
• remember
1. Center your body.
2. Generate power from your hips.
3. Keep your base foot firmly Planted.
4. Keep your head as straight as possible.
• avoid
1. Kicking too far past target
2. Breaking the vertical center
Twisting Power
Twisting power comes from coiling and uncoiling the muscles of the torso. Before practicing twisting outward with your kick, you can develop your twisting skills by kicking upward first.
1. Bring your knee forward at a tilted angle. Your arms swing to the left initially and return to this position naturally.
2. Raise your knee to your chest, while pivoting your standing foot.
3. Kick at a target directly on your centerline. Note the difference between this target and the one on page 146, which lies about one foot outside the body.
PUSHING KICK
purpose
Pushing kick is used to stop an incoming opponent or to push away an opponent to create space for a stronger attack. Chamber your knee quickly and push the opponent's lower abdomen or thigh. You can also raise your nearly straight leg and push straight forward. The goal of this kick is to destroy the opponent's equilibrium.
key points
Attack the opponent's center of gravity. Don't kick too high; kick the middle of the body. Maintain your balance by bringing your arms and torso forward with your kicking momentum.
striking area
Ball, bottom of the foot
targets
1. lower abdomen
2. femoral crease
3. thigh
4. solar plexus
how to
1. From fighting stance, shift your weight to the front leg.
2. Bring your knee forward toward the target.
3. Raise your knee to chest level and push your foot forward. Keep your upper body erect and bring your body weight forward with the momentum of the kick.
* You may push with the ball or bottom of the foot.
variations
Pushing kick is a timing-sensitive technique. Use it to take advantage of an opponent who rushes in recklessly or to push away an indecisve opponent as he hesitates. If you need to react quickly to an opportunity, use a front foot pushing kick; for a powerful push, use the rear foot. The former creates speedy action; the latter an agressive attack.
Hopping front foot pushing kick as a quick counter
Rear foot pushing kick for a powerful attack
applications
Pushing kick is a tactical skill used to better control distance in a fight by unbalancing the opponent's center of gravity. Push a joint, such as the knee or hip crease. According to the situation, you can push straight forward with a front kick, horizontally with a side kick, or diagonally with a hybrid of the two. The fundamental principles of execution for all of these applications are the same.
SELF-DEFENSE APPLICATION: Use a diagonal pushing kick to the knee to intimidate an opponent.
SPORT APPLICATION: Use a long pushing kick to the opponent's torso to create space for a follow up attack.
SELF-DEFENSE APPLICATION: Use a side pushing kick to the rib cage while avoiding the assailant's lunging attack.
SELF-DEFENSE APPLICATION: A stomping push kick to the groin can incapacitate a downed assailant allowing you to escape to safety.
SPORT APPLICATION: a long pushing kick with the ball of the foot negates the opponent's punches.
common mistakes
MISTAKE 1: Losing balance during the kick.
SOLUTION: Keep your standing knee slightly bent and don't overextend your kicking leg. Also, bring your upper body forward with the kick and keep your arms close to your body; don't lean backward.
MISTAKE 2: Telegraphing your movement.
SOLUTION: Begin your kick like front kick and push at the last moment. Don't make your initial movement too large.
• remember
1. Push through the target.
2. Bend supporting knee.
3. Use your bodyweight.
4. Control your torso.
• avoid
1. Opening your arms
2. Leaning backward
3. Overextending your kick
4. Kicking above the waist
weight shifting for pushing kick
Pushing kick requires coordination and flexibility to deliver power. Here are examples of exercises for rear (1-3) and front (4-6) foot pushing kicks using a heavy bag.
1. From fighting stance shift your weight to the front leg.
2. Bring your rear knee to the front at chest level and lean your body forward as you come into contact with the bag.
3. Push your foot as far into the target as you can without breaking your posture. Be sure to bring your upper body forward with the kick and keep your guard up.
4. From fighting stance, hop your rear foot to your front foot.
5. Upon arrival of the rear foot, quickly raise your front foot, keeping your body turned slightly sideways.
6. Stretch your kicking leg into the target and lean your body forward as you push the bag.
FLYING SIDE KICK
purpose
Flying side kick is used to strike a runaway or charging opponent. The dynamic appearance of flying side kick makes it popular for demonstrations as well. The ultimate goal of practicing this kick is to attain maximum jumping distance and height. It requires muscular strength and coordination to achieve complete synchronization of the body while in the air.
key points
The main force of flying side kick comes from momentum and the extension of the leg on impact. When you jump, make your body compact and thrust your leg at the target just prior to impact.
striking area
Blade, ball, bottom of the foot, heel
targets
1. head
2. neck
3. chest
4. solar plexus
how to
1. From fighting stance, size up the distance and height of the target.
2. Step forward and take as many steps as you need to pace yourself.
3. At the point of jumping, pivot your body 180° and elevate your torso as high as you can, while raising your kicking knee to chest level.
4. Just before reaching the peak of the jump, bring both knees close to your body and let your compact body "fly" toward the target.
5. At the peak, snap your primary foot out in a side kick. Keep your other leg tucked in close to your buttocks to augment the flying momentum.
Upward flying side kick is more advanced than conventional side kick and requires outstanding flexibility and strength of the leg muscles. The torso leans back more than horizontal flying side kick to allow the kicking leg to strike upward.
applications
Flying side kick is powerful because it delivers the weight of the entire body, which is propelled by the momentum of running and jumping. When used with control and coordination, it can knock an opponent to the ground. Flying side kick requires a long approach, so it is more commonly used for demonstrations and board breaking than in practical applications.
A short flying side kick against a lunging opponent can take his breath away and establish dominance.
Flying side kick is an excellent technique for demonstrations and board breaking. Practice in front of mirror to create a proper form and with a light target (such as a handheld target) to develop accuracy.
* When executing a flying side kick without a firm target to strike, for example in a demonstration, jump, extend your leg and hold it in the air rather than kicking to full extension. Putting too much power in a flying side kick without a proper target to absorb the impact can result in joint injuries.
PERFECTING YOUR FORM
• Build strong muscles in the legs through squats, lunges and jumping drills.
• Strengthen the stomach muscles with sit-ups and crunches.
• Jump rope or run in sand to strengthen your knees and ankles.
• Practice running rhythmically and jumping after a consistent number of steps.
• Align your body in a triangular shape (head, kicking foot and supporting knee as shown below) at the peak of your jump.
common mistakes
The most common mistake occurs in jumping—either in lowering the torso to the side or rushing the sequence of movements. To correct this, follow a natural sequence: run to pick up momentum, lift your rear knee forward as high as you can, pivot your torso using the momentum of the rising knee, stretch the kicking leg toward the target while bringing up the supporting leg up. Begin from a low jump and build up in height. Be sure to build your leg strength sufficiently before attempting this kick.
• remember
1. Elevate your torso first.
2. Raise your kicking knee high.
3. Pivot your body to the side and align your torso and legs on one plane.
4. Make your body compact while in the air and unfold it when kicking.
• avoid
1. Jumping too far or too high
2. Dropping your non-kicking leg
3. Rushing your jump or kick
4. Kicking with too much power
5. Lowering your torso to the side or rear
6. Opening your arms too wide
SPINNING & TURNING KICKS
The spinning and turning kicks in this section are based on what you have learned so far. Spinning or turning kicks involve quick rotation of the body while maintaining proper form.
Three important elements for successful spinning kicks are focus, balance and agility. Focus is enhanced by spotting the target before and after turning your body and following a direct kicking path to the target. For balance, keep your center of the gravity lower than normal and stay relaxed. For agility, concentrate on moving intuitively rather than analytically. To succeed, be in tune with your natural awareness rather than worrying about your mistakes.
BACK KICK
purpose
In competition back kick is very often used for counterattacking against roundhouse kick or as a surprise attack. Back kick's power comes from the rotation of the body, however deceptive execution adds to the kick's effectiveness. Since turning the body is risky, you should spot the target immediately after turning and be on guard at all times.
key points
Start from a sideways stance to reduce the initial movement of your body and initiation time. Rotate your hip first and then throw your foot directly to the target, delivering your entire bodyweight, which is magnified by the spinning motion.
striking area
Blade, bottom of the foot, ball, heel
targets
1. abdomen
2. solar plexus
3. neck
4. face
5. kidney
how to
1. From fighting stance, shift your weight to the neutral position.
2. Turn your head and spot the target over your shoulder as you rotate your body so your back is to the target. Chamber your knee with your kicking foot next to the standing knee.
3. Shoot your foot out while watching the target over your shoulder and protecting your vital targets with your arms. After kicking, return to the original position or step down into fighting stance.
Keep your head upright for optimal balance. Spot the target over your shoulder while maintaining your equilibrium. Utilize your arms to center your body and to protect your torso while turning. Keep your foot under your hip along the centerline to protect your groin against a counterattack.
variations
Variations of the back kick are most useful when applied in the least expected manner. Back kick from the ground, for instance, is an effective technique to surprise or confuse an opponent who expects you to stand and fight. As soon as you are done kicking, use your opponent's momentary distraction to get up and protect yourself from standing position.
From the kneeling position, put your hands on the floor so that you have a firm triangular base of one knee and two hands to support a back kick to the lower stomach of an advancing opponent. Do not turn your body; keep it sideways so that your groin and face are covered.
Keep your kicking foot tilted so your toes are lower than your heel.
Keep your head up and your eyes on the target. Bend your kicking knee slightly to facilitate balance and the ability to move quickly after kicking.
BODY ALIGNMENT
To hit a moving opponent, you need to spin and kick quickly. For accuracy in such a fast and complex movement, your sight line, foot, and target should be aligned on one plane. To perfect your kick, slow supported-kicking drills (right) can help you develop muscle memory. The more familiar your body is with the sequence of the kick's movements, the more precise your kick will be.
Three point stabilization (1 foot and 2 hands) enhances posture and accuracy while developing muscle memory.
applications
Back kick is a versatile kick that is primarily used to stop an incoming opponent or counterattack against a circular kick (roundhouse kick) or a vertical kick (axe kick). You can use it at short or medium distance, but avoid using it from long range. Retract your foot immediately and stay away from or move in close to your opponent after kicking.
SELF-DEFENSE APPLICATION: Left: When an assailant grabs you from the rear, take one step forward and kick while leaning forward to gain space.
SPORT APPLICATION: Below: As the opponent lands after jumping roundhouse kick, counter with a short back kick.
SPORT APPLICATION: As soon as the opponent completes his left turn kick and drops his body forward, counter with a right back kick.
SPORT APPLICATION: Against a left axe kick, when the opponent drops his head forward, counter with a back kick.
A powerful back kick is an audience pleasing demonstration skill. If you time your kick and the reaction of your partner well, the result is awesome, artistic and entertaining.
Back kick on the ground is frequently used in arranged step sparring in traditional schools and is a good way for students to practice the correct angle, distance and force control for back kick.
common mistakes
MISTAKE 1: The most common mistake is exposure of the hip, on which your opponent can land a pushing kick counter.
SOLUTION: This problem can be solved by turning your body more to the side and aligning the kicking foot, hip and shoulder on one plane.
MISTAKE 2: Another common problem is exposure of the groin and chest areas to the opponent's counterattack.
SOLUTION: This can be solved by closing your chest a little bit (turning it away from the target) and keeping the toes of the kicking foot slightly lower than the heel.
To kick higher it is acceptable to lower your torso, however, you should still look at the target, align your body on one plane and keep your weight neutrally centered over your standing leg.
• remember
1. Turn your body to the side.
2. Shift your weight to neutral position.
3. Spot the target over your shoulder.
4. Align your body for maximum length.
5. Withdraw your foot in reverse of the way you kicked.
• avoid
1. Exposure of your hip
2. Exposure of your face and chest
3. Misalignment of the body
4. Over-rotation
TURN KICK
purpose
Turn kick is used to attack deceptively, to add the power of spinning to roundhouse kick, or to counterattack while retreating. It is powerful enough to knock down an opponent and effective for penetrating defense. It is usually used in combination with double or triple kicks, a roundhouse kick or a back kick.
key points
Once you spot an opening, don't hesitate; turn and kick in one smooth committed motion.
Stalling during the turn hinders balance and naturalness. Keep your torso upright and lower your center of the gravity.
striking area
Instep, ball, shin
targets
1. stomach
2. solar plexus
3. kidney
4. liver
5. neck
6. face
7. temple
how to
1. From fighting stance, shift your weight to the front leg.
2. Pivot your front foot, rotate your hip to the rear, toward the target, and then turn your torso in the same direction. Spot the target over your shoulder. Move your rear foot forward.
3. As your rear foot crosses your body, rotate your hip quickly toward the target, and chamber your kicking leg in the air.
4. Release your foot to the target and rotate your hip fully into your kick for maximum delivery of power.
Ideally, at the moment of impact, your torso and legs should make a triangular shape with the primary force traveling to the kicking leg. In the illustration above your standing foot and leg are the power base, your front side shoulder is the power stabilizer, your hip is the power hub, and your kicking leg and foot are the power transmitters.
IN MOTION, LITTLE THINGS MATTER:
1. Avoid crossing your feet.
2. Keep your knees slightly bent.
3. Avoid long strides. Use short, quick footwork when turning.
applications
Turn kick is mostly used in competition, either moving forward or while retreating. It is powerful and deceptive when done correctly, however, it is reserved for advanced practitioners due to the speed and coordination required. If you lose your balance while turning or misjudge the distance to the target, you'll find yourself in trouble. To avoid problems, maintain good posture, turn your body quickly and smoothly, and complete the kick in less than a second, preferably in a half second or less.
SPORT APPLICATIONS
Moving faster than your opponent is the key to scoring with turn kick. To do so, you should try to perceive your opponent's intent to attack. Look for cues such as blinking, a sharp inhalation, lowering of the shoulder, movement the front foot, twitching the hip, etc. The above photo demonstrates a well timed turn kick, striking when the opponent's rear leg axe kick is at maximum height and his torso is vulnerable to a counterattack.
Before the opponent completes his high roundhouse kick, counter with a quick turn kick.
When an opponent hesitates, attack with a turn kick.
Counter with a retreating turn kick as soon as your opponent lands after kicking.
common mistakes
MISTAKE 1: Leaning the torso or head to the side.
SOLUTION: Avoid jerking your head when turning. Rotate your head and torso around the vertical axis that connects the top of your head and the center of the spine. Keep your head height level.
MISTAKE 2: Turning too widely or stepping in a zigzag pattern.
SOLUTION: If your turn is not made on a single line directly toward the target, you will find yourself out of position to hit the target. Rotate your hip first, spot the target over your shoulder and then turn so you have the target in sight at all times. Move on a straight line with your torso erect.
• remember
1. Rotate your hip first and let the other parts of the body follow.
2. Keep your head straight and rotate it smoothly to spot the target.
3. Relax your shoulders and allow your center of the gravity to guide you.
• avoid
1. Leaning the torso or head off the target line (
2. Jerky turning of the torso or head
3. Large turning motion
4. Zig zag stepping, stepping off the force line, poor distance control
5. Banging knees with the opponent
SPIN WHIP KICK
purpose
Spin whip kick is used to counterattack against roundhouse kick or axe kick. It can also be a powerful deceptive knockout attack. The knockout power of spin whip kick comes from the turning momentum of the body. It is stunning and devastating in effectiveness because it has the greatest range of motion of any kick.
key points
Timing and distance are the primary factors for a successful spin whip kick. Power is secondary because without proper timing and distance, a powerful spin whip kick is risky and useless.
striking area
Ball, heel, bottom of the foot
targets
1. temple
2. jaw
3. face
4. neck
how to
1. From fighting stance, shift your weight to the neutral position.
2. Rotate your body to the rear, chamber the rear leg and look over your shoulder.
3. Raise your kicking leg and keep your body as compact as possible before kicking.
4. Shoot your foot toward the target, about 6-12 inches in front of it, then hook across and through the target. Continue the spin and the trajectory of the kicking leg, returning to fighting stance.
Spin whip kick follows a fan shaped path, lashing the target as if with a whip. The power of this kick comes from a combination of the large range of motion of the foot and the central axis of the torso, around which the foot moves. The torso functions as a stabilizer and your standing foot acts as the pivot.
Diagonal sit-ups build the muscles of the torso used in spin whip kick. As your muscles become stronger, try a larger range of motion during the sit-up, which will increase the range of motion of your kick.
applications
The primary target for spin whip kick is the head, which means you should be flexible and fast. Generally, you can kick higher when you kick fast; you can kick fast when you have muscular stability. Through strength and flexibility training, you can attain both. That said, all you need to do is stay alert yet relaxed and look for or make an opportunity of perfect timing. Avoid hitting the torso with spin whip kick to prevent knee and back injuries.
SPORT APPLICATION: Spin whip kick is popular in Taekwondo competition. It is effective as a counterattack when your opponent attacks with an axe kick or roundhouse kick. The best time to counter is when his technique is at its peak. If you fail to do so, strike as his foot lands and his head leans forward.
SPORT APPLICATION: Use spin whip kick to strike the temple as your opponent steps in to attack.
SELF-DEFENSE APPLICATION: After the assailant completes a wide lunging attack, counter with a spin whip kick to the head, meeting force with force.
SELF-DEFENSE APPLICATIONS
A spin whip kick across the back of the knees can be used from the ground to take down an opponent.
If you've been knocked down, counter your opponent's charge with a spin whip kick from the ground.
360 ° BALANCE:
Spin whip kick is one of several advanced kicks that requires a 360° turn of the body. If you have practiced the other turning and spinning kicks in this chapter, you should have developed the foundation skills for a full turn.
To futher perfect your turning, focus on the following:
1. Stand in a neutral fighting stance (weight equally on both legs).
2. Rotate your hip, toros and head in sequence.
3. With the turning momentum let the rear foot "slide" around your body.
4. Repeat #1-3, getting progressively faster. Aim for less than 1/2 second.
target kicking Drills:
Practicing spinning whip kick on a heavy bag or a fixed target can result in a serious knee or back injury. Instead, use a flexible target like a handheld target or speed bag to enhance your accuracy and timing.
Use a hand held kicking target to practice accuracy at different target heights. You can kick at a horizontal, upward or downward angle.
The speed bag resembles a human head, providing realistic feedback when kicked.
ISOTONIC/ISOMETRIC Whip Kick Drill:
Using a combination of isotonic and isometric exercises, you can increase the flexibility, strength and range of motion in your legs and hip.
This combination of isotonic (muscular contraction where the muscle maintains relatively constant tension while the length changes) and isometric (muscular contraction against resistance while the length of the muscle remains the same) training methods is recommended for intermediate and advanced practitioners.
1. Hold the bar with both hands. Shift your weight to the left leg. Raise the right leg to the rear slowly and hold at a 45° angle. Point the toes and stretch the muscles in the front of the leg while tensing the muscles in the right buttock and rear of the leg. Hold for 1 minute.
2. Bring the right leg forward with the toes pointed and the muscles in the front of the thigh tensed while the muscles in the rear of the leg rest. Hold for 1 minute.
3. Raise your right leg to the rear while lowering your torso. Keep the toes pointed and feel the tension in the right buttock and right side of the torso. Rest the muscles in the front of the thigh. Hold for 1 minute.
4. With your right knee bent, move your leg horizontally back and forth. Increase the range of motion of the hip joint as you progress. Repeat 20 times.
Practice this exercises on both sides, 3 times a week.
common mistakes
The most common mistakes are in centering and balancing. When your kick has too much power, it becomes out of control. Your body loses balance and thus misses the target. *You may rotate your torso first, then the hip and leg. Choose what works best for you.
• remember
1. Center your body.
2. Spot the target before kicking.
3. Relax while moving.
4. Focus your power at impact.
• avoid
1. Kicking too powerfully without accuracy
2. Tilting the head
3. Over-rotation of the body
4. Kicking in a wide arc
HOPPING KICKS
Hopping is a graduated tactic between stepping-in and jumping. Hopping should be short, light, and quick and may be done with a straight, circular or spinning kick. Adding a hop to a kick generates powerful penetrating impact. Be sure to synchronize your entire body as one unit and hop-kick in one count. Ultimately, you should be able to hop and kick in less than one second.
HOP KICKS
purpose
Hop kicks are used to stun the opponent, to decisively break into his defense, or to counterattack against a fast attacker. Your movement should be short and quick, hitting the target accurately. Hop kicks should be springy in action and stinging in impact. On the following pages are three examples of hop kicks. Experiment in your practice to develop others.
key points
Kick with clear-cut precision and lightning speed. Coil your body as you move into the hop and uncoil it into the target as you kick. Always keep your supporting knee bent and charge your hop with aggressive energy.
striking area
Ball, blade, instep, toes, bottom of the foot
targets
1. groin
2. lower abdomen
3. solar plexus
4. chest
5. mouth
6. temple
7. rib cage
8. thigh
9. knee
HOP ROUNDHOUSE KICK
how to
1. From fighting stance, shift your weight to the front (right) leg.
2. Bring the rear foot forward, crossing in front of the front leg, and simultaneously twist the torso to the right and "coil" the body as you hop forward.
3. Rotate your hip into the target, uncoil your body and release your foot into the target. Strike the target before your bottom leg touches the ground. Land in fighting stance after kicking.
HOP SIDE KICK
how to
1. From fighting stance, shift your weight to the front leg, and bound forward bringing your back leg to your front leg as you leave the ground.
2. As your body elevates, snap your foot toward the target, striking with the bottom or blade of the foot before the other foot touches the ground. Keep your body perpendicular to the target throughout your movement. Land in fighting stance.
HOP WHIP KICK
how to
1. From fighting stance, shift your weight to the front leg while bringing the rear foot next to the front foot, bounding forward and elevating your body.
2. Raise your front leg and throw your whip kick, striking with the bottom of your foot before your other foot touches the ground. Land in fighting stance.
JUMPING KICKS
Jumping requires strong leg and abdomen muscles and complete commitment to the technique. To succeed, you need to jump suddenly and kick powerfully.
While hopping kicks are speedy techniques, jumping kicks are powerful attacks or counterattacks. However, jump kicks are impractical unless you have mastered the technique and have experience with a variety of situations and opponents. You should be able to sense, not just see and think, the perfect moment for a jumping kick. Be bold and focus on the timing of your kick. Generally, the best chance at success with a jumping kick is when your opponent is rigid, planning his next move, or tired. Avoid using jumping kicks against opponents who have superior footwork or agility, because they will take advantage of their superior skills to avoid your kick and counterattack.
JUMPING KICKS
purpose
Jumping kicks are used to overpower, to counterattack, or to demonstrate advanced skills, such as a board breaking. The key to success is total body coordination and control of your bodyweight while in the air. Since they are fully committed techniques you should be absolutely certain in their execution. A half-hearted jumping kick results in disaster. You can improve your chances of success by reducing your opponent's mobility and stamina prior to a jump kick.
key points
Once you commit to a jumping kick, never give up in the middle. If you've developed a jump kick to the point where you feel confident enough to use it against an opponent, be bold and fearless.
striking area
Knee, ball, shin, toes, bottom of the foot
targets
1. face
2. temple
3. neck
4. chest
5. solar plexus
6. kidney
JUMPING FRONT KICK
how to
1. From fighting stance, flex your knees slightly then start your jump by bringing your rear foot forward as you rotate your body 180° into the kick.
2. Bring both legs up with your knees bent, your arms close to your body and your kicking leg in front. Your feet should come up under your buttocks and your torso should be perpendicular to the ground to maintain your balance in the air.
3. At the peak of your jump, snap your front foot at the target while pulling your other leg up toward your buttocks. Hunch your head and torso slightly forward into the kick. Land in fighting stance.
JUMPING DROP KICK
how to
1. From fighting stance, flex your knees slightly then start your jump by bringing your rear foot forward as you rotate your body 180° into the kick.
2. Bring both legs up with your knees bent, your arms close to your body and your kicking leg in front. Your feet should come up under your buttocks and your torso should be perpendicular to the ground to maintain your balance in the air.
3. At the peak of your jump, snap your front foot downward into the target while keeping your torso upright. Land in fighting stance.
VARIATION: Drop the ball of the foot downward into the abdomen or solar plexus on impact. Land in fighting stance.
JUMPING AXE KICK
how to
1. From fighting stance, flex your knees slightly then start your jump by bringing your rear foot forward as you rotate your body 180° into the kick.
2. Bring both legs up with your knees bent, your arms close to your body and your kicking leg in front. Your feet should come up under your buttocks and your torso should be perpendicular to the ground to maintain your balance in the air.
3. At the peak of your jump, raise your front knee and foot as high as possible then drop it on the target, striking with the heel or bottom of the foot. Land in fighting stance.
JUMPING ROUNDHOUSE KICK
how to
1. From fighting stance, lower your posture to prepare to jump.
2. Jump, twist your hips, and kick with the instep or toes in one smooth motion. As with other jumping kicks, keep your upper body and head upright and move your arms in sync with your body rotation.
3. For power, rotate your hip beyond the target. Land in fighting stance.
JUMPING SPLIT KICK
how to
1. From a natural stance, jump straight upward, bringing your knees close to your chest. At the peak of your jump, split your legs to the sides and pull your toes back, kicking in an upward direction. Extend your hands at shoulder height as targets.
2. VARIATION: You may also drop both hands between your legs to gain additional height in your kick.
Keep your head up and look forward, rather than at the ground, throughout the jumping split kick. This will help you keep your torso upright and jump higher.
JUMP SPINNING & TURNING KICKS
Jump spinning and jump turning kicks are combinations of three elements: jump, spin or turn, and then kick. As complicated as it sounds, the principles are the same. Practice each segment separately at first and then practice how to transition between them seamlessly. When you use a jump spinning kick, don't procrastinate. Think and visualize before kicking. While you're kicking, only kick. When attacking with a jump spinning kick, you must surprise the opponent. When counterattacking, your goal is to make the opponent feel helpless by penetrating his defense either before, during and after his attack. Let him commit to his attack, and then strike. Be quick and agile.
JUMP SPINNING & TURNING KICKS
purpose
The main goal of jump spinning and turning kicks is to strike and penetrate deeply into the target. They are used for counterattacking, however, they may be used for attacking with a high success rate due to the novelty of their application. The target for circular kicks like jump turn kick and jump spinning whip kick is the head. For linear kicks like jump back kick, the target is the torso. Due to their riskiness, jump spinning kicks and jump turning kicks are reserved for advanced practitioners.
key points
Accurately anticipate the height and distance of the final target. Fully commit to your technique and then follow through.
striking area
Ball, heel, blade, toes, bottom of the foot
targets
1. solar plexus
2. rib cage
3. chest
4. neck
5. face
6. jaw
7. temple
JUMP BACK KICK
how to
1. From fighting stance, shift your weight to neutral position.
2. Jump, rotate your hips 180° to the rear, spot the target over your shoulder and chamber your legs by bending your knees and pulling your feet up toward your hips. Keep your torso upright.
3. Release your rear foot toward the target while keeping your torso upright and your other knee bent. Land in fighting stance.
PRACTICE TIP: Keep your kicking side shoulder, knee and foot aligned on one plane so your body travels directly to the target and doesn't swing around on a circular path.
JUMP TURN KICK
how to
1. From fighting stance, shift your weight to the front leg and bend your knees slightly.
2. Rotate your body 180° to the rear and lift your rear foot, crossing your front leg as you begin to jump.
3. Continuing to turn and elevate, keep your torso upright and your knees bent.
4. Release your foot to the target while maintaining balance by finishing your rotation directly over your other leg. Land in fighting stance.
PRACTICE TIP: Practice on the ground slowly without jumping at first to get the mental image of this kick. Then turn and kick fast with a minimal jump. As you acquire precision and confidence, jump progressively higher.
JUMP SPIN WHIP KICK
how to
1. From fighting stance, shift your weight to the neutral position.
2. In place, jump vertically, rotate your body to the rear, and either swing your rear leg in a circle to the rear or bend your knee and prepare to snap it at the target as you complete your rotation.
3. At the peak of your jump, snap or throw your foot at the target while keeping your torso upright and your supporting leg bent under you. Land in fighting stance.
PRACTICE TIP: Jumping requires leg strength. Spinning requires coordination to create torque in your hips. Kicking is a release of force created by both jumping and spinning. Visualize these three elements before kicking and when you kick, perform all three simultaneously.
EXPERIMENTAL TRAINING:
Experiment with ways to enhance awareness of your body and how it works. Here are some suggested training experiments:
1. Verbalize the movement sequence:
For example, for side kick:
1) Turn sideways.
2) Knee up.
3) Stretch the leg.
4) Hold it.
5) Adjust my stance.
6) Hold it as long as I can.
7) Return to fighting stance.
2. Discover what makes your body tick. Is it your torso, your head, your brain, or the angle of your chambered leg that is most influential for initiating each kick?
3. Empty your mind: Does your consciousness hinder your performance? Are you attached to negative feelings about what's going to happen? Then, forget them. Start fresh. Don't think. Pay attention only to each kicking sequence and fill your mind with the feeling of what happens in your muscles.
4. When you find something that works in your training, immediately repeat it at least 100 times to create muscle memory.
5. Have a quiet, non-active post training moment, such as walking or meditation, to synthesize all you have learned.
MULTIPLE KICKS
A multiple kick is any kick that consists of more than one impact, often with the same leg. The key to successfully
executing a multiple kick is distributing your energy properly as a result of understanding which kick is the primary technique. Generally, the initial technique is used to set up or distract the opponent and the final kick is the one that impacts the target powerfully. Practice diligently so you can maintain your balance throughout the kicks, especially if you are standing on leg for the entire technique. At this skill level, a flawless execution is critical. If you lose your balance in the middle of the technique, withdraw immediately and start fresh.
MULTIPLE KICKS
purpose
Multiple kicks are used to overwhelm or confuse an opponent. To be successful, control the distance and pace of the attacks. Your rhythm should be fast but unpredictable. Often flashy techniques like these are ineffective in a hard hitting fight, so know their limits and capitalize on the strengths of multiple kicks, especially the strength of striking multiple targets in quick succession.
key points
Coordinate your entire body to promote agility, balance and versality. Center yourself over your standing leg, keep your torso erect and control your arms for balance.
striking area
Ball, toes, blade, heel, bottom of the foot, bottom of the heel, instep, knee
targets
1. groin
2. lower abdomen
3. solar plexus
4. chest
5. neck
6. face
7. temple
8. rib cage
9. knee
SINGLE LEG LOW-HIGH ROUNDHOUSE KICKS
how to
1. From fighting stance, shift your weight to one leg (front or back according to the situation) and chamber the other leg for roundhouse kick.
2. Throw a low section roundhouse kick, striking the groin or thigh with the instep.
3. Chamber your knee back to position 1 then quickly throw a roundhouse kick to the face, lowering your torso to gain height.
PRACTICE TIP: Focus on the lightning speed of the second kick. The first kick is used to set-up or distract the opponent. To be effective, you need to appear relaxed and disguise your intent.
SINGLE LEG ROUNDHOUSE-WHIP KICKS
how to
1. From fighting stance, shift your weight to one leg (front or back according to the situation) and chamber the other leg for roundhouse kick.
2. Throw a middle section roundhouse kick to the groin or rib cage,
striking with the instep.
3. Immediately hook your leg up into a whip kick, striking the face or head with the bottom of your foot while lowering your torso for added height.
PRACTICE TIP: The first kick needs to be horizontal or near-horizontal in order to reverse the kicking direction for the second kick without straining your back. This also helps you stabilize the center of your body for a quick, powerful transition between kicks.
SINGLE LEG DOUBLE SIDE KICKS
how to
1. From fighting stance, shift your weight to one leg (front or back according to the situation) and chamber the other leg for side kick.
2. Throw a side kick to the neck or chest, striking with blade of the foot.
3. Briefly chamber you knee and throw a side kick to the neck while lower your torso to facilitate a high section kick.
AWARENESS EXERCISE:
Blindfold training heightens your awareness, because the absence of visual cues limits your movement potential. You'll be forced to internalize the movement path of a skill.
Find a safe area, free of obstacles, and try this exercise:
1) Choose one kick to practice (begin with front kick) and observe your movement carefully with your eyes open. You can do this in front of a mirror or simply by visually checking the various parts of your body as you practice.
2) Pay careful attention to the physical patterns of your movement and try to memorize the way you move.
3) Close your eyes (or use a blindfold) and visualize the kick while standing in ready stance.
4) Execute the kick with your eyes closed or blindfolded. Go slowly at first because the lack of visual cues will impair your balance.
5) When you feel comfortable, try kicking a handheld target while blindfolded. Face the target in fighting stance, note its position and then close your eyes or put your blindfold on. See what happens: is your kick landing where you think it is? Keep practicing until you can hit the target consistently, then try other kicks in the same way.
SINGLE LEG HIGH ROUNDHOUSE KICKS
how to
1. From fighting stance, shift your weight to your rear leg and chamber the front leg and throw a roundhouse kick to the face.
2. Retract your leg to a high chamber position and immediately kick to the same target again.
*This technique intimidates opponent. Follow up with a cross elbow strike to the face as you step down into fighting stance after kicking.
PRACTICE TIP: Inhale before kicking and then execute both kicks in one exhalation; the first kick is quick and the second kick is powerful.
SINGLE LEG MID-HIGH ROUNDHOUSE KICKS
how to
1. Against a high section roundhouse kick, counter with a middle section roundhouse kick, then drop your foot between you and your opponent.
2. As soon as your opponent's foot lands after kicking and his head rises, throw another roundhouse kick to his face with the same leg.
PRACTICE TIP: Even though you drop your kicking leg between kicks, keep your weight entirely on your supporting leg and create power through your hip rotation between kicks.
SINGLE LEG BACK KICK-WHIP KICK
how to
1. Attack with a back kick to the middle section. Lightly drop your kicking foot, stay close to the opponent and keep your body pivoted slightly away from him.
2. Before he launches his kick, lower your torso and throw a whip kick to the face very quickly. If the opponent moves backward between your kicks, you can slide in before whip kicking.
CAUTION: This technique is not practical unless you are a good in-fighter and very flexible.
SINGLE LEG ROUNDHOUSE-WHIP KICKS
how to
1. Attack with a front or rear leg roundhouse kick to the middle section.
2. As the opponent moves backward, use the same leg to throw a whip kick to the face, striking with the bottom of the foot. There are 3 options between the kicks: you can simply rechamber your leg, drop your foot in front and kick in place, or drop your foot in front and slide in and kick.
TRAINING TIP: The first kick is an intercepting technique. Kick as the opponent moves in to attack.
SINGLE LEG AXE KICK-SIDE KICK
how to
1. Against an opponent who moves to clinch or crowd you, strike the face with an axe kick.
2. Lightly drop your kicking leg in front, keeping your weight to the back of your stance, and when he moves back to evade your kick, pivot your body slightly and quickly thrust out a side kick to his jaw.
TRAINING TIP: When you attack with axe kick, assume your opponent will counterattack with a rear leg roundhouse kick. However he responds, as long as you attack his centerline with a long side kick, you can score or disrupt his attack.
SINGLE LEG ROUNDHOUSE-PUSHING KICKS
how to
1. Counter your opponent's high section roundhouse kick with a middle section roundhouse kick. After kicking, lightly drop your kicking foot right in front of you, keeping your weight to the back of your stance.
2. The instant he tries to kick again, use the same foot to push his rib cage or pelvis with the bottom of the foot.
TRAINING TIP: When you succeed with the pushing kick, knock him down with a jump side kick or jump back kick as he stumbles backward.
SINGLE LEG AXE KICK-ROUNDHOUSE KICKS
how to
1. When your opponent is stalling, throw an axe kick to the face to back him up. After kicking, drop your kicking leg in front of you lightly, keeping your weight more to the back of your stance.
2. As he retreats, pivot your torso and follow up with a powerful middle section roundhouse kick with the same leg.
TRAINING TIP: If he counterattacks with a rear leg back kick after your axe kick, push his hip with the same leg instead of doing a roundhouse kick.
SINGLE LEG TRIPLE KICKS
how to
1. This is a combination that requires agility, strength and coordination. First throw a quick roundhouse kick to the face to create confusion.
2. Re-chamber your kicking leg and immediately do whip kick to the face.
3. Briefly re-chamber your kicking leg then thrust a side kick to the jaw to complete the sequence. Keep your front knee high and your weight on your back leg throughout.
COMBINATION KICKS
A combination is a set of predetermined or random techniques designed to help you unlock the opponent's guard and penetrate his vital areas. It can be orderly or disorderly. In general, a methodical approach, after keen observation of the opponent's behavioral patterns, has a better chance of success than a random or habitual approach. Avoid using repetitive combinations. If you have a favorite combination, change the timing and keep your opponent guessing what the next change might be. That way, you can sustain your superiority.
COMBINATION KICKS
purpose
Combination kicks are used to put the opponent on the defensive, to set up a calculated response for re-countering, or to compensate for an error in your initial attack. The goal is to continue to have options for initiative attacks so that you can sustain your superiority and win.
key points
Choose techniques and targets that build momentum in successive kicks. Be unpredictable. Approach from diverse angles and attack varied targets.
striking area
Ball, toes, blade, heel, bottom of the foot, bottom of the heel, instep, knee
targets
1. groin
2. lower abdomen
3. solar plexus
4. chest
5. neck
6. face
7. temple
8. rib cage
9. knee
AXE KICK-SPIN WHIP KICK
how to
1. Against a stalling opponent, throw a rear leg axe kick to the face and set your kicking leg down in front.
2. As your opponent backs up, immediately follow up with a rear leg spinning whip kick to the face.
*If he does not back up, punch to the chest or rear elbow strike to the jaw.
TRAINING TIP: When you step down after the axe kick, use your stepping momentum to produce additional force in the spinning whip kick. Perform this combination aggressively, without hesitation.
DOUBLE ROUNDHOUSE KICKS
how to
1. When your opponent attacks with a rear leg roundhouse kick, throw a counter rear leg roundhouse kick and set your kicking foot down at a 45° angle to the opponent's torso.
2. As soon as he drops his foot and prepares to kick again (while he is inhaling), throw a quick roundhouse kick to the rib cage before he lifts his leg to kick.
TRAINING TIP: To succeed with techniques like this involving diagonal movements, you have to outplay your opponent with agile footwork.
HOP DOUBLE ROUNDHOUSE KICKS
how to
1. When your opponent retreats, hop in, bringing your rear foot forward to close the distance and simultaneously kick the closest target (thigh or ribs).
2. Before your kicking foot lands, throw a roundhouse kick to the lower abdomen with your other leg. The power in this combination comes from pivoting your full body weight around your first kicking leg while in the air.
TRAINING TIP: In double kicking, your legs work like a scissors. Use your body's falling momentum to create maximum power in the second kick.
AXE KICK-ROUNDHOUSE KICK
how to
1. Throw a rear leg axe kick to the face and step down in front. When you attack with axe kick, the opponent has two options: retreat or retaliate.
2. If he retreats, follow up with a powerful rear leg middle section roundhouse kick to his abdomen.
TRAINING TIP: If your opponent doesn't retreat and instead counters with his rear leg, block it with your forearm and punch to the chest to make space between you. Then follow up with a rear leg roundhouse kick to the trunk or face.
ROUNDHOUSE KICK-JUMP SPIN WHIP KICK
how to
1. Throw a powerful rear leg roundhouse kick to the stomach and step down in front. Generally an opponent will crouch in pain or stand up suddenly to expand his ribs and try to catch his breath.
2. If he stands up tall to breathe, jump and throw a rear leg spinning whip kick to the head.
TRAINING TIP: If your opponent bends forward after your first kick, throw a roundhouse kick to the face or neck with the same leg.
TRIPLE ROUNDHOUSE KICKS
how to
1. Execute all three kicks in one breath. First, counterattack with rear leg roundhouse kick against rear leg roundhouse kick. Step down at a 45° angle to the opponent.
2. As the opponent's foot lands and he inhales, throw a roundhouse kick to his rib cage with your other leg.
3. Before setting your kicking leg down, launch a roundhouse kick to the solar plexus with your standing leg (hopping double roundhouse kick).
TRAINING TIP: The purpose of the first kick is to penetrate your opponent's defense and make him move backwards or go on defense. The second and third kicks should be done before he can recover and counterattack.
RULE OF ELIMINATION:
A lack of balance in your combination kicks indicates two possible problems: something is disrupting your gravitational center or your base is weak.
Solutions: Eliminate bad habits and make minute changes to establish better habits.
Try the following exercises:
1. From a squat, jump up, turn around and land in the same position in one motion.
2. Draw a straight line on the floor and practice combination kicking moving along the line to enhance your sense of your center of gravity.
3. Practice the combination in extreme slow motion, holding each kick extended for 30 seconds or until you lose your balance. When you lose balance, move on to the next kick in the combination. Notice how your standing leg and your posture affect your kicks
4. To focus on improving your transitions between kicks, practice the combination using only kick chambers instead of full kicks. For each kick, quickly raise your knee to the chamber position and then step down and quickly proceed to the next movement in the combination.
PUTTING IT ALL TOGETHER:
1. Keep your head upright.
2. Look at the target from start to finish.
3. Check your body alignment.
4. Practice a variety of stances for different techniques.
5. Pay attention to details while you are moving to eliminate unnecessary motions that hinder performance.
6. Keep your knees relaxed and flexed— don't lock them out.
7. Stay relaxed to conserve energy.
8. Utilize gravity wisely.
9. Maximize the principle of reaction force: left vs. right and upward vs. downward movement for balance and power.
10. Always stabilize your initial movement for the best leverage, then let it go.
11. For spin kicks, pivot both feet before kicking, chamber the knee, spot the target, keep your standing knee slightly bent, and let the kick be released instead of kicking with a lot of force. Let things happen and go along with them. When you begin well, you'll finish well.
COMBINATION KICK SAMPLES
1) Right foot roundhouse kick to the trunk
\+ left foot roundhouse kick to the trunk
\+ left foot roundhouse kick to the face
\+ left foot axe kick to the face
\+ right foot roundhouse kick to the trunk
\+ left foot back kick to the trunk
\+ left foot turn kick to the trunk,
\+ forward footwork + right foot back kick to the trunk
2) Forward footwork (one step forward)
\+ back kick to the trunk
\+ roundhouse to the trunk/face
\+ spin whip kick to the face
\+ axe kick to the face
3) Backward footwork (one step back)
\+ back kick to the trunk
\+ roundhouse to the trunk/face
\+ spin whip kick to the face
4) Lateral footwork (one step to the side)
\+ roundhouse to the trunk/face
\+ back kick to the trunk
\+ axe kick to the face
5) Turning footwork (turn your rear foot backward around to the front)
\+ roundhouse kick to the trunk/face
\+ whip kick to the face
\+ spin whip kick to the face
6) Drawing footwork (pull your front foot toward you)
\+ front foot axe kick to the face
\+ front foot whip kick to the face
\+ roundhouse kick (either leg) to the trunk/face
about the author
Sang H. Kim is the creator of Power Breathing for Life. He is the author of international bestsellers Ultimate Flexibility, Ultimate Fitness Through Martial Arts, and 1001 Ways to Motivate Yourself and Others. His books have been translated into 22 languages. He has been featured or reviewed in Hartford Current, San Francisco Sun Reporter, Inner-self Magazine, The Observer, The New York Times, El Nacional, Dallas Observer, Donga Newspaper-Seoul, Chosun Daily-Seoul, Delta Sky, Fighter's Magazine-UK, Cumbernauld Gazette-Scotland, Memphis Business Journal and hundreds more publications.
An internationally respected authority on health and fitness and martial arts training, Sang H. Kim has taught tens of thousands of students in seminars and workshops in North America, Europe, and Asia. He is a certified 8th degree black belt and martial arts instructor as well as the holder of an MS degree in Sports Science and Ph.D. in Sports Media Studies. As the founder of Power Breathing for Life, he is recognized as one of the most innovative masters of healthy living. He divides his time among Southwest, East coast and Asia, writing and practicing martial arts and Power Breathing.
| {
"redpajama_set_name": "RedPajamaBook"
} | 5,140 |
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"redpajama_set_name": "RedPajamaGithub"
} | 2,036 |
Q: Solr Replicas Usually Recovering and Recovered Faild I have deployed a SolrCloud in AKS with 5 Nodes, each Node with 32Gb RAM. In Solr, I create a collection with 7 shards and 3 replicas. Usually, my pod is restarted. So I can see replicas to recovering and recovered failed. I checked the log and nothing.
What can I do to fix that?
I try to set autocommit with 5 min and 10 min. However, it does not work well.
I have a Statefulset solr with 5 replicas. Each replica requests 20Gi - 2 CPU and limits 32Gi - 4 CPU. I set JAVA_MEM with xms12g; xmx16g and each replica are on a node of Kubernetes. Sometimes, I see one or two solr pods restarts. So replicas on a shard will usually go from active to recovering and finally recovered failed.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,598 |
{"url":"https:\/\/map-rfun.library.duke.edu\/exercises.html","text":"Please note: These exercises assume you have downloaded the mapping-with-R repository to your local installation of RStudio and are running this as a unique project within RStudio. You should also download the practice code repository \u2013 find that link on the main page of this guide.\n\nExercises\n\n1. Georeference XY coordinates. In your RStudio editor, open the exercise-1_xy.Rmd file. Plot XY (latitude\/longitude) coordinates.\n\n2. Choropleths with tidycensus. Open exercise-2_tidycensus.Rmd. Plot a new variable and shade that variable by county.\n\n3. Simple Features. Open exercise-3_thematic-map_with_federal_data.Rmd`. Make a choropleth of the data using either ggplot2, tmap, or leaflet. Use the knowledge you gained in the Thematic Mapping module on this site.\n\nAnswers\n\nAnswers can also be found in the practice code repository.","date":"2019-05-19 22:35:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.22735416889190674, \"perplexity\": 8406.472929264082}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232255182.37\/warc\/CC-MAIN-20190519221616-20190520003616-00263.warc.gz\"}"} | null | null |
Jakob Adolf Seitz (February 14, 1898, Meitingen, Germany – April 6, 1970, Switzerland) was a German–Argentine chess master and journalist.
Career
In 1920, he tied for 2-4th in Canterbury, took 10th in Berlin, and tied for 4-5th in Kulmbach. In 1921, he tied for 8-9th in Hamburg. In 1922, he tied for 2nd-3rd in London (Major Open). In 1922/23 he tied for 6-9th in Portsmouth/Southsea. In 1923, he tied for 6-7th in Triest. In 1923/24 he tied for 5-6th in Hastings (Max Euwe won). In 1924, he took 13th in Győr. 1924/25 he took 3rd in Hastings (Géza Maróczy won). In 1925, he tied for 6-10th in Debrecen. In 1925, he took 3rd in Bologna (Mario Monticelli won). In 1925/26 he tied for 3-4th in Hastings (Alexander Alekhine and Milan Vidmar won).
In 1926, he tied for 6-7th in Milan, and tied for 3-4th in Scarborough. In 1927, he took 2nd, behind Stefano Rosselli del Turco, in Naples, and tied for 3-4th in London. In 1928, he tied for 3-4th in Cheltenham, tied for 4-5th in Dortmund, and tied for 1st-2nd with George Koltanowski in Tenby. In 1929, he took 10th in Paris (Savielly Tartakower won), tied for 4-7th in Duisburg (Carl Ahues won), and tied for 1st-2nd with Vajda in Ramsgate (B tourn). In 1930, he tied for 3-4th in Scarborough, and took 7th in Nice (Tartakower won).
In 1931, he tied for 5-8th in Nice (Brian Reilly won). In 1933, he tied for 11-12th in Bad Pyrmont (1st GER-ch; Efim Bogoljubow won). In 1934, he tied for 3-6th in Bad Salzuflen. In 1935, he took 2nd, behind Samuel Reshevsky, in Great Yarmouth. In 1935, he took 7th in Helsinki (Eero Böök and Rudolf Spielmann won). In 1938, he took 16th in Łódź (Vasja Pirc won).
In September 1939, when World War II was broke out, Seitz along with all German players (Eliskases, Michel, Engels, Becker, Reinhardt) and many other participants of the 8th Olympiad in Buenos Aires decided to stay permanently in Argentina.
After World War Two, he returned to Europe. In 1953 he represented Italy whilst living in Norway. He later lived in Switzerland and West Germany.
References
External links
Passengers of the Piriápolis
Visa with photo 1939
Visa with photo 1940
1898 births
1970 deaths
German chess players
Argentine chess players
German expatriates in Argentina
People from Augsburg (district)
Place of birth missing
20th-century chess players | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,562 |
Big and large are the same thing! why is the answer contranyms?
It isn't. Those people are wrong. Look at how many voted for 'synonyms'. Not everyone is perfect or correct.
Is it ok if someone tell me,,what is contranyms?? | {
"redpajama_set_name": "RedPajamaC4"
} | 4,918 |
Bu yıl İnönü Üniversitesi Bilgisayar Mühendisliği Bölümü "International Artificial Intelligence and Data Processing Symposium'16" adı altında bir konferans düzenledi. "A Study Based on Gray Level Co-Occurrence Matrix and Neural Network Community for Determination of Hypoxic Fetuses" başklı bir çalışma ile konferansa katılım sağladık. Sunduğumuz çalışmaya Research Gate üzerinden erişebilirsiniz.
Yapay zeka ve uygulamaları, makine öğrenmesi, derin öğrenme, örüntü ve nesne tanıma, biyomedikal işaret ve görüntü işleme gibi pek çok konuya dair bilimsel çalışmaların sunulduğu konferansın geleneksel olarak her Eylül ayının ilk haftasonu düzenlenmesi planlanıyor. Etkinliğin zamanla çok büyüyeceğini ve çok etkili olacağını düşünüyorum.
Abstract—Cardiotocography (CTG) is a monitoring technique used routinely during the pregnancy and labor and including the analysis of fetal heart rates and movements with uterine contractions. The fact that CTG signals are interpreted by experts generally with eye and CTG has high false positive rate results in intra- and inter-observer conflicts and causes the observers to frequently notice real pathological cases. Therefore, various computer-aided methods supporting diagnosis process have been developed. In this study, a new approach is suggested based on signal and image processing techniques in order to provide classification of CTG signals. In particular, morphological, spectral and statistical properties of CTG signals are obtained with the way defined conventionally. A spectrum of the signals containing time-frequency information was transformed into 8-bit gray-level image and it was enabled to build grey-level co-occurrence matrix (GLCM). In the final step, a combination of morphological, statistical, spectral and image-based properties was applied as input to the artificial neural network (ANN). In order to measure the performance of the proposed method, accuracy, sensitivity, specificity and quality indexes were used. The obtained results revealed that image-based features increased the classification success and they gave the best results when they were used with the conventional features. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,216 |
Q: Flex backkeypressedhandler not launching with states I'm developping an mobile program with the Flex 4.5 SDK, this for my thesis for school. I have a bug in my program that I can't fix.
I have a view where I have 2 states. When I change from state my program doesn't listen to the backKey event anymore.
I allready tried this:
backKeyPressed.mystate = "backKeyPressed(event)"
But this doesn't help.
I made a small program to show you:
<?xml version="1.0" encoding="utf-8"?>
<s:View xmlns:fx="http://ns.adobe.com/mxml/2009"
xmlns:s="library://ns.adobe.com/flex/spark" title="Home"
initialize="init()"
backKeyPressed="view1_backKeyPressedHandler(event)">
<fx:Script>
<![CDATA[
import mx.collections.ArrayCollection;
import mx.events.FlexEvent;
[Bindable]
private var m_list:ArrayCollection = new ArrayCollection();
private function init():void
{
}
protected function button1_clickHandler(event:MouseEvent):void
{
currentState = 'state2';
}
protected function button2_clickHandler(event:MouseEvent):void
{
currentState = 'state1';
}
protected function view1_backKeyPressedHandler(event:FlexEvent):void
{
event.preventDefault();
}
]]>
</fx:Script>
<fx:Declarations>
<s:State name="state1"/>
<s:State name="state2"/>
</fx:Declarations>
<s:Group id="stateOneGroup" includeIn="state1">
<s:Button x="30" y="112" label="To State 2"
click="button1_clickHandler(event)"/>
<s:Label x="72" y="216" text="State1"/>
</s:Group>
<s:Group id="stateTwoGroup" includeIn="state2">
<s:Button x="267" y="112" label="To State 1"
click="button2_clickHandler(event)"/>
<s:Label x="303" y="216" text="State2"/>
</s:Group>
</s:View>
I also searched on adobe if it's a bug but I don't seem to find it.
I hope someone can help me.
Kind regards,
Thibault Heylen
A: I had the same issues today. At last i got it. In my issues, if it is the first view, the backkeypressed doesn't work, but for the other views, the backkeypressed works well.
Hope this helps
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,516 |
Q: ionic3 app stuck on splash screen using build --prod flag I am successfully building my ionic 3 application using --prod flag:
ionic cordova run android --device --prod
When application is launched, it is stucked on splash screen and using chrome inspector I am seeing this error:
ERROR Error: StaticInjectorError[function(){}]:
Using regular build or build --release the app launches as expected.
What is the meaning of this error?
Thanks
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,406 |
Want to be different than anyone else? Then here is your chance. The Diablo is made from 14 gauge stainless steel. We have included smoking cool Kydex case along with a neck and key chain. Get yours today. You won't be disappointed! | {
"redpajama_set_name": "RedPajamaC4"
} | 245 |
namespace v8 {
namespace internal {
class HeapObject;
class Object;
class Isolate;
enum class RootIndex : uint16_t;
// Base class for serializer that iterate over roots. Also maintains a cache
// that can be used to share non-root objects with other serializers.
class RootsSerializer : public Serializer {
public:
// The serializer expects that all roots before |first_root_to_be_serialized|
// are already serialized.
RootsSerializer(Isolate* isolate, RootIndex first_root_to_be_serialized);
bool can_be_rehashed() const { return can_be_rehashed_; }
bool root_has_been_serialized(RootIndex root_index) const {
return root_has_been_serialized_.test(static_cast<size_t>(root_index));
}
bool IsRootAndHasBeenSerialized(HeapObject obj) const {
RootIndex root_index;
return root_index_map()->Lookup(obj, &root_index) &&
root_has_been_serialized(root_index);
}
protected:
void CheckRehashability(HeapObject obj);
// Serializes |object| if not previously seen and returns its cache index.
int SerializeInObjectCache(HeapObject object);
private:
void VisitRootPointers(Root root, const char* description,
FullObjectSlot start, FullObjectSlot end) override;
void Synchronize(VisitorSynchronization::SyncTag tag) override;
const RootIndex first_root_to_be_serialized_;
std::bitset<RootsTable::kEntriesCount> root_has_been_serialized_;
ObjectCacheIndexMap object_cache_index_map_;
// Indicates whether we only serialized hash tables that we can rehash.
// TODO(yangguo): generalize rehashing, and remove this flag.
bool can_be_rehashed_;
DISALLOW_COPY_AND_ASSIGN(RootsSerializer);
};
} // namespace internal
} // namespace v8
#endif // V8_SNAPSHOT_ROOTS_SERIALIZER_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 850 |
home Technology Amphibious Vessels New Reconnaissance Vehicle Seems to Have it All
New Reconnaissance Vehicle Seems to Have it All
Amphibious Vessels
A new 4×4 armoured reconnaissance vehicle is now in service at the Armed Forces of Belarus. The Cayman vehicle developed by JSC '140 repair plant' can be used in reconnaissance and sabotage missions, patrol, escort, peacekeeping and police operations, as well as in emergency response missions.
According to army-technology.com, the armoured car can accommodate a crew of six, who can enter and exit through either the side door or two roof hatches above the driver and passenger seats. A large hatch is also provided in the centre of the fighting compartment for firing weapons. The on-board cooling system enables the vehicle to operate at ambient temperatures up to 50°C. The front armoured shield of the crew cab is equipped with a dual heating circuit to prevent icing and fogging. The forward and rear sections have cameras that offer an enhanced field-of-view. The vehicle is also installed with an on-board information control system and a built-in navigation system.
The Cayman armoured vehicle is 6m-long, 2.82m-wide and 2.07m-high and has a gross vehicle weight of 7,000kg. The wheel track and wheel base of the vehicle are 2,400mm and 3,100mm respectively. The base variant of the Cayman is installed with a mount for a firearm. The weapon mount is surrounded by an armoured shield, which protects the gunner against splinters and small-calibre rounds. Three smoke grenade launchers are fixed on either side of the vehicle. The vehicle can be armed with a range of weapon stations and armament according to the requirements of the customers. It can be fitted with a 7.62mm Kalashnikov PKMB medium machine gun, a 12.7mm NSV Utyos heavy machine gun, or a 30mm AGS-17 Plamya automatic grenade launcher.
The front of the protects the crew from 7.62mm rounds fired from a sniper rifle, while the sides offer protection against 5.45mm and 7.62mm rounds. The Cayman vehicle also offers anti-mine protection. The bottom deflects the impact of landmine or improvised explosive devices (IED), while the floor in the landing areas is reinforced with add-on armour. The Cayman armoured vehicle is powered by a D-245.30E2 engine coupled to a mechanical transmission through a five-speed gearbox.
The power-plant provides a maximum road speed of 100km/h, a swimming speed of up to 8km/h and a cruising range of 1,000km. The fully amphibious vehicle is propelled in water by two propellers mounted on either side at the rear of the chassis. The vehicle can swim in the water without any preparation as the propellers can be activated on the move by the toggle switch within the crew cab. The all-wheel drive vehicle offers a ground clearance of 490mm at full load. It is fitted with independent torsional-lever-type suspension and a centralised tyre inflation system (CTIS) for high mobility across rugged terrains.
US Army Responds To Increased Riots
Self-Protection System Capable Of Detecting Danger
Watch: Military Operations Game Changer
New Hybrid Search and Rescue Platform Unveiled
New Egyptian IFV Offers Maximal Protection
New Military Solution for GPS-Denied Environment
New Rifle for Infantry Squads
Innovative Capability for Warfare in Low Light Conditions | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,147 |
\section{Introduction}
The remarkable discovery of Bekenstein \cite{Bekenstein:1973ur} and Hawking \cite{Hawking:1974sw} in the seventies laid the foundation of black hole thermodynamics, which has been the subject of ardent research in the following decades till date. Identifying the thermodynamic parameters (such as entropy, temperature, energy {\it etc.} ) from the geometrical quantities of the black hole sapcetime (such as the area of the horizon, surface gravity of the black hole horizon {\it etc.}), four laws of black hole mechanics were formulated in 1973 \cite{Bardeen:1973gs}. These works clearly imply the existence of thermodynamic structure of the black hole horizon. Since then, many thermodynamic phenomena have been observed in black hole spacetime. The study of phase transition, which is an important phenomenon in ordinary thermodynamics, has also been explored in black hole mechanics since seventies. It was introduced by Davies \cite{Davies:1978mf} and subsequently followed by many other researchers \cite{Lousto:1993yr, Lousto:1994cz, Lousto:1994jd, Muniain:1995ih}. Davies endorsed that a black hole goes through a second order phase transition when it passes through a point (Davies' point) where the heat capacity becomes infinitely discontinuous. However, later Kaburaki {\it et al.} \cite{Kaburaki:1993ah, Katz:1993up, Kaburakipla, Kaburakigrg} claimed that Davies' point is not a critical point. Instead, it is merely a turning point, where stability changes.
Although, Davies' claim was later falsified, other groups argued that a second order phase transition indeed takes place when a non-extremal black hole transforms to an extremal one and the extremal limit was identified as a critical point. It was first concluded by Curir in \cite{curir1, curir2}. Later Pav$\acute{\textrm{o}}$n and Rub$\acute{\textrm{i}}$ \cite{Pavon:1988in, Pavon:1991kh} calculated second order moments of fluctuation of mass, angular momentum {\it etc.}~using Landau-Lifshitz hydrodynamic fluctuation theory (see chapter 17 of \cite{Landaufluid}) and have shown that those second order moments diverge in the extremal limit of Kerr and Reissner-Nordstr$\ddot{\textrm{o}}$m (RN) black holes but those moments are finite in the non-extremal limit and for the Schwarzschild black hole. Also, those second order moments remain finite at the Davies' point. Both the analysis are in agreement with each other and suggest that the extremal limit of the black hole is a critical point, and the divergence of second order moments of fluctuation should signal a second order phase transition of the black holes which are changing from its non-extremal phase to the extremal phase. Later, this phase transition in the extremal limit has been rigorously studied for different (Kerr-Newman \cite{Kaburakigrg}, BTZ \cite{Cai:1996df, Cai:1998ep, Wei:2009zzf} {\it etc.}) black holes and critical exponents were obtained. These exponents satisfy the well known scaling laws \cite{Stanley1, book} of thermodynamics.
The works, which are mentioned above, are performed in different spacetimes to come to the same central conclusion that the extremal limit is a critical point and the transformation from a non-extremal to an extremal black hole is a second order phase transition. Moreover, in those cases, the information of the spacetime has directly been used to obtain the results. One question naturally appears: is it really necessary to start with a particular spacetime to reach this conclusion? The results present in different papers suggest us to believe that probably the conclusion is true irrespective of spacetime metric and its dimension. But till now there has not been any such proof. Moreover, there are few major questions which has not been addressed properly. Some of these are: Are the critical exponents universal? Is the effective spatial dimension one in every extremal black holes {\it etc.} \ In this paper we address all these issues systematically.
Our analysis is valid for all the black holes which are extremal at certain limit. Without introducing any particular spacetime we show that the transformation of black hole from non-extremal to extremal is a second order phase transition with the extremal limit being the critical point. To prove that, we calculate the second order moments of fluctuation modes of some thermodynamic quantities using equilibrium fluctuation theory of statistical mechanics \cite{Landau, Kaburakipla, Kaburakigrg} and show that those moments diverge in the microcanonical ensemble. Thereby we show that the phase transition is well described only by the micro-canonical ensemble instead of the canonical or the grand canonical ensembles. Later, we proceed our analysis to obtain the values of critical exponents in a general way. These exponents match with the results, obtained earlier by considering the explicit form of the spacetime. Also these have been shown to satisfy the scaling laws. We emphasize that in our whole analysis the only underlying information one requires is: { \it one should consider the particular class of black hole spacetimes which exhibit such non-extremal to extremal transition at certain limit} and additionally, {\it the thermodynamics of those black holes are governed by the usual first law of black hole mechanics at the non-extremal limit}.
We also analyze another interesting aspect in our paper. It is known for a long time that classical thermodynamics can also be studied by geometric method. This is the GTD formulation. In Weinhold's approach the metric is defined as the Hessian of the internal energy and in the Ruppeiner's approach the metric is defined as the Hessian of the entropy. It has been shown that Ruppeiner curvature scalar diverges at the extremal limit of the BTZ black hole\cite{Cai:1998ep, Wei:2009zzf}. In the present paper we have proved this result for any arbitrary black hole which has an extremal point.
Very recently it has been claimed that neither the Weinhold nor Ruppeiner formulation is Legendre invariant and, hence, they are inappropriate to analyze the thermodynamics. So, we proceed one step further to find the thermodynamic behaviour at the extremal point using Legendre-invariant metric. We do this for two Quevedo GTD metrics and find that the Ricci scalar for both of those metrics are finite at the extremal point. Thus, our work connects all the previous diverse conclusions about extremal phase transitions, all of which are black hole specific. In this sense, our work is unique and fills an important gap in the literature.
Before we proceed further, let us mention the organization of our paper. In the second section we discuss the black hole thermodynamics at the extremal point without using any particular form of spacetime. Second order moments of fluctuation are calculated for microcanonical, canonical and grand canonical ensembles in three subsections. It is observed that the phase transition is compatible with the first ensemble. Next section is dedicated to calculate the values of different critical exponents. Then in section \ref{sec4}, thermogeometric analysis has been performed separately for Weinhold, Ruppeiner and two Legendre invariant metrics. It is shown that the curvature scalar diverges only for the Ruppeiner metric. Finally, in the last section, we draw conclusions of our work.
\section{Thermodynamic analysis of extremal point in different ensembles}
We have already mentioned that, the extremal phase transition is regarded as a second order phase transition. This was first claimed by Curir \cite{curir1, curir2}. According to Pav$\acute{\textrm{o}}$n and Rub$\acute{\textrm{i}}$ \cite{Pavon:1988in, Pavon:1991kh}, the divergence of the second order moments of fluctuations of thermodynamic quantities is a signature of this phase transition. Following this argument, here we calculate these second order moments in different ensembles. We show that, only in microcanonical ensemble extremal limit of black hole (if it exists) is a second order phase transition.
Here, we calculate the second order moments using the well-defined equilibrium fluctuation theory of statistical mechanics. In that case, the required thermodynamical quantities are obtained from Massieu function, which are the Legendre transformations of the entropy. In that formalism, the state of a given environment is completely characterized by the Massieu function \cite{Kaburakipla, Kaburakigrg} $\Phi$, whose variation is given by
\begin{align}
d\Phi=\mathcal{X}_id\mathcal{Y}^i~. \label{MASSIEU}
\end{align}
Here, the summation convention has been adopted. In the above relation, Massieu function is a function of the intrinsic variables $\mathcal{Y}^i$. $\mathcal{X}_i$ which is the conjugate variables of $\mathcal{Y}^i$, is defined as $\mathcal{X}_i=(\partial \Phi/\partial \mathcal{Y}^i)_{\bar{\mathcal{Y}}^i}$~. In our notation, $\bar{\mathcal{Y}}^i$ is the set of all intrinsic variables excluding $\mathcal{Y}^i$~. {\it Throughout our analysis, a bar overhead will imply similar thing}. Now for a given environment, the spontaneous fluctuation from the equilibrium occurs only in the conjugate variables $\mathcal{X}_i$. This is because, the reservoirs are considered to be large compared to the system and as a result, the intrinsic variables are fixed. Then the probability of the deviation from the equilibrium is proportional to $\textrm{exp}[-\Sigma\lambda_i(\delta \mathcal{X}^i)^2/(2k_B)]$ \cite{Kaburakigrg}, where $k_B$ is the Boltzmann constant. The eigenvalues of the fluctuation modes are defined as
\begin{align}
\lambda_i=\frac{\partial \mathcal{Y}_i}{\partial \mathcal{X}^i}\Big|_{\bar{\mathcal{Y}}^i}=\Big(\frac{\partial^2\Phi}{\partial \mathcal{Y}^{i^{2}}}\Big)^{-1}_{\bar{\mathcal{Y}}^i}~.
\end{align}
Here it should be mentioned that the probability is accurate only up to the second order. The averages of modes of fluctuations always vanish \cite{Landau} and the second order moments are given by
\begin{align}
\mathcal{M}_{ij}=\langle\delta\mathcal{X}_i\delta\mathcal{X}_j\rangle=k_B\Big(\frac{\partial^2\Phi}{\partial \mathcal{Y}^{i2}}\Big)_{\bar{\mathcal{Y}}^i}\delta_{ij}=\frac{k_B}{\lambda_i}\delta_{ij}~. \label{SECMOM}
\end{align}
In the following analysis, we investigate the behaviour of these quantities in each ensemble. Since the extremal limit is not a turning point \cite{Kaburakigrg}, the divergence of the second order moments will imply the presence of second order phase transition.
\subsection{Microcanonical ensemble}
Let us consider an isolated black hole by definition which exchanges nothing with the environment. In this case, the proper Massieu function $\Phi_1$ is the entropy $S$. Its change is given by the first law of black hole mechanics{\footnote{This is one of the inputs of our present discussion; whereas the other input is the existence of extremal limit in the black hole thermodynamics.}}:
\begin{align}
dS=\beta dM-\tilde X^idY_i~, \label{ENTROPY}
\end{align}
where $\beta=1/T$ and $\tilde X^i=\beta X^i$~. According to our notations $X^i$ are potential, angular velocity {\it etc.}, whereas $Y_i$ are charge, angular momentum {\it etc.}~Then the eigenvalues of the fluctuations are given by
\begin{align}
\lambda^{(1)}_M=\Big(\frac{\partial^2 S}{\partial M^2}\Big)_{Y_i}^{-1}=\Big(\frac{\partial M}{\partial \beta}\Big)_{Y_i}=-T^2C_Y~ \label{LAMBM}
\end{align}
and
\begin{align}
\lambda^{(1)}_{Y_i}=\Big(\frac{\partial^2 S}{\partial Y_i^2}\Big)_{M, \bar Y_i}^{-1}=-\Big(\frac{\partial Y_i}{\partial \tilde X^i}\Big)_{M, \bar Y_i}=-TI_M^{(i)}~. \label{LAMBYI}
\end{align}
Here we used the following definitions: $C_Y=(\partial M/\partial T)_{Y_i}=-\beta^2(\partial M/\partial \beta)_{Y_i}$ and $I_M^{(i)}=(\partial Y_i/\partial X^i)_{M, \bar Y_i}=\beta(\partial Y_i/\partial \tilde X^i)_{M, \bar Y_i}$~. Therefore the second order moments are given by
\begin{align}
\langle\delta\beta\delta\beta\rangle=k_B\Big(\frac{\partial^2 S}{\partial M^2}\Big)_{Y_i}=-k_B \frac{\beta^2}{C_Y}~ \label{DELBDELB}
\end{align}
and
\begin{align}
\langle\delta \tilde X^i\delta \tilde X^i\rangle=k_B\Big(\frac{\partial^2 S}{\partial Y_i^2}\Big)_{{M, \bar Y_i}}=-k_B \frac{\beta}{I^{(i)}_M}~. \label{DELXDELX1}
\end{align}
In the following section, where we obtain the critical exponents in a general way, we show that both $(\partial^2 S/\partial M^2)_{Y_i}$ and $(\partial^2 S/\partial Y_i^2)_{{M, \bar Y_i}}$ diverge at the extremal limit (see \eqref{28} and \eqref{33}). Therefore, we can conclude from \eqref{LAMBM} and \eqref{LAMBYI} that all the eigenvalues $\lambda_M^{(1)}$ and $\lambda_{Y_i}^{(1)}$ vanish. As a result, from \eqref{DELBDELB} and \eqref{DELXDELX1} we see that all the second order moments diverge, which is the signature of phase transition. Thus, in the microcanonical ensemble, an extremal phase transition is a second order phase transition with the extremal limit being the critical point.
\subsection{Canonical ensemble}
In canonical ensemble, black hole can exchange only energy with the environment. The proper Massieu function ($\Phi_2$) in this ensemble is $\Phi_2=S-\beta M=-\beta F$, where $F=M-TS$ is the Helmholtz free energy. Note that $dF=-SdT+X^idY_i$ and $d\Phi_2=-Md\beta-\tilde X^idY_i$. Therefore, in this case, the intrinsic variables are $\beta$ and $Y_i$ whereas the conjugate quantities are $(-M)$ and $(-\tilde X^i)$. The eigenvalues are given by
\begin{align}
\lambda_{\beta}^{(2)}=\Big(\frac{\partial^2 \Phi_2}{\partial \beta^2}\Big)_{Y_i}^{-1}=-\Big(\frac{\partial \beta}{\partial M}\Big)_{Y_i}=\frac{\beta^2}{C_Y}~ \label{LAMBB}
\end{align}
and
\begin{align}
\lambda^{(2)}_{Y_i}=\Big(\frac{\partial^2 \Phi_2}{\partial Y_i^2}\Big)_{\beta, \bar Y_i}^{-1}=-\Big(\frac{\partial Y_i}{\partial \tilde X^i}\Big)_{\beta, \bar Y_i}=-TI_{\beta}^{(i)}~. \label{LAMBYI2}
\end{align}
In the above, we have used $I_{\beta}^{(i)}=(\partial Y_i/\partial X^i)_{{\beta}, \bar Y_i}=\beta(\partial Y_i/\partial \tilde X^i)_{{\beta}, \bar Y_i}$. The second order moments, in this case, are found to be
\begin{align}
\langle\delta M\delta M\rangle=k_B\Big(\frac{\partial^2 \Phi_2}{\partial \beta^2}\Big)_{Y_i}=k_B T^2C_Y~ \label{DELMDELM}
\end{align}
and
\begin{align}
\langle\delta \tilde X^i\delta \tilde X^i\rangle=k_B\Big(\frac{\partial^2 \Phi_2}{\partial Y_i^2}\Big)_{\beta, \bar Y_i}=-k_B\frac{\beta}{I_{\beta}^{(i)}}~. \label{DELXDELX}
\end{align}
In the appendix \ref{APPENCANO}, we show that $(\partial^2 \Phi_2/\partial \beta^2)_{Y_i}$ vanishes and $(\partial^2 \Phi_2/\partial Y_i^2)_{\beta, \bar Y_i}$ diverges. As a result $\lambda_{\beta}^{(2)}$ in \eqref{LAMBB} diverges and $\lambda^{(2)}_{Y_i}$ in \eqref{LAMBYI2} vanishes. Also, the nature of the second order moments are evident: $\langle\delta M\delta M\rangle$ of \eqref{DELMDELM} vanishes and $\langle\delta \tilde X^i\delta \tilde X^i\rangle$ of \eqref{DELXDELX} diverges. Therefore the extremal limit is not a critical point in the canonical ensemble.
\subsection{Grand canonical ensemble}
Finally we consider the black hole in grand canonical ensemble. It means, the black hole not only exchanges energy with the environment but also performs work on the surroundings. The proper Massieu function in this case is $\Phi_3=\Phi_2+\tilde X^iY_i=S-\beta M+\tilde X^iY_i=-\beta G$. Wher, $G=M-TS-X^iY_i$ is Gibbs free energy. The variation of $G$ is $dG=-SdT-Y_idX^i$ and the variation of Massieu function $\Phi_3$ is $d\Phi_3=-Md\beta+Y_id\tilde X^i$~. Therefore in this ensemble, the intrinsic variables are $\beta$ and $\tilde X^i$' whereas the conjugate variables are $(-M)$ and $Y_i$. The eigenvalues of the fluctuation modes are
\begin{align}
\lambda_{\beta}^{(3)}=\Big(\frac{\partial^2 \Phi_3}{\partial \beta^2}\Big)_{\tilde X_i}^{-1}=-\Big(\frac{\partial \beta}{\partial M}\Big)_{\tilde X_i}=\frac{\beta^2}{C_{\tilde X}}~ \label{LAMBB2}
\end{align}
and
\begin{align}
\lambda^{(3)}_{\tilde X^i}=\Big(\frac{\partial^2 \Phi_3}{\partial \tilde {X^i}^2}\Big)_{\beta, \bar{\tilde X}^i}^{-1}=\Big(\frac{\partial \tilde X^i}{\partial Y_i}\Big)_{\beta, \bar{\tilde X}^i}=\frac{\beta}{I_{\beta}^{(i)}} ~. \label{LAMBXI}
\end{align}
In the above, we have used $C_{\tilde X}=(\partial M/\partial T)_{\tilde X^i}=-\beta^2(\partial M/\partial \beta)_{\tilde X^i}$~. The second order moments in grand canonical ensemble are
\begin{align}
\langle\delta M\delta M\rangle=k_B\Big(\frac{\partial^2 \Phi_3}{\partial \beta^2}\Big)_{\tilde X^i}=k_B T^2C_{\tilde X}~ \label{DELMDELM2}
\end{align}
and
\begin{align}
\langle\delta Y_i\delta Y_i\rangle=k_B\Big(\frac{\partial^2 \Phi_3}{\partial \tilde {X^i}^2}\Big)_{\beta, \bar{\tilde X}^i}=k_B TI_{\beta}^{(i)}~. \label{DELYDELY}
\end{align}
In the appendix \ref{APPENGRAND}, we show that both $(\partial^2 \Phi_3/\partial \beta^2)_{\tilde X^i}$ and $(\partial^2 \Phi_3/\partial \tilde {X^i}^2)_{\beta, \bar{\tilde X}^i}$ vanish. As a result, we conclude that both the eigenvalues of the fluctuation modes $\lambda_{\beta}^{(3)}$ and $\lambda^{(3)}_{\tilde X^i}$ diverge. Naturally both the second order moments $\langle\delta M\delta M\rangle$ and $\langle\delta Y_i\delta Y_i\rangle$ vanish. As a result, the extremal limit is not a second order phase transition in the grand canonical ensemble.
\section{Obtaining the critical exponents in a general way}
In the earlier section, we have generally shown that the extremal phase transition is indeed a second order thermodynamic phase transition in the microcanonical ensemble. In this section we obtain the values of the critical exponents in a general manner. There are several works which studied extremal criticality and obtained the critical exponents case by case. For example, in \cite{Kaburakigrg} the extremal phase transition of Kerr-Newman black hole was studied and critical exponents were obtained. Similar studies were done for BTZ black hole in \cite{Cai:1996df, Cai:1998ep, Wei:2009zzf}. In our general framework, we obtain the values of critical exponents in a metric independent way.
The critical exponts are defined for the response coefficients and for the order parameters to show how those quantities diverge near the critical point \cite{Kaburakipla2}. The response coefficients are defined as the inverse of the eigen values $\lambda_i$'s \cite{Kaburakipla}. For the extremal phase transition and in the microcanonical ensemble, the response coefficients are defined as:
\begin{eqnarray}
&&\zeta_{Y}=\Big(\frac{\partial^2S}{\partial M^2}\Big)\Big|_{Y_i}, \label{JAIY}
\\
&&\zeta_{M}^{i}=\Big(\frac{\partial^2S}{\partial Y_i^2}\Big)\Big|_{M, \bar Y_i} \label{JAIM}
\end{eqnarray}
In the first definition, $Y_i$ includes all the charges present in the theory, whereas, in the second definition, $\bar Y_i$ includes all the charges except $Y_i$. In classical thermodynamics, the order parameters are the difference of some extensive quantities of the two different phases. For black hole, the order parameters are defined as the difference of the conjugate quantities on the inner and the outer horizon \cite{Kaburakipla2, Su:1994zz, Cai:1996df, Cai:1997cv}. For the presence of multiple charge and angular momentum, we define the order parameters in a general manner,
\begin{align}
\eta_{Y_i}=\tilde X^i_{+}-\tilde X_{-}^i~ \label{ETA}
\end{align}
where, $\tilde X^i=(X^i/T)=-(\partial S/\partial Y_i)_{M, \bar Y_i}$ as we have defined earlier. The subscripts ``$+$'' and ``$-$'' stands for the outer horizon ($r_{+}$) and inner horizon ($r_{-}$) respectively. Now, the critical exponents are defined as \cite{Kaburakipla2}
\begin{eqnarray}
&& \zeta_{Y}\sim m^{-\alpha} \ \ \ \ \ \textrm{(for $Y_i=Y_{ic}$)} \label{ALP}
\\
&& \zeta_{Y}\sim y_i^{-\phi_i} \ \ \ \ \ \textrm{(for $M=M_c$ and $\bar Y_i=\bar Y_{ic}$)}\label{PHI}
\\
&& \zeta_{M}^{i}\sim m^{-\gamma_i} \ \ \ \ \textrm{(for $Y_i=Y_{ic}$)}\label{GAM}
\\
&& \zeta_{M}^{i}\sim y_i^{-\s_i} \ \ \ \ \ \textrm{(for $M=M_c$ and $\bar Y_i=\bar Y_{ic}$)} \label{SIG}
\\
&& \eta_{Y_i}\sim m^{\beta_i} \ \ \ \ \ \ \textrm{(for $Y_i=Y_{ic}$)} \label{BET}
\\
&& \eta_{Y_i}\sim y_i^{\delta_i^{-1}} \ \ \ \ \ \textrm{(for $M=M_c$ and $\bar Y_i=\bar Y_{ic}$)} \label{DEL}
\end{eqnarray}
Here we use the notation $m=1-M/M_c$ and $y_i=1-Y_i/Y_{ic}$, whereas $c$, in the subscript, signifies the corresponding values at the critical point. Remember that the critical point, in our present discussion, is the extremal point where temperature $T$ vanishes.
Now we expand the mass as a function of entropy $S$ and charge $Y_i$ near the critical point. Then
\begin{eqnarray}
\nonumber
&& M=a_{00}+a_{20}s^2+a_{30}s^3+a_{40}s^4+...
\\
\nonumber
&& \ \ \ \ \ \ +a_{01}^{(1)}y_1+a_{02}^{(1)}y_1^2+a_{03}^{(1)}y_1^3+a_{04}^{(1)}y_1^4+...\\
\nonumber
&& \ \ \ \ \ \ +a_{01}^{(2)}y_2+a_{02}^{(2)}y_2^2+a_{03}^{(2)}y_2^3+a_{04}^{(2)}y_2^4+...\\
&& \ \ \ \ \ \ \ \ +...+a_{11}^{(1)}s y_1 +a_{11}^{(2)}s y_2. . . + a_{ij}^{(k)}s^iy_k^j...\label{MEXP}
\end{eqnarray}
Note that here $a_{10}\sim(\partial M/\partial S)_c=T_c=0$. Therefore, it has not appeared in the expansion of the mass. Now the contribution up to first order is
\begin{align}
\Big(\frac{\partial M}{\partial s}\Big)_{Y_i}\sim A_{10}s+A_{01}^{(k)}y_k ~. \label{kl}
\end{align}
Here we have rescaled the coefficients as $A_{ij}^{(k)}=(i+1)a_{i+1~j}^{(k)}$~. One can keep higher order terms in the above equation without any change of conclusion. Thus first order contribution serves our purpose. Now, we calculate $(\partial^2S/\partial M^2)_{Y_i}$ in the following way.
\begin{align}
\Big(\frac{\partial^2S}{\partial M^2}\Big)_{Y_i}\sim \Big(\frac{\partial}{\partial M}\Big(\frac{\partial M}{\partial S}\Big)^{-1}_{Y_i}\Big)_{Y_i}\sim \Big(\frac{\partial}{\partial M}\Big[\frac{1}{A_{10}s+A_{01}^{(k)}y_k}\Big]\Big)_{Y_i}~.
\end{align}
Therefore using \eqref{kl} we finally obtain,
\begin{align}
\Big(\frac{\partial^2S}{\partial M^2}\Big)\Big|_{Y_i}\sim \frac{1}{(A_{10}s+A_{01}^{(k)}y_k)^2}\frac{\partial s}{\partial M}\sim \frac{1}{(A_{10}s+A_{01}^{(k)}y_k)^3}~. \label{P2SPM2}
\end{align}
When $Y_i=Y_{ic}$ we find $s\sim m^{1/2}$ (from \eqref{MEXP})~. Thus from \eqref{P2SPM2}, taking the leading order contribution we get,
\begin{align}
\Big(\frac{\partial^2S}{\partial M^2}\Big)\Big|_{Y_i}\sim m^{-\frac{3}{2}} \ \ \ \ \ \textrm{(for $Y_i=Y_{ic}$)}\label{28}
\end{align}
Therefore from the definition of the critical exponent $\alpha$ (see \eqref{ALP}), we find $\alpha=3/2$~.
Again when $M=M_c$ and $\bar Y_i=\bar Y_{ic}$, we obtain $s\sim y_i^{1/2}$ (from \eqref{MEXP}). Thus, from \eqref{P2SPM2} we get $(\partial^2S/\partial M^2)_{Y_i}\sim (A_{10}y_i^{1/2}+A_{01}^{(i)}y_i)^{-3}$. This implies that the quantity diverges as
\begin{align}
\Big(\frac{\partial^2S}{\partial M^2}\Big)\Big|_{Y_i}\sim y_i^{-\frac{3}{2}} \ \ \ \ \ \textrm{(for $M=M_c$ and $\bar Y_i=\bar Y_{ic}$)}.
\end{align}
Therefore from the definition \eqref{PHI}, we get $\phi_i=3/2$~.
Next we expand $Y_i$ as a function of $S$, $M$ and other charge $\bar Y_i$:
\begin{eqnarray}
\nonumber
&& Y_i=a_{000}+a_{200}s^2+a_{300}s^3+a_{400}s^4+...
\\
\nonumber
&& \ \ \ \ \ \ +a_{010} m+a_{020} m^2+a_{030} m3+...
\\
&& \ \ \ \ \ \ +...+ a_{jkl}^{(p)}s^jm^k y_{p}^l+...\label{YEXP}
\end{eqnarray}
Similar to the earlier case, here $a_{100}\sim T_c=0$. Note that $Y_p$ includes all the charges except $Y_i$. Therefore, from \eqref{YEXP} we obtain up to the first order
\begin{align}
\frac{\partial Y_i}{\partial s}\Big|_{M, \bar Y_i}\sim A_{100} s+A_{010}m+A^{(p)}_{001} y_{p}~. \label{PYPS}
\end{align}
Again, we have rescaled the coefficients as $A_{jkl}^{(p)}=(j+1)a_{j+1~kl}^{(p)}$~. It should be mentioned that first order contribution is enough to serve our purpose. Now, following the similar approach as was done earlier, we obtain
\begin{align}
\frac{\partial^2S}{\partial Y_i^2}\Big|_{M, \bar Y_i}\sim\frac{1}{\Big(\frac{\partial Y^i}{\partial s}\Big)^3}\Big|_{M, \bar Y_i}\sim\frac{1}{(A_{100} s+A_{010}m+A^{(p)}_{001} y_{p})^3}. \label{P2SPY2}
\end{align}
Now, for all $Y_i=Y_{ic}$, we obtain from \eqref{YEXP} $s\sim m^{1/2}$. This when substituted in \eqref{P2SPY2} gives $(\partial^2S/\partial Y_i^2)_{M, \bar Y_i}\sim (A_{100} m^{1/2}+A_{010}m)^{-3}$~. Therefore, the leading order contribution gives
\begin{align}
\frac{\partial^2S}{\partial Y_i^2}\Big|_{M, \bar Y_i}\sim m^{-\frac{3}{2}} \ \ \ \ \ \textrm{(for $Y_i=Y_{ic}$)}~.\label{33}
\end{align}
Therefore from the definition of $\gamma_i$ (see \eqref{GAM}), we find $\gamma_i=3/2$~.
Again when $M=M_c$ and $\bar Y_i=\bar Y_{ic}$, we obtain from \eqref{YEXP} $s\sim y_i^{1/2}$. Therefore from \eqref{P2SPY2} we get the result
\begin{align}
\frac{\partial^2S}{\partial Y_i^2}\Big|_{M, \bar Y_i}\sim y_i^{-\frac{3}{2}} \ \ \ \ \ \textrm{(for $M=M_c$ and $\bar Y_i=\bar Y_{ic}$)}~.
\end{align}
Therefore, from the definition of the critical exponent $\s_i$ (in eq. \eqref{SIG}) we obtain $\s_i=3/2$~.
Again from \eqref{PYPS}, the leading order contribution provides
\begin{align}
\tilde X^i\sim \frac{\partial Y_i}{\partial S}\Big|_{M, \bar Y_i}^{-1}\sim \frac{1}{A_{100}} m^{-\frac{1}{2}} \ \ \ \ \ \textrm{(for $Y_i=Y_{ic}$)}~.
\end{align}
The above equation implies
\begin{align}
\eta_{Y_i}=\tilde X^i_{+}-\tilde X^i_{-}\sim \Big(\frac{1}{A_{100}}\Big|_{+}-\frac{1}{A_{100}}\Big|_{-}\Big) m^{-\frac{1}{2}} \ \ \ \ \ \textrm{(for $Y=Y_c$)}~.
\end{align}
Thus, from the definition of $\beta_i$ (see \eqref{BET}), we get the value $\beta_i=-1/2$.
Furthermore, when $M=M_c$ and $\bar Y_i=\bar Y_{ic}$, we obtain
\begin{align}
\tilde X^i\sim \frac{1}{A^{(i)}_{001}}y_i^{-\frac{1}{2}} \ \ \ \ \ \textrm{(for $M=M_c$ and $\bar Y_i=\bar Y_{ic}$)}~.
\end{align}
In that case,
\begin{align}
\eta_{Y_i}\sim y_i^{-\frac{1}{2}} \ \ \ \ \ \textrm{(for $M=M_c$ and $\bar Y_i=\bar Y_{ic}$)}~.
\end{align}
Therefore from the definition of $\delta_i$ in \eqref{DEL}, we get $\delta_i=-2$.
The numerical values of critical exponents obtained so far is given in the following table
\begin{table}[ht]
\begin{center}
\caption{Values of first set of critical exponents}
\label{table1}
~\\
\begin{tabular}{|c|c|c|c|c|c|c}
\hline \hline
$\alpha$ & $\phi_i$ & $\gamma_i$ & $\sigma_i$ & $\beta_i$ & $\delta_i$\\ \hline
$\frac{3}{2}$ & $\frac{3}{2}$ & $\frac{3}{2}$ & $\frac{3}{2}$ & $-\frac{1}{2}$ & $-2$\\ \hline
\hline
\end{tabular}
\end{center}
\end{table}
One can easily check above exponents satisfy the following scaling laws of ``first kind''.
\begin{align}
\alpha +2\beta+\gamma=2~, \label{SCAL1}
\\
\beta(\delta-1)=\gamma~, \label{SCAL2}
\\
\phi(\beta+\gamma)=\alpha~. \label{SCAL3}
\end{align}
The same values of the critical exponents were obtained earlier in \cite{Kaburakigrg, Cai:1996df} considering specific form of metrics. On the contrary, here we obtained those without the explicit information of the black hole spacetime by taking into account two inputs: (a) the black holes we considered here belong to the class, which exhibit extremal phase transition and (b) those black holes satisfy the first law of black hole mechanics. {\it This shows the universality of this type of critical phenomenon}.
Apart from these critical exponents which were obtained above, there are a few others which are studied in the context of the extremal criticality. In the following, we shall discuss those critical exponents and shall obtain their values in a general manner.
Near the critical point, the asymptotic form of the two point correlation function for large $r$ is defined by \cite{book},
\begin{align}
G(r)\sim \frac{e^{ (-r/\xi)}}{r^{d-2-\eta}}~. \label{CORR}
\end{align}
Here, $\eta$ is called as the Fisher's exponent, $d$ is the effective spatial dimension and $\xi$ is called the correlation length. Near the critial point, the behaviour of $\xi$ is given as
\begin{eqnarray}
&& \xi\sim m^{-\nu}\ \ \ \ \ (\textrm{for all}~ Y_i=Y_{ic})~; \label{NU}
\\
&& \xi\sim y_i^{-\mu_i}\ \ \ \ \ (\textrm{for $M=M_c$ and $\bar Y_i=\bar Y_{ic}$})~. \label{MU}
\end{eqnarray}
In the theory of quantum gravity, we do not have much knowledge about the two point correlation function defined in \eqref{CORR}. However, for extremal Reissner-Nordstrom black hole, the inverse of the surface gravity is argued to play the role of the correlation length\cite{Traschen:1994md}. This result also holds for BTZ black hole \cite{Lifschytz:1993eb, Ichinose:1994rg, Cai:1996df} and black $p$-branes \cite{Cai:1997cs, Cai:1997cv}. If we assume, this to be true in the presence of multiple charges in arbitrary dimensions, we get $\xi\sim 1/\kappa\sim 1/T$. Using \eqref{kl}, we can further conclude $\xi\sim (\partial M/\partial s)_{Y_i}^{-1}$. Therefore, from \eqref{MEXP}, the leading order contribution gives
\begin{align}
\xi\sim m^{-\frac{1}{2}} \ \ \ \ \ (\textrm{for all}~ Y_i=Y_{ic})~.
\end{align}
From the definition of $\nu$ in \eqref{NU}, we get the value $\nu=1/2$~. Now, when $M$ and all $Y$ are at their critical values except the $i^{th}$ charge $Y_i$, we obtain from \eqref{MEXP}
\begin{align}
\xi\sim y_i^{-\frac{1}{2}} \ \ \ \ \ (\textrm{for $M=M_c$ and $\bar Y_i=\bar Y_{ic}$})~.
\end{align}
Therefore, from \eqref{MU} we see that all $\mu_i$'s are the same and $\mu_i=\mu=1/2$.
Now, these critical exponents are supposed to satisfy the scaling laws of ``second kind'', which are given by \cite{Stanley1, book},
\begin{align}
\nu(2-\eta)=\gamma~, \label{SCAL4}
\\
\nu d=2-\alpha~, \label{SCA5}
\\
\mu(\beta+\gamma)=\nu~.
\end{align}
Using the obtained value of $\alpha$, $\beta$, $\gamma$, $\mu$ and $\nu$ in the scaling law of the second kind, we get the value of remaining critical exponent $\eta$ and effective spacetime dimension $d$ . These are $\eta=-1$ and $d=1$~. Following table shows these values of exponents.
\begin{table}[ht]
\begin{center}
\caption{Values of remaining critical exponents}
\label{table2}
~\\
\begin{tabular}{|c|c|c|c|c|c|c}
\hline \hline
$\nu$ & $\mu_i$ & $\eta$ \\ \hline
$\frac{1}{2}$ & $\frac{1}{2}$ & $-1$ \\ \hline
\hline
\end{tabular}
\end{center}
\end{table}
Remember, in the above analysis we have assumed that the correlation length is given by the inverse of the surface gravity. This has been checked and accepted for several instances \cite{Traschen:1994md, Lifschytz:1993eb, Ichinose:1994rg, Cai:1996df, Cai:1997cs, Cai:1997cv}. However, we are not sure if this is true in general. Therefore, it would be interesting if the same conclusion can be drawn from a general argument. For the time being, we leave that analysis for future.
\section{\label{sec4}GTD in extremal phase transition}
The concepts of differential geometry is used in thermodynamics for a long time. The underlying motivation to pursue in this direction is to study various thermodynamic phenomena in terms of the geometric properties of the phase space of the system. For non-extremal black holes, there are two major approaches of studying the phase transition of black hole-- one approach deals with the divergence of heat capacity and inverse of isothermal compressibility \cite{Banerjee:2011cz, Banerjee:2012zm, Majhi:2012fz, Lala:2012jp, Ma:2014tka, Azreg-Ainou:2014gja, Liu:2013koa, Mandal:2016anc}. The other approach \cite{Kubiznak:2012wp, Kubiznak:2016qmn, Majhi:2016txt, Bhattacharya:2017nru} is for the black holes in the AdS background, in which the cosmological constant is treated as the thermodynamic pressure. The latter approach exactly resembles the phase transition of the van der Waals fluid system. It must be mentioned that both these phase transitions have been studied extensively under the light of the GTD \cite{Banerjee:2016nse, Bhattacharya:2017hfj, Dehyadegari:2018pkb}. Here people have formulated thermogeometrical metrics in the thermodynamic phase space of black hole and have shown that the corresponding Ricci-scalar diverges at the phase transition point.
In this section, we incorporate those ideas to study the extremal phase transition. Here, we comment that there are several ways to formulate the thermogeometrical metric. First Weinhold \cite{WEINHOLD} introduced a metric, the components of which are given by the Hessian of the internal thermodynamic energy. Later, Ruppeiner \cite{RUPP, Ruppeiner:1995zz} introduced another metric, which is defined as the negative of the Hessian of the entropy, and is conformal to the Weinhold metric with the conformal factor being the inverse temperature. Later, Quevedo \cite{Quevedo:2006xk, Quevedo:2007mj, Quevedo:2008xn, Quevedo:2008ry, Alvarez:2008wa, Quevedo:2011np, Quevedo:2017tgz, Quevedo:2016cge} came up with the idea of defining the thermogeometrical metric in a Legendre-invariant way.
In our general procedure of analyzing the extremal phase transition, we study the behaviour of Ricci-scalar near the critical point for all these metrics.
\subsection{The Weinhold metric}
To write the Weinhold metric, one has to write mass (which plays the role of internal energy) as the function of entropy and the charges i.e., $M\equiv M (S, Y_i)$. Now for the sake of simplicity we consider the dependence of mass on a particular charge $Y$ and keep all other charges fixed. Therefore the first law of thermodynamics is written as,
\begin{align}
dM=TdS+XdY~. \label{1st Law}
\end{align}
Here $T=(\partial M/\partial S)_Y$ and $X=(\partial M/\partial Y)_S$.
Now the Weinhold metric is given by,
\begin{align}
ds_W^2=\frac{\partial^2 M}{\partial x_i\partial x_j}dx_idx_j\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \{x_1=S,\ x_2=Y\}~. \label{WEIN}
\end{align}
The expanded form of the Weinhold metric is
\begin{align}
ds^2_W=-f(S,Y)dS^2+g(S,Y)dY^2+2h (S,Y)dSdY~, \label{WEINEXT}
\end{align}
where $f(S, Y)=-M_{SS}$, $g(S, Y)=M_{YY}$ and $h(S, Y)= M_{SY}=M_{YS}$~. The Ricci scalar corresponding to the Weinhold metric \eqref{WEINEXT} is given by,
\begin{align}
\nonumber
R_{(W)}=\frac{1}{2(fg+h^2)^2}\Big[f(f_Y g_{Y}-g_S^2+2g_Yh_S)+g\Big\{f_Y^2+f_S(2h_Y-g_S)-2f(f_{YY}+h_{SY}-g_{SS})\Big\}
\\
+h\Big\{-g_Yf_S+f_Y(2h_Y+g_S)+4h_Yh_S-2g_Sh_S-2h(f_{YY}+2h_{SY}-g_{SS})\Big\}\Big] ~,\label{RICCIWEIN}
\end{align}
where $f_J=\partial f/\partial J$ and so on. Now, from the expansion of $M$ (given in \eqref{MEXP}) we can conclude that $f$, $g$, $h$ and their derivatives are finite. Therefore, the Ricci scalar of the Weinhold metric is a finite quantity near the critical point.
\subsection{The Ruppeiner metric}
We first write the first law of thermodynamics\eqref{1st Law} as $dS=\beta dM-\tilde X dY$. In this form, the conjugate quantities are taken as $\beta=(\partial S/\partial M)_Y$ and $\tilde X=-(\partial S/\partial Y)_M$. Now, the Ruppeiner metric is defined as
\begin{align}
ds^2_R=-\frac{\partial^2S}{\partial x_i'\partial x_j'}dx_i'dx_j'\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \{x_1'=M,\ x_2'=Y\}~. \label{RUP}
\end{align}
Here, $g_{11}=-S_{MM}$, $g_{22}=-S_{YY}$ and $g_{12}=g_{21}=-S_{MY}$~. It implies that the expansion of the Ruppeiner metric is
\begin{align}
ds^2_R=-f'(M,Y)dM^2+g'(M,Y)dY^2+2h' (M,Y)dMdY~, \label{RUPEXT}
\end{align}
where $f'=S_{MM}$, $g'=-S_{YY}$ and $h'=-S_{MY}$~.
The Ricci scalar of the metric \eqref{RUPEXT} is found to be,
\begin{eqnarray}
&& R_{(R)}=\frac{1}{2(f'g'+h'^2)^2}\Big[f'(f_Y' g_{Y}'-g_M'^{2}+2g_Y'h_M')\nonumber
\\
&&+g'\Big\{f_Y'^2+f_M'(2h_Y'-g_M')-2f'(f_{YY}'+h_{MY}'-g_{MM}')\Big\}\nonumber
\\
&&+h'\Big\{-g_Y'f_M'+f_Y'(2h_Y'+g_M')+4h_Y'h_M'-2g_M'h_M'-2h'(f_{YY}'+2h_{MY}'-g_{MM}')\Big\}\Big] \label{RICCIRUP}
\end{eqnarray}
Now, we have to calculate each term of the Ricci scalar of \eqref{RICCIRUP} to see its dependence on $s$. To do that, we find out the leading order contribution of $f'$, $g'$ and their derivatives.
From \eqref{P2SPM2} we see that $f'=-(\partial^2S/\partial M^2)_{Y}\sim 1/s^3$. Therefore, $f_M'=(\partial f'/\partial M)_Y\sim (1/s^4)(\partial s/\partial M)_Y$. Using \eqref{kl}, one obtains $f_M'\sim s^{-5}$. In a similar way, $f_{MM}'\sim s^{-7}$. Now, $f_Y'=(\partial f'/\partial Y)_M\sim (1/s^4)(\partial s/\partial Y)_M$. Again, using \eqref{PYPS} one gets $f_Y'\sim s^{-5}$. The same arguments yield $f_{YY}'\sim s^{-7}$ and $f_{MY}'=f_{YM}'\sim s^{-7}$~. Following the same procedure, one similarly obtains $g'\sim s^{-3}$, $g'_{x_i'}\sim s^{-5}$ and $g'_{x_i'x_j'}\sim s^{-7}$. Also, $h'\sim s^{-3}$, $h'_{x_i'}\sim s^{-5}$ and $h'_{x_i'x_j'}\sim s^{-7}$. As a result, we see that the denominator goes as $\sim s^{-12}$ and each term in the numerator goes as $\sim s^{-13}$. Therefore, the Ricci scalar diverges as
\begin{align}
R_{(R)}\sim s^{-1}~.
\end{align}
The property of the Ruppeiner metric has also been studied in a different way \cite{Cai:1998ep, Wei:2009zzf} while studying the extremal phase transition of BTZ black holes. It has been there argued that the Ruppeiner metric should diverge as $R_{(R)}\sim \xi^{d}$. Since, in our case $\xi\sim s^{-1}$ near the critical point, we obtain $R_{(R)}\sim \xi^{1}$. Therefore, we can again conclude that the effective spatial dimension $d=1$ for any extremal black hole, which is in agreement with the claim of the recent papers \cite{Horowitz:1996ac, Ghosh:1998cc}. Thus, from the thermogeometric approach, we can again generally prove that the effective spatial dimension of an extremal black hole is one.
\subsection{Legendre invariant metric}
Above two thermogeometrical metrics, namely the Wienhold and the Ruppeiner metric are not Legendre-invariant. Moreover in some cases, conclusions derived from the Weinhold metric and the Ruppeiner metric are not consistent with each other. Later Quevedo {\it et al.} claimed that those inconsistencies appear because these metrics are not Legendre-invariant and hence they came up with Legendre invariant metric formalism \cite{Quevedo:2006xk, Quevedo:2007mj, Quevedo:2008xn, Quevedo:2008ry, Alvarez:2008wa, Quevedo:2011np, Quevedo:2017tgz, Quevedo:2016cge}. In the following, we discuss two types of Legendre invariant thermogeometrical metric. One of them (Quevedo metric: 1) is mostly used as a Legendre-invariant metric. Here, we see that the Ricci scalar of the first type of the Legendre-invariant metric is a finite quantity at the critical point. So we discuss another type of Legendre-invariant metric (Quevedo metric: 2). The second metric is not that familiar but we see that the Ricci scalar corresponding to this metric vanishes. The formalism which we adopt here was originally developed by Hermann \cite{Hermann} and Mrugala \cite{Mrugala1, Mrugala2}, which was later followed extensively by Quevedo.
\subsubsection{Quevedo metric: 1}
We define a thermodynamic phase space $\mathcal{T}$ with coordinates $\mathcal{Z}^A=\{S, q^a, p^a\}$ where $q^a=\{M, Y\}$ are the variables and $p^a=\{S_M=\beta, S_Y=-\tilde X=-\beta X\}$ are the conjugate variables. Therefore, in the entropy representation, the fundamental one form in $\mathcal{T^*}$ (where, $\mathcal{T^*}$ is the cotangent space of $\mathcal{T}$) is given by,
\begin{align}
\Theta_S=dS-\beta dM+\tilde X dY~, \label{FUNDA}
\end{align}
which is invariant under the Legendre transformation
\begin{align}
M(q)=\tilde M(\tilde q)-\delta_{ab}\tilde{q}^a\tilde{p}^b \label{LEGENDRE}
\\
\nonumber
\textrm{with}\ \ \ \ \ \ \ \ \ \
\ \ q^a=-\tilde p^a \ \ \textrm{and}\ \ p^a=\tilde q^a~.
\end{align}
Now, following Quevedo's formalism, one possible form of the Legendre invariant thermogeometrical metric (on $\mathcal{T}$) is (Eq. (39) of \cite{Quevedo:2006xk})
\begin{align}
G_1=\Theta_S^2+ (\beta M+\tilde XY)(d\beta dM+dYd\tilde X).
\end{align}
Expanding the conjugate quantities ($\beta$ and $\tilde X$) as a function of the variables ($M$ and $Y$), one finds the the expression of $G_1$ in the space of equilibrium ($\Theta_S=0$) as
\begin{align}
G_1=-f_1(M, Y)dM^2+g_1(M, Y) dY^2~, \label{METLEG}
\end{align}
where $f_1(M, Y)=-(\beta M+\tilde XY)S_{MM}$ and $g_1(M, Y)=-(\beta M+\tilde XY)S_{YY}$~. The Ricci scalar of the metric \eqref{METLEG} is given by,
\begin{align}
R_1=\frac{1}{2(f_1g_1)^2}\Big[f_1(f_{1Y} g_{1Y}-g_{1M}^{2})+g_1\Big\{f_{1Y}^2-f_{1M}g_{1M}-2f_1(f_{1YY}-g_{1MM})\Big\}\Big] ~. \label{RICCI1}
\end{align}
Again, we check the order of each term in the Ricci scalar. $f_1\sim \beta S_{MM}\sim (\partial S/\partial M)_Y(\partial^2 S/\partial M^2)_Y$. This implies $f_1\sim s^{-4}$. Similarly $g_1\sim s^{-4}$. Following the same procedure as was done in the Ruppeiner case, we obtain $f_{1x_i}\sim s^{-6}$, $g_{1x_i}\sim s^{-6}$, $f_{1x_i x_j}\sim s^{-8}$ and $g_{1x_ix_j}\sim s^{-8}$. Therefore, we see that the denominator goes as $\sim s^{-16}$ and the numerator also goes as $\sim s^{-16}$. Therefore, the Ricci scalar is finite in this case.
\subsubsection{Quevedo metric: 2}
As the choice of Legendre invariant metric is not unique, we can formulate other Legendre invariant metric. Following Quevedo's formalism (Eq. (37) of \cite{Quevedo:2006xk}) we see
\begin{align}
G_2=\Theta_S^2+c_1\beta M d\beta dM+c_2 \tilde X Yd\tilde XdY+d\beta^2+dM^2+d\tilde X ^2+dY^2 \label{METR2}
\end{align}
is Legendre invariant for any value of the real constants $c_1$ and $c_2$. For the simplicity of calculation, we take $c_1=c_2=1$~. Now using $d\beta=S_{MM}dM+S_{MY}dY$ and $d\tilde X=-S_{YM}dM-S_{YY}dY$ in \eqref{METR2} we get, in equilibrium space
\begin{align}
G_2=-f_2(M,Y)dM^2+g_2(M,Y)dY^2+2h_2 (M,Y)dMdY~,
\end{align}
where $f_2=-[1+\beta M S_{MM}+S_{MM}^2+S_{MY}^2]$, $g_2=1-\tilde X YS_{YY}+S_{YY}^2+S_{MY}^2$ and $h_2=\frac{1}{2}(\beta M-\tilde X Y) S_{MY}+S_{MM}S_{MY}+S_{YM}S_{YY}$. Thus the Ricci scalar is given by,
\begin{eqnarray}
&&R_2=\frac{1}{2(f_2g_2+h_2^2)^2}\Big[f_2(f_{2Y} g_{2Y}-g_{2M}^{2}+2g_{2Y}h_{2M})
\nonumber
\\
&&+g_2\Big\{f_{2Y}^2+f_{2M}(2h_{2Y}-g_{2M})-2f_2(f_{2YY}+h_{2MY}-g_{2MM})\Big\}
\\
\nonumber
&&+h_2\Big\{-g_{2Y}f_{2M}+f_{2Y}(2h_{2Y}+g_{2M})+4h_{2Y}h_{2M}-2g_{2M}h_{2M}-2h_2(f_{2YY}+2h_{2MY}-g_{2MM})\Big\}\Big] \label{RICCI2}
\end{eqnarray}
Now, $f_2=\mathcal{O}(s^0)+\mathcal{O}(s^{-4})+\mathcal{O}(s^{-6})$. The leading order contribution near the critical point will be $f_{2}\sim s^{-6}$. As a result, $f_{2x_i}\sim s^{-8}$ and $f_{2x_ix_j}\sim s^{-10}$~. Leading order contributions of $g_{2}$ and $h_2$ are same as $f_2$. Therefore, the denominator goes as $\sim s^{-24}$ and the numerator goes as $\sim s^{-22}$. As a result,
\begin{align}
R_2\sim s^2~.
\end{align}
Consequently, we see that the Ricci-scalar vanishes near the critical point.
In this section, we have studied the behaviour of the Ricci-scalar for different thermogeometrical metrics and have shown that the Ricci-scalar of the Ruppeiner metric diverges at the extremal limit. On the contrary, the Ricci-scalar of other thermogeometrical metrics remains finite (or vanishes) at that point. Therefore, we conclude that the extremal phase transition shows the behaviour of the second order phase transition not only in the specific ensemble of thermodynamics ({\it i.e.~} the microcanonical ensemble), but also for a specific thermogeometric manifold as well (the Ruppeiner one). Note that the Legendre-invariant thermogeometrical metrics, which are mostly used nowadays, cannot confirm the second order phase transition in the present case. A plausible explanation to that might be as follows. Remember that the Legendre-invariant metrics are constructed on the line of arguments that a proper thermogeometrical metric should be Legendre invariant as the thermodynamic features are invariant in all ensembles. Since one thermodynamic potential, by which an ensemble is characterized, is connected to the same in the other ensemble by the Legendre transformation, the entire thermodynamic description is invariant due to the Legendre transformation, which should reflect on the thermogeometrical metric. However, as we have noticed in the present case, the identification of the non-extremal to extremal transformation with the second order phase transition is valid only in the microcanonical ensemble. As a result, the present thermodynamic description is not invariant across all ensembles. Therefore, the use of a Legendre-invariant metric might not be suitable in this case. Nonetheless, we have checked the behaviour of the Ricci-scalar of all the thermogeometrical metrics which are popular in GTD and from that analysis we found that the Ruppeiner metric is the ideal one for the thermogeometric description of the extremal phase transition. {\it Interestingly, here entropy $S$ plays the central role both in microcanonical ensemble ($S$ is chosen as the Massieu function) and in Ruppeiner geometrical description (the metric is constructed by considering $S$ as the thermodynamic potential)}.
\section{Conclusions}
In this work, we have studied the extremal phase transition of black hole in a general framework. There are several works \cite{Kaburakigrg, curir1, curir2, Pavon:1988in, Pavon:1991kh, Cai:1996df, Cai:1998ep, Wei:2009zzf, Cai:1997nb, Ma:2013eaa} to show that the extremal phase transition is a second order phase transition. These earlier works were done case by case for a particular spacetime and dimension. The obtained results in different spacetimes (such as the critical exponents, scaling laws {\it etc.}) are in accordance with each other and strongly suggest that there must be a metric independent way to establish those earlier results. This has been the major motivation for this work.
We have proved that the transformation of the black hole from a non-extremal to an extremal one is a second order phase transition. For that, we have calculated the second order moments of fluctuations in different ensembles and have shown that those moments diverge for a black hole in microcanonical ensemble, which is a sign of a second order phase transition as per the prescription of Pav$\acute{\textrm{o}}$n and Rub$\acute{\textrm{i}}$ \cite{Pavon:1988in, Pavon:1991kh}. Afterwards, we have generally obtained the critical exponents for this phase transition and have shown that the critical exponents satisfy the scaling laws. While proving those results, we have not accounted any particular spacetime, which implies our results are valid for all the black hole spacetimes which become extremal at certain limit. Thus, the universality of results, which were predicted by earlier works, is proved by our analysis and hence from now on one need not check the critical behaviour case by case.
Finally, we have extended our analysis to GTD, which is a recent formalism to describe the phase transition geometrically. We have shown that the extremal critical point of black holes can be identified as a particular point where the Ricci scalar corresponding to the Ruppeiner metric diverges. In addition, we have also shown that the Ricci scalar of the Weinhold metric and of one type of Legendre-invariant metric (Quevedo metric: 1) is a finite quantity and does not show any special behaviour. In another Legendre invariant metric (Quevedo metric: 2), the Ricci-scalar vanishes on the critical point. In this analysis we observed that extremal phase transition is properly explained in microcanonical ensemble and by Ruppeiner geometry. Note that in both the descriptions, entropy plays the central role: $S$ acts as Massieu function in microcanosical ensemble and thermodynamical potential in GTD. At this moment, the actual reason for this is not known to us; hope we shall be able to find the precise reason in future.
Thus our paper covers different thermodynamics aspects of extremal black hole. Other previous works in this field confined their analysis to specific cases and hence can not explain questions regarding universality. The novelty of our work is, it is very general and does not require any specific metric. In this sense our paper unifies all other work on extremal phase transition in an elegnant manner. At last we shall conclude by making the following comments on our observations we made here on the extremal phase transition.
In this work, we have examined whether any phase transition occurs during the transition of a black hole from a non-extremal to an extremal one. For that, in our general framework (i.e. without using the explicit expression for black hole metric), we have taken the help of the fluctuation theory. It has been observed that the presence of a second order phase transition naturally occurs only in the microcanonical ensemble, while the other ensembles (canonical and grand canonical) fail to show that. This has also been observed earlier in several case by case studies (i.e. explicitly using the black hole metric expression) \cite{Kaburakigrg, Cai:1996df, Cai:1998ep, Wei:2009zzf}. The possible reasons for that can be stated as follows.
In this context, let us first mention why not all ensembles agree upon the same result in the fluctuation theory. Usually, we see that the mean values of different thermodynamic quantities are the same in different ensembles for a given system in equilibrium. However, it must be noted that the different ensembles predict different fluctuations of a thermodynamic parameter around its equilibrium value \cite{Kaburakigrg}. In other words, average value of thermodynamic quantities are same in all ensembles, but fluctuations are not. Thus, the usual notion of the equivalence of the different ensembles can break down while investigating the physics with the help of fluctuations in the macroscopic parameters. We also have observed the same in the present analysis as well. Only in the microcanonical ensemble all the second order moments of the relevant quantities are divergent and implies the presence of the critical point. While in other ensembles (canonical and grand canonical) one cannot confirm the presence of the critical point at $T=0$ as all the second order fluctuation modes do not diverge in those cases.
Let us now understand why the microcanonical ensemble appears to be so special in this case. Remember, in several cases of black hole thermodynamics, one particular ensemble (specially the microcanonical ensemble) can be more preferred then the other ensembles. For example, the microcanonical ensemble is the most suitable one for the discussion of the fluctuations of stellar mass or more massive black holes. This is because the time scale of particle exchange is much larger than the present age of the universe in such cases \cite{Kaburakigrg}, which means the black hole hardly exchange any particle with the environment. On the contrary, if the black hole is small, more particle exchange can take place and the grand canonical ensemble becomes more suitable for the thermodynamic description. Another example is that the microcanonical ensemble is the proper ensemble for the thermodynamic description of the microscopic black holes which are not in equilibrium, such as the radiating black holes \cite{Casadio:2011pd}. This example is particularly important in this case because we have accounted the temperature and entropy of the black holes, which is obtained only when one considers the quantum (microscopic) effect in the theory. Thus, it can be concluded that in certain cases, one particular ensemble can be more favourable then the others in black hole thermodynamics. From that line of argument, it can be said that the microcanonical ensemble can be the appropriate or a proper ensemble for the thermodynamic description of extremal phase transition of black holes.
Later from our thermogeometric analysis, we have found that the divergence of the Ricci-scalar at the critical point occurs only for the Ruppeiner metric, whereas the scalar curvature is either finite or vanishing for the Weinhold and Quevedo (I and II) metrics. Firstly, we mention why the Ruppeiner metric is unique in this study. It would be interesting to note that the Ruppeiner metric is the Hessian of the Massieu function of the microcanonical ensemble (the entropy), which, as we have observed earlier, can be regarded as the proper ensemble for the thermodynamic description of the extremal phase transition of black holes. From that viewpoint, the Ruppeiner metric is special in this case, in spite of the fact that this metric is not formulated in a Legendre invariant way.
Now, we mention why the Legendre-invariant formalism by Quevedo has not been able to reflect the extremal phase transition through the divergence of corresponding Ricci scalar. We have already seen, our analysis can predict the criticality only in the microcanonical ensemble. On the other hand, the Legendre-invariant way of defining thermogeometrical metric implies the result should be valid in all the ensembles. Since there is a pre-existing in-equivalence among the ensembles in the extremal phase transition, it is not surprising that the Legendre-invariant formulation is not suitable in the present case. Again, the root lies in the fact that we are looking at the average value (here it is Ricci scalar), not on the moments of the fluctuations (like $<\delta R \delta R>$) which can be different in different Legendre invariant metrics. Having the feel that the fluctuations in Ricci scalar can be good quantity in explaining the extremal phase transition in the context of thermogeometric study of phase transition, we calculated $<\delta R \delta R>$ for both the Quevedo metrics. The details of this is presented in Appendix \ref{AppC}.
We found that the moments of fluctuation of the Ricci-scalar diverges at the critical point for Quevedo-I metric, which is mostly used in the thermogeometric description. {\it Thus, it can be conjectured that instead of the Ricci-scalar, from the study of the fluctuation of the Ricci-scalar the presence of the criticality can be well determined}.
\section*{Appendix}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,180 |
Ralph E Stewart, amerikansk tonsättare. Han finns representerad i ett flertal psalmböcker bland annat EFS-tillägget 1986 och Frälsningsarméns sångbok 1990 (FA) med tonsättningen av en psalm.
Tonsättningar av psalmer
Frälsare på korsets stam (EFS nr 762 och FA nr 340) komponerad före 1951.
Amerikanska koralkompositörer
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\subsection{Introduction}\label{s:int}
Quantum physics, in a similar way to Probability and (mostly historically) Differential and Integral Calculus, has been riddled with major arguments about how to interpret it, with multifaceted paradoxes to tackle. That despite the fact that all three cases stand on a clear-cut established mathematical formalism -- a term applied to contrast with `deep' essence -- which tells precisely when formulas and modes of solution to problems are valid and what the (mathematical) theory predicts experimentally.
But let us follow closely the mathematics/formalism, the nuts and bolts of what quantum physics says. Then, basically well-known considerations seem to highlight some rather bold points about the `logic' aspect in quantum physics, necessarily restricting when and which logic may be admissible (see below). And one may understand why that path is hardly followed in the literature.
That mathematics/formalism can be fairly viewed as characterized, compared with classical physics, by non-commutative algebras replacing commutative. These classically appearing, in fact, in dealing with systems of possibilities (say, all possible planetary motions under gravity of which one is the actual one).
In particular, contrary to too common usage, the quantum non-commutativity should make it impossible to simply `transcend' the `system of possibilities' aspect into a `yes-no' logic essential for an `actual world'. One may have the latter only in a `haven' of approximately commutative algebras of `quasi-classical macroscopic observables', and moreover that `yes-no actual world' would plainly be an `extra ingredient' to the base quantum theory itself.
I recapitulate here some points in \cite{Levy-Occ}.
\subsection{The Advent of Quantum Physics}
Adhering thus to the mathematics/formalism, one may fairly say that \textit{the advent of quantum physics came when observations and experiments forced physicists to replace the classical commutative algebra related to a system of possibilities (to wit, the algebra of bounded magnitudes) by a non-commutative algebra (to wit, the algebra of the (complexified) bounded observables)}.
Which turned to be just the right thing so that things fit.
Bluntly, as per the formalism, our case is basically closed -- we know how to oblige to Nature the Autocrat. Mathematically=formalistically speaking, \textit{to understand the base of quantum physics we have to cope with that non-commutativity}.
That classical commutative algebra refers, of course, to a system of `possible worlds'. Very often in classical physics `the actual world' is approached as one `state' or `possible world' in a system of possibilities. Thus there are the laws of planetary motion under gravity, which allow many possible scenarios, one of which is the actual case.
And mathematically we have the algebra of scalar \textbf{magnitudes}, say complex-valued (usually with some qualification such as `bounded') -- these having, in general, different numerical values at different possible worlds. Define the algebraic operations there \textit{pointwise}, i.e., say, addition of magnitudes by adding the values they take at each possible world. They form an \textit{algebra} -- a set where operations like addition, multiplication, multiplication by a complex scalar and (complex) conjugation are defined and satisfy standard requirements.
In such system the logic is a `logic of possibilities' -- logical statements (`events' in the parlance of probability theory) cannot be said to be `true' or `false'. Rather, they are true in some possible worlds and false in others. They form a \textbf{Boolean Algebra}, bigger than just the truth values $\mathbf{2}=\{\mathbf{'True'},\mathbf{'False'}\}$.
The algebra of magnitudes is obviously \textbf{commutative}, here meaning that the commutative law for multiplication $ab=ba$ holds (while mathematics has important examples for non-commutative algebras -- notably the algebras of matrices or of linear operators in linear spaces).
And as very common in such mathematics, one may start from either of the structures: the set of `possible worlds'; the boolean algebra of `events'; the algebra of magnitudes, and retrieve = define the others. In this sense each can be the primary notion.
In particular, the possible worlds (here with the role of `states', even `points'), can be retrieved as functionals on the algebra, i.e.\ scalar-valued functions, assumed to be \textit{homomorphisms}, i.e.\ to respect all the operations%
\footnote{Considering, of course the, say complex, scalars as naturally also an algebra with operations.}
-- let us refer to these as \textbf{hom-functionals}.
Indeed, identify a `point' $a$ with the \textit{hom-functional whose value at a magnitude $\phi$ is the value $\phi$ takes at $a$}!
And with assumptions that often hold, one proves sorts of theorems%
\footnote{I have in mind theorems like Wedderburn's theorem that a finite-dimensional \textit{semisimple}
non-commutative algebra is a cartesian product of simple algebras, in the commutative case the latter must
be the scalars; Gelfand's theorem that a commutative $C^{\ast}$-algebra is always (isomorphic to) the algebra
of continuous scalar functions on a Hausdorff compact topological space; that a commutative von Neumann
algebra is necessarily $L^\infty(X)$ for $X$ some measurable space; etc.}
which say that \textit{giving a (properly qualified) set of points is `the same' as giving a (qualified) commutative algebra: the points are (in one-one correspondence) with the hom-functionals on the algebra as above; and the algebra is (isomorphic) to that of (qualified) magnitudes on the points, with operations defined pointwise}.
Bluntly speaking, when that is the case \textit{a commutative algebra is just another way to say a (totally classical) many-world scenario}.
And keep in mind that all that refers to a \textit{system of possibilities} where a statement (= event) cannot be said to be simply true or false! That is, of course, untenable as a final conclusion. Science, which has to tell us about `the real world' should assert: yes or no.
Still, when commutative we know that the algebra can very often be viewed as embodying a set of possibilities/possible worlds/states, if you wish those being (identified with) the hom-functionals on the algebra, of which, in the commutative case, there are very often plenty, enough to determine and build the algebra.
And in each particular one of these `possible worlds' statements \textit{are} `true' or `false'!
So in the classical/commutative scenario we may try to `fool ourselves' by pretending
to have in mind some \textit{particular} possible world, with statements having truth values, `yet letting it vary'.
Returning to the quantum case, we then have a non-commutative algebra, which, in analogy to the commutative case, should refer to a set of possibilities, where statements/events certainly are not just `true' or false.
The usual way that the quantum non-commutative algebra is described is, of course, as the (in fact, von Neumann) algebra $B(\mathcal{H})$ of bounded operators on a Hilbert space $\mathcal{H}$. But for us the algebra has the first place, the vectors in the Hilbert space etc.\ might be said to serve a role of indices (cf.\ the discussion of `states' below).
One may say that the non-commutative algebra makes, primarily, a description of \textit{a new and strange logic} on \textit{the system of possibilities}.
In that, \textit{an accompanying Boolean-like algebra is missing}. Indeed, one cannot base the logic, as in the commutative classical case, on operations among the logical statements/events themselves.%
\footnote{From the point of view of the algebra, these should be the analogs to the numbers $0,1$ ($=\{\mathbf{'True'},\mathbf{'False'}\}$) among the scalars, the numbers $a$ satisfying $a^2=a$. For magnitudes that equation would characterize a magnitude which takes values in $\{0,1\}$,
the characteristic function of a set $E$ of points = an event, (i.e.\ the magnitude taking value $1$ in $E$ and $0$ in its complement).
For the non-commutative scenario, to wit the algebra $B(\mathcal{H})$ of operators in a Hilbert space,
these are \textit{orthogonal projections}, members $p$ of the algebra satisfying $p^2=p$ (adding the requirement $p^{\ast}=p$ -- Hermitian -- the analog automatically holding for $0,1$-valued), orthogonal projections on closed subspaces in the Hilbert space, thus in one-one correspondence with these.}
Boolean operations among events = subspaces = projections such as conjunction and disjunction are naturally partially defined (just for compatible events, equivalently commuting projections).%
\footnote{One may be tempted to extend the definition of union and intersection to non-compatible events as the sum and intersection of the relevant subspaces of the Hilbert space. Note, however, that these depend highly non-continuousely on the subspaces -- when an `angle between subspaces' turns to zero the intersection and sum spaces both `jump', which seems to definitely disqualify that.}
The logic should rather be thought of as \textit{given by the non-commutative algebra itself}, whose operations are always defined. (Thus there is no harm if the original algebra, which serves to define the logic, contains only \textit{bounded} observables although in general observables are unbounded.)
\begin{remark}
So, noting the mathematical difference in properties, we see when this `novelty' may be dealt in some analogy with the classical/commutative case, and when things are totally different.
Referring, in particular, to \textit{probability measures}, the transit to quantum is rather smooth. In the commutative case probability measures on the `points' = `possibilities' would, from the point of view of the algebra of magnitudes, be given by \textit{positive linear functionals on the (commutative) algebra of magnitudes mapping the unit element of the algebra to $1$} -- the functional which gives to a magnitude its expectation and in particular to the characteristic function of an event $E$ the probability of $E$.
By analogy, the role of probability measures will be played, in the non-commutative case too, by positive functionals with value $1$ at the unit element $\mathbf{1}$.%
\footnote{To be mathematically correct, in the infinite-dimensional case, only positive linear functionals belonging to the \textit{predual} of the von Neumann algebra (such as $B(\mathcal{H})$).}
Thus when the algebra is $B(\mathcal{H})$ -- the (von Neumann) algebra of the bounded operators in a Hilbert space, these will be the \textit{density matrices}, i.e.\ positive semi-definite operators $\tau$ with trace $1$. Relative to such $\tau$, an element of the algebra, i.e.\ an operator $A\in B(\mathcal{H})$, will have the expectation
$\text{tr }(\tau\cdot A)=\text{tr }(A\cdot\tau)$. In particular an event -- projection $p$ (i.e.\ $p^2=p$ and $p^{\ast}=p$) will have the probability $\text{tr }(\tau\cdot p)=\text{tr }(p\cdot\tau)$ which will be in the interval $[0,1]$.
\end{remark}
\subsection{States? What Can or Cannot Be}
Recall again, that crucially, the above non-commutative algebra=logic should define just the quantum counterpart of the \textit{logic of possibilities}.
In the classical/commutative case one often has the algebra as just another way to say `a set of (possible) worlds', retrieved as the set of hom-functionals from the algebra to the complex scalars.
Then certainly only such states/possible worlds could (as a genuine world with usual logic obviously must) endow statements/events of the system of possibilities with a truth-value `true' or `false', so that these events occur or not.
In the system of possibilities itself these are just elements of a Boolean algebra, and saying that they occur or not is meaningless. The logic consists of manipulating them, if one wishes as members of that Boolean algebra,
The same should hold for the non-commutative quantum system of possibilities. To go beyond possibilities -- to be able to assert some statement as `true' or `false', we would need to refer to a state/possible world.
A full, genuine such state/possible world should thus be a \textit{bona fide} hom-functional. Then every observable will get its value and every event will get the value $1$ or $0$ $=$ `true' or `false'. And in the commutative case we had them plenty and sufficient.
Yet alas, one knows mathematically that \textit{for a non-commutative algebra that is out of the question}.
Such hom-functionals to the scalars are scarce and insufficient, sometimes nonexistent, in a non-commutative algebra -- it definitely does not make a many-world scenario (totally classical!) as the commutative case did.
The (pure) states spoken of in quantum physics, given by wave-functions = elements of a Hilbert space up to multiplication by a scalar (thus by definition always abundant) are something else, definitely not where statements are `true' or `false'. In the above sense they are not states at all.
These are events characterized by being minimal (one may use the word `atomic' in the mathematical sense), i.e.\ they have no proper sub-events. Equivalently, there is only one `probability measure' supported in each of them. Physically, they give the maximum specificity that one can have, what in the classical/commutative case had characterized single states/possible worlds.
But a general event need not either happen in them (contain the atomic event) or not happen (its complement contains the atomic event), equivalently their probability distribution gives to a general event values in the interval $[0,1]$ different from $0$ or $1$. In particular, future events have just probabilities with respect to such mathematically atomic `states'.
In this sense these pure states, which give the maximum specificity that one may conceive, are like general events -- sets of states -- and general probability distributions on them in the (deterministic) classical systems. Thus one should not wonder that this maximum specificity does not determine the future (yielding only probabilities).
Again, as we have emphasized, the quantum non-commutative structure should refer just to a \textit{system of possibilities}, thus it is meaningless to say that statements are true or false, events occur or not. What the Theory does is just manipulating them, as members of the non-commutative logic -- to wit subspaces of the Hilbert space (equivalently orthogonal projections) using its clear-cut formalism.
Otherwise put, as we have seen, the mathematics/formalism definitely tells us that:
The Quantum Theory does of course richly speak about particles, fields, physical systems etc.\ and about statements = (events = subspaces of the Hilbert space), such as `the electron is here or there'; `it has this or that property'; `the system is (or was) in that state'. \textbf{But to stress again}: these being actors in the non-commutative logic of \textit{the system of possibilities}, the Theory aptly manipulates them according to its clear-cut formalism, but \textbf{they in no way happen, are true (or not happen, are false)}, in spite of the tempting wording -- asking that is meaningless.
That was the case also with the commutative/classical scenario. Yet as said above we then could, on the back of our mind, think that we are dealing with a set of `possible worlds', so they are true or false \textit{for each one of them} and judge ourselves as `having in mind some particular possible world, yet `letting it vary'. As we saw, for the quantum/non-commutative even that small consolation is denied us.
\subsection{Our Actual World (an Extra Ingredient)}
But, in this quantum picture, we still have to recover our actual world -- where events occur or not -- and our usual logic. Otherwise put, we have to locate the as if \textit{haven}, inside Quantum, where statements \textit{can} be true or false.
It seems clear where to find that. We need a commutative sub-algebra, and something like that presents itself: \textit{the algebra of the macroscopic, quasi-classical observables which almost commute}. (In fact, even its members should be viewed as only approximate -- they cannot be handled in greater precision than the uncertainty that makes them commute.)%
\footnote{One may object that our approximate notions here are hardly given a mathematical definition. Maybe one can try some mathematically rigorous ways to do that. I still do prefer this kind of discourse, which seems to be within the scope of the ways of physics, to picking some mathematically correct definitions, but otherwise arbitrary from the point of view of the situation at hand itself.
Roughly, that approximate discourse should mean that `one can say something if and only if one stays within the allowed approximation'.}
These, and the events they define, are what we have in our old classically behaving world.%
\footnote{Note that the evolution of a system in Time is defined by conjugation with imaginary exponents of the energy (Schr\"odinger's equation), So the energy, itself quasi-classical, cannot exactly commute with other quasi-classical observables, otherwise there would be no time evolution there. Similarly with the momentum which induces variation in space.}
And our actual world is described by an \textit{approximate} hom-functional on this (approximately commutative) quasi-classical algebra. It will give values $0$ or $1$ to projections/events/closed subspaces of the Hilbert space \textit{which belong to this quasi-classical algebra}, (again, a $p$ satisfying $p^2=p$ must map by a hom-functional to a scalar doing the same!) i.e.\ truth values `true' or `false' to these quasi-classical statements. Only these events occur or not in our actual world, and only with them we can use our usual logic.
Indeed, as far as the approximation goes, having an (approximately) commutative algebra, we naturally have a `many-world' scenario -- the worlds correspond to (`are') all possible (approximate) hom-functionals from this algebra to the complex scalars.
Which our world is one of, its choice being \textbf{an extra ingredient} -- from the point of view of the quantum theory other ones could have equally been chosen. (One might wonder whether we could not deduce everything in our actual world from `probability close to 1' arguments. This seems not to be the case. It seems that in many cases quantum fluctuations have been magnified to macroscopic consequences, making many different outcomes each with small probability, of which just one is asserted in the actual world. And moreover there are so many details in our actual world that seem entirely erratic.)
\subsection{Paradoxes?}
And a great origin of paradoxes in thinking about quantum theory is our reluctance to obey its `strange' logic.
As we must contend, we have, on the one hand, the quantum theory, for us the non-commutative algebra, a description of a new and strange logic on the \textit{system of possibilities}, which the Theory aptly manipulates as members of the non-commutative logic -- mathematically, operators, orthogonal projections in the Hilbert space -- using its clear-cut formalism. But saying there that a statement = (event = subspace of the Hilbert space) is true or false would be meaningless. That will have meaning only per our actual world, and just for an event belonging to the quasi-classical almost commutative algebra. Only then we have our logic.
But the Quantum Theory does speak about particles, fields, physical systems etc., and one is so tempted to say, in its frame, that `the electron \textbf{is} here or there'; `it \textbf{has} this or that property'; `the system \textbf{is} (or was) in that state' as if events there happened or not, which the non-commutative logic of the `system of possibilities' forbids.
And then one runs straight into paradoxes.
\subsection{Some Interplays Quantum -- Classical, as per Our Actual World}
The approximate nature of our actual world is usually unnoticed by us, since we ourselves come from this approximate world.
But it will limit the number of different events (statements) that we can meaningfully conjunct or disjunct -- without totally leaving the quasi-classical almost commutative algebra. Thus it limits the number of things -- amount of information -- that we can speak about (to something like a `mundane' action measured in Planck's Constant -- something like $10^{34}$);
It limits the amount of time to the past or future that can have meaning for our actual world -- because the non-commutativity with observables transformed by Hamiltonian evolution, although small and negligible for our mundane intervals of time, becomes big for enormous intervals, hence one cannot include presumably quasi-classical quantities pertaining to enormously distant times or distances -- something like multiples of mundane times or distances by the ratio of a mundane action to Planck's constant -- in the same approximate commutative algebra;
All that making our physical actual world \textit{finite to a delimited extent}.
One may even say that as per the `actual world', the infinite space or time models used in physics serve, in this respect, a similar role as the infinite plane of coordinates in which a map includes the grounds of a city.
Of course, we can investigate non-quasi-classical systems only by making them bear on our almost-commutative quasi-classical world, i.e.\ by measuring them.
Moreover, our world is protected from `stray non-commutativity', such as carrying conclusions of former measurements to the future via the Hamiltonian (Schr\"odinger's equation) evolution, by \textit{decoherence} \cite{JoosZeh} \cite{Zurek} which will wipe out any such conclusions, preserve only what is quasi-classical and (almost) commuting and thus create the separating wall between the quasi-classical and the truly quantum worlds.
Any Schr\"odinger cat (or `Schr\"odinger physicist or mathematician, for that matter) is either in the quasi-classical domain, hence one may assume in principle that in our actual world the question: is (s)he alive? is settled, or is in the truly quantum domain, where superpositions are routine, but can then be investigated by us only via measurements.
Usually, the quasi-classical world is governed by the deterministic laws of classical physics, to be derived, in principle, from the quantum theory. But the fact that everything is approximate has consequences. Thus when the deterministic classical equations are chaotic we have true non-determinism in the quasi-classical system: between assertions about far enough time-moments one may have only probabilistic relations. Another case of non-determinism comes from measurements and measurements-like phenomena, where `truly quantum' elements bear on the quasi-classical world.\par\medskip
\subsection{The `Measurement Problem'}\label{s:mea}
One may say that the original version of quantum theory has put itself in a physics laboratory.
Atomic systems are prepared (by measurements, selecting the cases where a favored outcome occurred, say electrons moving in a specific direction) and measured. Such systems (more exactly, their way of preparation) always have a state, described by a wave function, or even by a density matrix (in case the preparation had a classically random element). Doubtlessly, these states and their measurements are governed by Copenhagen quantum theory, as repeated in numberless textbooks, with collapse of the wave function and everything.
All that works perfectly well as long as one sticks to the laboratory scenario, in which the division of labor, so to speak, among the quantum system, the experimenter(s) and the laboratory apparatus goes unquestioned. (A great part of the problematic nature of the Schr\"oedinger Cat example in Copenhagen quantum theory seems to stem from the unnatural role of a sentient being as the quantum system.)
So let us look at the `measurement problem' in view of what we noted above.
Consider a proverbial quantum measurement. A quantum system is prepared, by making a measurement and taking only the cases with suitable outcomes (say, `electrons that move in a certain way'), Then, maybe after some development in the quantum system, another measurement is made, then, maybe, more development and measurements.
We assume that each measurement distinguished between all vectors in a basis of the Hilbert space of the quantum system to be measured (there is no classical randomness).
We, of course, live in the quasi-classical world (in fact in our actual world -- an extra ingredient), where in our actual world the measuring apparatuses recorded results of the measurements. Here, contrary to a quantum `non-commutative system of possibilities', assertions are true or false -- it is true at the measurements gave these results.
Anyhow, an essential ingredient of that being a measurement, is recording that the quantum system that one measures now was indeed prepared (by a former measurement) as required. The total Hilbert space must have room for keeping records of all these.
In fact, as per the quasi-classical almost commutative algebra, roughly speaking, each subsequent measurement tensors (qua Hilbert space, qua algebra of operators) the already recorded results with the results of the new measurement.
And as per our actual world -- all these quasi-classical events either occur of not.
Note also that, as per the quasi-classical almost commutative algebra, if one assumes knowing that a quasi-classical event = subspace occurred, one might work only in (`condition to', in the probability parlance) that subspace as the Hilbert space.
\subsection{The Copenhagen Recipe Follows Smoothly}
Then, maybe to one's surprise, it seems that (restricting ourselves to `allowed logic' as above) the Copenhagen recipe follows smoothly -- no extra assumptions.
This is because mathematically, each subsequent measurement, by adding to the quasi-classical record, basically makes our entire Hilbert space tensor with the Hilbert space of the new (quasi-classical) record. In writing the probability distribution -- density matrix -- that we should take, always in a basis that (approximately) diagonalizes the quasi-classical observables, each entry $w_{ij}$ of the density matrix will be replaced, by the above tensoring, with a sub-matrix $W_{ij}$ with trace $w_{ij}$, (a positive Hermitian sub-matrix $W_{ii}$ in the case of a diagonal element $w_{ii}$), and, \textit{speaking in our world -- only there could we assert that anything is true or false} we condition to the quasi-classical event of the specific result of the new measurement (true in the actual world), hence pick the corresponding particular entry in the diagonal of that sub-matrix.
For the next measurement this entry itself is to be expanded into a sub-matrix, etc. Note the remarkable fact that every subsequent measurement as if imposes its pure state as the new density matrix after the conditioning -- we can `forget' what density matrix we used before. In other words, we must compute as if each measurement had induced a collapse of the state of the measured quantum system (which has nothing to do with the group of time-shifts -- Hamiltonian evolution -- which of course, always acts by Schr\"odinger's equation).
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«Хатина дядька Тома» () — роман американської письменниці Гаррієт Бічер-Стоу антирабовласницького спрямування, опублікований 1852 року. На думку історика Вілла Кауфмана роман «сприяв утвердженню підґрунтя громадянської війни».
Стоу була вчителькою й активною аболіціоністкою. Головним персонажем свого роману вона зробила дядька Тома, старого чорношкірого раба, відданого своїм білим господарям. Роман, написаний у дусі сентименталізму, зображує реалії життя рабів, водночас проводячи думку, що християнська любов здатна здолати навіть таке негативне явище, як рабство.
Роман став одним із бестселерів XIX століття, другою за кількістю продажів книгою після Біблії. Вважається, що він сприяв зростанню руху аболіціоністів у 50-x роках XIX століття. За перший рік після публікації було продано 300 тис. примірників тільки в США, понад мільйон у Великій Британії. Через три роки після публікації його вже називали «найпопулярнішим романом нашого часу»
. Існує легенда, що на початку громадянської війни в США Авраам Лінкольн зустрівся із Стоу і проголосив: «Ви і є ця сама маленька жінка, що почала цю велику війну?». Однак відомості про цю зустріч і слова Лінкольна з'явилися в пресі тільки в 1896 році й, можливо, були зумовлені бажанням тогочасних інтелектуалів підкреслити роль літератури в сприянні суспільним змінам.
Книга та поставлені за її мотивами п'єси допомогли утвердженню й популяризації стереотипів чорношкірого населення США: годувальниці-негритянки, чорношкірих дітей та дядька Тома як сумлінного багатостраждального слуги, вірного своєму господарю чи господині. З часом ставлення американців до таких стереотипів змінилося, вони стали сприйматися як образливі, що дещо затьмарило історичне значення книги як «важливого антирабовласницького знаряддя».
Джерела
Стоу народилася в Коннектикуті й викладала в Гартфорській жіночій академії, водночас беручи активну участь в русі за скасування рабства. Свій роман вона написала у відповідь на другий акт про рабів-втікачів 1850 року. Більша частина книги була написана в Брансвіку, штат Мен, де в Боудін-коледжі викладав її чоловік Калвін Елліс Стоу.
Частково натхненням для написання «Хатини дядька Тома» стала книга «Життя Джозаї Генсона, колишнього раба, тепер мешканця Канади, розказане ним самим», яку написав Джозая Генсон, у минулому раб на тютюновій плантації Айзека Райлі в Північній Бетесді, Меріленд. Генсон утік у Канаду, де став допомагати іншим утікачам облаштуватися й розпочати незалежне життя. Саме там він написав свої мемуари. 1853 року Стоу визнала, що спогади Генсона підштовхнули її до написання роману. Коли її твір став бестселером, Генсон перевидав свою книгу під назвою «Мемуари дядька Тома» й став подорожувати США та Європою із лекціями. Будинок Генсона в місті Дрезден, Онтаріо, 1940 року став музеєм і носить назву «Історичного місця хатини дядька Тома». Будинку, в якому жив Генсон у рабстві, більше нема, але графство Монтгомері, Меріленд помилково купило іншу хатину, яка стала частиною національного парку, присвяченому підпільній залізниці до свободи, як називали свого часу канали втечі рабів із рабовласницьких штатів у штати, де рабство було заборонене законом, й до Канади.
Іншим джерелом для роману була книга «Американське рабство, як воно є: покази тисячі свідків» Теодора Двайта Велда й сестер Грімке. Стоу стверджувала, що написанню роману сприяли численні розмови з рабами-втікачами, з якими вона зустрічалася, коли мешкала в Цинциннаті, Огайо, місті, розташованому на протилежному березі річки Огайо від рабовласницького штату Кентуккі. У Цинциннаті місцеві аболіціоністи симпатизували підпільній залізниці й активно допомагали рабам-утікачам.
У книзі «Ключ до Хатини дядька Тома» 1853 року Стоу наводить низку інших джерел з метою підтвердити свої твердження щодо рабства. Однак ретельне вивчення показало, що чимало з цих книг Стоу прочитала вже після публікації роману.
Публікація
«Хатина дядька Тома» почала виходити частинами в аболіціоністському журналі «National Era» з 5 червня 1851 і продовжувалося протягом 40 випусків. Зважаючи на популярність роману, видавник Джон Джуїтт запропонував Стоу оформити його у вигляді окремої книги. Стоу сумнівалася, чи читатимуть роман у вигляді книги, але зрештою згодилася.
Переконаний у популярності книги Джуїтт прийняв незвичне на той час рішення додати 6 ілюстрацій на цілу сторінку, ескізи яких для першого видання виконав Гамматт Біллінгс. Книга побачила світ 20 березня 1852 року, і дуже швидко весь тираж розійшовся. Книгу кілька разів перевидавали, зокрема побачило світ делюкс-видання зі 117 ілюстраціями Біллінгса.
За перший рік після публікації розійшлося 300 тис. примірників книги, однак надалі попит «несподівано впав, книга не перевидавалася багато років». Попит почав знову зростати 1862 року, коли роман перевидали у «Ticknor and Fields».
Книгу переклали багатьма мовами. Деякі з ранніх видань містили вступ, написаний преподобним Джеймсом Шерманом, проповідником-конгреціоналістом, відомим своїми аболіціоністськими поглядами. «Хатина дядька Тома» добре продавалася і в Англії: перше лондонське видання розійшлося тиражем 200 тис. примірників. Упродовж кількох наступних років було продано понад півтора мільйона примірників книги, більшість з яких були піратськими (як і в США).
Сюжет
Еліза з сином утікає, Тома продають «униз річкою»
Книга починається з того, що кентукському фермеру на ім'я Артур Шелбі загрожує втрата ферми через борги. Хоча він та його дружина Емілі Шелбі добре ставляться до рабів, Шелбі вирішує зібрати потрібні кошти, продавши работорговцю двох із них: дядька Тома, чоловіка середніх років із дружиною та дітьми, та Гаррі, сина покоївки Емілі Елізи. Емілі Шелбі ця ідея не до вподоби, оскільки вона обіцяла покоївці, що ні вона сама, ні її син ніколи не будуть продані. Син Емілі,Джордж Шелбі теж проти втрати Тома, бо бачить у ньому друга й ментора.
Еліза підслухала розмову містера й місіс Шелбі, в якій вони обговорювали продаж Тома й Гаррі. Вона вирішує утекти, щоб урятувати сина. У романі стверджується, що рішення Елізи зумовлене страхом утрати єдиної живої дитини (уже дві її дитини померли немовлятами). Тієї ж ночі Еліза покидає ферму, залишивши господині записку з поясненнями й вибаченнями.
Тома продають, і він опиняється на судні, що пливе вниз річкою Міссісіпі. На борту Том знайомиться із білою дівчиною Євою. Вони швидко стають друзями на основі глибокої християнської віри. Батько Єви, Августин Сент-Клер, купує Тома й забирає його в Новий Орлеан.
Родину Елізи переслідують, Том живе у Сент-Клерів
Втікаючи, Еліза зустрічається зі своїм чоловіком Джорджем Гаррісом, який утік раніше. Вони вирішують спробувати дістатися до Канади. Однак їх вистежує мисливець за рабами-втікачами Том Локер. Врешті Локер зі своїми людьми наздоганяють Елізу з родиною і Джордж вимушений скинути його зі скелі. Стривожена тим, що Локер може померти, Еліза переконує Джорджа принести мисливця до найближчого поселення квакерів.
У Новому Орлеані Сент-Клер дискутує питання про рабство зі своєю двоюрідною сестрою Офелією із Півночі, яка, засуджуючи рабство, має упередження щодо чорношкірих людей. З другого боку, попри те, що він рабовласник, Сент-Клер ставиться до чорношкірих без будь-якого упередження. Щоб доказати Офелії помилковість її поглядів, він купує молоду рабиню Топсі й просить Офелію виховати її.
Том жив у Сент-Клерів уже два роки, коли Єва серйозно занедужала. Перед смертю вона має видіння, в якому бачить небо, де її оточують близькі люди. Під впливом смерті Єви та її видіння інші персонажі вирішують змінити своє життя. Офелія обіцяє відкинути свої упередження щодо чорних, Топсі обіцяє буди чемною, а Сент-Клер дає клятву звільнити Тома.
Тома продають Саймону Легрі
Сент-Клер не встиг виконати свою клятву. Він помер від ножового удару, якого зазнав на виході з таверни. Його дружина не стала виконувати волю чоловіка й продала Тома на аукціоні плантатору з поганою славою Саймону Легрі. Легрі, виходець із півночі, забирає Тома в луїзіанську глибинку, де Том знайомиться із іншими рабами, зокрема з Еммелін, купленою водночас із Томом.
Легрі зненавидів Тома, коли той відмовився виконати наказ власника й відшмагати батогом іншу рабиню. Він жорстоко карає непокірного раба й вирішує зламати його віру в Бога. Незважаючи на жорстоке покарання, Том продовжує читати Біблію й робити все, щоб допомогти іншим рабам зносити страждання. На плантації Том знайомиться із рабинею Кессі. Їй довелося пережити втрату сина й доньки, проданих іншим рабовласникам. Не бажаючи знову бачити, як продають її дітей, вона вбила свою третю дитину.
Розповідь повертається до Тома Локера. Квакери вилікували його й він змінився духовно. Джордж, Еліза і Гаррі зуміли пробратися до Канади та отримати свободу. А в Луїзіані дядько Том на межі відчаю. Страждання на плантації ледь не підірвали його віру в Господа. Однак до нього приходять два видіння: Ісуса й Єви, які зміцнили його рішення залишатися вірним християнином навіть перед лицем смерті. Він підтримує план Кессі втекти. Врешті Кессі втікає, захопивши з собою Еммелін. Том відмовляється сказати власнику плантації, куди пішли Кессі та Еммелін і Легрі велить своїм наглядачам убити його. Помираючи, Том пробачає наглядачів, які, присоромлені шляхетністю людини, яку щойно вбили, каються. Перед самою смертю Тома на плантацію прибуває Джордж Шелбі з метою викупити раба, але вже надто пізно.
Завершальний розділ
На судні, яким Кессі й Еммелін скористалися для втечі, вони зустрічають сестру Джорджа Гарріса й разом із нею добираються до Канади. Кессі довідується, що Еліза — її втрачена донька. Усією родиною вони їдуть до Франції, а звідти до Ліберії, країни в Африці, створеної колишніми американськими рабами. Джордж Шелбі повертається на ферму в Кентуккі й звільняє всіх своїх рабів. Він розповідає їм про жертву Тома й просить не забувати про справжнє значення християнської віри.
Основні персонажі
Дядько Том
Спочатку дядько Том вважався благородним багатостраждальним рабом-християнином. В останні роки ставлення до цього персонажа в США змінилося. Дядьками Томами називають чорношкірих, які запродалися білим. Сама Стоу бачила Тома «шляхетним героєм», людиною, гідною похвали. Упродовж усієї розповіді Том, не дозволяючи себе використовувати, твердо стоїть на захисті своїх переконань, і навіть його вороги змушені, хоча й неохоче, захоплюватися ним.
Еліза
Еліза — покоївка місіс Шелбі, яка зважується втекти разом з п'ятирічним сином Гаррі на північ. Її чоловік Джордж зумів розшукати Елізу й Гаррі в Огайо й емігрувати з ними спочатку до Канади, потім до Франції й, врешті, до Ліберії.
Стоу створила цього персонажа на основі історії, яку аболіціоніст Джон Ренкін розповів її чоловікові в Цинциннаті. Ренкін згадував, що у лютому 1838 року молода рабиня перебралася з дитиною на руках через замерзлу річку Огайо до міста Ріплі й зупинилася в його домі, перш ніж піти далі на північ.
Єва
Єванджеліна Сент-Клер — донька Августина Сент-Клера. Читач уперше знайомиться з нею, коли дядько Том пливе на пароплаві до Нового Орлеана, де його повинні продати. Єві тоді було 5 або 6 років, вона ледь не втопилася, але дядько Том витягнув її з води. На прохання доньки містер Сент-Клер купує Тома й робить його головним кучером у своєму помісті. Том та Єва проводять багато часу разом. Єва — маленьке янголятко, вона багато розмірковує про любов та прощення, переконує навіть скривджену дівчинку рабиню Топсі, що вона заслуговує на любов. Їй пощастило достукатися навіть до серця тітки Офелії.
Пізніше Єва смертельно занедужала. Перед смертю вона подарувала пасмо свого волосся кожному рабу, заповідаючи їм вести християнське життя, щоб вони могли знову зустрітися на небі. На смертному одрі вона благає батька звільнити Тома, але через збіг обставин цього не трапилося.
Схожий персонаж на ймення маленька Єва фігурує в романі Філіпа Дж. Козанса «Маленька Єва: квітка півдня», хоча цей роман, написаний у відповідь на «Хатину дядька Тома», належав до літератури цілком протилежного звучання, захищаючи інститут рабства.
Саймон Легрі
Саймон Легрі — жорстокий рабовласник, за походженням із півночі, ім'я якого стало синонімом жадоби. Він основний негативний персонаж роману. Прагне деморалізувати Тома й зламати в ньому християнську віру. Врешті, розлючений незламною вірою раба в бога, він велить зашмагати Тома до смерті. У романі розповідає, що ще молодим він покинув хвору матір і відправився в море, а на її листи з проханням повернутися до неї, щоб вона могла його ще раз побачити перед смертю, не відповідав. Він сексуально експлуатує Кассі,попри її зневагу до нього і має ті ж наміри щодо Еммелін.
Не відомо, чи образ Легрі мав прототипа серед реальних людині. Після 1870 року була поширена думка, що Стоу використала як прообраз Легрі особу багатого бавовняного й цукрового плантатора Мередіт Калгун, плантація якого була розташована на річці Південна Ред-Рівер на північ від міста Александрія. Однак Калгун був «високо освіченою і вихованою людиною», що не асоціюється з неохайністю й жорстокістю Легрі. Калгун навіть видавав власну газету. Однак наглядачі на плантаціях Калгуна могли відповідати зображеним у романі характеристикам Легрі, використовувати ті ж методи й мати ті ж прагнення.
Інші персонажі
Серед інших персонажів роману слід згадати:
Артур Шелбі – хазяїн Тома в Кентуккі, «добрий» рабовласник, типовий для американської літератури образ південного джентельмена.
Емілі Шелбі – дружина Артура Шелбі, набожна жінка, яка намагається бути доброю і бути прикладом для рабів. Для неї рішення чоловіка продати рабів — важкий удар. Як жінка вона не має права завадити йому, оскільки за законом уся власність належить чоловікові.
Джордж Шелбі — син Артура й Емілі, для якого Том є другом і взірцевим християнином.
Августін Сент-Клер — другий власник Тома, батько Єви. Це складний персонаж, часто саркастичний, у нього завжди знайдеться влучне слово. Довго впадаючи за своєю майбутньою дружиною, після років спільного життя він став зневажати її, але ввічливо намагається це приховати. Сент-Клер розуміє, що рабство — зло, але не бажає відмовитися від достатку, яке воно йому приносить. Після смерті доньки його релігійні почуття стають щирішими, він починає читати Тому Біблію. Наприкінці він зважується на рішучу дію проти рабства, вирішивши звільнити своїх рабів, але передчасна смерть завадила цьому плану
Топсі — вередлива й непокірна дівчинка-рабиня. Коли її запитали, звідки вона взялася, вона відповіла: «Думаю, виросла (американське діалектичне growed замість grew). Не думаю, що хто-небудь мене створив», демонструючи повне незнання таких понять, як бог або матір. Під впливом Євиної любові вона змінюється. Фраза «growed like Topsy» (віросла як Топсі) утвердилася в англійській мові зі значенням вирости як бур'ян, а пізніше — дуже вирости.
Міс Офелія — набожна кузина Августина Сент-Клера із Вермонту, працьовита, з аболіціоністськими переконаннями. Однак вона ставиться до темношкірих людей із упередженням, характерним для тогочасних північан. Вона виступає проти інституту рабства, але, принаймні спочатку, сахається рабів як людей.
Квімбо та Самбо — раби Саймона Легрі, які виконують функції наглядачів на плантації. За наказом Легрі вони зашмагали Тома до смерті, але після того, як Том вибачив їх перед смертю, щиро каються.
Основні теми
Основна тема роману — зло й аморальність рабства. Стоу вплітає в канву твору й інші теми, наприклад материнську відповідальність та перевиховну силу християнства, але вони тільки допомагають у розкритті теми жаху рабства. Іноді вона вкладає осуд руйнівної природи рабства в уста епізодичних персонажів, на кшталт того, коли біла жінка на пароплаві говорить: «Найжахливіше в рабстві те, як на мене, що воно глумиться над почуттями й прив'язаностями людини, наприклад, коли розлучаються сім'ї». Саме цю рису цього «дивного інституту» Стоу виставила на передній план.
Оскільки Стоу бачила в материнстві «етичну й структурну модель для всього американського життя» й вірила, що тільки жінка має моральний силу врятувати Сполучені Штати від демона рабства, іншою головною темою «Хатини дядька Тома» є моральна сила й святість жінки. Образи Елізи, яка втекла з рабства разом із сином і зуміла об'єднати родину, та Єви, як «ідеальної християнки», Стоу використовує, щоб показати, як жінка на її погляд може врятувати своїх ближніх навіть від найжостокішої несправедливості. Хоча пізніше критики зауважували, що жіночі образи Стоу не реалістичні, що вони є тільки кліше жінки-домогосподарки, роман Стоу «ще раз стверджує важливість жіночого впливу» і прокладає шлях до боротьби жінок за свої права в наступні десятиліття.
Пуританські релігійні переконання Стоу звучать в ще одній темі — дослідженні природи християнства. Стоу намагається показати, що християнська теологія принципово несумісна з рабством. Ця тема звучить найбільш очевидно тоді, коли Том закликає Сент-Клера після смерті доньки «шукати Ісуса». Коли помирає сам Том, Джордж Шелбі промовляє «Ось що значить бути християнином». Оскільки тема християнства настільки важлива для роману, й тому, що Стоу часто прямо від імені автора розмірковує над проблемами віри й релігії, «Хатина дядька Тома» «набуває звучання проповіді».
Стиль
«Хатина дядька Тома» написана в сентиментальному й мелодраматичному стилі, характерному для романів 19 століття та для жіночої літератури. В часи Стоу ці романи користувалися великою популярністю. Зазвичай головними героями в них були жінки, а написані вони були так, щоб викликати у читача симпатію та почуття. Хоча роман Стоу відрізняється від більшості сентиментальних романів тим, що висвітлює ширшу тему рабства, й тим, що головним героєм у ньому є чоловік, все ж вона опирається на сильні почуття читача. Сучасні роману читачі реагували відповідно. Джорджіана Мей, приятелька Стоу, у листі до автора зауважує: «Минулої ночі я зачиталася до першої години, аж доки не закінчила "Хатину дядька Тома". Не могла відірватися так, наче від дитини на смертному ложі.». Інша читачка настільки перейнялася романом, що планувала змінити ім'я своєї доньки на Єва. Вочевидь Євина смерть настільки вплинула на читачів, що в 1852 тільки в Бостоні 300 новонароджених дівчаток отримали її ім'я.
Попри схвалення читачів літературні критики впродовж кількох десятиліть зверхньо дивилися на стиль «Хатини дядька Тома» й інших сентиментальних романів, оскільки ці книги були написані жінками й опиралися на «плаксиві жіночі емоції». Один критик сказав, що якби роман не був про рабство, то «він був би ще одним з низки сентиментальних романів», а інший описав роман «перш за все вторинним шматком халтури». В «Історії літератури США» Джордж Ф. Вічер назвав «Хатину дядька Тома» «літературою в стилі недільних шкіл», «повною масштабної мелодрами, гумору й пафосу».
1985 року Джейн Томпкінс у книзі «Сентиментальний дезайн: Культурна робота американської літератури» глянула на «Хатину дядька Тома» по іншому. Вона похвалила стиль, який багато інших критиків відкидали, зауваживши, що сентиментальні романи показали як жіночі почуття мають силу перетворити світ на краще місце. На її думку популярні домашні романи 19 століття, включно з «Хатиною дядька Тома», відзначаються складністю, амбіціями й винахідливістю, а «Хатина дядька Тома» критикує американське суспільство набагато гостріше, ніж відомі його критики на кшталт Натаніеля Готорна чи Германа Мелвілла.
Така точка зору залишається спірною. 2001 року юрист Річард Познер охарактеризував «Хатину дядька Тома» як одну з тих канонічних робіт, які випливають на поверхню, коли літературі силоміць нав'язуються політичні критерії.
Значення та критика
Небагато романів в історії літератури можуть зрівнятися за значенням із «Хатиною дядька Тома». Прихильники рабства зустріли його бурею протестів. У відповідь було написано низку книг, що трактували питання рабства по іншому. Аболіціоністи зустріли книгу із захопленням. Роман мав велике значення для становлення літератури протесту в наступні десятиліття.
Реакція сучасників у США та світі
«Хатина дядька Тома» обурила Південь Сполучених Штатів. Прихильники рабства шалено критикували книгу. Відомий романіст Вільям Гілмор Сіммс оголосив роман абсолютно брехливим, інші називали його злочинним і наклепницьким. Власник книжкового магазину в місті Мобіл, Алабама, змушений був покинути місто, тому що продавав роман Стоу. Письменниця отримувала листи із загрозами, в одному з них їй прислали відрізане вухо раба. Чимало південних письменників, зокрема Сіммс, незабаром написали книги у відповідь.
Стоу закидали необізнаність із життям на плантаціях, що призвело до помилок. Вона одного разу була на плантації, але за її власними словами, при написанні книги вона опиралася на розповіді рабів-утікачів, які чула в Цинциннаті. Вона на власні очі бачила випадки, який і спонукав її написати свій знаменитий роман. Серед них продаж чоловіка та жінки у різні руки. Вона також читала журнальні нариси та інтерв'ю і на цьому матеріалі розбудовувала свій сюжет.
У відповідь на цю критику Стоу опублікувала 1853 року «Ключ до хатини дядька Тома», де намагалася підтвердити свій опис рабства документально. Для кожного з головних персонажів «Хатини дядька Тома», вона наводить «приклади з реального життя», роблячи ще «агресивнішу атаку на рабство на Півдні, ніж у самому романі». «Ключ до хатини дядька Тома» теж став бестселером. Хоча Стоу стверджувала, що «Ключ» наводить джерела, які вона використала при написанні роману, чимало з цих книг вона прочитала вже після його опублікування. Значну частину «Ключа» займає критика юридичної системи, що підтримувала рабство й давала рабовласникам право знущатися над рабами. Цим Стоу виставила на суд не тільки рабство, вона виставила на суд закон. Ця тема, тема тіні закону, що витає над інститутом рабства й дозволяє рабовласникам безкарно кривдити людей, звучить уже в самому романі. Стоу зазначає, що в окремих випадках закон навіть забороняв власникам відпускати своїх рабів на волю.
Попри критику роман сильно вплинув на американців і сприяв посиленню аболіціоністського руху. Юніоністський генерал і політик Джеймс Берд Вівер говорив, що саме цей роман спонукав його до активної участі в русі за звільнення рабів.
«Хатина дядька Тома» мала також великий успіх у Сполученому Королівстві. Перше лондонське видання побачило світ у травні 1852 й розійшлося тиражем 200 тис. примірників. Частково цей успіх був зумовлений антипатією британців до США. Як пояснював відомий британський письменник: «Злорадство, яке "Хатина дядька Тома" розбудила в Англії було зумовлене не ненавистю [до рабства], не бажанням помститися за нього, а національною заздрістю й марнослав'ям. Американський обман нас давно дратував - ми втомилися чути, як Америка вихваляється тим, що вона найвільніша й найпросвіченіша країна, яку світ коли-небудь знав. Наш клір ненавидить їхню волюнтаристську систему — наші торі ненавидять її демократів — наші віги ненавидять її парвеню — наші радикали ненавидять її самопроголошену правоту, наглість та амбіційність. Усі партії прославляють місіс Стоу як бунтівника в лавах ворога». Чарлз Френсіс Адамс, посланець США до Британії під час війни, пізніше зазначав, що «"Хатина дядька Тома" або "Життя серед скривджених", опублікована 1852 року, мала в світі, значною мірою через сприятливі обставини, набагато швидший, значніший й драматичніший вплив, ніж будь-яка рашіне надрукована книга».
До 1857 роман було перекладено 20 мовами, словенською було зроблено два різних переклади вже через рік після появи оригіналу. Згодом роман переклали майже кожною мовою світу, зокрема китайський переклад побачив світ 1901 року, а переклад амхарською мовою 1930 року. Популярність книги була такою, що Зигмунд Фрейд повідомляв про випадки садомазохістских схильностей, поштовх яким дали сцени шмагання рабів у «Хатині дядька Тома».
Літературне значення й критика
«Хатина дядька Тома» була першою книгою політичного звучання, що здобула масову популярність у Сполучених Штатах, а тому вплинула на розвиток не тільки американської літератури, а а й літератури протесту загалом. Впливу «Хатини дядька Тома» завдячують, зокрема «Джунглі» Аптона Сінклера й «Мовчазна весна» Рейчел Карсон.
Попри беззаперечне значення «Хатину дядька Тома» називали «мішаниною дитячої казочки й пропаганди». Низка критиків зневажливо відкидала роман як «просто сентиментальний», а Джордж Вічер в «Історії літератури Сполучних Штатів» написав: «Ніщо з рукоділля місіс Стоу не може пояснити надзвичайний фурор роману: ресурси автора як представника літератури для недільних шкіл нічим не вирізнялися. Максимум, що в неї було, це готовність до широко замисленої мелодрами, гумору, пафосу і тих популярних слізливих почуттів, якими вона заповнила книгу».
Інші критики хвалили роман. Едмунд Вільсон зазначав: «Коли у зрілому віці зустрічаєшся з "Хатиною дядька Тома", це може... виявитися приголомшливим досвідом». Джейн Томпкінс стверджує, що роман належить до класики американської літератури й задає питання, чи не відкидають критики книгу тільки тому, що вона була аж занадто популярна свого часу.
З роками дослідники запропонували кілька теорій про те, що Стоу намагалася сказати своїм романом попри очевидні теми, такі як засудження рабства. Наприклад, як палка християнка та аболіціоністка, Стоу вклала в роман свої релігійні переконання. Деякі дослідники стверджують, що Стоу бачила свій роман як пропозицію способу розв'язання моральної та політичної дилеми, яка хвилювала значну частину противників рабства: чи виправдовує протистояння злу заборонену законом діяльність? Чи можна захистити з точки зору моралі використання насильства проти насильства й порушення законів, що захищали рабство? Якого із персонажів Стоу треба брати за взірець: пасивного дядька Тома чи бунтівника Джорджа Гарріса? Відповідь Стоу аналогічна позиції Ральфа Волдо Емерсона: божа воля виконуватиметься, якщо кожна людина щиро перегляне свої принципи й діятиме відповідно до них.
Дослідники бачили також у романі вираження цінностей та ідей руху за вільну землю. Якщо глянути на книгу з цього боку, то образ Джорджа Гарріса втілює ідеали вільної праці, тоді як складний образ Офелії відображає погляди тих північан, що намагалися знайти компроміс із рабством. На противагу Офелії Діна опирається в своїх вчинках на почуття. Образ Офелії з розвитком сюжету трансформується так, як через три роки змінила свою позицію Республіканська партія, проголосивши необхідність стати на позиції протистояння рабству.
Виразно звучать у книзі Стоу й феміністські теорії. В цьому ракурсі роман є критикою на патріархальну природу рабства.
Для Стоу в основі родини лежать кровні зв'язки, а не патерналістські зв'язки між господарем і рабом. Більше, Стоу вбачає в національному єднанні продовження родинних зв'язків, тому для неї почуття національності пов'язане з єдиною расою. Відповідно до цього вона пропонує переселення звільнених рабів в Африку, а не їхнє вростання в американське суспільство.
Книгу також розглядали як спробу змінити погляд на риси чоловічого характеру, необхідні для боротьби за відміну рабства. Епоха завоювання й колонізації висунула на перший план агресивний, домінантний чоловічий характер, ставлення до якого деякі аболіціоністи почали переглядати. На перший план вони висували інші риси: здатність до співробітництва, співчуття, громадянський дух і відповідальність. На думку аболіціоністів такий чоловічий характер міг би протистояти рабству, не ставлячи під загрозу почуття гідності та положення людини в суспільстві. Водночас, інша частина аболіціоністів вважала ідеалом звичні рішучість та мужність. Усі чоловіки в романі Стоу представляють один із цих двох типів характерів.
Створення й популяризація стереотипів
Сучасні критики й читачі критикують роман за зверхній опис чорношнірих персонажів, особливо стосовно зовнішнього вигляду, мови, поведінки, а також пасивності в сприйнятті своєї долі, характерної для дядька Тома. Використання в романі звичних стереотипів має тим більше значнення, оскільки «Хатина дядька Тома» була бестселером у всьому світі. Як наслідок своєї популярності книга, ілюстації до неї, сценічні постановки за її сюжетом сприяли вкоріненню цих стереотипів у психології американців.
Серед стереотипів чорношкірих у «Хатині дядька Тома» тип «щасливого чорного» в образі лінового й безтурботного Сема, світлошкірої «трагічної мулатки» як об'єкта сексуальних домагань в образах Елізи, Кассі й Еммелін, доброї чорної годувальниці — тут одразу кілька образів, серед них Маммі, куховарка на плантації Сент-Клерів, стереотип чорної дитини, який отримав назву «піканіні», в образі Топсі, та образ дядька Тома, африканця, який надто запобігає перед білими. За задумом Стоу Том мав бути «шляхетним героєм». Стереотип «запобіжливого дурня, що кланяється білим», вочевидь склався як наслідок сценічних постановок, які Стоу не могла контролювати.
Негативні асоціації значною мірою затьмарили історичне значення «Хатини дядка Тома», як протесту проти рабства. Сприйняття роману почало змінюватися, починаючи з есе Джеймса Болдвіна «Роман протесту для кожного». Болдвін назвав «Хатину дядька Тома» «дуже поганим романом», по расистськи образливим і естетично грубим , хоча й зробив у своєму аналізі кілька помилок. У 1960-х і 1970-х рухи Чорна влада й Чорне мистецтво нападали на роман, стверджуючи, що образ дядька Тома є «расовою зрадою», і що рабська покора Тома роблять його гіршим від рабовласників. З часом стала гострішою критика стереотипів, змальованих у романі. Втім, останнім часом критики на зразок Генрі Луїса Гейтса мл. почали знову переглядати значення роману, стверджуючи, що він є «центральним документом з історії взаємних стосунків рас в Америці і важливим політичним дослідженням характеру цих стосунків».
Антитомова література
У відповідь на «Хатину дядька Тома» письменники Півдня США написали чимало книг, у яких намагалися заперечити роману Стому. Це література отримала назву антитомової. Вона здебільшого стояла на захисті інституту рабовласництва і стверджувала, що зображення рабства в романі Стоу занадто драматичне й не відповідає правді. Романи цього жанру типово зображали доброго хазяїна, батька для своїх рабів, та його чисту дружину, які вкупі опікувалися схожими на дітей рабами, таким чином уся плантація ставала наче великою дружньою родиною. В цих книгах або розумілося або й прямо говорилося, що афроамериканці як діти, що вони не здатні вижити без нагляду білої людини.
Відомими книгами антитомового спрямування були «Меч та посох» Вільяма Гілмора Сіммса, «Хатина тітки Філліс» Мері Гендерсон Істмен та «Північна наречена плантатора» Каролін Лі Герц. Герц була подругою Стоу, коли та мешкала в Цинциннаті. Книга Вільяма Сіммса побачила світ через кілька років після роману Стоу, чимало розділів у ній присвячено дискусії з поглядами Стоу на рабство. Роман Герц, який свого часу користувався популярністю, але його вже майже забули, наводить аргументи на захист інституту рабства з точки зору північанки, доньки аболіціоніста, яка одружилася із південним рабовласником-плантатором.
Упродовж десятиліття між публікацією «Хатини дядька Тома» й початком громадянської війни вийшло двадцять-тридцять антитомових книг. Серед них дві книги під назвою «Хатина дядька Тома як вона є» — одна пера В. Л. Сміта, інша К. Г. Вайлі та книга Джона Пендлтона Кеннеді. Здебільшого антитомові книги писали білі жінки, на що Сіммс зауважив: «є поетична справедливість у тому, що північанці (Стоу) відповідають жінки з півдня».
Сценічні постановки
Хоча «Хатина дядька Тома» була бестселером у 19 ст. більшість американців того часу знали історію завдяки сценічним постановкам і мюзиклам, не читаючи книгу. Ерік Лотт у книзі «Дядькотомики: расова мелодрама й манера постановки» стверджує, що ці п'єси бачили принаймні три мільйони людей, що в 10 разів перевищує тираж перших років.
Тогочасне законодавство щодо авторських прав не було строгим, і п'єси за мотивами «Хатини дядька Тома» почали з'являтися ще в той час, коли роман виходив частинами. Стоу відмовляла в дозволі на драматичні постановки, оскільки не любила театр, хоча вона відвідала постановку Джорджа Ейкена і за словами Френсіса Андервуда, їй сподобалася Топсі у виконанні Каролін Говард. Постановка Ейкена протрималася на сцені в Англії та Америці 75 років. Відмова Стоу дати дозвіл на певну постановку відкрила поле для появи численних п'єс. Деякі з них переслідували свої політичні цілі, інші ставилися як чисто комерційні.
Міжнародних законів про авторське право тоді зовсім не існувало. Книга й п'єси були перекладені кількома мовами, за що Стоу не отримала грошей, які «могли б скласти три чверті всіх її прибутків».
Усі п'єси мали мелодраматичний характер і виконувалися білими акторами із наваксованими обличчями. За політичним спрямуванням вони дуже різнилися: деякі притримувалися сентиментальної позиції Стоу, інші були поміркованішими, а деякі навіть відстоювали рабство. Чимало з постановок виводили принизливі карикатурні образи чорношкірих. В частині п'єс звучали пісні Стівена Фостера (зокрема «My Old Kentucky Home», «Old Folks at Home» та «Massa's in the Cold Ground»). Найвідомішими були постановки Джорджа Ейкена та Г.Дж. Конвея.
Мабуть, найбільш відомою серед сценічних постановок є версія Ейкена. Її прем'єра відбулася через кілька місяців після виходу книги. Це була грандіозна постановка на шість актів, вона першою із бродвейських шоу гралася на сцені сама, їй не передували і після неї, жодних інших розваг для глядачів не було. Більшість діалогів у постановці Ейкена були розширеними текстами із роману, крім них були ще чотири музичні номери, написані продюсером Джоржем Говардом. Відзначало цю постановку ще й те, що на одній сцені змінювалися декорації, відображаючи різні місцевості. Пізніше цьому наслідував ранній кінематограф із своїми великими знімальними майданчиками. Ейкен грав на почуттях глядачів, підкреслюючи страждання і безвихідь головних героїв. Комбінація мелодраматичного підходу зі змістом роману Стоу дозволила Ейкену створити сильний звинувачувальний документ проти рабства.
Українські переклади
Хатина дядька Тома: Ґаррієт Бічер-Стоу: Навчальна книга - Богдан, 2006 — 368. ISBN 966-692-510-9.
Виноски
Романи США
Англомовні романи
Романи, перекладені українською мовою
Рабство в США
Романи 1852 | {
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} | 3,912 |
Thought I start this thread to share your favorite classical music works, by any composer you like... Note that I'm refering to "classical music" in the common sense that everyone uses, refering to everything from early Mozart to Penderecki, or more! So, feel free to post any pieces you like and would love everyone else to hear!
Just pick yourself up a copy of Classical Thunder Volumes 1 and 2 for starters.
Does anyone else know and like Alan Hovhaness?
Yes, indeed. Introduced via Williams's bassoon concerto album.
I'm crazy about the guy's work!
Some great works pointed out here. But I'm surprised no Mahler got mentioned.
And one for Incanus: have you ever noticed the nice little quote from Vala-laulu(sua suojelemme,verin varjelemme) at 1:12?
Actually I hadn't noticed that quote before! Thanks for pointing it out. A gorgeous piece of work this symphony.
Oh, useful thread. and I wantedf to start one of my own.
Can anyone recommend me please something?
I'm searching for classical music pieces, kind of minimalistic and dramatic of the last 30 years. (eg. i listened to the 3rd movement of Philip Glass' 8th symphony and I liked it a lot).
I've listened to such soundtracks (eg all the Korzeniowski ones, that I love), but I would like to expand my listening to classical music. I'm not so aquainted with such recent works, but more with older works up until Stravinsky..
by the way: My favourite piece of all time - regarding classical music - is Barber's Adagio for strings.
Thank you Maurizio. Yes, i've listened to them (these days I'm listening to a lot of John Adams).
I'm looking for something in slower tempo.
I really liked also the 2nd movement of Naive and Sentimental Music.
here the first horn part in Schumann's konzertstück for 4 horns. 1st. mov.
Holy chops of steel, Batman! Ha, if that piece doesn't sell people on classical, I don't know what can!
Gives Wagner's Ring Cycle a run for its money!
I don't know much about classical music, but I do love a lot of stuff from Shostakovitch. I was introduced to him through Fantasia 2000 (in that regard, the film served is purpose well). To this day, his Piano Concerto No. 2 remains a favourite of mine.
The second movement of Shostakovich's Piano Concerto n°2 is achingly beautiful.
If, for a change, you'd like to listen to something calm and soothing, you should try Eric Satie.
Also I must say there have been almost none but excellent suggestions already been done in the posts above.
I would recommend Ravel's Violin Sonate #2. It's a beautiful work.
I'm going to go small on this.
Also last year I discovered Ludovico Einaudi who composes contemporary classical with piano. All of his pieces revolve around a motif of some sort, and usually feature very interesting use of chords, in which I find more emotion.
But really, I can enjoy anything that's got a melody. Rob Dougan wrote a beautiful string piece called Instrumental, which I discovered after hearing another song on Top Gear some years ago.
You're talking about classical music. "Horn concerto" isn't a title. It's just what it is: A concert for horn and orchestra. Just like Williams wrote flute, violin, cello etc. concertos. There are tons of those out there, often several by a single composer.
The same is true of "Minuet" and "Rondo." A minuet is just a type of dance of French origin, usually in 3/4 time, while a rondo can mean different things, either a fast and vivacious piece of music or a type of music that repeats itself.
The fact that such generic musical terms are used for the titles of various pieces makes naming classical music files in a tidy music collection very challenging. What constitutes a title? Simply the composer's description, or also the name of the piece and its op. number? What is the album -- the name of the piece, which makes for a lot of small albums, or do you group them based on the composer and type of work, like "Beethoven Sonatas" and "Mozart Symphonies," or simply the CD you bought? Who is the artist -- the composer, the conductor, the performing group?
Take the third movement -- Alla Turca: Allegretto in A minor and major -- of Mozart's Piano Sonata No. 11 in A major, K 331. "Turkish Rondo" sounds a lot more familiar.
So whose minuet and rondo did you mean?
Personally, I prefer the Minuet as played by Carolyn McCormick.
Organizing my classical collection is practically the most difficult task in the world. Particularly those albums that feature the works of multiple composers and more than one conductor or orchestra.
I tend to organize by composer instead of conductor or orchestra, at least until I hit the saturation point when I find different performances of the same piece or a special reason to favor the specific album.
For example, a five-disc sampler of "classical music" by anybody from JS Bach to Gershwin was eventually split so that the tracks were grouped into folders by artists, which eventually may become albums of "Pieces by _______."
For albums like the Classical Thunder set, on the other hand, or the Fantasia albums, or albums that intentionally feature music conducted by Lenny Bernstein or the Pittsburgh Symphony Orchestra, I don't split them up by composer -- because I probably have the pieces elsewhere by virtue of miscellaneous downloads over the years or other albums that I dissected -- because the albums are noteworthy in and of themselves.
Now that I have at least two different performances of Scheherazade, or Rite of Spring (outside of Fantasia), or the Nutcracker (again, outside of Fantasia), it becomes a little more important to distinguish by composer or orchestra, but I can put that information into the album title or even on the album art if it's high enough resolution, since I won't be searching by such data until I get X number of albums by that entity.
I don't really want my iPod "Artist" tag to be cluttered with composers for some pieces/albums and conductors for others, because I'm generally not interested in who conducted certain pieces, especially if they only conduct one album. Take "Alien," for example. It's not classical, but do you make the artist "Jerry Goldsmith," "Lionel Newman," or "National Philharmonic Orchestra?" Which is the most interesting to sort by, that adds value to a sort as opposed to that which you can find out by a Google search or a walk to the CD shelf? If I had a dozen albums conducted by Lionel Newman, maybe, but I know I don't. I don't keep tabs on who conducts soundtracks if it's not the composer, unless it's somebody unique like Charles Gerhardt. For his albums, I make him the Artist that the iPod sorts by instead of someone I "know," like JW for Gerhardt's Star Wars albums.
Another big question is, do I tag them as "First Name Last Name" or "Last Name, First Name." Because most auto-tagging software and online vendors sell "Billy Joel" and "John Williams" and "Wolfgang Amadeus Mozart," but the CD stores file by the last name. Sorting by last name is logical, but when the collection spans so many genres -- rock, classical, pop, film scores, and a whole lot of other -- it becomes a pain to overhaul at any point. I mean, do you want both Johann and Richard Strauss to be beside each other in the artist list, or do you want Richard Strauss and Richard Wagner to be together? And do you have the cranial capacity to remember that Rimsky-Korsakov's first name is Nicolai if you need some Capriccio Espagnol? Sting and Vangelis make sorting easy.
I have observed that if I connect my iPod to a computer without iTunes to charge, the library will often re-sort itself so that all albums that begin with "A," "An," or "The" will now find themselves in the A or T section, instead of where you thought they should go. iTunes seems to be smart enough to ignore those words, but other programs aren't, which compels me to manually put those articles after the keywords of album titles and artist names just so I don't have to use iTunes.
This is why I don't use Listen.FM or upload to the CDDB, because I don't want to be "that guy" accused of messing up the way other people tag music with little or no thought. My tagging is a closed system.
All of Mozart's Horn Concerto's are brilliant to listen to - very similar in their material but always different enough to offer another catchy motif. To think he wrote them all for a friend who played the French Horn - imagine what he could have done if he'd lived another 20 years?!
I listen to a lot of 'classical' (hate generalising) and I have to say that one of my all time favourite composers is Johann Strauss II. Die Fledermaus, The Blue Danube, Artists Life, Tale from the Vienna Woods - all absolute classics that everyone's heard and that are brilliant.
Thank you...the discovery of the month!
I just read about this composer and am very interested.
This music is faultless!! Better than anything Rimsky Korsakov, Mussorgsky or Glazunov have ever composed, and better than a lot Tschaikovsky wrote. The orchestration is perfect. Rimsky Korsakov is considerably poor in comparison. I can totally understand Teneyev's criticism towards his contemporaries of the Mighty Five, given his his own vast talent.
Raise the Roof, great pick!
One of my absolute favorite composers is Paul Hindemith. His music is so distinct, you really can't mistake him for someone else. That quartal writing in particular gives his melodies a very unique sound. His chamber works - the sonatas and quintets are excellent. And of course his horn quartet...brilliant work.
Here's the whole inaugural concert conducted by Gustavo Dudamel when he was appointed Music Director of the LAPhil a couple of years ago. | {
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\section{Introduction}
\input{introduction}
\section{Faster Retranslation Model}
\input{model}
\section{Experiments}
\subsection{Datasets}
\input{datasets}
\subsection{Metrics}
\input{metrics}
\input{figures_tex/pos_nonpos}
\input{results}
\section{Related Work}
\input{related_work}
\section{Conclusion}
\input{conclusion}
\subsection{Preliminaries} We briefly describe the simultaneous translation system to define the problem and set up the notations. The source and the target sequences are represented as $\mathbf{x}=\{x_1, x_2, \cdots, x_S\}$ and $\mathbf{y}=\{y_1, y_2, \cdots, y_T\}$, with $S$ and $T$ being the length of the source and the target sequences. Unlike the offline neural translation models (NMT), the simultaneous neural translation models (SNMT) produce the target sequence concurrently with the growing source sequences. In other words, the probability of predicting the target token at time $t$ depends only on the partial source sequence ($x_1, \cdots, x_{g(t)}$). The probability of predicting the entire target sequence $\mathbf{y}$ is given by:
\begin{equation}
p_g(\mathbf{y}|\mathbf{x}) = \prod_{t=1}^{T} p(y_t| \mathcal{E}(\mathbf{x}_{\leq g(t)}), \mathcal{D}(\mathbf{y}_{<t})),
\end{equation}
where $\mathcal{E}(.)$ and $\mathcal{D}(.)$ are the encoder and decoder layers of the SNMT model which produce the hidden states for the source and target sequences. The $g(t)$ denotes the number of source tokens processed by the encoder when predicting the token $y_t$, it is bounded by $0 \leq g(t) \leq S$. For a given dataset $D = \{\mathbf{x}_n, \mathbf{y}_n\}_{n=1}^{N}$, the training objective is,
\begin{equation}
\ell_g = -\sum_{(\mathbf{x}, \mathbf{y}) \in D} \log p_g(\mathbf{y}|\mathbf{x}).
\end{equation}
In a streaming based SNMT model \cite{ma2018stacl}, whenever new source information arrives, a new target token is generated by computing $p(y_t| \mathcal{E}(\mathbf{x}_{\leq g(t)}), \mathcal{D}(\mathbf{y}_{<t}))$ and is added to the current partial translation. On the other hand, the re-translation based SNMT models compute $\prod_{i=1}^{t} p(y_i| \mathcal{E}(\mathbf{x}_{\leq g(t)}), \mathcal{D}(\mathbf{y}_{<i}))$ from scratch every time the new source information arrives.
Let us assume that the source tokens are coming with a stride $s$, i.e., $s$ input tokens arriving at once, and $O(T \times C)$ is the computational time required to generate the sequence, where $C$ is the computational cost of the model for predicting one target token, then the re-translation based system takes $O(\frac{S}{s} \times T \times C)$ due to the repeated computation of the translation.
\input{tables/ne-cne}
\subsection{Faster Re-translation With NAT}
To address the issue of high computation cost faced in the existing re-translation models, we design a new re-translation model based on the non-autoregressive translation (NAT) approach.
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
The encoder in our model is based on the Transformer encoder \cite{vaswani2017attention} and the decoder is adopted from the Levenshtein Transformer (LevT) \cite{gu2019levenshtein}. We choose the LevT model as our decoder since it is a non-autoregressive neural language model and suits the re-translation objective of editing the partial translation. The overview of the proposed system, referred to as \textit{FReTNA}, is illustrated with a German-English example in the Figure \ref{fig:arch}. We describe the main components of LevT and proposed changes to enable the smoother re-translation in the following paragraphs.
The LevT model parallelly generates all the tokens in the translation and iteratively modifies the translation by using insertion/deletion operations. These operations are achieved by employing Placeholder classifier, Token Classifier, and Deletion Classifier components in the Transformer decoder. The sequence of insertion operations are carried out by using the placeholder and token classifiers where the placeholder classifier is for finding the positions to insert the new tokens and the token classifier is for filling these positions with the actual tokens from the vocabulary $\nu$. The sequence of deletion operations are performed by using the deletion classifier. The inputs to these classifiers come from the Transformer encoder $(T_E)$ and decoder blocks$(T_D)$ and are computed as:
\begin{equation}
\tiny
\mathbf{e_{0}^{l}}, \cdots, \mathbf{e_{g(t)}^{l}} = \begin{cases}
\mathbf{E_{x_0}}+\mathbf{P_0}, \cdots, \mathbf{E_{x_{g(t)}}}+\mathbf{P_{g(t)}},& l = 0\\
\mathrm{T_E}(\mathbf{e_{0}^{l-1}}, \cdots, \mathbf{e_{g(t)}^{l-1}}), & l = \{1, \cdots, L\}
\end{cases}
\end{equation}
\begin{equation}
\tiny
\mathbf{h_{0}^{l}}, \cdots, \mathbf{h_{t}^{l}} = \begin{cases}
\mathbf{E_{y_0}}+\mathbf{P_0}, \cdots, \mathbf{E_{y_t}}+\mathbf{P_t},& l = 0\\
\mathrm{T_D}(\{\mathbf{h_{0}^{l-1}, e_{\leq g(t)}^{l-1}}\}, \cdots, \{\mathbf{h_{t}^{l-1}, e_{\leq g(t)}^{l-1}}\}), & l = \{1, \cdots, L\}
\end{cases}
\end{equation}
where $E \in \mathbb{R}^{|\nu| \times d}$ and $P \in \mathbb{R}^{N_{\mathrm{max}} \times d}$ are word and position embeddings of a token.
The decoder outputs from the last Layer $(h_{t}^{L})$ are later passed to the three classifiers to edit the previous translation by performing insertion/deletion operations. These operations are repeated whenever new source information arrives.
\paragraph{Placeholder classifier:} It predicts the number of tokens to be inserted between every two tokens in the current partial translation. As compared to the LevT's placeholder classifier, we incorporate a positional bias which is given by the second term in the Eq. \ref{eq:pc}.
As the predicted sequence length grows, the bias becomes stronger, and the model inserts lesser tokens at the start, reducing the flicker. The placeholder classifier with positional bias is given by:
\begin{multline}
\small
\label{eq:pc}
\pi_{\theta}^{pc}(p|i, \mathbf{x}_{\leq g(t)}, \mathbf{y}_{i}) = \mathrm{softmax}({\color{ForestGreen}\alpha} * \mathbf{h}.\mathbf{B}^T + {\color{ForestGreen}(1-\alpha)\mathbf{q}} ), \\ {\color{ForestGreen}q_k = \frac{\gamma_i}{k+1}}, \\ \quad i = \{1, \cdots, t-1\}, \quad k=\{0, \cdots, K-1\},
\end{multline}
where $\mathbf{h}=[\mathbf{h}_i^L:\mathbf{h}_{i+1}^L]$, $\mathbf{B} \in \mathbb{R}^{K \times 2d}$, and {\color{ForestGreen}$\gamma_i=\frac{t-i}{t}$}. Based on the number of $(0 \sim (K-1))$ of tokens predicted by Eq. \ref{eq:pc}, we insert that many placeholders ($<plh>$) at the current position $i$ and it is calculated for all the positions in the (partial) translation of length $t$. Here, $K$ represents the maximum number of insertions between two tokens and $\alpha$ is a learnable parameter which balances the predictions based on the hidden states and (partial) translation length.
\paragraph{Token Classifier:} The token classifier is similar to LevT's token classifier, it fills in tokens for all the placeholders inserted by the placeholder classifier. This is achieved as follows:
\begin{equation}
\label{eq:tc}
\pi_{\theta}^{tc}(v|i, x_{\leq g(t)}, y_i) = \mathrm{softmax}(\mathbf{h_i^L}.C^T), \quad \forall y_i =\phi,
\end{equation}
where $C \in \mathbb{R}^{|\nu| \times d}$ and $\phi$ is the placeholder token.
\paragraph{Deletion Classifier:} It scans over the hidden states $(h_{0}^{L}, \cdots, h_{t}^{L})$ (except for the start token and end token) and predicts whether to \textit{keep}(1) or \textit{delete}(0) each token in the (partial) translation. Similar to the placeholder classifier, we also add a positional bias to the deletion classifier to discourage the deletion of initial tokens of the translation as the source sequence grows. The deletion classifier with positional bias is given by:
\begin{multline}
\label{eq:dc}
\pi_{\theta}^{dc}(d|i, x_{\leq g(t)}, y_i) = \mathrm{softmax}({\color{ForestGreen}\beta}*\mathbf{h}_i^L.\mathbf{A}^T + {\color{ForestGreen}(1-\beta)\gamma_i * \mathbf{l}}), \\
{\color{ForestGreen}\mathbf{l}=[0, 1]} , \quad i = \{1, \cdots, t\},
\end{multline}
where $\mathbf{A} \in \mathbb{R}^{2\times d}$, and we always keep the boundary tokens ($<s>, </s>$).
The model with these modified placeholder and deletion classifiers focuses more on appending the partial translation whenever new source information comes in, which results in smoother translation having lower textual instability. Here, $\beta$ is a learnable parameter.
The insertion and deletion operations are complementary; hence, we combine them in an alternate fashion. In each iteration, first we call the \textit{Placeholder classifier} followed by \textit{Token classifier}, and the \textit{Deletion classifier}. We repeat this process till a certain stopping condition is met, i.e., generated translation is same in consecutive iterations, or MAX iterations are reached. In our experimental results, we found that two iterations of insertion-deletion operations are sufficient while generating the partial translation for the newly arrived source information. To produce the partial translation, the model incurs $2*Z$ cost, where $Z$ is the cost for insertion-deletion operations equals to $C+\epsilon$ since we also have similar decoding layer. The overall time complexity of our model (\textit{FReTNA}) is $O(\frac{S}{s} \times Z)$ \footnote{The time complexities provided for \textit{ReTA} and \textit{FReTNA} models are for comparison and do not represent the actual computational costs}, since all the target tokens are generated parallelly. The \textit{FReTNA} computational cost is $\sim T$ times less than the \textit{ReTA} model.
\input{figures_tex/BLEU-AL_de-ed}
\input{figures_tex/TIME-AL_de-ed}
\subsection{Training}
\label{sec:lev-training}
We use imitation learning to train the \textit{FReTNA} similar to the Levenshtein Transformer. Unlike \citet{re-translation}, which is trained on prefix sequences along with full-sentence corpus, we train the model on the full sequence corpus only. The expert policy used for imitation learning is derived from a sequence-level knowledge distillation process \cite{knowledge_distillation}. More precisely, we first train an autoregressive model using the same datasets and then replace the original target sequence by the beam-search result of this model. Please refer to \citet{gu2019levenshtein} for more details on imitation learning for LevT model.
\subsection{Inference}
\label{sec:inference}
At inference time, we greedily (beam size=1) apply the trained model over the streaming input sequence. For every set of new source tokens, we apply the insertion and deletions policies and pick the actions associated with high probabilities in Eq. \ref{eq:pc}, \ref{eq:tc}, and \ref{eq:dc}. During the re-translation based simultaneous translation, the partial translations are inherently revised when a new set of input token arrives; hence, we apply only two iterations of insertion-deletion sequence on the current partial translation. We also impose a penalty on current partial translation to match the prefix part of the previous translation by subtracting a penalty $\eta$ from the logits in eq. \ref{eq:pc} and eq. \ref{eq:dc}.
\subsection{Measuring Stability of Translation}
\label{sec:stability_metrics}
One important property of the re-translation based models is that they should produce the translation output with as few textual instabilities or flickers as possible; otherwise, the frequent changes in the output can be distracting to the users.
The \textit{ReTA} model \cite{arivazhagan2019retranslation} uses Normalized Erasure (NE) as a stability measure by following \citet{Niehues2016,NiehuesPHSW18,arivazhagan2020retranslation}, it measures the length of the suffix that is to be deleted from the previous partial translation to produce the current translation. However, the metric does not account for the actual number of insertions/deletions/replacements, which provide a much better measure to gauge the visual instability. In the Table \ref{table:ne-cne}, the NE gives same penalty to both the current translations, however, the \textit{current translation 2} would obviously cause more visual instability as compared to the \textit{current translation 1}. In order to have a better metric to represent the flickers during the re-translation, we suggest a new stability measure metric, called Normalized Click-N-Edit (NCNE). The NCNE is computed (Eq \ref{eq:cne}) using the Levenshtein distance \cite{distance_Levenshtein}, which computes the number of insertions/deletions/replacements to be performed on the current translation to match the previous translation. As shown in the Table \ref{table:ne-cne}, the NCNE gives higher penalty to the \textit{current translation 2} since it has a higher visual difference as compared to the \textit{current translation 1}. The NCNE measure aligns better with the textual stability goal of the re-translation based SMT models. The metric is given as
\begin{equation}
\label{eq:cne}
\mathrm{NCNE} = \frac{1}{T} \sum_{i=2}^{S} \mathrm{levenshtein\_distance}(o_i, o_{i-1}),
\end{equation}
where $o_i$ and $o_{i-1}$ represent the current and previous translations.
\subsection{Implementation Details}
The proposed \textit{FReTNA}, \textit{ReTA} \cite{arivazhagan2020retranslation}, and \textit{Wait-k} \cite{ma2018stacl} models are implemented using the \textit{Fairseq} framework \cite{ott2019fairseq}. All the models use Transformer as the base architecture with settings similar to \citet{re-translation}. The text sequences are processed using word piece vocabulary \cite{subwords}. All the models are trained on 4*NVIDIA P40 GPUs for 300K steps with the batch size of 4096 tokens. The \textit{ReTA} is trained using prefix augmented training data and \textit{FReTNA} uses distilled training dataset as described in \citet{knowledge_distillation}. The hyperparameter $\eta$ described in the Section \ref{sec:inference} is set to $0.2*k$, where $k=\{1, \cdots, K\}$. The Appendix contains more details about the implementation and hyperparameters settings.
\input{tables/offline_bleu}
\subsection{Results}
In this section, we report the results of our experiments conducted on the DeEn, EnDe and the EnFr language pairs.
In order to test our \textit{FReTNA} system, we compare it with the recent approaches in re-translation (\textit{ReTA}, \citet{arivazhagan2019retranslation}), and streaming based systems (\textit{Wait-k}, \citet{ma2018stacl}). Unlike traditional translation systems, where the aim is to achieve a higher BLEU score, simultaneous translation is focused on balancing the quality-latency and the time-latency trade-offs. Thus, we compare all the three approaches based on these two trade-offs: (1) Quality v/s Latency and (2) Inference time v/s Latency. The latency is determined by the AL. The inference time signifies the amount of the time taken to compute the output (normalized per sentence).
\input{tables/example_table}
\paragraph{Quality v/s Latency:} The Figures \ref{fig:de_ed_bleu}
and \ref{fig:en_fr_bleu} shows the quality v/s latency trade-off for DeEn, EnDe, and EnFr language pairs.
We report the results on both the NE and NCNE stability metrics (Section \ref{sec:stability_metrics}). The Re-translation models have similar results with NCNE $< 0.2$ and NE $< 0.5$ metrics. However, with NE $< 0.2$, the models have slightly inferior results since it imposes a stricter constraint for stability.
The proposed \textit{FReTNA} model performance is slightly inferior in the low latency range and better in the medium to high latency range compared to \textit{Wait-k} model for DeEn and EnFr language pairs. For EnDe, our models perform better in all the latency ranges as compared to the \textit{Wait-k} model.
The slight inferior performance of \textit{FReTNA} over \textit{ReTA} is attributed to the complexity of anticipating multiple target tokens simultaneously with limited source context.
However, \textit{FReTNA} slightly outperforms \textit{ReTA} from medium to high latency ranges for EnDe language pair.
\paragraph{Inference Time v/s Latency:} The Figure \ref{fig:al_time}
shows the inference time v/s latency plots for DeEn, EnDe, and EnFr language pairs.
Since our model simultaneously generates all target tokens, it has much lower inference time compared to the \textit{ReTA} and \textit{Wait-k} models.
Generally, the streaming based simultaneous translation models such as \textit{Wait-k} have lower inference time compared to re-translation based approaches such as \textit{ReTA}, since the former models append the (partial) translation whereas the later models sequentially generate the (partial) translation from scratch for every newly arrived source information. Even though our \textit{FReTNA} model is based on re-translation, it has lower inference time compared to the \textit{Wait-k} and \textit{ReTA} models since we adopt a non-autoregressive model to generate all the tokens in the (partial) translation parallelly.
For the comparison purpose, we also trained offline \textit{ReTA} and \textit{FReTNA} models for the three language pairs, and the results are reported in Table \ref{table:offline}. The BLEU scores of SMT and offline models of \textit{ReTA} and \textit{FReTNA} are comparable.
Thus, we can conclude that our proposed \textit{FReTNA} approach is better than \textit{ReTA} and \textit{Wait-k} in terms of inference time in all the latency ranges, while maintaining the property of superior translation quality of re-translation over the streaming based approaches.
\paragraph{Impact of positional bias:}
We evaluate the \textit{FReTNA} model with and without including positional bias (FReTNA\_pos vs FReTNA\_non\_pos) introduced in Eq. \ref{eq:pc} and \ref{eq:dc} to see whether positional bias can help the model to generate smoother translations. As shown in Figure \ref{fig:pos-nonpos} FReTNA\_non\_pos has more flickers compared to the FReTNA\_pos model since it's not able to cross the NCNE cutoff of 0.2 in the low latency range. The lower performance of FReTNA\_non\_pos in the low latency range is due to predicting more tokens (insertion policy) than required with less source information. Later, when more source information is available, then some of the tokens have to be deleted (deletion policy), causing more flickers in the final translation output. From Figure \ref{fig:pos-nonpos}, we can see that positional bias reduces flickers in the translation and very useful in low latency range.
\paragraph{Sample Translation Process:} In the Table \ref{table:ex}, we compare the process of generating the target sequence using \textit{ReTA} and \textit{FReTNA} models. The examples are collected by running inference using these two models on the DeEn test set. The \textit{ReTA} generates the target tokens from scratch at every step in an autoregressive manner which leads to a high inference time. On the other hand, our \textit{FReTNA} model generates the target sequence parallelly by inserting/deleting multiple tokens at each step. We included only one example here due to space constraints; more examples can be found in the Appendix.
| {
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using System;
using System.Collections.Generic;
using System.Linq;
using System.Web;
using MSD_Umbraco_Portal.UI.Models.Shared;
namespace MSD_Umbraco_Portal.UI.Models.Master
{
public class FooterViewModel
{
public FooterViewModel()
{
Links = new List<UrlViewModel>();
RelatedLinks = new List<RelatedLinkViewModel>();
}
public string HomeUrl { get; set; }
public UrlViewModel RegisterUrl { get; set; }
public IEnumerable<UrlViewModel> Links { get; set; }
public IEnumerable<RelatedLinkViewModel> RelatedLinks { get; set; }
public FooterDictionaryViewModel Dictionary { get; private set; }
internal void SetDictionary(FooterDictionaryViewModel dictionary)
{
Dictionary = dictionary;
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,518 |
{"url":"http:\/\/clay6.com\/qa\/39066\/boltzmann-curve-is-given-below-","text":"Browse Questions\n\n# Boltzmann curve is given below :\n\n$(A)\\; T_1 > T_2 > T_3 \\\\ (B)\\; T_1 < T_2 < T_3 \\\\ (C) \\; T_2 < T_1 < T_3 \\\\ (D)none\\; of\\; these$\n\nSince velocity increases with temperature, Same fraction of molecules will have greater velocity at greater temperature.\n=> $T_3 > T_2 > T_1$\nHence B is the correct answer.","date":"2017-03-30 08:54:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.929069459438324, \"perplexity\": 2182.747323067468}, \"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-2017-13\/segments\/1490218193288.61\/warc\/CC-MAIN-20170322212953-00288-ip-10-233-31-227.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/mathematica.stackexchange.com\/questions\/95246\/efficient-creation-of-lists-from-other-lists","text":"# Efficient creation of lists from other lists\n\nGiven the following lists:\n\nv1 = {a, c, e};\n\nv2 = {b, d, f};\n\n\nhow does one efficiently create the following \"combination lists\"?\n\n{a, c, e};\n{a, c, f};\n{a, d, e};\n{a, d, f};\n{b, c, e};\n{b, c, f};\n{b, d, e};\n{b, d, f};\n\n\nHow can one generalize the algorithm to lists v1 and v2 of arbitrary but equal length $n$?\n\n\u2022 Tuples[Transpose@{v1,v2}] should do it. That will also generalize to any length for v1,v2. Assuming the lengths of v1 v2 are the same. \u2013\u00a0N.J.Evans Sep 22 '15 at 19:49\n\u2022 I am speechless, really. Perfect answer! I erect a statue of you! Thanks a lot. =) \u2013\u00a0TeM Sep 22 '15 at 19:55\n\u2022 Congratulations on your first Statue Badge, @N.J.Evans! \u2013\u00a0march Sep 22 '15 at 20:26\n\nI figured I should make this an answer so it can be closed.\n\nWhat you're looking for is all combinations of three elements such that the first element is either {a,b}, the second element is either {c,d}, and the last element is either {e,f}. Where the choices for each element come from two given vectors v1={a,c,e}, and v2={b,d,f}. This is what Tuples is meant to do.\n\nIn order to get the answer you're after you need to manipulate the vectors a bit.\n\nUsing Transpose@{v1,v2} gives a list {{a,b},{c,d},{e,f}}. Passed to Tuples in this form, the function will create all possible combinations of the three sublists. This is the second form given in the documentation.\n\nTuples[Transpose@{v1,v2}]\n\nThis can also be generalized to any number of vectors of any length, as long as the vectors have the same length. Tuples[Transpose@{v1,v2,v3}].\n\nThe only problem with vectors of different lengths is the Transpose operation. So with some assumptions about the vectors, vectors with different lengths can be handled using a trick from @The Toad for transposing uneven lists, Transpose uneven lists.\n\nTuples[{v1,v2,vSmaller}~Flatten~{2}]\n\nFor instance:\n\nv1 = {1,2,3};\nv2 = {6,7};\nTuples[{v1,v2}~Flatten~{2}]\n\n\nGives:\n\n{{1, 2, 3}, {1, 7, 3}, {6, 2, 3}, {6, 7, 3}}","date":"2019-10-19 23:19:05","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\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3222724199295044, \"perplexity\": 1280.1381044915588}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986700435.69\/warc\/CC-MAIN-20191019214624-20191020002124-00245.warc.gz\"}"} | null | null |
typedef m2::CellId<19> RectId;
m2::PointU PointD2PointU(double x, double y, uint32_t coordBits);
inline m2::PointU PointD2PointU(m2::PointD const & pt, uint32_t coordBits)
{
return PointD2PointU(pt.x, pt.y, coordBits);
}
m2::PointD PointU2PointD(m2::PointU const & p, uint32_t coordBits);
int64_t PointToInt64(double x, double y, uint32_t coordBits);
inline int64_t PointToInt64(m2::PointD const & pt, uint32_t coordBits)
{
return PointToInt64(pt.x, pt.y, coordBits);
}
m2::PointD Int64ToPoint(int64_t v, uint32_t coordBits);
std::pair<int64_t, int64_t> RectToInt64(m2::RectD const & r, uint32_t coordBits);
m2::RectD Int64ToRect(std::pair<int64_t, int64_t> const & p, uint32_t coordBits);
uint32_t DoubleToUint32(double x, double min, double max, uint32_t coordBits);
double Uint32ToDouble(uint32_t x, double min, double max, uint32_t coordBits);
| {
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How Tasty Was My Little Frenchman (1971)
August 7, 2007 admin Comments 1 comment
"The natives are barbarous savages — different from us, and without any religion."
In 16th century Brazil, a French mercenary (Arduano Colassanti) is mistaken for Portuguese and captured by a tribe of Indians, who tell him he has eight months to live before being eaten.
Historical Drama
This uneven yet compelling film by director Nelson Pereira dos Santos is a classic of Brazil's cinema novo movement, which emphasized a reliance on native Brazilian aesthetic sensibilities and a break with cinematic conventions. It's a cynical and subversive look at colonialism in 16th century South America, told through the non-idealized story of Indians dealing with the rape of their land in the only way they know how: through native traditions. This includes capturing and eating their enemies (the Portuguese), in order to literally ingest their "strength" — in other words, cannibalism. It's to Pereira dos Santos's credit that this element of the film is never sensationalized. In fact, he makes every effort to present the Indians' lifestyle as "natural" — including their near-absence of clothing. Not surprisingly, Brazilian censors had a problem with this lack of modesty, and prevented the film from being shown for a year after it was made; but it's a testament to the film's ethnographic authenticity that the nudity quickly seems commonplace, and never exploitative. While How Tasty is a provocative and disturbing film in many ways, however, it's not uniformly successful. This is primarily due to the opening montage sequence, which misrepresents the film as a comedy; though it certainly possesses satirical elements (the title alone is evidence of this), it's not really a farce. Once this brief sequence is over, however, it's remarkably easy to get caught up in the travails of Arduano Colassanti's "Frenchman" — whose fate remains uncertain until the very end.
A compelling pseudo-ethnographic look at the Tupinambas tribe of Brazil
The Frenchman's new wife describing to him what his cannibalism ritual will be like
Beautiful natural settings
The haunting final images
Yes, as an acknowledged classic of Brazilian cinema novo.
Foreign Gem
Mystery of the Wax Museum (1933)
Five Star Final (1931)
One thought on "How Tasty Was My Little Frenchman (1971)"
First viewing. A frustrating once-must, for its place in cinema history.
This one must have been an art-house hit. And someone like Bunuel would probably have liked this very National Geographic travelogue a lot more than I did.
There's little set-up; the film throws the viewer into a reality and seems to require considerable prior knowledge of its subject. Myself, I'm not up enough on this chapter in history – however, on its own terms, '…Tasty…' doesn't seem to fully illuminate. And, for the story it's trying to tell, it's somewhat tiresome.
I'll agree that it's too unique to not be a must. And perhaps a second viewing helps. And perhaps I'm too out-of-the-loop for the material to appreciate it more. But I went into it as an interested outsider – I'd been hunting the film down for years – and didn't find it as compelling as I'd hoped. | {
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} | 500 |
\section{Introduction}
A final answer about the origin and the composition of ultra-high energy cosmic rays (UEHCR) is still missing. Several models have been proposed for the acceleration of UHECR \cite{hillas1984origin, hill1987ultra, berezinsky1997cosmic, berezinsky1997ultrahigh, venkatesan1997constraints, farrar1998correlation, fargion1999ultra, arons2003magnetars} (see \cite{nagano2000observations, bhattacharjee2000origin} and Ref. therein for a review) and it is generally accepted that the candidate sources are extragalactic and trace the distribution of luminous matter on large scales \cite{waxman1997signature}. The recent result reported by the Pierre Auger Collaboration, from observations in the southern hemisphere, experimentally supports compact sources with a number density in the range $10^{-5}$-$10^{-3}$~Mpc$^{-3}$\,\cite{dedomenico2011bounds,auger2013bounds}, showing a correlation between the observed data with energy above 57~EeV and the distribution of nearby mass distribution \cite{auger2010correlation}. Observations in the northern hemisphere by the HiRes Collaboration, but with smaller statistics and a different energy scale, do not confirm this result \cite{abbasi2008search}, while more recent measurements by the Telescope Array Collaboration, based on 25 observed events with energy larger than 57~EeV, suggest a correlation with nearby Active Galactic Nuclei with chance probability of 2\% \cite{abu2012search}.
Even the observed suppression of UHECR, due to their propagation in the Universe, is still debated: in fact, UHECR of extragalactic origin with energy above 100\,EeV (1\,EeV $= 10^{18}$ eV) could be subjected to a strong attenuation because of their interaction with relic photons of the extragalactic background radiation. Recently, the Pierre Auger Collaboration reported a suppression of the spectrum above 40\,EeV with significance greater than 20 standard deviations \cite{settimo2012spectrum,abraham2010measurement}, improving previous measurements \cite{abraham2008observation,abbasi2008first}. Such results are compatible with the existence of the GZK effect \cite{greisen1966end,zatsepin1966upper}, although not providing a definitive evidence. In fact, alternative suitable scenarios, compatible with the same observations, involve a spectrum cutoff directly at the source.
It is evident that both modeling and realistic simulations of production and propagation mechanisms are required to shed light on the nature of UHECR \cite{puget1976photonuclear,allard2005uhe,harari2006ultrahigh,hooper2007intergalactic,allard2008implications,globus2008propagation,allard2009interactions}, trying to avoid the limitations \cite{kachelriess2009gzk} given by the continuous energy loss approximation adopted by some authors to simplify calculations.
In this study, we present the general structure of our propagation code (HERMES) \cite{dedomenico2011thesis}. We show the simulated diffusion of charged particles in both turbulent and structured magnetic fields for energy values ranging from $10^{17}$~eV to $10^{21}$~eV and we provide an estimation of mean free paths and energy-loss lengths of UHE nuclei. The expected GZK horizon is reported together with a comparison with existing results and an estimation of the expected spectrum at Earth is compared against recent observations.
\section{Simulating the propagation of UHECR with HERMES: background radiation and magnetic fields}
In this section, we describe the HERMES propagation code, presenting the modeling adopted for i) the cosmological framework, ii) the cosmic background radiation (microwave, infrared/optical and radio), iii) the regular component of the Galactic magnetic field and the irregular component of both the Galactic and the extragalactic magnetic fields, iv) the cross sections describing the interactions between UHE nuclei and photons of extragalactic background radiation, v) the production of secondary particles because of such interactions. In the following, we will briefly describe such a framework, to provide the reader with the necessary tools to understand the parameterizations and the energy-loss equation adopted in our Monte Carlo code.
\subsection{Cosmological framework}\label{sec-cosmology}
Motivated by up-to-date observations, we have chosen a general Friedmann's Universe, defined by a Friedmann-Robertson-Walker metric, to be the cosmological framework in HERMES. Let us consider the Einstein equation in the classical General Relativity framework to describe the gravitational field. Under the assumptions of an isotropic and homogeneous Universe, we also consider the Friedmann-Robertson-Walker (FRW) metric
\begin{eqnarray}
ds^{2}=c^{2}dt^{2}-a^{2}(t)\[ \frac{dr^{2}}{1-\kappa r^{2}}+r^{2}\( d\theta^{2}+\sin^{2}\theta d\phi^{2} \) \],
\end{eqnarray}
where $a(t)$ is the scale factor, such that $a(0)=1$ is its present value, while the parameter $\kappa$ accounts for the spatial curvature: $\kappa=-1$ denotes an open metric, $\kappa=0$ a flat metric and $\kappa=1$ a closed metric. Indeed, we consider the Universe as a perfect fluid with energy density $\varrho$ and pressure $p$, described by the stress-energy tensor $T_{\mu\nu}=\(\varrho+\frac{p}{c^{2}}\)u_{\mu}u_{\nu}+pg_{\mu\nu}$, where $u_{\mu}$ denotes the 4-velocity. Friedmann equations can be derived from such assumptions.
HERMES is able to propagate particles in a $\Lambda$CDM Universe, with several tunable parameters expressed in terms of the critical density $\varrho_{c}=3H^{2}/8\pi G$. More specifically, we consider $\Omega_{b}$ due to baryonic matter, $\Omega_{c}$ due to cold dark matter, $\Omega_{\Lambda}$ due to dark energy, $\Omega_{r}$ due to radiation and $\Omega_{\kappa}$ for the spatial curvature. If we define the redshift $z$ by $1+z=a^{-1}(t)$, the first Friedmann equation can be written in terms of $z$ and of density parameters as
\begin{eqnarray}
\label{def-Hz}
\frac{H^{2}(z)}{H_{0}^{2}}=\Omega_{r}(1+z)^{4}+\Omega_{M}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda},
\end{eqnarray}
where $\Omega_{M}=\Omega_{b}+\Omega_{c}$ is the total density of matter and $H_{0}$ is the Hubble parameter at the present time. By taking into account that the radiation density contributes only in the early Universe, i.e. at high redshifts, whereas in practice it is negligible in the late Universe, the constraint $\Omega_{M}+\Omega_{\kappa}+\Omega_{\Lambda}=1$ for the density parameters can be obtained from very general considerations. We will describe further in text the role of Eq.\,(\ref{def-Hz}) in the numerical simulation of the propagation of UHECR.
It is worth noticing that a particle with energy $E(z)$ at redshift $z$, propagating through the Universe and not subjected to energy loss processes, will adiabatically lose its energy because of the expansion of the Universe (of course, by assuming a cosmological model where the Universe is expanding), and it will be observed with energy $E_{0}=E/(1+z)$ at the Earth.
The values of all relevant parameters discussed so far, as the Hubble constant, the density of matter and energy, can be freely varied in our simulator to reproduce very different cosmological models, and a study of the impact of cosmology on the GZK horizon of UHECR protons has been recently published \cite{dedomenico2012influence}.
\subsection{Spectrum of UHECR and Evolution of sources}
Let $Q(E)$ indicate the injection spectrum of UHECR at the source, representing the number of particles injected per unit energy and time, and let us indicate the source luminosity by
\begin{eqnarray}
\mathcal{L}=\int_{E_{\text{min}}}^{E_{\text{max}}}Q(E)EdE,
\end{eqnarray}
quantifying the energy emitted from the source in terms of UHECR per unit time. Here, we are assuming that UHECR at the source can be produced from a minimum energy $E_{\text{min}}$ to a maximum energy $E_{\text{max}}$. There are some arguments predicting a power-law injection spectrum of both Galactic and extragalactic CRs \cite{fermi1949origin,bell1974upper,torres2004astrophysical}. Under such an assumption, we can rewrite the injection spectrum as a function of the source luminosity by $Q(E)=\mathcal{L}\mathcal{N}E^{-\gamma}$, being $\gamma$ the injection index and $\mathcal{N}$ a normalization factor. The source luminosity may increase with redshift, as well as the comoving density of sources: in general such a cosmological source evolution depends on several factors, related to the class of astrophysical sources under consideration. If the source evolution is present, the luminosity should include an additional factor $\mathcal{H}(z)=(1+z)^{m}$, giving $\mathcal{L}(z)=\mathcal{H}(z)\mathcal{L}$. It is worth remarking that the source evolution factor can play a significant role for the study of the energy spectrum of UHECR at Earth. Thus, in general, the injection spectrum simulated in HERMES is given by $Q(z,E)=\mathcal{H}(z)Q(0,E)$. The following evolution factors are available in our simulator:
\begin{enumerate}
\item Star formation rate (SFR) \cite{hopkins2006normalization}:
\begin{eqnarray}
\mathcal{H}_{\text{SFR}}(z)=\left\{\begin{array}{ll}
(1+z)^{3.4} & z<1,\\
2^{3.7}(1+z)^{-0.3} & 1<z<4,\\
2^{3.7}\times5^{3.2}(1+z)^{-3.5} & z>4;
\end{array}\right.
\end{eqnarray}
\item Gamma-ray burst (GRB) \cite{yuksel2007enhanced}: $\mathcal{H}_{\text{GRB}}(z)=(1+z)^{1.4}\mathcal{H}_{\text{SFR}}(z)$;
\item Active galactic nuclei (AGN) \cite{stanev2009ultra,hasinger2005luminosity}:
\begin{eqnarray}
\mathcal{H}_{\text{AGN}}(z)=\left\{\begin{array}{ll}
(1+z)^{5} & z<1.7,\\
2.7^{5} & 1.7<z<2.7,\\
2.7^{5}\times10^{0.43(2.7-z)} & z>2.7;
\end{array}\right.
\end{eqnarray}
\item Quasi-stellar object (QSO) \cite{engel2001neutrinos}:
\begin{eqnarray}
\mathcal{H}_{\text{QSO}}(z)=\left\{\begin{array}{ll}
(1+z)^{3} & z<1.9,\\
(1+1.9)^{3} & 1.9<z<2.7,\\
(1+1.9)^{3}e^{1-z/2.7} & z>2.7.
\end{array}\right.
\end{eqnarray}
\end{enumerate}
In the case of a uniform evolution $\mathcal{H}_{\text{unif}}(z)=(1+z)^{3}$, whereas in the case of no evolution $\mathcal{H}(z)=1$ can be assumed.
\subsection{Modeling the extragalactic background radiation}\label{sec-ebr-parameterization}
The propagation of UHECRs is affected by their interactions with photons of the extragalactic background radiation (EBR). While the relevant energy losses will be discussed successively in the text, we briefly describe here the models of background radiations simulated in HERMES.
In fact, EBR modeling is rather difficult, if the well known cosmic microwave background is excluded. Such a radiation should be produced by the assembly of matter into stars and galaxies, as well as by the evolution of such systems which releases radiant energy powered by gravitational and nuclear processes. Absorption of large frequency radiation by dust and re-emission at small frequency considerably increase the infrared component of the background light, whose investigation should shed light on structure formations processes. In the following, we indicate with $\epsilon$ the relic photon energy in eV, $n(\epsilon)$ the photon spectral number density in units of photons cm$^{-3}$~eV$^{-1}$ and $\epsilon^{2}n(\epsilon)$ the energy density in units of eV~cm$^{-3}$.
EBR spans over almost 20 decades, according to observations and models, from radio waves around $10^{-7}$~eV up to the high energy $\gamma-$ray photons of several GeV, with cosmic microwave background (CMB), the relic blackbody radiation from the Big Bang, being the dominant form of electromagnetic energy followed by ultraviolet/optical (CUVOB) and infrared backgrounds (CIRB).
For the propagation of UHECR nuclei, in HERMES we adopt the blackbody model with temperature $T_{0}\simeq 2.725$~K for CMB. The semi-analytical ``model D'' proposed by Finke et al \cite{finke2010modeling}, modeling the star formation rate recently introduced by Hopkins and Beacom \cite{hopkins2006normalization} is adopted for CIOB (for a more detailed treatment of infrared and optical background radiations we refer to \cite{hauser2001cosmic,malkan2001empirically,lagache2003modelling,stecker2006intergalactic,stecker2008spectrum,franceschini2008extragalactic,finke2010modeling} and Refs. therein). The model proposed in \cite{protheroe1996new} is adopted for the universal radio background (URB).
Some models of extragalactic background radiations are shown in the top panel of Fig.\,\ref{fig:ebl-model}, as a function of the photon energy $\epsilon$ in the laboratory frame. The red solid line indicates the EBL parameterization included in HERMES, and it should be considered the default, where not specified otherwise. For sake of completeness, we also show the common parameterizations by Puget, Stecker and Bredekamp (PSB76) for COB, lower and higher IRB (LIR and HIR, respectively) \cite{puget1976photonuclear}, and other IRB models, derived from theoretical arguments or experimental observations \cite{epele1998propagation,funk1998upper,uryson2006ultra,fixsen2011probing}. The bottom panel of the same figure shows the evolution with redshift for different values of $z$, ranging from 0 to 2.
\begin{figure}[!t]
\centering
\includegraphics[width=0.70\textwidth]{HERMES_EBR.pdf}
\caption{Top panel: different parameterizations of extragalactic background radiation as a function of relic photon energy: CMB, (Far, Low and High) IRB and COB. The red line indicates the EBL parameterization included in HERMES. The other parameterizations, shown for reference, are taken from PSB76 \cite{puget1976photonuclear}, FIRAS \cite{fixsen2011probing}, ER98 \cite{epele1998propagation}, Mkn501-98 \cite{funk1998upper}, U06 \cite{uryson2006ultra}. Lower panel: evolution of EBR for different values of redshift. Photon energy is considered in the laboratory frame.}
\label{fig:ebl-model}
\end{figure}
By assuming that the cosmological model of gravitation is described by general relativity and electromagnetism by Maxwell theory, a theoretical consequence of the adiabatic expansion of the Universe is that photons should propagate along null geodesics and that the CMB temperature should evolve with redshift as $T(z)=T_{0} (1+z)^{1-\beta}$, with $\beta=0$. From the same arguments, it can be shown that the energy of CMB photons evolve as $E(z)=E_{0} (1+z)$, whereas their number density evolve as $n(\epsilon,z)=n(\epsilon,z=0)(1+z)^{3}$.
The evolution of the density of CIRB photons is still debated and depends on the adopted scenario for the luminosity evolution. Two models, included in HERMES, have been recently suggested by Stecker et al \cite{stecker2006intergalactic}:
\begin{enumerate}
\item \textbf{Base-line model}:
\begin{eqnarray}
\mathcal{E}(z)=\left\{
\begin{array}{ll}
(1+z)^{3.1} & z\leq 1.3\\
(1+1.3)^{3.1} & 1.3< z\leq 6\\
0 & z>6
\end{array}\right.
\end{eqnarray}
\item \textbf{Fast model}:
\begin{eqnarray}
\mathcal{E}(z)=\left\{
\begin{array}{ll}
(1+z)^{4} & z\leq 1\\
(1+1)^{4} & 1.3< z\leq 6\\
0 & z>6
\end{array}\right.
\end{eqnarray}
\end{enumerate}
In the current cosmological epoch and at the IRB maximum epoch, which is around $z=2$, the fast evolution model provides an higher density than base-line model. In any case, it is worth remarking that the cosmological evolution of the infrared background density is much slower than that of CMB.
Finally, the evolution of the density of CRB photons included in HERMES is the one proposed by Protheroe and Biermann, who modified the luminosity evolution to fit the source counts \cite{protheroe1996new}:
\begin{eqnarray}
\mathcal{E}(z)=\left\{
\begin{array}{ll}
(1+z)^{4} & z< 0.8\\
(1+0.8)^{4} & z\geq 0.8
\end{array}\right.
\end{eqnarray}
where the value $z_{0}=0.8$ has been obtained from the best fit for both normal galaxies and radio galaxies.
We will see further in this chapter that the radio background is negligible when the propagation of high energy nuclei is considered: conversely, it plays an important role during the propagation of high energy photons.
\subsection{Modeling magnetic fields}
The presence of magnetic fields, both in the intergalactic space and in our galaxy, has a non-negligible impact on the propagation of charged nuclei. It is thus of fundamental importance to investigate the structure of galactic and extragalactic magnetic field (GMF and EMF, respectively), that have a direct impact on the energy spectrum, the strength of the anisotropy signal and the correlation with candidate sources. In order to simulate the diffusion of particles with charge $q=Ze$ in magnetic fields, we adopt in HERMES a standard approach, based on the numerical integration of the equation of motions obtained in the ultra-relativistic approximation, in the case of nuclei.
If the electric field is absent (or negligible) and we assume the case of a particle in ultra-relativistic regime, i.e. the particle travels at the speed of light in the direction $\hat{v}(t)$ at time $t$ subjected to a magnetic field $\vec{B}(\vec{r})$ along the trajectory $\vec{r}(t)$, the Lorentz equation reduces to the set of six ordinary differential equations defined by
\begin{eqnarray}
\frac{d\vec{r}(t)}{dt} &=& c\hat{v}(t)\nonumber\\
\frac{d\hat{v}(t)}{dt} &=& \frac{qc^{2}}{E}\hat{v}(t)\wedge\vec{B}(\vec{r}).
\end{eqnarray}
In practice, charged particles accelerating in a magnetic field lose energy because of the emission of synchrotron radiation: in the case of light particles as electrons or positrons, such energy loss should be taken into account during the propagation, whereas for heavier particles as protons it is negligible.
While the trajectory of a charged particle along the regular field is deterministic, i.e. for a given initial condition only one solution to the equations of motion exists, the trajectory of a particle through the turbulent field is stochastic, thus not unique, and it depends on the features of the irregular field as its r.m.s. strength and its coherence length. Unfortunately, we have no exact knowledge of both galactic and extragalactic magnetic fields and, as a consequence, the investigation of charged particles propagation through our galaxy and intergalactic space, respectively, should be based either on empirical or theoretical models and numerical simulations. For the simulation of the irregular component of the magnetic field, we adopt in HERMES the approach proposed by Giacalone and Jokipii \cite{giacalone1994charged,giacalone1999transport}, based on a local step-by-step simulation of the turbulent field.
\subsubsection{Simulating a turbulent magnetic field}
The randomness of the irregular component of the magnetic field is probably due to the evolution of stochastic fluctuations which are correlated up to a given correlation scale. In fact, such an irregular component should show the features typical of correlated flows undergoing turbulent evolution, characterized by a minimum and a maximum scale of turbulence, $\ell_{\text{min}}$ and $\ell_{\text{max}}$, respectively. The particles scatter off the magnetic irregularities and change their pitch angle $\theta$, but not their velocity. The pitch angle scattering is principally dominated by the inhomogeneities with scales of the order of the Larmor radius, i.e. by resonance, providing an effective mechanism of isotropization as long as $r_{L} < \ell_{\text{max}}$.
Our simulation of such an irregular behavior is based on the following approach. The turbulent magnetic field $\vec{B}(\vec{r})$ satisfies two main requirements: i) it is a zero-mean field $\langle \vec{B}(\vec{r}) \rangle=0$ with ii) non-vanishing fluctuations $\langle\vec{B}^{2}(\vec{r})\rangle=B_{\text{rms}}>0$. Let $\vec{k}$ be the wave vectors with modulus $k$, power spectrum $\mathcal{P}(k)\propto k^{-5/3}$ and amplitudes $\vec{B}(\vec{k})$ of its Fourier modes following the Kolmogorov spectrum $|\vec{B}(\vec{k})|^{2}\propto k^{-11/3}$: such a field defines a turbulent Kolmogorov 3D magnetic field\footnotemark\footnotetext{The spectral index is 8/3 and 5/3 for 2D and 1D magnetic fields, respectively.}. In the Fourier space the wave vectors satisfy $\frac{2\pi}{\ell_{\text{max}}}\leq k\leq \frac{2\pi}{\ell_{\text{min}}}$, where the correlation length of the field is equal to \cite{harari2002lensing}
\begin{eqnarray}
\Lambda_{c}=\frac{1}{2}\ell_{\text{max}}\frac{\gamma-1}{\gamma}\frac{1-(\ell_{\text{min}}/\ell_{\text{max}})^{\gamma}}{1-(\ell_{\text{min}}/\ell_{\text{max}})^{\gamma-1}},
\end{eqnarray}
where $\gamma$ is the spectral index of the Kolmogorov spectrum. The approach, proposed by Giacalone and Jokipii \cite{giacalone1994charged,giacalone1999transport} considers the field as the sum of $N_{m}$ modes, physically corresponding to the superposition of a finite number of plane waves:
\begin{eqnarray}
\vec{B}(\vec{r})=\sum_{n=1}^{N_{m}}A_{n}\hat{\varepsilon}_{n}e^{i\vec{k}_{n}\cdot\vec{r}+i\beta_{n}},
\end{eqnarray}
where $\hat{\varepsilon}_{n}=\cos\alpha_{n}\hat{x}_{n}+i\sin\alpha_{n}\hat{y}_{n}$ and the amplitude $A_{n}$ of the $n-$th plane wave is given by
\begin{eqnarray}
A^{2}_{n}=\mathcal{A}B^{2}_{\text{irr}}G(\vec{k}),
\end{eqnarray}
with
\begin{eqnarray}
G(\vec{k})=\frac{\Delta V_{n}}{1+(k\Lambda_{c})^{\gamma}},\quad \Delta V_{n}=4\pi k^{2}\Delta k,\quad \mathcal{A}=\(\sum_{n=1}^{N_{m}}G(\vec{k}_{n})\)^{-1}.
\end{eqnarray}
In this last equation, the index $\gamma$ is equal to 11/3, 8/3 and 5/3 for 3D, 2D and 1D turbulent magnetic fields, respectively. The direction of the $n-$th wave vector $\hat{k}_{n}$ is randomly chosen: the unit vectors $\hat{x}_{n}$ and $\hat{y}_{n}$ are chosen in order to form an orthogonal basis with $\hat{k}_{n}$ and the real numbers $\alpha_{n}$ and $\beta_{n}$ represent random polarizations and phases, respectively. For practical applications, the spacing $\Delta k$ between $k_{\text{min}}=\frac{2\pi}{\ell_{\text{max}}}$ and $k_{\text{max}}=\frac{2\pi}{\ell_{\text{min}}}$ should be constant in logarithmic scale and the number of modes $N_{m}$ should be large enough to obtain the expected results in the small-angle regime. The main advantage of such an approach is the definition of the turbulent field at any point in space with arbitrary precision at the price of a much slower computation than other methods. Where not otherwise specified, in the following we will make use of the isotropic model, although the simulation of the composite model is also allowed by our code. Moreover, we will consider a total magnetic field $\vec{B}=\vec{B}_{\text{tot}}=\vec{B}_{\text{reg}}+\vec{B}_{\text{irr}}$ and, following Ref.\,\cite{casse2001transport}, we define the turbulence level by
\begin{eqnarray}
\eta=\frac{\langle\vec{B}^{2}\rangle}{B_{\text{reg}}^{2}+\langle\vec{B}^{2}\rangle}.
\end{eqnarray}
The above arguments can be used to simulate the turbulent component of both the extragalactic and the Galactic magnetic fields. For instance, the deflection $\delta_{\text{irr}}$ due to the irregular component of the Galactic magnetic field can be estimated by assuming that the particle undergoes a brownian motion at the scale of the coherence length $\Lambda$ of the field and that the ratio between the traversed distance $D$ and $\Lambda$ provides an estimation of the number of magnetic regions traversed \cite{roulet2004astroparticle,giacinti2011ultrahigh}:
\begin{eqnarray}
\delta_{\text{irr}}=\frac{1}{\sqrt{2}}\frac{ZeB_{\text{rms}}}{E}\(D\Lambda\)^{\frac{1}{2}}\simeq 0.6^{\circ}\frac{10^{20}~\text{eV}}{E/Z}\frac{B_{\text{rms}}}{4~\mu\text{G}}\(\frac{D}{3~\text{kpc}}\)^{\frac{1}{2}}\(\frac{\Lambda}{50~\text{pc}}\)^{\frac{1}{2}},
\end{eqnarray}
being $B_{\text{rms}}=\langle B^{2}_{\text{irr}}\rangle$. Similarly, in the case of the extragalactic magnetic field, by considering the appropriate coherence length and by neglecting energy loss processes \cite{bhattacharjee2000origin} we obtain
\begin{eqnarray}
\delta_{\text{irr}}=\simeq 0.8^{\circ}\frac{10^{20}~\text{eV}}{E/Z}\frac{B_{\text{rms}}}{1~\text{nG}}\(\frac{D}{10~\text{Mpc}}\)^{\frac{1}{2}}\(\frac{\Lambda}{1~\text{Mpc}}\)^{\frac{1}{2}}.
\end{eqnarray}
\subsubsection{Simulating the Galactic magnetic field}
In spiral galaxies, the turbulent component of the magnetic field is almost always strongest within the spiral arms, following the distribution of cool gas and dust, whereas the regular component is generally weak within spiral arms, except for rare cases like M51 with strong density waves. However, the regular field also extends far into the inter-arm regions. Observations suggest that the large-scale spiral field produce an halo, extending outside the galactic disks. In cylindrical coordinates, the distribution of the magnetic field $B(\rho, \phi, z)$ in the galaxy can be described by the product of three separated components, related to pure radial dependence $R(\rho)$, spiral ``winding'' modulation $S(\rho, \phi)$, and halo extinction $H(z)$, respectively. Several models have been proposed to describe the regular component of the magnetic field in our galaxy. A detailed description of all models is beyond the scope of the present paper, therefore we limit to mention the models included in HERMES.
The structure of the magnetic field obtained by dynamo mechanisms can be described by modes of different azimuthal symmetry in the disk, and vertical symmetry perpendicular to the disk plane: bisymmetric (BSS) or axisymmetric (ASS), depending on $\pi$ or $2\pi$ symmetry, respectively. Along the vertical dimension, the field can change direction while traversing the disk plane (odd or A-parity) or keep it fixed (even or S-parity). Thus, the possible patterns of the spiral field are four, indicated with the notation BSS-S, BSS-A, ASS-S and ASS-A, and they are all present in HERMES, coupled with galactic magnetic field models proposed by Stanev \cite{stanev1997ultra}, Harari, Mollerach and Roulet (HMR) \cite{harari1999toes}, and Tinyakov and Tkachev \cite{tinyakov2002tracing}. For the sake of completeness, we refer to Refs. \cite{prouza2003galactic,kachelriess2007galactic,page2007three,sun2008radio,jansson2009constraining} for other models, which we plan to include in HERMES, describing the galactic magnetic field.
\begin{comment}
Let us assume a Cartesian space centered at the Galactic Center (GC) of coordinates (0,0,0), and let $(-R_{\text{sun}},0,0)$ the position of the sun, with $R_{\text{sun}}=8.5$~kpc. The cylindrical coordinates are $(\rho,\phi,z)\equiv(\sqrt{x^{2}+y^{2}},\tan y/x,z)$. Below, we will briefly describe\footnotemark\footnotetext{We would like to thank Serguei Vorobiov, Mustafa Hussain and Darko Veberic for their nice review about galactic magnetic field models, circulating within the Pierre Auger Collaboration as an internal note.} some common galactic magnetic field models proposed by Stanev \cite{stanev1997ultra}, Harari, Mollerach and Roulet (HMR) \cite{harari1999toes}, and Tinyakov and Tkachev \cite{tinyakov2002tracing}. For the sake of completeness, we refer to Refs. \cite{prouza2003galactic,kachelriess2007galactic,page2007three,sun2008radio,jansson2009constraining} for other models describing the galactic magnetic field.
\vspace{0.5truecm}\hspace{0.25truecm}\textbf{Stanev model.} From the investigation of the current data available in the early 1990s, Stanev proposed the following functions to model the field:
\begin{itemize}
\item Radial dependence:
\begin{eqnarray}
R(\rho)=\left\{
\begin{array}{ll}
B_{0}\frac{R_{\text{sun}}}{\rho} & \rho>\rho_{0}\\
B_{0}\frac{R_{\text{sun}}}{\rho_{0}} & \rho\leq\rho_{0}
\end{array}\right.,
\end{eqnarray}
where $B_{0} = 3$~$\mu$G and $\rho_{0} = 4$~kpc. The magnetic field is $6.4$~$\mu$G in the central region of the Galaxy (at $\rho < 4$~kpc), and decreases like $1/\rho$ at larger $\rho$ values.
\item Spiral modulation:
\begin{eqnarray}
S_{\text{BSS}}(\rho,\phi)&=&\cos\(\phi-\beta\ln\frac{\rho}{\rho_{1}}-\phi_{\text{sun}}\),\\
S_{\text{ASS}}(\rho,\phi)&=&\left|\cos\(\phi-\beta\ln\frac{\rho}{\rho_{1}}-\phi_{\text{sun}}\)\right|,
\end{eqnarray}
where $\beta=1/\tan p$, being $p$ the pitch angle. The parameters have been chosen according to observations, with $p=-170^{\circ}$, $\phi_{\text{sun}}=\pi$ and $\rho_{1}=10.55$~kpc, the latter related to the distance to the two possible close reversals (in the BSS model) or locations of the field zeroing (in the ASS model), which occur in the Galactic Center direction at 0.5 kpc and $\sim$3 kpc.
\item Halo extinction:
\begin{eqnarray}
H_{\text{S}}(z)&=&\left\{
\begin{array}{ll}
\exp\(-\frac{|z|}{z_{1}}\) & |z|<z_{0}\\
\exp\(-\frac{|z|}{z_{2}}+\frac{z_{0}}{z_{2}}-\frac{z_{0}}{z_{1}}\) & |z|\geq z_{0}
\end{array}\right.,\\
H_{\text{A}\pm}(z)&=& \pm \text{sign}(z)H_{\text{S}}(z),
\end{eqnarray}
for the even and odd parity, respectively, where $z_{0}=0.5$~kpc, $z_{1}=1$~kpc and $z_{2}=4$~kpc.
\end{itemize}
\vspace{0.5truecm}\hspace{0.25truecm}\textbf{HMR model.} The model proposed by Harari, Mollerach and Roulet (HMR), is a modification of the model proposed by Stanev, smoothed out in order to avoid the discontinuities of the field and its derivatives:
\begin{itemize}
\item Radial dependence:
\begin{eqnarray}
R(\rho)=B_{0}\frac{R_{\text{sun}}}{\rho}\tanh^{3}\frac{\rho}{\rho_{2}},
\end{eqnarray}
where $B_{0} = 3$~$\mu$G, $\rho_{2} = 2$~kpc and $R(\rho)$ goes smoothly to zero at the Galactic Center.
\item Spiral modulation:
\begin{eqnarray}
S_{\text{ASS}}(\rho,\phi)=\cos^{2}\(\phi-\beta\ln\frac{\rho}{\rho_{1}}-\phi_{\text{sun}}\),
\end{eqnarray}
where $\rho_{1}=10.55$~kpc.
\item Halo extinction:
\begin{eqnarray}
H_{\text{S}}(z)&=&\frac{1}{2}\( \frac{1}{\cosh(z/z_{1})} + \frac{1}{\cosh(z/z_{2})} \),\\
H_{\text{A}\pm}(z)&=& H_{\text{S}}(z)\tanh(z/z_{3}),
\end{eqnarray}
for the even and odd parity, respectively, where $z_{1}=0.3$~kpc, $z_{2}=4$~kpc and $z_{3}=20$~pc.
\end{itemize}
\vspace{0.5truecm}\hspace{0.25truecm}\textbf{Tinyakov and Tkachev model.} In their model, Tinyakov and Tkachev have considered only the spiral disk component with bisymmetric field and different parity with respect to the $z-$axis. Moreover, they have provided a new expression for the in-plane field $R(\rho)S(\rho,\phi)$ in terms of parameters related to the local field, including an additional parameter related to the distance $d$ to the nearest field reversal (being negative if it occurs in the direction of the Galactic Center, being positive otherwise):
\begin{itemize}
\item Radial dependence:
\begin{eqnarray}
R(\rho)=B_{0}\frac{R_{\text{sun}}}{\rho\cos\phi_{\text{add}}},
\end{eqnarray}
where $B_{0} = 1.4$~$\mu$G and $d=-0.5$~kpc and $\phi_{\text{add}}=\beta\ln\(1+d/R_{\text{sun}}\)-\pi/2$.
\item Spiral modulation:
\begin{eqnarray}
S_{\text{BSS}}(\rho,\phi)=\cos\(\phi-\beta\ln\frac{\rho}{R_{\text{sun}}}+\phi_{\text{add}}-\phi_{\text{sun}}\).
\end{eqnarray}
\item Halo extinction:
\begin{eqnarray}
H_{\text{S}}(z)&=&\exp\(-|z|/z_{1}\),\\
H_{\text{A}\pm}(z)&=& \pm\text{sign} H_{\text{S}}(z),
\end{eqnarray}
for the even and odd parity, respectively, where $z_{1}=1.5$~kpc.
\end{itemize}
\end{comment}
In the left panel Fig.\,\ref{fig:mf-gal-bss3d} we show the HERMES simulation of the HMR model for the regular component of the magnetic field in our galaxy (at $z=0$). In the right panel of the same figure, the two-dimensional projection of the corresponding backtracked trajectories\footnotemark\footnotetext{A backtracked trajectory is the path traveled by the antiparticle, and it is obtained by substituting the charge $Z$ with the charge $-Z$ in the equations of motion.} of UHECR are shown for different values of the rigidity $E/Z$, ranging from $10^{17}$~eV (0.1~EeV) to $10^{20}$~eV (100~EeV). It is evident that at the lower energy particles tend to move along helical trajectories around the field lines, whereas for increasing energy particle tend to be less deflected.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=7.5cm]{MF_Galaxy_BSS-S_3D.pdf}
\includegraphics[width=7.cm]{MF_Traj_Deflections_BSS-S.pdf}
\caption{\textbf{Left:} HERMES simulation of the HMR model for the regular component of the magnetic field in our galaxy (at $z=0$), where the color indicates the intensity of the field. \textbf{Right:} Two-dimensional projection of the corresponding backtracked trajectories of UHECR for different values of the rigidity $E/Z$, ranging from $10^{17}$~eV (0.1~EeV) to $10^{20}$~eV (100~EeV)}
\label{fig:mf-gal-bss3d}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=14cm]{MF_GJI_p_B3_R1_l1_L100.pdf}
\caption{A random realization of nuclei trajectories in a uniform magnetic field ($B_{0}=3$~$\mu$G, parallel to the $z-$axis) plus a Kolmogorov 3D turbulent field ($\gamma=11/3$), for three different values of the ratio $E/Z$, namely $10^{17}$~eV, $10^{18.5}$~eV and $10^{20}$~eV. We have simulated the turbulent field ($\ell_{\text{max}}=100$~pc, $\langle B^{2}_{\text{irr}}\rangle=1$~$\mu$G, $\eta=0.1$) according to the Giacalone-Jokipii 3D isotropic approach.}
\label{fig:mf-gal-turb-forwtrack}
\end{center}
\end{figure}
For what concerns the irregular component of the GMF, as previously discussed, observations suggest a r.m.s. intensity of the order of the regular one, although no precise information is currently available. In Fig.\,\ref{fig:mf-gal-turb-forwtrack} we show the HERMES simulation of the trajectory (forward in time) of a particle with $E/Z$ ranging from $10^{17}$~eV to $10^{20}$~eV (left, middle and right panel, respectively), propagating in a magnetic field with an uniform component of intensity $B_{0}=3$~$\mu$G, parallel to the $z-$axis and a 3D turbulent component, characterized by maximum coherence length $\ell_{\text{max}}=100$~pc, r.m.s. strength $\langle B^{2}_{\text{irr}}\rangle=1$~$\mu$G and Kolmogorov index $\gamma=11/3$. The turbulence level is $\eta=0.1$. At the lowest energy the particle undergoes a brownian motion, being the Larmor radius of the order of turbulence scale $\ell_{\text{max}}$, whereas for increasing energy the particle only partially ``feels'' the turbulent component. At the highest energy the particle is subjected to the regular component only.
\section{Simulating the propagation of UHECR with HERMES: modeling interactions between UHECR and EBR photons}
\label{sec-modeling-interact}
We have shown the impact of magnetic fields on the propagation of UHE nuclei, without considering the energy-loss processes relevant for a complete study. This is the main subject of this section, where we show the impact of energy-loss processes on the propagation of UHE nuclei, photons and neutrinos. We will define the parameterizations we have chosen for the cross sections of the interactions between propagating UHECRs and photons of the background radiation and we will discuss all the relevant energy-loss processes included in our simulator as the adiabatic loss (due to the expansion of the Universe), the pair and photo-pion production, and, in the particular case of heavy nuclei, the photo-disintegration processes. The creation of secondary particles, produced by UHE nuclei undergoing pair and photo-pion production during their propagation, is also described: the development of the resulting UHECR cascade, including neutrinos and photons, will be briefly described to underline the complexity of simulating a realistic propagation.
During their propagation, photons, neutrinos and nuclei $(A,Z)$ (electric charge, mass) with injection energy $E_{i}$, generally undergo interactions with background photons. UHECR that reach the Earth are therefore detected with a degraded energy $E_{f}<E_{i}$, depending on the type of interactions they were subjected to and on the distance between the source and the Earth. In HERMES, we describe the energy loss of non-stochastic processes in a unit interval of $z$ in terms of equations like
\begin{eqnarray}
\frac{1}{E}\frac{dE}{dz}=-\beta(z,E)\frac{dt}{dz},
\end{eqnarray}
where
\begin{eqnarray}
-\frac{dt}{dz}&=&\frac{1}{H_{0}(1+z)}\[ \Omega_{M}(1+z)^{3} + \Omega_{\Lambda} + (1-\Omega_{M}-\Omega_{\Lambda})(1+z)^{2} \]^{-\frac{1}{2}}
\end{eqnarray}
is the general metric element accounting for the cosmological expansion \cite{engel2001neutrinos, ave2005cosmogenic, stanev2009high}, and the involved cosmological parameters have been introduced in Sec.\,\ref{sec-cosmology}. The function $\beta(z,E)$ is related to the cooling rate of the UHE particle and it depends on the particular energy loss process considered. As we will see further in this section, $\beta(z,E)$ is proportional to the inverse of the mean free path and depends on the density of background photons and their energy, on the energy of the UHECR and on the cross section of the interaction under investigation. In the case of nuclei, it also depends on the nuclear mass and charge. Thus, the total energy loss rate is obtained by
\begin{eqnarray}
\label{def-energylosseq}
\frac{1}{E}\frac{dE}{dz} = -\frac{dt}{dz}\sum_{\text{process}}\beta_{\text{proc}}(z,E),
\end{eqnarray}
where the sum is extended to all interactions acting during the propagation. In HERMES, we include only those interactions which have a significant impact on the propagation of UHECR:
\begin{itemize}
\item \textbf{Adiabatic loss:} it is due to the expansion of the universe; it is considered for all nuclei with $A\geq 1$, photons and neutrinos;
\item \textbf{Pair production:} it involves the creation of a positron/electron pair; it is considered for all nuclei with $A\geq 1$ and photons;
\item \textbf{Photo-pion production:} it involves the creation of one or multiple pions; it is considered for all nuclei with $A\geq 1$;
\item \textbf{Photodisintegration:} it involves the fragmentation of the original nucleus, with the creation of lighter nuclides (generally referred to as \emph{fragments}); it is considered for all nuclei with $A\geq 2$;
\item \textbf{Inverse Compton and synchrotron emission:} it is considered for photons and pairs which are part of the electromagnetic cascade generated by nuclei, and we refer to \cite{settimo2012eleca} for further details.
\end{itemize}
In the following we will take into account the interactions of nuclei with cosmic microwave background (CMB) and cosmic infrared/optical background (CIOB) radiations, by adopting the parameterization described in Sec.\,\ref{sec-ebr-parameterization} (see Fig.\,\ref{fig:ebl-model}) for the extragalactic background radiation. Eq.\,(\ref{def-energylosseq}) and mean free paths corresponding to the above interaction processes can be used to obtain an analytical approximation of the total energy loss. However, in order to obtain more realistic results, a Monte Carlo approach should be adopted for those processes where stochasticity is relevant, as in the case of photo-pion production and photodisintegration of heavier nuclei.
In the following, for the sake of simplicity, we will omit to specify that results shown in the following plots have been obtained from HERMES. At the end of this section we will describe the propagation of UHECR with no regards of magnetic fields: such an approach is generally known as ``1D propagation''.
\subsection{Adiabatic loss}
In order to take into account the energy loss due to the expansion of the universe, we use
\begin{eqnarray}
\label{def-betarsh}
\beta_{\text{rsh}}(z)&=& H_{0}\[\Omega_{M}(1+z)^{3} + \Omega_{\Lambda} + (1-\Omega_{M}-\Omega_{\Lambda})(1+z)^{2} \]^{\frac{1}{2}}
\end{eqnarray}
for the adiabatic term, as previously explained in Sec.\,\ref{sec-cosmology}.
\subsection{Cross section of $A\gamma$ nuclear interactions}
\label{sec-cross-A}
The probability of UHE protons to interact with background photons rapidly increases with proton energy. If $E$ and $\epsilon$ are the energies of the UHE proton and the photon in the observer rest frame, respectively, the interaction is equivalent to a collision with a high energy photon with energy $\epsilon'=\Gamma\epsilon(1-\cos\theta)$, being $\theta$ the collision angle. For instance, when the energy $\epsilon'$ equals at least the pion mass $m_{\pi}c^{2}\approx 140$ MeV, the proton undergoes photo-meson production and loses energy. Such a process is known as Greisen-Zatsepin-Kuzmin effect and dominates above $50-60$ EeV \cite{greisen1966end,zatsepin1966upper}. The two main channels for $p+\gamma_{\text{EBR}}$ interaction, involving the resonance $\Delta(1232 \text{ MeV})$ close to the threshold energy, are $\Delta(1232~\text{MeV})\lto p+\pi^{0}$ and $\Delta(1232~\text{MeV})\lto n+\pi^{+}$ (with the consequent channel $n\lto p +e^{-}+\bar{\nu}_{e}$). At higher energies, heavier resonances and multi-pion production channels are likely. Just above the threshold, baryonic resonances dominate and protons are subjected to photo-meson production, mainly through the $\Delta(1232)$-baryon resonance, whereas heavier resonances (up to $\Delta(1950)$-baryon) play a more marginal role. We parameterize the cross-section for baryonic resonances by
\begin{eqnarray}
\sigma_{\text{BR}}(\epsilon)&=& \sum_{i=1}^{4}\sigma_{i}\sigma_{L}(\epsilon;\epsilon_{i},\Gamma_{i})\nonumber
\end{eqnarray}
where $\sigma_{L}$ is the Lorentzian function, ($\epsilon_{i}$ (GeV), $\Gamma_{i}$ (GeV), $\sigma_{i}$ ($\mu$b)) $=(0.34, 0.17, 351)$, $(0.75, 0.50,159)$, $(1.00, 0.60, 21)$ and $(1.50, 0.80, 26)$ for $i=1,2,3$ and 4, respectively. For all other processes participating in photo-meson production, including multipions (MP) or direct particle production involving $\pi$, $\eta$, $\Delta$, $\rho$, $\omega$ and strange-particle channels (RP), we use Rachen's parameterizations \cite{rachen1996thesis}. In the following we will use the abbreviation ``BR'' to refer to baryonic resonances, direct particle and multi-pion production, where not specified otherwise.
As in the case of protons, the probability of heavier UHE nuclei to interact with background photons rapidly increases with nucleus energy. The processes involved in such interactions are the same that we have previously described in the case of protons, namely pair and photo-pion production. However, in the case of heavy nuclei we have to take also into account the photo-disintegration (or photo-erosion) process
\begin{eqnarray}
^{A}_{Z}X+\gamma \lto ^{A'}_{Z'}Y + m\alpha + [(Z-Z')-2m]p + [(A-A')-(Z-Z')-2m]n,
\end{eqnarray}
resulting in the emission of subatomic particles, with the creation of lighter nuclides. Here, $m$ is the multiplicity of $\alpha$ particles, $p$ indicates the proton and $n$ the neutron. In general, in order to describe the changes in abundance of the heavy nuclei as a result of the interaction of the UHECR with the background radiation, a nuclear reaction network including all interactions of interest should be used. Such a network is described by a system of coupled differential equations corresponding to all the reactions affecting each nucleus, i.e. mainly photo-disintegrations and $\beta-$decays. Such an approach has been recently proposed, and adopted in many successive works, in Ref.\,\cite{allard2005uhe} for the study of UHE nuclei propagation by using up to date measurements of cross sections \cite{khan2005photodisintegration}. Instead of direct measurements, other recent works related to this topic \cite{kampert2009propagation,ahlers2010analytic,ahlers2011cosmogenic} make use of TALYS \cite{koning2005talys,TALYSweb}, a software for the most likely simulation of nuclear reactions.
We adopt the simplest approach to the treatment of the photo-disintegration channels, by following the chain of stable nuclei (stability chain), as suggested for the first time by Puget, Stecker and Bredekamp (PSB) \cite{puget1976photonuclear}. The relative contribution of all decay channels corresponding to nuclei with $A\leq 56$ are taken from Ref.\,\cite{puget1976photonuclear} and \cite{stecker1999photodisintegration}. However, in order to produce more realistic simulations of the photo-disintegration process, we have obtained from TALYS reactions the branching ratios associated to the most relevant exclusive channels, including one nucleon, two nucleons and multi-nucleons emission on CMB and CIOB, similarly to recent studies \cite{kampert2009propagation,ahlers2010analytic}. Hence, in HERMES, we have included different models for the photo-disintegration of nuclei, with cross sections corresponding to: i) the PSB Gaussian approximation; ii) the Rachen's parameterizations; iii) the TALYS reactions. See the corresponding referenced works for further details.
In Fig.\,\ref{fig:prop-cross-p-fe} we show the comparison between the total cross sections estimated for iron (left panel) and proton (right panel) nuclei, together with the contribution of each single process separately.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=12cm]{PROP_cross_p_fe.pdf}
\caption{Comparison between the total cross section for Fe$\gamma_{\text{EBR}}$ (left panel) and $p\gamma_{\text{EBR}}$ (right panel) interactions as a function of the background photon energy $\epsilon'$ in the nucleus rest frame, obtained from our HERMES, following Rachen's parameterizations\,\cite{rachen1996thesis,stanev2007icrc}. Different contributions due to baryonic resonances (BR), direct particle (RP) and multi-pion (MP) production are shown as reported in Refs. (left panel) and as obtained from HERMES, following Rachen's parameterizations (right panel).}
\label{fig:prop-cross-p-fe}
\end{center}
\end{figure}
For the sake of completeness, it is worth remarking that in HERMES the inclusion of some additional processes, not depending on the background radiation, are currently under development:
\begin{eqnarray}
^{A}_{Z}X\lto ^{A}_{Z+1}Y+e^{-}+\bar{\nu}_{e}&\quad&\beta^{-}-\text{decay},\nonumber\\
^{A}_{Z}X\lto ^{A}_{Z-1}Y+e^{+}+\nu_{e}&\quad&\beta^{+}-\text{decay},\nonumber\\
^{A}_{Z}X+e^{-}\lto ^{A}_{Z-1}Y+\nu_{e}&\quad&\text{electron capture}.\nonumber
\end{eqnarray}
\subsection{Interaction lengths for $A\gamma$ interactions}
The adiabatic loss is considered during the whole propagation as a continuous energy loss process. Instead, the interaction length (or, equivalently, mean free path) corresponding to different processes is used as an input to the Monte Carlo algorithm to randomly sample the next point where the nucleus will undergo one of the interactions described at the end of Sec.\,\ref{sec-modeling-interact}. Such interactions are treated as competitive processes, except the pair production which is treated as a continuous energy loss in the current version of HERMES (see further in the text). Therefore, the estimation of the interaction lengths is fundamental and allows to simulate the production of secondary UHECR (lighter nuclei from photo-disintegration, neutrinos and photons cascades). The interaction length is given by
\begin{eqnarray}
\label{def-lambda}
\mathcal{\lambda}^{-1}_{A}(z,E)=\mathcal{E}(z)\frac{c}{2\Gamma_{A}^{2}}\int_{\epsilon_{\text{thr}}/2\Gamma_{A}}^{\epsilon_{\text{max}}}d\epsilon\frac{n(\epsilon)}{\epsilon^{2}}\int_{\epsilon_{\text{thr}}}^{2\Gamma_{A}\epsilon}d\epsilon'\epsilon'\sigma(\epsilon')
\end{eqnarray}
where $\Gamma_{A}=(1+z)\frac{E}{Am_{p}c^{2}}$ is the Lorentz factor of the nucleus at redshift $z$, $\epsilon_{\text{thr}}$ is the energy threshold of the considered process in the nucleus rest frame, $n(\epsilon)$ is the density of background photons with energy $\epsilon$ in the observer's rest frame, $\epsilon'$ is the energy of the photon in the nucleus rest frame and $\mathcal{E}(z)$ is the evolution function of the ambient photon field. It is straightforward to show that $\lambda_{A}(z,E)=(1+z)^{-3}\lambda_{A}[z=0,(1+z)E]$ when the CMB is considered \cite{stanev2000propagation,berezinsky2006astrophysical}, whereas for other background radiations a more complicated evolution should be used. By following Stanev et al \cite{stanev2000propagation}, we define the average energy loss length by
\begin{eqnarray}
\chi_{\text{loss}}(z,E)=\frac{E}{dE/dz}=\frac{\lambda_{A}(z,E)}{\kappa(E)},
\end{eqnarray}
where $\kappa(E)=\langle\Delta E\rangle/E$ is the mean inelasticity, i.e. the average fraction of energy lost by the nucleus because of the interaction. The inelasticity for pair production is $\kappa\approx 2m_{e}/(Am_{p})$ (being $m_{e}$ and $m_{p}$ the masses of electron and proton, respectively), i.e. around $10^{-3}$ in the case of protons, and even smaller for heavier nuclei. Conversely, for photo-pion production by protons, the inelasticity ranges from 0.2 to 0.5, depending on the energy.
In the case of heavy nuclei, the differences in the cross section (with respect to the case of protons) are reflected in the interaction length. In Fig.\,\ref{fig:prop-cross-p-fe}, the available channels above the threshold for single pion production ($\epsilon'\approx 145$~MeV) involve baryonic resonances and direct particle production, with multi-pion production playing a significant role at the highest energies ($\epsilon'>700$~MeV). In the case of iron, the additional channels due to photo-disintegration process are evident at lower energies ($1<\epsilon'<150$~MeV). The energy loss due to the pair production, Eq.\,(\ref{def-betapair}), and to the adiabatic loss, Eq.\,(\ref{def-betarsh}), occurs in any case, with significant contributions only in a small range of energies. We treat the photo-pion production similarly to the case of protons by using Eq.\,(\ref{def-lambda}) and the $\Delta-$baryon decay channels. The energy loss equation, defined by Eq.\,(\ref{def-energylosseq}), still applies but coupled to the nuclear mass loss rate
\begin{eqnarray}
\frac{1}{A}\frac{dA}{dz}=-\frac{dt}{dz}\beta_{\text{dis,eff}}(z,E;Z,A),
\end{eqnarray}
leading to
\begin{eqnarray}
\frac{1}{E}\frac{dE}{dz}=\frac{1}{\Gamma}\frac{d\Gamma}{dz}+\frac{1}{A}\frac{dA}{dz}.
\end{eqnarray}
An analytic approach for the estimation of the spectra at Earth, based on the numerical integration of such an equation, has been recently reported in \cite{aloisio2008analytic}.
In the rest frame of the nucleus, the pair production process $A+\gamma_{\text{EBR}}\lto A+e^{+}+e^{-}$ occurs at the threshold energy $2m_{e}c^{2}\approx 1$~MeV and it plays an important role only when CMB is considered, the CIOB participating marginally \cite{puget1976photonuclear}. We can treat the process as a continuous energy loss, because the loss per interaction is very small. In HERMES, the energy loss accounting for the pair production, due to the Bethe-Heitler interaction with ambient photons with density $n(\epsilon)$, is given by \cite{blumenthal1970energy}
\begin{eqnarray}
\label{def-betapair}
\beta_{e^{\pm}}(z,E;Z,A)\simeq S(Z)\frac{\alpha^{3}Z^{2}A^{2}}{4\pi^{2}\hbar}\frac{m_{e}^{2}m_{p}^{2}}{E^{3}}\int_{2}^{\infty}d\xi\frac{\varphi(\xi)}{\exp\[\frac{m_{e}Am_{p}}{2E(1+z)k_{B}T_{0}}\xi\]-1},
\end{eqnarray}
that is similar to the parameterization adopted in \cite{cuoco2006footprint}, where the auxiliary function $\varphi(\xi)$ is obtained from \cite{chodorowski1992reaction} and masses are in units of eV/$c^{2}$. However, there is no parameterization in the case of CIOB and, in our code, we estimate the corresponding energy loss rate by using Eq.\,(\ref{def-lambda}). In Eq.\,(\ref{def-betapair}), $\gamma\approx E/(Am_{p}c^{2})$ is the Lorentz factor of the nucleus, $m_{e}$ is the electron mass, $\alpha=e^{2}/\hbar c$ is the fine-structure constant, $r_{e}=e^{2}/m_{e}c^{2}$ is the classical electron radius, $T_{0}=2.725$~K and $k_{B}$ is the Boltzmann constant. The factor $S(Z)$ is a correction term to agree with experimental data for nuclei with $Z>1$ \cite{rachen1996thesis}, even if it has been pointed out that Coulomb corrections to the Born approximation have a negligible effect on the pair production loss rate of ultra-relativistic heavy nuclei as $^{56}$Fe \cite{stecker1999photodisintegration}.
Concerning the photo-pion production process (see Sec.\,\ref{sec-cross-A}), in the particular case of protons propagating in the CMB, Eq.\,(\ref{def-lambda}) at present time reduces to
\begin{eqnarray}
\beta_{\pi}(E)=-\frac{k_{B}T_{0}}{2\pi^{2}\hbar}\frac{m^{2}_{p}}{E^{2}}\int_{0}^{\infty}d\epsilon \kappa(\epsilon)\sigma(\epsilon)\epsilon\times\ln\[1-\exp\(-\frac{m_{p}}{2Ek_{B}T_{0}}\epsilon\)\],
\end{eqnarray}
where $m_{p}$ is the proton mass in units of eV$/c^{2}$, $\sigma(\epsilon)$ is the cross-section for pion production in terms of the photon energy $\epsilon$ and $\kappa(\epsilon)$ is the inelasticity factor. In order to avoid further numerical integrations, we parameterize the contribution of this term as in Ref. \cite{cuoco2006footprint} by
\begin{eqnarray}
\beta_{\pi}(z,E;1,1)\simeq\left\{
\begin{array}{ll}
A_{\pi}(1+z)^{3}\exp\[\frac{B_{\pi}}{E(1+z)}\] & E\leq E_{\text{match}}(z)\\
C_{\pi}(1+z)^{3} & E> E_{\text{match}}(z)
\end{array}\right..
\end{eqnarray}
The function $E_{\text{match}}(z)=6.86 e^{-0.807z}\times 10^{20}$ eV ensures the continuity of the function $\beta_{\pi}(z,E;1,1)$ and $\{A_{\pi},B_{\pi},C_{\pi}\}=\{3.66\times10^{-8}\text{yr}^{-1},2.87\times10^{20}\text{eV},2.42\times10^{-8}\text{yr}^{-1}\}$ are taken from Ref. \cite{anchordoqui1997effect}.
We treat the case of neutron in a similar way, by considering the additional process of the $\beta-$decay. The neutron decay rate is given by $m_{N}/(\tau_{n}E)$, with $\tau\simeq 888.6$~s the laboratory lifetime, providing a range of propagation $\lambda_{\beta}=\tau_{n}\frac{E}{m_{N}}\simeq 0.9\(E/10^{20}~\text{eV}\)~\text{Mpc}$, which becomes competitive with photo-pion production only at the highest energy, above $10^{21}$~eV.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=7.cm]{PROP_lambda_proton_detail.pdf}
\includegraphics[width=7.cm]{PROP_lambda_iron_detail.pdf}
\caption{Our estimation of the mean free path $\lambda$ at $z=0$ as a function of the energy of the nucleus in the observer rest frame. The contributions due to different processes (adiabatic, pair and photo-pion production, as well as giant dipole resonance and quasi-deuteron effect for photo-disintegration) in CMB and CIOB are shown separately. The total interaction length $\lambda_{\text{tot}}$ and the total energy loss length $\chi_{\text{loss}}$ are shown as well in the case of proton (left panel) and Iron $^{56}_{26}Fe$ (right panel) nuclei: for the latter, the $\lambda_{\text{tot}}$ estimated by Allard et al \cite{allard2006cosmogenic,allard2009propagation} is reported for reference.}
\label{fig:prop-lambda-pfe}
\end{center}
\end{figure}
In Fig.\,\ref{fig:prop-lambda-pfe} we show the interaction length $\lambda$ of proton (left panel) and iron (right panel) nuclei, in the CIOB and the CMB, for each process separately and for all processes together, as well as the energy loss length $\chi_{\text{loss}}$, as a function of the energy $E$ of the nucleus in the observer rest frame at present time ($z=0$).
In the case of proton, it is evident that the pair production on CIOB is negligible with respect to other processes, because occurring on time scales larger than the adiabatic expansion, for all energies above $10^{18}$~eV. A similar argument applies for the photo-meson production in the CIOB, which, below $10^{20}$~eV, contributes less than pair production in the CMB, whereas above $10^{20}$~eV the production of pions in the CMB dominates up to the highest energy. In the energy interval between $2\times10^{18}$~eV and $\sim5\times10^{19}$~eV, the main energy loss process is the pair production in the CMB. The obtained results are in perfect agreement with recent literature \cite{berezinsky2006astrophysical,harari2006ultrahigh,stanev2009propagation,kotera2011astrophysics}, with small differences related to the different CIOB adopted.
In the case of iron, the figure shows that the main energy loss below $10^{19}$~eV is due to the adiabatic expansion of the universe, whereas photo-disintegration process through the giant dipole resonance dominates above $10^{19}$~eV and photo-meson production becomes dominant above $10^{22}$~eV.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=12cm]{PROP_lambda_xloss_nuclei.pdf}
\caption{\textbf{Left:} Total mean free path $\lambda_{\text{tot}}$ in CMB and CIOB for several nuclei at $z=0$, from proton ($A=1$) to iron ($A=56$), as a function of the Lorentz factor $\Gamma$. \textbf{Right:} Same as in the left panel, but for the total energy loss length $\chi_{\text{loss}}$.}
\label{fig:prop-xloss-nuclei}
\end{center}
\end{figure}
The estimation of the total interaction (left panel) and energy loss (right panel) lengths at $z=0$ obtained with HERMES are also shown in Fig.\,\ref{fig:prop-xloss-nuclei} as a function of the Lorentz factor $\Gamma$, for several nuclei, from proton to iron. Both quantities decrease for increasing nuclear mass and for any value of the energy, although the energy loss length tends to become constant for all nuclei above $\Gamma=10^{11}$, approximately the value where baryonic resonances occur.
\subsection{Propagation of secondary neutrinos and photons}
We have discussed the production of electron/positron pairs and of secondary pions. Produced UHE photons and pairs interact with the extragalactic background photons, participating to the electromagnetic cascade generated by the primary nucleus. Conversely, in the case of photo-meson production, pions have small lifetime, of the order of $10^{-16}$~s for $\pi^{0}$ and $10^{-8}$~s for $\pi^{\pm}$: thus, we neglect their propagation because they quickly decay to new secondary particles, which can decay to other particles (as in the case of secondary muons) generating a cascade of electrons, positrons, photons and neutrinos. In HERMES, we consider all the main decay channels involving the production of a single pion and we include the $\beta-$decay of neutrons.
Additionally, channels with multi-pion production are present. As shown in \cite{mucke124sophia}, close to the threshold and for $\epsilon'<1$~GeV, the dominating processes involve single pion production only, whereas at the highest energies channels with two or three pions are available. The inclusion in HERMES of such channels is currently under development.
The propagation of UHE photons produced by neutral pions, and the consequent pairs, are performed with EleCa and its description is beyond the scope of the present work. We refer to \cite{settimo2012eleca} for further details.
Neutrinos, produced by the decay of charged pions and $\beta-$decay of neutrons, are chargeless particles with negligible mass, undergoing interactions only through the weak nuclear force (and gravity, if they are considered massive particles). Because of such features, neutrinos are likely to traverse the extragalactic space, even for cosmic distances, without interacting with background photons or interstellar medium, and without being deflected by magnetic fields: characteristics that makes neutrinos the ideal candidates for particle astronomy. On the other hand, the flux of cosmogenic neutrinos is relatively small if compared to the flux of charged particles, at the highest energy. Propagation and energy loss of neutrinos, can be easily described by energy loss equation (\ref{def-energylosseq}), considering only the adiabatic energy loss rate defined by Eq.\,(\ref{def-betarsh}).
\section{Applications}
In this section we briefly discuss some applications to show the potentiality of HERMES for studying UHECR, including the comparison between results obtained with HERMES and those either from other propagation codes available in the UHECR community or from observation.
First, we investigate the surviving probability $\omega_{\text{GZK}}(z,E_{\text{thr}})$ of protons, i.e. the probability that a proton produced by a sources at redshift $z$ could reach the Earth with an energy above a given threshold. We consider an homogenous distribution of equal-intrinsic-luminosity sources in the nearby Universe, up to $\approx300$~Mpc: each source emits protons following a power-law injection spectrum with spectral index $2.4$ and energy cutoff $10^{21}$~eV. Hence, we estimate $\omega_{\text{GZK}}(z,E_{\text{thr}})$ for different energy threshold $E_{\text{thr}}$ at Earth, ranging from 60~EeV to 100~EeV. The result is shown in Fig.\,\ref{fig-hermes-comp2}, where a comparison between HERMES, CRPropa~v1.4~\cite{armengaud2007crpropa} and D. Allard \emph{et al} \cite{allard2005uhe}, are reported. The resulting curves are in good agreement, putting in evidence the goodness of our simulator.
\begin{figure}[!t]
\centering
\includegraphics[width=11cm]{hermes_wgzk_comparison_s2,4.pdf}
\caption{Surviving probability of protons (see the text) produced by an homogenous distribution of sources within 300~Mpc, as a function of the propagation distance and for different energy threshold $E_{\text{thr}}$ at Earth. A power-law injection with energy cutoff $10^{21}$~eV and spectral index $2.4$ is used. We show the result of our simulations performed with HERMES (solid line), compared to those obtained with CRPropa~v1.4 (dashed line) and by Allard (dotted line).}
\label{fig-hermes-comp2}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=11.0cm]{hermes_GZK_horizon_comparison_s2,7_Emax1e21eV.pdf}
\caption{GZK horizon estimated in the case of protons (left panel) and iron nuclei (right panel) injected with spectrum $E^{-2.7}$, as a function of the energy threshold at Earth. Results from CRPropa and Harari \emph{et al} \cite{harari2006ultrahigh} are shown for reference.}
\label{fig-hermes-comp4}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=11cm]{HERMES_vs_hiresPRL.pdf}
\caption{Expected all-particles energy spectra obtained from HERMES for different astrophysical scenarios, compared to observations reported by HiRes Collaboration (see the text). The legend indicates the spectral index at the source and the source evolution adopted (only star formation rate, in this case).}
\label{fig:HERMES-spectra}
\end{figure}
Successively, we estimate the GZK horizon for both protons and iron nuclei, as a function of the energy threshold at Earth. In particular, we compare against well-known results in literature~\cite{harari2006ultrahigh} and CRPropa~v2.0$\beta$\footnote{The version used here is dated September 2011.}, the up-to-date version of the Monte Carlo code simulating the 3D propagation of nuclei in a magnetized Universe~\cite{sigl2011icrc,kampert2013crpropa}. In Fig.\,\ref{fig-hermes-comp4} we show the GZK horizon of protons (left panel) and iron nuclei (right panel). In both cases, the horizons obtained by HERMES are in agreement with those of CRPropa over the whole energy range under consideration, although for iron nuclei some differences are present at the lowest energy.
Moreover, we estimate the expected energy spectra of UHECR at Earth in different astrophysical scenarios, involving evolution of sources, different spectral indices and mass composition at the source. The result, shown in Fig.\,\ref{fig:HERMES-spectra}, are compared against recent observations reported by the HiRes Collaboration \cite{hires2008spectrum}. For sake of simplicity, we show only some representative spectra: a study of their goodness in reproducing the observed UHECR spectrum is beyond the scope of the present paper and it will be the subject of a future study.
\begin{figure}[!t]
\centering
\subfigure[2MRS Catalogue]
{
\includegraphics[width=6.6cm]{auger_mock_2MRS.pdf}
\label{fig:skymap2MRS}
}
\hspace{5mm}
\subfigure[SWIFT58 Catalogue]
{
\includegraphics[width=6.6cm]{auger_mock_SWIFT58.pdf}
\label{fig:skymapSWIFT}
}
\subfigure[2MRS Catalogue + Isotropic]
{
\includegraphics[width=6.6cm]{auger_mock_iso_2MRS.pdf}
\label{fig:skymap2MRSiso}
}
\hspace{5mm}
\subfigure[SWIFT58 Catalogue + Isotropic]
{
\includegraphics[width=6.6cm]{auger_mock_iso_SWIFT58.pdf}
\label{fig:skymapSWIFTiso}
}
\caption{Skymaps, accounting for the Pierre Auger Observatory non-uniform exposure, of simulated UHE protons produced by nearby sources (within 200~Mpc) experiencing deflections due to an intervening extragalactic magnetic field. Galactic coordinates are shown. 2MRS (\ref{fig:skymap2MRS} and \ref{fig:skymap2MRSiso}) and SWIFT-BAT 58-months (\ref{fig:skymapSWIFT} and \ref{fig:skymapSWIFTiso}) are considered. See the text for further details.}
\end{figure}
As a final application, we simulated protons from real candidate sources in the nearby Universe, with distance between 4 and 200~Mpc. We included the effect of deflections due to an intervening Kolmogorov-like extragalactic magnetic field with r.m.s. strength of 2~nG and coherence length of 1~Mpc. Moreover, we consider the case of absence of isotropic contamination and the case where simulation are contaminated with 56\% isotropic events, according to recent measurements of the Pierre Auger Collaboration \cite{auger2010correlation}. The resulting skymaps of simulated events, as they would be observed by accounting for the non-uniform exposure of the Pierre Auger Observatory, are shown in Fig.\,\ref{fig:skymap2MRS} and Fig.\,\ref{fig:skymap2MRSiso}, for candidate sources of UHECR from 2MASS Redshift Survey \cite{huchra20112mass} with magnitude ranging from -27.5 to -9.8, and in Fig.\,\ref{fig:skymapSWIFT} and Fig.\,\ref{fig:skymapSWIFTiso} for active galactic nuclei from SWIFT-BAT 58months \cite{swift2010}.
Although a deeper analysis of correlation and intrinsic clustering is out of the scope of this paper, the results show how HERMES can be used for such purposes. Moreover, it is possible to investigate the compatibility between simulated scenarios and observation by coupling HERMES with other methods. For instance, it is possible to quantify the clustering signal in the arrival direction distribution \cite{dedomenico2011multiscale} or to perform multi-messenger analysis including photons propagated with EleCa \cite{settimo2012eleca}. Another interesting application is to use the parameterization based on the generalized Gumbel distribution \cite{dedomenico2013reinterpreting} to perform detailed mass composition studies, as comparing the expected first and second momenta of the $\text{X}_{\text{max}}$ distribution from different scenarios against observations.
\section{Conclusions and outlook}
Realistic simulations of the propagation of UHECR might help to shed light on their origin and their nature. In this work, we presented HERMES, the \emph{ad hoc} Monte Carlo code we have developed to propagate UHECR in a magnetized Universe. We have briefly discussed the theoretical framework behind HERMES, involving the modeling of cosmology, magnetic fields, nuclear interactions between UHECR and relic photons of the extragalactic background radiation, and the production of secondary particles. The distribution of sources, their intrinsic luminosity, injection spectrum and evolution are tunable parameters in HERMES, allowing to simulate a wide variety of astrophysical scenarios and to investigate the impact of propagation on physical observable as the flux, or the chemical composition observed at Earth.
We showed some representative applications validating the suitability of HERMES for astroparticle studies at the highest energies. More specifically, we estimated the surviving probability of UHE protons, the GZK horizons of nuclei, the all-particle spectrum observed at Earth in different astrophysical scenarios and the expected arrival direction distribution of UHECR produced from different catalogues of nearby candidate sources.
The major advantage in using HERMES is in its modularity, allowing high customization of involved physical and astrophysical parameters. In fact, it is possible, for instance, to add new models of extragalactic background radiations or nuclear interactions, according to up-to-date measurements.
In the near future, we will release a stable version of our simulator for public use and, in the meanwhile, we will make available for the community libraries of propagated nuclei useful for mass composition and energy spectrum analysis.
\begin{acknowledgement}
The author is in debt with the Pierre Auger Collaboration for the invaluable discussions and acknowledges the financial support of the Scuola Superiore di Catania, the Department of Physics and Astronomy of the University of Catania and INFN (Sez.\,Catania). The author would like to thank P.L. Ghia for invaluable support and precious suggestions, and H. Lyberis and M. Settimo for useful discussions before, during and after the realization of HERMES.
\end{acknowledgement}
\bibliographystyle{jcappub}
\providecommand{\href}[2]{#2}\begingroup\raggedright | {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,957 |
'Heartbreaking, powerful, clear-sighted. Powered by the clarity and force of a mother's great love for her children, and the author's reflective capacity honed by years of legal practice and research, this memoir faces fundamental questions about life itself. Hannah Robert experienced an immense tragedy and has given others a true and guiding light.'
Zoë Morrison
'Hannah Robert's _Baby Lost_ is a stunning rumination on the minutiae of loss and grief, and the epic struggle involved in putting yourself back together after unspeakable tragedy.
_Baby Lost_ is a courageous and beautiful memoir. With devastating honesty, Hannah offers up her grief, presenting her story with a deft, insightful touch, allowing the reader to bear witness to a loss that is far too often unspoken.'
Monica Dux
'A gutsy, vivid and unflinching book about unspeakable loss. Hannah Roberts's book will resonate with anyone who has known the darkness of grief, and the gradually returning light.'
Hilary Harper
baby
lost
baby
lost
A story of grief and hope
HANNAH ROBERT
MELBOURNE UNIVERSITY PRESS
An imprint of Melbourne University Publishing Limited
Level 1, 715 Swanston Street, Carlton, Victoria 3053, Australia
mup-info@unimelb.edu.au
www.mup.com.au
First published 2017
Text © Hannah Robert, 2017
Design and typography © Melbourne University Publishing Limited, 2017
This book is copyright. Apart from any use permitted under the _Copyright Act 1968_ and subsequent amendments, no part may be reproduced, stored in a retrieval system or transmitted by any means or process whatsoever without the prior written permission of the publishers.
Lyrics by Katy Steele (Little Birdy) are reproduced with permission, via Graham McLuskie. Extracts from 'When Things Fall Apart' by Pema Chödrön are reproduced with permission, courtesy of the Pema Chödrön Foundation. Zainab's blanket, pictured on the cover, was knitted and crocheted by Joanna Robert.
Every attempt has been made to locate the copyright holders for material quoted in this book. Any person or organisation that may have been overlooked or misattributed may contact the publisher.
Text design and typesetting by Cannon Typesetting
Cover design by Klarissa Pfisterer
Printed in Australia by McPherson's Printing Group
National Library of Australia Cataloguing-in-Publication entry
Robert, Hannah, author.
Baby lost: a story of grief and hope/Hannah Robert.
9780522869439 (paperback)
9780522869446 (ebook)
Robert, Hannah.
Mothers—Australia.
Fetal death—Psychological aspects.
Grief in women.
Hope.
If the content of this book brings up issues for readers, for help or information call: SANDS (Stillbirth and Neonatal Death Support): www.sands.org.au
Stillbirth Foundation (A charity raising money for research and education on stillbirth and stillbirth prevention): stillbirthfoundation.org.au
An online community of baby lost parents, which includes resources on how to plan a funeral for a baby, how to help a friend through baby loss etc.: www.glowinthewoods.com Lifeline: 131 114
Contents
Part I: Impact
**1** Sunday, 27 December 2009
**2** The second-best blanket
**3** Shavasana
**4** The torture booties
**5** The crazy lady in ward four
**6** Frida and me
**7** Tabloid tragedy
**8** Permission to bend
Part II: Re-entry
**9** Zombieland
**10** The 'born alive' rule
**11** Sun salute with bedpan
**12** The posthumous godfather
**13** Matryoshka
**14** Histopathology
**15** Proof
**16** Scar tissue
**17** Funeral appreciation
**18** I have a dark-haired daughter
**19** Dr No-Sperm-for-You
**20** Heartbeat
**21** Making the judge cry
**22** Close up with hope
**23** The charnel ground
**24** Fat Tuesday
**25** Undone
**26** Tsunami
**27** Earth and sun
Part III: Ripples
**28** Both my babies
**29** Zoe's Law
**30** Holding the torch
Acknowledgements
Notes
_For Zainab and Mia and all the other babies gone too soon_
**Promises**
I will greet you with hands smelling of oranges.
I will kiss your mouth in your sleep.
I will let you surprise me
Over and over again.
I will curse that my hands can't bat away all the things that will hurt you.
I will remember—despite the shock—that no matter how many times I have dreamt you
You are your very own dream
From your very own flickering head.
I will breathe you in and mingle you with my familiar cells.
I will breathe you out and let you mingle amongst the hard and soft particles of the air.
I will bring you home,
And I will open the door.
And as much as I delight
In the still unreal thought
of seeing the light bounce
from your face onto mine
I will not hurry you.
(September 2009)
Part I
IMPACT
1
Sunday, 27 December 2009
There is only one place to start with this story—the point where all the ripples start, the moment of impact. Everything circles around that.
I replay this moment often. There we were, buckled in and travelling north on a suburban arterial road at around 5.40 p.m. two days after Christmas. We were not a conventional family for all kinds of reasons—two mums (one Lebanese, Rima; one a 'skip', me), with Rima's teenage daughters from her previous marriage (Jackie and Jasmin), and our long-awaited donor-conceived baby on the way—but it was the most ordinary of family car trips. We were heading home in the station wagon after visiting my cousin to drop off belated Christmas presents. I was driving, with Rima next to me in the front passenger seat; Jasmin in the back seat on the left, reading her book; and Jackie behind me. She had been leaning on the window gazing out, but leaned forward to ask Rima something.
We'd been listening to the cricket, and I said to Rima, 'Hon, can we change this? Listen to some classical music for Haloumi?' Haloumi was our name for the baby that bulged in my eight-months pregnant belly, that had been hiccuping all morning.
But Rima didn't reply, and didn't change the station, because in front of us we could see exactly what this moment was—in the shape of a four-wheel drive, which had hit the car in front of it in the southbound lane and was now swinging sideways onto our side of the road. I'd started an annoyed query, 'What is he doing?', but finished with a yell, 'FUCK OFF!!'
And I braked. I pushed with my arms and my legs, and the tiny hairs on my arms and legs, to try to push that car away from my family and me, and the little one curled in my belly.
The impact smacked two moments—before, and after—together so forcefully that I was left puzzling about what they were doing next to one another. All I know of it was its loudness, and the shudder it left in our bones. We know it happened, we had the evidence before us, in torn metal pieces and CT scans, but it was too quick—too much to fit into one tiny moment, so that everything broke, and the normal boundaries of our lives split apart.
We stopped moving instantly and I could still hear myself yelling and thought, 'Too late for that,' and shut my mouth so hard that my teeth chipped against one another. I made a decision—this was actually happening, and since I couldn't undo it, I'd better deal with it.
I turned the engine off and looked at Rima—lovely, alive Rima, though she was screaming too by now. I could hear the girls screaming behind us, and though I couldn't turn and see them, I knew they would be hurt but okay.
I looked to my right, where the four-wheel drive had come to a stop, as if we were just parked cars in some wrecking yard. A clear liquid was gushing from the other car's mangled engine. I thought, 'If that's petrol, we could be blowing up any moment now.' I had visions of an action-movie scene—a billow of flame, and bodies moving in slow motion. I couldn't move—the car was crushed in around my legs. 'Rima, get out of the car. Tell the girls to get out.'
Later, in the hospital, Rima mused, 'I opened the car door, but then I realised I was too hurt to actually move. Why did I open the car door?'
'Because I told you to get out. Because I thought the car would explode.'
I felt calm. I drew great draughts of air and tried to send some of that calmness towards Rima, who was still screaming. My thoughts sliced through the slow-moving time around us. If we could just be calm and reasonable, it would all be okay—the ambulances would come, they would unfold this car around me, my baby might have to arrive a little early but would be okay. Thirty-four weeks—this child would already be so strong. 'Viable'. _Isn't that right, Haloumi_?
•
Seven months before, on a Friday night, I had got off the train from Newcastle, and walked up the hill from Central Station in Sydney's gritty heart and into the pub where Rima was having work drinks.
'Hannah!' Nan and Veronica beamed at me, arms open. 'Here she is!' Rima turned and gave me a bigger smile and a tighter squeeze than usual. And then, in my ear, 'You still feeling nauseous?'
'Yep—still queasy.'
'Good—I'll get you a lemonade then!'
We hugged our secret to ourselves; it was still early days. But we were each allowed to tell one person, and Rima's was Chantal. She found us later that evening, gave me a big hug and whispered, 'So! I hear there's a Mazloumi-haloumi on the way! A baby haloumi!'
I bit my lip and a smile split across my face. 'Yeah—just a tiny little haloumi cheese so far, but definitely a little haloumi!'
I made cryptic Facebook posts: 'I love haloumi cheese' or 'haloumi in my belly!' I fretted about the logistics. Two-and-a-half years before, I'd made the leap from commercial litigator to lecturer at Newcastle Law School—I loved my work, but commuting from Sydney was complicated. I'd just received an offer to move to La Trobe Law School in Melbourne, where I'd grown up, and where my family would be close by. We'd decided on a move, but there was nothing simple about uprooting ourselves.
•
Where the impact had strangely calmed me, it had done the opposite to Rima. She was sobbing, 'My children, my children.' I held her hand; ' _Habibi_ , please, we are going to be okay.'
'Can you feel the baby move?' She looked at me hard, and asked the question again. I didn't want to answer and engage with the universe of doubt that surrounded it.
'I don't know, my love. I've got a few other issues to think about right now.'
I listed these in my head, concisely and calmly: explosion, being cut out of the car, whether my legs or spine were crushed, Jackie, Jasmin. Inside my body felt calm, safe—it was the outside that was in trouble. Don't worry, Haloumi, I'll get us out of here.
•
When we sat in the obstetrician's office six weeks later for our review appointment, I asked him about the heartbeat the paramedic said he had heard in the ambulance. 'Is there anything on my file about that? Could she still have been alive in the ambulance?'
It took a good twenty minutes on the phone for him to get to talk to the person in charge of the medical records department.
'There's nothing on your file about a fetal heartbeat of 155 in the ambulance. We'll probably never know, but I have to say it took me many years of practice as an obstetrician before I could accurately measure a fetal heartbeat with a stethoscope, so there's a good chance they got it wrong. And, from the look of your placenta when we did the caesar, it had completely abrupted, probably very quickly on impact.'
I sat there looking at the dots on his bow tie and wished I could slap my coolly calm self as she sat in that wrecked car and say, 'Your child is dying right now—anything you want to say to her, you need to say it now.'
•
In that calm space, it didn't take long for people to come to us. A man appeared at my window. 'I'm thirty-four weeks pregnant,' I told him matter-of-factly.
He said, 'Here, hold this to your head,' and put a cloth in my hand, pressing it against the side of my head above my right ear. It didn't hurt there; it was just warm and wet. He was already on the phone. 'Two women, one of them is thirty-four weeks pregnant'—looking at me for confirmation.
I nodded.
'Are you in pain?'
'I'm okay; I just feel squashed. I can move my toes but I can't get my legs out. I need to be cut out of here.'
I looked down—my legs were pulled up protectively around my bulging belly, my toes flexed a bit further back than I had thought possible. Metal and plastic were bent around my legs, but they felt whole and okay, just trapped. My toes were obediently wiggling—painted toenails (for Christmas), new bronze metallic Birkenstocks bent at angles. 'I'll have to get new Birkenstocks,' I thought.
He kept moving around the car, relaying Rima's injuries to the operator, then Jasmin's, then Jackie's. Another guy, younger, came to my door. 'Are you okay?' he started, and then said, 'Oh fuck.' I could feel the panic sweating off him as he looked at me and the car bent around me. I turned away and looked at Rima instead, letting my calm roll towards her.
•
In the trauma ward afterwards, I listened over and over to a particular song by Little Birdy:
I haven't seen no place like this
I haven't seen no place like this
No one will see, no one will see, what I do now, what I do now, oh it's just us moving
I haven't seen a place so ghost-like
a place that's seen some of the best in my eyes
Pages will turn, sirens will sing, words will be said, words that will hurt,
oh it's just us drifting
I was stuck there, drifting mid-impact, in the moment where my daughter's whole lifetime folded concertina-like into nothing, where she became a ghost, and perhaps I did too. My body suddenly contained life and death at the same time, like babushka dolls nestled within one another. In that eerie place, the sound of ambulance and fire engine sirens is stuck on repeat; time stretched out, to create a new reality, abruptly disjointed from our previous one.
The first time I heard sirens after being released from hospital, my body shook with sobs before I could register what was happening. The taste of nausea on my tongue, a feeling of my blood draining out through my legs, my stomach dropping sideways.
•
In the time it took for the paramedics to come, for the firefighters to bring the giant can opener to release me from the car, I breathed. I held Rima's hand until the paramedics took her, and then one of the firefighters got in her seat and held my hand. 'You're an ideal patient—very calm,' he said.
'I don't see any point in making this worse,' I replied. We didn't really need to make small talk. The others were working hard, concentrating on bending the metal without hurting my soft body. I kept my hand on my belly— _Come on, Haloumi, stay with me, little one_.
•
I was fierce about our little family; for so many years, I hadn't allowed myself to think it was possible. As a teenager, I'd stood in my school uniform at the tram stop, radiating shame as I thought about the dream that had woken me that morning. I'd dreamt I was struggling with a snake that grew bigger and bigger, and at the very moment I thought it would constrict me, it became a woman, and the struggle changed into something else, which made me gasp because it felt so incredibly good. _Oh no_ , I said silently, solemnly, to the mannequins in the shop window. _I must be one of them_. I couldn't even say the word 'lesbian' in my head. It was a taunt, an insult hurled after all the others had been exhausted. The worst of the worst. At the school I went to, you might as well tattoo 'Bully me' on your forehead if you were going to admit to anything but vociferous heterosexuality.
I'd had boyfriends, and genuinely loved, and sometimes desired, them. That was how I could recognise the power of these feelings and responses—though they had a distinctly different social value. Having a boyfriend had won me a level of acceptance, of approval; the feeling of growing up how I was supposed to grow up. These feelings, though, threatened to mark me out, to contaminate me as abnormal, unacceptable, clearly destined for a sad, lonely, embarrassing existence. No one I knew was in a same-sex relationship—not a cousin, teacher, family friend, no one on any of the TV shows or films I'd seen. (No, that's not quite true, there was the gay lawyer in _Philadelphia_ , who died.) I knew people 'like that' existed, but in a universe so shadowy and far from my own that I had no desire to go there. Yet, that morning at the tram stop, I confronted the solemn knowledge that wherever that universe was, I was already an expatriate citizen whether I liked it or not. And the fact that I was having this imaginary conversation with womanly mannequins wearing foundation garments (no racy lingerie for Camberwell shop windows, thank you!) was only further proof of my guilt.
Fast-forward several years, to university, and I was having a very different conversation, at least with a live, human woman this time.
'Have you ever been in love?' she asked.
We were making noodles (literally, not metaphorically, I'm afraid) in her room at college. This was what happened after beers and dancing if we didn't a) pass out or b) pick up boys. I preferred option c), even if I had to run the risk of a) or b) to make it a possibility.
I was carefully flippant in replying: 'Oh, I'm always in love with at least one person at any given time.'
She hopped into bed with the bowl of noodles on her lap, and patted the space next to her. I squished in and stole some noodles with my fork, acutely aware of the warmth of her leg parallel to mine on the bed.
'Really? So you're in love with someone now then? Who?'
I imagined a voice bubble that said, 'You, silly!' then carefully dismantled it and said, 'Oh, no one you know.' She pressed me further, and I came up with a boy in one of my law classes who I strongly suspected was gay. 'Anyway, he doesn't seem interested. What about you—have you ever been in love?' She said no, she didn't think so.
Before we finished the noodles, I asked, 'What would you do if your brother or sister were gay?'
'But they're not. You know them—there's no way either of them are.'
'No, but just _say_ —what if one of them were?'
'But they're not, and they never would be.'
I sighed. I couldn't tell whether she was being stubborn or misunderstanding me. 'No... I mean, just imagine a hypothetical brother or sister, not J or C but another imaginary sibling, who was gay—how would you be?'
She got up and crossed the room to find her toothbrush. 'This is a stupid question—who knows how I'd be with a hypothetical imaginary sibling, anyway? I'm tired; I'm going to go to bed.'
It was a few months later, when I woke up alone and disorientated in bed after a big night out with no memory of how I'd arrived there, that I realised no amount of drinking was magically going to turn her gay, or make me brave enough to make a move. It was time for me to face up to the heartbreak, move out of college, and find a woman who might actually be interested.
The other factor, apart from my cowardice, that had prevented me from coming out was the knowledge in my gut that I wanted to have children. I was tortured by the thought that I had an impossible choice before me: to be true to myself, or to have the children I longed for. But other women were already challenging that impossible choice. In the Queensland Anti-Discrimination Commission, in early 1997, a lesbian couple had succeeded in challenging laws preventing them from accessing assisted reproductive services, as had several de facto couples in Victoria. I walked through campus with this revolutionary information rattling around my head. Maybe I didn't have to choose. Maybe I could be with someone I loved and desired _and_ have a child with them. My heart felt as if it had grown wings and was flapping through the sky a few metres ahead of me, and I smiled a big, goofy smile to myself.
•
The emergency workers surrounding me—levering the car open, cutting metal, lifting me out—were so diligent in their work, it was as though I were some ancient Etruscan vase being extracted from an archaeological dig. A screen like a photographer's reflector was fitted around the shattered windscreen so the glass didn't hit me while they prised the car open. The world folded in around me, narrowing to this small space and the faces that came into it. I held their solid arms while I was lifted, like a circus girl being passed through a hoop, letting my eyes focus on the heavy blue cotton weave of their overalls. I was strapped to the smooth plastic of a spinal board and slotted into the ambulance, a paramedic still holding my hand.
While we travelled, sirens blasting, I asked the paramedic whether he could try to find a fetal heartbeat. There was no Doppler machine in the ambulance, but he tried with a stethoscope. Things were hazy but I clearly remember the number—155 beats per minute. It confirmed what I thought I knew: Haloumi would be okay. I repeated that number to the doctors when I arrived in emergency— _155, 155_.
•
Before that little heart started beating, there was just a tiny dot—a scarcely believable little thing somewhere below my belly button. And before that, the sticky plastic cup our friend, and sperm donor, Jorge left on our dresser on a Tuesday night at the beginning of May.
'We're having ice-cream; do you want some?'
We were all a bit awkward. This was the first time we'd tried a fresh donation—all of our previous eight months' worth of attempts had been at the hospital, using his frozen samples.
'No, I'd better get on home; this isn't a social visit this time.' Jorge smiled sneakily, kissed us and left, his magazine under his arm.
•
When we arrived in emergency, I was parcelled from ambulance to examining table in a series of clicks, rolling wheels, and an efficient one-two-three. Cool surgical scissors slid under my bra straps and up my trouser legs, slicing through fabric and elastic so that my clothes fell away, creating a clear workspace—my damaged body—for the nurses and doctors who moved around me. My limbs were distant, faces moved in and out of focus. I was asked my name, my age, today's date, what had happened. I repeated these facts diligently. But, like Alice wondering 'Do cats eat bats, do bats eat cats?', I started to wonder whether I was thirty-three years old and thirty-four weeks pregnant, or thirty-four years old and thirty-three weeks pregnant.
I could hear the doctors talking over in the corner. They had wheeled the ultrasound machine in, after every man and his dog had tried to get a heartbeat with the Doppler machine, and then another Doppler with new batteries. I knew that if they'd seen a heartbeat, there would be reassurances, smiles. I was still waiting.
'Okay, so that would be the explanation...' was the only bit of the conversation I caught.
I still had my hand on my belly, now sticky with ultrasound gel. They'd had to move my hand during the scans. They did it gently, and I edged my fingers back each time, feeling softly for those little heels. _Come on, Haloumi, now is your moment, my beautiful one. You weren't so shy at your last ultrasound, four days ago_. I didn't want to hear an explanation, only a heartbeat.
A doctor came to me and introduced himself. He had thick white hair, a bow tie, and worst of all, a concerned look. 'I understand you've been told?'
'No, I haven't.' _Don't tell me. Don't tell me_.
But I found myself saying helpfully, 'You haven't found a heartbeat, have you?' It mustn't be easy to have to break this kind of news to women. I felt sorry for him.
'No.' There were words after that, coming out of his mouth like a speech bubble—about being induced, about labour—but I couldn't match them with any meaning.
Someone had called my mum; I could hear her voice coming from the phone held to my ear. 'We've lost our Haloumi,' I said into space.
Rima had been sent to a different hospital. Finally, I could speak to her on the phone. ' _Hayet_ , Haloumi didn't make it.'
I could make these words come out of my mouth; I knew I had to say them, but that doesn't mean I believed them.
Once they'd established they only had one life left to save within my body, they started rearranging the various cords and tubes attached to me, so that I could be wheeled away for a CT scan. I asked, 'Is it okay to have this scan when I'm this pregnant?'
'It's okay now,' I heard.
I was arranged like a posable doll on the narrow table, still tilted so that the full weight of my womb and baby didn't cut off the blood to my legs. My belly sloped downhill—I asked to be strapped onto the table because I felt as if I could roll off at any moment. And then, suddenly, everyone left the room, and it was me and the futuristic white donut of the scanner. The table buzzed into motion, and took me slowly in and then out again, making silent and invisible slices through my marbled flesh.
•
When I'd done the pregnancy test the first time, I'd been a bit too enthusiastic and flooded it, so that it was impossible to get any reading from the test. 'You'll just have to be patient, and wait until you need to go again,' Rima laughed. I was working from home that day, but she had to leave to catch her train.
In the nine months that we'd been trying, my period had never been this late. After that many attempts, I'd tried to discipline myself about obsessing over possible symptoms, but this time felt different. When I'd played hockey the night before, I'd gagged with nausea when I put my mouthguard in.
With the second pregnancy test, I made little rules for myself. No sitting here and staring and staring at it. I would put it on the side of the bath and come back at the designated three minutes, not a moment earlier. Within two-and-a-half minutes, I was back in the bathroom. I stared at the test and then stared at the wall. _Oh man! Wall, this looks like two lines. Two lines!_ What a beautiful piece of wall was in front of my eyes, and I'd never noticed it before.
I couldn't get through to Rima, who was in a meeting. I left a hyperventilated message, and called my best mate from high school, Penny. 'I think it's positive!' She squealed, while I caveated my joy: 'I know this doesn't necessarily mean anything, I know it's early days, but it looks like it's positive. We've never had that before.'
Finally, I got Rima on the phone. ' _Hayet_ —I think there are two lines!'
I knew what our odds were at that early stage. We breathed out a little when my blood test came back positive; and exhaled properly a fortnight later, when we had our six-week scan, and saw a tiny, pulsing jellybean of a creature. I curled my neck up to look, and then looked at Rima. 'It looks like Yoda!' I said.
She grinned. 'Yep. And it's saying, "Pregnant, you are!"'
2
The second-best blanket
Where my memories of the immediate aftermath of the accident are acid clear, those first hours in the emergency department are morphine-clouded. Things jump from scene to scene. Suddenly, I was back from the CAT scan and my mum was washing blood from my face. 'I knew your dad would be coming soon, and he didn't need to see you like that,' she told me weeks later.
They had stapled my head wound, but it was still bleeding. 'She's lying in a pool of her own blood,' my mum said, matter-of-factly. 'Can it be re-stitched?' They re-stitched it twice before it stopped bleeding.
By the morning, my hair was bright red from the blood, and stood up in stiff curls where the blood had caked. It looked as if I'd been to some demonic hairdresser. Penny later confided that when she came to see me the next day, her first thought was that this was a bit of a funny time for me to be dyeing my hair.
I was reassured when a midwife, Jen, arrived and introduced herself. Here was something I was prepared for. 'At least I can give Haloumi a good birth,' I said. No tears had come yet. I still couldn't imagine this baby's face, alive or not. Everything was in the abstract until this child was born. I had been given a prostaglandin gel to start the induction process, but had no sense of how long it might take. While I waited, I thought of my friend Brigette.
•
Brigette was a family friend who unexpectedly fell pregnant when she was nineteen and got married. That alone was surprising enough, but then we got news that her daughter was stillborn at full term. I didn't send a card or call because I didn't know what to do. We'd always conducted our friendship in person, when we met up at the beach once or twice a year as kids and teenagers. Once we were adults, and she had moved interstate, I was unsure how to keep up the friendship and even more clueless about how to respond to her loss.
I made contact again about six years later when I was living in Sydney, and she came to visit me. She'd since had another daughter, had broken up with her husband, and was facing an acrimonious family law battle. I asked whether she sometimes wished she'd never gotten pregnant that first time. And she said, 'No. No, I'm really glad we had the time we had with Sacha—I'm glad she was my daughter.' I was surprised, because I had imagined that stillbirth was like a mathematical equation—plus one, minus one—leaving you more or less where you were before. She gave me an inkling that there was something completely unquantifiable about the loss, and the gain, which a stillborn child brought.
•
'This may be a silly question,' I said to Jen, 'but can I play the relaxation CD from our birthing course?' She nodded, and, without any fuss, my beautiful, practical sister, Erica, drove all the way across town and back to bring me the CD and CD player.
Rima and I used to joke that the key relaxation technique the birthing CDs provided was the giggle we got from hearing the hippy windchimes and the woman with a corny accent drawl out the word 'Aaaaaaffirmations'. I think of it now, and wonder that the medical staff were happy for my sister to plug the CD player into sockets used for lifesaving equipment, and to let the softly spoken platitudes of hypnobirthing wash across the hard surfaces of the emergency room. People humour you when death is this close by.
And, somehow, the affirmations worked. Somehow, I unzipped myself from the hospital drama scene, from all the tubes, monitors and bandages, and I was back, floating in the Coogee sea baths, cupping light-filled water softly in my hands, the salt water and sunlight washing against my belly. Haloumi was big then—it was only a week or so before we left Sydney. I remember thinking at the time, 'I need to hold onto this; this will be my good place to come back to when I'm in labour.' Not complicated, not difficult, just light so bright I could close my eyes and still feel it all around me, as though it were the water I floated in.
•
I first discovered the Coogee baths when I was staying in Sydney for an internship; before I had graduated, before I had torn away from Melbourne. Even then, I ached for a baby. That was the whole point of the internship—to get a graduate-lawyer position, so I could save up for a baby and, hopefully, find a partner in the process. I had ducked under the water and surfaced with new knowledge: that I would swim here pregnant. That knowledge latched tightly somewhere in my belly, pulling me towards Sydney and back onto that plane. I'd felt so vindicated when it came true, when I immersed my rounded body in the same salt water and felt a beautiful squirming that was not my own—a small swimming body within my own swimming body. It was an odd sense of infinity, of being one small bit of this Mandelbrot universe—a pattern within a pattern, with another, smaller, pattern inside.
•
The calm of the impact was still there, but now everything really hurt, especially my hardened womb. Jen was trying to monitor my contractions, which were lasting over six minutes each.
After the CT scan, and after the possibility of spinal injury was ruled out, they released me from the neck brace, which had been starting to feel like a sarcophagus. I immediately wanted to turn over, to rest on all fours, and my mum and the midwife started to help me, but I quickly realised that my body wasn't responding the way it used to. I couldn't move smoothly into the yoga poses I had practised every day, and that I had hoped would help me deliver this baby. My body was heavy, tight with quickly appearing bruises, and when I tried to bend my bandaged left knee, there was a queasy pain. I managed to sit on the edge of the bed and swing my legs down, but was afraid of collapsing forwards. Gently, Mum lifted my arms, with their vines of tubes and monitors, and ducked under them, so that I could hug her and melt into her shoulder.
I was so thirsty. I kept asking for water, but all I was allowed was chunks of ice from a polystyrene cup. I would roll the ice in my mouth, choke slightly, spit it out; or, if I was feeling particularly gutsy, crunch it into many small icebergs. Erica held the cup for me, quietly and calmly. This was a pain I needed to hold in my gaze, with my full concentration. I wanted to deliver this child, to see him or her for myself, before accepting any bad news from a fuzzy ultrasound machine. This pain would get me there.
'Hannah.' We'd been relatively unprodded for an uncountable number of hours, while I crunched ice and worked my way, hand over hand, through the pain. It was the obstetrician, back again. 'We'd like to do an internal examination again and check your progress.'
I was startled. 'The midwife did one not so long ago and she thought I was at about 3 centimetres.'
'I know, but we need to do another one, to see how things are going.'
It hurt, and he wasn't impressed with my progress.
'The problem is, with all your other injuries, this needs to happen pretty quickly. I think we're going to need to do a caesar.'
•
'The problem,' the obstetrician explained six weeks later, 'was that we were concerned about you going into a state called DIC—disseminated intravascular coagulopathy. It's when you've got a big bleed—like the placental abruption—which pulls in all the clotting agents in your blood, so that the rest of your blood loses its ability to clot.'
'Which is a bad thing,' I added helpfully.
'Yes, particularly when you have other wounds. No matter how much blood we transfuse in, you can still bleed to death. We were testing your clotting factors to monitor you for DIC, and by about 3 a.m. they were starting to fall, which meant you were starting to go into DIC. Those six-minute contractions were probably not contractions. It's more likely that they were an indication of Couvelaire uterus—bleeding into the wall of the uterus. If I was dealing with your case again, I would send you straight for a C-section.'
•
Suddenly, I was being prepped for surgery.
'This is very unlikely, but there's a possibility that we may need to do a hysterectomy.'
He saw my face, and moved from disclosure into reassurance mode.
'We'll try to do a lower-section caesarean, but if we have concerns about the liver and spleen bleeding, we may need to embolise them, and that would require a bigger incision. At this stage, we're hoping we can conservatively manage the liver and spleen bleeding, and not operate on them.'
'I only have one kidney,' I announced to the obstetrician. I'm sure I'd already told that to someone in the emergency room, but I wanted to make sure he knew now, so that he didn't just have to guess from my silent, scarred body.
'Why was your kidney removed?' He was genuinely interested. This was nice, but also a little bit worrying.
'I had an ectopic ureter, so I kept getting recurrent kidney infections. I had surgery here,' I reached around my still-big belly and touched a thin scar just above my pubic bone, 'to fix where the ureter went into my bladder, and then they went to remove the damaged bit of my kidney.' I touched my side, where a thick scar ran below my ribs. 'But when they operated on my kidney, they accidentally hit an artery that wasn't supposed to be there, and I lost a lot of blood. So they just took the whole kidney, and closed me up and did a blood transfusion.'
'When was this?'
'In 1983. I was seven.'
What I didn't say was _Please be careful operating on me_.
Before they wheeled me into theatre, Jen asked if we wanted to pick an outfit and blanket for the baby. It still seemed very unreal. My sister had brought in a bag full of baby things that my mum had made or bought for Haloumi. My hand hovered between two blankets Mum had knitted. One was just too lovely to be cremated—I wanted to save it for a living baby (callously, I think now)—so I picked the other one.
•
There was no night on Sunday, 27 December. When I closed my eyes from the general anaesthetic, it felt as if only a moment had passed before I was opening them again; but this time, the weight and pain of my pregnant womb were gone. My fingers crept to my side—my belly felt soft, flat, bandaged. For a second I felt relief, and then remembered why I felt so much lighter. A face moved into my field of vision and came close enough to be less blurry. It was Jen. She handed me a photo. 'You had a little girl.'
Everything else faded into background and all I could see was her—our daughter—with her round cheeks, dark hair and a pointy little Rima-chin. Our daughter, wrapped in the second-best blanket, which instantly became the best, most beautiful blanket for being wrapped around her. I couldn't be stingy with my love now that she was born; I couldn't hold off on loving my child because she was no longer alive.
3
Shavasana
We had one day with our baby daughter—Monday, 28 December. One day to name her, to give her a bath, to hold her and sniff her head and memorise every millimetre of her. People asked me questions while she was in my arms, but their voices sounded distant; I couldn't look away from her face. She had a tiny bruise on her right eyebrow, lots of dark hair, big chubby cheeks, a serious, expressive mouth. I could imagine teenage attitude coming out of that mouth. I unfurled her little fist. A small, strong hand.
Before I had my own dead baby, I couldn't imagine anything more macabre than a dead child. A dead baby was a plot device, pure metaphor, something frightening precisely because it was so unthinkable. But my dead baby, you would have loved her! On any objective scale, she was clearly the most beautiful baby girl ever born. She had a world-weary, uber-cool way about her. Not for her the corporeal indignities of living, screaming, pooing babies. Here she was, the mysterious one who'd been kicking me all along, now terminally mysterious and unknowable, yet so specifically herself. Birth is the 'big reveal' of motherhood, when a hypothetical, potentially generic baby becomes your own child. And yes, my heart split forever, with her part of it not setting off to walk around outside my body but heading for a stupidly small coffin and a private cremation.
My brother, Jeremy, arrived just as I was being prepared to be sent from ICU for another CT scan to check on the internal bleeding. He brought with him the soft grey rabbit he and his fiancée had given us for our baby at Christmas. I entrusted both rabbit and baby to my mum and the midwives, as the orderlies wheeled my bed out the door and towards the lifts. Jeremy took my hand, and stayed with me all the way down to the scanner. He is known for being the least chatty in a family of extreme chattiness. But there was nothing awkward about being quiet with him. I was reluctant to relinquish his hand when I was moved onto the bed of the scanner, but knowing he was there, solid and waiting for me as the machinery sent me into the tunnel of the CT scanner and out again, enabled me to breathe quietly. When I emerged, I took his hand again for the journey back to ICU—my new alien, but life-sustaining, universe.
•
On Christmas night, I'd driven to the airport with my sister and mum to pick up Jeremy. He was back from Germany because he'd been offered some lucrative rigging contracts. His German fiancée was having knee surgery and rehabilitation, but would soon follow him to Australia. We were staying at my dad and stepmum's house over Christmas, before we moved into our new house after relocating from Sydney. I had missed my family so much in my seven years living in Sydney, and now it felt like things were gathering in again. We'd farewelled Jez early on Boxing Day morning when he caught a lift with a friend, heading down to work at the Falls Festival on the coast. He had hugged me, then ducked down to give my belly a quick peck. 'See ya, Haloumi.'
•
'Have you named her yet?' more than one person asked me. I know they were being kind, but the question and being hurried irritated me. I couldn't name her without Rima being there and, despite my pleading for her to be transferred, she was still in another hospital, being treated for broken ribs and a broken hand, and awaiting release.
'Haloumi' had worked well as our nickname for a growing, kicking bump, but the quiet baby girl in my arms needed a real name. We couldn't keep referring to her as a type of cheese. In the busy weeks before the accident, in our preparations for Christmas and the move to Melbourne, Rima and I had chewed over the topic of baby names many times. We were settled on a boy's name, but girls' names were more difficult. We wanted an Arabic name, a connection to Rima's Lebanese heritage, but so many of the names I loved provoked the response from Rima, 'Oh, _that_? That's an old lady's name! We can't call Haloumi that!'
At last, Rima appeared at the door of my ICU room, wearing her brother's tracksuit pants and a hospital gown, her broken hand in a sling. She had discharged herself from the Alfred. I was holding our still-unnamed baby in my arms. 'Rima, come meet your daughter,' I said. 'She looks like you, _hayet_ —look at her chin.' So much had happened since the ambulance officer had prised her hand out of mine so that he could remove her from the wreckage. All our haggling over baby names was irrelevant—here was our daughter and she needed us to name her.
Zainab was the first name that tumbled out of my mouth when I asked Rima what we should name her. It was one of the old-lady names we'd argued about, but now Rima said, 'Yes, Zainab,' without taking her eyes off our baby's face. And I thought of our friend Izzy, the way she would squeeze us and say, ' _Kha-li-la_!' (my darling) like a Lebanese _tayta_ (grandmother). It felt strange to declare her name, when she would never get to use it herself. But there she was, our Zainab Khalila. It was a big name for a small person, and particularly for one we knew so little about; so, often I would shorten it and think of her as little Z.
•
Shavasana. It's a fancy yoga word for lying on the floor. It's also called 'corpse pose', because imagining yourself as a corpse is a good way to relax every muscle in the body. When I started going to yoga classes a month or so after being released from hospital, the yoga teacher would introduce shavasana, and the tears would run sideways across my cheeks and make wet patches on the yoga mat near each of my ears. I would lie there, slack faced, heavy limbed, and think of little Z, her face soft with the absolute calm of death. Before our accident, I'd been safely insulated from death. It was something that mostly happened to old people, and corpses were horror-movie props.
Now, on the yoga mat, I would test out what it felt like to be dead, not in a melodramatic or suicidal way, but because it was so odd to be alive when I could feel so little continuity between my life 'before' and 'after'. It was hard to shake the thought that maybe, somehow, I had in fact died in the accident, that maybe I was a ghost. Death had found its way into my body, had carried off its victim under my very ribs, yet here I was, fraudulently breathing in and out. I was so alien to myself that I wondered if the old Hannah had died, and a new one had been born, like Venus, as a naked adult woman, appearing not on a shell from the ocean, but on an emergency-room trolley, the doctors and nurses around me like the four winds.
•
When I'd first held Zainab, she'd been warm. After Rima had come, after we'd bathed her, marvelled at her toes, dressed her in a small suit with brightly coloured stripes, I held her again and she was noticeably colder. _We don't have much time, little one_ , I thought. Did we put a nappy on her? It is a silly, small detail, but it bothers me that I can't remember. The suit was soft, soft cotton and had little giraffe faces on the feet. At one stage, one of the midwives or social workers brought in a tiny pink dress with lace and bows—exactly the kind of thing we would never have picked for her. 'Would this fit?'
We politely declined, but once the woman left, I was disparaging: 'She may be dead, but we're not dressing her in _that_.' And we laughed the brittle laughter of the bereaved. I realised afterwards that it had probably been donated by a family who knew this feeling first hand, and I felt unaccountably mean for mocking their kind gift. Zainab, though, looked solemn; _Nope, she wasn't going to wear that, thanks!_ We felt wicked as we laughed. Guilty for allowing our faces to crack wide with laughter, disloyal for producing anything but tears. Still, it was a release—an emptying-out of the dark waters of grief, like weeping out loud together but without the fear that our sadness co-mingled would drown us.
Sometime after Rima got there, Jackie and Jazzie arrived. They had been discharged from the Children's Hospital—Jazzie with a fractured hip, and a suspected fracture in her wrist; Jackie with a badly broken nose, stitches in her lip and mild concussion. Jackie had been sitting in the middle row of seats in our Nimbus, designed to fold down to give access to the third row of seats. In the impact, the seat had malfunctioned and folded, tilting her forwards, so that her face hit the back of my seat, or her own knees; we don't know which. The ambulance had taken them to the Children's Hospital as unaccompanied minors, while Rima and I had been sent to different hospitals. (Though, thanks to an extraordinary coincidence, the one and only person they knew who was employed at the Children's Hospital—Charlie, a longstanding family friend—was on duty in the ER and ended up caring for them.) My stepmum, Debbie, got there as quickly as she could, and stayed with them all night. A week later, Jasmin presented me with a drawing she had done with her left hand (her right was in a cast) of her and Jackie in hospital beds, tears running down their faces, Debbie on a camp bed between them.
When they arrived in my ICU room, they were wearing blue scrubs from the Children's Hospital, as their clothes had been cut off too. Jazzie was in a wheelchair; Jackie's face was swollen from the impact, with surgical tape and gauze on her cut lip and broken nose. They had come to meet their baby sister. And to reunite us as the human contents of TAZ 012, together once more after being splintered apart across three different hospitals.
Here is our family portrait—me holding Zainab in the centre, Rima leaning in on one side, and Jazzie and Jackie leaning in on the other. Our faces were shiny with tears, swollen with bruises. My hair was a mass of blood-red Medusa coils. I was too bereft to remember to close my mouth or even to look properly at the camera, and Rima was unsure whether she was supposed to try to smile. Jazzie's eyebrows were incredulous; _Is this really happening_? (This is not how she thought she'd meet her baby sister.) But Jackie's gaze was level, direct, reproachful—not vindictive, but heavy with truth. _This. This is what happened_. I want the person who caused our accident to experience that gaze. Or, really, for every driver to see it as they lay their hands on a steering wheel.
Jazzie, then Jackie, cradled their little sister in their arms. It was difficult with a broken arm. We took a photo of Zainab surrounded by our clasped hands. Her lips had gone a dark red, as though she were wearing lipstick for newborns. 'It's the delicate skin on her lips drying out,' a nurse explained. It meant she was looking less 'sleeping newborn' and more 'funeral photo'. I loved her still, but I was glad we'd got some photos early on.
I am not religious, but in Rima's faith, it was important to hold a funeral as quickly as possible. We planned something small. When Penny and I were in Year Nine at high school, Penny had confided in me that her mother, a hospital chaplain, often had to plan funerals for babies. As teenagers, we treated this as a weird horror-movie fact, remarkable mainly for the reaction it could elicit. I hadn't considered what it felt like for the parents of those babies. I asked my dad to contact Penny and her mum, Judith, to ask for their help. I imagine it is very hard to write a eulogy for a baby who has died before she is born, but Judith did that for us.
Penny came and sat vigil in my ICU room. It was her hands, along with my mother's and my sister's, that helped Rima and me give Zainab her first and last bath. ICU only allowed immediate family to visit, so we told them she was my sister—and, in a way, it was true.
We had been inseparable as thirteen year olds, writing long notes to one another in class, borrowing lines and characters from our favourite movies, songs and books. We'd later stolen one another's boyfriends and become estranged twice, but each time we reconnected through writing letters; a sorority in words that built a shared history bigger than any betrayal. In Year Ten, after I'd written something particularly angsty, my English teacher suggested I read _Cat's Eye_ by Margaret Atwood. 'It's one of the Year Twelve texts,' she explained, 'but I think you'll like it.' Somehow, it became a book whose storyline will always be entangled with my high school years, when both Penny and I read and re-read it, and with our friendship. Like Elaine and Cordelia in _Cat's Eye_ , Penny and I had lost each other for a while, but, unlike them, we found one another again and were kinder to one another than we had been as teenagers.
That night, Rima slept on two chairs pushed together next to my ICU bed. The next morning, she and my dad sat on the bed and I dictated a death notice for the paper. No one argued about the wording. I wondered aloud whether it should go in the 'births' or 'deaths' column, or both. Strictly, it was a double bill, and I wanted her birth _and_ death acknowledged, but I conceded when Dad said the funeral home would probably just put it in the 'deaths' column.
To prepare for the funeral that afternoon, the ICU nurses tilted my bed so that my head was lowered, and washed the blood from my hair. Warm rivulets ran up my scalp, as though I were showering upside down. The nurses washed gently around the wound on the right side of my head. They rinsed all that bloody water from my hair and rinsed it again, then blow-dried it; not the way a hairdresser does, but the way you do getting ready for work in the morning. It wasn't styled, but it was better than the Medusa coils.
My sister had been to see our car in the wrecker's yard and to collect our belongings from it. There, folded neatly on the dashboard, were my glasses. In the chaos of the emergency department and the fog of the morphine, I hadn't even realised they were missing. I was relieved to discover that the closed-in focus wasn't just my shell-shocked brain but also my own ordinary short-sightedness. I was also reunited with my handbag—such an ordinary object, but it was something from before, and its familiarity made our new circumstances feel all the more alien. I opened my little mirrored compact to put on some make-up, fuzzily remembering my routine, usually undertaken on the train to Newcastle. First, the silvery grey eyeshadow along the lid line, then getting lighter from the outer corner. Then the pearly cream on the inner hollow and under the brow. Finally, a lip gloss. The mirror only afforded a view of small slots of my face, which was just as well.
For the first time since they had cut my clothes off me in emergency, I wore clothes rather than hospital linen. Rima had brought in the nightie the girls had given me for Christmas—soft cotton, with pink and green florals, and (as requested) buttons at the front that would be practical for breastfeeding a new baby. Perhaps a bit more nanna than my usual style, but beautiful because the girls had chosen it for me. Still, nighties weren't quite funeral attire, even in these circumstances, so I needed something else over the top of it. My sister brought in a black top of hers and we stretched it gingerly over my head, cutting the waistband with scissors to get it over my post-partum belly.
I was to be transferred into a wheelchair with my broken leg raised in front of me. There was a moment of consternation when we realised the leg board I would sit on and that would support my broken left knee, held straight in a splint, was designed to support a right leg. A nurse was about to take it back and change it over, when we all realised, sheepishly, that it was the same on both sides; we just needed to flip it over and it could support a left leg.
A new ICU nurse—a young woman, Janelle, with a long, dark plait down her back—began her shift just as we were heading downstairs for Z's funeral. She was a watchful guest, sitting in the last row of seats. Our other guest at the funeral was the Le Pine woman, who was going to take our daughter to be cremated. She waited patiently after the short service while we said our goodbyes. Being in the wheelchair, I couldn't lean over to kiss Zainab, so I pulled the whole casket onto my lap and kissed her cool cheek, already wet with our tears. 'This will have to do,' I thought. This will have to suffice for all the child-care drop-offs, the first day at school, have-fun-at-camp goodbyes, airport-departure-lounge hugs—so many smaller farewells exchanged for this big one. 'Please take care of her,' I said to the Le Pine woman, and she squeezed my hand.
Afterwards, we sat in the hospital cafeteria. Numb. I drank real coffee, I chewed through a significant piece of cake, without tasting either. My sister came back to the table, and laid a black-and-white notebook in front of me with a pen. 'Write,' she said. 'It's what you do.'
4
The torture booties
That second night in ICU, it was just me, the ICU nurse and the torture booties. Rima had gone home at my insistence; her ribs were hurting her, and another night sleeping on chairs wasn't going to help. The torture booties were a clever invention designed to prevent me from dying of a blood clot, as I wasn't able to take anti-clotting medications due to the internal bleeding. They were strapped to my feet and legs, and inflated and deflated on a timer providing 'intermittent pneumatic compression'. What it really felt like was a robust whack to the soles of my feet every four minutes or so. Four minutes seemed to be just enough time for me to start drifting off to sleep before— _whack, whack_ —they went off again, and I jerked my broken knee, so that the staples pulled and the bone ached.
The new notebook from Erica was out of reach, but by the red glow of the heartbeat monitor on my finger, I rummaged in my handbag on the nightstand and found my old yellow notebook.
•
I had bought the yellow notebook on the Newcastle campus the week I'd got the phone call telling me my pregnancy blood test had come back positive. The nurse had been deliciously deadpan. 'Yep, it's all fine.' I had to prod her to get the exact level (of hCG or human chorionic gonadotropin, the pregnancy hormone, in my blood) and then for reassurance that the test was positive. I felt like Alice in Wonderland after she ate the mushroom, putting her hand on her head to work out whether she is shrinking or growing. My body was out of whack, unpredictable in a way it hadn't been since I was a teenager. I had been waking up in the early hours of each night since we'd begun to have real suspicions that this might be it—at first I thought it was nerves and over-excitement. But incessant googling also told me that the surging progesterone of early pregnancy tends to make you tired during the day and interrupt your sleep at night, so maybe that was playing a role.
I found an image of Leonardo Da Vinci's Annunciation on the internet, and added small speech bubbles:
Archangel Gabriel: 'You've got a hCG level of 2063!'
Mary: 'So, does that mean a yes or a no?'
•
With cannula-punctured hands I opened the yellow notebook carefully, holding in all the pieces of paper documenting this pregnancy: ultrasound results, blood tests, brochures for prenatal yoga. I opened it to a new page, after my notes from our birthing class, and started to write.
Tuesday, 29 December 2009
I look like heavily discounted supermarket stock—sticky from all the previous price stickers, and leaking in unexpected places. In ICU, they don't bother with the neat little patient ID plastic bangles—they stick your barcode ID directly onto your skin, securing it with a clear plastic dressing, like a piece of clingwrapped ham. My limbs feel foreign to me. My arms are bandaged and puffy with bruises, I'm only just reclaiming them for my own use. My legs I haven't quite remembered yet. They belong in the land below sheets. Somehow, I feel like I've just been born—uncertain of the sensations assaulting me and reliant on others for my basic needs. I am tentative about my body. It doesn't feel quite mine again yet. Indeed I'm not really sure whether I haven't been completely reborn with a new body that I will have to learn how to use again from the beginning.
I dreamt that the sun was rising as the pieces of a shipwreck floated into a beachy shore, the water sparkling innocently where only the night before it had been a violent breaker of things and bodies. Uneven chunks of wood were gently tipped over and over along the sand by foamy waves. The sea isn't malicious—it is just the sea and the weather is just the weather. It would be pointless to expect fairness.
I woke again with the _whack_ of the torture booties, and all that salt water spilled over into sobs—for my sore knee, for my tiredness, for my baby girl. Janelle, my ICU nurse, came, apologising that she couldn't turn the boots off without doctor's orders. She dragged over a stool so she could sit by the bed, took my hand in hers and asked me about the accident. I told her my opiate-smudged story, and wept while she leaned her head against the bed and listened, until I couldn't cry anymore and we both dozed off, only to wake again with a start when the torture booties went off.
When I woke up next, it felt like a proper awakening, one where a decent chunk of sleep had happened in the interim. It was nearly morning, and the new nurse was muttering to herself about faulty equipment and what the doctors would do if they saw that the compression booties had somehow been turned off in the middle of the night.
5
The crazy lady in ward four
The day after Zainab's funeral, I was transferred to the trauma ward. I was wheeled—bed and all—out of ICU, past unmoving patients attached to life support machines or breathing apparatus, past their worried relatives. I was being returned to the land of the living, but frail, bruised and dependent on others for everything. I had my first shower, I took my first steps (with the help of the hospital physio and a Zimmer frame), I ate my first hospital meal. Everything was new to me, everything felt different. I joked that instead of having a baby, I had become a baby. I had anticipated night wakings, changing nappies, first steps—just not my own. But here I was, being the helpless one needing assistance to move, to go to the toilet, to wash, to eat.
My life took on the pace of hospital routines. Big Tony came at 7 a.m., with a clean jug of iced water and a clean glass. He would take away the rubbish, and carefully attach a new bin liner to my hospital table with a bulldog clip. He moved quietly, trying not to wake me if I was sleeping. If I was awake, he was the first day-staff person I would see—a welcome reminder that the long, stretching night was over, that breakfast was coming and, soon after that, Rima, Mum, Dad or Penny. We made light of the hospital meals. When I grumbled over a rubbery croissant, Mum disappeared into the bathroom, and reappeared with a kooky grin and the hospital hair-dryer in hand. She plugged it in, and crisped the pastry up nicely.
In the evening, it was Little Tony bringing a new jug of iced water—sunset, in the hospital universe. He had similar hair and features to Big Tony but was as short and thin as Big Tony was big and wide. I wondered whether they were related.
I had a room to myself, with a bathroom, but there was nonetheless a curtain between my bed and the glass-panelled door. People would dance attendance behind the curtain, tweak it back with a finger, stick their heads around it to comical effect. When I needed to get to the bathroom, I would banish visitors behind it; things worked better without an audience.
During the day, the social worker came to see me with the Transport Accident Commission paperwork. She took me through the claim form, and I was surprised to find injuries listed that I hadn't known I had. The next time the doctors did their rounds, I asked about these extra injuries and discovered that some were incorrect (no fractured vault of skull, hooray) but some were correct. I had thought the ache in my breast was my heart breaking, but it turned out it was my fractured sternum.
The physio came to teach me how to go up and down stairs on crutches. We put her off for a day. When she came back, I was still weary. 'There are some steps at the end of the hall. We can practise there,' she said, enthusiastic and way, way too healthy looking. I shook my head. I was still not prepared to acknowledge the existence of the other end of the hall. The thought of stepping outside my own room made me swallow with fear. Resolute, she came back with a wooden step, so I could practise right outside my room. She helped me past the curtain and out the door, where lino-floored corridor stretched into the distance. I focused on my legs, which were sticking out from the bottom of my nightie, the left clothed in a red and black velcro splint, and both puffy with ripening bruises.
'Okay; so, when you're going up a step, you need to start with your good leg, then the bad leg, then the crutches.' We did a few steps, before I couldn't help but object.
'This leg isn't bad—it's just injured. And my "good" leg is sore too. Can we call them something else?'
'Um, okay. I guess so. What are you going to call them?'
I breathed in. 'This one is Alfred, and this one is Hillary.'
'Okay,' she smiled. 'So, when you go up a step, start with Hillary, then Alfred, then the crutches.'
'Okay.'
Despite my protests, the physios also took my Zimmer frame away. Where the frame had hovered helpfully, I would now lean two crutches against the bed, or try to balance them in the gap between bed and nightstand. In the night I woke and fumbled for the crutches, knocking first one then the other to the floor. I tried to reach for them but I was stuck, and had to press the assistance buzzer. The night nurse came and, after helping me to the bathroom and back to bed, disappeared with a conspiratorial look, reappearing shortly with my beloved Zimmer frame.
Other family and friends started coming to visit. My uncle and aunt sat on the end of my bed, and squeezed my hand while we had a good chat about nighties. My midwife, Jen, visited, and brought me homemade rhubarb and strawberry cheesecake. I wolfed it down, while we talked about _Where the Wild Things Are_. Matt, Steve, Sam and Sal all came in at once. Matt hugged me first and surprised me by sobbing into my neck. It was a relief. It meant I could sob too; I could bring out these pictures of our still, little daughter, invite old friends in to the weep-fest that had become my life. I had words I could say when people looked at these pictures. 'My hair was red and curly from all the blood. I look like another person. At least I know that red's not my colour.' _Ha ha_. I laughed—sometimes it helped. But it came out unnaturally, like an actor pronouncing the words, 'Ha ha ha'—high and voice-like. After a while, I worked out why and added the explanation to my hilarious repertoire: 'I can't laugh properly because my belly hurts from the caesar and my chest hurts from my broken sternum, so I just have to laugh in my mouth—ha ha ha.'
They had never seen me like this. _I_ had never seen me like this. I was a stranger to myself—a wounded, maddened woman in a hospital bed.
Tash, an old school friend, called. The last time we'd spoken, it was so I could get her recommendations for maternity hospitals. She'd just had her third baby, and I had been looking forward to being in Melbourne and on maternity leave at the same time as she was. Tash had lost a little brother before I met her, at the start of high school, and in Year Seven she had been hit by a car while getting off a tram. Now this knowledge meant something very different to me; like text that had suddenly become legible. 'There is no silver lining,' she said. 'Don't let anyone tell you "everything happens for a reason"—there is no upside to a child dying.'
As long as visitors were around, I could keep it together. Rima filled in the pink meal order form for me each day, and when dinner arrived, we would lift off the heavy plastic covers with great ceremony, announcing steamed fish and salad, or vegetable omelette. As eight o'clock neared, the ward got quieter, and I would kiss Rima goodbye and watch her disappear behind the curtain, listening for the sound of the door closing.
One night, a baby started crying in the corridor. My visitors all gone for the day, I sobbed along silently. _Oh, my little love, it's hard, I know_. It was an unfamiliar noise to me, that little cry, little eh-eh-eh sobs from a small, shaking chest. I wondered, 'What would Zainab have sounded like in full voice? Where was she?' I wanted to hear my child's voice, I wanted to hear her cry. But I also thought, 'Wherever she is, she must be so scared.' Crying that kitten cry with no one to hear her, to offer a breast, their arms, a heartbeat. I shook when I realised that she lost my heartbeat at exactly the time I lost hers, but while her heartbeat had been a novelty to me, mine had been her constant companion, from her very first stirrings as a few little cells. _Oh, my little one, it's okay, little one_ , I hushed that little stranger-baby in the hallway, from my bed, as well as my own sobbing self. I'd lost my moving, mysterious bump, as well as the still, little baby she'd become, so that it felt like an amputation as much as a bereavement—a piece of me severed and lost forever, leaving a gap that no amount of tears could fill.
My body still didn't remember the distinctions between nighttime and daytime, and while it felt tired, I still woke every few hours. There's no such thing as real darkness in a hospital ward. In the bluish glow, I found the button for the light, on the bed-goes-up-bed-goes-down controller, and opened my black-and-white notebook.
Wednesday, 30 December 2009
Now that I've started writing, suddenly it is hard to stop. Words soak out of me like blood. They spot my skin like measles and appear like a coating on my tongue. They leak out of every orifice, soaking into notebooks like the blood that haemorrhaged from my organs. I'm certainly talking too much but I always did that. This is different. Things are so heavy with meaning—there is so much I need to get across.
My trouble now is getting time to eat and sleep because I want to write so much. It used to be so hard to get the words out. Now, it is as though the impact crushed me like a blueberry and all these words came flowing out in dark staining lines of juice.
My handwriting was a fourteen year old's again, punctuated with love hearts and sad faces with lines of small teardrops. The black-and-white notebook became my journal while the yellow book filled with lists—names of every caregiver I encountered, movies I wanted to watch, books I needed to read, questions for the doctor, things I wanted my mum to buy on her shopping expeditions for me. 'Hannah's Rules for Hospital'. Suddenly life felt so short—there were so many things that I needed to do _right now_. Odd memories came back to me: the old ivory bangle of Mum's that I used to sniff as a four year old—where was it? My dad obligingly found it and brought it in. Thirty years later it still had the same comforting smell, and I wore it day and night. I collected other talismans to have around me—Zainab's blankets, knitted by Mum and Penny, and the soft grey bunny my brother and his fiancée had given us.
With Zainab's funeral over, her death (and birth) notice published and our sad news trickling out to friends, family and the world, I filled the long hospital days, and longer nights, with lists and projects: arranging my return to work, planning the garden at our new house, requesting pen refills, contacting my high school art teacher, negotiating relations between Rima and my family. 'You realise, don't you,' my psychologist remarked months later, 'that when you first called me from the trauma unit, you were quite manic.'
Thursday, 31 December 2009
It is so easy to get tricked into thinking that time is some substance that comes in measurable, equivalent units. As though we can compare surviving the last two days with surviving the next two days—as if your chances are as good. Instead every moment in time is its own catastrophe—its own miracle. A sibling to the moment which came before it, with its own personality, its own idiosyncracies and significance. Maybe it will be an average Joe Citizen moment, significant only to its loved ones—maybe it will be an Adolf Hitler moment.
If I could take my tiny sewing scissors and cut carefully around that moment—just cut along my line of sight when I saw the four-wheel drive swinging onto my side of the road. Just snip away that moment of impact, cut along to the quiet bit, where suddenly we were still, just before Rima started screaming. We'll lift that piece carefully out of the picture and put it down somewhere else. And then we'll pull those little edges together and stitch them, leaving a little more room around my legs and belly. Some blood noses and bruises we can deal with, but let's just skip that bone-breaking, placenta-abrupting, heart-stopping moment of impact. I'm happy even to bargain over a broken patella.
But when I lift my scissors and try to pluck that offending moment with my finger and thumb, it is all attached. It's that dilemma when you want to cut a picture from a magazine and on the back is the other picture you want to keep—terrible choices! So I find I can't cut out that nasty little moment of impact without cutting everything else. There are no neat edges.
Even if I could do the fine snipping and stitching, I still can't breathe life into my monster-moment. Because even if I wish for it without that ugly impact moment, I'd still be disfiguring time—amputating a moment, which, for someone else, might be the shining star of their lives. Time is not mine to snip.
Enough of the philosophising. My mission for today is to eat protein. Sharon, my nurse here in the trauma ward, is worried about my albumin levels and I can't just blame it on the hospital food. I'm thinking this might be a good excuse to ask Rima and Erica to revisit the Afghan chicken shop on Sydney Road up near Pentridge and bring me back some bbq chicken goodness.
And there shall be a festival of eggs. I think of the wallchart in Mrs Galt's home economics kitchen classroom—showing the egg in cross-section, yellow yolk suspended on its twisted umbilical-like cord with a neat arrow, 'Albumin'. Perhaps that is what I need—an explanatory diagram?
One of my windows is starting to go a light aqua blue—a relief to have morning draw a bit nearer with promised distraction and food.
For New Year's Eve, and with special permission from the doctors, Rima brought in a bottle of Bollinger, and, with my mum, cracked it open and tried to convince the nurses to have some with us. We cackled hysterically about Alfred and Hillary, bedpans and pain meds. 'You're asking the nurses for Endor,' said Rima. 'I don't think that's actually what it's called. Endor is the Ewok's home planet in _Star Wars_!' We phoned the girls. My dad was being super protective and wanted them to go to bed at ten, but it was New Year's Eve, after all, and we insisted that he let them stay up for the countdown. In the early hours of 2010, the champagne clearing from my head, I wrote an earnest letter to the new year.
Friday, 1 January 2010
Dear 2010,
Oh I had thought you would be so different already, and I haven't even seen two hours of you yet. But I promised my Haloumi that I would be open to surprises. Please let that include some happy surprises as well as the rough and unfunny one 2009 dealt us.
My hopes are too delicate and frankly I'm too scared to hope or wish for anything at the moment. But I can start with what I've got and from where I am now. Right now I have my beautiful Rima and our beautiful girls. Please don't go hurting them just yet, 2010?
During the day, I could hold onto the raft of family and friends and the distractions of food, doctors' rounds and the physio or social worker. Nighttime was harder—it was just me and the big salty sorrow, with the beam of light from the night nurse swinging around slowly every so often. Pauline, the night nurse who had returned my Zimmer frame, often heard me weeping and came in to see if I was okay. She would pull up a chair and we would talk. One night, to try to calm me, she offered to rub my back. I rolled onto my tummy and with the fancy-smelling oil I had bought to rub into my stretch marks, she laid her hands on me and smoothed out some of my sadness, so that I could sleep for a little while.
Tuesday, 5 January 2010 1.10 a.m.
I don't understand with all the buckets I am crying how I can also be peeing so much. There must be some great inland sea inside me. Perhaps when I move around you can hear me sloshing?
Oh I am in so much pain right now—a choir of pain, from the deep baritone ache of my knee to the sharp peaks of breathing with this broken chest bone. And over the top comes a weird melody of moans which I have never heard myself make before—a keening series of wavering 'ahhs' and alto 'ohs'. I remember reading Shakespeare plays in high school and laughing with Penny at all the 'Ah me!' and 'Alas!' exclamations. But that was the luxuriously naïve laughter of someone who had not felt sorrow so visceral that it makes you cry out in pain.
And please don't think that my analgesic needs are being neglected! Pauline and Pervin have just brought me my strong pain meds—signing them off in tandem as though I were a library book. No, I am being looked after well in that department—yet not too much, so I'm not living in a morphine fug.
No, while some of the other singers in this choir sing on the frequency of physical pain, most of them specialise in the other end of the spectrum—in sadness and madness. These are my two torture masters. They would quite like to drive me to a bad place of hurt and hate. They are taking me somewhere whether I like it or not, but I am pushing for a gentler destination—a silly puerile comic-grotesque direction in which people name their own legs. I know Alfred (left) and Hillary (right) are a bit old fashioned, but I really didn't get a lot of time to think about it. One must make snap decisions sometimes and live with those decisions.
I need to find a type of crazy in which you can laugh at the most humiliating and revolting things, because this is the reality of my life right now—stinking out the room with farts that make even the flowers weep.
6
Frida and me
By day five, they wanted to send me home.
'I have patients recovering from exactly the same knee surgery on the orthopaedic wards who are home already by day four,' said the nurse manager.
I stared at her for a second. I hadn't really comprehended that the rest of the world still existed beyond the hospital walls, let alone that I could be sent home.
'But have those patients had caesareans too?'
'Oh, I've had caesareans!' By the time I formed the words to ask if her babies were born alive, she was gone, out of the room and off to manage the rest of the ward.
I started writing lists of the reasons why I couldn't be discharged just yet:
* • because I can't get out of bed without assistance,
* • because my partner is also injured and unable to care for me,
* • because my able-bodied family members are already caring for my injured partner and two injured stepdaughters,
* • because we don't have a home to go 'home' to.
I still looked pregnant and I was scared that well-meaning people would say things that would inadvertently break my heart. I planned a little preventative sign to wear:
_No, I'm not pregnant anymore_.
_My baby has died_.
•
I dreamt of a broken bathtub, of roadkill bathmats, of a chandelier constructed from tiny, dried-out bats, like muscatel grapes. I dreamt that I was at Rima's work, and was holding my yellow book with all my pregnancy medical records. Someone who didn't know about the accident saw my Transport Accident Commission form, printed with two little human outlines on it where the doctors could indicate my injuries, front and back. She mistook it for an ultrasound scan.
'Oh,' she said. 'You can see two of them in there—are you having twins?' I wasn't angry; I just cried, 'No, no, no, I've lost my baby.' Then someone offered to drive me home and I gave a scared moan: 'No, no, no; I don't want to go in a car.'
I dreamt that my hand was safe in my mum's hand, strong and warm. With fingers interlocked, we stretched our fingers out—hers more wrinkled but so similar in size and shape to mine. As I turned our facing hands, I saw her ring, middle and little fingers on one hand had been amputated above the first knuckle. They were smoothly healed, and strong, but incomplete. I cried out, 'Mama! What happened to your hand?' She gave me a sad look. 'Oh, that happened at work the other day.' I want to shake her. _Mama! No! No, this is not okay. Mama, you can't let this happen to you_. But she was right, it was too late to make a fuss. Nonetheless, I woke up crying. 'Mama—no, no!'
I had a jungle of flowers growing from the top of the cupboard in my room, a bit like Max's forest in _Where the Wild Things Are_.
•
Sunday, 3 January 2010 nearly 1 a.m.
I was telling Matt the truth when I said I was not even thinking about the driver who caused the accident. He's an irrelevance to us now and to our recovery. What has happened has happened and getting angry at him will not change what we need to do to get better, and may make it harder.
I do remember that his name is Amrik—and so he must have a mother who brought him into the world, who held him and looked at his tiny face in wonder. And who gave him that name wishing only good things for him. I won't judge him (not that that is my job or in my power in any case) because I have driven stupidly too at times.
I know that on the Wednesday before we moved to Melbourne, I drove so fast and so angrily with the girls in the car that they were scared—just because I was angry and frustrated and impatient with Rima and with some difficulty with the move, I can't remember what. And Haloumi was in the car too, right there, below my angry heart, feeling all those stress chemicals circulate in my bloodstream. I thank God that the worst I did that day was upset my stepdaughters (that was bad enough) but I could have very easily caused an accident like the one which took our Khalila away. So, even if I can't control other people's driving, I promise never ever ever to drive when I'm that angry again.
I will stop the car, or not get into it, I will take a little walk or deep breaths, or smash plates, but I will not drive. I am scared to get in a car again—I will have to take that slowly.
•
My cousin Lou came to see me. We'd been to see her on the day of the accident. We'd kissed her goodbye at the gate, crossed the road to where our car was parked, pulled out from the kerb, turned left, then right, then left onto Warrigal Road, before being sucked into the black hole of the collision.
She hadn't told her kids yet that our baby died. They had come in while she was on the phone to her mum, my Aunty Helen. They heard her weep. 'Mummy, what's wrong?'
I pictured her in that lovely new kitchen, leaning on the bench where my pregnant belly touched the stone, where we ate Greek biscuits before getting in the car. She went down on her knees, gathered their little living bodies close, one in each arm, sniffed into their necks.
'There's been a car crash. You know Hannah and Rima and Jackie and Jasmin, who came over yesterday? Their car crashed and they were hurt.'
'Oh, Mummy!' said Connor. 'That baby must have been so scared.'
I hadn't wanted to think of this. I want to think that she felt loved all the way through, that she felt the bang, but knew I was still there with her, even if I couldn't protect her.
Tuesday, 5 January 2010 9 p.m.
Today was the day when everyone thought I was going crazy. No wonder really, when I tell them I'm writing a blockbuster novel which is going to be made into a movie starring Charlize Theron (as me, obviously) and Salma Hayek (as Rima) ('A heart-warming tale of tragedy, hope and incontinence' or 'Pollyanna on crack meets AB Facey'). Suddenly everyone was giving me worried looks—particularly when I insisted on writing down the names of every member of the hospital staff I met.
Here I was thinking that I had discovered a new post-accident Hannah with some inspiring new talents when everyone around me was thinking, 'She's dropped her bundle.' What shocked me was when the person from Epworth Rehabilitation came to assess me for a transfer to the rehabilitation hospital. I immediately wrote his name down in the yellow book—Kamal. He made jokes about being born in the seventies, but when we got down to it, told me that their primary concern for me was my neuro-psychological state.
We still had a holiday house on the Mornington Peninsula booked for the two weeks before our tenants were to move out. It was meant to be our babymoon—our reward for making it through the big move and Christmas, a little rest before we unpacked our house and got set up for the baby to arrive. When the doctors did their rounds, I asked whether we could still go. I wanted to feel the salt water wash over me. I wanted to hide a bit longer from Melbourne and any idea of 'normal' life.
'I don't think it's such a good idea,' the doctor said. 'This is the thing with internal bleeding—we take a three-three-three rule. For the first three days, there's a significant risk of further internal bleeding—that's why we kept you in ICU. For the next three weeks, the risk is reduced, but it's still quite serious. The risk reduces again after three weeks, but you've still got a higher-than-average risk of bleeding for the next three months.'
'How would I know if I was bleeding again? Would it hurt?'
'Not always. You can bleed to death without feeling much at all.'
A different nurse manager came to see me. His name was Ali. He admired the letter Jasmin had written to me, left-handed, as her right hand was still in a cast.
Dear Hannah
I'm so happy that your happy (ish) but just know that I'm here for you like how you were those past few years but yeah so is everyone else. I was so happy that I got to see you yesterday but yeah sorry I'm writing in my left hand!! My hand doesn't smell much anymore!! But it hurts. I love you see you chao from: Jasmin
He Blu-tacked it to the wall for me. 'I need to ask you a favour. We're looking at transferring you to rehab, but we've got a bit of an issue with rooms. Can I ask you to change to a shared room, just down the hall?'
I was grateful for any reprieve. My mum helped me pack up my things, piling bags and belongings on the bed, with me holding the vases of flowers. The photo the midwife gave me of Z was now in a glass frame Mum had bought from the hospital gift shop. I wrapped it carefully in her blankets, first one and then the other, and hugged it to my chest. The nurse released the brakes on the bed, and pushed us out of the room and down the hallway, like a little boat. Suddenly I felt the urge to document this. 'Mama, do you have your camera? Can we take some photos?'
The image of the bed piled with notebooks, colourful blankets and flowers, reminded me of Frida Kahlo, the surrealist artist I had been obsessed with as a teenager. When she was eighteen, Frida was on a bus on her way home to Coyoacán, Mexico City when it collided with a trolley car. She was thrown from the bus—and in the process was impaled on a handrail from the bus, which pierced her lower hip and came out her vagina. The impact fractured her pelvis, and broke her spine in three places, as well as her collarbone, two ribs and her right leg and foot. She was immobilised in a cast for a month, and underwent over thirty-five operations. Frida's life, too, was pierced through with the after-effects of the accident—her health, her fertility (her injuries meant she was unable to carry a pregnancy to term) and her artwork.
Prior to the accident, she had ambitions to become a doctor, but 'bored as hell' in a hospital bed, she began painting. During her frequent convalescences she would often decorate her bed and the spinal casts or orthopaedic corsets she had to wear. Stuck in hospital or in bed, her portraits were primarily self-portraits—her gaze, level and direct, her brows, serious and meeting in the middle like a pair of raven's wings. Her portraiture was realist, down to the fine black hairs on her upper lip, but she interwove or surrounded her portraits with more surreal scenes. Frida's face appeared on the body of a deer, or suckling from a Pre-Columbian statute, or surrounded by flowers and monkeys. This was exactly what I wanted—to meet the horrors and indignities head-on, to examine them, and myself, and somehow to find beauty without smoothing out the painful parts or finding neat answers for unanswerable questions.
I had been at a loss as to how to understand myself as injured and grieving while still maintaining some measure of dignity. So I took Frida as my patron saint, as my survival mentor, as inspiration to look my circumstances directly in the eye and to document my pain with an artist's curiosity. My life in Frida-vision suddenly seemed artistically surreal rather than dangerously crazy.
In the new room, I felt the need to keep the rails on the bed up, as though this bed-boat that had carried me here might at any moment lurch and roll, bellied by a big dark ocean. I drew a ship in my notebook, with Z as the anchor, and wrote the words of another Little Birdy song on the sail.
Who's going to love you now, baby?
Who's going to love you now, baby?
When you're done fighting that war at sea
Get on that ship and sail back to me!
Penny came to visit again. She climbed up onto the bed with me, and I showed her the ship drawing and together we wept.
I called my mum and wept again. 'Mama, I just need to stop for a minute. I need to stop and wipe the phone because I'm crying so much that I'm scared I'll electrocute myself.' This, for me, counted as a good laugh—that and announcements over the PA system about returning the bladder scanner to level five—and for a little while, I shook with both tears and laughter. 'Mama, I'm not a person at all, I'm just a force of nature,' I said. I pictured the _Pasha Bulker_ , the container ship washed up on Nobbys Beach in Newcastle in the big storm of June 2007. On our way north for a hockey carnival, we had stayed the night in an onsite van at a Newcastle motel, feeling the storm shake the thin walls. We woke and drove into town, passing cars made flotsam by the floods, with high-tide marks of dirt and leaves reaching their windows. And there on the beach, tall as an eight-storey building, was the _Pasha Bulker_ , the surf lapping it, but dwarfing the beach pavilions, and commanding an audience of surprised-looking adults and kids still in pyjamas.
My hysterical laughter slowed to sobs. 'Mama, I don't know how I'm going to sleep.'
On the other end of the phone, my mum spoke softly. 'When I was in the Alfred and had trouble sleeping, the women there gave me something to say which I could repeat to make myself feel safe and calm.'
She meant the psych unit at the Alfred Hospital, where she had spent uncounted weeks in late 1987 after a particularly bad bout of depression. This was not a usual topic of conversation for her.
My brother and I had been in primary school, and while Mum was in hospital, our art teacher, Miss Komis, would drive us home from school. 'Call me Fifi,' she said, and sometimes took us out for ice-creams on the way home, as a treat. I would sit in the front passenger seat of her cobalt-blue car, touching my finger to each unfamiliar object: the perforated leather of the door handle, the chrome door latch, the glove box. The more practical details of that time are fuzzy. How long was Mum in hospital for? Who looked after our little sister, who was not at school yet? But I clearly remember the piles of cicada shells my brother and I collected each afternoon after Fifi dropped us home, and the vinyl chairs in the hospital's TV room when we visited. Everything was hushed there, and Mum was even quieter than she'd been in the weeks leading up to her hospitalisation. She looked half-transparent, as though she had lost the will to be seen. At home, I was a bossy, responsible big sister. I helped Dad cook dinner and do the shopping, and I referred to my brother and sister, five and seven years younger than me, as 'the littlies'.
'The phrase they gave me,' Mum continued, 'was, "I wrap this pure white cloak around me through which nothing can harm me and only goodness may pass".'
I repeated it back to Mum several times. I liked the idea of a cloak filtering out harm, permeable only to goodness. But I wasn't convinced that anything could protect me, or those I loved, anymore.
I was conscious that, now that I was sharing, I shouldn't keep my poor roommate awake, but I couldn't figure out how to stop the weeping and laughing and writing. I asked the nurses again whether I could see a counsellor. Several hours later, a volunteer arrived. Just as well—on one of my many lists for the day, I'd written, 'If no psych assessment by 5pm, DRAW ON MONOBROW.'
Pat was older, quietly spoken and wore a smart suit with a brooch. She wasn't sure whether she could help, but encouraged me onto my crutches and out into the hallway, further than I'd been in my whole time there. Pushing back the boundaries of the known universe, we found a corner with two chairs, and she patted my arm. We formulated a plan that didn't involve weeping all night and she walked me back to the ward. I pulled up the sides on the bed, invoked Frida, covered myself with hand-knitted baby blankets, pushed the ivory bangle up my arm and set sail for the night.
7
Tabloid tragedy
My sister let slip that there was a story about us in the paper. I could always trust her not to filter information for me.
I asked for a copy and Rima obliged, buying one from the hospital newsagency and bringing it up to the ward. I wasn't prepared for it to be the front-page headline. Our story wasn't just our story anymore; suddenly, it was a 'tragedy', for people to tut over on their tea break. Part of me felt vindicated. I wanted our loss to be important—everyone should know we had lost our baby daughter. How could anything hurt this much and not be front-page news?
But it also felt very odd to have everything that had happened to us in the past days reduced to twelve characters of thick black headline. They had also got it wrong. 'LOST IVF ANGEL'. She wasn't IVF conceived. And Rima was not, as was reported, my sister-in-law. I felt ill at ease. I was conscious that the pedestal for tragic, wronged mothers is a narrow and unsteady one, surrounded by a sea of condemnation for 'bad' mothers. I knew from growing up with seeing Lindy Chamberlain on the television that it all could turn very quickly. I didn't have an unusual religion; I had the benefit of white, middle-class privilege; and now the _Herald Sun_ had incorrectly extended me the benefits of presumed heterosexuality. Did I really want to poke that bear?
When I had a gap between visitors at lunchtime, I called the _Herald Sun_ news desk and asked to speak to the journalist: 'The front-page story from today is about me, and it contains factual errors. I want a correction published.' They put me through to the journalist, and when I told him off for his inaccuracies, he was contrite, and wanted to make it right.
'I'll do better than a correction,' he said. 'I can do another story.'
Suddenly he seemed a little too eager to put it right—and he was offering to come in and talk to me. 'I need to think about this,' I said. I gave him Mum's phone number, deliberately creating a buffer zone.
When I told family members I'd been in touch with a journalist, there was some alarm, particularly given my recent crazy-lady antics. Apparently, a journalist had turned up at my dad's house, pressing him for comment, and my dad and stepmum had had to keep the phone off the hook. Over the next few days, I consulted with my friend Matt, a journo, about what to do.
In the meantime, I had to pack. I was being transferred to a rehabilitation hospital. I was triumphant about not being sent home yet. Mum helped me shower. It was complicated. I covered my left leg, splint and all, in a garbage bag, propped it on one chair and sat on another. Mum left to let me dry myself and start getting dressed. I took my time. There was a lot to look at, seeing my body naked.
Black and dark purple bruises blossomed, like tropical botanical illustrations, from my elbows to my fingertips, cascaded down my thighs, and nestled, deepest of all, in a seatbelt stripe across my chest. The dark, dead blood collected in strange patterns—sheet marks, the creases of my wrist, the waffle pattern of a wound dressing, in the lines of the old scar from my kidney operation. This was a kind of bodily print-making practice. My left breast hung heavy, covered in one big bruise that faded to yellow at my cleavage.
When Rima came in, I had a new favour to ask her. 'Can you take photos of all the bruises? They'll disappear soon—I need to remember this.' She was systematic about it, starting with my feet and working her way up. 'There's one behind your right ear; maybe from your glasses?' I posed, pointing to the bruises, like the saints in Renaissance paintings touching their wounds. _Look. Here. Proof_. What did it mean, to photograph this naked, bruised body? Was it pornography? Crime-scene photography? Art? I didn't know what to do with my face in these photos. Sometimes I smiled. Sometimes I looked away.
To catch the full colours of the bruising, I had to hold my arms out in front of me, hands up. When Rima showed me the photos, I saw a woman trying to defend herself.
When the nurses came to change the dressings, I got the camera out. I wanted to see what was under all those patches. This body, which had been mine for thirty-three years, was unfamiliar to me. On my right ankle, two staples held shut a small but deep incision, alongside a U-shaped cut. In the fleshy bit of my forearm, a small wound was still weeping where they had pulled out a long shard of glass. Several big dressings covered my left knee, and I could feel the staples underneath making three quarters of a circle around the knee cap. The surgeon had joked that I'd have a hammer and sickle scar. My communist left leg. Ha ha. Rima took fuzzy photos of the lumpy scar on my head—pink, and still stitched with thick black thread.
As systematic as Rima's photos were, they were not enough. I borrowed my mum's camera, and shut myself in the bathroom. My supply of Rima's t-shirts had run out, so I had been wearing two hospital gowns, one on forwards and one on backwards, to avoid the unflattering gap. It was the sartorial equivalent of a kidney dish—without my own clothes, jewellery or make-up, and in the starkness of the hospital bathroom, my body looked like a medical specimen.
The morning of my transfer, I picked out a kaftan-like green and white dress from the bag of Mum's clothes she had brought in for me so I didn't need to wear my maternity clothes. I tied the hot pink belt from my dressing gown above my still-big post-partum belly. I tied my hair back and put on make-up. I held Mum's ivory bangle to my lips, inhaling its smell and invoking the elephant whose life was claimed in making it. _Elephant—I'm so sorry. Please haunt me, please protect my baby girl, wherever she is_. I sticky-taped a sign to my belly: 'I'm not pregnant anymore. My baby has died.' I knew it was unlikely that anyone would comment, but the thought that people might see me and think I was still pregnant was unbearable. I needed to open a window on this stifling pain, let others peer in.
•
The week before we'd come down to Melbourne for Christmas, Mum had stayed with us in Sydney and we'd gone shopping in Newtown. In a small shop full of beautiful things, she had bought an oval enamel brooch with a matryoshka face on it, like a little doll. At the time, I'd felt covetous. It was a beautiful, expensive object, exactly the kind of thing I couldn't justify buying when we were relocating interstate and about to have a baby. But the matryoshka brooch reappeared while I was in the trauma unit, Mum slipping it wordlessly into my hand. The enamel was solid yet silky-soft, and the small baby face, with a blue asterisk on each cheek, smiled at me. In the shop, it had just looked cute and smiley. Now I assumed the asterisks were tears. I pushed the pin through my clothes, and then through my bra, so I could feel the cool metal of the clasp against my breast, always on the left side, over my heart. _Here you are, my little one_.
•
When the orderlies arrived to transfer me by ambulance, I was chipper. 'We're working on the theory that you're taking me to rehab in a hovercraft,' I said. 'So it can just float up and out of the way of any accident. Does that sound okay?' They were reassuring. They thought I was having a joke with them.
Mum came with me. I held her hand as the trolley clicked into place and the doors were closed. I blinked slowly as we emerged into daylight and the world beyond my hospital bubble. Through the venetian blinds of the ambulance, I could see streets, trees, traffic, all behaving as though nothing had happened. This time we travelled without the sirens. This time, the outcome was known. There was no great rush.
•
In my room in the rehab hospital, I met the nurses and told my story. Then the doctor, then the physio; lunch; then the psychologist, the occupational therapist, and the other doctor. By 5 p.m., I had told our accident six times over, and by the time Rima arrived, my compliance was gone. Everything felt wrong, and suddenly I was white-hot with anger—at this stupid hospital, at the other driver, at myself for thinking this was all somehow fixable if only I behaved correctly, at Mum and Rima for being there and not being able to fix any of it. I was contrary, inconsolable. The air-conditioning was on too high, the mattress was too soft and I missed the nurses from Royal Melbourne. Mum and Rima didn't know what to do. Take photos, I told them, and then I gave them the finger when they did. I was sorry, though, when visiting hours were over, and they had to go, so that it was just me and the photos of Z in my new room. I stayed up late drawing a diagram of myself, with arrows explaining my injuries and my family relationships, to be stuck onto the door of my room so that I wouldn't need to explain it all to anyone else.
At 7 a.m. my phone buzzed with a text message. It was from my friend Kana, announcing the safe arrival of her baby girl. I wept, and plucked tissue after tissue from the box on the nightstand, with the 'robot arm' the occupational therapist had given me, just to hear the tearing sound they made as they came out of the box. The information echoed around my heart like a loose spanner in an empty toolbox. I logged onto Facebook for the pictures. Here were the photos of Kana and me at her baby shower only a few weeks ago, standing belly to belly. And here was her baby daughter, alive and safe in her arms; my baby, dust. I wrote my own Facebook post.
7 January 2010
What happened is still so raw and new that we are wrapping our heads around a new bit of it every day. And also failing to wrap our heads around, and howling at that failure and our loss and the fact that our daughter will never open her beautiful eyes to see the people who love her more than anything. An inaccurate version of what happened was on the front cover of a tabloid newspaper. This is my response to it:
Letter to the Editor
Eleven days ago I was 34 weeks pregnant and driving home with my defacto wife in the passenger seat and my beloved stepdaughters in the back seat. A four-wheel drive came onto our side of the road and hit us. We were all badly hurt but our baby daughter died in my womb from the impact. My defacto wife (I can't call her my wife because in this country we cannot marry) and I had spent nearly four years getting to know our sperm donor, undertaking tests and trying to get me pregnant using assisted conception (fortunately we did not need IVF). When the _Herald Sun_ reported the accident and our loss on 28 December 2009, ('LOST IVF ANGEL'), it mistakenly called her my 'sister-in-law' and referred to my stepdaughters vaguely as 'two children'. Many people reading your article must have been wondering about the relationships between a pregnant woman, her sister-in-law and these two children who were all hurt in the one car. I just need to clarify—we are a family. My defacto wife (what a clunky phrase that is) and I lost our little girl, and our big girls lost their baby sister. I don't want our family to be invisible—we have enough pain and injuries to deal with at the moment.
But in this strange movie which is apparently now my life, the most genuine and real thing is the love we have felt around us from family, friends, and also people who may not know us in real life but have very real compassion for us (or worse, have been through equally heartbreaking things themselves). It is huge, and we feel so warmed by your love but at the moment we are still so broken—physically, and in our hearts, that we can't respond to all the messages. I am out of ICU (yay), out of the trauma ward (yay), and in rehab. I hate being here—but am doing my best to heal and learn how to do basic things so I can get home and be with my beautiful girls and do the rest of my healing there. Rima & girls are out of hospital (yay), but there are still various stages to go.
We will never ever be the same after this. I could never have imagined that the Haloumi who kicked and hiccuped inside me could be such a beautiful little baby girl. If the impact had not abrupted the placenta she would have been fine and things would be so different. I am so proud of her and so so heart-broken. Thank you for your thoughts and love.
We will be having a 40 days memorial at some stage—possibly early Feb when she was due. Forty days is significant within Rima's faith. For forty days the soul wrestles with itself and its good and bad deeds in the desert, before moving on (though we're pretty sure our girl didn't have a bad deed to her name). If you are able to join us, we would love it so much. I warn you it is going to be very sad. We will have a weeping competition and I will win. The pay-off is you will get to see pictures of the most beautiful grumpy baby girl in the world.
•
One morning in rehab, I had a phone call from the police officer handling our case. He'd spoken to a number of witnesses and it was clear that it hadn't been the Pajero driver's fault at all. The collision had started, he said, with an older-model Commodore, whose driver was involved in some kind of road-rage altercation with another car. Several kilometres back, the Commodore driver, a uni student, had slowed down and stopped in a no-stopping zone to drop off a friend. The driver behind objected, hooted his horn, overtook the Commodore in an aggressive way, and was seen shaking his fist and yelling at the Commodore driver. In response, the Commodore driver sped up and tried to overtake aggressively in turn, but he didn't get the chance to yell anything, because he didn't check his blindspot when overtaking and failed to see the Pajero in the next lane. The flanks of the two cars made contact, knocking the Pajero onto our side of the road.
It took me a few moments to process this news. The Pajero driver was not at fault. His car was just a big, heavy billiard ball tapped in our direction.
In the aftermath of the collision, witnesses saw the Commodore slow down and pull over briefly, but while other drivers around us were stopping to render assistance, the Commodore driver drove off, and returned the car to his brother's house. When the police traced the car's number plate to that address and arrested him, he told them he had fled the scene of the accident because he was fearful of the other driver involved in the road-rage incident. He was charged with five counts of dangerous driving causing serious injury (one each for me, Rima, Jasmin, Jackie and for the driver of the Pajero) and one count of failing to stop at the scene of an accident, and was released on bail. Because she was stillborn, there was no charge for dangerous driving causing Z's death. 'Serious injury' didn't come close to describing our loss, but I was okay with her coming under my jurisdiction rather than that of the _Crimes Act_.
I sent my letter to the journalist, and, as promised, he spun it into another front-page story. I sticky-taped the front page to my door in rehab, with a drawn-in apostrophe correcting it from 'Mum's Pain' to 'Mums' Pain'. The article also reported on the bail application, at which the police prosecutor argued there was a risk the defendant would try to leave the country while on bail, citing the example of another Indian student, Puneet Puneet, who fled in June 2009 after pleading guilty to culpable driving causing death and negligently causing serious injury.
Bail was granted but, nonetheless, our case fed into the mounting tension between the Victoria Police and the Indian community in Melbourne, over perceptions that police were failing to respond adequately to a spike in violence and race hate incidents against Indian people. A string of murders and assaults on Indians in Melbourne through 2008 and 2009 had prompted the Indian government to issue a travel warning for citizens considering studying in Australia, and sparked protests in Melbourne and New Delhi. In this climate, many journalists described the driver who caused our accident not as 'a student' but as 'an Indian national'. I had a queasy feeling as I realised that our story played neatly into racist stereotypes of a white woman (and, worse still, a pregnant one) harmed by a brown man.
•
Each day in rehab, I was booked to work with the physio for an hour or two. I would go on my crutches down the corridor and to the 'gym', which was less like a gym and more like a surrealist kindergarten. There was play equipment for adults, parallel ballet barres, a play kitchen, and a rail with clothes hanging on it (dress-ups?), along with fit balls and yoga mats. And then there was the thing I averted my eyes from. I knew it was there because I had seen it the first time I walked into the gym and felt a jolt of fear through my gut. A weepy, elongated sigh escaped from me. I knew that, at some point, the physio would move me towards it. It was a dismembered car—just the front half, so that you could practise getting in and out of it on either the driver or passenger sides. As the physio taught me how to tackle stairs on crutches, I arranged my ascent and descent so that I didn't have to look in that direction. The steps led up to a clothes line, and with each ascent I triumphantly moved a peg from one line to the other, with an absurdist flourish.
To add to the surrealist effect, one day I arrived in the gym to find my Year Twelve English teacher there, also being coached by a physio. We were like racehorses, being nudged along by our minders. We couldn't chat properly but I got enough of my story out for her to get it.
'I hope you're writing, Hannah.'
I assured her I was, and she made me memorise her email address so that I could write to her.
8
Permission to bend
The accident, and the long hours that followed of ambulance, emergency room, labour and then waking from surgery to see our daughter, had obliterated the normal markers of daytime, nighttime or mealtimes. Time started again from zero—everything was measured in hours, and then days, since the accident. In rehab, I started reconnecting myself with 'outside' time, looking at the newspaper each day and marvelling that people's lives went on regardless of our catastrophe.
One of the elephants at Melbourne Zoo was due to give birth to a calf any day now, and I followed the story, terrified that her baby too might die. The article recounted a statistic that made my heart ache for elephant mothers. For elephants giving birth in captivity, there was an extraordinarily high stillbirth rate, particularly for first babies. I thought of mother elephant eyes, weary with sorrow, and of a mother elephant trunk, searching out and touching on the still, hairy form of her baby, and wept my own elephant tears.
While I was eating my breakfast one morning, I drew rows of wonky boxes in my diary so that I could count the days. It was a double calendar. Each box had two numbers: the date and how many days since our own ground zero. Just writing the number '27'—the date of the accident—made me shudder and cry. I dripped strawberry jam on that box and drew a sad face beneath. It didn't quite capture the violence of that day, but I couldn't leave it unmarked. I'd had plans for these dates. These were to be days in a rented holiday house, lazy walks to and from the beach, with time to decide on a name for this baby, and to recover from the interstate move before we moved into our house and set up a nursery. By numbering those boxes, by stitching together these beautiful summer dates with our new messed-up reality, I inserted myself back into time, and was no longer in the suspended, timeless world of hospital.
I'd been discharged from Royal Melbourne with various letters (so much paperwork, getting hit by a car!), including one confirming that I had an appointment with the orthopaedic surgeon who had operated on my knee. That and my hoped-for discharge date were the only goalposts I could work towards at that moment.
I was so sick of rehab that the thought of an excursion, even if it involved more doctors, was vaguely appealing. Mum came with me, holding my hand in the back of the ambulance. Cars and traffic still terrified me, but the hovercraft delusion was holding up reasonably well. The ambulance driver was friendly, and I felt more like a well-treated parcel than an injured person by the time he delivered Mum and me to the patient transport lounge at Royal Melbourne. Appointment letter in hand, we followed coloured lines painted on the lino floor to find the right lift, the right floor, and eventually the right waiting room.
I sat in it with Mum and a variety of recently operated-on people, while the doctors and hospital staff moved around us. 'I am a patient,' I thought. 'This is what we do: we wait patiently, we move slowly.' I had a moment of horror when I recognised one of the surgeons. He'd been in the same year as me at college but studying medicine. I'd felt a tiny bit of pride on getting to Royal Melbourne without a major freak-out, despite the horror I still felt at cars, but suddenly I imagined how he might have seen me—as another patient on crutches, dressed in loud colours, or maybe just as a fairly standard knee injury. Thankfully, it wasn't he who called out my name, but another young surgeon, with a dapper suit and piles of curly hair.
Before he would see me, the surgeon sent in a nurse to undo the staples that had held my skin together. She spent some time hunting around in drawers and cupboards until she emerged triumphant with what looked like a giant pair of plastic preschool scissors with the tips bent in sinister-looking ways. She angled a spotlight at my knee and, one by one, removed the staples, leaving little red dots either side of the thick pink scar that curved around my kneecap. I asked Mum to take photos. They came out more theatrically than I expected, a rubber-gloved hand hanging in the spotlight above my newly unstapled knee.
The nurse piled the staples in a yellow plastic kidney dish. I eyed them off and had visions of taking them to a jeweller, and asking them to make a short chain from all these wonky 'W's—maybe enough for a bracelet? They'd held me together for nearly three weeks. I wasn't sure I was ready to let them go.
'Is it okay if I keep the staples?' I asked the nurse.
She wrinkled her nose slightly and smiled. For her, they were medical waste.
'I don't think so, but I can ask for you.'
She bustled out, and Mum and I sat there, waiting for the doctor.
I engaged in a short thought experiment, based on a popular reality TV show that I used to call 'my life'. It involved not catastrophic injuries, grief and prodding by medical professionals but standing up in front of my contracts law class, talking to them about implied duties of good faith. How to bring colour and movement to the crucial High Court case of _Hospital Products v United States Surgical Corporation_? Why not show them some photos of my knee, with thirty-three surgical staples, just like the kind the defendant had promised to distribute on behalf of the US manufacturer but decided to repackage as his own product? There was something macabre about this thought, about plucking a teachable moment from the wreckage that the accident had wrought on our lives. At the same time, I felt ruthless. The accident took so much from us, why couldn't I take a little back?
Suddenly, the surgeon was there, in his slim-tailored suit, the fabric creasing expensively around his own highly operational knees as he sat down on the swivel chair and riffled through my file. My leg was laid out on the bench, the velcro straps of the brace undone around it, but I had to swing it down so I could turn and face him.
'There wasn't a scan from before the surgery, was there?' he asked, without introducing himself.
'No, no... I don't think so.'
'Okay then; well, let's have a look.'
He briefly examined my knee, made a quick note and pronounced he shouldn't need to see me again.
'And you shouldn't need that brace anymore,' he added.
'But, um, will I be able to bend it again?' I queried.
'Yes, yes, of course—the physio will help you with that.'
And that was it.
As I gathered my crutches and bags and returned to the waiting room, I felt a little like Dorothy at the end of _The Wizard of Oz_. 'You mean all I needed to do all along was click my heels together and say, "There's no place like home"?'
It was somewhere in another queue—after we'd taken a wrong turn down another winding corridor—that I wept. The lino changed colour here. Corridors of one building joined to an older building, with old-fashioned skirting boards. It was as though the new hospital had swallowed the old one whole.
I'd circled this day on the calendar. It had pulled me forward as some kind of marker, some way to differentiate one babyless week from another. I'd started the day chipper. This was a step on my way out of rehab, on my way home. I'd dressed smartly, felt like I was carrying off the pretence of being a normal, capable adult. Once these staples were out, I had reasoned, I would be able to have a bath or go for a swim. In rehab I had been able to put myself together each morning—piece new outfits together from my new 'Crippled but Quirky' collection and propel myself out into the thin simulacrum of the world that was rehab's public spaces. And today we had braved vehicular transport, and navigated the long corridors of the Royal Melbourne Hospital to find the very dapper orthopaedic surgeon who granted me permission to bend.
I'd wanted the staples out, but the thought of bending, of walking unaided, of returning to this noisy, busy world that was so sharp with reminders of our loss, felt exhausting. My carefully gathered energy had brought me this far, but the lino corridors defeated me. I crumpled and wept. Mum held my hand, gathered me in a little, and then we went and found a wheelchair. We asked for directions, found another waiting room, and made it through the last appointment with the trauma consultant on the promise of a cuppa. I was relieved when Mum navigated us back to the caf. She went to buy the tea, and I looked around and realised that just over there was where we'd sat and drunk coffee after Z's funeral; and there, just down that corridor, not 20 metres away, was the room where I'd pulled that small, solid box onto my lap and kissed her cold face.
Part II
RE-ENTRY
9
Zombieland
Coming 'home' to our new house on Bayliss Street in the flat expanses of Melbourne's northern suburbs, nothing was familiar. In a matter of weeks, we had lost our home, our city, our jobs, our baby daughter and all our plans and hopes for her. We'd also lost our car, each of us had experienced significant physical injury, and our surviving children had lost their school and contact with their friends and extended family. No wonder I felt like an alien, as I crutched my way around our new neighbourhood. And if I, who was originally a Melbourne person and had all my family and old friends here, felt like an alien, how much more alien did Rima and the girls feel?
While I was in rehab, they had been staying at my dad and stepmum's place. Mum was living in her mobile home, which was parked in their driveway, and my brother Jeremy was sleeping in the rumpus room. It was like some extreme blended-family in-law reality TV challenge. Not only did they have to live with one another, grieve and recover, but my parents also needed to ferry Rima and the girls around to numerous medical appointments, visit me in rehab, and help us get ready to move into the Bayliss Street house.
Rima's day surgery to insert a metal plate into her hand was helpfully scheduled for the same day as surgery for Jackie's broken nose. Dad and my stepmum, Deb, arranged for my brother to drop Rima off at hospital on his way to a job, while they stayed with Jackie. Concerned that Rima would wake up from the anaesthetic alone and with no transport, I rang my friend Bins, who answered the phone at 7 a.m. and rushed out of bed to be there, Diet Coke in hand, when Rima woke up. As soon as Rima was released from post-op recovery, Bins ferried her over to my rehab hospital, both of them in good spirits. Rima, still dozy from the anaesthetic, fell asleep on my bed, and, mistaking her for me, the nurses tried to give her my painkillers.
In rehab, I was taking more and more photos; fewer of my bruises, and more of my face. I couldn't quite recognise this face. Something had changed. I pored over my face in the photos of me holding Zainab the morning she was born. The puffiness and blood-red hair were now gone, but when I looked in the mirror, my eyes had not returned to normal. There was a weariness, a knowledge, there that I didn't have before. I felt haunted—my body had been the scene of a death. I had carried death within me and birthed her. I needed to catch what traces remained on my face of the moment when her small soul flickered out from within my living one. How did her soul escape my body? Where did she go, and how much of my own soul did she take with her? Scrolling through the photos on Rima's camera, I saw something. Two shots where I had accidentally held the button down and closed my eyes. I flicked back and forth. Eyes open, eyes closed. Eyes open, eyes closed. Like an old-fashioned doll that you tilt to make her blink, lifelike but lifeless, blinking in disbelief. The image jumped slightly. This was not real life, just a photographic trick.
I'd never found horror movies to be up my alley but I now felt an empathy for the zombies, dazed and numbed by death. They don't know what has happened, and are still trying to work it out. Something terrible befell them, but they're still here, moving and apparently alive. Every now and then, I would wonder if I hadn't also died in the accident; that this was some dream or neural flicker.
After my clothes had been cut from me in emergency, and I'd spent my ICU days in half-undone hospital gowns, being able to wear proper clothes was re-humanising. The act of choosing my outfits became a ritual. I couldn't wear the same things I'd worn before—my post-partum body still couldn't fit into my non-maternity clothes—but I couldn't bring myself to wear the same maternity outfits that I'd worn only a few weeks ago. Instead, I tied scarves or soft fabric belts around my hair, like Alice bands; I put bright colours together, Frida-style. My mum lent me clothes from her eclectic wardrobe. I sought out big jewellery. I wore big red plastic earrings made from recycled monkeys in a barrel, my aqua scarf around my head, a soft grey shirt dress tied with the hot pink belt from my bathrobe. With make-up, I reclaimed my face from grief and the hospital environment.
Mum sallied forth with my absurdist shopping lists:
* • doorstop
* • bra
* • underwear (high waisted)
* • sticky tape or glue
* • _Where the Wild Things Are_
* • coloured textas or pencils
* • more fruit
* • Charlize Theron
I needed high-waisted underpants that wouldn't irritate my caesar scar; bras that were soft-cupped, but not maternity bras. Nanna underwear, it turns out, is the gentlest. Mum found bras for me that were designed for women undergoing breast cancer treatment. As I dressed, I felt solidarity for those women, sent them my love.
If I was demanding of family members, I was positively bolshie with the nursing staff. In retuning to this baby-like state, I had realised that it matters a lot how gently (or otherwise) caregivers provided their care. When you are dependent on others, small things can upset your routines, and familiar ways of doing things are inordinately comforting. I'd gone from a public teaching hospital with high nurse–patient ratios to a private suburban rehab centre with low ratios. Many other patients here were elderly and for them this was probably a step on the way into a nursing home. The night nurse often told me off for staying up late, and one night, without asking, she turned the lights out when I was awake and writing. I was outraged, and asked my mum to bring in a small lamp so that I had my own light source.
I'd been given a mattress with an extra-soft section at the foot, designed to avoid heel pressure sores for elderly people with paper-thin skin. But it meant that the weight of my foot pulled at my broken knee. When an ordinary mattress wasn't forthcoming after a few days, I picked up my bedding and napped on the floor; much more comfortable. I awoke to the nurse manager freaking out. Incident reports had to be filed, I got a stern talking to, and a new mattress arrived that afternoon.
The police officer handling our case had visited me in hospital once. He was an older bloke—small but wiry, and with an unironic moustache. 'Were you wearing a seatbelt?' he'd asked in an accusing tone.
My hand shot up to my chest, and the bruises that the seatbelt had left. 'Yes; yes, I was.'
When he returned to see me again in rehab, I was in bed. I asked when we might be able to do my police statement. I wanted it done before I was discharged, so we could start a new chapter in our new home with, at least, our part in the criminal investigation resolved. His preference was to do it once I was at home, so that we'd be uninterrupted. He was testy. 'I've already got people on my case, complaining about keeping him in custody over a week because he's an Indian student.'
His remark got my lawyer hackles up—the driver who caused our accident had to answer for his actions, not for his cultural heritage. But I wasn't going to engage in that debate with the police officer in front of me; we just needed to sort out how this interview was going to happen. I asked what information he'd need, so I could make notes, and he spat back, 'You should know, you're a lawyer.' We went back and forth, and suddenly he was yelling, telling me that we'd do this interview on _his_ terms, not mine. I agreed—anything to get him out of my space. I was shaken, and hot with shame.
The friends who were renting our new house had arranged to vacate it a few days early. Our furniture was already there, waiting for us in the shed, but neither Rima nor I were physically able to move it. So I put the call out on Facebook, and suddenly we had a schedule of helpers, assembling furniture, moving beds, stocking our fridge. Moving day coincided with my second-last day in rehab, so that I would have a properly assembled bed to come home to the next day.
While friends shifted our furniture, Jeremy sat with me in an interview room at the rehab hospital for my police interview, this time with the head of the collisions unit. After the incident with the leading officer on our case, I'd called and requested that someone else interview me. Jeremy sat with me and listened, giving a silent hand squeeze when I needed it.
On my last night in rehab, I stayed up late writing, despite the night nurse making firm and sensible suggestions about getting some sleep. The light from the small lamp Mum had brought in for me bounced off the mirrored frame around Z's picture, sending small spirals and leaves across the wall.
Thursday, 14 January 2010
Oh, my little one. I wanted to be setting up our house to be babysafe, not Zimmer-frame safe. I wanted your cot in our lovely front bedroom, where the bay window lets the light in, not physiotherapy equipment to help me lift my poor leg and reach for things. I was sick of rehab, and delighted at the thought of being in our own space, but I was also terrified of leaving my bubble and re-entering 'real life' without a Haloumi in my belly or a baby in my arms.
The next morning, my dad and stepmum came to take me home. I sat in the back of the car, my sore leg lying across the bench seat. The traffic made me nervous. All these death-machines, pacing in our midst like semi-tamed tigers, and everyone but me oblivious to their teeth and claws. I held on, and we made tense little jokes about hovercrafts. When we arrived, Dad helped me out of the car, giving me my crutches so I could meet Rima on the threshold, with tears and an embrace.
I sat on a chair in the garden—our garden—in the northern sun, and our little dog, Eddie, leapt up onto my lap. Big dog Atari would have leapt up too, were it not for Rima protecting me from his enthusiasm. The dogs and cat had spent a much longer holiday than expected at the kennels where we'd dropped them off just before Christmas. In the sterile, shiny hospital space, I had missed their abundant furriness, and their wordless, boundless affection.
That first day at home was razor sharp. It was so perfect in every way except one. Here was the room we'd dreamt of, but where was the cot? Here was the garden I'd always wanted, but no small feet to take their first steps on the grass. It felt like a grisly nursery rhyme from an era when death was mundane enough to feature in children's games: 'down will fall baby, cradle and all.' When friends came to help with the unpacking, weariness hit me and I slunk back into bed to hide and weep. I'd always been relentlessly sociable, but now I prioritised my own need for quiet. I couldn't explain this and I couldn't make it nice.
When I started unpacking, I found the jumper I'd worn on the day of the accident. Unlike all the other clothes I'd worn that day, it hadn't become bloodstained or been cut up, because I'd taken it off before we got in the car. I held it up to my body and looked in the mirror. The jumper was slack and flabby from where my big, taut belly had been. _This_ , I told my weeping self, _this is proper crazy-lady grief_. A small, mean voice in my head wanted to call it histrionics, to slap myself out of it, but this grief was bigger and more elemental than anything I could put into words.
That night was the first since the accident that Rima and I could sleep in one bed together. Unlike the surreal one in ICU when she'd slept on two chairs pushed together beside my bed, we could embrace properly. We held onto one another, shipwreck survivors in a salty ocean of tears.
10
The 'born alive' rule
Home didn't quite feel like home those first few weeks—here were our familiar things but oriented differently in a new, unfamiliar space. Our bed in Sydney had faced the door, but here friends had assembled our bed next to the door. My entire life had been turned 90 degrees; everything was off-kilter. I had to remember the way to the bathroom. I noted down each medication I took. It felt remiss suddenly to live unmonitored after all the heart rate monitors and oxygen saturation clips that we'd relied on for proof that I was alive.
Saturday, 23 January 2010
It's 4 a.m. and I'm awake again—thinking of her being cut from my body. Four weeks ago today was Boxing Day. A day to sleep in—or in my case, wake up, eat enormous pregnant woman breakfast, then go back to bed.
At that point, we hadn't even decided whether or not to go to that stupid picnic on the 27th—our fate was still open. I know I'm torturing myself, but I still wish we'd decided not to go. All I wanted from that day was to hang out with my mum for a bit, do as little as possible and enjoy being pregnant after all the crazy moving interstate and Christmas business. Why did I feel the need to drive around seeing people?
Four a.m. would still find me awake most nights. I'd started reading again, frightening myself with the dystopian future Margaret Atwood created in _The Year of the Flood_ and making things worse. Some nights I'd call friends living overseas to quell the rising tide of my own fear and panic. There I was, limping around the bathroom floor in my summer nightie at four in the morning, ranting like a madwoman to my dear friend Will while he was at work in a London law firm.
Our days were structured around appointments: physio, psychologist, occupational therapist. In between, we unpacked and worked towards two dates: the girls' first day at their new school, and Z's memorial on 7 February, to mark _al Arba'een_ —forty days since her death and the end of the initial period of intense mourning. As it turned out, it also corresponded with her due date; the week I had imagined I'd be waddling around, impatient for her to be born.
I was avoiding the _why_ question whenever it popped into my head. The newspapers Dad had brought me in hospital were full of pictures of the Haiti earthquake—more senseless trauma and loss. I could remember that feeling of being crushed. Among the people buried in rubble, was there a pregnant woman? For me, it had only been a matter of minutes of being crushed and trapped before I knew that fire and ambulance crews were on their way to free me. Meanwhile, earthquake victims were trapped, half-crushed, for days, and with no certainty of good medical care if they were freed. Their pain and loss, unlike mine, were compounded by poverty, and indifference from those who could afford to help.
At home, I started creating more of the blinking portraits, using Photoshop to create animated portraits punctuated with sighs, swallows and blinks. The image would jump unnaturally, disclosing that it was a trick, a sleight-of-hand substitute for the real thing. These moving portraits were my own little robots, helping me carry out the weary waiting and time-marking of grief. I created loops of thirty-three, thirty-four, twenty-seven or twelve seconds, putting to work the numbers that still buzzed in my head—my age, Z's gestation, and the date of the accident.
I made a chronology of the photos, starting with the ones Mum took at the picnic before our accident, then the CT scans on the night of the accident, then the bruise photos, and then the self-portraits. In that sequence, it was as though the accident turned me inside out, and then right way out again.
•
I was making small excursions out into the world, but everything felt other-worldly. I wondered at people in the supermarket—'These are the ones who survived,' I thought. For all these people walking around, how many babies were there who didn't make it? I was incredulous at all this rude life in front of me. The sight of sleeping babies made me panic. How did the parents calmly pushing a pram know whether their child was still breathing? Could I see the little chest rise and fall?
Friends lent us a car, and Mum drove us around to our many appointments—GPs, hospital, counselling. I had wanted to take public transport but was told the risk of a fall (and of more internal bleeding) was too high. I was getting used to cars again, angling my broken knee across the back seat, holding onto the door handle of the taxis I took to and from rehab. But getting into the car with Rima, Jac and Jaz for the first time, to go to a movie, my slow-earned composure was suddenly gone. The sight of my legs in front of me, my feet in the same Birkenstocks I'd worn at the time of the accident, made my mouth twist into unfamiliar shapes.
'Hannah.'
My chest was heaving and it was hard to place Rima's voice.
'Hannah. You're freaking the girls out. Stop it.'
Rima's voice and her hand on my leg brought me back, and somehow I caught up with my own racing breath.
•
I was doing my physio exercises one day when the door slowly swung open, ghostlike.
'Hello?'
No one answered, but a moment later the cat slunk in. He was cautious in this part of the house. Until very recently, he'd been confined to the laundry, so he could acclimatise slowly.
Like the cat, I was gently sniffing out the expanding boundaries of my grief. At first, grief (and my bodily injuries) confined me to a white bed, then to a room, then a hospital corridor. Now the doors had all been opened and I was still cautious about where I set down each paw—wary of outside spaces and terrified of running into the territory of someone else's grief. Besides, I still had so many tender spots to investigate right here: cupboards to stick my whiskers into, beds to slink under, suitcases to peer into. The boxes of baby things that had come down with all our Sydney belongings had been repacked, into the furthest reaches of the shed.
In my dreams, cars claimed more people I loved. My best friend and her husband, crossing the road. It was sunset, and the western sun blinding. Birds swooped and confused the driver. I couldn't stop it happening, no matter how much I screamed. I dreamt I stood on a crowded platform and as I was trying to get on the train, a woman came up and whispered in my ear, 'Excuse me, are you forty-one weeks pregnant?' I shook with anger and sobs, and recited my line, 'I'm not pregnant anymore.' I dreamt that I birthed Z naturally but she slipped out and hit the floor. No one was there to catch her. And she was as white and unmoving as when I'd last seen her. I woke up weeping. When I looked over at the clock, it was 4 a.m. again, the time she was born. She and I, both cut from the wreckage.
In that time between coming home from rehab and the memorial, we hovered. I hummed Dusty Springfield—I didn't know what to do with myself. Some mornings were harder than others. After a particularly bad night, I took a photo of myself with each item of clothing I took off, casting off the nightmares and capturing the now yellow-greenish bruises, before a long, weepy shower. As I dressed, I took another photo for each item I put on, layering myself against the day. I imagined flicking through the photos, a jerky stop-motion film, or a nature documentary of a snake shedding its skin.
The newspaper was not quite a highlight of the day, but it was evidence that time was moving, that each new day was, apparently, distinct from the previous one. One Monday, we again found ourselves the subject of a newspaper article. Our case was one of two road accidents in Victoria that summer where a baby in utero had died. The other woman had lost not just her child in utero but also her partner. I thought of her every time I held Rima tight.
Our cases had reignited debate about the criminal law's response. Australian law followed the UK common law approach, in that criminal charges could only be laid regarding a 'death' if the person had been 'born alive'. Birth, therefore, was the moment when a legal person came into existence. Harm incurred to a fetus in utero could sometimes give rise to criminal or civil liability, but only where the baby had been born alive. This meant that, for me and for the woman who lost both her partner and her baby that summer, the charge of dangerous driving causing death could not apply because neither of our babies had shown signs of life when they were born. I had vague memories from law school of debriefing over coffee with friends after a criminal law lecture on the 'born alive' rule. But the Hannah who had sat through the lecture and could discuss death and birth in the abstract now felt as remote as a minor character in a book.
When I looked up the cases, they were grisly. In a key UK case, for example, Mr B had stabbed his partner, Ms M, with a long-bladed kitchen knife when she was around twenty-four weeks pregnant. The knife penetrated her uterus and nicked the fetus. Ms M survived the attack, had surgery to repair her wounds and was discharged from hospital, still pregnant, and with the fetal heart making its reassuring swishy beat. But seventeen days after the attack, she went into premature labour and her daughter, S, was born, 'grossly premature'. Baby S had surgery to repair the knife wound to her abdomen and lived 121 days before dying of bronchopulmonary dysplasia. There was no evidence that the knife wound contributed to her death. Rather, she died as a result of her premature birth, which in turn was caused by the injuries inflicted on her mother when Mr B stabbed her.
Mr B was convicted of wounding Ms M with intent to cause grievous bodily harm, and after S died, was charged with murder. The matter was appealed all the way to the House of Lords, which held that 'a foetus was neither a distinct person separate from its mother nor merely an adjunct of the mother, but was a unique organism'. In these circumstances, this meant that a murder conviction could not be sustained, but that Mr B could be guilty of manslaughter on the basis that his act in stabbing Ms M was 'both unlawful and dangerous because it was likely to cause harm to some person' and that it had caused the death of S. Had S been stillborn instead, however, the 'born alive' rule meant that the only criminal charge would have been for causing grievous bodily harm to Ms M.
The newspaper article pictured Nancy Asani, standing (bravely, I thought, given the circumstances) in front of a car. Nancy's story was horribly familiar. In 1999, she was thirty-seven weeks pregnant (also with a little daughter) when someone driving without his headlights on ploughed onto the wrong side of the road and into her car, injuring her and killing her baby. A ripple of feeling went through me—anguish that anyone else should have been through what we were going through, but also a weird relief, of knowing that we were not the only ones. I was torn between wanting to find Nancy, so I could give her enormous, weepy hugs, and throwing the article in the bin, so I didn't have to think about it.
At the time of Nancy's accident, there were two possible driving offences: culpable driving causing death, which carried a sentence of up to twenty years, and dangerous driving, a charge under the _Road Safety Act_ , which carried only a maximum two-year term. Because Nancy's baby, Meriem, was not born alive, the driver who caused the accident could only be convicted of dangerous driving in relation to the injury to Nancy, and was given a twelve-month sentence, two-year suspension of his licence and a $2500 fine. Since then, Nancy had been campaigning for law reform.
And, in fact, the law had changed since she'd lost Meriem. Legislation in 2004 had created a new offence of dangerous driving causing serious injury or death, which was designed to fill the 'gap' between culpable and dangerous driving, and carried a maximum term of five years prison. Then, in 2008, the same legislation that took abortion out of the _Crimes Act_ also clarified the definition of 'serious injury' to include 'the destruction, other than in the course of a medical procedure, of the fetus of a pregnant woman, whether or not the woman suffers any other harm'.
This amendment made the Victorian legislation consistent with the law in New South Wales, where the parliament had followed the lead of the Court of Criminal Appeal in the case of _R v King_. In 2002, Phillip King had discovered that Kylie Flick, the young woman he'd had a sexual encounter with, was pregnant. He was unable to convince her to terminate the pregnancy, and had allegedly offered friends $500 to punch her in the stomach, but with no takers. In August 2002, when Flick was six months pregnant, and was moving house, King contacted her, saying he wanted to 'say goodbye to her before she moved'. He came over to her place, and Flick's evidence was that they spoke for about an hour, before 'everything went silent, she heard a rushing noise and as she turned she felt pressure hit her stomach'. Flick testified that King punched her to the ground, then stomped on her stomach 'six or seven times'.
In King's evidence, he recounted that he had been angry with Flick for threatening to tell his girlfriend about the pregnancy, but the trigger for the attack was Flick's action in lighting up a cigarette:
Q. Did you intentionally aim for her belly?
A. I wasn't—at the time I wasn't thinking, I just—everything, we were talking—everything was fine, then after—
Q. Everything was fine was it?
A. We were talking—we were talking about things and then—then things got brought up about she was going to tell my girlfriend and then I seen her light up a smoke and... then I just lost it and punched her in the stomach, she fell over and then I stomped on her right arm.
...
Q. So you cannot be any clearer about why it was that you punched her in the stomach?
A. After that she lit up a smoke, I just—
Q. And did the cigarette smoking enrage you did it?
A. That and other things she was saying to me.
Q. Because you were concerned that it was going to damage her baby?
A. Yes.
Q. And you went along and damaged her baby instead?
A. Yes but I just—it just happened, it's not like I—I just clicked and done it, I didn't have any—yes that is.
I had to read that part of the judgment several times, just to make sure I hadn't read it incorrectly. When I'd been pregnant, I'd joked with other women about the death stares you sometimes got if you were seen holding a glass of something alcoholic. But here was a direct line between the notion that pregnant women were to be policed—that their bodies were open for public judgement and advice—and a savage act of violence.
Flick was taken immediately to Bankstown Hospital, but no fetal heartbeat could be found, and her son was stillborn several days later. King was charged with maliciously inflicting grievous bodily harm with intent to do grievous bodily harm. A district court judge, however, ordered a permanent stay on the charges because she found that the Crown 'could never prove that the demise of the fetus itself and/or the abruption of the placenta amounted to grievous bodily harm to the complainant Kylie Flick' because a fetus was a 'unique organism' and therefore not part of the mother. The Criminal Court of Appeal overturned this decision, with Chief Justice Spigelman finding that:
The close physical bond between the mother and the fetus is of such a character that, for purposes of offences such as this, the fetus should be regarded as part of the mother.
The 2008 Victorian _Crimes Act_ amendments also split 'dangerous driving causing death or serious injury' into two distinct offences—one for death (a maximum ten-year sentence) and one for serious injury (a maximum five-year sentence). Yet, for Nancy, 'serious injury' couldn't quite sum up the enormity of her loss, and she was lobbying for something more: something 'to cover the death of unborn babies on the road'. She had a point—losing Zainab was qualitatively different from suffering internal bleeding—but I was hesitant about any proposed law that was directed at a baby in utero without addressing how it might affect the person in possession of the uterus. Aside from telling Nancy's story, the article quoted various legal experts, alongside the 'Australian Family Association', a conservative lobby group that sought to exclude us, and any other families that didn't feature a husband and wife, from the definition of 'family':
Australian Family Association spokesman John Morrissey said the inconsistencies in the state's laws appeared to exist because of 'fairly permissive abortion laws'. He supported a new law to cover the death of unborn babies in road collisions. 'An unborn baby, we all know, is no different really from a baby who was delivered maybe two days later and equipped with a birth certificate. It's nonsense to distinguish between the two,' he said.
I was incensed. Not only did Mr Morrissey reduce the pregnant woman surrounding any 'unborn baby' to 'a nonsense', but he had the nerve to set this up as an either/or choice for women. Either we can access safe, legal abortion, or we can have the law recognise the harm caused when someone else's violence or carelessness ends a pregnancy, but not both. I sat down and wrote my second letter to the editor that month.
Wednesday, 27 January 2010
One month ago (though it feels like another lifetime ago), I was 34 weeks pregnant and was driving home with my family. A four-wheel drive hit us head-on, and (among our other injuries) caused my placenta to abrupt and killed my little daughter before she was born. I was so sad to read ('Mother vows to fight on for law change over road death of unborn child' _The Age_ 25 Jan 2010) that Nancy Asani suffered a similar loss in December 1999, and that another woman also lost her baby this 'holiday' period.
We would support Nancy in her campaign to have the law changed to recognise that dangerous driving causing the death of an unborn baby is not just an injury to the mother. It was an injury to me, but in a much more profound way than my other injuries. Our baby, had she been delivered before the accident, would have had excellent prospects of survival—she was already 2.5kg (around 5lbs) and 48cm long. I can't put into words what we have lost and what she has lost.
What I find offensive is that anyone could try and twist our tragedy into some kind of argument against safe, legal abortion. John Morrissey, spokesman for the 'Australian Family Association', has done this in your article on Monday. How dare he try to appropriate our loss and turn it to his own political/religious ends. We were lucky not to be in a position where we had to consider abortion, but I have had a number of friends who have been in that awful position, and it is not something any woman considers lightly. Women are not stupid—we know that pregnancy is the process of turning a potential life into a living breathing child. That is what makes our loss so heartbreaking. To try and draw some connection between abortion laws and recognition of my and Nancy's loss as a loss of life is offensive and ignorant.
Hannah Robert, Preston
The newspaper published my letter, but edited it slightly, replacing 'my little daughter while she was still in utero' with 'my unborn daughter'. Nine characters rather than forty-seven. I could see the logic, but I had deliberately avoided using the word 'unborn' for Z. 'Unborn' sounded like a horror movie, _Return of the Undead_. It was a looking-glass word—the opposite of 'born', but creepier. And it was incorrect. She had been born, just not while she was alive. She wasn't un-anything. 'Unborn' sounded like a train cancellation, an almost-something, a hypothetical loss; as though you could just press the 'rewind' button and become magically unpregnant. I preferred 'dead' because it was solid, it had a weight like her two-and-a-half kilos in my arms. I knew the legal technicalities, but they were just that, about as relevant to my grief as the lot number of a land title was to the home where we lived.
'Unborn' was a grisly image on an anti-abortion protester's poster; it was an intractably polarised moral debate; it was a word I could not reconcile with my beautiful, whole, soft-cheeked daughter. I knew what my views on that debate were. Even when I'd been pregnant and in wonder at the life within me, I knew the difficult decisions that came with this amazing potential could only properly be made by the person whose body contained it. That decision—to continue or to end a pregnancy—was both a medical one about your own body and a fundamental parenting one about the quality of life you could give your potential child. I wasn't about to be forced to trade my basic beliefs about equality and autonomy for the right to grieve my child. That our loss could be churned into political capital by people who sought to impose their moral or religious views on women facing extraordinarily difficult decisions about their pregnancies made me feel sick.
I'd started seeing a psychologist at the Royal Women's Hospital, next door to the hospital where I'd spent those long days in emergency, ICU and the trauma ward. I mentioned to her that coming to the hospital for my appointments was upsetting. By this, I meant having to walk past the room where we'd held Z's funeral, and the cafeteria where we'd sat afterwards. But she gave it a very different spin.
'I know; it is hard. Did you know, there are more pregnancies terminated in this hospital than there are babies born? All those people aborting when you've just lost your child. I can imagine that is very hard.'
I was too stunned to reply on the spot. It hadn't occurred to me to begrudge others the ability to end a pregnancy for whatever reason they saw fit. I didn't want _their_ pregnancies or the babies that might have resulted—that was their business. I missed _my_ baby. The value I placed on her was not that she could be categorised as a fetus of a particular gestation. She was not a generic fetus, she was our daughter, and it was our relationship with _her_ specifically that I grieved. Had things gone differently, who knows what decisions we might have had to make, even though she was a desperately loved and wanted baby. That love and longing to hold her in our arms did not cancel out my concern for my own (and other women's) bodily autonomy. I had started to connect online with other women grieving their babies. One had needed a second-trimester termination to prevent pre-eclampsia from taking her life; another had taken the decision to still her baby's heart in utero in the third trimester after learning that he had a condition that made him 'incompatible with life'. These were pregnancy decisions with no happy options, but they belonged unquestionably with the pregnant person, informed and supported but not dictated by medical advisers, partners and loved ones.
•
On a Thursday afternoon four weeks post-accident, a few days after I'd been released from hospital, we were sitting on the front porch behind the rose bushes when the postie came. This was as public as I could handle being at that time, with a thick row of thorny bushes between me and anyone I didn't know. Rima collected the mail straight away. This was about the only time in our lives together we'd actually both been home on a weekday to see the postman. Among the mail was a post-pack with six CDs of images from the Royal Melbourne radiology department.
We loaded up the first one ('CT–Trauma series 1/3') and were told it had 2166 images on it. We didn't get past the first image. It showed my body, from my neck down to about my shins, with my arms held above my head.
I remembered this being taken. It hurt so much to lie like that, and you could see the pain in the awkward, lopsided way I was lying to try not to put pressure on the sorest, most broken bits of my body. The time on the image was '20:26'. It was after I'd found out that Haloumi had died, but before they had operated to take her out and to repair my knee. And you could see her there, curled within me. To my un-medically trained eye, she looked for all the world like a beautiful, healthy living baby. My flesh and hers looked the same—how could mine be living and hers not?
That Thursday night, I had so many dreams it was hard to believe they could all fit into one night. But, best and hardest of all, Z visited me in my dreams. I was on the CountryLink train to Newcastle and she lay on the tray table before me; as cold and still as when I'd last seen her. I was watching over her. I had to get her safely to her destination, but the sun coming through the window was warm and the lull of the train moving made my eyelids heavy. I realised with a shock that I'd fallen asleep and guilt shot through me for breaking my vigil. But as I awoke, her eyelids fluttered open, just as mine had. I embraced her and wept, _You're alive! Oh, my little love, my little one!_ This time, constant waves of baby expressions animated her face. She looked directly at me and I heard her say, in an adult voice, _Are you okay?_ Before I could reassure her, I was waking up again, this time for real.
11
Sun salute with bedpan
The hardest and the best bit for me about yoga is when you are doing something difficult, when it feels like your bones just cannot move the way you're asking them to. It's uncomfortable and your initial reaction is _No—enough; I can't do this_. But then you notice the discomfort, acknowledge it, breathe in, and then, as you breathe out, move past it. You ask an open question of your body, and sometimes it responds in surprising ways. Things unfold, settle, stretch. And you realise that the thing you had thought was unimaginably difficult... well, you've already been doing it for thirty seconds. The answer was there all along, within you; you just needed to ask the right question, and to listen patiently for an answer. It doesn't happen all the time, but when it does, it is good. It restores my faith, it reminds me that sometimes my body knows more than my mind.
When I'd still been in the trauma ward, I had been desperate to move my creaky body, all gummed up with bruises. What I really wanted to do was yoga—some kind of yoga that was possible with my broken bones and wounds.
I left a long, garbled phone message for a woman in the physiotherapy unit at Royal Prince Alfred Hospital, where I had attended a yoga session only a few weeks before we left Sydney. This wasn't the usual yoga environment, with candles, pictures of lotus flowers or other hippy accoutrements. There was hospital carpet, a sensible colour scheme; it was institutional, rather than inspirational, décor. I hadn't been to this class before and didn't know anyone, so I shrank into myself a little. The room was packed with very pregnant women, all taking up more space than they were used to, and with no non-pregnant people to make room for them. So entrenched was I in solitary mode that it took me a moment to realise I did in fact know someone—an old friend—and that she was waving at me.
'Hello, Renee!'
'Hannah!'
We hugged, making an awkward A shape over two big bellies. 'When are you due?' she asked.
'February! How about you?'
'January! And you know Karin is pregnant too? Due in December.'
'No, I didn't know. That is hilarious! Three in three months. Are they still in Paris?'
'Yes, having a little French _bébé_! Um, we'd better...' The instructor, an older woman with a loose grey bun, had come in and class was about to start, so we found our places and smiled at one another in a 'Let's talk later' way.
•
The RPA instructor didn't get back to me, so it wasn't until I'd been released from rehab that I ventured out to a yoga class. I'd spoken on the phone with the owner of the studio, as I needed to deal with someone who understood the backstory. Penny and my sister came with me, flanking me as I, with my crutches, worked my creaky way up the long staircase to the yoga studio.
•
Yoga was my answer to my official rehabilitation plan, which mostly involved taking Panadeine Forte and screaming into a pillow while the physio put his body weight into forcing my knee to bend. Afterwards, he would measure the new degree of flexion with a giant Perspex protractor as though I were a Year Nine maths project.
'Why does it hurt this much?' I asked him.
'Your knee was healing straight for three weeks, so we need to break down a whole lot of scar tissue in the joint.'
'And when you say "break down", do you mean tear it? Because that's what it feels like.'
'I guess so. But it has to happen if you want your full range of motion again.'
He was a nice enough bloke, but I hated him for his casualness, for the fact that he didn't take my pain personally, and that he was more concerned with refining the details of the TAC paperwork than with my suffering.
•
Before the yoga class began, the teacher approached us, noticing that we were new.
'Hi, I'm Jess,' she smiled.
I explained that I'd called in advance to say that my knee could only bend so far, that I was also recovering from a C-section, that our baby died. It is hard to remember that first conversation with Jess without the overlay of all my subsequent conversations with her, and all her gently worded yoga instructions. But I know that she contained her shock, that she didn't do the 'tragedy recoil' that so many people unconsciously do. And after offering her condolences, she smiled—not to trivialise what I'd just told her, but as pure kindness; as a reassurance that she would watch out for me, that this was a good place to be, even if I came here broken.
•
At home, in our new bedroom, I laid out my yoga mat, and stood, feet hip-width apart. I flexed my toes, spread them as wide as I could and, from little toe to big, re-placed them on the mat. _Breathe in_ , hands to heart; then, breathing out, I swung my right leg back behind me, making a broad-based triangle. No, nothing was really triangle-shaped. I was rusty and bruised, still regaining sovereignty over the remodelled territory of my body. My body facing to the side, I raised my arms to stretch out my hands to either horizon, and turned my head to face my left hand. I breathed in; and then, with the out-breath, I bent that once-broken knee as close to ninety degrees as I could, and focused my gaze along the middle finger of my left hand, like a magic laser beam. _Warrior Two_. Rima and my mum stood in the doorway and cheered. My gaze took a direct line, into space and into the path of a hurtling silver four-wheel drive. _Bring it on, universe—if you want to mess with me, I will take you on_.
12
The posthumous godfather
Thursday, 21 January 2010
Dear Joan,
I hope this finds you and the rest of your family well. I've been thinking of you often since David's funeral, but particularly in the last three and a half weeks. Three and a half weeks ago something awful happened—while I don't want to distress you, I feel like I need to tell you because I have an odd request to make of you. If it is too much, please just let this letter go by the by and I promise never to pester you again, and apologise for pestering in the first place. But grief is a strange thing which can make you behave quite oddly, so I feel compelled to write on the off-chance that you don't mind.
Three and a half weeks ago I was involved in a serious car accident. I had been driving to my dad's house with my partner Rima (who you met briefly at David's funeral) and two of Rima's daughters (my stepdaughters) in the car. I was 34 weeks pregnant—with a baby we had been planning and trying for for about four years via assisted conception.
Four of us survived the crash, but our baby did not. We were sent to three different hospitals—all injured but thankfully with no serious brain or spinal injuries. We are still recovering now, and will be for some time. I am writing because in the days after our little daughter died, the thought which was driving me mad with grief was the thought of her being alone—crying and not understanding why we were not there to comfort her. No baby should be alone and uncared for.
And then, after I thought my heart would burst with tears, a little comforting thought came to me. I thought, this is why parents give their children godparents—to care for their child when they are not able to. And that, wherever our baby is, we would need to find godparents in that place to hold her and to explain to her how much we love her and how badly we wished we could care for her ourselves, but that a terrible thing had happened—not her fault and not our fault—which had stranded us in different worlds.
So we needed to think of godparents who were in the same place as our little girl. We first thought of Rima's cousin, who was like a brother to Rima, and passed away very suddenly in April 2008. We also thought about a dear family friend of mine in the UK, who had died of breast cancer nearly two years ago, and who had been like a mother to me at several times in my life when I really needed her.
And I thought of course of David. Because David really was more like a godparent than a mentor to me in his generosity with his time, energy and guidance. He shaped my teaching, my research and my writing more than I realised at the time, but was also incredibly supportive in his personal capacity.
I imagine he might make a slightly gruff, but very loving and extremely knowledgeable and protective godfather for our daughter. We would be honoured if you felt able to give your consent as earthly guardian of David's memory.
I'm not sure what your beliefs are on what happens after death, so I sincerely hope this request is not an offensive one. Up until now I'd had the luxury of never having to think too hard about it. I'm still quite fuzzy about it but I realise I do believe that souls go somewhere, and that where that somewhere is really doesn't have too much to do with what religion the person has followed during their life. I really hope that this is the case, because the three people we would like to have as Z's godparents are each of different faiths—Ahmed Muslim, Rosie Christian and David Jewish. If this godparent arrangement 'works', at the very least she will be well educated in the religions of the book!
My partner Rima has similar beliefs to mine, but within the framework of the faith she was raised within—Islam. It was very important to her that our child be Muslim (at least until she could make a decision for herself) so we have given her Muslim rites and an Arabic name.
This is all a lot to take in. Please have a think about our request—if there is any more information you need to know please let me know. Or, if it is easier, you may wish to talk with Rosalind (who also came and spoke at David's funeral).
We are having a multi-faith memorial service for our little girl on Sunday, 7 February down at Somers on the Mornington Peninsula. We have included an invitation and you would be very welcome, but please don't feel obliged to come. I realise interstate travel is expensive and disrupting. If you are happy for us to nominate David as a godparent, we will include something brief in the ceremony to this effect. We're not sure how cryptic we will be about it—the whole concept (as you can see from this letter) takes a bit of explaining.
If you could let us know either way before the 7th of February—either directly or via Rosalind—that would be much appreciated.
With love and the fondest of memories of David,
Hannah
•
Prior to our accident, my longest stay in hospital had been as a seven year old. I'd always complained of 'tummy aches', but the winter my baby sister was born, the tummy aches got worse, and were eventually diagnosed as recurrent kidney infections. My mum realised that this wasn't an ordinary tummy ache when I couldn't leave the couch, squirming and weeping with the pain. Even the novelty of having a doctor come to our house was only a minor distraction from the dull but intense ache in my side. Once the kidney infection was diagnosed and being treated, I underwent an unpleasant test involving a catheter, an isotope and an X-ray machine, and it was revealed I had a congenital defect. The ureter on my right side was misplaced, and was allowing urine to flow back up to the kidney; hence the recurrent infections and damage to that part of the kidney. I was going to need an operation to correct the ureter and to remove the damaged part.
At seven, I was delighted by this news. In primary-school currency, a broken arm or leg brought instant popularity; the thought of hospital and an Operation (just like the 1980s electric board game!) made my small mind explode with the possibilities. Indeed, when the Operation was delayed a week, I turned up to school, and everyone was making get-well-soon cards for me and hastily had to hide them away. I still have the photo of all those cards Blu-tacked around my hospital bed, with me grinning smugly in my new pyjamas in the middle of them all, clutching my Care Bear—another hospital trophy.
It was in that hospital bed that I learned to read properly. Not to spell out letters or say the words; I'd learned how to do that at school. But to read—to breathe in a story, to weave your own dreams from its dangling threads, to leap wholeheartedly and without realising you've leapt into another person's world. At the start of my week in hospital, my parents took turns reading to me; by the end of the week, I was reading to them.
The book was _The Brothers Lionheart_ by Astrid Lindgren—it still rates as one of my favourite books of all time. It didn't strike me until recently that perhaps a book that starts with two little boys dying might be considered morose reading for a seven year old in hospital. But it wasn't morose, not in the least, because dying was just the kicking-off point for marvellous adventures for these kids, in a world where they could fight dragons, and lead revolutions, and learn that sometimes, even people you loved failed you. This wasn't 'heaven' and it certainly wasn't a cushy affair with clouds, harps and eternal life. In my pink pyjamas and with my seven-year-old certainty, I wasn't scared of dying, and I didn't find out until years later what a close thing it had been, for a moment there. But when my grandparents died, I thought of them as there, in Astrid Lindgren's Nangijala, going on with their slightly more adventurous lives and sending us a dove every now and then.
I hit a snag, though, in another hospital room twenty-six years later, when the buzz of medical people doing things to various remote parts of me had stopped at last, and I was left alone, and with-it enough, to think for the first time since the accident. My belly was still so swollen, but not with Z—so where was she? We'd just farewelled her cold little face, so where was my moving, hiccuping baby? What exactly did I believe happened after death? Rubber, meet Road.
I'd been to Sunday school and to church with my parents for a short period, but none of it rang true for me. If I had to sit through a Christian service, I would find myself going in argumentative circles in my head. So if I couldn't bring myself to believe in heaven or hell, could I believe in Nangijala? Another life, just as mortal and complex as our own, with some familiar characters but different props? It seemed to work okay for my grandparents, but what about Z? She was too small for adventures, too small for riding horses and fighting dragons. She was still too small to be away from me and my heartbeat, or even to know how the whole communication-via-doves thing works. If small babies have trouble knowing that their parents still exist when they play peek-a-boo, then what hope did Z have of knowing how much we loved her, stranded as she was from us by death?
All I could think of was her wailing, in a rustic-looking basket on someone's stone doorstep, and her little hands searching, and rustling the swaddling clothes. Someone would come, of course, but who? Some anonymous pre-modern wet nurse? I couldn't work out which was worse—to think of her annihilated and stopped forever, or to think of her continuing on without us, lost and disconnected. Both scenarios made me howl and choke.
This was why I had to invent the idea of godparents, to populate her imaginary world with people we loved, who knew us and who could tell her how much we loved her, who could sort out doves for her. It kind of worked, but it still felt like an invention, a delusion to make things feel okay. And it still tore my heart to imagine her crying, and not being able to pick her up.
The week after I had been released from rehab, I sat down to write my rather odd letter to Joan Philips, the mother of the late David Philips, who had supervised my master's thesis, and who had employed me in the history department for four years as a sessional tutor while I was finishing my MA and law degree. David was South African. After doing his first degree, at the University of Witwatersrand, he'd won a Rhodes Scholarship to study for his PhD at Oxford. He was passionately involved in the anti-apartheid movement and decided he did not want to return to South Africa while that regime prevailed. So he took a job at the University of Melbourne, where (a couple of decades on) I encountered him in a first-year subject on comparative colonial history. He was an imposing man—an associate professor by that stage—and he unapologetically took up space both physically and intellectually, and demanded that you, in turn, stand your ground and explain your position. He was fierce, but funny and good hearted. In a year when I was heartbroken from breaking up with my first girlfriend, David and my co-supervisor, Pat Grimshaw, helped me refocus on the thesis and get it written.
In August 2008, David had just retired, and was on holiday in Broome, when he died suddenly of a heart attack. Rosalind Hearder, a friend and colleague who'd taught with David and me, called. I was delighted to hear from her, but when I heard her tone, my heart dropped. It hadn't occurred to me that David wouldn't be here forever, and there were so many things I'd neglected to tell him. Rosalind and I asked David's family if we could attend his funeral, and offered to say something on behalf of his students and university colleagues. David's mother, Joan, welcomed us, and so we went along to the funeral, and spoke briefly about David and his significant impact on our lives and the lives of his students and colleagues.
Eighteen months later, the January after our accident, I was thinking a lot about David. He appeared in one of my dreams, in the crowds outside Shea Stadium in New York, which we'd visited in the months after he died. I was shocked to see him and said, 'David, I thought you were dead.' He guffawed at the idea and said, in his characteristic style, 'Hannah, you are clearly incorrect!'
Joan wrote back, giving her blessing for us to appoint David as a posthumous godfather to our daughter, and recommending we go see the movie _Invictus_ , which she thought he would have loved. The night before the memorial, I dug through the filing cabinet in the garage to find a page with his handwriting on it—the first page of the final draft of my master's thesis. Seeing his lead-pencil handwriting, I blinked. I could see his office; I'd always have to move a pile of books so I could lean my notepad on the desk to take notes during our meetings.
We took the page with us to Somers. There, my dad folded it into a paper aeroplane and gave it several maiden flights before the service, when we buried it with Z's ashes.
•
For a while after the memorial, the sadness made me paper-thin. Just breathing, opening my eyes and looking at my surviving loved ones felt so hard. I was glad of the automatic breathing reflex, because I certainly couldn't have bothered doing it consciously. The rich smell of lillies pervaded our house. Not everyone had got the 'no flowers, just donate to Oxfam' memo, and I couldn't just throw them out. They were beautiful. They were tangible expressions of love and sorrow. But watching them open, spill their pollen and slowly die, was less heartwarming. Still life, indeed. We'd had enough of that.
The memorial was hard but good—in a painfully satisfying way. We felt so loved, by everyone who came, and by everyone who didn't come but sent messages or cards. It felt strangely like a wedding (perhaps because my dad and stepmum got married there nearly twelve years before), except for the volume of the weeping. We may not have been allowed our own wedding in this country, but Rima and I were now wedded in this grief.
An hour before the service started, I left things in everyone else's hands and hobbled off to the beach with Rima, my sister and my brother. Jez and I went in the water—him rapidly, like an otter (his stubbly beard adding to the otter impression), and me slowly, letting the water lap its way up my broken body. It was warmer than usual, crystal clear and with very little seaweed.
I dived down and opened my eyes, feeling for the bottom with my hands. I came up, rolled onto my back and let myself float. How many days and hours since I last did that—but in the ocean baths in Sydney, and with Haloumi also floating inside me? And I thought of the spectacle I presented then, with my belly popping above the water like a fleshy island. The girls had thought it was hilarious when I took them to the pool and did backstroke; my belly sinking and rising with each stroke.
Now my fleshy island was just a wrinkly belly below the surface. I sobbed and let my tears mingle with the big, salty sorrow of the sea.
And, as always happens when I float like that, I realised that I'd stopped being aware of time, and was startled back into myself. When I opened my eyes and rolled over, Jez was floating right there beside me.
Somehow, time disappeared and, although we'd arrived about two hours early, we didn't get a chance to test the music system, with the result that none of the music played properly. We would get the first few stanzas, and then it flickered in and out and was awful to hear. The songs I'd listened to on repeat in the hospital, which had come to feel as if they were written about us and our loss, were reduced to crackling static and snatches of a tune. I was cranky about it, but Rima was philosophical and calmed me with little pats on the arm.
Afterwards, we dried our salty cheeks as we walked back from the bush chapel, and ate and laughed and hugged people we hadn't seen for months. I had made platters and platters of haloumi and zucchini fritters. The zucchinis had gone crazy in the vegie patch we'd inherited from our tenants. A friend had asked, half-seriously, while I was still in hospital, 'Does this mean you can never eat haloumi again? Because it will be kind of like eating her?'
'No, no,' I'd chided. 'We will eat it in remembrance of her!'
The next day, there were plates of the remaining haloumi zucchini fritters in the fridge. I didn't want to throw them out; doing so felt sacrilegious. Our friends and family had gone home with full bellies and sore eyes, and now this leftover grief was ours to continue eating, day after day, magically renewed every time we finished it, like a weepy magical pudding.
An old friend who lived interstate called—could she come and visit? Was it okay if she brought her baby daughter?
People had been extra thoughtful about not bringing babies into our presence, as though I might be allergic to them. 'No, please bring her. That would be lovely.'
She did, and the minute she walked in the door, asked, 'Do you want to hold her?' I did, and we looked at the pictures of Z and talked, while fat tears dropped onto her baby's wispy head.
13
Matryoshka
On the one-month anniversary of the accident, Rima and I were at the doctor's again. Our GP, Kelvin, was our new best friend in Melbourne. We'd picked an inner-suburban clinic that friends had recommended as LGBTI-friendly. None of the lesbian doctors we asked about were taking on new patients, so we reluctantly made an appointment with one of the male doctors. I presumed that he'd be expert on sexual health and drug interactions, but perhaps not so comfortable dealing with 'women's issues'—and we had plenty of those.
At the first appointment, though, Kelvin didn't bat an eyelid. He sat and listened as we gave him the matter-of-fact version of the accident, paying compassionate attention, but not reacting. Then he offered help with the pragmatics of referrals, prescriptions and insurance certificates. He quickly became expert at summing up our story in a sentence or less; and navigating the bureaucracy of the three hospitals we'd had to deal with, as well as the government insurer who covered all vehicle accidents. This time, he was on the phone to the histopathology department at Royal Melbourne, on the Trail of the Disappearing Placenta.
Not much of Z's birth had gone to plan, but the one request the midwives had thought they could help with was to keep Z's placenta, so that we could bury it under a tree. A tree with a transient body part buried under it was a lousy substitute for a daughter, but it was something. Or it would have been, if only we could convince the histopathologist to give us back the placenta.
While Kelvin was on the phone, Rima found a set of nested matryoshka dolls in the toy basket. She looked at me, opened the first doll, and made her goofy 'Surprise!' face as she pulled out the next doll, making me smile. She opened the next and the next. There was surprise all the way down, until, 'Ohhhhh'—sad face—when the last, tiniest doll could not open. The last of the line. My silent giggle turned into a sad face too, and I thought, 'I don't want to be that doll.'
On my next trip to the doctor, I discovered a hobby shop within hobbling-on-crutches distance, and bought several slabs of balsa wood and a small set of tools, so I could start whittling my own morose matryoshka set. The first and biggest doll was not a doll at all, but our car post impact, the front driver's corner crushed in. Our car had been a Nimbus, a puffy little white cloud of a station wagon with Tardis-like properties. I hollowed out four spots inside, for Rima, Jac, Jas and me, and made small dolls for each of us. And then I hollowed out my own rounded doll, and made a tiny one to fit inside. It was harder than I'd thought. My whittling skills were pretty ordinary and the balsa did not behave as I expected it to. It was probably the wrong kind of wood. But there it was—a chunky wooden approximation of our wrecked car, with us wrecked inside it, and our daughter, the most terminally wrecked of all, inside me.
I made a little black-bordered birth announcement card, with an open matryoshka doll in one corner, the two empty halves leaning against one another. She had a sweepy fringe like mine and a chubby tear rolling down her cheek. In the opposite corner was a tightly swaddled dark-haired babushka baby, long eyelashes resting on her cheeks. A thick black border connected the two. We had the cards printed with Z's name, weight, length, date of birth and a message in the centre. I had grand plans of sending them out to all the friends and family who had visited, helped and sent us cards, flowers and love, but every time I got out the list, I only managed a few before it all just felt too, too sad.
My Aunty Connie came to see us. When I was eleven, and her son was thirteen, he killed himself with a handgun—whether by accident or not, we'll never know. She was blunt. 'Your life is split now,' she said. 'There'll always be before, and...', she sighed, '... after.' I imagined the bit in between as a gaping chasm, so that reminders of before felt like stranded relics, completely irrelevant and alien in their new setting. Not only my clothes, and our things as we gradually unpacked, but even songs on the radio. Books did not pass through my hands without me flipping to the front to find the publication date, so I could work out if it was a naïve resident of 'before', or a wiser, sadder survivor of 'after'.
Tuesday, 16 February 2010
I visited the dentist this morning to have my front teeth repaired. They'd been chipped in the accident, when my jaws banged together like a nutcracker puppet. I'm usually one of those strange people who quite likes visiting the dentist, but today, holding my mouth open and seeing the dentist work above me, I was suddenly back in the ER, clothes sliced off and my body pinned down to the spinal board like a prize butterfly in a neck brace—a disinterested observer to the workplace that was my body.
Now my teeth are dulled again—the sharpness of chipped edges no longer catching on my tongue. I feel dulled too—shell shocked. The bomb has gone off—a good seven and a half weeks ago, and I'm still stunned, staring into space.
If I were a chimpanzee in the zoo, today would be the day I would spend with a blanket on my head, occasionally hitting myself (and others, if they came near) with a small tree branch. The equivalent human behaviour is staying in bed and eating 70% cacao chocolate, while listening to 90s grunge pop. Friends have made mixtapes for us—I played these and made my own lists to include the lyrics which echoed around my head about the 'saddest summer ever' and 'help I'm alive, my heart keeps beating like a hammer'.
After the dentist, we went back to the hospital to meet with the same bow-tied obstetrician who had told us Haloumi had died. It was as though the movie was over and we were chatting with one of the actors on how the movie played out—reflecting on the motivations and plot, the 'makings of'.
I feel slowed. I still don't get it. There were women walking out of the hospital with babies tinier than mine. Living, breathing babies. Some with less hair than mine, some with more. Where is my baby? Where is she? Why can't she be here in my arms? Why can't we be fussing over her carseat so we can take her home? I know these thoughts are not productive, whatever that means. I still want to know. I was that close to having a living child in my arms.
I dreamt that I was at Z's memorial service again, but I was riding a child's ride-on toy. It was too small and whenever I stopped moving it would slide out from under me and ignominiously dump my bottom on the ground. A work colleague was there and remarked, 'Oh Hannah! So good to see you getting around!'
In another dream, I was cutting rosebuds, each with a dead rosehead in the centre. In another, I had to walk through a pool of shining wet eels, a dark slithering against my skin. Things were lost, landscapes disoriented, obstacles stood in my way. I had another baby and she lived, but I couldn't remember her date of birth. Was she Z's twin? Or had another whole pregnancy elapsed within seven weeks?
The other night I dreamt that I walked into a Victorian terrace house. A slick-looking bloke put his hands together in greeting: 'It's already started but what I'd suggest is that you join in with the group upstairs.' I could hear voices raised and chairs pushed across the floor in the room above. Before I could climb the stairs, he caught my sleeve: 'The scenario is—it is 1939 and you are requested to go on a secret mission to get sixteen European leaders to sign a pact undermining the Treaty of Warsaw, pre-empting it and giving them an out.'
It didn't make sense to me. I felt foolish—I didn't know enough to ask a meaningful question.
'The main obstacle is this Austrian bloke who wouldn't sign because he'd been circumcised.'
'Circumcised? What did that have to do with it?'
'He objects to the subterfuge. He wants it all out in the open. We need to convince him...', he looked at me meaningfully, '... that this is the only way'.
I wanted to ask why, but he's gone, and there are stairs I must climb.
It is chaos upstairs. I'm frightened. I know this is pretend, but it feels like a very serious sort of pretend.
I was still waking at 4 a.m. most nights. The sadness would ball up in my stomach so much that I wanted to throw it up, to get it out of my system so I could somehow go back to the land of 'before'. I could see now the appeal in imagining things like this as being punishment by a vengeful god. Once you are punished, the balance is levelled and you can't be punished again for it—apparently even God operates under a concept of double jeopardy. But if it's not punishment, if it's all random universal cruelty, then it could happen again at any time, and to anyone I love.
Sunday, 21 February 2010
Seven weeks. Seven times seven. 49 bare little squares between us and her last heartbeat. 49 days later I can walk, I can go to the toilet by myself, I'm even contemplating going back to work. When people told me that time would heal, I didn't realise that all this healing also takes me away from her—our ship is sailing on without her. I'm still not sure I can let go. I know she'll always follow us; no longer as a living passenger—maybe as a gull we can see from a distance, who sometimes lights on the bow. Sometimes she'll swoop in close and I'll think she's still with us, other times we might not see her for days at a time. I feel like my brain is a machine which keeps spitting out metaphors for this grief—(here I go again) one error message after another. It still does not compute.
Even her birth date and death dates feel like a mathematical error. She died on the 27th and was born on the 28th—birth and death folded in on one another so that they come in the wrong order.
My phone has mysteriously reset its own date and time to 9 a.m. on 1 January 2007. Imagine that—three and a bit years ago we were at a friends' holiday house, greeting a new year. It feels so distant, but part of me knows it wasn't an idyll. It was an ordinary life. I still have moments of that and I know the moments will come closer together with time.
14
Histopathology
When my dad had visited me in rehab, he'd brought little offerings to make me feel better: a newspaper clipping about the zoo, little sachets of miso soup, a small posy of violets. And, one day, a pomegranate. Everything about it was exquisite. Once my visitors had gone home for the day, I held it in my hand—a crimson magical orb. It felt too beautiful to cut up.
The next morning, I was just waking up when one of the catering ladies brought in my breakfast. She asked how I was and, as I struggled to get a sentence out of my pasty, still-asleep mouth, she sighted the pomegranate on the bedside table. 'This! Good antioxidants! Very good for your healing!' It was both an admonishment and a command, but kindly ones, so I was happy to comply.
I'd eaten a pomegranate before, but never dissected one like I did that day. The process of cutting into the fruit and cracking it open felt like some kind of brutal surgery, the seeds bleeding into my fingers. I peeled away chunks of pith and peel, at once leathery and delicate—football red on the outside and a soft cream on the inside. Row on honeycombed row of translucent seeds were lodged into the pith like teeth in gums, each compartment veiled from the others with a filmy rose–yellow silk membrane. I prised them out one by one, and took photos of all of it: the broken scraps of peel, the membrane, the seeds. Here was something that, even when split apart and broken, only revealed more beauty.
A few days after I was released from rehab, and could join Rima and the girls in our new house, a pomegranate tree was delivered. It was a gift, in Z's name, from dear friends who lived overseas. It stood, green and hopeful, on the porch as the summer days and weeks wore on—hot and dry. I feared it would die there. I almost willed it to die there, and then was torn by guilt at the idea that I could kill my daughter's memory in plant form. Nonetheless, we waited.
At first we were waiting for the placenta. I wanted to bury those cells that belonged to both Z and me underneath the tree. Our midwives at the hospital had dutifully saved the placenta. The histopathologists at the hospital were holding it, after having examined it to confirm the cause of Z's death. I had to look up what histopathology meant. It was from the Greek: _histos_ 'tissue', _pathos_ 'suffering', and _logia_ 'study of'—the study of suffering tissue. The histopathologists had not met Z or me, but there they were, making a study of our suffering by examining the bloody organ that had joined us; at least, until it came unplugged.
When I tried to follow up the placenta issue, we were invited to a meeting at the hospital. The people there put on their understanding faces, and made 'Sorry for your loss' noises. The doctor seemed horrified by our predicament. The placenta had been treated with formaldehyde, making it toxic. I imagined it floating in a jar and asked, 'Does that mean you want to keep it?'
'No, no, but it has to be disposed of as medical waste. Not so great to plant in your garden, especially if you want to grow food there.' It was a very long way of saying, 'No, you can't have it.'
The doctor carefully watched us absorb this information. I wasn't sure what she was expecting; perhaps a hysterical grieving woman screaming, 'Give me back my placenta!' I was tempted, but didn't have the energy for staging a revolt this time.
There were pragmatic considerations too. If I went leaking this grief all over the place every time I was triggered, I'd be a big mess. We'd had to live with what had happened for over seven weeks now and were weary of it. It was not a surprise anymore; this was our banal, everyday horror. 'Forgive me,' I wanted to say. 'Forgive me if I don't seem as shocked and as saddened as you—the person who has just heard this awful news. Believe me, we still feel it, and there are plenty of moments when I turn a corner and bump into a new aspect of the horror and feel the shock all over again. But most of the time we have to keep a lid on it, for our own sanity. To mix metaphors, we can't keep picking at our scabs just to demonstrate our wounds.'
I remembered going to see a dear friend, F, maybe a week and a half after her brother's funeral, when we were both in second year at university. I was distraught—for her, at the thought of losing my own brother, and at the idea of death itself. Her calm surprised me, and now it made sense—that weary familiarity when you've been wearing grief for a while, so that it begins to feel normal, when you've cried all you can for the moment.
So, by that time the fight had gone out of me. The poor histopathologists; I think it was probably quite odd for them to have the owner of some tissue they had preserved and examined show up and demand it back. From then on, we were no longer waiting on medical bureaucrats but on my own battered ability to make decisions and to dig a hole.
The drought had killed a small tree in the front yard. It stood, unrepentantly ugly, between our bay window and the front fence. I didn't know what kind of tree it was. Much as I liked the idea of a garden, gardening itself was still something I thought old people did. It was nearly March by the time we started digging the tree out, when the Preston clay was at its hardest. I threw the pick at the ground, over and over again, carving out the rough outline of a circle around the dead tree. The arc of the pick swinging up, the rush down and the 'thuck' of contact, the sheer solidity of the earth, was a relief. I didn't need to weep, or think, or speak. Just dig. My convalescent limbs were sore and sweaty from the work. I took a long bath with some luxury bath powder my sister had given me for Christmas, just two days before the accident.
The next day I carried bucket after bucket of cold, milky water across the porch and out to our hole. I gave the dead tree a relaxing bath in my second-hand bathwater. The clay held the water almost as well as the enamel bathtub. The digging, to my regret, had to be postponed while the water level slowly soaked lower and lower until I braved the mud and worried away at the dead tree's root system, carving away the stiff mud. My dad and, occasionally, Rima took turns, but I was alone for the last bit, when the tree developed a tantalising wobble, like that of a loose tooth. Even then, it took nearly an hour for it to give way with a satisfying crunch, the small dead tree suddenly lurching, so that it looked more dead and more out of place than before. Remembering what it was like to feel strong in my unfamiliar, resurrected body, I lifted it partway out of the hole before calling for help.
It left a crater in the front yard; a crater I tended lovingly with clay-breaker and compost, before we finally eased the sickly looking pomegranate tree into the hole. Promptly upon arriving in its new home, the tree dropped the rest of its leaves for autumn, leaving us to wonder about its survival until spring. Miraculously, come August, there were tiny red buds. Having eschewed the colour red for autumn yellows, our little pomegranate tree wore red for spring instead.
I would prune the miniature roses at the front of the house, making a tiny posy to bring inside, and then carrying the loose petals and dead flowerheads over to the pomegranate tree. I would sprinkle the petals at the base of the tree, giving it a composting carpet of pink, red and yellow–gold–pink. It became a ritual. It was a chance to have a natter with my beautiful girl, to feel the leaves brush at the side of my face like small hands. _I miss you, my little love. I wish you were in the house, being loud_. I would kneel in the front yard, chatting to a pomegranate tree. I was okay with being the crazy grieving mother of the neighbourhood if it meant I could chat with my daughter. Or maybe they thought I was just a very attentive gardener?
15
Proof
Things are moving towards 'normal'. We are all home from hospital, I'm walking unassisted, there is talk of returning to work. One day, we grab the mail on our way out of the house to have coffee with an old friend. I tear an envelope open and can tell from the feel of the paper that it is not a bill. This is thicker, watermarked paper, like that of a bank cheque or a passport page. When I stare at it, I can't tell whether it is just my eyes or whether the colour of the paper changes softly towards the centre—from creamy white to pinky cream.
Here is my name, and Rima's; here is the name we chose for her and her date of birth. This paper certifies me as a 'mother', and certifies Z's birth; that she was here, a human child, even if she never drew breath. Part of me wonders why they produce these certificates. Is she ever going to need it to get a passport? To get her driver's licence? Will we ever need it to enrol her in school? No, this certificate is for us, to make us feel better, to offer administrative proof of our child's existence. A child was here. She must be recorded.
On paper, I am a mother, but there is no pram here; no noisy, squirming baby. I feel like one of those flat felt figures we had at kinder. You can peel me off this situation and stick me onto another. It makes a soft ripping sound as you do it, quieter than velcro. Here is my picture-baby, here is my piece of paper. I love her so much, but she's now my two-dimensional child—stilled, flattened out on the page like a rare flower. I didn't dream her three-dimensional little life, she was definitely here ( _right here_ ), moving and being. But all the remaining evidence I have of that fact is unsatisfying.
The next envelope I open is an overdue fine from the library: _Sheila Kitzinger, Rediscovering Birth_. We have to go, to move on; we'll be late for coffee with Aron. I fold these pieces of mail together, and worry that I'll mix them up or lose them—confuse the proof of my daughter's existence with a library fine.
•
About a month after I'd been released from hospital, my dad and stepmum decided to take me to see the Melbourne Symphony Orchestra because, as my dad put it, 'music is good for the soul'. They could tell my soul needed all the help it could get. In some strange cosmic joke, the first piece was Fauré's Pavane—one of the pieces we'd played at Z's funeral. It was on a classical music CD that Rima liked to play to my pregnant tummy.
After the funeral, the CD player had stayed with me in intensive care and then in the trauma ward, and I played that CD over and over. It drowned out the sound of my weeping, and somehow the weeping felt less pathetic with a majestic orchestral backing track. There in the concert hall, shiny program on my lap, the same notes from the trauma ward came flooding back at me, live and properly oceanic. Silent tears wet my cheeks, and my dad squeezed my hand.
•
Reinstating 'normal' meant we needed to buy a new death-machine (ahem, car), and get it insured. We'd received a 'Bereavement Payment' from the government—they don't call it the 'Baby Bonus' when your baby dies. With that, and the money from the insurance payout from the old car, we could afford a much newer and safer car, with enough airbags to demand their own collective noun. A cloud of airbags? A reassurance of airbags? I was sulky about paying good money for another car when the last one had stolen our child from us. I was less angry with the human error and thoughtlessness that had caused our accident than I was with these dangerous machines, and the extent to which we were reliant on them and complacent about their propensity to kill us.
Still, in order to see family, and get to and from the shops and various medical appointments, we needed a car. Back in the trauma ward, I had made a rule for myself: that my decisions would be led by what my family and I needed to heal and recover, not by fear or shame. So, even though the thought of driving still gave me the shakes, we started shopping for a new car.
My dad took us on expeditions to inspect cars, carefully navigating so that we didn't need to travel along the road that had been the scene of our accident. We avoided the car yards full of four-wheel drives and the sense of queasy doom they gave me. And, finally, we found something—a creamy-white station wagon with plenty of airbags; auto emergency braking; and, best of all, a GPS system with a calm and reassuring voice. We called her Pearl, as though having a humanised name for our car could somehow immunise us against the risk of another accident.
Each person I spoke to as I called around getting insurance quotes had to ask whether we'd had any previous accidents in the past three years, 'regardless of fault'. I would tell them, 'Yes, we had a serious crash, just in December. A four-wheel drive came onto the wrong side of the road and hit us head-on. Yes, the car was written off.' Inevitably, they said something like, 'That sounds awful. I hope everyone was all right?'
I didn't know what to say to that, so would usually just say, 'Mostly,' in a tone that (I hope) firmly communicated 'Do not ask me any more about this'. If they did ask more, I blathered on a bit about broken knees, ribs, spleens, liver, etc etc. That made them uncomfortable enough.
I didn't say, 'No, we are not all right. My baby daughter died.' I wanted to be correct and accurate and honest, and I wanted our loss acknowledged, but I had to make a number of these phone calls, get a number of quotes. My composure was stretched thinly enough already. I had functions I needed to perform before disintegrating into a weepy pulp. I couldn't go there; not for a flipping insurance quote, not with someone who would only know me as a voice from a call-centre shift. I couldn't risk the random responses the truth might evoke.
It felt ridiculous, shopping around for insurance when something like this had happened. Everything felt ridiculous, flippant. To continue to live and breathe was a mean joke. I didn't realise I could become so bitter. I didn't really know the meaning of it. But bitter and interesting I could handle, maybe; bitter and boring—trapped in this repetitive, ongoing grief—was much harder.
Even with our new car in the driveway, I hadn't yet brought myself to drive again, let alone lay my hands on a steering wheel. A week later, my brother came to visit, and I asked if he wanted to see our new car. We got in, and he shifted the passenger seat back to make room for his long legs and reclined the seat. I laughed. When he was still living at home, you could always tell when Jeremy had borrowed my dad's car by the ultra-relaxed seat. I got into the driver's seat so I could turn on the stereo and show him the GPS, and, somehow, just like that, I drove him around the block.
•
In the heady, queasy days after that positive pregnancy test in June 2009, I'd started a blog, calling it 'Sesame seed sized dreams'. Rima and I had lain in bed and looked up images of a five-week embryo. At that stage, it was just three layers of cells forming into a neural tube and, all up, approximately the size of a sesame seed. I liked the tangential connection with Lebanese food. _Tahini_ (sesame seed paste) is a basic ingredient of many dishes, and sesame seeds appear whole in many other recipes, particularly in _zaatar_ , a mix used on the pizza-like _manoushe_ , which was one of our favourite weekend breakfasts. I knew that we still faced about a 25 per cent miscarriage rate, so I wasn't about to start building big dreams on this tiny wisp of life within me. Or maybe I was, but I wanted to hedge my bets.
The blog documented the slightly nervous, ridiculous and exciting aspects of being pregnant: having a sizeable bust for the first time in my life, and having to find maternity bras to contain it; strange pregnancy dreams; the frankly bizarre, but wonderful, sensation of feeling someone else's hiccups within your belly. The blog also enabled me to connect with other 'rainbow' families—mainly, lesbian mums—across Australia, the US and UK. Those of us at a similar stage of pregnancy gravitated to one another, wanting to hear how others were faring with the discomforts, dilemmas and delights of building a new human being within your entrails. Knowing that each week of pregnancy was moving us closer, not just to the birth, but also to our move interstate, I was also delighted to discover online a thriving community of Melbourne rainbow families, some of whom would become long-term friends online and in real life.
When I put up my 'what happened' post telling the awful news of our accident, my demographic shifted within days. My post had been linked to on 'Lost and Found Connections Abound', a blog aggregator for those experiencing infertility and pregnancy loss. Some of my regular commenters on the blog left their shocked condolences, never to be heard from again. I was sad but I sympathised. If I had been the one who was still pregnant, I would probably have felt awkward and unsure of what to say. Others moved closer, hearing, empathising, and sharing their own experiences of grief. Of those local to Melbourne, a few offered practical support, creating a depth of friendship that I am still grateful for, nearly eight years on. And in my new demographic, I found a whole 'baby lost' community struggling to make sense of babies dying, and of their own role as bereaved parents.
I sought out their stories. I wanted to know how other people survived this, what was 'normal'. What were you supposed to do when the catastrophic thing—glanced at in all the pregnancy books but never discussed—happened? And what did you do with your days when you'd been all lined up to shush, and wipe bottoms, and barely have a moment to yourself? Conversations with friends and family were mostly about our progress—the girls at school, me getting ready to return to work, settling into our new home and suburb. 'We're getting there,' was my refrain. But where exactly was that? And how could I navigate 'there'?
I was diligent in my grief. I sought out my own homework and devoured it. I hunted out baby lost blogs. I wanted a manual. I wanted practice guidelines, some kind of rules to follow. Anything that would tell me there was a solution.
I wanted to hear other women's stories, but with those stories came all their pain and trauma. Rima would find me at the laptop, tears falling onto the keys. ' _Habib_ , don't,' she would say. 'You've got enough sadness of your own.' Indeed, there were points when I had to stop reading, turn away, distract myself with TV or Facebook instead. I could feel something shattering in me. This massive ocean of grief—mine and everyone else's—was cracking open the small bathtub I'd allowed for sadness in my life and was leaking out in an unstoppable flood. I had thought that the sad, hard bits of life could be contained—that was what optimism and psychiatric hospitals were for.
I would click away, thinking, 'This stuff will drown me.' How could I hang onto my basic beliefs about the world as an essentially fair and good place in the face of all this sadness, injustice and cruelty?
Yet, a small part of me was relieved to know that it wasn't just me being picked on, that loss and grief were catastrophically normal and common. Somehow, though, I clung onto the idea that there was some maximum amount of pain any one person could suffer. The bathtub was gone, but I thought a full-length, above-ground plastic pool might do the trick.
Sunday, 21 March 2010
We're getting close to three months since the accident, and it suddenly occurred to me that maybe I wanted to write a three-month letter to Z. I'm not having a dig at those who write letters to their living children—god knows if she'd lived I would have been right on the bandwagon. It is a beautiful idea, that's why I just wanted a little taste of it, even though it isn't quite the same when your baby isn't here to record all the new amazing things they learned and you learned about them each month. But this is part of my task here, to accept that I don't get any more time here on earth with her. It could go like this:
My darling girl,
I'm trying to work out how big you might be, if this was your three month birthday rather than three months since you died. We saw a baby today on our way back from the market, probably a bit more on the newborn side than you would be by now. You'll be happy to know that I still haven't seen any baby that comes near you in the looks department, and we seem to be surrounded by them at the moment. They're lovely, they're sweet, but they're not you.
I'm hoping that wherever you are, in the non-denominational, vaguely agnostic Good Place where I like to think you might be 'living the dream', you are growing and learning. Those little legs would be filling out, and maybe you are giving your godparents some smiles, starting to focus on their faces and grin gummily at them. God, I wish we were there to see you and hold you, my love. I wish I could be feeding you and feeling some pride and amazement in your increasing fatness. Rima would be making faces at you, doing her expert babymama thing, teaching you Arabic.
But enough about your milestones, let's talk about mine! I can now bend my knee well over 100 degrees. Woo hoo. And my quadricep muscle now responds when I want to move it. I can get in and out of bed without doing that weird robot-leg move I had to do before. We're going for big walks, to and from the shops, around the park, with only one crutch—and I won't need that for much longer. We're sleeping through the night a lot more than last month. I think I'll be starting my new job next month—beginning part-time and working my way up to full-time by July.
Your sisters miss you. They are making friends at their new school, and they've freaked them out showing them photos of our wrecked car. They were all geared up to be the best babysitters ever, I hope you know that.
I won't write you a letter every month, I hope you'll understand. But I love you and think about you every day.
With all my love, Mama
16
Scar tissue
In late February 2010 I had my review appointment with the trauma unit at Royal Melbourne. The doctor who saw me introduced himself as Ganesh, and I immediately thought of images of the elephant-headed god. He talked me gently through the CT scans that had been taken on the night of the accident. There was Z, curled in my womb, her hand up near her face. _Oh, my little one_. There was no need for any further scans. The liver and spleen damage seemed to be healing up well, and I wouldn't need to come back.
At home, I made a sketch of my trauma doctor. Like his namesake deity, he had the head and sad, serious eyes of an elephant, but a human body with two arms: one gesturing towards a CT scan, another holding aloft tweezers gripping a bloodied shard of windscreen glass.
I had dreams of walking with my one crutch and a heavy backpack, up hills, through endless train stations, around in circles. Somewhere along the way, I realised I'd left the crutch behind and I'd been walking without it—but, instead of being pleased, I was devastated. I woke and walked stiffly to the bathroom, realising how fond I was of this limp and my crutch as visible signals that I was still wounded. I understood now why people wore black for mourning. It is simpler and less confronting than having to continually convey the message, _'Someone I love has died. I feel irretrievably broken—please go gently with me_.'
Since the accident, I hadn't been able to lift my left leg from the knee. I would sit there, cajoling my foot to rise, but the best it could do was slide forward. The message to lift seemed to fizzle out somewhere between brain and leg. My physio had brusquely assured me this movement would come back, and had focused on breaking down the scar tissue so that I could bend my knee. One day my stepmum, Debbie, a retired physiotherapist, sat down with me.
'You still can't lift this leg?'
'Nope; I can't even remember what it feels like to lift it.'
She patted the couch, saying, 'Both legs up here,' and rolled a towel to prop under my left knee. With a tea towel, she made a little sling for my ankle, so that she could lift my foot.
'Okay. I'm going to help you lift it—but you lift too.'
She lifted it slowly, giving me time to respond. _Move_ , I told my leg. And, suddenly, my breath caught with a small sob of pain. When I slowed things down, I realised that my brain wasn't just issuing an unheard 'Move' command—it was also receiving a pain message. I had to receive and listen to the pain message before my leg could override it and lift through the pain. Once I did that, and allowed myself to feel the pain, my foot started lifting out of Debbie's hand.
'Yes; see, you're doing it now!'
It was tiny, but it was movement.
I tried again the next morning, on my own, getting bolder and lifting my whole leg forwards from the knee. It took a few tries, but suddenly my leg was lifting, and I was sobbing—at first with pain, but then from relief, sadness and the effort of it, that such a simple thing could be so hard. In waking up those nerves and calling them into action, I also had to tell them the bad news, and, in doing so, hear their shock and pain from the impact. Even my knee missed her.
Friday, 26 March 2010
I'm dreaming again. I'm in a bathroom like the Sydney Law School one—a long grey infinity of tiles with door after door after door. Margaret Atwood is here—eyes wry, her tight curls moving with her gaze. She is advising me right here in the ladies toilets.
'If you want it, you'll have to get up and do it every day. Even on the days you don't want it. And be honest.' She gives me a sharp look and I open my mouth to say something.
'No, not pretending to be honest—actually being honest.'
I feel like she is stripping the husk off me, roughly, but as though there was something there worth un-husking.
Again, she reads my face, and speaks to me through the mirror this time.
'Yes, indeed—but don't expect me or anyone else to find it...', she points at me with a slender writing-calloused finger, '... that's your job.'
And she pats my hand. Not in a nanna-like way but briskly, reassuringly.
'You'll get there,' she quotes my own words back at me, 'wherever that is.' She smiles with half-lidded eyes.
'Oh, I know. I'm no writer-goddess. But I am older, and that still counts for something.'
I don't know how to address her. 'Margaret' could refer to all kinds of women of her age.
'Of her age,' I hear her tutting under her breath, still in camera with my thoughts.
'Ms Atwood' isn't specific enough either. I sense that people who know her call her something else, but I wouldn't fall into that category of familiars. Peg? Marg? Her eyebrows ratchet higher with each suggestion. And then she looks at her watch, 'Well, kiddo,' and here I get the first genuine smile—a glow that rolls out across the space between us and warms me. 'Enough fairy godmother time. You take care and don't be a scaredy cat.'
'Yes, ma'am!' I hadn't quite planned to say that, but the words click into place and she smirks an approving smirk.
'See? You'll be fine.'
•
With the girls back at school, Rima and I knocked around the house, untethered by work and with a diminishing number of medical appointments. One weekday, we drove down to Somers and found our way to Z's spot. The ground was still disturbed. It wasn't that long ago we had knelt here, and tapped the plastic bottom of the container to see if any more ashes would fall out. We stood there—unsure of what to do.
I'd never done the grave-visiting thing before. We'd wandered through graveyards out of curiosity, boredom, but never to visit anyone. My grandparents' ashes were somewhere here too, but that grief had been a much neater and more timely affair than the ache to hold our child, to be near her. When we'd chosen this place, I'd thought about the trees, the smell of the beach, the sandy dirt imprinted with our childhood footprints; not about a little hole in the ground. Rima hunted around and found a small chunk of stone to mark the spot. I wanted desperately to fall on the ground, to lay my cheek on the sandy soil, but I was still self-conscious about this grief. We hugged, wept, and, as we left, I patted the strong, curved bough of the gum tree bordering the chapel. _Take care of our gorgeous girl_.
•
Several times a week, I went to the hydrotherapy pool to do the exercises the physio had prescribed. The water was warmer than bath temperature, and I was usually the youngest one in the pool by at least three decades. There was something very comforting about being around the elderly and injured. Stripping down to bathers and exposing my scars was easier around others who were similarly lumpy, limpy and scarred. We had been marked by life in more extreme ways than those in the 'normal' pool. Where once we had wounds, now we had scar tissue telling the history of those wounds, and their healing. Mine were no longer raw cut edges, but new, reddish skin—difficult to stretch and unrecognisable as normal skin. Over time, the redness would fade, but the seams remain; reminders of being broken and being mended. In the hydrotherapy pool, no one really looked one another in the eye, or wondered about the repetitive motions and odd contortions we had to perform, or the adult bath toys we needed to perform them. At the same time, I felt a little like a fraud because my disabilities were (I hoped) temporary. Sometime soon, I could turn my back on the hydro pool—at least until age or injury sent me back.
One day I turned up, and an arthritis aquatherapy class was on in the hydro pool. Reluctantly, I walked back to the lap pool and started doing my exercises in the shallow end. I wasn't the only refugee from the hydro pool—two older Italian women were marching up and down, talking quietly. I was doing okay, even when a heavily pregnant woman started swimming in the next lane. But when she turned and started doing backstroke, her belly submerging and emerging with each stroke, I had to dive deep and close my eyes.
•
At the Preston Market I went to pay for our fruit and vegies. The woman at the cash register turned. There, under her apron, was her broadly pregnant belly. She reciprocated my look at her belly and looked at mine, asking, 'Are you pregnant?'
My tongue rolled back in my head. 'No.' I wanted to say more but the words could not fit out of my mouth and, instead, I stared down at the net bag she had just put my tomatoes in, willing myself through the fine holes.
'Well, have a nice day,' she said.
•
My mum has a thing for new babies. There's a photo of her at a cousin's Christmas party; the usual gathering of my dad's extended family. It's not a posed photo—it captures various conversations between cousins, the slight awkwardness of the biannual catch-up. And there, tucked in the middle, is my mother, holding my cousin's baby son, not quite a week old, locking eyes with him as though no one else existed. A big, raw chunk of intimacy in the midst of the public family–Christmas business.
You hold a baby differently once you've lost one of your own. For my mother, like me, it was her firstborn, a baby girl. Lost, not to stillbirth but to adoption in an era of choicelessness—no reliable information about conception (or contraception); no access to abortion; a family who kicked her out, a boyfriend who didn't want to know, and social workers who told her the best thing she could do for her child was to permanently remove herself from its life. It is hard to coax my mum to talk about what happened. I know she stayed with a friend in Melbourne, and looked after her friend's kids in return for board and lodging. When labour began, she walked to the Women's.
Things often get shaky in November for my mum. November is the season for quiet concern. For the squeak of vinyl armchairs in the TV room of the mental health unit. A birthday that goes unmarked, that floats unspecified over the entire month. Conversations that peter out, questions left hanging while we divert to other, more concrete topics.
So much of my mother disappeared into the dark spaces of her unspeakable grief, windows papered over with layers of shame, of silence; not just hers, but that of family members, and the ones who shamed her. I know there are whole rooms there, filled with specific things. We hear noises from in there. Pipes that rumble, appliances that fizzle in the dark, unknowable, inscrutable. I have given up asking. But I knew that for me, the windows had to be wide open on my grief. This was a loss that anyone who loved me had to know, and know well. Nothing matched, for me, the horror of the locked room, the claustrophobia of a sealed-up grief.
Monday, 3 May 2010
We're sick of the house, sick of our own misery and sick of each other's company. So what is the best remedy for this malcontent? Clearly, wandering around Ikea with legions of pregnant women and parents holding small children lurking behind every Billy bookcase is a fabulous idea.
Things started badly this morning when I woke early, and read the last few chapters of _Northern Lights_. Nothing like young adult sci-fi for comfort reading, or so I'd thought. The book had begun with the irresistibly heartening premise that all humans have their own spiritually-connected talking animal companion—leading me to imagine that I could expect a happy ending, or at the very least a Harry Potter-esque happy-and-safe-for-now ending. But no. Apparently author Philip Pullman has other ideas, which don't include rounding off my escapist bout of children's sci-fi in a gentle enough way so that I can start my Sunday morning without feeling like Armageddon is around the corner.
We entered the Ikea play-house with two very simple objectives, and neither of them was to be reminded that even if we buy all this stuff, our house will never look like an Ikea showroom. I think it must be a genetic thing—either you have the tidy-decluttering-clean-lines-matching-furniture-Ikea gene or you don't, and Rima and I clearly don't. We're not complete grots—we do Make An Effort, and temporarily fight back the jungle on a regular basis—but with three pets and two teenagers, as well as our own messy selves, there is quite a bit of jungle to deal with.
If you've been to one of those water theme parks which has a canal section where everyone floats around the same circular route on giant inflatable donuts, then you may as well have been at Ikea with us, floating along a twisting series of Ikea-ised rooms, bumping up against pregnant tummies and living babies at every turn. I'm not mortally offended by all this evidence of everyone else's fecundity, but it is hard to concentrate on finding soft furnishings while I'm constantly playing games of 'Would she have been about that big by now? Or fatter?'
Eventually, the current brought us along to the cashiers, and we piled our small pieces of pleasantly-smelling wood and Nordic-looking fabric into our tiny reusable carry bags. By then, shopping centre fatigue had set in, and it only took one song to make me weep in the car. From there it was only a short hysterical step to melt-down-land when I got home and realised that there was no tofu in the fridge for the one meal I could imagine making—green Thai curry. It is a sad thing when you feel like you are useless at everything, including feeding your stubbornly vegetarian self some kind of protein on a regular basis.
Somehow, the lack of tofu, and consequent nutritional failure, was the last straw on top of the giant haystack of things I'm not managing to do very well lately, including finding decent work clothes to wear, cleaning the house, being an academic and being a likeable stepmother. It is the state I've come to call Everything is Broken. _But, you've just lost your child, only four months ago—give yourself a break_ , as a beloved friend was telling me this morning. Yes, yes. Four months. How long will it take before I can function normally? I was doing it okay two days ago, or at least creating the appearance of it. If things fall apart only every second day, is that progress?
Rima was lovely—Ikea and shopping centres don't seem to have quite the same enervating effect on her. She let me weep all over her in the kitchen, and suggested we order in pizza. Instead, I marched off damp-eyed into the dark to hunt and gather tofu from the supermarket just to prove to myself that I could do the adult thing and make dinner.
•
During my time in hospital and rehab, Mum had been solid—visiting every day, ferrying Rima and the girls around, procuring exactly what was needed before we knew we needed it. We had asked her to stay with us in Preston and help once I'd been discharged, but things were harder than we could have imagined. Mum was sleeping in her campervan in our driveway, but mainly living in the house with us, along with her exuberant red heeler puppy. Sharing the same space and relying on Mum to drive us around and help with the house put new pressures on our relationships. Nothing was right and nothing would be right for some time.
As our situation eased from sharpened emergency to the duller, slower stuff of grief, I could see Mum growing quieter. We wanted help but we also needed privacy. The vague plan had been for her to stay in Melbourne after Haloumi's birth and to enjoy being an _oma_ , but now it wasn't clear what she would do. Mum had never been one for winter—the cold and the darkness triggered her depression, turned her inwards towards that dark, sad room. When my brother and sister-in-law told us they wanted to head up to Cairns to train as diving instructors, Mum suggested driving them there in her campervan. And so, she was off, fleeing the advancing Melbourne autumn.
17
Funeral appreciation
Two days after my birthday, I was at work in the law school. I was eating lunch at my desk, a bad habit, and flicked onto Facebook. There was a post from my friend Karin.
She was living in Paris, with her partner, Ned, and their baby, Albert, who had been born in mid-December 2009. We had seen the first photos of him on Christmas Eve at my dad's house, resting the laptop on my huge Z-filled belly, and cooing at his creased feet and dark, thoughtful eyes. They were planning a visit home to Australia in July, and we had plotted a meeting of the babies—hers, mine and Renee's. After the accident I had been so heartbroken that Z would never meet Albert, that they would never be playmates. Karin and Ned were devastated for us. Among their busy-ness with their baby and Paris, Karin stayed in contact, remembered my birthday. But her post on 17 June was brief:
We lost Albert yesterday. they say SIDS—La mort subite—we cant breath. sorry for telling you this way.
News like this has a rushing sound, like a vandal, like a destructive wind. At first I didn't understand and had to read it again. I wept and called Rima, sent this awful news in her direction. I wept for Karin, for Ned, for little Albert, for us and for Z. I wept because I knew the words to say, because it was all too familiar. I wept for the thought of him growing cold, and for the little voice that said, _It is not just us now_.
In the days and nights that followed, Karin's words pulsed through my head, tapped out like a telegram, over and over again. _La mort subite_. And with it, the thoughts of his little form, lying cold somewhere in a Paris hospital. How could we undo this? What is it about death, that it has to be so damn permanent and non-negotiable? There is no 'maybe' left, only 'never'. I was unable to work, to write, I frittered away time on the blogs, and the baby lost forums, weeping for all these babies. And on a friend's blog, I came across the concept of tonglen meditation:
When things are painful and difficult, the quality of difficulty should remind us to have the thought, 'Other people feel this.' Isolation in our pain and the loneliness of our burden reminds us of our shared humanity.
I'd found solace in meditation before; in the short meditations within yoga classes, and in a meditation group at Newcastle University. Often in yoga, the instruction was to breathe in light or energy, and to breathe out anything you didn't need; a pranic version of the tiny transfers of oxygen and carbon dioxide our lungs performed with every breath. Tonglen meditation ran counter to this logic. It instructs you instead to breathe in the hot, dark, heavy feelings of sadness, anger, of feeling stuck—not just for you but anyone else feeling the same way—and then to breathe out a sense of spaciousness, of light, of safety, of relief from suffering. There was a generosity to it that went beyond our own biologically acquisitive nature of bringing nutrients in, sending waste out. In biological terms, this was photosynthesis; taking in the unwanted carbon dioxide, sending out the oxygen that was needed—and in the process generating something new, an unexpected sweetness.
With tonglen I could breathe in those hammering words and the dark, sickly fever of _la mort subite_ —for us, and for Karin, Ned and Albert. I could let the sadness shake me and be absorbed into my cells. And I could breathe out relief from the hammering, and the wish for his little spirit to be somewhere warm, somewhere good. And that he might just bump into Z and make knowing baby eyes at her. 'You too, huh?'
My birthday present to myself was a t-shirt printed with a graphic of an African elephant in full charge. Dokkoon, the Indian elephant at Melbourne Zoo, had safely delivered her baby in February, but in following her progress I'd also discovered that zoo elephants have much higher infant mortality than do wild or semi-captive populations. I felt a solidarity for the elephant mothers left behind. Wearing my elephant t-shirt enabled me to get out of bed, to make decisions without tears, to feel maybe I could be like a mama elephant—all the more fierce with love because of my loss.
We flew to Sydney for Albert's funeral. After the ceremony in the chapel, we followed Karin and Ned, carrying their boy to his grave. The Port Botany wind blew up the cliffs to meet us. It was a big crowd for a small person. On the way back up the hill, I saw Renee. We hugged, this time shaking with sobs, no big bellies between us. My brain knew that I couldn't turn this into a statistic—that it is not normal, or usual, for two out of three babies to die. Certainly not in countries with modern medical care, certainly not for women educated to postgraduate level. Later, Karin and I stood looking at Renee's baby, Anna, and held onto one another. 'She'll be our measuring-stick baby,' said Karin. 'That's how big our babies would be. She will always be the right age.'
Sunday, 11 July 2010
It's been a season of grief. Our own grief is becoming worn and supple, though it still catches at our heels, constrains the way we walk. We limp as veteran mourners into the new fresh grief of Karin and Ned for their tiny son, not quite six months old, and my cousin for her husband of 22 years. I'd never been to a funeral of someone younger than 60 before (how fortunate! what lucky planet was I living on?)—but we've been to two within a week now, and three in the last seven months if you count our baby daughter's.
I now have a new appreciation for funerals—the time that goes into them, the importance of the small details, the careful, deliberate laying-down of ritual and memory. We sit on hard wooden benches, search for tissues, give much longer than usual hugs.
How precious it is to have a festival for the lost one you treasured—to put them in the centre, include them in the party one last time. At the funeral, two incompatible realities collide: 'He is gone forever, we must say goodbye' and 'He will always be here with us, in our hearts'. Nothing can stitch those two opposites into a cohesive story, but this is the heaving, fractal reality. Instead we have to re-stitch our hearts around it—or let them break, brittle, on the floor. It hurts to stitch our hearts like this, when the needle goes in we think we cannot bear it a moment longer. We think everything our hearts are made of will shatter. Yet the laws of physics bend once again, and so do our hearts—sore and tortured by the thread, pulled painfully back into something heart-shaped.
I wished so much I could offer some sage advice to Karin and Ned on how to survive this loss, but the truth is that I'm no closer to finding 'the secret' than they are. Surely, it isn't helpful to say, 'If your loss is anything like ours, you will struggle to stay sane and find meaning, you will feel broken for a long time and your loss will creep its way into everything—your work, your sex life, your friendships, and into the minutiae of what you wear and how you cook.'
It's the truth—but it is also true that we have times when things feel good, when it feels like the edges are coming together and we can laugh. And even if we did know the secret (I'm still hopeful) it would nonetheless be our secret to our loss, and would likely be helpless in the lock of their sadness.
I was so hesitant about coming back to Sydney—to look at these 'before' places where the imprint of being pregnant and pre-accident is so fresh. But once we were here, things were nowhere near as raw as I had imagined. Yes, I was here before—we drove these streets in our now-dead car, my tummy rounded and living, full of our now-dead baby. But things were wound-back then, and it felt surprisingly good to remember that witless hopefulness and presumption that everything would be okay. We drove past our old house and parked across the road, and I thought of the last time we pulled the front door shut behind us.
We were sweaty and gritted with the dust that emerged from behind the furniture. It had been humid since I'd woken up in the summer morning dark, to keep filling boxes. From eight weeks of pregnancy, Haloumi had been waking me early, but that morning it was my excitement as well as hers that propelled me out of bed—this was the day, the day we moved. What kind of insanity was this—to be moving an entire family interstate while I was nearly eight months pregnant? My own insane optimism, and the broad-braided rope of homesickness, were pulling me back to Melbourne. After so many hours, our hands papered with corrugated cardboard and the sweetness of packing tape on our teeth, the truck had finally left. Inside it, our things were packed tight like tetris blocks—an entire family life condensed into so many square metres. We'd tiptoed across the damp-mopped floors with the real estate agent to sign off on the condition report. The electricity company man had come and read the meter, and switched off the power. We'd wiped the place clean of our existence there.
The car was loaded with all our holiday things—I took it to fill up on petrol while Rima and the girls walked to the chicken shop, then parked it across the road, its nose pointing west to the M5, and Melbourne beyond that. We couldn't go back into the house—we'd just handed the keys in. So we sat there eating takeaway chicken and chips on the nature strip across the road from our house (no longer our house) having an impromptu picnic—relishing for a moment all the work of packing and the relief of finishing it.
I had worked so hard—to convince Rima to make the move at all, to get the job leading us back, to organise the move, and in the last weeks, to sift and pack our things while finishing up at work and dashing to pre-natal yoga. At times my mum or Rima or the removalists had told me to sit down and have a break, but I felt strengthened by my very-pregnant state, not weakened by it. My belly was heavy, but even in my sweaty dirty state, sitting on the nature strip eating takeaway, I felt like a trailer-trash goddess—beautiful and potent.
Not that potent, as it turned out. But sitting in the car nearly seven months later with a saggy belly and our baby girl reduced to ashes, I could still feel the humidity of that December afternoon on the grass and remember my optimism.
18
I have a dark-haired daughter
I started my new job not as the promising recruit, but grief-shattered and limping. I had a 'return to work' coordinator, who asked me about my physical limitations and fatigue levels but didn't canvas how to interact professionally when you felt like a walking piece of roadkill.
Ours was a quiet corridor, doors mostly shut. I took to wandering the campus at lunchtime, and soon found a garden enclosed by a quadrangle, lush with wide-leafed cherry trees, where there was a memorial stone for a woman who'd died in 2008, Kathleen. Here at least was someone I had something in common with, I joked with myself darkly. But still, I often joined Kathleen for lunch, and discovered that if you sat still long enough, her other friends appeared. There were tiny, spherical blue wrens and sometimes other birds—magpies, of course, and clear-eyed ravens, and sometimes a New Holland honeyeater, with a streak of yellow on its brow. I still couldn't quite convince myself of the idea that Z was somewhere else, but I found the birds a comfort. _Tell her I love her_ , I would tell them, _if you happen to see her_. I had always dismissed birdwatching as the daggiest of hobbies, but now I found myself looking up bird identification websites, and calling my dad to describe a new sighting.
I cried the first time I saw a pair of red-rumped grass parrots, and was immediately mortified. Part of it was the shock. They were so well camouflaged that when my eyes focused and I realised what I was seeing, it was as though they had appeared out of nowhere. But with that was a miraculous sense of coming home to beauty, right under my nose. As though Z were saying, _What? I've been here all along_.
•
At the end of November 2009, I was giving my last lectures and seminars before starting my maternity leave, waddling around the university and feeling enormously, deliciously pregnant. In among the rush of teaching and busy preparations at home for the move to Melbourne, I dashed off an abstract for a conference in Singapore in July 2010. Haloumi would be nearly six months old, surely old enough for us to have gotten the hang of baby-wrangling, and for me to have enough brain space to present a conference paper. The future with a baby in it was impossible to imagine; making plans for 2010 felt like being blindfolded and sticking a pin in a map. But it sounded like a great conference, and I loved the idea of travelling overseas with our baby.
It was March 2010 when I got the email letting me know my paper had been accepted. It was like a letter from outer space that took light years to arrive—the Hannah who had written that abstract was long dead. And yet here I was, answering to the same name, but the future we had expected had never arrived; instead I was stuck in this alternate reality of being a babyless mother. At first I dismissed it. I couldn't even summon the energy to reply, or to decide how much of our sad tale to tell the organisers when declining the invitation. But as autumn set in, and I gritted my teeth and started my new job, the thought of a trip to Singapore in the midst of a Melbourne winter looked more appealing. It wasn't too late to apply, and maybe it could be a good thing for Rima and me to have a break, and for me to get my research moving again. I couldn't get funding for the trip, but I needed a date to work towards, so I accepted and started making travel plans.
As June and July dissolved into more grief and sadness, I thought, 'I've made a terrible mistake—I won't be able to do this. The bit of me that could draw research together and write is gone.' I'd forgotten how much work it was to write up a conference paper, and as the departure date approached, I felt more and more panicked. But I had to come up with something.
And, somehow, I did. It was only after I became too tired to be tired, and too panicked by the deadline to panic, that from weariness came something that was there all along. For a little while, I had my concentration back, and I could look at all my work and pull the threads together, say what I needed to say. It was such a relief, to get a taste of that pre-accident me, to remember that I was still there, that the sadness hadn't wiped away everything.
When we arrived in Singapore, we stripped away the layers of Melbourne winter clothes we had worn on the plane. Each layer of clothing seemed to wind us back in time, towards summer and the black hole that was 27 December 2009. Early the next morning, while Rima was still sleeping in, I snuck out of our room and made my way to the pool on the hotel rooftop. I wore the same one-piece bathers that my sister had lent me for my hydrotherapy sessions to strengthen my knee; they'd become saggy with all the chlorine at the local indoor pool. I was nervous relinquishing my towel and walking to the pool—my body still felt like hospital property, all staple marks and scars and wasted muscle. But the water greeted me with its familiar silkiness and I discovered I could still comfort myself with slow, deliberate movement. I closed my eyes and summoned back the feeling of swimming with Haloumi swimming in my belly—a buoyant equilibrium between inside and outside. 'I have a dark-haired daughter.' My hands pushed through the water. She can't be erased. 'I have a dark-haired daughter.'
•
At the conference, it felt odd to be mingling with people in my professional capacity again, rather than as a patient or as a shy colleague. In every conversation, there was a little wall I needed to step over as I decided whether or not to disclose that, just seven months ago, I had held my dead child in my arms. I had worried that when it was time to give my paper, my voice would fail me, that it might betray my brokenness. But in the immediacy of the conference, and my desire to do justice to my research and the paper I had prepared, I felt less broken than I had in months.
After the conference, we took a bus and then a boat out to a tiny Malaysian island. We were adrift not only in the South China Sea, but when nighttime came, in the middle of the Milky Way. I've never seen so many stars.
One of the lovely things friends did for us after Z died was to band together and name a star after her. They gave us a chart, a certificate and the coordinates. After a few unsuccessful attempts at finding her particular star, with zero astronomical knowledge (and without a telescope), we took to appropriating whichever star we liked as 'her' star. Usually, for me, it was the first star I would see in the west as I was walking home from the tram after work. But that night on the island, our heads together and our toes in the sand, Rima and I saw a shooting star, and it felt as if she'd sent it for us—a tiny, solitary Haloumi firework.
19
Dr No-Sperm-for-You
After I got out of hospital, after the memorial service, after all the physio appointments, after I had started my transition back to work, after all the news and drama had settled down—I hit a wall. I had been so focused on 'getting there'; coping, dealing with the basic survival tasks in front of me. I had imagined that this grief was something to be got through—a swamp, if not a mountain to be got over. I didn't expect a great big wall of 'my baby died' staring me in the face every morning.
Worse, what I had thought was the whole wall was only one particular bit of it, because I'd been standing so close. I took a step back and blinked, and there was the wall; higher and wider than I could see, with no edge in sight in any direction. 'Right. Wall. Wall as far as the eye can see. Shit.' I thought of cartoons of people imprisoned for life, carving lines in the wall to mark the years, finishing with a skeleton leaning back against it.
Now that I could amble up the road to the library by myself, I decided this was my project. Surely others had scaled this wall before, or found a loose brick, a secret door. I would research my way right over this wall and away from it. 'Stillbirth', I typed into the catalogue. 'Neonatal death'. 'Grief'. I ordered books from neighbouring libraries, I waited for parcels of books ordered online. _Tell me your secret_ , I whispered desperately into their dust jackets, _how do I fix this?_ I had started researching and reading with the thought that I would find an answer, that I would 'get religion', or hear the secret from mothers who had somehow been 'healed' after their child's death. This was what kept me going: the thought that there was a way through, that there was an answer to all my sadness out there somewhere—a key, a secret formula. I worked on it like it was a maths problem. I followed each line of reasoning carefully, and tried not to cry when they all led back to another spot with a good view of the wall.
I found three main answers. The first was boring old Time. Fat use that was. I needed to fix this pain right now! What was I supposed to do—put myself into a deep freeze until the requisite amount of time passed and I could wake up feeling human again? 'Time' did not involve me doing anything; it was beyond my control. The second answer I didn't like either, because it suggested that, actually, this grief wasn't fixable. Rather, it was me who would have to adapt to it and learn to live with it in time—taking me back to my objections to answer number one.
The third wasn't exactly presented as 'the answer', but I surmised it from the fact that every book or story I read about perinatal death featured a subsequent child. Here was a nice, practical solution: have another baby. And while I couldn't guarantee results, this was the only option I felt I could actually do anything about. I knew, of course, that one baby couldn't replace another, that Z would always have a special place in our hearts, but I wanted a child I could parent, in the earthly, messy way of living children. I was impatient with the slow repetition of my grief. I needed a sweetener, a happy ending, resolution of the narrative. I stopped writing and went into waiting mode—waiting for a happy ending on which to finish our story.
•
I was at the Korean jewellers in a small arcade in Preston, handing over a small ziplock bag containing a fine, but heavy, chain. The woman opened the bag and laid the chain on a velvet board, poking it into a line with one little finger. This was the spot where the ambulance officer's scissors bit, snapping the gold chain between their silver blades. The chain had run like a rivulet into his cupped hand. And, in its place, he negotiated a foam and plastic neckbrace behind my head, and velcroed it into place, careful not to catch my hair.
The chain had been broken for seven months now. I explained what happened, and that I'd like it mended but with a small gold heart to mark the mending spot. 'I was in a car accident—they cut my chain—our baby died.' It was enough for her to get it and she gasped, 'Awww!' and put her hand on mine. 'Awww! We will make you happy!'
•
Happy and mended is what we wanted, and so, after waiting the minimum six or so months recommended by the obstetrician, we were in hot pursuit of the one thing that we imagined could deliver that state: a positive pregnancy test. But before we could have a shot at pregnancy, there was paperwork to fill out, and administrative hoops to leap through—police checks, child welfare record checks, applications to the treatment authority to import the frozen sperm from New South Wales to Victoria, counselling and consents. Many of these hoops we had already jumped when we started the process in Sydney, but because each state had its own fertility treatment laws, we had to repeat our efforts in Melbourne. When we explained our story and Z's loss, the staff apologised but there was nothing they could do.
In the calendar, we'd circled August as the month where we could leap from the grief roller coaster onto the trying-to-conceive roller coaster; or, better still, perform some Evel Knievel feat of riding both roller coasters at once. August was the month that had been pulling me forwards, getting me through. We had diligently submitted all our paperwork; arranged for our donor, Jorge, to fly down from Sydney for repeat counselling; submitted to blood tests both for me and Rima. But when day one of my cycle arrived, and I called the clinic to work out the treatment schedule, I discovered that our euphemistically titled 'samples' had arrived from New South Wales, but without the requisite paperwork and without the requisite tests having been carried out on them. We carefully packed up our hopes and bundled them into the diary for September. By then, the tests would have all been carried out and we'd be able to press ahead with an insemination.
Sunday, 29 August 2010 Inside-out Day
Friday was the 27th—eight months since our accident. I was trying to figure out why it felt so much harder than seven months. We were in Singapore at the seven month mark, and somehow felt like we were 'on holiday' from the grief. I'd just given my conference paper and we had a little holiday ahead of us. I felt close to Z, but the grief felt distant, smoother. Eight months isn't half a year, it didn't make sense for it to be any harder than seven months. The answer was so obvious it took me a while to realise. She lived eight months in my belly, and from now on she would have been dead longer than she existed. I spent eight months gearing up to be a mother, and then the pendulum swung back, and I feared that my whole pregnancy has now unwound—that I'm back to where I started. We've now spent more time grieving her than I was pregnant. Babies that were conceived on the night ours died will be born soon.
I dreamt last night that someone was giving away a baby car seat and pram for free, and Rima and I were discussing—is it too soon to start buying baby things again? I woke, and she'd had a very similar dream—that we'd won baby things in a competition, and were toying with the idea of bringing them home.
Maybe this means we are ready to start again, to push the pendulum back in the direction of hope.
•
September arrived, and with it, dire test results about our frozen sperm samples. The same samples that had given our Sydney fertility doctor cause to wax lyrical about their vitality and motility were, when defrosted, only 7 per cent motile. The freezing process, it seemed, had turned our little Usain Bolts into Grampa Simpsons. I wasn't giving up, though, and continued to call the clinic—even with lousy odds, could we still go ahead with an insemination, if for no other reason than so that I could feel like we were doing something? While the rest of the country was waiting to see whether Julia Gillard would be able to form government after a hung election, I was waiting for a phone call from our clinic.
When the phone call came, we eked out a little more hope. We would be able to go ahead with an insemination, but first we needed to make an appointment with the doctor, so she could explain in person exactly how lousy our chances were. I was happy to leap through another hoop, but when I called to try make the appointment, we found she was booked out for another four weeks.
It felt like torture by bureaucracy. No matter how many people I called or how many times I trotted out our sad saga, I couldn't speak directly with any of the people who could change the decision. I knew there was something a little unhinged about my desperation, yet still I left messages for our doctor, both at her clinic and at her private rooms, furious that the one thing I'd been surviving for over the past eight months could be derailed by something as trivial as appointment availabilities. I was ready for a miracle, any moment now. I thought I could hope it into existence. I was furious with these delays. _Don't you know you are standing between us and our miracle baby!_
Sure enough, an appointment magically opened up, and the next day, Rima and I found ourselves face to face with Dr No-Sperm-for-You. Yes, she knew how important this was to us, given our recent loss. Yes, there was sufficient sperm. But no, we couldn't have it—not for a clinic insemination (because of the low motility) and not for a take-home insemination (because our donor had not specifically consented to that at the time of the donation, back in 2007). There would be no September cycle. Our consolation prize was a medical certificate. Given our history, she was willing to class me as medically sub-fertile, and therefore eligible for a Medicare rebate on IVF and ICSI treatment—something she would recommend, given the low motility of the thawed sperm. It was as though we'd taken our old Commodore to the mechanics for repairs, only to be told, 'This one won't work again, but we've got a very nice Mercedes we could sell you.' I didn't want a Mercedes, and I certainly didn't want IVF; not if there was still a chance we could conceive via simpler means.
I moped for a while, but sparked up after a message from our donor that he was going to be back in the country in October, and was willing to make a fresh donation for an at-home insem, just as he had back in May 2009. There would be an October cycle, regardless of the pronouncements of Dr NSFU. The hope that we'd stretched back like a rubber slingshot, from August to September to October, could finally be released, catapulting our hopes into movement. _Dear Universe, it happened once. Please let it happen again_. I just wanted that feeling of small knees and elbows tapping out a message, that warmth and potential. I wanted to finish the story this time—not with a memorial service, and condolence cards and a small amount of ashes falling through our hands, but with a new little voice crying, and baby eyes that opened and moved.
By the time we flew home from Sydney after an October weekend of turkey baster-related activities, the broad beans I'd pushed into the dirt in the cool, weepy days of April were towering with flowers and ripening bean pods. We made our first harvest, slipping the bright green beans from their pods and wishing the fecundity might rub off on us too.
Tuesday, 26 October 2010
It's day one again, and even though it is the first month we've tried since losing Z, it still feels like Groundhog Day. As philosophical as I can be in my head about percentages and buying our lottery ticket, flipping our coin and whatever stupid metaphor you want to use, I'm still crushed because I'm a dirty hope addict, and I really did think something miraculous might happen.
20
Heartbeat
There was a day when my heart started beating again.
Once I was back at work, Penny and I signed up for a Thursday night beginners yoga class. We'd both done yoga before, but this was a back-to-basics approach—a careful and precise dissection of all the familiar asanas and the movement into and out of them. The teacher was a former dancer; tiny, delicate, but exact and rock-solid. I'd come to know and love the sun salute as my five-minute wake-up routine for the early mornings when I took the 4.30 a.m. train to Newcastle for work, back in the misty land of 'before'. It shook the zombiness of deep sleep from my limbs. I could shower the night before and I could eat breakfast on the train, but without those five minutes of yoga I couldn't gather the wakefulness to put clothes on and get out the door.
But now we took apart each step of the sun salute and pored over the mechanics, cleaned each component, moved and oiled it, and then slowly started reconnecting the pieces. We found the moments within the cycle to inhale into—opening, stretching—and the ones that need an exhalation, pushing with our breath, as well as with our muscles, to exact an alignment, to create momentum, to press our palms into one another or the floor, and release into a deeper stretch. It took us eight weeks of classes to complete the cycle, to return to an upright stance, hands in prayer position, bodies warm from the movement.
'Press each point of your fingers and palms together, like mirror images,' the teacher said. 'Let your thumb bones press into one another and let the knuckles cosy into your breastbone, so you can feel your heart beating.'
I did, and there, against all expectations, was my heart—knocking warmly against my thumbs. Big, hot tears spilled out and scorched down my cheeks.
_This is my heart_. My big, noisy heart, which kept banging on as Z's small, fast one slowed and then stopped. The enormity of that engulfed me, with anger, that my heart could so callously continue, and thankfulness that it did; that its familiar beat held her and sang to her as hers faded. And there was surprise; that I was, indeed, alive—that, despite the massive impossibility of continuing on without my child, my heart just kept on at its work, wearily, faithfully, persistently.
•
As I re-engaged with my PhD work, in reading my way back into the literature on DNA paternity testing, I came across a mention of a biological phenomenon called maternal-fetal microchimerism—the persistence of fetal DNA within the mother's body for decades after a pregnancy. These were not just fragments, but whole new stem cells. One study found fetal cells replicating within the mothers' bone marrow over fifty years after their children had been born. In mother mice with liver disease, fetal cells were found to 'contribute to the repairing process'. Tears dripped onto my keyboard. Z's cells were right here—pumping through my heart, replicating in my very marrow—and may very well have helped heal the liver and spleen lacerations I had sustained in the crash.
The question that had been dogging me, of where Z went; what if I could answer it literally? Rather than having to decide on a spiritual story for her, in some religion or other, or imagine her in a heavenly elsewhere, what if I could trace the different elements of her—physical and psychological—here in this world? What if she literally still existed in all of those places? As a memory in my brain; as dark and light specks of ash in the sandy soil at Somers; as a fragment of DNA still replicating in my cells; or as small units of energy, unleashed in the heat of her cremation and still bumping their way through the universe? I had half-expected that, in the wilds of my grief, I might find God. What I hadn't imagined was that I could find solace in science.
There was something about the persistence of atoms and energy that I found comforting. Heaven, perhaps, was not a separate elsewhere, but a continuity in elemental form—a photon sparkling on water, or a molecule transforming from dirt to food to living thing and back again.
21
Making the judge cry
Rima and I had made a deliberate decision after the accident not to invest too much in the criminal proceedings against the driver who caused our accident. I'd seen enough in my time as a litigator that I didn't expect a lot in the way of resolution, let alone healing, from the court process. That work lay elsewhere.
All I'd seen in the moments before our accident was the four-wheel drive. Like a drunk at a party, it had bumped up against a sedan travelling alongside it in the next lane, before lurching directly into our path. I'll admit that I wasn't fond of the large, petrol-guzzling vehicles before our accident, but afterwards, that dislike transformed into something else. Just walking past the bulky frame of a four-wheel drive, I could feel my shoulders tensing, my teeth clenching, and the sight of one on the road triggered tears and an open-mouthed horror.
On a windy September day, I armed myself with two of my oldest and dearest friends and attended the Commodore driver's committal hearing. Unusually, the defendant's lawyer wanted to cross-examine several witnesses, including me. We arrived early at the Office of Public Prosecutions building and were seen into a waiting room. The solicitor handling our matter warned us, 'We're not sure if the defendant's solicitor will turn up'; there was some confusion about whether he'd ceased acting. The waiting room had a large, opaque window that faced the street, and blurred figures of people walked past while we sat turning pages of magazines, as if that would make time move faster.
The solicitor returned with a smile. 'Okay, we're on—looks like he's showed up, after all.'
I grabbed awkwardly at my bag and fumbled for my phone, and we were bundled into the hallway. We joined the other solicitors and police officers in that small space, and one of the men put out his hand to me and introduced himself.
I smiled politely; it was a familiar name. 'Oh, hi. You're one of the police officers I spoke to on the phone?'
'No, no. I was driving the Pajero.'
His words moved through my ears and solidified like ice through my bloodstream. In the eight months since the accident, I'd gritted my fear in my teeth in order to get back in a car, to cross a road, and, finally, to drive. But I still couldn't see a four-wheel drive without flinching. I knew intellectually that he wasn't at fault, that he was as much a victim of this accident as I was. My body and some older, more survival-related part of my brain, however, only knew what it had experienced—the sight of several tonnes of Pajero hurtling towards me and my family, and the bone-breaking, placenta-tearing sensation of impact.
'I just wanted to say I'm so sorry.' He leant in for a hug.
I felt myself recoil, and then try to correct it. 'Oh.' _Pause. Gulp. Breathe. Remember that I need to respond_. 'You were hurt too. How are you going?'
'Not so good. I haven't worked since the accident.'
Somehow, we were able to fill the space between us with words again, but he'd felt me recoil. And I'd pulled away from another human being at the very moment he sought my forgiveness.
The cross-examination itself was less charged. The defence barrister was vaguely familiar, but it wasn't the time for figuring out how we knew one another. He quizzed me on the TomTom GPS system I'd given Rima for Christmas, to help her navigate Melbourne, and that we had been using for the first time on the day of the accident. Had I been looking down? How long was I looking at the GPS for? He seemed to fret over my answers. None of this was helping. I had never appeared in this courtroom, but in that formal space felt more like my lawyer self; able to be clear about what I could and couldn't remember, and to tune out the emotional static that usually accompanied talking about 27 December.
It was several weeks later, when Sydney friends were visiting and we'd taken them to a pub, that the prosecutor rang with some surprising news: 'The driver has pled guilty.' The next step was sentencing. The prosecutor got in touch with us and let us know that we could prepare victim impact statements. We could decide whether we wanted to read them ourselves during the sentencing hearing or have the prosecutor read them for us.
Rima decided she didn't want to be there on the day. She didn't want to see the defendant, and wasn't certain how she would react. But if my statement was going to be read out, I wanted to be the person reading it. There was so much in our situation that I had zero control over, so, at the very least, I wanted to tell it in my own words. I wasn't going to enter into the debates about whether our driver should instead have been charged with dangerous driving causing death. What mattered to me was to make our daughter visible, not as a creepy generic 'unborn child' or an object lesson for debates about abortion, but in our relationship with her changeling form: as a long-held hope; as a tiny leap in my belly; as a mysterious, moving bump; and, finally, as a serious-looking but unbreathing baby.
The sentencing hearing was set for early November.
•
Tuesday, 14 September 2010
I was nearly four years old when Azaria Chamberlain disappeared. The controversy surrounding her mother, Lindy Chamberlain—who was accused of murdering 9-week-old Azaria—formed such an interwoven part of the cultural carpet of growing up in Australia in the 1980s, that it took me a while to realise, first, what an extraordinary woman Lindy Chamberlain is, and second, that I now have several things in common with her. It bothers me that I have some kind of cultural cringe in saying these two things. But when I read her letter, published in the newspaper to mark thirty years since Azaria's disappearance, it hit me like a tonne of bricks:
'It is hard to believe it is thirty years today since my darling baby was taken.
'For some odd reason everyone says you will soon forget.
'Why is it that people expect me to forget a part of myself? Why would you? Loss of a loved one, particularly a child, is not something you forget any more than you can get out of your mind that you once attended school.
'That does not mean you dwell on it all the time. It is simply there in the fabric of your life and history. In some ways it seems forever and in others it is like yesterday still.'
For all the movies and telemovies and tabloid newspapers and magazine coverage, I had somehow forgotten that this woman lost her baby—lost her beloved 9-week-old daughter. And suddenly I thought, here is a woman who has survived babyloss—and she seems so functional. Not just losing a daughter (and never having the chance to say goodbye, because her body was never found), but on top of that, being accused of killing your baby as part of some cult ritual, enduring more trials, inquests and royal commissions than have ever been held on the one issue in Australia, being jailed and separated from her living children for over three years, and being at the centre of a media circus for much of two decades. And yet, here she is, self-possessed and able to articulate her position clearly and passionately. I think she deserves some credit for that.
I also think of Lindy whenever I have one of those awkward moments when I'm out somewhere and I've pulled it together and am actually enjoying talking to people, but then I have to tell someone what happened to us, and their natural reaction is shock and sadness. They look at me, and I'm not weeping and falling to pieces, in fact, I want the conversation to move on, and I wonder whether they think I'm some monster who doesn't care that her baby died. I need a little sign that says, 'Yes, I do care. This is the saddest thing that has ever happened to me. My grief is huge and voracious and has eaten huge amounts of my time and energy and personality. But right now, I've got it on a leash and feel like I'm in control of it. Don't start poking at it now, or it will chew my leg off before your very eyes. I need my grief to behave in public, for my own sanity and dignity.'
I wonder—why do I care so much about whether people think I'm grieving 'enough' or in the ways that they would expect? What standard am I trying to perform to here? If I don't fit within one stereotype ('good grieving mother, tragic, weeping') does that automatically push me into another stereotype ('bad, uncaring mother')? And this is where it comes back to Lindy, and back to the way she was demonised by the media for appearing to be 'cold' when she had to give evidence at her trial for her daughter's murder. We grieve in the shadow of all these myths surrounding Lindy Chamberlain. For me it is a reminder of why we need feminism—to remember the force that stereotypes have over women, the way in which our bodies and stories are so often appropriated for other people's purposes. That sometimes we need to claw away all the stereotypes and speak for ourselves.
I have to give evidence tomorrow. Unlike Lindy, I won't be on trial for killing my daughter (someone else will be, though he's charged with dangerous driving causing serious injury, not with murder). But I'll be thinking of Lindy, and wearing sunglasses on the steps of the courthouse in her honour, and in memory of Z and Azaria and all the babies that we wish were here with us.
On the morning of the sentencing hearing, I wore my babushka brooch over my heart, and slipped Z's photo into an envelope placed in the pages of my statement, so that I could touch it while I was reading.
Victim Impact Statement (Nov 2010)
I found it hard to sit down and write this statement, because it is impossible to fit into words the impact of the defendant's dangerous driving and the resulting crash on my life. For a long time I really wondered whether I had died and started a new, different life, because everything was so unrecognizable, including my own body and personality.
Just seeing a car accident on TV grips me with terror and leaves me crying. Big four-wheel drives still frighten me. The sound of ambulances, lying on my back, sitting in a car, putting my right hand to my head—all these things trigger shock and trauma. It is a marathon effort to get dressed, to get to work, to try and bring my mind to a task. The scars on my legs, arm, head and abdomen remind me every day of what it feels like to be crushed by broken metal. But the biggest impact is the least measurable—it is the growing space that our daughter would have had in our lives, had she survived.
Let me tell you about our daughter. She was conceived in Sydney in May 2009 after nine months of unsuccessful fertility treatments, and several years before that of finding a sperm donor and going through counselling processes and quarantine periods as part of the fertility process. We weren't going to tell the girls about the pregnancy until we'd had a scan confirming the pregnancy, but they guessed because we were smiling so much.
At fifteen weeks of pregnancy I first felt her move, like a little goldfish in my belly. We nicknamed her 'Haloumi'. Soon we were getting lots of kicks, and when she stretched, you could see my whole belly move. Her favourite (or, I was hoping, least favourite) music was Rod Stewart—every time he came on the radio she would kick. I was lucky enough to have a healthy, smooth pregnancy. Each night, Jackie and Jasmin would pat my tummy and say 'goodnight Haloumi'.
About a week before the accident we had just moved to Melbourne so that we could be close to my family, and so I could take up a job after my maternity leave finished that wouldn't require so much commuting. The interstate move combined with both Rima and me preparing to go on leave had made it a busy, stressful time, and we were looking forward to spending Christmas with family, and had booked a holiday house for two weeks in January so we could have some quiet family time before the birth.
Four days before the accident, I had an ultrasound and Rima and the girls and my mum and sister came along, and we saw her, very squished up by now, but heart beating strongly and headed in the right direction. At my last midwife visit, the midwife showed me how to feel her back, her legs, her little hand, through my belly. On the morning of the accident, we were at a picnic, and she was hiccuping.
When the four-wheel drive hit our car, and we came to a stop, Rima kept asking, 'Can you feel Haloumi moving?' I didn't answer because I could see what I thought was petrol pouring from the other car and I was scared it would blow up. I told Rima and the girls to get out of the car. I couldn't get out myself—the car was crushed around my legs.
Rima and the girls went to different hospitals, so I was left by myself while the doctors tried and tried to find her heartbeat, and when they finally told me she'd died I had to tell Rima over the phone that our baby hadn't made it. I was induced, and was having six minute long contractions while I was having blood transfusions, while the nurses extracted the glass from my arm, and while my head wound was stapled, re-stapled, and finally stitched because it kept leaving me drenched in blood. Early the next morning, my daughter was born by caesarean section. My dad and mum and sister held her while the surgeons repaired my broken knee. When I came around from the anaesthetic, the first thing I saw was my midwife coming towards me with a photo, saying, 'You had a little girl.'
She was 2500g (5 pound 8) and 48cm long—the length I was when I was born. She had dark curly hair and skin softer than rose petals. She looked calm, but with a slightly worried brow.
I am so sad for Jackie, Jasmin and Mariam who lost their little sister, for my parents, who lost their first grandchild, for my sister and brother, who lost their first niece. Z was so loved and we were all so ready to welcome her.
We have thirty-six photos of our daughter. This is all we will ever get. We have her hand print in black ink, and her footprints. There is a drop of her blood on a blanket we wrapped her in. We didn't take a lock of her hair. I wish we'd known at the time to do that.
What breaks my heart is all the things I can't tell you about my daughter, the things I will never know—the colour of her eyes, the sound of her voice, the things that help her get to sleep, how her sisters make her laugh. So many things we can only ever hypothesise about—would she be crawling by now? What foods would she have reached for? Would she be keeping us up all night? Would she have enjoyed her granddad's singing? Would she be comforted by her Oma's kisses? What kind of little girl might she have been? What kind of young woman?
It seems so strange that all these possibilities—a whole lifetime's worth of them—could have disappeared in one stupid moment on Warrigal Road on 27 December last year. I don't know if I'll ever get my head around it. To the defendant—I sincerely hope you never have to go through sorrow like this, but I also hope that you never again cause anyone else such sorrow.
•
Afterwards, the prosecutor let me know there would be journalists on the steps as we left the court complex, and that I didn't have to talk to them unless I wanted to. I'd said most of what I needed to in the courtroom. The only point left to make was an obvious one about cars and their capacity to become death-machines.
Out on the court steps, I resisted the instinctive urge to smile at the camera. I thought about Lindy Chamberlain, and wondered if I'd been naïve choosing to do this. How much grief was enough? What was my face supposed to look like? The grinding grief was suddenly elusive, and instead I felt a strange elation, like the relief when your ears pop as the plane goes up and the pressure equalises in your head. At the very moment my sadness was to be publicly broadcast, I felt lighter—almost fraudulent for making such a fuss. But I held Z's photo inside the envelope and thought, 'She's worth making a fuss over.'
22
Close up with hope
All through 2010, the calendar mocked me. We lived in the shadow of two ghost calendars—the year before, when we'd been unbroken, and the year that might have been. February, the month Haloumi would have been born. June, the birthday that was my first as a mother, but spent without my baby. The June before, I'd just done a pregnancy test and, for the first time ever, encountered that second faint blue line. We went out dancing with friends, and I surreptitiously sipped soft drink, and Rima and I exchanged secret (or probably not so secret) goofy smiles. September 2009, we'd started telling people, and I had trouble doing up my jeans. By November 2009, I was well and truly showing, and my students and colleagues were getting excited for us. November 2010 was a very different story. The criminal proceedings were over and the young man who'd set the billiard balls rolling was in jail, but our daughter was no less dead. We were still having no luck trying to conceive, and the black hole of 27 December was looming.
And then, two weeks after the media bubble that surrounded the sentencing hearing, I was driving somewhere, and while we were waiting to turn right, a silver four-wheel drive turning left swung wide and nearly hit us, but corrected in time. I saw the driver's face—he was young, maybe still on his L-plates—his own eyes as scared as mine. In the choking tears that followed, I felt a weird confusion. How could this be as terrifying as the moments leading up to our accident, when there was no impact? I knew I was safe this time, but my body and mind remained tensed for the impact, my jaw locked.
The days after were panicky, with headaches and jaw pain from grinding my teeth in my sleep. Another trip to Sydney for an insemination left me wiped out and weepy. I had gone back to full-time work within a few months of the accident, but suddenly I couldn't hold things together anymore. I took sick leave, begged work colleagues to finish some of my marking and cancelled giving a paper at a local conference.
Thursday, 16 December 2010
There is a new sensation I've discovered in the past few months, which I've nicknamed 'the hard swallow'. It happens when I'm driving, and see a four-wheel drive coming towards me, or when I see a baby and try to estimate—11 or 12 months old? Or when the IVF administrator tells me the dollar amount we have to pay to start IVF. A ball of fear or sadness, or something of a similar texture, rises in my throat and I have the urge to run, scream and hide. But I know I can't, so instead I swallow it down, and get on with the business of moving through the world.
I'm not pregnant this time. I didn't really think I was, but when I got to the 27 day mark, I just started entertaining little thoughts, 'maybe Christmas would feel good after all' etc. But no. And while IVF felt like a relatively positive Plan B when I went to visit Dr Lovely last week, it doesn't feel like such a fun path now.
I'm a big hippy, you see. I don't like the idea of doctors taking control of my cycle, forcing my ovaries to blister with artificially stimulated ova, vacuuming out my eggs, and coercing them to germinate with a selected sperm. It all feels a bit too much like high school dancing classes where we were supposed to hold our dancing partner tight enough so that a vinyl record put between us couldn't fall to the floor. It's as though doctors are telling my body, 'Oh, just get out of the way and let us do this properly!' I know what it feels like to have medical experts take over my most basic bodily functions—I'm lucky they did, otherwise I would be dead, but that doesn't mean I like it.
We don't *have* to do IVF. As a dear friend has pointed out, our lack of luck so far is probably more about timing than anything else. But given our issues with frozen inseminations, and the difficulty and stress involved in travelling interstate every month for fresh inseminations, and 'advancing maternal age', it is making sense. What I don't like most about IVF is that I feel corralled into it by fear—fear that maybe Z will be the only baby I have, that it is all too late, that if Christmas 2011 were to roll around without a pregnancy in sight, I'd lose what scrap of sanity I've got left. So it is a pragmatic choice, but a very reluctant, sulky one. And it makes me even sulkier to know how much we have to pay for procedures which I don't want anyway (or wish I didn't need). But this is where the hard swallow comes in.
•
Christmas in Melbourne minus a pregnancy, however, felt too big and too hard to swallow, so we hatched a plan to run away from the whole thing. Like Max in Maurice Sendak's picture book, we would sail off to rumpus with the wild things, or at least, with some friendly wombats. We packed up the car with tents, bags and the girls, drove onto the ferry to Tasmania and sailed off to rumpus with the wombats. And what about the small matter of 27 December—the date that had been hovering like a four-wheel drive half a second before impact? I wanted to look into its beady eyes and remember what it felt like on the other side; to feel whole and unharmed and hopeful. I wanted a whole day where we didn't need to get in the car, preferably with a beach and a big, salty ocean nearby.
We woke up on Christmas morning in a tiny cabin near Cradle Mountain, and marked the day with small presents, a big walk, and a fancy lunch at the lodge. By 27 December, we were camped near Wineglass Bay. I had been so scared of the day itself, but in the end, it was just an ordinary day—arguments, half-successful pancakes, a picnic lunch, peacemaking. We walked all the way from our campsite to Wineglass Bay and back again (with swimming in between), Rima insisting that we stop at the bar in the lodge for a drink in Z's honour. A superb blue wren joined us on the balcony.
It was almost dusk as we walked back to our campsite via the beach, and in the wet sand, the girls drew our family—depicting Z still in my belly. I wrote her name too, with a Zorro-like 'Z' right at the edge of the waves, where the sand is not solid or liquid but some other matter. The thought that her name would wash away felt like some kind of anti-memorial. Rather than her name persisting, set in stone, it would merge with the grains of sand, with the gallons of ocean and with the movement between them.
•
Tuesday, 11 January 2011
Recently, I've started feeling queasy about my own hope in the same way that I do with really corny advertising. It doesn't feel true. I know from experience that hopes can be shattered, even when you are being cagey, trying not to hope too much. I know at some level, that was why I hadn't finalised a name for Haloumi before she was born, why I hadn't found out her sex. I was trying to arm myself against hope. But I still believed in it—and attempting to guard against it was really just replacing overt hope with secret hope.
I've gone through life with a naïve idea that things will work out, that if I'm calm and careful, it will all be okay. In that moment when the car stopped moving after the impact, when people had arrived on the scene and were helping us, when I'd been able to wriggle my toes, and didn't feel any pain in my uterus, I was so certain that Haloumi would be okay. I was good, I stayed calm, I did everything I could to cooperate with the paramedics and firefighters. I didn't even let the idea that she'd died enter my head—I kept my hand there, on my belly, inconveniencing all the doctors and nurses wielding Dopplers and ultrasound wands, because I was trying to keep her alive by hope alone.
And I was so so wrong—she was so so dead, even by the time they got me in the ambulance. That doesn't mean that screaming and losing it would have been a better response—but it has taken me a long time to try to get my head around my own broken hope. I know it makes no sense, but I'm so sad that my hope wasn't strong enough to save her, that it failed when put to the test.
So, I'm exploring a bit about hope, and Pema Chödrön's suggestion—'if we're willing to give up hope that insecurity and pain can be exterminated, then we can have the courage to relax with the groundlessness of our situation'. That sounds quite stark, but I know very well now that there are no guarantees, that the ground can fall out from under you at any moment. This approach is realistic at least—there really is no hope that you could live your life with nothing bad ever happening to you. But is it a healthy approach to take? Wouldn't it be morbid and negative to be continually mindful of your complete lack of any security? It seems counter-instinctive that you could be both thinking about your 'groundlessness' and 'relaxing' at the same time. So far, though, trying out this groundlessness has been calming in an odd way. It is helping me drive the panics back a bit—or rather to acknowledge them and sit with them rather than run around looking for something I hope might 'make it better'.
But I still find it very hard to embrace the idea that giving up hope is a good thing to do—or that it is a part of appreciating that life is full of impermanence and change. I like hope! I'm always hoping this, hoping that—for myself and for others I care about. So much of the culture I have grown up in is based on the idea that 'things will get better', that the good life is normal. But there is also a big sense of relief in accepting that what happened to us wasn't an aberration from the happy life that everyone else gets—that as sad as it was, loss, injury and grief are part of the human condition. Yes, there are heart-achingly beautiful, good things in the world, but they don't last forever, and death and cancer, and embarrassment and disappointment are just as normal and as common.
This bit especially made sense to me: 'Hope and fear come from feeling that we lack something; they come from a sense of poverty. We can't simply relax with ourselves. We hold onto hope, and hope robs us of the present moment... Rather than letting our negativity get the better of us, we could acknowledge that right now we feel like a piece of shit and not be squeamish about taking a good look...'
Now that I've started giving myself the tiny injections of the IVF drugs every morning, like DIY acupuncture, I know that we're getting close to the extreme hope-dance that is an egg pick-up, an embryo-transfer. I haven't coped very well with hope for the last three cycles of inseminations. So I'm going to try this groundlessness—to sit with the complete uncertainty at the heart of baby-making and do a bit less grabbing onto the hope of some other future moment making things better. Of course I want it to work. But I'm curious about how different things might feel if I just take each moment for the groundless, uncertain thing that it is.
Fear is the other one I've been experimenting with, trying to get close up with. On the first day of our holiday, we went white water rafting, and half an hour in, our guide pulled into a deep, still corner of the river, and pointed up. 'See that little cliff?' he said, grinning. 'You're going to jump off that.' And we did. I felt the panic grip me and tell me to turn around, and I hesitated once, well, twice. But then I jumped, and the panic jumped with me and I screamed like a big girl and flapped my arms all the way down. The girls laughed their heads off. And I got to know my fear a little better.
23
The charnel ground
With the intensity of the first anniversary of our accident behind us, January felt like a relief. I would still look at the date and know that a year ago on that day, I was transferred from ICU to the trauma ward, or transferred home from rehab, but these were dates I was familiar with, here on the 'after' side of the river.
I'd decided to sign up to staff one of the January camps held at Somers, where Z's ashes were buried in the bush chapel—partly (if I were honest) to spend some time near her. We were starting our first IVF cycle, and I would wake up early with the summer sunrise, swab my soft belly with alcohol and watch the needle push against my skin, then pop through. It felt momentous. I'd had so many objections to IVF, to the artificiality of it—making your ovaries swell up like cantaloupes, only to be, yes, harvested. But months of flying back and forth to Sydney to meet our donor to conduct inseminations (or asking him to fly down to Melbourne) had been complicated and ultimately fruitless, and I was terrified at the thought of another Christmas with no living baby.
Most mornings, I snuck out to the beach and to Z's spot, to touch the sandy earth and bring her a little flower or a washed-up shark egg. And, after all the rushing about of the past year, it felt like I finally had time to think properly, with a good beach to do it on.
•
One of the things I had found hard about the idea of being 'in the moment' was the fact that some moments are awful. If you completely focused on that particular moment, wouldn't you drown in the sheer awfulness of it? Wouldn't it be too depressing to survive it? Wasn't it better to just edit those moments out? I had one particular opportunity to test run this theory, because there was quite clearly one moment I would have loved to cut away from the fabric of my life—the moment of impact and everything it set in motion. It couldn't be undone, I understood that, but was it really a moment to focus on?
What if I had taken that moment, where I was sitting in the wreckage—trapped, bleeding and so afraid that the car next to us would explode—and let my fears and hopes dissolve, so that I was no longer being tugged forwards into a better or worse imagined future? What might I have experienced right there? With hindsight, I could have been present for the last moments of my daughter's life. She was doing the hard work of dying while I was fervently wishing I were somewhere else: in an imaginary future where she was okay.
When Rima tried to call me into the moment and asked me, 'Can you feel Haloumi moving?', I was so angry. I stubbornly wanted to avert my attention, to avoid the uncertainty. I look at this now with tenderness. It was a futile denial, like when a cranky three year old holds their hand up so they can't see you. I didn't want to be engulfed by fear, but I couldn't imagine doing anything other than fearing or hoping. I couldn't imagine that I could just sit with the huge, frightening uncertainty of the situation; that I could treasure a moment with my daughter when it was possibly her last.
I thought that by denying the possibility she might die, I could magically save her by force of hope alone. I know there is no way I could really have known what was happening with her, but I wished I'd been a bit more present for those last little beats of her heart. Instead I was demanding something of her she could no longer do. ( _Please, Haloumi, please be okay. Please be okay, my little one_.) I'd sworn off regret, because I didn't think dragging myself into the past helped either, but perhaps there was something in this idea that even the worst moments deserve attention.
It seemed odd to me that such an awful, traumatic moment could also be such a precious one. But it rings true with my other experiences—with the preciousness of seeing her little, still face, and the pride I felt in labouring for her. And, bizarrely, this realisation that paying attention to a moment couldn't make it any worse—and, indeed, that running away from it (into fear, hope or denial) could cause further suffering—made me feel calmer in my grief. I finally felt as though I were learning something from all this grief: that I didn't have to keep grasping for some kind of solution; that I could sit with this discomfort and uncertainty; that I could feel something impossibly painful and still experience it, be alive to it.
On the last night of the camp, all three hundred or so of us filed into the darkness of the bush chapel, and, with the native creatures noisily putting themselves to bed around us, I held a torch and told my story.
This is my daughter's first big camp. She's not in any of the grouper huts, or in the staff huts or tents. She's here, in the bush chapel—her ashes are buried just between the altar and the bush.
I was eight months pregnant with her when we had a head-on collision with a Pajero. She died on impact from a placental abruption. Apart from that, she was a beautiful, healthy baby with a tiny bruise on her right eyebrow. I was hospitalised for three weeks with a broken knee, broken sternum, lacerations to my liver and spleen and various cuts to my legs, arms and head.
In the last year since the accident, I have had to do the impossible every day. I have planned my baby's funeral from an intensive care bed. I have learned how to walk with a broken knee. I have held the people I love the most while their hearts are breaking and there was nothing I could do to fix it. And every day, I live, while she is dead.
For a long time I was desperate to escape my grief—I thought there would be some 'solution' to it, a time when I might feel some ground under my feet again. But like it or not, this is the nature of being a human being. We know that we are fragile, and we know that we will all die, but it all seems pretty theoretical until you lose someone you love. It seems impossibly cruel that a baby could die when we loved her so much and we hadn't even had a chance to see her open her eyes. But, this is what life throws at us—impossible miracles like babies, and impossible losses.
And while I now know there are no guarantees, this is what gives me a little peace—that what we have experienced is not a terrible aberration from the good life that we are all entitled to, but that the sadness and wretchedness of grief is part and parcel of the love and inspiration I still feel for my daughter.
I have been reading _When Things Fall Apart_ by a Buddhist nun called Pema Chödrön who puts it this way: 'Inspiration and wretchedness are inseparable. We always want to get rid of misery rather than see how it works together with joy. The point isn't to cultivate one thing as opposed to another, but to relate properly to where we are.
'Inspiration and wretchedness complement one another. With only inspiration, we become arrogant. With only wretchedness, we lose our vision. Feeling inspired cheers us up, makes us realise how vast and wonderful our world is. Feeling wretched humbles us. The gloriousness of our inspiration connects us with the sacred elements of the world. But when the tables are turned and we feel wretched, that softens us up. It ripens our hearts. It becomes the ground for understanding others.'
And this is the strange thing. As this loss has carved my heart out so painfully, I've also felt an intensity of joy beyond anything I felt before—often mingled together. Where I thought this pain would crush me, it has transformed me, and by feeling it, and gently observing it, rather than trying to escape it, my heart has expanded beyond my imagination.
It was a relief to let the words out, to make visible my grief. It was all true—there was a new intensity to everything. It was like the rawness where my knee had healed up; the nerves were still not quite sure what messages to send, so they sent them all in loud capitals. As much as I wanted to be okay with the awful moment, to 'lean into the sharp points', as Chödrön would say, I still wanted to transform it into something, to make it worthwhile—to tack on a happy ending. Hope still had its hooks in me.
•
In January 2011, I flew to Cairns to visit my mum. It was as hot as Melbourne but stickier. The first night I slept in Mum's swag outside her campervan, an arc of tent pole holding the mosquito netting away from my face. At one point in the night, a possum investigating my bed woke me up. I fumbled with my tiny torch—I was pretty sure it was a possum. All the same, the next night I slept alongside Mum above her campervan's cab.
I was still reading Pema Chödrön and was struck by her description of the hospital emergency room as the 'closest thing to a charnel ground in our world'. I had a vague, gory idea of what a charnel ground was—a place where human bodies were left to decompose—but it made a weird sort of sense that a charnel ground would be an ideal place to meditate on the nature of death and impermanence. Indeed, it felt like my body had been the charnel ground—housing the dead, being sliced open, pieces of chipped bone and fluids being discarded. So many people, laying their hands on me, holding me together, slicing me up; gory and sacred all at once.
Jen, one of my midwives, had visited us at home in Preston a few weeks after I was released from rehab. She wasn't there in any official capacity, just to say hello and pick up the plastic container that she'd given me in the trauma ward, complete with its big wedge of homemade rhubarb and strawberry cheesecake.
'You're looking good,' she said, coming onto the verandah.
'Much better than when we met in the ER, I bet,' I said.
'Yeah. You were looking very grey for a while there; you'd lost a lot of blood.'
'From the liver and spleen?'
'Maybe, but I think it was mostly from the abruption. When the surgeon did the incision for the C-section, there was blood everywhere...' She winced and smiled. 'On the walls, on us—bit of a horror-movie scene!'
And we laughed, because this was my squeamish body producing horror-movie effects that made seasoned midwives wince. Yet, her reaction was not horror or disgust for my charnel ground of a body—just a pragmatic bearing witness, a tenderness.
24
Fat Tuesday
While I continued the IVF injections and my ovaries swelled, Tropical Cyclone Yasi was building in the Pacific Ocean off Fiji. As Yasi neared the Queenland coast, and shifted direction towards Cairns, where my mum, brother and sister-in-law were living, it was upgraded from category 3 to 4 and then to 5, putting it in the ranks of Hurricane Katrina. I'd just finished reading _Zeitoun_ , by Dave Eggers, about one family's experience of the chaos that was Katrina, and I was terrified. Mum was holed up with my brother and sister-in-law in their Cairns apartment; thankfully, in a cyclone-proof building. They loaded mattresses against the windows, and were well stocked with food and water.
When text messages came through, letting us know that they were okay, I could finally exhale. The storm had been noisy, but not as destructive as had been feared, at least in Cairns. Between them, my mother and brother had survived four car crashes, the Black Saturday bushfires, a Bolivian uprising, and now, a tropical cyclone, the lucky buggers.
Whether it was the IVF hormones or general cyclone anxiety, I still felt off-balance and tender, with a weird low-level nausea. If my ovaries made noises, they would have been submarine-depth-sounding 'pings'. So, it was a further relief when the clinic rang after a scan to say they were scheduling me for an egg retrieval.
Our egg pick-up was on a Friday. I slipped so easily back into patient mode; _Here, you drive_. I surfaced from the sedation in a recovery room, alongside two other women. A nurse came along to see how we were, whispered in each of our ears the results of the egg retrieval—it felt very _The Handmaid's Tale_. They'd harvested an even dozen of my eggs, whisked away to be firmly introduced to their spermy boyfriends. I was reunited with Rima in the waiting room, wearing a groggy smile, and hugged her with what felt like big, fuzzy gloves, and spent the rest of the day at home under a doona, half-watching the midday movie.
We were back at the clinic two days later, for the embryo transfer. No sedation this time, just the waking indignity of chatting while the doctor navigated a fine catheter through my cervix. The embryologist appeared, and asked us to confirm our names and the name of our donor, and we saw on the screen a magnified blob: two plump cells within a circular cell wall. _Welcome aboard, tiny speck!_
•
In the quiet waiting space, we took a daytrip down to Somers to visit Z and the bush chapel. It was a whole year now since we'd farewelled Z's gritty ashes into a sandy hole. We took her some baby roses from the front garden, listened to the birds and the crash of waves, and let the dogs run on the beach.
The ten days before any pregnancy could possibly be detected inched past. I dreamt that I was driving but I couldn't see properly, that I drove a truck off a cliff, that we survived and stole towels from a great aunt. The first home pregnancy test was negative. I was deflated, but held out a little hope that I'd just tested too early. But then, a day later, my period arrived and I felt silly for believing that such a tiny thing could actually turn into a baby. I was sulky about turning up to the clinic for the official blood test. What was the point, when it was so clearly going to be negative?
Except that it wasn't. The test came back with a HCG of thirty-seven—a fraction of the 2063 I'd had with Z, but, still, a positive. 'Come back on Tuesday,' said the nurse over the phone. 'I'm sorry; it is really just too early to tell as yet.'
On Tuesday, my HCG level was seventy-eight; still very sluggish and probably an indicator of an implantation, then early miscarriage, explained the nurse. 'Come back on Friday—but if you get any sharp abdominal pains, go straight to emergency, just in case it's ectopic.'
Friday, 25 February 2011
When I arrive at the clinic for my blood test this morning it hasn't opened yet. A queue of people stretches a good fifteen metres down the hallway. Usually we only cross paths in ones and twos in the waiting room. We glance at one another and drop our gaze, respectful of our mutual privacy; we studiously read trashy magazines.
But here we all are, leaning against the wall, relieved and embarrassed to see just how many of us there are. Some women are alone, but most are accompanied by a male partner. 'Here to support her' are the looks they give one another, not so much 'I just want to be a dad'. Either way, they move sheepishly when the queue starts to advance. We didn't giggle about this type of baby-making in high school.
Seated in the waiting room, I start reading a newspaper article about the Christchurch earthquakes. An earthquake begins in me, sobs catching in my ribs, tears steaming to the surface. I want to close the paper, to stop all these sad things from happening, to un-read the headline, 'Mother dies with baby in her arms'. I try to divert, read something else. But still, my body freezes, remembers the sensation of being pinned by twisted metal, the realisation that I can't get out on my own, someone has to come and help me. This fear is here even without the newspaper. I take a deep breath and open the flimsy pages again and let myself weep for the mother, for her baby, and for me and my baby. All that sadness muddied together—if I could just breathe it all in, soak it all up and breathe out a sense that it will all be okay, that there is something connecting my child and I that cannot be crushed by a falling building or by the impact of a 4-wheel drive.
•
On Friday, my hCG level was up to 201. This time, I got the call from the specialist early pregnancy nurse. She asked me again about the bleeding. I explained that it had been heavy, like a normal period. That it had stopped now.
The nurse hmphed.
'It's clear that there's been implantation, and that you've got pregnancy hormones being produced; it's just that those levels are much lower than we'd like to see. You'll just have to come back on Monday morning for another blood test.'
'So, have you ever seen a viable pregnancy with those kinds of numbers?'
'I have, but it's important to be realistic about your chances here—realistically, it is looking very unlikely at this stage.'
I put down the phone and felt like laughing out loud. _Ha ha_. Realistic! The realistic view was that they had no idea what was happening, and neither did I. We were all spectators to the unfolding soap opera that was my uterine environment.
By Sunday night, though, my Zen was running out. I had fretful dreams of dodgy hotels, where one room connected to another and another and another. I kept waking, thinking it was time to get up and go in for my blood test, even though it was 3 a.m., 4 a.m., 5 a.m. I'd run my situation past my online IVF veteran friends, and heard story after story of low HCG results that were now snoring happily in the cot down the hall. A little speck of hope had got under my skin, and with it, the fear that it would be dashed.
By the Monday morning, my HCG had risen to 630. And although it was the same Nurse Realistic giving me the news, she was much more upbeat this time. She'd spoken with my doctor, and he didn't see any need for a further blood test, just a scan in a week's time.
'So, this might mean that it may actually be viable?'
There was a big, realistic intake of breath. 'Look. With all early pregnancies, but especially when your betas started low, we can't really confirm anything until the scan; but, yes, congratulations. But if you do have any sharp pains or bleeding, don't ignore it, go straight to emergency.'
I couldn't help it—underneath all the caveats, the one word I heard was 'congratulations'. _Welcome, uncertainty—come on in_.
A week earlier, when I'd been certain I wasn't pregnant, I had booked a flight up to Sydney as a consolation prize. I could catch up with friends, enjoy the Gay and Lesbian Mardi Gras parade, and if the timing was right, attempt a fresh insemination with our donor. But now, the trip was a welcome distraction while I waited out the days until the next scan.
Sydney, when I touched down, was thick with memories of 'before'. They were heavy on the ground and mostly still undisturbed because I had spent so little time there since December 2009. The sight of kids in the uniforms from the girls' old school, the shops where I had bought ordinary, inconsequential things, the road that led to our house—all of these things couldn't fade into the background until my brain had trotted through its 'Last time I saw x, Haloumi was here' routine.
I returned to one of my favourite Sydney spots: the women's baths at Coogee. The last time I was there, my Haloumi-filled belly had stuck out obscenely between my bikini top and bottom. I had greeted other swimmers with my stretch marks. That belly was remarkable. Everyone remarked on it, speculated on Haloumi's gender and wished me well.
I had so many pregnant and unpregnant swims there over the years. Some involved mildly athletic laps; some, snorkelling and marvelling at the starfish, shellfish and, once, even an octopus under the surface; some were splashy and noisy, with the girls; some quiet and contemplative, with no one else in the water. I'd seen it in a storm, with the waves crashing over the rock wall; I'd seen the surface sparkle with a beating sun; and I'd eyed off the greeny-blue depths when it was far too cold to swim.
And now—who knows? I was egging on this tiny speck of potential, hoping it was in the right spot, hoping it wasn't ectopic, chemical, blighted, all kinds of words for 'lost already'. To swim there felt like an act of love—towards my tentative self and this little question mark of cells.
I met up with an old friend who had also been going through IVF, and was now ten weeks pregnant, that odd early stage of pregnancy where you just feel fat and off-colour; a tenuous, uncertain state. We arranged a spot in the mardi gras parade on my usual float. I'd been to the workshop earlier that day, and made loopy green headdresses and tutus for us. And we danced all the way up Oxford Street, spangled with glitter and shining from the cheers of the crowd. I dared to imagine two small children who might delight in the idea they'd marched in the mardi gras even before they were born. It was our Fat Tuesday (quite literally; from the French, _mardi gras_ )—we feasted on hope while we could.
Flying back to Melbourne, that hope solidified a little. Just being around someone else in those strange early days of pregnancy made me feel like maybe, maybe, we'd see a heartbeat too this time. Lo and behold, at the scan on Monday, there was a fetal sac in the uterus, where it should have been, but nothing else. No fetal pole, no heartbeat.
Our doctor was philosophical. 'There are a number of possibilities here. It could be that you've just got a slow starter. Sometimes embryos drop a few cells before implanting, and that puts them a little behind. It's still early days, so I think we need another scan, next week, before we make a call either way.'
On the Tuesday, I was back at work, wearing a summery outfit—new dress, new shoes—when the bleeding started. There was a heaviness, but it came suddenly, like something loosening. In the time it took for me to half-run from my office to the toilet, my new strappy white sandals and the lino floor were marked with big polka dots of blood, perfectly circular.
25
Undone
In the emergency waiting room at the Women's, we ran into my friend Sophie, tall as always, but rounder than usual. She was coming in with her partner for her 38-week check-up. Soph had been at the picnic on the day of our accident. She had just been starting IVF, and there I was, heavily, smugly pregnant. When we said our goodbyes, I gave her a hug and said, 'Good luck!' She squeezed me back and smiled, 'You too!' And as I'd walked to the car, I thought, 'I don't need luck, I'm already there.' How these little memories mocked me. Now, it was the best I could do to acknowledge Sophie and indicate that we couldn't really talk.
The hospital took bloods and let us wait around for the results. My hCG levels were still rising, but we'd need a scan to get any real information on what this bleeding meant.
We went to our fertility clinic the next morning, and, despite the horror-movie bleeding, the scan showed that the little fetal sac was still there. There was still no sign of a fetal pole but the sac was persisting nonetheless. When we went back four days later and saw the same sight on the ultrasound screen—no growth, no embryo—our doctor was pragmatic. 'Looks like nothing is going to happen with this one; I'm sorry. I can book you in for a D&C on Friday, if that suits?'
I wanted to be just as pragmatic. I would work, as planned, on Wednesday and Thursday, then go in for the D&C on Friday. Miscarriage managed—a Harry Potter-esque tap of a wand and the slate (uterus?) would be wiped clean. _This is just a blip_ , I told myself. _We will get there_.
That Wednesday morning, I forced myself into some clothes and out the door. I was too shaky to drive, so I took the tram, hoping that none of my students would see me sobbing behind my sunglasses. By the time I reached work, my tears had crystallised into a hard, mean anger. Walking to my building from the tram stop, I slammed each foot down. I deliberately grazed my knuckles against walls. I locked myself in the bathroom, and wept against the wall, tapping and then knocking my forehead against the bricks. While my head was screaming expletives, there was another voice there too; the one I would speak to Z in when I looked up at her star, or leaned in close to open roses and breathed them in. _Oh, honey, this is so hard_. It was that voice that stopped me from punching the wall, that gently took me back outside so I could retrace my steps, get back on the tram, walk home and get into bed, send an email to my colleague that I wouldn't be able to teach the next day, that I was having a miscarriage. Today there would be no forward motion. This pregnancy had to be unravelled before we could go on; things were not just paused, but would have to go in reverse for a while.
My colleague responded compassionately. 'Take this week and next week off,' she urged. I did that, and, instead of booking in for the D&C, called up my doctor's office and asked for recommendations of where to go for the less interventionist medical version. Part of that white-hot anger, I'd discovered when I took the time to listen to it, was anger with myself, for thinking that I could just timetable my grief around my work responsibilities. But the whitest white-hot anger focused on the planned procedure on Friday and the thought of more surgery, more prodding. I had handed my body over to the IVF people so many times already. I'd never wanted a C-section, I'd never wanted to go through IVF, and the thought of more surgery triggered a kind of roar within me of _Leave me alone!_ I was furious that my body couldn't just sort this out on its own; yet, I still wanted to defend this small bit of turf, to retain some kind of control.
After a few hours, I heard back from our IVF doctor. Yes, a medical termination was probably possible, but you needed to be specially registered in order to prescribe it, and he wasn't. He gave me the name of another provider, and I drove for an hour to find myself in a dodgy-looking carpark with vague signage, and, soon after that, a dingy waiting room full of brochures about 'options'.
So this was what an abortion clinic looked like. It was a far cry from the soft lighting and tasteful art that decorated the IVF clinic where this pregnancy had started. The accusing fingers of the anti-abortion lobby had marked out pregnancy termination care as something controversial and distasteful—in the process, stigmatising the women who needed these services and the health professionals who provided them. Services like this were construed as the exact opposite of the optimistic health business of making people well and delivering living babies. Yet, here I was, straddling those categories—desperate for a living child, but needing help to evict a pregnancy.
This doctor wasn't used to seeing people who wanted to be pregnant. When I asked if he was sure that the pregnancy wasn't viable, he said, 'I don't know—if you're not ready, wait a week and have another scan.' I chickened out and drove home, cursing and crying at the thought of another week's purgatory.
At home, I spoke to a midwife friend, who informed me that the Women's Hospital could admit me as a day patient for 'medical management of miscarriage'. I called their clinic, and when I went in for another scan, we saw the same little heartbeatless oval floating there. 'Come back tomorrow,' they said, 'at 7 a.m.'
I took comfortable clothes and a good book. After they'd given me the medicine, I stared earnestly out the window. _Little one, it's okay to go now. I know, you tried so hard, it wasn't your fault that things didn't work out. It's okay, little one_. This time, when I bled, I was relieved rather than scared. I didn't want to be unconscious for this. I needed to be there to know for myself that it was over. When the pains became bad, the nurses gave me some pethidine, and, when it was all over, a cup of tea.
•
On the Monday morning after my hospital visit, Rima was driving me to work when my phone rang. It was one of the pathologists from the Women's. They'd tested the 'products of conception' from the miscarriage and there was a chance it might have been a molar pregnancy, a disorder with the placenta. 'We don't know yet whether it was molar, but testing will take about four weeks, so it's a good idea to make sure you don't get pregnant in the meantime.' No chance of that, I assured her.
At work, I googled 'molar pregnancy'; the Women's Hospital had a helpful fact sheet. This type of pregnancy involved a genetic defect with the embryo, which meant that it was all placenta, no baby. And worse, in a science-fiction twist, sometimes it could develop into a mole-like growth that burrows into your uterus, cells dividing and multiplying and, all the while, pumping out more and more of the hCG hormones that make blue lines blithely appear on pregnancy tests. Undetected, a malignant molar pregnancy could become cancerous and spread to other organs, though it apparently responded well to chemotherapy.
'Chemotherapy'. This was not a word I was expecting to come across in the process of trying to get pregnant. I took deep breaths, and made a cup of tea. 'All they're doing is further testing. And the results will take four weeks. The most likely outcome is that I don't have this.' _Exhale_. I would worry about this if and when I was diagnosed. In the meantime, I had work to do.
On the Thursday, I was working from home. There was a staff seminar scheduled for lunchtime that I would go in for but, meanwhile, I had teaching prep to do. My 'work space' wasn't ideal. The secretaire in the hallway had a small drop-down desk, so I sat there, on a folding chair. The doorbell rang, and it was the postie holding an enormous brown-paper parcel the size of a small TV. I wasn't expecting anything, so my heart lifted a little—I thought it might be from a friend. The label was typed, though, and the parcel wasn't nearly as heavy as its size suggested. When I tore away the paper, I discovered first one empty 4-litre plastic bottle with a medical patient ID label; then another; then two more, along with a heavy letter from the Women's Hospital. It was addressed to me, but was a form letter referring to my 'recent diagnosis' of a molar pregnancy.
Here was a helpful pamphlet on the different kinds of molar pregnancies. Here was a flyer for the molar pregnancy support group. Here were detailed instructions on how to collect all your urine in one of the 4-litre containers over a twenty-four-hour period, and a little map showing the location of the hospital pathology unit where the samples needed to be dropped off within twenty-four hours—one each week. I had visions of wandering nonchalantly through the hospital sliding doors, lugging a sloshing container of my own wee. I backtracked to the letter— _diagnosis_? 'What? I'd only had the call from the hospital pathologist a few days ago. I thought it was going to take four weeks to get a result?'
My stomach dropped. I frantically dialled the number on the letter. I needed to sort this out—had I been diagnosed? The first number led me to an answering machine, as did the next two, so I tried going through the switchboard and searching the hospital website for clues. An hour later, I was none the wiser but flicked back into my email account to discover an increasingly frantic series of emails from the co-convenor of the seminar series.
_The seminar_. The floor suddenly seemed unsteady beneath me. I'd completely forgotten about the seminar, leaving my co-convenor to somehow locate the guest speaker and chair the event in my place. I called and left a breathless message, apologising for not being there and letting her know that I'd had some upsetting medical news.
I didn't hear anything until I received an email from my head of school, indicating that my colleague had spoken to her about my worrying news, and that they had decided to employ a casual to cover my teaching for the rest of the semester. For the third time in one day, things crumbled beneath my feet, but this time I had an overwhelming sense of shame and stupidity. They were collapsing not because of a medical problem, or an administrative error. This was a crumbling I had created directly, from my own blind panic. I wanted to disappear, to shrink into a tiny ball and roll underneath the couch. Yet there was also a small, burning fury with myself and the situation. I wanted this job, I didn't want to be sidelined.
•
Despite the cool autumn morning, I was in the sweat of an energised panic when I arrived at work just after 8 a.m. the next day. The afternoon before, I'd done what I could to repair the damage, and had set up a 9 a.m. meeting with my boss, to plead my case to continue teaching for the semester.
I'd slept fitfully, resisting further consultations with Dr Google on molar pregnancies and what they could mean, and furious with myself for taking my work for granted, for letting these medical dramas cloud my work. Where my body had felt slowed for weeks, as the possibility of the new pregnancy leached out of me, now I was suddenly taut with tension. Nothing concentrates the mind like threat.
I bounced up the lino steps two at a time, silently punching out the lines of my reasonable email to my boss from the afternoon before. As I touched my hand to my office door, my mobile rang, and I answered it with the same efficiency. It was Penny.
'You have a nephew!'
'What?'
'You have a nephew—he was born at five-fifteen this morning!'
I let myself into my office, locked the door behind me, and stepped over to the window. 'Oh, Pen, wow... wow.' I looked at the calendar. I knew Penny's due date was any day now—I'd been on stand-by to be there at the birth for her and Kent. 'I'm sorry I wasn't there for the birth—did you call me? Did everything go okay?'
'Yes; it was all too late, and then too quick, to call you. It was intense—the worst bit was being separated from him for an hour while I waited on a trolley for a doctor to come and do some stitches. I was in a hospital gown, and didn't have my glasses, and I think they forgot about me. I had to go wandering up to the nurses' station to see what was happening—half-naked, blind as a bat—and then they sent me back down to the birthing centre for the stitches, anyway.'
'Oh, love; ouch. That's not good. But you're back in the birth centre now? Can I come in and see you? And meet your son—your son—oh, wow. I've got to see my boss at nine, but can I come right after that?'
'Yes, come! Is stuff okay with your boss?'
'Yeah, yeah; it's all okay. I should be there before ten.'
Gratitude rushed over me. He was here, he was okay, and he was a boy. I could love him, and I could congratulate Penny and Kent wholeheartedly, clean of the sadness and conflicted feelings that I feared might have come with a baby girl.
And, despite all my brokenness, I met with my boss, apologised for missing the seminar and negotiated my workload. Then I drove over to the same hospital where we'd had Z's last scan, a few days before the accident, and met my tiny new nephew, awash with love for him and his parents, and the honour of being his Aunty Hannah.
That night, with two seemingly impossible things already under my belt for the day, Rima and I went with friends to a Melbourne Comedy Festival show, and I laughed so hard I bumped my front tooth on the seat in front of me, chipping the cap the dentist had so carefully used to repair my teeth after the accident. My laughter morphed first into sobs at another thing gone wrong, but then into a deeper laughter that stretched the entire space between the utter devastation and sheer ridiculousness of my situation.
26
Tsunami
On Easter Friday 2011, I got a call from V, a mutual friend of Karin and Ned, our friends who had lost their baby son, Albie, just six months after our accident. They were still living in Paris, Karin was pregnant again, and their baby was due any day now.
'Hi, Hannah, just wanting to let you know that Baby Esther was born yesterday,' said V. Before I could squeak my congratulations, she added, 'It's not good news, though, I'm afraid.'
'Oh.' My brain started moving in circles—how could this not be good news?
'There was a lot of blood when she was born, and she's needing a lot of help to breathe. It is really not looking good.' V wasn't teary, just solid and serious.
I had a lot of questions. I had so many happier possibilities that I needed to put forward—'Sometimes babies just need some help to start breathing, I guess?'—only for V to gently pack them away.
Karin and Ned's parents were flying to Paris to be there, and to meet baby Esther. Even I could join the dots. They had to meet her while they could, because she was a very ill little baby.
'I'll keep you posted,' said V.
I asked her to pass on my love, and tell Karin and Ned that I was sending all the good thoughts I could muster. I put the phone down and wept.
I dug through my knitting things. I found a silky-feeling navy blue alpaca yarn and a hot pink wool of similar weight, and cast on. I was furious. Furious with myself for tossing our dear friends into the category of those lucky people whose seemingly effortless pregnancies rubbed salt into our own painfully unsuccessful attempts to conceive.
Back in October 2010, in the same week that Karin had told me she was pregnant, two couples in our Thursday night SANDS (Stillbirth and Neonatal Death Support) group had given their own glowing news, along with the ongoing trickle of Facebook announcements. I'd started to think my psychologist was also pregnant, and, after having to cancel a number of appointments due to illness, she confirmed my suspicions. Penny had also let me know that she was pregnant. Her news stung a little less because she had grieved Z with us. Our loss was hers too, and her news was our good news.
Nonetheless, it was starting to feel as if everyone but us could get pregnant. (Apparently, there is a made-up term for this state of mind: _preganoia_.) I didn't say anything to Karin (how could I?), but in my own head, a small, resentful voice muttered and felt betrayed, as though Karin and Ned had skipped the baby queue. I hadn't wished them ill, but my heart had closed a fraction, and refocused on my own misery.
As my knitting needles worked their way around—pink, navy, pink, navy, pink, navy—they repeated my mantra: _Please let her be okay, please let her be okay, please let her be okay_. _Please_ , I implored all the gods I didn't believe in, and the universe I'd sworn at for the last fifteen months, _please let something biblical happen_ —'and on the third day, she breathed on her own, and resumed normal brain activity'.
We spent the long weekend around the house, in a state of suspended animation. I was supposed to be marking essays but found myself staring out the window or reading the same line over and over. I gave up, and just kept knitting.
On Easter Monday, V rang again. 'I'm sorry, it's not good news. They took Esther off life support this morning, and she died in Karin and Ned's arms.'
I shuddered.
'Oh no. Oh, V, no! That is just not bloody fair. For fuck's sake.'
V was quiet. I didn't know how many of these calls she had to make that day, how many times she'd already had to calmly tell this news, to be on the other end of the phone making it real over and over again. I apologised. I asked more questions.
My fingers pressed into the points of my knitting needles, while we touched on funeral plans (they would bring her home, to be buried with her brother). I hung up. The beanie wasn't finished yet. If I'd knitted it faster, would she have lived? If I'd been a good enough friend to have knitted it before she was born, would she have been okay? I was reading Joan Didion, and knew that this was magical thinking, as was my fuzzy presumption that pregnancies and living babies could be doled out on the basis of an orderly queue. But that didn't stop the hypotheticals whizzing faster and faster around my head.
If losing one baby were enough to break you, what would losing two do to you? To survive losing two in a row was inconceivable. But Karin and Ned were not in a lonely category all their own. Of course, there were women like my great-grandmother, who, I knew, had mourned three babies and one six year old—heartbreakingly normal for turn-of-the-nineteenth-century Melbourne. Now, when I looked at the black-and-white photos of her, I saw not just weariness but sorrow and strength. I was dimly aware that women living in other decades and in countries with poorer health systems had much higher infant mortality statistics than ours, but until now I had failed to imagine properly the babies or the mothers embedded in those statistics.
The only other person I then knew to have lost two babies was my friend Jude. In a friend's Newcastle kitchen, she had given me mates' rates haircuts. After our accident, she got in touch on Facebook to offer her condolences and disclosed that she'd lost two babies in a row to stillbirth. Her son and daughter would have been twenty-nine and twenty-eight. She and her then husband lived in a small town at the time, and after the second stillbirth, her husband had to visit all the local shops and tell everyone, so that she could go to the shops without people asking, 'What did you have? Boy or girl?'
More than a year out from losing Z, I was barely hanging on by my fingernails. I tried to extrapolate my loss, to multiply it by two, and then fast forward several decades, but I still couldn't see myself ever becoming as Zen, as kind, or as genuinely funny and cheery as Jude. If I lost another baby, I imagined, I would spontaneously combust. But Karin and Ned didn't combust, and it wasn't for any lack of love or feeling for their children. Their ability to keep breathing, to continue moving through time and space, and even to find the headspace to ask how we were going, blew my mind. 'Freaking superheroes,' I thought. But also beautiful human beings. As Karin and I kept up our correspondence, my desperation to fix things for them, to magically knit it all back together, ebbed away. All I could do was respond, bear witness to their grief, and let the conversation move lightly between picking colours for baby-sized caskets, knitting, funeral arrangements, our ongoing IVF saga, and laughing about weeping on public transport and the horrible realness of burying babies.
Somewhere in those conversations, my idea of a 'fair' or fixed amount of grief and misfortune got washed out to sea. I knew Karin and Ned well. They are two of the most generous, open-hearted and hilarious people I know. If anyone deserved a living child, it was them. The fact that their universe had been destroyed not once but twice was irrefutable proof that life was not fair. My fantasy that there was some kind of balance sheet, or someone out there to add up the columns, was gone, swept away like the fragile bit of debris that it was.
Just six weeks before, we'd watched the 40-metre waves of the Tōhoko tsunami sweep across the Japanese countryside, insensible to the homes, farms, nuclear power plants and 18 000 people in its path. We'd sat agog, awed by the power, the destruction, and, God forbid, the beauty of the ocean. Not only were my fantasies of fairness now dissolved, but also my tenuous idea that we could map everything into categories: good/bad, happy/sad, fair/unfair. It wasn't just the ocean that could be both indescribably beautiful and heartbreakingly destructive, that simultaneously fostered life and wrought suffering and death. When I looked closely, I reluctantly had to put more and more things in that awesome/devastating category: human relationships (including parenting), technology, cars, food, substances, law, democracy, nature—in short, anything that mattered. There was no 'safe' zone, there was no unmediated 'good', just a whole lot of awesome/devastating chaos.
That didn't mean I accepted it all, or that I was indifferent to whether it came out as heads or tails, awesome or devastating, in any particular instant. Of course, I had a preference. Of course, I still prefer peace to war, love to fear, alive to dead, fairness to injustice, friendliness to cruelty, safety to harm. But I recognise these as my preferences, not universal truths.
I would have understood if Karin and Ned had become bitter, with the manifestly unfair hand they'd been dealt. But they didn't. Instead, Karin's mantra was _See beauty_ , _see beauty_. That meant all of it. The devastation, the grief, those short hours of holding our dead babies in our arms, were just as tender and beautiful as the prettier, smoother stuff I'd previously taken as beauty.
27
Earth and sun
Late July 2011 found me in Queensland for a conference. After all the molar pregnancy drama, the tests had come back negative, and I could happily put the 4-litre urine containers in the recycling and move on with IVF. Two unsuccessful frozen embryo transfers later, the grinding greys had returned. There was no date to look forward to in the calendar—we were already throwing everything at Project Bump, and getting nothing back. After refusing to miscarry a 'blighted ovum' in March, my body seemed to have gone into a sulk or lost interest. In May and June, there hadn't been even a whiff of a pregnancy symptom.
Saturday, 23 July 2011
There's a special art to running through crowded city streets. Speed up, sideways step, watch for a gap. My heart expands to knock at my ribs and nearly bowl over the people in my path, until I'm all heart—messy, beating, puffing and suddenly seeing all these messy human hearts around me. A woman sees me running towards her and fear blanks across her face briefly—she looks wildly behind me, her own steps a little quicker. I have a good reason to run—I don't want to be late for my osteo appointment, but I feel like I've just woken, as though my blood is reaching cells that have been slowly greying.
Things have been really grey lately. Everything is a big effort. I'm kind of embarrassed to write about it because this kind of sadness is dull. I bore myself. It's as though I'm stuck at the bottom of a big hole in the ground. Poem by poem, I'm digging myself out, and I know from the voices of loved ones which way is up, but I can't really pretend to be anywhere else at the moment. I have to make reluctant friends with this situation.
So what are you trying to tell me, deep dark hole? To stop dreaming of the stars (and one particularly bright little star)? That my slow-crafted words will come to nothing? That I am one and the same as the slippery grey-black clay on every side of me? Come on, hole, teach me your lesson and then we can be done. I'm not going to be bullied into silence and self-pity. Enough of that.
I'm not at all prejudiced against holes in the ground—in fact, my daughter lives in one, as do many of my favourite trees, earthworms and root vegetables. If dirt is my destiny, then bring it on, dirt. Show me your microbes, let me remember what dirt smells like, let me feel the grit of it between my fingers.
Time moves slowly under the earth. Things are hidden, processes work slowly but powerfully. Minerals are crushed, underground rivers carved, liquids percolate drip by drip, continental plates grind past one another millimetre by millimetre—all monumental changes occurring at a pace measured in centuries rather than minutes. What else is down here? Things unwanted or forgotten, buried and mourned—so many things lost and wasted which are slowly being turned back into the earth itself. Nothing goes away down here, but is slowly transformed, releasing water and nutrients to feed patient tree roots, or our lawn. This is where rivers are born. Nothing flashy or spectacular, just cold humble earth.
Dear hole in the ground, that's what I'd like—some of that persistence, slow elemental momentum. The ability to slowly work through this sad stuff with earthworms and use it to grow something good.
•
After tears and long conversations with friends who were IVF veterans, we went back to our doctor with a proposal—to stop the frozen transfers of two-day embryos and start again on a new protocol, of growing the embryos to blastocyst stage (about five days), so that if they made it that far, they'd have a stronger prospect of success. Our doctor was willing, and so we were all set to start a stimulation cycle in July, until I realised that the egg pick-up dates would have conflicted with a conference I'd agreed to speak at. A few months before, frustrated from all the waiting and delays, it would have been unimaginable to put fertility treatment on hold. But now, it felt like a small assertion of control over the process. I was more than just a pincushion; or, at least, I was a pincushion with things in her life other than the IVF process. And as wonderful as it would be to become pregnant again, to try our hand at parenting a living child, I didn't feel the same desperate grasping for it as I had before.
Rima didn't join me on the conference trip, so I was alone in what turned out to be an enormous apartment—and the biggest chunk of solitude I'd had since those weeks in hospital and rehab. The unfamiliar Queensland sun streamed loudly into the bedroom and woke me; a revelation after becoming accustomed to dark Melbourne mornings. I laid a towel on the floor and began a sun salute, relishing the uncluttered space and lifting my face to the morning light.
Even with eyes closed, the light made everything glow. And I missed it when I folded at the hips for _Uttanasana_ , to bend my knees and lay my hands flat on the floor on either side of my feet. I felt the sun again when I stepped my right leg back into a lunge, then brought the other leg back and dropped my hips for cobra pose, indulging my face in the sun's warmth; only to miss it again, when I lifted my hips and dropped my head for _adho mukha shavasana_ (downward-facing dog). And it occurred to me that this was exactly the pattern of things—alternating between time with your face to the sun, feeling exalted; and time facing the darkness, feeling humbled. Day and night, life and death, love and loss. And my job was not to chase one or the other but to move freely between the two, to honour both and to keep breathing all the while.
•
I had been waiting for a happy ending for this book. I couldn't bear the thought of leaving my story here, in what I'd thought of as the 'waiting place' between one (silent) baby and the next, hopefully more noisy, one. But this 'waiting place' is all I have right now—it's all anyone has when they've lost one baby and are hoping for another. There's no fast-forward button we can push to speed through to the 'good bits'. And even if there were, I'm less and less sure I would have wanted to push it. As painful as this grieving has been, it is mine, and it connects me to my daughter, as well as to everyone else who has suffered loss.
In that dark year after our accident, the thing that made me choke with fear and sadness was the idea of no more Z. I thought, 'That's it. My whole relationship with my daughter was over and done with before it had hardly started.' Thirty-six photos, some inky footprints and handprints, a tiny amount of ashes, and a drop of her blood on a blanket my mum had made; the countable, finite remains of my child. It wasn't enough, it would never be enough. It was such a sad, awful, unfillable hole of 'no more' that, to live in this world, I had to close that drawer, to look elsewhere for the 'more' I needed.
What I hadn't imagined was the feeling of my soul tearing in two as I tried to hurtle away from her. It was an impossible choice: to go back and sit at that point in the road where she disappeared, or to move on. In that linear frame, it was either backwards, towards death and sorrow; or forwards into life and good things, but away from my child.
Specifically, the 'more' that I was looking for was to get pregnant again. I was so sure it would happen. I don't think I had really let go of feeling pregnant. This was my pregnancy and I would finish it, even if I had to mourn a daughter along the way. As I had danced uphill, waving my mardi gras pompoms, tenaciously pregnant, it had felt like forward motion. In spite of everything, I would move on. Now that we were moving, I could feel gracious about the pauses along the road the IVF process had imposed, even though I'd been impatient at the time.
But when I'd unwrapped the parcel from the Women's Hospital, with its generic 'you're diagnosed' letter, something broke in me. I thought, initially, that it was my sanity, that I'd finally popped a crucial cog and I'd be completely broken. But what was broken was my idea of a future without her—a future where that empty space was filled by another, living, baby. A molar pregnancy meant having to wait at least six months to a year before I could try to conceive again. And maybe this wasn't just the pause button; maybe this would mean Z was my only child. Just having to think about that question made me stare very hard at the train tracks while I waited to cross at the level crossing.
The molar pregnancy diagnosis also gave me plenty of time to think. I was scared that if I really looked hard at my grief for Z, if I opened that drawer, the big, sad black hole of 'no more' would suck me in and swallow me whole. Because there was no solution to it. There was no way my logical brain could think a way around the big, stark reality of no more Z.
I had been holding my breath for so long waiting for another baby that I wanted to vehemently push each babyless minute past me and away from me—just throw it away. As long as there was some prospect of another pregnancy to look forward to, I could mark a date in the calendar and set my eyes on it, hold on for that date, even if it were shifting away from me cycle after cycle. But now I no longer had that option; at least until I got the all clear on the molar pregnancy front.
I was furious with myself for being tricked by my own body, with that duplicitous bundle of non-baby cells, and with everyone and everything. I came closer to psychiatric-hospital madness than I ever wanted to come. This was not the quiet, weary disinclination to continue existing that I'd seen in my mum and had glimpses of myself. Instead it was a crackling-hot rage, to smash up myself, my situation and anything in my path. Most of all, what broke was the hoped-for Hannah, who I'd been preparing my life for, who had everything in order but who never seemed to arrive, despite my best efforts. Hoped-for Hannah's baby didn't die, she didn't need IVF, and she could choose when she got pregnant, while managing a glittering career. No wonder she was so infuriating. I had hoped so hard and for so long that to smash up those hopes felt like it would break me. But, in breaking, I also exhaled, and felt what it might be like to live without hope dragging me forwards into an imaginary future moment. Exhausted with my own drama, I lay on the floor with all that sadness, and we breathed and looked at one another. And I breathed in all the scary things that a molar pregnancy might mean: not knowing whether I could get pregnant again for six months, a year or ever; chemo; having to do stupid 24-hour urine tests and carry 4-litre plastic containers of my own wee into the Women's Hospital every week. And I breathed out, because I wasn't there yet, and every second standing between me and a 4-litre urine container was a precious, precious thing.
Breathing in an uncomfortable spot like that can be hard, but I'd had lots of practice at it by then. I take great pride in the fact that when my brother and sister-in-law (both dive instructors) took me for my first-ever ocean scuba dive that January, I used less oxygen than either of them, despite freaking out underwater about how to clear my mask. It's not a remarkable talent—breathing—but it is a useful one.
It surprised me to find that if I didn't run away from the awfulness of my situation, I could breathe into it, explore it, feel exactly what it was like to be mother to a child who has died. This was different from being mother to a hypothetical Z, who would have been this-many months old by now, or to a hoped-for new baby, who I might conceive sometime in the future. Like it or not, neither of those babies was in my life now. All I had was Z. In that space, I thought to her, 'Well, my love. I wish you hadn't gone and died. But there's not much you can do about it now.' And in the spirit of parents whose kids have been conscripted to the army, I thought, 'I wish you didn't have this job (being dead/being "one with the universe" or whatever it is that baby souls do after they die) but I still love you and I wish you'd send me a postcard or call me sometimes.' Then I felt silly, because there was her star, which was always there twinkling at us, and the camellia tree, which burst into bloom just when my heart was breaking, and her pomegranate tree, and her roses, and the leaves in the river in Cairns, and slow-moving clouds, and the sea at Somers, and the bird noises in the bush chapel where her ashes were, and I realised I was being a pretty demanding mama.
It dawned on me that I actually know Z better now than when she was born. And if somehow my knowledge of her, and love for her, has expanded, then there _is_ more Z. She is still growing, she is finding her feet in the world, even if I don't (in the way of all parents) really understand what her job entails. It isn't how I wanted my daughter to be in the world, but I know now that whatever she is doing is important, because it is important to her, and therefore to me. If I just keep demanding that she fit in with what I need (which I know she can't do anymore) then we will both feel awful, and maybe I'll miss seeing what she can do. I wish we'd had more time together in the conventional sense, but I can't be churlish about it, because it isn't her fault. And if I want to love her exactly as she is, then I have to be open to receiving her little hippy-style postcards of brightly coloured leaves and odd cloud formations.
I have a better sense now that part of my job in parenting Z is to trace where she went when she died—to resolve for myself where her little soul went, so that I can keep loving her and learning about her. When you prepare for parenthood, they don't tell you that you may need some existential philosophy. But I think that is one of my main tasks for Z. And, as far as I can tell, she is here in this world. In fact, she is in the process of reconnecting me with the world I felt so lost in after the accident.
For so long after the accident, everything felt wobbly, groundless, precarious. I was terrified about setting up rituals, in case I made promises to her that I couldn't keep: a grave neglected over the years, a name unspoken. But that was back when I thought of time as a line, and of our grief as a spot fading in the distance.
In those months after the miscarriage, something big shifted, so that I felt more settled with my grief. Where before, when I had heard people say that Z would 'always be with us', I would nod and vaguely agree; now I genuinely feel as though she is always with me. She is not stuck in the past, not defined by the trauma of the accident or the delicacy of her newborn form; nor trapped elsewhere in a 'heaven' that I don't believe in, or a far-off future moment of reunion. Instead, I carry her and my grief for her in my heart, in my cells, and I find her everywhere I go. She is woven into things right here—in the clothes and jewellery I choose each morning, the leaves of her pomegranate tree, the starry night, the little words I say quietly to myself and to her. I no longer have to choose between embracing her and being here in this life because I can exist now as her mother; as someone partially constituted by her and the love we still feel for her. I don't have to choose between life and death either, because they too are intertwined. She skips between both, playing on and around them, as though on a giant Möbius strip.
There's still sadness that she's not here in the fleshy, noisy way of other children, but I recognise that as my own small sense of not getting what I want, rather than as any failing on her part. The sadness at losing her and the joy at having her as my daughter have become stitched together, so that I can hardly tell which is which. It's specific to her, and my love for her, rather than being measurable as happy or sad. So, I'm still a bit of a weepy mess, but in an alive way rather than a broken or depressed way. This is what it means to love a dead child. You can expect nothing back in return. Yet, in accepting this, I feel like she has schooled me on living and dying.
It sounds trite to say, 'I feel more grounded now,' but I mean it literally. My child, flesh of my flesh, is buried in the ground, and so that ground is part of me too. When the soles of my feet touch the earth, I say hello to her, I tell her where I'm going. At first she was localised to that particular spot, but the rain has leached her essence and the worms have exchanged her particles, so that now I'm not sure exactly where she is—which means she exists everywhere, in a state of possibility.
Slowly, my sadness for having no more Z in my arms—and in our house, in her fleshy realness—is mingling with wonder that I can still get little peeks of her. I was walking to the shops the other day and it hit me that she might have been walking with me by now. I suddenly thought, 'Here; this is where her little hand would be, tight in my hand. Walking together.' And I could just about feel her chubby fingers, the softness of her skin, and could suddenly feel both the no more and the more at the same time. _I love you exactly as you are, my darling girl_.
•
I feel surprisingly sane, for all that has happened. I think it helped to give myself permission to go insane with grief when I needed to. Just small bits of mundane madness: smelling roses and muttering, _I love you, my little one_ , into their centres, naming our new car and bestowing her with magical protective powers. And is that really insanity? Or just permission to feel the full range of human emotion—to refuse to pack certain emotions off to the loony bin?
As for my desperate searching for a solution to my grief, here is my answer: there is no answer ( _sharp intake of breath_ ) but you are infinitely more capable of surviving and, indeed, flourishing in this groundless state than you give yourself credit for ( _exhale_ ). This groundlessness, this suffering, this feeling that your heart will explode and that _this_ is unbearable, is about as normal as it gets, and you are, in fact, able to bear it, even when it doesn't feel that way. If you can stop trying to escape long enough to pay attention, you'll notice that not only are you bearing the unbearable, but what feels monumental and unchangeable nonetheless does change moment to moment; sometime subtly, sometimes radically. Even the mountains are not static. The shifting ground beneath us will take away everyone we love, but it will also (eventually) end all suffering. This suffering doesn't mean that something is broken or wrong with you—this is the state of being human, of being a fragile living thing. This is what it feels like to be stretched between being born and dying.
Part III
RIPPLES
28
Both my babies
When I was in the depths of grief, it seemed as though all the books I could find on perinatal death traced a similar narrative arc: baby dies, sadness ensues, then the birth of new baby restores faith in life, the universe and everything. It's a nice framing device, babies as bookends with the grief neatly contained in between. My own experience was more unruly.
Hearing all those 'happy ever afters' just felt cruel—particularly when we were trying so hard to replicate the formula, with no success. In that babyless space, I had to find something else. Mostly, it was a willingness to experience and sit with the things I wanted to run a mile from. I am now old friends with grief. I treat it with a healthy respect. On some days, it is still enormous and crushing, but at the same time has become ordinary—we fold it up with our washing and rinse it out when we brush our teeth. When I laugh, I want to know that her little cells are laughing within mine, and that when I see something beautiful, it is all the more beautiful because it feels like she is a part of it, and all the more heartbreaking because she is not here to see it.
If you are not yet in a state to hear about subsequent children, stop reading here.
•
This is our happy ever after. Three IVF cycles after the miscarriage, we got some good news. There was a positive pregnancy test, a scan with a heartbeat, and then, in May 2012, a beautiful baby boy, Ali. We still can't believe our luck. He is, as a dear friend puts it, a 'cube of joy'. Often a cube of joy that doesn't want to go to bed, or put his pants on, but a cube of joy nonetheless. He is reassuringly robust. He bellowed the moment he was born, and continues to be—which is a comfort to me—a noisy sleeper. I hope he lives to tell his own story. That sounds a morbid way to think of your child (and, indeed, the thought of losing him makes me weep regularly) but it also gives me some perspective and sharpens my gratitude for him.
I have to be honest here. Having another baby did help. As the grief settled like muddy water in a jar, I could see there were different components of differing weights within it. The specific grief for our specific daughter was sedentary, solidifying at the bottom. It is with us wherever we go, it has become part of the ground underlying everything else. I am still heartbroken that she is not here, that we will never know what she would have been like as a schoolkid, as a big sister. But the grief for not being a mother of a living child—the hankering for the ephemera of tiny socks and small seashell ears, the patting and the rocking, playing peek-a-boo, making the train-cake—I have been slowly and happily pouring away since the wintery Monday afternoon when Ali emerged screamingly alive.
And while Ali is the happy ever after to our story, as always, the story has taken some unexpected turns. Rima and I have separated—mostly for reasons pre-dating our accident—but we remain friends, and co-parents to Ali and the girls. Our universe has fractured again, but no one has fallen down the chasm this time. And there is a freedom in calling a truce, and in deciding to stop pacing through the old dance steps that sent us round in painful circles.
Thursday, 28 June 2012
Ali has changed so much already since he was born. Even within 24 hours of his birth, his head was no longer the soft squished newborn head, and the cord which was so plump and pulsing at his birth was quickly drying up and turning into a belly button. Too many tiny changes to catalogue—new skills, new habits, growth in every direction. He's now over a month old, and yet his birth still feels so close—the surprise of having a living baby hasn't worn off yet for me.
And it hit me that this is what being a parent is, to bear witness and care for another human being through their most intense period of growth and change—where their existing self is constantly slipping like mercury through your fingers, becoming a new baby, a new little person every day as they grow and change. As much as I want to grasp onto who Ali is this very minute, I know that this current version of him is just a snapshot—that he is the process rather than the minute by minute product of himself.
When I had that thought, it made me cry because I'm only just starting to grasp how much we missed out on with Z. Does that mean I completely missed parenting her, because, by the time I held her in my arms, she was still—she was not going to grow or move anymore? I felt lost for a moment as her mother. But not only did I love her through the constant transitions and growth of pregnancy—from a tiny cellular possibility to a kicking, hiccoughing, nearly six pound baby—I also loved her and held her through that other big transition, from life to death. I was there surrounding her as her heart slowed and then stopped as we sat in the wreckage, but I was also there after she was born, holding her as the living warmth ebbed away from her body and her little soul stretched away to begin its travels.
I asked Rima the other day whether she thought Z could hear my dad singing her a lullaby when he held her after she was born, and she said, 'Yes, the soul hangs around for a while, at least a day—that's why we stay with someone who has died, with their body for the first day.' That second transition—from someone you love whose heart has just stopped beating, to a cold body—has always frightened me a bit, thanks to all those cultural phobias of dead bodies and deterioration. There was a moment on the day we spent with her, when I had slept briefly and I woke and asked to hold her again, and the cold on her cheeks was noticeable. I knew we didn't have much time with her—that the little baby soul we loved so much was mingling back into the atmosphere and gradually relinquishing the atoms of her body back to the elements.
I look at all the beautiful cards and gifts that family and friends have sent congratulating us on Ali's birth, and it feels so unfair that Z got condolences instead. It will always be unfair. But now she exists in a state beyond fair and unfair. And to hold her as she crossed into death and to love her even all the way into death was all I could do as her mama.
Last night I dreamt that I was out shopping with Rima and the girls. We were in a toyshop, and Z was with us—she was a curly-headed toddler about fifteen months old. One moment she was looking at toys in our aisle, and the next I asked Rima where she'd gone—we couldn't see her anywhere. We were searching all over the shop, calling out her name, and when it was clear she wasn't there, we ran out to the street and were looking for her. I saw Rima run across the road and I was so scared that I'd see her pick up Z from the road—I wanted to find her but please god, not on the road, not hurt or killed.
Then a tram came, and I realised it was our tram home. I felt compelled to get on. In my head all sorts of arguments were tested and rejected—maybe she would know it was our tram, maybe someone took her on it. I had no idea, but I just needed get on that tram. Somehow we were now looking for both Ali and Z. I stepped up onto the tram, searching—and there she was, running into my arms. I hugged her to me and breathed her in, simultaneously looking around for Ali. 'Who found her?' I asked. 'Was there a little boy with her?' I asked. Some lanky teenagers sitting opposite waved at me to indicate it was them who had found her. They pointed and there was Ali—himself but a toddler only a few months younger than Z. I drew him and Z in—a solid little person in each arm—sobbing with relief. 'Oh my babies,' I cried, 'I'm so sorry. I'm so sorry I took my eyes off you!'
I woke to my own crying, and found my arms around someone warm—Rima. I listened for Ali's snuffling breath in the co-sleeper next to our bed, and when I heard him, I exhaled—grateful for him; grateful for Z visiting my dreams; and so, so grateful for that feeling, however brief, of holding both my babies in my arms.
29
Zoe's Law
In mid-2013, several Sydney friends got in touch about the proposed 'Zoe's Law' bill before the New South Wales Parliament. Zoe was the name Brodie Donegan and Nick Ball gave their baby daughter, stillborn on Christmas Day 2009 after Brodie was hit by the drug-affected driver of a van, just two days before our accident. A fetal heartbeat was still present when Brodie arrived at hospital, but by the time doctors could stabilise her, it was fading, and Zoe was stillborn at thirty-two weeks gestation. As in our case, because Zoe did not show any signs of life when she was born, the driver could only be charged with dangerous driving causing grievous bodily harm, not dangerous driving causing death. For Brodie and Nick, this quirk of legal personhood meant that they felt that Zoe's life was never 'acknowledged or taken into account'.
When Brodie and Nick took to the media to campaign for a change to the law to recognise Zoe and other stillborn babies as legal persons for the purposes of the criminal law, it was Christian Democrat politician Reverend Fred Nile who acted first. Nile had made a name for himself by proclaiming homosexuality a 'lifestyle choice' that is 'immoral, unnatural and abnormal', seeking a moratorium on Muslim people migrating to Australia, and using the suicide of TV personality Charlotte Dawson to publicise his (completely unfounded) theories about a link between depression and abortion. Classy politics, indeed.
Without consulting Brodie and Nick, Nile used their daughter's name to introduce a private member's bill that would create a new offence of 'causing serious harm to or the destruction of a child in utero'. To claim back their daughter's name, Brodie and Nick worked with their local member, Chris Spence, to launch their own private member's bill. It was more circumspect than Nile's with its pro-life undertones, but nonetheless sought to define a fetus as a legal person for the purposes of particular offences. Nile's bill would have extended legal personhood to a 'child in utero' defined as 'the prenatal offspring of a woman'—vague enough to cover an embryo from conception, and conspicuously avoiding use of the term 'fetus', which only applies from about nine weeks after conception. Spence's bill would have applied once the fetus was at least twenty-weeks gestation; or, if gestational age could not be determined, where the fetus had a body mass of 400 grams or more.
I'd avoided getting involved in the debate because I still wasn't sure exactly what I thought. Rima and I were in the midst of a divorce (when you've got kids, pets and a mortgage, I think separation still counts as a divorce, even though we'd never been able to get legally married). Ali and I were living with friends and family while we sorted out our post-separation accommodation. I was back at work, teaching a new subject, and often sleep deprived thanks to a night-waking toddler. But I knew that this was something I had to engage with—and that it was only by writing about it that I could figure out exactly what I thought about it. I'd been livid in January 2010 when a pro-lifer had sought to use our case to suggest that 'permissive abortion laws' were to blame for the law's failure to recognise the distinct harm of losing a wanted pregnancy due to someone else's violence or reckless driving. I had been deep in my grief then, and wasn't ready to cross the chasm between my grieving heart and my brain's capacity for legal reasoning.
When I revisited it all in late 2013, the thing that bothered me about the debate so far was that it had been reduced to polarised understandings of pregnancy. Either you were pro-choice, and opposed Zoe's Law because it sought to extend legal personhood to fetuses; or you saw the fetus as a 'baby', and, therefore, all babies, whether in or outside the womb, should be 'worthy' of legal personhood, even at the expense of the legal personhood or basic human rights of their mothers. In that shallow dichotomy, I could be a feminist or a grieving mother, but not both. But as I read and wrote my way around the topic, it became clear that this dichotomy had at its core some shady unstated assumptions.
First was the assumption that everything turned on the inherent characteristics of the fetus: was it human enough, or similar enough to a newborn baby, to count as a legal person? Or did women's rights to make decisions regarding their own bodies while pregnant depend on defining the fetus as 'just a bundle of cells'? Given the diversity of religious and philosophical beliefs in the world, and the diversity of situations in which pregnant women find themselves, it is unlikely that any kind of consensus will ever be reached on the inherent value of any particular fetus, let alone 'the fetus'.
This whole line of inquiry about the 'value' or 'sanctity' of the fetus as an archetype required a disembodied kind of thinking, a stripping away of the pregnant woman and the maternal–fetal relationship that makes fetal life possible. In assessing the status of 'the fetus', the mother was dissected out of the picture, reduced to mere geography. I thought of Da Vinci drawings—disembodied wombs sliced open like seedpods.
This was the kind of logic that led Alabama lawyers and judges to interpret 'environment' to include the womb—repurposing laws designed to address children being exposed to meth labs, in order to charge and convict women whose babies have been found with drugs in their system. The same logic led an Alabama district attorney in 2015 to file a motion to terminate a pregnant prison inmate's parental rights over her embryo when she was in early pregnancy and wanting an abortion. The case never went to trial, as the woman eventually withdrew her request for an abortion. Whether that represented her choice or a concession to the pressure placed on her will never be known. If a woman's 'parental rights' to an embryo or fetus within her own uterus can be terminated and handed over to welfare authorities, what rights are left for the rest of her? If the food she eats or drugs she inhales affect the embryo, then do the authorities also have 'parental rights' over her mouth, stomach and lungs? Her blood vessels? Imagining the fetus as a separate legal person suddenly propels us into a world like that of Margaret Atwood's dystopian novel _The Handmaid's Tale_ , where women are treated as 'two-legged wombs'.
Framing the debate in terms of the inherent features of the fetus also conceded important ground. It suggested that the reason we could 'let' women make decisions about pregnancy and abortion was because those decisions were relatively inconsequential ones, which were merely about a 'bundle of cells'. When those decisions became weightier, as fetuses have a tendency to do, the arguments about women's rights to make those decisions appeared shakier.
The key difference between a late-term fetus in utero and a newborn baby is not size or stage of development; plenty of premmie babies have been born alive who were smaller and less developed than Z. Rather, it is the fetus's situation within its mother's womb and person, and its reliance on her for every possible need, that makes the question of fetal legal personhood complicated. Z had died not from any wound caused from the impact, but from the placenta abrupting and detaching from the wall of my uterus. She was left drifting, like a small astronaut whose oxygen supply has come unplugged. Her death did not make sense outside the context of my womb and my body bearing the brunt of two vehicles colliding.
I found myself agreeing with Sydney Law School academic Kristin Savell, who suggests that birth is significant because it is the moment when the baby becomes an embodied individual. At birth, the baby's skin encounters not amniotic fluid or the inside of the mother's uterus, but air, and other people, and with them the social interactions that create our need for law and legal personhood. Birth is itself the division that creates a new individual, and in which the baby transitions from the universe of the uterus to our worldly, social universe. Here we interact via laws, language, social conventions and sometimes violence, rather than via the flow of nutrients across the placenta, and the muffled sounds of heartbeat, voice and bodily functions.
Savell's approach resonated with me, because in my belly, Z had been Haloumi—a mysterious baby-fetus, known only by the sensations she created within my body. We loved her, even as we guessed whether she was a she or a he, or whether that was a knee or an elbow moving across my belly. But once she was born, she solidified into herself. Yes, her face was always hers, but we didn't know it as hers until she was born and we became acquainted with the whole particularity of her. And as much as we loved Haloumi as a nickname, she was now a little person, and needed a proper person name. The fact that she died before that transition occurred—from fetus-baby to person—may have meant that the law couldn't treat _her death_ as the death of a legal person, but it didn't mean the law couldn't treat _her_ as a legal (but deceased) person once she _was_ born, requiring a funeral and certification as our child.
I also wanted to address the pervasive assumption that legal personhood amounted to some kind of societal certification of human worth. In an increasingly secular society, law is one of the last bastions of ritual. Judges and barristers wear ceremonial robes and wigs, and the court space is a semi-sacred one where there is a time for bowing, a time for speaking, and a time for silence. And what are legal documents but magical incantations, which, when prepared, signed, sealed and certified in the correct manner, can change the unseen properties of people and things? Yet, I wanted to challenge the recourse to law, and to suggest that the task of making sense of pregnancy loss or termination is an intensely personal one, unsuited to the universal definitions and mechanisms for consistency that are the hallmarks of legal processes. I was hesitant to invite the law into my uterus—to let it mediate the shifting relationships between the unfolding process of fetal life within my body, and my developing emotional and cognitive sense of my child within my heart and mind.
When I started researching other jurisdictions that had created criminal offences around fetal deaths in utero, what shocked me was how often such laws were used to prosecute the pregnant women themselves. These were women like Melissa Rowland. Pregnant with twins in Utah in 2004, she had initially refused to consent to a caesarean section delivery, and was subsequently charged with murder when one of her twins was stillborn. On a plea bargain, the prosecutor dropped the murder charge in return for her pleading guilty to child endangerment. By expanding the legal meaning of 'child' to include a fetus in utero, the Utah prosecutors construed a pregnant woman's decisions about her medical treatment and about birth as criminal acts.
Nina Buckhalter and Rennie Gibbs were Mississippi women also charged with homicide offences when their babies were stillborn. Prosecutors argued that the women's illegal drug use amounted to manslaughter by culpable negligence, in Buckhalter's case, and 'depraved heart murder', in Gibbs case. Both indictments were eventually dismissed, but only after consuming years of these young women's lives.
Indiana woman Bei Bei Shuai was 33-weeks pregnant when her lover left her. Distraught and suicidal, she swallowed rat poison. She was hospitalised, survived and the pregnancy continued, but, a week later, an obstetrician noticed an unusual fetal heart rate, and Shuai consented to an emergency caesarean section. Her baby daughter, AS, was born alive, but gravely ill, due to a brain haemorrhage. Three days later, Shuai consented to removing AS from life support and she died. The coroner's view was that the brain haemorrhage was due to the rat poison. Prosecutors treated the transmission of the poison from Shuai's bloodstream, across the placenta and into her baby's bloodstream as a voluntary act of poisoning, and charged Shuai with murder and attempted feticide.
I thought of these women and the hospital rooms where they had held their babies, feeling their cheeks get a little colder. I thought of how broken we were in that moment, and imagined that vulnerable space suddenly crowded by police with accusing looks, taking away our child's body as 'evidence' against us. To heap on punishments when these women had already lost so much felt more like cruelty than justice.
The acts that these women were being punished for also felt very different from the violence in cases where men stabbed or punched their pregnant partners. These were acts that concerned the women's own bodies: decisions about medical treatment, or addictive or self-harming behaviours. It was only due to the involuntary physiological processes of their own pregnant bodies that their acts affected their fetus. They couldn't insulate their fetus from their actions, their addictions or their mental illnesses any more than they could separate their minds from their bodies.
When laws are promoted as 'protecting' mothers like me and our babies but end up being used to prosecute women like Melissa Rowland, Nina Buckhalter, Rennie Gibbs or Bei Bei Shuai, it contributes to polarised views of 'good mothers' and 'bad mothers'. And it is not just the so-called 'bad mothers' who are policed by these prosecutions. These cases send a powerful message to all pregnant women that if they are not compliant when doctors advise an intervention during their pregnancy, they risk criminal charges and the shaming that comes with the 'bad mother' tag. And for those women who are pregnant and struggling with addictions or mental health difficulties, evidence from the US shows that taking a punitive approach means that these women are less likely to seek prenatal care or to disclose these complicating factors to their health care providers—resulting in worse outcomes for their babies.
I sat down and nutted out a short article for _The Conversation_ website.
**Why losing my daughter means I don't support Zoe's Law**
18 November 2013
A bill currently before the NSW Parliament attempts to criminalise harm to late-term fetuses that die due to injuries inflicted on their mother. But is fetal legal personhood the best way to recognise the particular harm of losing a much-wanted pregnancy due to someone else's violent or careless act?
As a feminist legal academic, I have professional insight into the prospective law; I also have personal insight as a mother who lost her unborn child in a car accident.
It reads like a macabre riddle—someone died within my body, and yet I live. It's one that I've puzzled (and wept) over many times since a four-wheel drive hit our station wagon when I was eight months pregnant in December 2009, causing our baby daughter to die before she could be born.
The current law's attempt to answer this riddle is a clumsy one. It characterises our daughter's death as one of my 'injuries', because she died in utero, and was not a legal 'person' with a separate existence from me at the time she died.
It is exactly this riddle which Zoe's law (No 2) attempts to resolve.
**What the bill does**
The bill defines a fetus over twenty weeks or weighing more than 400 grams as a legal 'person', for the purposes of dangerous driving causing grievous bodily harm and a number of other criminal offences.
Zoe's law is named after Brodie Donegan's daughter, Zoe, who also died in utero due to dangerous driving in December 2009.
For Zoe's parents, this bill seeks to close what they see as a gap in the law:
We have never felt that Zoe's loss of life was acknowledged or taken into account... I couldn't reconcile that the child I'd applied for a stillbirth certificate for, held a funeral for, received the baby bonus for, received paid parental leave from work for; wasn't recognised separately to me.
The law's response touches a raw nerve because so much of the grieving process is about developing an understanding of _who_ you are mourning—a process already complicated when your child dies before (or around the time of) birth.
Calling our loss an 'injury' fails to acknowledge the depth of sorrow involved in grieving a child. But is fetal legal personhood the best way for the law to recognise our loss?
**Legal personhood**
A key difficulty here is that legal personhood has been interpreted as the definition of human life and worth. Legal personhood is a technical category that sometimes includes non-humans, such as corporations. Its purpose is not to define human life but to enable an autonomous interaction with the law.
Legal personhood doesn't make sense for a baby in utero. The physical reality of pregnancy means that the baby is the opposite of autonomous—it depends completely on the mother and is completely contained within her body until birth.
While inside the mother, a baby is covered by her legal personhood. Birth is the moment of separation when the baby is no longer contained within the mother and her legal personhood.
I have no doubt that my daughter was a person—but I am comfortable with the idea that, at the time she died, she was protected by my legal personhood rather than her own.
Once the fetus is defined as a legal person, the law has a direct relationship with it, and the mother's consent becomes irrelevant. She becomes invisible in the eyes of the law, despite the physical realities of pregnancy meaning that any interaction with the fetus necessarily involves her.
Zoe's bill is drafted to create exceptions for anything done to the fetus by the mother, with her consent or by a medical professional. But this creates a situation where it is legal to take the life of some legal persons, but not others, depending on the consent of a third party (the mother).
And it opens up the prospect of human rights claims being brought on behalf of a fetus. With that comes the prospect of challenges to the pro-choice exceptions built into Zoe's law.
**Reproductive autonomy**
My own view is that the fetus is a life. But because it is a life completely contained within a legal person (the mother), any interests or rights it could have can only be advanced through the consent of the mother.
As a pregnant woman, what you choose to do with the life within you is a huge moral decision. But because that decision is completely contained within your body, and because you are already the mother to any child that might potentially be born, you are the best person to make that decision.
You are the most qualified person, the most concerned person, the person most at risk, and the most interested person.
For anyone to take that decision out of your hands, whether to insist that you continue with a pregnancy or to terminate a pregnancy against your will, is a violation. It goes beyond pain and physical injuries—it violates the mother's decision for her own body and for any potential child.
It is _this_ violation that I would suggest would be a much more effective base for a law recognising the harm Brodie Donegan and I experienced.
This could involve, for example, a specific offence addressing conduct that ends the life of a fetus without the mother's consent.
Our current laws misconstrue forced pregnancy loss as just another type of bodily injury, rather than recognising it as a violation of reproductive autonomy. Like rape, forced pregnancy loss deserves its own offence centred on the notion of violation, rather than injury.
A law that frames forced pregnancy loss as a specific offence could acknowledge the family's suffering in cases like mine and Brodie's—and protect reproductive autonomy.
It doesn't resolve the heart-breaking riddle of losing a child, but I'm not sure that any law could.
•
Just three days after my piece in _The Conversation_ was published, the bill passed the lower house by sixty-three votes to twenty-six. Suddenly, fetal legal personhood was no longer a speculative idea put forward by the fundamentalist pro-life fringe, but a viable prospect, and I was fielding interview requests from local and national radio and TV stations. Again, I thought of Lindy Chamberlain, but this time I had to speak both in my personal and academic capacity. I suspect many journalists initially found me a contradiction in terms: a bereaved mother who _didn't_ want her child recognised as a legal person (at least while in utero). They were compassionate, though, and curious, which I was thankful for. And this was exactly the paradox I wanted to talk about—the limited usefulness of legal personhood for defining who was 'human', and the gap between diverse understandings of the emotional or relational personhood of the fetus, and the law's need to set more universal definitions and minimum standards.
In the months after the lower house vote, I was invited to speak with a number of members of the New South Wales upper house, along with the Australian Medical Association and Family Planning NSW. We sat on couches in MPs' offices, while I told my story, and the stories of women in juridictions allowing for fetal legal personhood—women like Melissa Rowland and Bei Bei Shuai.
We urged caution in passing legislation that could have unpredictable impacts on women's access to all kinds of medical treatment while pregnant, and on their ability to make decisions regarding pregnancy and birth. We talked about Angela Carder, a pregnant woman dying of cancer in Washington DC in 1987, who against her will was subjected to a caesarean at 26-weeks gestation because a court found that 'the state has [an] important and legitimate interest in protecting the potentiality of human life'. Her daughter died within two hours of the operation, and Angela two days later.
We talked about Marlise Muñoz, who in November 2013 was declared brain dead in Texas, but was kept on life support machines against her express wishes and those of her family because she was fourteen weeks pregnant at the time. The hospital believed it was bound to do so by the _Texas Advance Directives Act_ , which states that, 'A person may not withdraw or withhold life-sustaining treatment... from a pregnant patient.' It was over two months before Marlise's husband was able to obtain a court order both requiring the hospital to turn off the machines and holding that 'lifesaving measures' could not apply to brain-dead patients. Like my body, Marlise's had held both life and death, but in the reverse order. A 'fetal personhood' approach would suggest that the two scenarios were the same, that pregnancy involved two persons, and just because one person died, that did not require the death of the other. This was a version of pregnancy alien to that I had experienced, in which the embryo/fetus makes itself known in the early stages much as a virus would (via queasiness and weariness) and gradually accretes into a presence—not of an equal or a rival for control of my body, but of a small vulnerable being, for which I was completely responsible. When my daughter's heartbeat stopped and her brain went quiet in my womb, my survival did not require the artificial continuation of her breathing and blood circulation.
There are so many kinds of unenviable decisions to make, and being pregnant means it is impossible to separate a decision made for yourself from a decision made about your fetus. It is easy to feel cynical about politicians from a distance, but the representatives we met with took their parliamentary roles seriously, and were intent on learning as much as they could about the potential repercussions of treating a baby in utero as a legal person.
In early 2014 there were rumours there would be an upper house vote on the Zoe's Law bill, but numbers seemed to be shifting against it, and the bill's sponsors held off. In February, the bill's sponsor in the lower house, Chris Spence, stood down from parliament to face allegations before the Independent Commission Against Corruption, shortly to be followed by the bill's sponsor in the upper house, Marie Ficarra.
Meanwhile, in October 2013, a drug-affected driver misjudged the lanes on a bridge in Dapto, New South Wales, and ploughed into a hatchback containing the 32-weeks pregnant Jacqueline Sparks and her brothers. Jacqueline's uterus ruptured on impact, resulting in her daughter being stillborn and the loss of Jacqueline's uterus. After the driver who caused the accident was sentenced in October 2014, Jacqueline and her partner, Chi Nguyen, called for the New South Wales Parliament to pass the Zoe's Law bill, which had been languishing in the upper house for nearly a year. Despite their appeals, in November 2014, the Zoe's Law bill lapsed without ever having been voted on in the upper house.
In the years since the vote, the accidents, and the headlines keep coming. Brodie Donegan continues to agitate for legislative change, stating in 2016, 'I believe that the current laws must be improved to adequately reflect the loss of the unborn child due to a serious criminal or violent act and acknowledge the impact that has on the entire family, rather than keeping the loss within the mother's injuries.' Fred Nile introduced a new 'Zoe's Law' Bill to the New South Wales Upper House in April 2017, again without permission from Zoe's parents to use her name. Meanwhile, the decriminalisation of abortion, which was delivered in Victoria in 2008, appears stalled in New South Wales. While abortion remains on the criminal code, a Zoe's Law-type bill is likely to throw the legal status of abortion in New South Wales into confusion, with the possibility of health professionals withdrawing their provision of services, for fear of criminal charges. Meanwhile, a Queensland mother whose daughter was stillborn in similar circumstances is agitating for a Queensland version—'Sophie's Law'. She wants 'all babies past 30 weeks gestation to have the right to be classed as a human being', but specifies that she does not 'want to affect a woman's right to abortion or have a pregnancy terminated for any medical reasons'.
There is a persistence here—in the women who want their and their family's loss acknowledged, in strong public support for their campaigns; and in the heartbreaking frequency of pregnancy loss resulting from criminal behaviour, particularly driving offences. This issue is not going to disappear of its own accord. Their babies' names haunt us. And this, I guess, is partly the reason these families persist with their campaigns to change the law. Our babies' names did not end up on child care waiting lists, on school rolls, on excursion slips or birthday invitations. If anyone but our loved ones is to remember them, it is up to us. We do not want their names, their little lives and the big dreams we had for them to die with us. If we have to live without them, we want something to come of this grief—that it might reduce the chances of other babies and families suffering a similar fate, or at least of suffering the indignity of the loss of their child being treated as an 'injury'. But the last thing I would want my daughter remembered for would be for laws that had the effect (intended or not) of winding back women's reproductive rights and ability to access medical treatment during pregnancy. Or, worse still, laws that flattened the complexities of pregnancy, maternal health, and addiction into two categories: 'bad mothers', who should be punished for potentially harming their babies in utero; and 'good mothers', who were willing and compliant gestators, always prioritising fetal life above their own.
30
Holding the torch
I had such good intentions in writing this book. I wanted to give something to newly grieving parents, to say, _You are not alone in this devastated landscape_. There are so many of us here with you, and, despite the devastation, this is still a beautiful place. We have walked it. Here is my fragile map, here is where the path emerges, here is how the parts relate to one another. It is frightening, it is painful; yet, it also inspires awe in its devastation. Unexpectedly exquisite things grow here, things you would never know about had you not set foot in the devastation. And if you can keep putting one foot in front of the other, the mud firms into a path beneath your feet, rains wash away the dirt from your body, you find debris from which to build a shelter, and, after a time, you start to notice small signs of life.
What I'd forgotten, of course, is that in the beginning, you can't see any of that because your eyes are so puffed with tears that they can hardly open. Even if you can open them and blink away the salt, you'd have to peel your hands from your face, the hands that are the only things keeping your head from hitting the floor with a dull thump. I am remembering all of this because we have, this week, been ordered back to that devastated landscape.
My sister's firstborn baby girl blows bubbles in her sleep, wrapped in layers of flannel, tubes and cords intruding on her soft skin. Her condition doesn't have a name yet; letters and numbers hover menacingly above her clear perspex box like a fatal bingo call. Whatever the ultimate order of the numbers, the real damage is being done in her cells, where her mitochondria—tiny molecular engines within each of her cells—are stalling. Not all of them, and not all at once, but for each one that stalls (one test says 78 per cent, but that may just be of her blood cells), a cell is deprived of the energy it requires for muscle activation, for digestion, or for production of proteins. Whatever the cell's purpose, it is short-changed, and her body has to work harder to achieve the same result.
She frowns, sighs wearily. Her blood shows high lactate levels, as though she is running a half-marathon rather than just lying there. Her heart is working so hard, it is enlarged, stretched. Sometimes it too gets tired, and slows. Alarms go off, the nurse comes. She presses the button quickly, then loosens the flannel layers, talks gently to her—'Hey, little one'—turns her on her side, and pats her small bottom firmly until the numbers on the heart rate monitor rise, and we all start to breathe again.
It has already been such a hard road for Mia, my tiny flannel-wrapped niece. An early evacuation from my sister's womb due to her alarmingly small size, then a transfer to the Children's Hospital with a gut infection and a heart murmur; so much prodding and poking as the doctors cajoled the secrets of her illness out of her body.
And now that they are decoding those secrets, it looks like it will be a short road too. At first they said she might live to two; then twelve months, if she's lucky. Even that may be optimistic.
So here I am, back in the wasteland, making the same wordless sounds, stepping through the familiar script of denial, anger, sadness, magical thinking. The pain doubles when I think of my sister and her pain, of her and my brother-in-law's tenderness with their fragile little daughter. Could this really be coincidence? That my mother, my sister and I all lose our firstborn daughter? I know that this is magical thinking, I know that these losses are pure dumb misfortune, but that doesn't stop my brain spinning theories, trying to construct sense from the senseless, to find a plot line or a moral for the story.
I hold my sister tight. Sob into her shoulder and try to hold firm for her sobs. I want to give my sister all that she gave me in my most broken hour, I want to give her everything I wish I'd known. But, despite the familiarity, this is a different wasteland. My map turns to soggy tissue in my hand.
I tell my law students that the value of a good set of study notes isn't the content. You can have the most accurate and concise notes but they won't help you a bit if you haven't developed them (or at least worked through them) yourself. The value is in the process of boiling down notes into summaries, of banging your head against the law, getting confused and then getting a little bit clearer. Grief is the same. I can spill the beans on what makes sense for me—on the mechanics of my beliefs about where Z is, how I honour her in my day-to-day life, how I let the grief move through me, like a sneeze, when it appears these days, intense but short-lived. Whatever your loss, I want to make it easier for you, to help you short-cut through the pain and the banging of one's head against the finality of death. But my mechanics work because I have crafted them for my own head, custom-built them around my irregularly shaped heart. Yours will be different. Just know I am here with you, holding the torch.
**She was cremated, you see**
Not many people
have a baby daughter who is a star.
The light and heat released
with her little five pound eight body
is still travelling
through the heavens
will bounce,
and one day light upon
someone's eye
as the light from a star.
Not many people (I like to think)
have a baby daughter who is an ocean
(and at the same time, rain).
Her water atoms
went up like a mist
found new friends
among the atoms
of other babies
grandmas
well-loved dogs.
And though it was scary to fall
(as rain)
when they hit the ocean
it felt like home.
(I like to think) Not many people
have a baby daughter who is a ballerina-shaped fuschia bud.
Her nutrients—every molecule that made
her soft skin
her fingers grip
has gifted itself to the earth
(I wouldn't have been so generous)
except for a few of the most beautiful
which circulate still in my blood.
'You have a daughter,' they say
each time they get pumped through my heart.
'You have a daughter,' they say
as they tend my broken cells.
The others
(there are millions)
find themselves
pulsing along a green stem
willing a bud to open,
feeding the thing that colours the petal,
scenting the pollen dust,
unfurling the leaf.
Are chewed on or breathed in by
living things,
And find a new home in them.
She is here.
Here.
Here a thousand times but also everywhere.
She makes me weep
at how clever and beautiful she is
And at my own small flimsy wish
For a more conventional baby.
(Still her—but here in the more conventional way)
It takes a very still
clear nightful of stars
or a big stormy oceanful of ocean
for me to know
(again, as I've always known)
how many babies it takes
to make up the sky.
Cremated and uncremated.
Missed and kissed.
Sung to and unsung to.
Innumerable,
visible
and each such a particular
little pinprick of light.
(August 2011)
Acknowledgements
With each ripple that the accident sent into our lives, I was lucky to have loved ones and strangers offering kindness and support—from the people who stopped to help and called the ambulances, the emergency workers who freed me and got us all to hospital, all the health care providers (even the physiotherapists) who worked hard to help us survive, heal and grieve, to the friends who moved furniture for us, brought us food, sent gifts and good thoughts and who continue to remember our daughter with us. My midwives (and now dear friends) Jen and Mandy—what incredible work you do, and what a difference you made!
Special thanks go to my family, particularly Mum, Dad, Erica, Jeremy, Deb, Jazzie, Jackie and Rima, as well as Penelope Goodes, for being there both in the living and the writing, every step of the way. I love you all so much and I'm so thankful you are in my life.
Belinda Quantock, Steve Macmillan, Matt Drummond, Rosalind Hearder and Samantha Kimpton and Sabdha and Cristi Pink-Charlton—thank you for providing friendship, encouragement, practical help and sage advice.
So many of the insights that made our loss bearable came from other baby lost parents who I've come to think of as my 'dead baby parents group', both at the SANDS Thornbury group and online, particularly Angie, Sally, Catherine, Merry, Jeanette, Kate, Sarah and Kate. As sorry as I am that we have this in common, I am so glad I found you. Karin and Ned, thank you for letting me tell a piece of Albie and Esther's story, and Brigette, for sharing your experience with Sacha.
This book would not have happened without the Somers friends who encouraged me to be brave and realise I wanted to really write it. And it was Maria Tumarkin's excellent Creative Non-Fiction course put on by Writers Victoria which helped kickstart it. That course also led me to my fabulous writing group—Debi Hamilton, Frank Golding, Kath McKay and Anastasia Kanjere. Thank you for providing a structure and a space for me to build this book, one chapter at a time, and for unfailingly helpful questions, feedback and advice. Thanks also to Madeline Hamilton, for reading early fragments and encouraging me to take it further, and to the amazing Monica Dux, for taking a chance on my writing in _Mothermorphosis_ , and for wise advice and playing fairy godmother to the project ever since.
Colleagues and the head of the Law School at La Trobe, Patrick Keyzer have been incredibly supportive in helping me find a middle space between what I had experienced and my legal academic writing, particularly Laura Griffin, Kirsty Duncanson and Fiona Kelly. Thank you to the various 'Shut up and Write' groups at La Trobe, Melbourne Law School and RMIT who egged me on, provided valuable feedback about the title and other key decisions about the book, and who allowed me to literally shut up and write.
For bringing it to publication and making it a better manuscript, thank you so much to Dina Kluska, Sally Heath and Louise Stirling at MUP, Sarina Rowell for sensitive and careful copyediting, Klarissa Pfisterer for working with us to develop such a beautiful cover, Nicholas Purcell for the profile photography, Paul Smitz for your proofreading, Hilary Harper, for reading closely and for her own beautiful writing, and my literary agent Clare Forster for wonderfully astute advocacy and advice.
Last but not least, thank you to my beautiful children—to Ali, for shining such light into our lives and persistently harassing me about 'finishing my chapter', and to Zainab, for teaching me about life, death and being a mother.
Notes
**Chapter 1 Sunday 27 December 2009**
_JM v QFG, GK and State of Queensland_ [1997] QADT 5 (31 January 1997); _MW & Ors v Royal Women's Hospital & Ors Mw, Dd, Ta & Ab v Royal Women's Hospital, Freemasons Hospital and Victoria_ [1997] HREOCA 6 (5 March 1997). The Queensland decision on direct discrimination was, however, overturned by the Queensland Supreme Court in October 1997, and the decision on indirect discrimination was remitted back to the Queensland Anti-Discrimination Commission: _QFG and GK v JM_ (Ambrose J, Supreme Court of Queensland, 24 October 1997).
**Chapter 6 Frida and me**
Hayden Herrera, _Frida Kahlo: The Paintings_ , Bloomsbury, 1992.
**Chapter 10 The 'born alive' rule**
_Attorney General's Reference No 3 of 1994_ [1998] AC 245.
_Attorney General's Reference No 3 of 1994_ [1998] AC 245, 269.
Adrian Lowe, 'Mother vows to fight on for law change over road death of unborn child', _The Age_ (25 January 2010), 5.
_Crimes Act 1958_ (Vic) s15.
[2003] NSWCCA 399.
_R v King_ [2004] NSWCCA 444 (7 December 2004) at [30] per McColl JA.
_R v King_ [2004] NSWCCA 444 (7 December 2004) at [31] per McColl JA.
_R v King_ [2004] NSWCCA 444 (7 December 2004) at [32] per McColl JA.
_R v King_ [2004] NSWCCA 444 (7 December 2004) at [39] per McColl JA.
_R v King_ [2003] NSWCCA 399 (19 December 2003), [14] Tupman DCJ quoted in [10] and [14] of Spiegelman CJ's judgment.
_R v King_ [2003] NSWCCA 399 (19 December 2003), [96].
Adrian Lowe, 'Mother vows to fight on for law change over road death of unborn child', _The Age_ (25 January 2010), 5.
ibid.
**Chapter 20 Heartbeat**
<http://www.sciencedirect.com/science/article/pii/S0006291X04023824>.
**Chapter 22 Close up with hope**
Pema Chödrön, _When Things Fall Apart_ — _Heart Advice for Difficult Times_ , Shambhala, Boston 2000, pages 44 and 41.
Pema Chödrön, _When Things Fall Apart_ — _Heart Advice for Difficult Times_ , Shambhala, Boston 2000, page 53.
**Chapter 23 The charnel ground**
Pema Chödrön, _When Things Fall Apart_ — _Heart Advice for Difficult Times_ , Shambhala, Boston 2000, page 124.
**Chapter 29 Zoe's Law**
In Victoria, the charge is referred to as 'dangerous driving causing serious injury': _Crimes Act 1958_ (Vic), s 15, as amended by _Abortion Law Reform Act_ 2008 (Vic) s 10(2). For NSW, see: _Crimes Act 1900_ (NSW), s 4, as amended by _Crimes Amendment (Grievous Bodily Harm) Act 2005_ (NSW).
Brodie Donegan, 'In the eyes of the law, her daughter's death doesn't count' (18 September 2013), _Mamamia_ , <http://www.mamamia.com.au/social/zoes-law/>
Fred Nile (27 May 2013), 'Religion, marriage & euthanasia', _Q &A_ (transcript). Australian Broadcasting Corporation; 'Stop Muslim immigration, NSW Christian Democrats say', _ABC News_ , Australia, 11 March 2007, <http://www.smh.com.au/lifestyle/celebrity/fred-nile-under-fire-for-facebook-post-about-charlotte-dawson-abortion-20140224-33bol.html>.
_Crimes Amendment (Zoe's Law) Bill 2013_ (NSW), proposed s 41AA(1).
It was only in 2016 that the law was amended to clarify that this law excluded women who had taken legally prescribed drugs. Nina Martin, 'Alabama lawmakers limit drug prosecutions in pregnancy', _ProPublica_ (4 May 2016), <https://www.propublica.org/article/alabama-lawmakers-limit-drug-prosecutions-in-pregnancy>.
Victoria Law, 'Alabama case illustrates difficulties women behind bars face when seeking abortion', _Rewire_ (5 August 2015), <https://rewire.news/article/2015/08/05/alabama-case-illustrates-difficulties-women-behind-bars-face-seeking-abortion/>.
Kristin Savell, 'Is the "born alive" rule outdated and indefensible?' (2006), 28 _Sydney Law Review_ , 625, 664.
_State v Rowland_ , No. 041901649 (Utah Dist. Ct.-3d Apr. 7, 2004) (Fuchs J).
Nina Martin, 'A stillborn child, a charge of murder and the disputed case law on "fetal harm"', _ProPublica_ (18 March 2014) <https://www.propublica.org/article/stillborn-child-charge-of-murder-and-disputed-case-law-on-fetal-harm>. See also the Alabama Supreme Court's decision in _Ex parte Ankrom No. 1110176_ , 2013 WL 135748 (Ala. January 11, 2013).
_Bei Bei Shuai v State_ , 966 N E 2d 619 (May, Najam, Riley JJ) (Indiana Court of Appeals, 2012). After significant public protests, in August 2013, the prosecutor offered Ms Shuai a plea bargain. She pleaded guilty to criminal recklessness and was immediately released, having already served more than the maximum sentence. National Advocates for Pregnant Women, 'Thank you! Bei Bei Shuai is free!' (6 August 2013), <http://advocatesforpregnantwomen.org/blog/2013/08/thank_you_bei_bei_shuai_is_fre.php>.
Barry M Lester and Jean E Twomey, 'Treatment of substance abuse during pregnancy' (2008), 4 _Womens Health_ 67, 74. Similarly, the Intergovernmental Committee of Drugs found that punitive attitudes were an 'impediment' to responding more effectively to fetal alcohol spectrum disorders: Lucinda Burns et al. (eds), _'Fetal Alcohol Disorders in Australia: An update' (Monograph of the Intergovernmental Committee of Drugs Working Party of Fetal Alcohol Spectrum Disorders, June 2012)_ [2012], 76.
Angela Carder's family, via her estate, successfully appealed the decision after her death, in the District of Columbia Court of Appeals 573 A.2d 1235 (1990).
_Texas Health and Safety Code_ s 166.049.
<http://www.perthnow.com.au/news/western-australia/hundreds-farewell-crash-victim-shona-caley-and-her-unborn-child-at-corrigin-funeral/news-story/08b1cd0892a0215130d36dd515203652>; <http://www.smh.com.au/nsw/marco-silvestri-jailed-for-crash-that-killed-unborn-baby-20141022-119rpg.html>; <http://www.news-mail.com.au/news/unborn-baby-killed-car-crash/2638355/>; <http://www.news.com.au/lifestyle/parenting/babies/sophies-law-mother-fights-for-unborn-babies-to-be-protected-after-a-car-crash-claimed-her-daughters-life-days-from-birth/news-story/cf6f9424b2522fec0b05ba32bd3cd494>; <http://www.northernstar.com.au/news/unborn-baby-killed-queensland-traffic-crash/2370046/>.
<http://www.dailytelegraph.com.au/newslocal/central-coast/we-owe-it-to-zoe-to-fight-for-criminal-law-change/news-story/7e1365ac002321acf979911ad75473dd>.
<https://www.change.org/p/attorney-general-please-bring-in-sophie-s-law-to-protect-all-unborn-babies-post-30-weeks-gestation>; <https://www.9now.com.au/a-current-affair/2016/clip-cijyzzrok00bxb5p5r9mm81nj>
| {
"redpajama_set_name": "RedPajamaBook"
} | 9,813 |
Q: python gif size on disk bigger after saving I am using the bird animated gif from this stackoverflow question about resizing gifs. The image size on disk is about 2.1MB. Now if I do (using python3.5)
import PIL.Image
PIL.Image.open('bird.gif').save('output.gif', save_all=True, optimize=True)
I obtain an image of size 7.5MB.
Using imageio, I am not getting any luckier:
import imageio
im = imageio.mimread('bird.gif')
imageio.mimsave('output.gif', im)
I obtain the exact same size of gif.
Does anyone know how to save an animated gif so that it remains as small as possible, which means at least as small as the input gif ?
Thanks!
| {
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"http:\/\/math.univ-lille1.fr\/~cempi\/activites_scientifiques\/FR\/groupe_travail\/gdtnlse.php","text":"# Dynamique non-lin\u00e9aire dans les syst\u00e8mes complexes : aspects d\u00e9terministes et stochastiques\n\n## Organisation: J.C. Garreau (PhLAM)\n\nLes personnes qui souhaitent \u00eatre inform\u00e9es des activit\u00e9s de ce groupe de travail peuvent s'inscrire dans la liste d'envoi en envoyant un courrier \u00e9lectronique (\u00e0 partir de l'adresse \u00e0 laquelle ils souhaitent recevoir l'information) \u00e0 nlse-cempi-subscribe@univ-lille1.fr . La d\u00e9sinscription se fait en envoyant un courrier \u00e9lectronique \u00e0 nlse-cempi-unsubscribe@univ-lille1.fr .\n\nmercredi 27 septembre 2017, 11h00, salle 172, b\u00e2timent P5 bis\nDynamiques complexes dans les syst\u00e8mes atomiques et optiques Abstract\nGuillaume Berthet Laboratoire Charles Fabry, Institut d'Optique Palaiseau\n\n12 juin 2017, 11h00, salle 172, b\u00e2timent P5\nRatchet currents in parameter space\nMarcus Werner Beims Universidade Federal de Curitiba (Br\u00e9sil)\nIn this talk I present the parameter space of a discrete ratchet model which gives direct connections between chaotic domains and a family of isoperiodic stable structures with the ratchet current. The isoperiodic structures, where larger currents are usually observed inside, appear along preferred direction in the parameter space giving a guide to follow the current. Currents in parameter space provide a direct measure of the momentum asymmetry of the multistable and chaotic attractors times the size of the corresponding basin of attraction. Results about the thermal and vacuum fluctuations effects on classical and quantum ratchet currents, respectively, are presented.\n\n20 f\u00e9vrier 2015, 14h00, Amphi Pierre Glorieux, CERLA\nBose-Einstein Condensates of Photons in a Dye-Filled Microcavity affiche\nRob Nyman Imperial College de Londres\n\n20 janvier 2015, 9h30, salle 172 b\u00e2t. P5\nRoute to thermalization in the Fermi-Pasta-Ulam system\nMiguel Onorato Universita di Torino\nI will present some new theoretical results on the original Fermi-Pasta-Ulam (FPU) system with N=16,32 and 642 masses connected by a nonlinear quadratic spring. The approach is based on resonant wave-wave interaction theory, i.e. I assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the FPU equation of motion, I will show that the first non trivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. The time scale of such interactions is extremely large and it is of the order of $1\/\\epsilon^8$, where $\\epsilon$ is the small parameter in the system. The results are supported by extensive numerical simulations.\n\n5 d\u00e9cembre 2014, 11h, salle 172 b\u00e2t. P5 bis\nAnderson localization loop spectroscopy\nCord M\u00fcller Universit\u00e9 de Constance (Allemagne) et Institut Non Lin\u00e9aire de Nice\nIn this talk, I will introduce a new framework to study weak localization as well as the onset of Anderson localization in disordered systems. The idea is to expose the waves propagating in a random scattering environment to a sequence of short dephasing pulses. The system responds through coherence peaks forming at specific echo times, each echo representing a particular process of quantum interference. We have suggested a concrete realization for cold gases, where quantum interferences are observed in the momentum distribution of matter waves in a laser speckle potential. Our proposal has been recently realised at Institut d\u2019Optique in the weak localisation regime. Eventually, we envisage the probing of higher-order processes like coherent-forward scattering, which has been established as the momentum-space signature of Anderson localisation. Reference: T. Mickliltz, C.A. Mueller, A. Altland, arXiv:1406.6915\n\n24 novembre 2014, 11h, CERLA\nDispersive Shock Waves : a perspective from nonlinear optics\nStefano Trillo Universita di Ferrara (Italie)\nDispersive shock waves (DSWs) entails the breaking of a wave disturbance through the spontaneous emission of fast oscillations (wavetrains) caused by a weak dispersion in response to the tendency of a nonlinear system to form a shock wave. Optics offers an unsurpassed potential for studying the features of DSWs in repeatable lab experiments. Such experiments are accurately described by the (integrable) semiclassical nonlinear Schroedinger (NLS, also known as Gross-Pitaevskii in the context of cold atoms) equation. We review the properties of such fascinating objects in the framework of such model. Weak or strong perturbations to NLS also accounts for new phenomena such as radiation from dispersive shocks, competition of shock and other breking phenomena (modulational instability), and formation of shock from a completely random field (incoherent shock). These new directions will be also briefly discussed.\n\n20 juin 2014, 11h\nSalle 172, B\u00e2. P5 bis.\n\"Propagation of Collective Pair Excitations in Disordered Bose Superfluids\"\nSamuel Lellouch Laboratoire Charles Fabry de l\u2019Institut d\u2019Optique\nCollective excitations govern most dynamical properties of many-body quantum systems, such as propagation of correlations and thermalization processes. In disordered systems, the transport properties of collective excitations may however be strongly altered, mainly due to Anderson localization. The case of interacting bosons is expected to be particularly involved, since repulsive interactions in Bose systems can compete or cooperate with disorder, inducing nontrivial localization effects. In this talk, we will discuss the effect of disorder on the propagation of collective excitations in a disordered Bose superfluid in dimension d>1. We incorporate local density depletion induced by strong disorder at the meanfield level, and formulate the transport of the excitations in terms of a screened scattering problem. We show that the competition of disorder, screening, and density depletion induces a strongly nonmonotonic energy dependence of the disorder parameter, which governs the localization behaviour. While in low dimensions, all excitations are localized, we find that in three dimensions, the excitation spectrum can split into alternating bands of localized and extended states, with up to three mobility edges. Implications on experiments with disordered ultracold atoms are discussed.\n\n12 mai 2014, 11h\nSalle 172, B\u00e2. P5.\n\"Shock and rogue waves in nonlinear systems\" slides\nMatteo Conforti IRCICA\nExtreme wave events, such as Rogue Waves and Dispersive Shock Waves, are encountered in several physical settings, ranging from oceanography to optics, form physics of the atmosphere to Bose-Einstein condensation. I will show the development of extreme waves in different nonlinear systems, of particular relevance to nonlinear optics, such as vector NLS and three-wave interaction.\n\n8 avril 2014, 14h\nSalle de r\u00e9union, B\u00e2. M2.\n\"Inverse Scattering Transform\" Lien\n\n7 avril 2014, 14h\nSalle 172, B\u00e2. P5.\n\"On a coupled system of Schr\u00f6dinger Poisson equations\"\nMaxime Gazeau Inria-Lille\n\n14 mars 2014, 15h\nSalle 172, B\u00e2. P5.\n\"Dynamique des paquets d'\u00e9lectrons relativistes circulant dans les anneaux de stockage\" Serge Bielawski PhLAM-Universit\u00e9 Lille 1\nNous pr\u00e9sentons un ensemble de r\u00e9sultats sur la dynamique spatio-temporelle des paquets d'\u00e9lectrons relativistes, circulant dans les anneaux de stockage (comme SOLEIL, UVSOR, etc). Lorsque la charge paquet d'\u00e9lectrons d\u00e9passe un certain seuil, une instabilit\u00e9 dynamique fait apparaitre une structure spatiale (avec une longueur d'onde de l'ordre de millim\u00e8tre), pouvant \u00e9voluer de mani\u00e8re erratique. D'un point de vue th\u00e9orique, le mod\u00e8le est bas\u00e9 sur une \u00e9quation de type Vlasov-Fokker-Planck, et l'ingr\u00e9dient principal de l'instabilit\u00e9 est l'interaction entre les \u00e9lectrons. D'un point de vue exp\u00e9rimental, jusque r\u00e9cemment, les signatures de l'instabilit\u00e9 \u00e9taient extr\u00eamement indirectes, et consistaient en la d\u00e9tection d'un rayonnement terahertz intense. Depuis 2013, il nous a \u00e9t\u00e9 possible d'\u00e9tudier de mani\u00e8re directe, et en temps r\u00e9el, la forme des structures spatiales, avec une r\u00e9solution de l'ordre de la picoseconde. Cette nouvelle possibilit\u00e9 permet \u00e0 pr\u00e9sent de comparer de fa\u00e7on extr\u00eamement directe les r\u00e9sultats exp\u00e9rimentaux et num\u00e9riques, en particulier en ce qui concerne les \u00e9volutions d! es structures spatiales en fonction du temps. Les premi\u00e8res observations exp\u00e9rimentales ont \u00e9t\u00e9 effectu\u00e9es \u00e0 UVSOR (Japon, en mars 2012 et 2013), et \u00e0 SOLEIL (France, en octobre-novembre 2013).\n\n18 f\u00e9vrier 2014, 11h\nSalle 172, B\u00e2. P5.\n\"The noise is the signal\": thermometry and exotic correlations from the momentum distribution noise of cold atoms\"\nTommaso Roscilde Laboratoire de Physique, ENS de Lyon\n\n6 d\u00e9cembre 2013 Amphi du CERLA\nHydrodynamic Rogue Waves Lien\nslides\nMiguel ONORATO (Universit\u00e9 de Turin)\n\njuin 2013\nModulational instability in the non-linear Schr\u00f6dinger equation Lien pptx\nA.Taki (PhLAM)\n\nnovembre 2012\nWave turbulence in integrable and non-integrable optical fiber systems Lien ppt\nP. Suret (PhLAM)\n\n9 novembre 2012\nUne nouvelle d\u00e9composition pour l'\u00e9quation de Schr\u00f6dinger non lin\u00e9aire semiclassique menant \u00e0 des sch\u00e9mas pr\u00e9servant l'asymptotique Lien pdf\nChristophe Besse (Painlev\u00e9)\n\n6 septembre 2012\nLa physique du rotateur frapp\u00e9 quasi-p\u00e9riodique, son rapport avec le mod\u00e8le d'Anderson et sa r\u00e9alisation exp\u00e9rimentale Lien pdf\u00a0\/\u00a0Lien ppsx ; visite de la manip.\nJean-Claude Garreau (PhLAM)\n\n18 juillet 2012\nLa localisation d'Anderson\nStephan De Bi\u00e8vre (Painlev\u00e9)","date":"2017-12-16 03:30:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5392749309539795, \"perplexity\": 10848.198864346215}, \"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-2017-51\/segments\/1512948581053.56\/warc\/CC-MAIN-20171216030243-20171216052243-00090.warc.gz\"}"} | null | null |
Q: Java Iterating an object of custom class I have no idea about iterating an object in java, however I've tried to create a custom class and try to get a result from it but I got a same iteration result whereas the data set into object are different.
and here are my class.
package property;
import java.util.ArrayList;
public class obj_error {
public Integer error_code;
public String error_desc;
public Integer error_line;
public String error_modul;
public Integer total = 0;
public ArrayList<obj_error> list = new ArrayList<obj_error>();
public void set(Integer code, String desc, String module,Integer line ){
this.error_code = code;
this.error_desc = desc;
this.error_modul= module;
this.error_line = line;
this.total = total+1;
//list.add(this);
set_list();
}
public Integer error_code() { return error_code; }
public String error_desc() { return error_desc; }
public Integer error_line() { return error_line; }
public String error_modul() { return error_modul; }
public Integer total() { return total; }
public ArrayList<obj_error> get_error_list() { return list; }
public void set_code(Integer param) { this.error_code = param; }
public void set_desc(String param) { this.error_desc = param; }
public void set_line(Integer param) { this.error_line = param; }
public void set_modul(String param) { this.error_modul = param; }
public void set_total(Integer param) { this.total = total+1; }
public void set_list(){ list.add(this); }
}
and this is where I call that class
package testcase;
import java.util.ArrayList;
import java.util.Iterator;
import property.obj_error;
public class set_error {
public static void main(String[] args){
obj_error error = new obj_error();
error.set(1001,"Not Valid","Loand",1);
error.set(1002,"Not Validsafasd","Loan",2);
System.out.println(error.list);
for(Iterator<obj_error> i = error.get_error_list().iterator(); i.hasNext(); ) {
obj_error item = i.next();
System.out.println(item.error_code+" " + item.error_desc+ " " +item.error_line+ " " + item.error_modul);
}
//print out result
//1002 Not Validsafasd 2 Loan
//1002 Not Validsafasd 2 Loan
}
}
A: What you have here is a case of frankendesign. You're using a single instance of your obj_error (which you should probably rename to ObjError to make it easier on Java eyes to read), when you really seem to want 2 instances.
You should take the list out of your obj_error class and put it in your main method instead. If you convert your set method into a constructor, the design starts to look a lot better (artist's rendering):
public static void main(String[] args){
ArrayList<obj_error> list = new ArrayList<obj_error>();
obj_error error1 = new obj_error(1001, "Not Valid", "Loand", 1);
list.add(error1);
obj_error error2 = new obj_error(1002, "Not Validsafasd", "Loan", 2);
list.add(error2);
System.out.println(list);
for(obj_error item : list)
System.out.println(item.error_code+" " + item.error_desc+ " " +item.error_line+ " " + item.error_modul);
A: There is so much to say... First, applause that it compiles.
As tip, you should use different classes for the main object that contain the list of errors from the objects in this list. Currently, you are dealing with a sort of tree structure.
For example:
class ErrorContainer{
// ...
private List<ErrorMsg> = new ArrayList<>;
// ... continue
It should be clearer.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,310 |
<?php
namespace test\Mockery;
use Mockery\Adapter\Phpunit\MockeryTestCase;
class MockClassWithFinalWakeupTest extends MockeryTestCase
{
protected function mockeryTestSetUp()
{
$this->container = new \Mockery\Container();
}
protected function mockeryTestTearDown()
{
$this->container->mockery_close();
}
/**
* @test
*
* Test that we are able to create partial mocks of classes that have
* a __wakeup method marked as final. As long as __wakeup is not one of the
* mocked methods.
*/
public function testCreateMockForClassWithFinalWakeup()
{
$mock = $this->container->mock("test\Mockery\TestWithFinalWakeup");
$this->assertInstanceOf("test\Mockery\TestWithFinalWakeup", $mock);
$this->assertEquals('test\Mockery\TestWithFinalWakeup::__wakeup', $mock->__wakeup());
$mock = $this->container->mock('test\Mockery\SubclassWithFinalWakeup');
$this->assertInstanceOf('test\Mockery\SubclassWithFinalWakeup', $mock);
$this->assertEquals('test\Mockery\TestWithFinalWakeup::__wakeup', $mock->__wakeup());
}
public function testCreateMockForClassWithNonFinalWakeup()
{
$mock = $this->container->mock('test\Mockery\TestWithNonFinalWakeup');
$this->assertInstanceOf('test\Mockery\TestWithNonFinalWakeup', $mock);
// Make sure __wakeup is overridden and doesn't return anything.
$this->assertNull($mock->__wakeup());
}
}
class TestWithFinalWakeup
{
public function foo()
{
return 'foo';
}
public function bar()
{
return 'bar';
}
final public function __wakeup()
{
return __METHOD__;
}
}
class SubclassWithFinalWakeup extends TestWithFinalWakeup
{
}
class TestWithNonFinalWakeup
{
public function __wakeup()
{
return __METHOD__;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,104 |
#ifndef _ERRNO_H
#define _ERRNO_H
extern int errno;
/*
* Error number definitions
*/
#define EPERM 1 /* not permitted */
#define ENOENT 2 /* file or directory not found */
#define EIO 5 /* input/output error */
#define ENOMEM 12 /* not enough space */
#define EACCES 13 /* permission denied */
#define EFAULT 14 /* bad address */
#define EBUSY 16 /* resource busy */
#define EEXIST 17 /* file already exists */
#define ENODEV 19 /* device not found */
#define EINVAL 22 /* invalid argument */
#define EDOM 33 /* math argument out of domain of func */
#define ERANGE 34 /* math result not representable */
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,407 |
FROM ubuntu:12.04
MAINTAINER Remy Sharp <remy@leftlogic.com>
RUN rm /bin/sh && ln -s /bin/bash /bin/sh
RUN apt-get update && apt-get install curl npm -y
ENV NVM_DIR /usr/local/nvm
ENV NODE_VERSION iojs
ENV NVM_VERSION 0.26.1
ENV TRAVIS TRUE
# # Install nvm with node and npm
RUN curl https://raw.githubusercontent.com/creationix/nvm/v$NVM_VERSION/install.sh | bash \
&& source $NVM_DIR/nvm.sh \
&& nvm install 0.10 \
&& nvm install 0.12 \
&& nvm install iojs-v1 \
&& nvm install iojs-v2 \
&& nvm install iojs-v3 \
&& nvm install 4.0 \
&& nvm alias default $NODE_VERSION \
&& nvm use default
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,670 |
Q: Creating Android AAR file from open source and add it to my own project Hey all I am having a difficult time figuring out why the export of an .aar file does not seem to be working when I add it to my own project.
I am using the Expanable-tab project and it compiles the library it has to an .aap file via the build.gradle:
apply plugin: 'com.android.library'
apply plugin: 'kotlin-android'
ext.versionMajor = 1
ext.versionMinor = 2
ext.versionPatch = 1
android {
compileSdkVersion 30
defaultConfig {
minSdkVersion 15
targetSdkVersion 30
versionCode versionMajor * 100000 + versionMinor * 100 + versionPatch * 1
versionName "$versionMajor.$versionMinor.$versionPatch"
setProperty("archivesBaseName", "expandable-fab")
// Changes the name of generated AAR artifacts to be more descriptive
libraryVariants.all { variant ->
variant.outputs.all { output ->
if (outputFileName.endsWith('.aar')) {
outputFileName = "${archivesBaseName}-${versionName}.aar"
}
}
}
}
buildTypes {
release {
minifyEnabled false
proguardFiles getDefaultProguardFile('proguard-android-optimize.txt'), 'proguard-rules.pro'
}
}
sourceSets {
main.java.srcDirs += 'src/main/kotlin'
}
}
dependencies {
implementation fileTree(dir: 'libs', include: ['*.jar'])
implementation"org.jetbrains.kotlin:kotlin-stdlib-jdk7:$kotlin_version"
implementation 'com.google.android.material:material:1.4.0'
implementation 'androidx.appcompat:appcompat:1.3.1'
}
apply from: 'generate-docs-and-update-website.gradle'
apply from: 'publish-artifacts-to-maven-repo.gradle'
When I rebuild the project I navigate to the expandable-fab-master\library\build\outputs\aar directory where expandable-fab-1.2.1.aar file has been produced.
I then copy the .aap file to my libs folder in my own project:
Then I proceed to add this to the build.gradle:
All of this compile when selecting the rebuild option - no errors.
However, when it gets to the point in my app that uses the expandable-fab code it crashes with this as the error:
E/AndroidRuntime: FATAL EXCEPTION: main
Process: com.myapp, PID: 19960
android.view.InflateException: Binary XML file line #36 in com.myapp:layout/_fragvideoplay1: Binary XML file line #36 in com.myapp:layout/_fragvideoplay1: Error inflating class com.nambimobile.widgets.efab.ExpandableFabLayout
Caused by: android.view.InflateException: Binary XML file line #36 in com.myapp:layout/_fragvideoplay1: Error inflating class com.nambimobile.widgets.efab.ExpandableFabLayout
Caused by: java.lang.reflect.InvocationTargetException
And the code its referencing (videoPlay.java:99):
If I comment out the implementation files('libs/expandable-fab-1.2.1.aar') and uncomment the original implementation 'com.nambimobile.widgets:expandable-fab:1.2.1' and then run the app it works as it should...
What am I missing here?
A: This is because of how you're using the aar file as a gradle dependency. Since you already have
implementation fileTree(dir: 'libs', include: ['*.jar', '*.aar'])
You don't have to do
implementation("libs/library_name")
instead do this:
implementation(name: "library_name", ext: "aar")
I hope this helps.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,072 |
Occupation / Country
Usage Citation
Bea Arthur
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Beatrice Arthur (born Bernice Frankel; May 13, 1922 – April 25, 2009) was an American actress and comedian. Arthur began her career on stage in 1947. She won the 1966 Tony Award for Best Featured Actress in a Musical for playing Vera Charles in Mame. Read more on Wikipedia
Since 2007, the English Wikipedia page of Bea Arthur has received more than 8,365,325 page views. Her biography is available in 36 different languages on Wikipedia (up from 34 in 2019). Bea Arthur is the 1,620th most popular actor (up from 2,171st in 2019), the 2,853rd most popular biography from United States (up from 3,653rd in 2019) and the 797th most popular American Actor.
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Fabio Testi
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Among people born in 1922, Bea Arthur ranks 145. Before her are Anker Jørgensen, Sōsuke Uno, Jackie Cooper, Georges Cottier, Yuri Averbakh, and Dilip Kumar. After her are Paul Scofield, Princess Elisabeth, Duchess of Hohenberg, Vasko Popa, Stanley Schachter, Dorothy Dandridge, and Seymour Martin Lipset. Among people deceased in 2009, Bea Arthur ranks 103. Before her are Willy DeVille, Johannes Mario Simmel, Inger Christensen, David Eddings, Lino Lacedelli, and Al Martino. After her are Samak Sundaravej, Eleanor F. Helin, Manea Mănescu, Meir Amit, Friaça, and Walter Cronkite.
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Anker Jørgensen
Sōsuke Uno
Georges Cottier
Yuri Averbakh
Paul Scofield
Princess Elisabeth, Duchess of Hohenberg
Vasko Popa
Stanley Schachter
Seymour Martin Lipset
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Lino Lacedelli
Al Martino
Samak Sundaravej
Eleanor F. Helin
Manea Mănescu
Meir Amit
Friaça
Among people born in United States, Bea Arthur ranks 2,853 out of 15,968. Before her are Ronald Inglehart (1934), Adrian Smith (1944), Michelle Monaghan (1976), Lillian Moller Gilbreth (1878), Nicholas Negroponte (1943), and Allen Newell (1927). After her are Chuck Lorre (1952), Ralph Waite (1928), Miley Cyrus (1992), Barbara O'Neil (1910), Thomas Curtis (1873), and Sue Lyon (1946).
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Ronald Inglehart
Lillian Moller Gilbreth
Allen Newell
Chuck Lorre
Thomas Curtis
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Among actors born in United States, Bea Arthur ranks 797. Before her are Maggie Gyllenhaal (1977), Octavia Spencer (1972), Dita Von Teese (1972), Eva Longoria (1975), Julie Harris (1925), and Michelle Monaghan (1976). After her are Ralph Waite (1928), Barbara O'Neil (1910), Sue Lyon (1946), Denise Richards (1971), Robert Davi (1953), and Mary Elizabeth Mastrantonio (1958).
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\section*{Introduction}
Let $\g$ be an affine Kac-Moody Lie algebra, $\g_{fin}$ be the
corresponding simple finite-dimensional algebra, $$\g=\g_{fin}\T\C[t,t^{-1}]
\oplus\C K\oplus\C d,$$ where $K$ is a central element and
$[d,x\T t^i]=-ix\T t^i$ for $x\in\g.$
We fix the Cartan decomposition $\g=\n\oplus\h\oplus\n_-,$ with
$$\h=\h_{fin}\oplus\C K\oplus\C d,$$ where
$\h_{fin}$ is the Cartan subalgebra of $\g_{fin}$.
We also denote $$\g'=[\g,\g]=\g\T\C[t,t^{-1}]\oplus\C K, \ \
\h'=\h_{fin}\oplus\C K.$$
Let $P^+_k\hk \h^*$ be the set of all dominant integrable level $k$ weights
of $\g$. We denote by $P^{'+}_k\hk {\h'}^*$ the image of $P_k^+$ with respect
to the restriction map $\h^*\to {\h'}^*$, $\la\mapsto\la'$. For $\la\in P_k^+$
we denote by $L_\la$ the corresponding irreducible highest weight $\g$ module
and by $L_{\la'}$ a $\g'$ module which coincides with $L_\la$ as a vector
space and the action of $\g'$ is a restriction of the action of $\g$.
For $\la_1\in P_{k_1}^{'+}, \la_2\in P_{k_2}^{'+}$ consider a decomposition of
the tensor product of $\g'$-modules
\begin{equation}
\label{LL}
L_{\la_1'}\T L_{\la_2'}=\bigoplus_{\mu'\in P_{k_1+k_2}^{'+}}
L_{\mu'}\T C_{\la_1'\la_2'}^{\mu'}.
\end{equation}
Note that $C_{\la_1'\la_2'}^{\mu'}$ can be considered
as a space of highest weight vectors of $\h'$-weight $\mu'$ in
$L_{\la_1'}\T L_{\la_2'}$.
To define a character of $C_{\la_1'\la_2'}^{\mu'}$ we assume
$\la_1(d)=\la_2(d)=\mu(d)=0$ (note that for any $\la'\in P_k^{'+}$ there exists
$\bar\la\in P_k^+$ such that $\bar\la(d)=0$ and $\bar\la|_{\h'}=\la'$).
Then we obtain a grading by the operator $d$ on $L_{\la_1'}\T L_{\la_2'}$. Set
$$c_{\la_1'\la_2'}^{\mu'}(q)=\ch_q C_{\la_1'\la_2'}^{\mu'}=
\Tr\ q^d|_{C_{\la_1'\la_2'}^{\mu'}}.$$
These functions are called $\g$ branching functions.
We note that in the conformal field theory branching functions appear as
characters of spaces of states of coset theories. These characters differs
from $c_{\la_1'\la_2'}^{\mu'}(q)$ by an extra factor
$q^{\triangle_{\la_1'}+\triangle_{\la_2'}-\triangle_{\mu'}}$, where
$\triangle_{\la'}$ is a conformal weight of $\la'$ (see \cite{DMS}).
There exist different approaches to the study of $c_{\la_1'\la_2'}^{\mu'}(q)$
(see for example \cite{BNY, KMQ, R, DJKMO, S1, S2, SS, F, FOW}).
These approaches give
different types formulas for some particular cases of branching functions.
In our paper we use homological technique to derive two bosonic formulas for
$c_{\la_1'\la_2'}^{\mu'}(q)$ for general affine Kac-Moody algebras
(note that similar approach is utilized in \cite{FFJMT, FF, F}).
Let us briefly describe our results. Recall the Garland-Lepowsky theorem:
\begin{equation}
\label{G-L}
H_p(\n_-,L_{\mu})\simeq\bigoplus_{\substack{w\in W\\ l(w)=p}}\C_{w*\mu},
\end{equation}
where $W$ is a Weyl group of $\g$, $w*\mu$ is a shifted action of W and $l(w)$
is the length of $w$.
Note that $(\ref{G-L})$ is an isomorphism of $\h$-modules and $\C_{w*\mu}$ is
one-dimensional $\h$-module of the weight $w*\mu$.
From (\ref{G-L}) we obtain that for $\mu,\nu\in P_k^+$ homology
$H_p(\n_-,L_{\mu})^\nu$ (superscript denotes the corresponding $\h$-weight
subspace) vanishes if $p>0$ or $\mu\ne\nu$. In addition
$$H_0(\n_-,L_{\mu})^\mu\simeq\C_{\mu}.$$
Therefore from (\ref{LL}) we obtain
$$H_p(\n_-,L_{\la_1}\T L_{\la_2})^{\mu'}\simeq C_{\la_1'\la_2'}^{\mu'}
\delta_{0,p}$$
and so
\begin{equation}
\label{mu'}
\sum_{p\ge 0}(-1)^p \ch_q H_p(\n_-,L_{\la_1}\T
L_{\la_2})^{\mu'}= c_{\la_1'\la_2'}^{\mu'}(q).
\end{equation}
We now compute the same Euler characteristics using the BGG-resolution of
$L_{\la_1}$:
\begin{equation}
\ldots\to F_p\to\ldots\to F_0\to L_{\la_1}\to 0,
\end{equation}
where $F_p=\bigoplus_{l(w)=p} M_{w*\la_1}$ and
$M_{w*\la_1}$ is the corresponding Verma module.
Tensoring the BGG-resolution by $L_{\la_2}$ we obtain the $\U(\n_-)$-free
resolution of $L_{\la_1}\T L_{\la_2}$:
\begin{equation}
\label{GG}
\ldots\to F_p\T L_{\la_2}\to\ldots\to F_0\T L_{\la_2}\to
L_{\la_1}\T L_{\la_2}\to 0.
\end{equation}
Then the homology $H_p(\n_-,L_{\la_1}\T L_{\la_2})^{\mu'}$ can be counted as
homology of a complex
\begin{equation}
\ldots\to \left[\C\T_{\U(\n_-)} (F_p\T L_{\la_2})\right]^{\mu'}\to\ldots\to
\left[\C\T_{\U(\n_-)} (F_0\T L_{\la_2})\right]^{\mu'}\to 0.
\end{equation}
Therefore the Euler characteristics $(\ref{mu'})$ is given by the formula
\begin{equation}
\label{fhf}
\sum_{p\ge 0}(-1)^p \sum_{l(w)=p}q^{(w*\la_1)d}\ch_q
(L_{\la_2})^{(\mu-w*\la_1)'}.
\end{equation}
We thus obtain our first bosonic formula for $c_{\la_1'\la_2'}^{\mu'}(q)$.
To get the second formula we replace the "product" $L_{\la_1}\T
L_{\la_2}$ by the "fraction" $L_{\mu}\T L_{\la_1}^*$ and consider the
homology $H_p(\n_-,L_{\mu}\T L_{\la_1}^*)$, $\la_1\in P_{k_1}^+$,
$\mu\in P_{k_1+k_2}^+$.
We prove that
$$H_p(\n_-,L_{\mu}\T L_{\la_1}^*)^{\la_2'}=0 \text { for } p>0$$
and
$$H_0(\n_-,L_{\mu}\T L_{\la_1}^*)^{\la_2'}\simeq (C_{\la_1'\la_2'}^{\mu'})^*.$$
We thus obtain that
$$\sum_{p\ge 0} (-1)^p \ch_q H_p(\n_-,L_{\mu}\T L_{\la_1}^*)^{\la_2'}=
c_{\la_1'\la_2'}^{\mu'}(q^{-1}).$$
Using the BGG-resolution of $L_{\mu}$ we again rewrite this Euler
characteristics in terms of the characters of spaces
$(\C\T_{\U(\n_-)}(M_{w*\mu}\T L_{\la_1}^*))^{\la_2'}.$
This gives the following formula:
\begin{equation}
\label{shf}
c_{\la_1'\la_2'}^{\mu'}(q)=\sum_{p\ge 0}(-1)^p \sum_{l(w)=p}q^{-(w*\mu)(d)}
\ch_q (L_{\la_1})^{(w*\mu-\la_2)'}
\end{equation}
(recall that we assume $\la_1(d)=\la_2(d)=\mu(d)=0$).
The specialization of this formula to the simplest case
$\g=\slth$ gives the formula from \cite{BNY, KMQ, R} in the form of \cite{F}.
We note that $(\ref{shf})$ looks like $(\ref{fhf})$, but the proof is
much more complicated.
Our paper is organized as follows.
In Section $1$ we fix affine Kac-Moody Lie algebras notations.
In Section $2$ we derive our first formula for branching functions
$c_{\la_1'\la_2'}^{\mu'}(q)$ using the homology
$H_p(\n_-,L_{\la_1}\T L_{\la_2})^{\mu'}.$
In Section $3$ we derive our second formula for $c_{\la_1'\la_2'}^{\mu'}(q)$
using the homology of the "fraction"
$H_p(\n_-,L_{\mu}\T L_{\la_1}^*)^{\la_2'}.$
In Section $4$ we specialize formulas from Sections $2$ and $3$ to the
simplest case $\g=\slth$.
{\it Acknowledgements.}\quad This work was partially supported by
RFBR Grant 06-01-00037 and LSS 4401.2006.2.
\section{Affine Kac-Moody Lie algebras}
In this section we fix our notations on the affine Kac-Moody Lie algebras.
The main references are \cite{Kac, Kum}.
Let $\g_{fin}$ be a simple finite-dimensional Lie algebra with the Cartan
decomposition $\g_{fin}=\n_{fin}\oplus\h_{fin}\oplus (\n_-)_{fin}.$
Consider the corresponding affine algebra
$$\g=\g_{fin}\T\C[t,t^{-1}]\oplus\C K\oplus\C d,$$
where $K$ is a central element and $[d,x\T t^i]=-ix\T t^i.$
We fix the Cartan decomposition $\g=\n\oplus\h\oplus\n_-,$
where
\begin{gather*}
\n=\n_{fin}\T 1\oplus\g_{fin}\T t\C[t],\\
\h=\h_{fin}\oplus\C K\oplus\C d, \\
\n_-=(\n_-)_{fin}\T 1\oplus\g_{fin}\T t^{-1}\C[t^{-1}]
\end{gather*}
and denote $\g'=[\g,\g]=\g_{fin}\T\C[t,t^{-1}]\oplus\C K,$
$\h'=\h_{fin}\oplus\C K\hk\g'$.
Let $\al_i^{\vee}\in \h,\ \al_i\in\h^*,\ i=1,\ldots,n$,
be simple coroots and roots. Note that $\al_i^{\vee}$ form a basis of
$\h_{fin}\oplus\C K.$
We denote by $\slt^{(i)}$ the $\slt$ Lie algebra spanned by $e_i,
\al_i^{\vee},f_i$, where $e_i,f_i,i=1,\ldots,n$ are the Chevalley
generators, $e_i\in\n, f_i\in\n_-.$
We note that
$$\n=\bigoplus_{\al\in \triangle_+}\g_{\al},\
\n_-=\bigoplus_{\al\in \triangle_-}\g_{\al},$$
where
$\triangle_+$ and $\triangle_-$ are the sets of positive and negative roots
and $\g_{\al}=\{x\in\g:[h,x]=\al(h)x\ \forall h\in\h\}.$
Spaces $\g_{\al_i}$ and $\g_{-\al_i}$ are spanned by $e_i$ and $f_i$.
Let
\begin{equation}
\label{u-}
\un_-^{(i)}=\bigoplus_{\substack{\al\in \triangle_-\\ \al\ne -\al_i}}\g_{\al}.
\end{equation}
Note that
$$\g_{-\al_i}\simeq\n_-/\un_-^{(i)}.$$
Let $P_k^+$ be the set of level $k$ integrable dominant $\g$-weights, i.e.
$$P_k^+=\{\la\in {\h}^*: \la(\al_i^{\vee})\in\Z_{\ge 0},\ \la(K)=k\}.$$
We also denote by $P_k^{'+}\hk {\h'}^*$ the image of $P_k^+$ with respect to
the projection $\h^*\to {\h'}^*$, $\la\mapsto\la'$.
For $\la\in P_k^+$ let $L_{\la}$ be an integrable
highest weight $\g$-module with highest weight vector $v_{\la}\in L_{\la}$
such that
$$\n v_{\la}=0,\ \U(\n_-)v_{\la}=L_{\la},\ h(v_{\la})=\la(h)v_{\la}, h\in\h.$$
Let $L_{\la'}$ be $\g'$ module which coincides with $L_\la$ as a vector space
and the action of $\g'$ is a restriction of the action of $\g$.
For any $\al\in {\h'}^*$ set
$$(L_{\la})^{\al}=\{v\in L_\la:\ hv=\al(h)v\ \forall h\in\h'\}.$$
Note that $L_\la$ is graded by an operator $d$. We set
$$\ch_q (L_{\la})^{\al}=\Tr q^d|_{(L_{\la})^{\al}}.$$
Fix $\la_1'\in P_{k_1}^{'+},\ \la_2'\in P_{k_2}^{'+},$
and consider the decomposition of the tensor product of $\g'$-modules:
\begin{equation}
\label{tp}
L_{\la_1'}\T L_{\la_2'}=
\bigoplus_{\mu'\in P_{k_1+k_2}^{'+}}
C_{\la_1'\la_2'}^{\mu'} \T L_{\mu'}.
\end{equation}
The space $C_{\la'_1\la'_2}^{\mu'}$
can be identified with a subspace of highest weight vectors of
$\h'$-weight $\mu'$ in $L_{\la_1'}\T L_{\la_2'}.$
To define a character of $C_{\la'_1\la'_2}^{\mu'}$ one needs to
fix an action of the operator $d$ on each $\g'$ module $L_{\la'}$.
Note that if $\la|_{\h'}=\bar\la|_{\h'}$ then $L_{\la'}\simeq L_{\bar\la'}$.
Therefore an action of $d$ on $L_{\la'}$ depends on the choice of $\la(d)$.
It is convenient for us to choose a normalization
$\la_1(d)=\la_2(d)=\mu(d)=0$.
This defines the characters of $L_{\la_1'}\T L_{\la_2'}$ and of
$C_{\la'_1\la'_2}^{\mu'}$. The character
$\ch_q C_{\la'_1\la'_2}^{\mu'}$ is called $\g$ branching function and
is denoted by $c_{\la_1'\la_2'}^{\mu'}(q)$:
$$c_{\la_1'\la_2'}^{\mu'}(q)=\Tr q^d|_{C_{\la_1'\la_2'}^{\mu'}}.$$
Recall that spaces $C_{\la_1' \la_2'}^{\mu'}$ appear in the conformal field theory
as spaces of states of coset theories (see \cite{DMS}).
Namely the Sugawara construction defines an action of the Virasoro algebra
with generators $L_n$ on each $L_{\la'}$.
In particular for the operator $L_0$ one has
$$L_0 v_{\la}=\triangle_{\la'} v_{\la},\ [L_0,x\T t^i]=-ix\T t^i$$
($\triangle_{\la'}$ is a conformal weight).
Now the GKO construction (see \cite{GKO}) defines an action of Vir on the
tensor product $L_{\la'_1}\T L_{\la'_2}$ which commutes with the diagonal
action of $\g'$.
Namely one puts
$$L_n^{GKO}=L_n^{(1)}\T \Id +\Id\T L_n^{(2)}-L_n^{diag},$$
where $L_n^{(1)},\ L_n^{(2)}$ and $L_n^{diag}$ are Sugawara operators acting
on $L_{\la_1'},\ L_{\la_2'}$ and $L_{\la_1'}\T L_{\la_2'}$ respectively.
Therefore, we obtain a structure of Vir-module on
$C_{\la_1'\la_2'}^{\mu'}$ and an equality
$$\Tr q^{L_0}|_{C_{\la_1'\la_2'}^{\mu'}}=c_{\la_1'\la_2'}^{\mu'}(q)
q^{\triangle_{\la_1'}+\triangle_{\la_2'}-\triangle_{\mu'}},$$
where the left hand side is a character of the space of states of the
corresponding coset model.
In the end of this section we recall the Weyl group notations,
the Garland-Lepowsky $\n_-$-homology
theorem and the BGG resolution.
Let $W$ be the Weyl group of $\g$, generated by simple reflections $s_i$.
We denote by $l(w)$ the length of an element $w\in W$.
Recall that the shifted action of $W$ on $\h^*$ is given by
$w*\la =w(\la+\rho)-\rho$, where $\rho(\al_i^{\vee})=1.$
We will need the following lemma:
\begin{lem}
\label{Wlem}
$a)$\ If $\la\in P^+_k$ and $w*\la\in P^+_k$ then $w=e$. \\
$b)$\ If $(w*\la)\al_i^\vee\le -1$ for some $\la\in P^+_k$ then
$l(s_iw)<l(w)$.
\end{lem}
\begin{proof}
To prove $a)$ we rewrite an equality $w*\la=\la_1$ as
$w(\la+\rho)=\la_1+\rho$. But
if $\la,\la_1\in P^+_k$ then
$$(\la+\rho)\al_i^\vee>0, \quad (\la_1+\rho)\al_i^\vee>0\quad \forall i.$$
Therefore both $\la+\rho$ and $\la_1+\rho$ are the elements of the Weyl dominant
chamber and so $\la=\la_1$. But from $w*\la=\la$ one gets $w=e$ (see
Lemma $3.2.5$ from \cite{Kum}).
We now prove $b)$. Note that $(w*\la)\al_i^\vee\le -1$ is equivalent to
$$(\la+\rho)(w^{-1} \al_i^\vee)\le 0.$$ This gives
$w^{-1} \al_i^\vee=\sum_{i=1}^n c_j \al_j^\vee$ with $c_j\le 0$. Therefore
$l(w^{-1}s_i)<l(w)$. Lemma is proved.
\end{proof}
The following theorem is proved in \cite{GL}:
\begin{theo*}
For any $\la\in P_k^+$ we have an isomorphism of $\h$-modules:
$$H_p(\n_-,L_\la)\simeq\bigoplus_{\substack{w\in W\\ l(w)=p}}\C_{w*\la},$$
where $\C_{w*\la}$ is one-dimensional $\h$-module of the weight $w*\la$.
\end{theo*}
We will also need the BGG resolution of integrable irreducible representations
$L_{\la}$ (see \cite{BGG, Kum}).
Namely there exists an exact sequence of $\g$-modules and $\g$-homomorphism
\begin{equation}
\label{BGGres}
\ldots\to\bigoplus_{\substack{w\in W\\l(w)=p}}M_{w*\la}
\to\ldots\to M_{\la}\to L_{\la}\to 0,
\end{equation}
where $M_{\mu}$ is the weight $\mu$ Verma module.
\section{First homological bosonic formula}
\begin{lem}
For any $\la_1\in P^+_{k_1}$, $\la_2\in P^+_{k_2}$ and
$\mu\in P^+_{k_1+k_2}$
we have:
\begin{gather*}
C_{\la_1'\la_2'}^{\mu'}\simeq H_0(\n_-, L_{\la_1}\T L_{\la_2})^{\mu'},\\
H_p(\n_-, L_{\la_1}\T L_{\la_2})^{\mu'}=0 \text{ for all } p>0.
\end{gather*}
\end{lem}
\begin{proof}
Recall that for any $\la\in P^+_k$ and $e\ne w\in W$ one has
$w*\la\notin P^+_k$ and so $(w*\la)'\notin P^{'+}_k$. Therefore, from
Garland-Lepowsky theorem we obtain that $H_p(\n_-, L_\la)^\mu=0$
unless $p=0$ and $\la=\mu$. Now our lemma follows from the decomposition
$(\ref{tp})$ and an equality
$$H_0(\n_-, L_{\la_1}\T L_{\la_2})^{\mu'}\simeq
\bigoplus_{\substack{\bar\mu\in P^+_k\\ \bar\mu|_{\h'}=\mu'}}
H_0(\n_-, L_{\la_1}\T L_{\la_2})^{\bar\mu}.$$
\end{proof}
\begin{cor}
\label{br}
For any $\la_1\in P^+_{k_1}$, $\la_2\in P^+_{k_2}$ and $\mu\in P^+_{k_1+k_2}$
one has
\begin{equation}
\label{ec}
c_{\la_1'\la_2'}^{\mu'}(q)=
\sum_{p\ge 0} (-1)^p \ch_q H_p(\n_-, L_{\la_1}\T L_{\la_2})^{\mu'}.
\end{equation}
\end{cor}
We now compute the Euler characteristics $(\ref{ec})$ using the
BGG-resolution of $L_{\la_1}$.
Tensoring $(\ref{BGGres})$ by $L_{\la_2}$ we obtain the $\U(\n_-)$-free
resolution of
$L_{\la_1}\T L_{\la_2}$. Therefore the following complex counts
$H_p(\n_-, L_{\la_1}\T L_{\la_2})$:
\begin{equation}
\label{coinv}
\ldots \to \C\T_{\U(\n_-)} (L_{\la_2}\T F_1) \to
\C\T_{\U(\n_-)} (L_{\la_2}\T F_0)\to 0,
\end{equation}
where $F_p=\bigoplus_{l(w)=p} M_{w*\la_1}$.
We can rewrite
($\ref{coinv}$) as
\begin{equation}
\label{cu}
\ldots \to (\C\T_{\U(\n_-)} F_1)\T L_{\la_2} \to
(\C\T_{\U(\n_-)} F_0)\T L_{\la_2}\to 0.
\end{equation}
\begin{lem}
\label{form}
\begin{multline*}
\sum_{p\ge 0} (-1)^p \ch_q H_p(\n_-, L_{\la_1}\T L_{\la_2})^{\mu'}=\\
\sum_{p\ge 0} (-1)^p \sum_{l(w)=p} q^{(w*\la_1)d}
\ch_q (L_{\la_2})^{(\mu-w*\la_1)'}.
\end{multline*}
\end{lem}
\begin{proof}
Recall that $F_p=\bigoplus_{l(w)=p} M_{w*\la_1}$. Therefore,
$$\C\T_{\U(\n_-)} F_p= \bigoplus_{l(w)=p} \C_{w*\la_1}.$$
Now our lemma follows from the equality of Euler characteristics of the complex
$(\ref{cu})$ and the right hand side of $(\ref{ec})$.
\end{proof}
\begin{prop}
\label{mainhom}
We have a bosonic formula for the branching functions:
\begin{equation}
\label{genhom}
c_{\la_1'\la_2'}^{\mu'}(q)=
\sum_{p\ge 0} (-1)^p \sum_{\substack{w\in W\\ l(w)=p}} q^{(w*\la_1)d}
\ch_q (L_{\la_2})^{(\mu-w*\la_1)'}
\end{equation}
\end{prop}
\begin{proof}
Follows from Corollary $\ref{br}$ and Lemma $\ref{form}$.
\end{proof}
\begin{rem}
\label{rem}
We can use the BGG resolution of $L_{\la_2}$ instead of
$L_{\la_1}$. This interchanges $\la_1$ and $\la_2$ in the right hand
side of $(\ref{genhom})$ and leads to another formula for branching
functions
$c_{\la_1'\la_2'}^{\mu'}(q).$
\end{rem}
\section{Second homological bosonic formula}
In this section we study homology $H_p(\n_-, L_\mu\T L_{\la_1}^*)$ replacing
the tensor product $L_{\la_1}\T L_{\la_2}$ from the previous section by the
"fraction" $L_\mu\T L_{\la_1}^*$. We note that though $L_\mu\T L_{\la_1}^*$
does not belong to the category $\EuScript{O}$ (the eigenvalues of the
operator $d$ are not bounded from below) it is still integrable. So we first
prove some statements about integrable representations.
Recall the definition $(\ref{u-})$ of the subalgebra $\un^{(i)}_-$.
\begin{lem}
\label{nu}
Let $M$ be an integrable $\g$ module. Then
$$H_n(\n_-, M)\simeq H_0(\g_{-\al_i}, H_n(\un^{(i)}_-, M))\oplus
H_1(\g_{-\al_i}, H_{n-1}(\un^{(i)}_-,M)).$$
\end{lem}
\begin{proof}
We consider the Hochschild-Serre spectral sequence associated with a pair
$\un^{(i)}_-\hk\n_-$.
Note that $\un^{(i)}_-$ is an ideal and
$\n_-/\un^{(i)}_-\simeq \g_{-\al_i}$. The second
term of this spectral sequence is given by
$$E^2_{p,q}=H_p(\g_{-\al_i}, H_q(\un^{(i)}_-,M)).$$
We prove our lemma by showing that $E^2_{p,q}=E^{\infty}_{p,q}$.
Because of the integrability condition $M$ is a direct sum of irreducible
finite-dimensional $\slt^{(i)}$ modules. Therefore the same is true for
$\Lambda^{q}(\un^{(i)}_-)\T M$ and also for $H_q(\un^{(i)}_-,M)$.
For any nonnegative integer $s$ we denote by $\pi_s$ an irreducible
$\slt$ module with highest weight $s$ ($\dim\pi_s=s+1$) and fix
highest and lowest weight vectors $v_s$ and $u_s$.
Let
$\pi_s\hk H_q(\un^{(i)}_-,M)$ be a direct summand and
$$\alpha_p\in \Lambda^p(\g_{-\al_i})\T \pi_s$$ be a chain
representing some class in
$H_p(\g_{-\al_i},\pi_s)\hk H_p(\g_{-\al_i}, H_q(\un^{(i)}_-,M)).$
We set
$$\al_0=v_s, \ \ \ \al_1=f_i\T u_s.$$
Let
$\be_p\in \Lambda^p(\g_{-\al_i})\T \Lambda^{q}(\un^{(i)}_-)\T M$
be the chains of the form
$$\be_0=x_0,\ \ \ \be_1=f_i\T x_1$$
which represent $\al_p$ (i.e. $x_0$ represents $v_s$ and $x_1$ represents $u_s$).
Now let $d_{\n_-}$ ($d_{\un^{(i)}_-}$) be the differential in the standard
complex for $H_n(\n_-,M)$ ($H_n(\un^{(i)}_-,M)$).
We state that $d_{\n_-} \be_p =0$. In fact, for
$p=0$ this just follows from $v_s\in H_q(\un^{(i)}_-, M)$. Now let
$p=1$. Then
$$d_{\n_-} \be_1= d_{\n_-} (f_i\T x_1).$$
We know that $d_{\un^{(i)}_-} x_1=0$ and $f_ix_1=0$ (because $x_1$ represents
the lowest weight vector). This gives $d_{\n_-} \be_1=0$.
But because of
$d_{\n_-}\be=0$ we obtain that differentials $d_2,d_3,\ldots$ in the
Hochschild-Serre spectral sequence
are trivial and $E^2_{p,q}=E^\infty_{p,q}$. Lemma is proved.
\end{proof}
\begin{cor}
\label{H0w}
Let $M$ be an integrable level $k$ $\g$ module. Then
$$H_0(\n_-,M)^\la=0 \text{ unless } \la\in P^+_k.$$
\end{cor}
\begin{proof}
Because of Lemma \ref{nu} we obtain
$$H_0(\n_-,M)\simeq H_0(\g_{-\al_i}, H_0(\un^{(i)}_-,M))$$
for all $i=1,\ldots,n$. We recall that $H_0(\g_{-\al_i},\pi_s)$ is
one-dimensional space of the $\al_i^\vee$-weight $s$. Therefore, because
$H_0(\un^{(i)}_-,M)$ is a direct sum of finite-dimensional $\slt^{(i)}$ modules, we
obtain $\lambda(\al_i^\vee)\in\Z_{\ge 0}$ for any weight $\lambda$ of
$H_0(\n,M)$. This gives $\la\in P_k^+$.
\end{proof}
\begin{prop}
\label{wla}
Let $M$ be an integrable level $k$ $\g$ module. Then\\
$a)$ $H_n(\n_-,M)^{w*\la}=0$ for $w\in W$, $\la\in P^+_k$ if $l(w)>n$.\\
$b)$ $H_n(\n_-,M)^{w*\la}\simeq H_{n-l(w)}(\n_-,M)^\la$ for any
$w\in W$, $\al\in P^+_k$ such that $l(w)\le n$.
\end{prop}
\begin{proof}
We prove $a)$ and $b)$ simultaneously using
$$H_n(\n_-,M)^\mu\simeq H_0(\g_{-\al_i}, H_n(\un^{(i)}_-,M))^\mu\oplus
H_1(\g_{-\al_i}, H_{n-1}(\un^{(i)}_-,M))^\mu$$
and the induction on $n$. The case $n=0$
follows from Corollary \ref{H0w}. Suppose our lemma is proved for $m<n$. We
assume $l(w)>0$ (otherwise $a)$ and $b)$ are trivial). Then there exists
$i$ such that
\begin{equation}
\label{notin}
(w*\la)\al_i^\vee \le -1.
\end{equation}
We have
$$H_n(\n_-,M)^{w*\la}\simeq H_0(\g_{-\al_i}, H_n(\un^{(i)}_-,M))^{w*\la}\oplus
H_1(\g_{-\al_i}, H_{n-1}(\un^{(i)}_-,M))^{w*\la}.$$
Because of the condition $(\ref{notin})$ we have
$$H_0(\g_{-\al_i}, H_n(\un^{(i)}_-,M))^{w*\la}=0$$
and therefore
\begin{multline}
\label{n}
H_n(\n,M)^{w*\la}\simeq H_1(\g_{-\al_i}, H_{n-1}(\un^{(i)}_-,M))^{w*\la}\simeq\\
H_0(\g_{-\al_i}, H_{n-1}(\un^{(i)}_-,M))^{(s_iw)*\la},
\end{multline}
because
$$H_1(\g_{-\al_i},\pi_s)^\al\simeq H_0(\g_{-\al_i},\pi_s)^{s_i*\al}$$
for any $\al$.
We also know that
\begin{multline*}
H_{n-1}(\n_-,M)^{(s_iw)*\la}\simeq
H_0(\g_{-\al_i}, H_{n-1}(\un^{(i)}_-,M))^{(s_iw)*\la}\oplus\\
H_1(\g_{-\al_i}, H_{n-2}(\un^{(i)}_-,M))^{(s_iw)*\la}
\end{multline*}
and because of $((s_iw)*\la)\al_i^\vee=-(w*\la)\al_i^\vee-2\ge -1$
(see ($\ref{notin}$))
we obtain
$$H_1(\g_{-\al_i}, H_{n-2}(\un,M))^{(s_iw)*\la}=0$$
(because $H_1(\g_{-\al_i}, \pi_s)^t=0$ for any $t\ge -1$).
Therefore
\begin{equation}
\label{n-1}
H_{n-1}(\n_-,M)^{(s_iw)*\la}\simeq
H_0(\g_{-\al_i}, H_{n-1}(\un^{(i)}_-,M))^{(s_iw)*\la}.
\end{equation}
From $(\ref{n})$ and $(\ref{n-1})$ we obtain
\begin{equation}
\label{reduce}
H_n(\n_-,M)^{w*\la}\simeq H_{n-1}(\n_-,M)^{(s_iw)*\la}.
\end{equation}
Note that because of $(\ref{notin})$ and Lemma $\ref{Wlem}$
$l(s_iw)=l(w)-1$.
Now suppose that $n<l(w)$. Then iterating
$(\ref{reduce})$ we obtain
$$H_n(\n_-,M)^{w*\la}\simeq H_0(\n_-,M)^{w'*\la}$$
for some $w'$ with $l(w')>0$. But this homology vanishes because of
Corollary $\ref{H0w}$. This gives $a)$. To obtain $b)$ one needs to iterate
$(\ref{reduce})$. Proposition is proved.
\end{proof}
Let $\omega:\g\to\g$ be the Chevalley involution defined by $e_i\to -f_i$,
$f_i\to -e_i$, $h\to -h$ ($h\in\h$).
For $\g$ module $V$ we denote by $V^\omega$ a $\g$ module which coincides
with $V$ as a vector space and the action of $\g$ is twisted by $\omega$.
\begin{lem}
\label{ng}
Let $M$ be some $\g$ module, $\la\in\h^*$. Then
$$H_n(\n_-, M)^\la\simeq H_n(\g,\h, M\T M_\la^\omega),$$
where $M_\la$ is the Verma module.
\end{lem}
\begin{proof}
We first rewrite
$$H_n(\n_-, M)^\la\simeq H_n(\bo_-,\h, M\T \C_{-\la}),$$
where $\C_{-\la}$ is one-dimensional $\bo_-=\n_-\oplus\h$-module with trivial
action of $\n_-$. Now our lemma follows from
$$\mathrm{Ind}_{\bo_-}^{\g} (M\T \C_{-\la})\simeq M\T M_\la^\omega.$$
\end{proof}
In what follows we study homology
$H_n(\n_-, L_\mu\T L_{\la_1}^*)^{\la_2}$ for the triple of weights
$\la_1\in P^+_{k_1}$, $\la_2\in P^+_{k_2}$, $\mu\in P^+_{k_1+k_2}$.
Note that $L_\mu\T L_{\la_1}^*$ is integrable $\g$ module.
Because of Lemma $\ref{ng}$ we have an isomorphism
$$H_n(\n_-, L_\mu\T L_{\la_1}^*)^{\la_2}\simeq
H_n(\g,\h, L_\mu\T L_{\la_1}^* \T M_{\la_2}^\omega).$$
\begin{prop}
\label{barE}
There exists a spectral sequence $\bar E^{r}_{p,q}$ with
$$\bar E^1_{p,q}=
\bigoplus_{w:\ l(w)=p} H_q(\n_-, L_\mu\T L_{\la_1}^*)^{w*\la_2},$$
such that
$\bar E^r_{p,q}$ converges to
$H_\bullet(\g,\h, L_\mu\T L_{\la_1}^* \T L_{\la_2}^*).$
In addition $\bar E^1_{p,q}=0$ for $p>q$.
\end{prop}
\begin{proof}
We first note that $\bar E^1_{p,q}=0$ for $p>q$ because of part $a)$ of
Proposition $\ref{wla}$.
Now
consider the BGG resolution
$$\ldots \to F_{\la}(2)\to F_{\la}(1)\to F_{\la}(0)\to L_{\la}\to 0,\
F_{\la}(p)=\bigoplus_{w:\ l(w)=p} M_{w*\la}.$$
Recall that for any $\la\in P^+_k$ $(L^*_{\la})^\omega\simeq L_\la$.
We thus obtain the dual BGG-resolution
\begin{equation*}
\ldots \to F^\omega_{\la}(2)\to F^\omega_{\la}(1)\to
F^\omega_{\la}(0)\to L^*_{\la}\to 0,\
F^\omega_{\la}(p)=\bigoplus_{w:\ l(w)=p} M^\omega_{w*\la}.
\end{equation*}
This gives the following resolution
\begin{equation}
\label{dBGG}
\ldots \to L_\mu\T L_{\la_1}^*\T F^\omega_{\la_2}(1)\to
L_\mu\T L_{\la_1}^*\T F^\omega_{\la_2}(0)\to
L_\mu\T L_{\la_1}^*\T L^*_{\la_2}\to 0.
\end{equation}
In order to establish a connection between
$$H_q(\g,\h, L_\mu\T L_{\la_1}^* \T L_{\la_2}^*)\text{ and }
H_q(\n_-, L_\mu\T L_{\la_1}^*)^{w*\la_2}$$
we use a bi-complex $K_{p,q}$ associated with the resolution $(\ref{dBGG})$:
$$K_{p,q}=
\left[ \Lambda^q(\g/\h)\T L_\mu\T L_{\la_1}^* \T F_{\la_2}^\omega(p)\right]
^0
$$
(the space of $\h$-invariants).
The first term of the corresponding spectral sequence is given by
$${\bar E}^1_{p,q}=
H_q(\g, \h, L_\mu\T L_{\la_1}^* \T F_{\la_2}^\omega(p))$$
and ${\bar E}^r_{p,q}$ converges to
$H_n(\g,\h, L_\mu\T L_{\la_1}^* \T L_{\la_2}^*).$
From the definition of $F_{\la_2}^\omega(p)$ and Lemma $\ref{ng}$ we obtain
$$\bar E^1_{p,q}=
\bigoplus_{w:\ l(w)=p} H_q(\n_-, L_\mu\T L_{\la_1}^*)^{w*\la_2}.$$
Proposition is proved.
\end{proof}
\begin{cor}
\label{H0}
$H_0(\n_-, L_\mu\T L_{\la_1}^*)^{\la_2'}\simeq (C_{\la_1'\la_2'}^{\mu'})^*$.
\end{cor}
\begin{proof}
Note that $H_0(\n_-, L_\mu\T L_{\la_1}^*)^{\la_2'}$ is isomorphic to
$$\bigoplus_{\substack{\bar \la_2\in P_k^+\\ \bar \la_2|_{\h'}=\la_2'}}
H_0(\n_-, L_\mu\T L_{\la_1}^*)^{\bar \la_2}.$$
Fix some $\bar\la_2$ with $\bar \la_2|_{\h'}=\la_2'$.
Then using the spectral sequence from the Proposition $\ref{barE}$ we obtain
$$H_0(\n_-, L_\mu\T L_{\la_1}^*)^{\bar\la_2}=\bar E^1_{0,0}$$
and $\bar E^1_{n,0}=0$ for all $n>0$. Therefore
$\lim_{r\to\infty} \bar E^r_{0,0}=\bar E^1_{0,0}$.
In addition
$$\bar E^\infty_{0,0}\simeq H_0(\g,\h, L_\mu\T L_{\la_1}^* \T L_{\bar\la_2}^*)
\simeq H^0(\g,\h, L_\mu^*\T L_{\la_1} \T L_{\bar \la_2})^*.$$
Now our Corollary follows from
$$H^0(\g,\h, L_\mu^*\T L_{\la_1} \T L_{\bar\la_2})^*\simeq
\mathrm{Hom}_{\g} (L_\mu, L_{\la_1} \T L_{\bar\la_2})^*$$ and
$$\bigoplus_{\substack{\bar \la_2\in P_k^+\\ \bar \la_2|_{\h'}=\la_2'}}
H^0(\g,\h, L_\mu^*\T L_{\la_1} \T L_{\bar\la_2})^*\simeq
\mathrm{Hom}_{\g'} (L_{\mu'}, L_{\la_1'} \T L_{{\bar\la_2}'})^*.$$
\end{proof}
We now study the special case of $H_n(\n_-, L_\mu\T L_{\la_1}^*)^{\la_2}$
with $\la_2=0$ and $\mu=\la_1$.
\begin{lem}
\label{mumu*}
$H_n(\n_-,L_\mu\T L_\mu^*)^0=0$ for $n>0$.
\end{lem}
\begin{proof}
We consider a filtration $(L_\mu^*)_m$ on $L_\mu^*$:
$$(L_\mu^*)_m=\mathrm{span}\bra e_{i_1}\ldots e_{i_s} v_\mu^*,\ s\le m,
1\le i_l\le n \ket,$$
where $v_\mu^*$ is a lowest weight vector of $L_\mu^*$.
This induces a filtration
$$(\left[\Lambda^n(\n_-)\T L_\mu\T L_\mu^*\right]^0)_m =
\left[\Lambda^n(\n_-)\T L_\mu\T (L_\mu^*)_m\right]^0.$$
For the associated spectral sequence one has
$$E^1_{n,m}=\left[ H_{n+m}(\n_-,L_\mu)\T (L_\mu^*)_m/(L_\mu^*)_{m-1}\right]^0.$$
Because of
$H_{n+m}(\n_-,L_\mu)\simeq\bigoplus_{w:\ l(w)=n+m} \C_{w*\mu}$ we obtain
$$E^1_{n,m}=\bigoplus_{w:\ l(w)=n+m}
\left[ (L_\mu^*)_m/(L_\mu^*)_{m-1}\right]^{-w*\mu}.$$
But
$$(L_\mu^*)^{-w*\mu}\simeq (L_\mu^*)^{-w\mu+\rho-w\rho}\simeq
(L_\mu^*)^{-\mu+w^{-1}\rho-\rho}=0$$
because $w^{-1}\rho-\rho < 0$ for $l(w)>0$. This gives
$E^1_{n,m}=0$ for $n+m\ne 0$. Lemma is proved.
\end{proof}
In the following Lemma we calculate
homology $H_n(\g,\h, L_\mu\T L_\nu^*)$ for two weights
$\mu,\nu\in P^+_{k_1+k_2}$.
\begin{lem}
\label{munu}
Let $\mu,\nu\in P^+_{k_1+k_2}$. Then
$$\dim H_{2n}(\g,\h, L_\mu\T L_\nu^*)=\delta_{\mu,\nu} \#\{w\in W:\ l(w)=n\}.$$
In addition $H_{2n-1}(\g,\h, L_\mu\T L_\nu^*)=0$.
\end{lem}
\begin{proof}
Because of the isomorphism $(L_\nu^*)^\omega\simeq L_\nu$ the BGG resolution
gives the following resolution:
$$\ldots\to \bigoplus_{\substack{w_1,w_2\in W \\ l(w_1)+l(w_2)=p}}
M_{w_1*\mu}\T M_{w_2*\nu}^\omega\to\ldots \to
M_\mu\T M_\nu^\omega\to L_\mu\T L_\nu^*\to 0.
$$
This resolution is $(\g,\h)$-free. Therefore a complex
$G_\bullet$ with
$$G_p= \bigoplus_{\substack{w_1,w_2\in W\\ l(w_1)+l(w_2)=p}}
\left[\C\T_{U(\g/\h)} (M_{w_1*\mu}\T M_{w_2*\nu}^\omega) \right]^0$$
counts $H_p(\g,\h, L_\mu\T L_\nu^*)$.
Note that $G_p=\bigoplus_{\substack{l(w_1)+l(w_2)=p\\ w_1*\mu= w_2*\nu}} \C_0$.
In view of $\mu,\nu\in P^+_{k_1+k_2}$ the condition
$w_1*\mu=w_2* \nu$ is equivalent to $w_1=w_2$, $\nu=\mu$. Therefore
$G_{2n-1}=0$, $n\ge 1$ and
$\dim G_{2n}= \delta_{\mu\nu} \#\{w\in W:\ l(w)=n\}.$
Lemma is proved.
\end{proof}
\begin{prop}
\label{triv}
The natural map
$$\tau: H_n(\g,\h, L_\mu\T L_{\la_1}^*\T M_{\la_2}^\omega) \to
H_n(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_2}^*)$$
is trivial for $n>0$.
\end{prop}
\begin{proof}
Because of an isomorphism
$$H_n(\g,\h, L_\mu\T L_{\la_1}^*\T M_{\la_2}^\omega)\simeq
H_n(\bo_-,\h, L_\mu\T L_{\la_1}^*\T \C_{\la_2}^*)$$
it is enough to prove that the map
$$\tau': H_n(\bo_-,\h, L_\mu\T L_{\la_1}^*\T \C_{\la_2}^*)\to
H_n(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_2}^*)$$
is trivial.
Note that $\tau'$ comes from the natural embedding
\begin{equation}
\label{imath}
\imath:
\left[ \Lambda^n(\bo_-/\h)\T L_\mu\T L_{\la_1}^*\T \C_{\la_2}^*\right]^0\hk
\left[ \Lambda^n(\g/\h)\T L_\mu\T L_{\la_1}^*\T L_{\la_2}^*\right]^0.
\end{equation}
Therefore it suffices to show that any chain
$$c\in \left[ \Lambda^n(\bo_-/\h)\T L_\mu\T L_{\la_1}^*\T L_{\la_2}^*
\right]^0\hk
\left[ \Lambda^n(\g/\h)\T L_\mu\T L_{\la_1}^*\T L_{\la_2}^*\right]^0$$
defines a trivial class in $H_n(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_2}^*)$.
Because of Lemma $\ref{munu}$ it is enough to show that any chain from
$$\left[ \Lambda^n(\n_-)\T L_\mu\T L_\mu^*\right]^0\hk
\left[ \Lambda^n(\g/\h)\T L_\mu\T L_{\la_1}^*\T L_{\la_2}^*
\right]^0$$
defines a trivial element in $H_n(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_2}^*)$
for $n>0$. But from Lemma $\ref{mumu*}$ we know that
$$H_n(\n_-, L_\mu\T L_\mu^*)^0=0 \text{ for } n>0.$$
Proposition is proved.
\end{proof}
\begin{theo}
\label{main}
For any $\la_1\in P^+_{k_1}$, $\la_2\in P^+_{k_2}$, $\mu\in P^+_{k_1+k_2}$
and $n>0$
$$H_n(\n_-, L_\mu\T L_{\la_1}^*)^{\la_2}=0.$$
\end{theo}
\begin{proof}
We use the spectral sequence from Proposition $\ref{barE}$:
$$\bar E^1_{p,q}=\bigoplus_{w:\ l(w)=p}
H_q(\n_-, L_\mu\T L_{\la_1}^*)^{w*\la_2}\simeq
\bigoplus_{l(w)=p} H_q(\g,\h, L_\mu\T L_{\la_1}^*\T M^\omega_{w*\la_2}).$$
From Corollary $\ref{H0}$ we obtain $\bar E^1_{0,0}\simeq
(C_{\la'_1\la'_2}^{\mu'})^*$ and therefore
\begin{equation}
\label{E1kk}
\bar E^1_{k,k}\simeq\bigoplus_{w:\ l(w)=k} (C_{\la'_1\la'_2}^{\mu'})^*.
\end{equation}
In addition Propositions $\ref{wla}$ and $\ref{barE}$ gives
\begin{equation}
\label{know}
\bar E^1_{p,n+p}\simeq \bar E^1_{0,n},\
\bar E^1_{p,q}=0 \text{ for } p>q.
\end{equation}
Our goal is to show that $\bar E^1_{0,n}=0$, $n>0$.
(Because of $(\ref{know})$ this is
equivalent to the proof of $\bar E^1_{p,q}=0$ for $p\ne q$).
Note that this agrees with a fact that $\bar E^1_{k,k}$ is isomorphic to
$H_{2k}(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_1}^*)$ (see Lemma $\ref{munu}$ and
$(\ref{E1kk})$).
We prove the statement $\bar E^1_{0,2n-1}=0$, $\bar E^1_{0,2n}=0$ by
induction on $n\ge 1$. First let $n=1$. Because of Proposition $\ref{triv}$
the map $$\bar E^1_{0,2}\to H_2(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_1}^*)$$
is trivial and therefore $\bar E^\infty_{0,2}=0$. This gives
$\bar E^\infty_{1,1}=\bar E^1_{1,1}$, because
$$\bar E^1_{1,1}\simeq H_2(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_1}^*)\simeq
\bar E^\infty_{1,1}\oplus \bar E^\infty_{2,0}.$$
So a differential
$d_1: \bar E^1_{1,1}\to \bar E^1_{0,1}$ is trivial and
$$\bar E^1_{0,1}=\bar E^\infty_{0,1}=
H_1(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_1}^*)=0.$$
According to Proposition $\ref{wla}$ we obtain $\bar E^1_{1,2}=0$, which
gives
$$\bar E^1_{0,2}= \bar E^\infty_{0,2}= 0.$$
Now suppose $\bar E^1_{0,s}=0$ for $s\le 2(n-1)$. This gives
$\bar E^1_{p,s+p}=0$ for $s\le 2(n-1)$, $p\ge 0$. Note that the map
$$\bar E^1_{0,2n}\to H_{2n}(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_1}^*)$$
is trivial and so $\bar E^\infty_{0,2n}=0$. Recall that
the differential $d_r$ acts from $\bar E^r_{p,q}$ to $\bar E^r_{p-r,q+r-1}$.
This gives
$$\bar E^r_{p,q}=\bar E^1_{p,q} \text{ for } p+q\le 2n.$$
Because of
$$H_{2n}(\g,\h, L_\mu\T L_{\la_1}^*\T L_{\la_1}^*)\simeq \bar E^1_{n,n}$$
we have $\bar E^1_{n,n}=\bar E^\infty_{n,n}$. So
the differential $d_n: \bar E^n_{n,n}\to \bar E^n_{0,2n-1}$ is trivial.
Therefore
$$0=\bar E^\infty_{0,2n-1}=\bar E^n_{0,2n-1}=\bar E^1_{0,2n-1}$$
($\bar E^\infty_{0,2n-1}=0$ because by induction assumption we know that
$\bar E^\infty_{p,q}=0$ for $p+q=2n-1$, $(p,q)\ne (2n-1,0)$).
The equality $\bar E^1_{0,2n-1}=0$ gives $\bar E^1_{1,2n}=0$. Therefore,
$$0=\bar E^\infty_{0,2n}=\bar E^1_{0,2n}.$$
Theorem is proved.
\end{proof}
\begin{cor}
For any $\la_1\in P^+_{k_1}$, $\la_2\in P^+_{k_2}$, $\mu\in P^+_{k_1+k_2}$
and $n>0$
$$H_n(\n_-, L_\mu\T L_{\la_1}^*)^{\la_2'}=0.$$
\end{cor}
\begin{theo}
Let $\la_1\in P^+_{k_1}$, $\la_2\in P^+_{k_2}$, $\mu\in P^+_{k_1+k_2}$. Then
\begin{equation}
\label{mform}
c_{\la_1'\la_2'}^{\mu'}(q^{-1})
=\sum_{p\ge 0} (-1)^p \sum_{\substack{w\in W\\ l(w)=p}} q^{(w*\mu)d}
\ch_{q^{-1}} (L_{\la_1})^{(w*\mu-\la_2)'}.
\end{equation}
Another expressions for branching functions can be obtained by interchanging
$\la_1$ and $\la_2$ in the above expression.
\end{theo}
\begin{proof}
Consider homology $H_n(\n_-, L_\mu\T L_{\la_1}^*)^{\la'_2}$. Because of
Corollary $\ref{H0}$ and Theorem $\ref{main}$ we know that
$$\sum_{p\ge 0} (-1)^p \ch_q H_n(\n_-, L_\mu\T L_{\la_1}^*)^{\la'_2}=
c_{\la_1'\la_2'}^{\mu'}(q^{-1}).$$
Using the BGG-resolution of $L_\mu$ we obtain that the following complex
counts $H_n(\n_-, L_\mu\T L_{\la_1}^*)^{\la'_2}$:
\begin{equation}
\label{Eu}
\ldots\to D_2\to D_1\to D_0\to 0,\ D_p=\bigoplus_{w:\ l(w)=p}
(\C\T_{\U(\n_-)} (M_{w*\mu}\T L_{\la_1}^*))^{\la'_2}.
\end{equation}
We note that
$$(\C\T_{\U(\n_-)} (M_{w*\mu}\T L_{\la_1}^*))^{\la'_2}\simeq
(\C_{w*\mu}\T L_{\la_1}^*)^{\la'_2}$$
and so we have
$$\ch_q (\C\T_{\U(\n_-)} (M_{w*\mu}\T L_{\la_1}^*))^{\la'_2}=
q^{(w*\mu)d}\ch_q (L_{\la_1}^*)^{(-w*\mu + \la_2)'}.$$
Therefore the Euler characteristics of the complex $(\ref{Eu})$ is given by
$$\sum_{p\ge 0} (-1)^p \sum_{w:\ l(w)=p} q^{(w*\mu)d}
\ch_{q^{-1}} (L_{\la_1})^{(w*\mu-\la_2)'}.$$
But the Euler characteristics of $(\ref{Eu})$ coincides with the sum
$$\sum_{n\ge 0} (-1)^n \ch_q H_n(\n_-, L_\mu\T L_{\la_1}^*)^{\la_2'}.$$
Theorem is proved.
\end{proof}
\section{The $\slth$ case}
In this section we specialize formulas $(\ref{genhom})$ and $(\ref{mform})$
to the case $\g=\slth$.
Let $h$ be the standard generator of the Cartan subalgebra of $\slt$.
Then $\h$ is spanned by $h_0=h\T 1$, $K$ and $d$.
Define $(i,k,m)\in\h^*$ by
$$(i,k,m)h_0=i,\ (i,k,m)K=k, \ (i,k,m)d=m.$$
Let $(i,k)=(i,k,m)|_{\h'}$.
Note that
$$P_k^{'+}=\{(i,k)\in {\h'}^*: i,k\in \Z_{\ge 0}, i\le k\}.$$
We denote by $L_{i,k}$ ($0\le i\le k$) the highest weight irreducible
representation of $\slth'$ with highest weight $(i,k)$.
For fixed levels $k_1,k_2$ let $c_{i_1i_2}^j(q)$ be the corresponding
branching functions defined as
characters of $C_{i_1i_2}^j$:
$$L_{i_1,k_2}\T L_{i_2,k_2}=\bigoplus_{j=0}^{k_1+k_2} C_{i_1i_2}^j \T
L_{j,k_1+k_2}.$$
Recall that for any $s>0$ there exist two elements
$w_{s,1},w_{s,2}\in W$ with $l(w_{s,i})=s$. In addition
\begin{gather}
\label{W}
w_{2n,1} * (i,k,m)=(i+2n(k+2),k,m+n(n(k+2)+i+1)),\\ \nonumber
w_{2n,2} * (i,k,m)=(i-2n(k+2),k,m+n(n(k+2)-i-1)),\\ \nonumber
w_{2n-1,1} * (i,k,m)=(-i-2+2n(k+2),k,m+n(n(k+2)-i-1)),\\ \nonumber
w_{2n+1,2} * (i,k,m)=(-i-2-2n(k+2),k,m+n(n(k+2)+i+1)).
\end{gather}
Let $V^l$ be the eigenspace of the operator $h\in\slt$ with
an eigenvalue $l$.
\begin{prop}
\begin{equation}
c_{i_1i_2}^j(q)=
\sum_{p\in\Z} q^{p^2(k_1+2)+p(i_1+1)}
\left(\ch_q L_{i_2,k_2}^{2p(k_1+2)-j+i_1}-\ch_q L_{i_2,k_2}^{2p(k_1+2)+j+i_1+2}
\right).
\end{equation}
\end{prop}
\begin{proof}
Follows from $(\ref{genhom})$ and $(\ref{W})$.
\end{proof}
\begin{prop}
\begin{multline}
\label{second}
c_{i_1i_2}^j(q)=\\
\sum_{p\in\Z}
q^{-(k_1+k_2+2)p^2-(j+1)p}\bigl(\ch_q L_{i_1,k_1}^{2(k_1+k_2+2)p+j-i_2}-
\ch_q L_{i_1,k_1}^{2(k_1+k_2+2)p+j+i_2+2}\bigr).
\end{multline}
\end{prop}
\begin{proof}
Follows from $(\ref{mform})$ and $(\ref{W})$.
\end{proof}
\begin{rem}
Note that formula $(\ref{second})$ coincides with bosonic formula from \cite{F}.
\end{rem}
\newcounter{a}
\setcounter{a}{1}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 789 |
Stanley Electricity Rates
Commercial Electricity in Stanley
^ The average commercial electricity rate in Stanley, ID is 6.8¢/kWh.[1]
Residential Electricity in Stanley
^ The average residential electricity rate in Stanley, ID is 7.96¢/kWh.[1]
Industrial Electricity in Stanley
^ The average industrial electricity rate in Stanley, ID is 3.36¢/kWh.[1]
Stanley, ID Electricity Statistics
Commercial electricity rates in Stanley
The average commercial electricity rate in Stanley is 6.8¢/kWh.[1]
This average (commercial) electricity rate in Stanley is 0.87% less than the Idaho average rate of 6.86¢/kWh.[2]
The average (commercial) electricity rate in Stanley is 32.61% less than the national average rate of 10.09¢/kWh. Commercial rates in the U.S. range from 6.86¢/kWh to 34.88¢/kWh.[2]
Residential electricity rates in Stanley
The average residential electricity rate in Stanley is 7.96¢/kWh.[1]
This average (residential) electricity rate in Stanley is 8.19% less than the Idaho average rate of 8.67¢/kWh.[2]
The average (residential) electricity rate in Stanley is 33% less than the national average rate of 11.88¢/kWh. Residential rates in the U.S. range from 8.37¢/kWh to 37.34¢/kWh.[2]
Industrial electricity rates in Stanley
The average industrial electricity rate in Stanley is 3.36¢/kWh.[1]
This average (industrial) electricity rate in Stanley is 38.69% less than the Idaho average rate of 5.48¢/kWh.[2]
The average (industrial) electricity rate in Stanley is 49.63% less than the national average rate of 6.67¢/kWh. Industrial rates in the U.S. range from 4.13¢/kWh to 30.82¢/kWh.[2]
Idaho Electricity Rates & Consumption
Commercial electricity in Idaho
Commercial electricity rates in ID [3]
The average commercial electricity rate in Idaho is 6.86¢/kWh, which ranks 51st in the nation and is 32.01% less than the national average rate of 10.09¢/kWh.[3]
Commercial electricity consumption in ID [3]
Commercial electricity consumption in Idaho averages 4,869 kWh/month, which ranks 42nd in the nation and is 21.95% less than the national average of 6,238 kWh/month.
Commercial electricity bills in ID [3]
The average monthly commercial electricity bill in Idaho is $334, which ranks 51st in the nation and is 46.9% less than the national average of $629.
Learn more about commercial electricity in Idaho
Residential electricity in Idaho
Residential electricity rates in Idaho [3]
The average residential electricity rate in Idaho is 8.67¢/kWh, which ranks 49th in the nation and is 27.02% less than the national average rate of 11.88¢/kWh.
Residential electricity consumption in ID [3]
Residential electricity consumption in Idaho averages 1,010 kWh/month, which ranks 19th in the nation and is 11.85% greater than the national average of 903 kWh/month.
Residential electricity bills in ID [3]
The average monthly residential electricity bill in Idaho is $88, which ranks 43rd in the nation and is 17.76% less than the national average of $107 per month.
Learn more about residential electricity in Idaho
Industrial electricity in Idaho
Industrial electricity rates in Idaho [3]
The average industrial electricity rate in Idaho is 5.48¢/kWh, which ranks 44th in the nation and is 17.84% less than the national average rate of 6.67¢/kWh.
Industrial electricity consumption in ID [3]
Industrial electricity consumption in Idaho averages 30,082 kWh/month, which ranks 50th in the nation and is 73.18% less than the national average of 112,158 kWh/month.
Industrial electricity bills in ID [3]
The average monthly industrial electricity bill in Idaho is $1,648, which ranks 50th in the nation and is 77.98% less than the national average of $7,483.
Learn more about industrial electricity in Idaho
Household Income in Stanley, ID
Stanley, ID Income Breakdown [4]
Income Summary for Stanley, ID
Electric bills: The average residential electricity bill in Idaho is about $88/month, ranking 43rd in the U.S. and 17.76% less than the national average of $107.[5]
Income: Stanley vs. Nation [4]
Additional information about Stanley
About Stanley
Stanley is a city located in Custer County in the state of Idaho, and has a population of approximately 63.[6]
More about Stanley utilities
For more information about Stanley electricity, or for additional resources regarding electricity & utilities in your state, visit the Energy Information Administration.
Seven southern states (AL, FL, GA, SC, TN, TX & VA) are often in the top ten in electricity consumption due to high cooling demand.
Additional Cities in ID
Bovill
Craigmont
Fernan Lake Village
Firth
Tensed
Local Electricity Rates in Stanley, ID | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,047 |
{"url":"https:\/\/worldbuilding.stackexchange.com\/questions\/28198\/what-kind-of-technology-would-be-better-than-nuclear-power-in-terms-of-energy-pr","text":"What kind of technology would be better than nuclear power in terms of energy production and destructive power? [closed]\n\nIn the year 2235 scientist discover a new type power that yields more energy than nuclear power and is more destructive than it as well. But what is it? I could make something up but I would like something based in reality.\n\nThe energy needs to be able produce large amounts of consumer safe & usable power, like nuclear power plants. But the energy itself is incredibly dangerous, destructive enough to reduce the state of Texas to nothing, the amount of radiation produced does not matter.\n\n\u2022 Can you describe more what you need from this energy type? As is, this would be idea generation to try to generate arbitrary sources of energy. How available should the energy be? How do you define \"destructive?\" Is this energy anyone can use, or do you need the combined efforts of a massive galatic civilization to build it? \u2013\u00a0Cort Ammon Oct 22 '15 at 21:24\n\u2022 How is the question to broad? \u2013\u00a0DohnJoe Oct 23 '15 at 2:02\n\u2022 What needs to be super-destructive? A cell phone battery? Or the power plant itself? Can the cell phones just operate off normal batteries (well, normal for 2235), or do they have to operate off some exotic type of matter? Any kind of localized power would have to be very stable, especially if it's capable of destroying Texas. An antimatter reaction as listed in several answers would work fine for a super-explosive power plant, but it's rather unlikely you'd put it in a motorcycle. \u2013\u00a0MichaelS Oct 23 '15 at 3:10\n\u2022 If the question is to be answered literally, its 'nothing' because if we could name something based in reality, it can't get discovered in 2235. If its something \"similar\" to reality, I can start from one end of the wikipedia pages on science and move to the other, and at each step of the way define something \"similar.\" Also, before you edited the post, it looked like you wanted a source of power, but from the edit, it looks like you want a way to store power, which is very different. Heinlein's \"Shipstone\" comes to mind. \u2013\u00a0Cort Ammon Oct 23 '15 at 3:12\n\u2022 @MichaelS I would want the energy to be able produce large amounts of consumer usable power, like nuclear power plants. But the energy itself is incredibly dangerous. \u2013\u00a0DohnJoe Oct 23 '15 at 4:10\n\nYour best bet for a new type of power is figuring out how to convert mass directly into energy. The only way we know how to do this currently is with antimatter, but then you run into the problem of storing the antimatter. It would be believable that in 200 years' time we figure out a way to convert mass into energy without needing to use antimatter to do it.\n\nSo how much better is direct conversion than nuclear power?\n\nAccording to a random source on the internet the bomb dropped on Hiroshima could have yielded 20 kilotons of explosion per kilogram of uranium if all of the uranium underwent fission.\n\nUsing $$E=mc^2$$, for $$m=1kg$$ we get $$E=8.987*10^{16}J$$. A 20 kiloton explosion is $$8.368*10^{13}J$$. So if you can turn the uranium directly into energy, you get at least 1000 times as much energy out of it.\n\nSeeing how everything that we know about in the universe is made of matter, you'll never1 have to worry about running out of fuel. Also, you can really ruin someone's day if you can suddenly turn even a small section of their spaceship's hull into energy.\n\nIf you're willing to go a little bit out on a limb, another possibility would be to make use of parallel universes. The explanation could be that we confirmed that at least some parallel universes exist, but these parallel universes have different rules of physics, making them uninhabitable. In one of these universes, conservation of energy as we know it simply doesn't apply. By expending a tremendous amount of energy, mankind was able to open some sort of connection to that universe. Then, using energy obtained from that other universe, the connection could be maintained indefinitely unless the equipment was destroyed. If you want it to be a potential weapon, you can just say that destabilizing the connection (by destroying the equipment, for instance) can be very bad.\n\nEven though this lets you have unlimited energy, it is not infinite \u2014 you can only retrieve energy at a certain rate and opening a larger connection requires more extra energy to maintain than it provides.\n\nOne nice thing about this route is that it's still somewhat plausible. The concept of parallel universes is widely known, so you only have to build a little bit on what people understand of them. For example, if your story reads as if someone from 2235 (right before the new power source starts being used) is trying to explain things to someone from our day:\n\nYou know the idea of parallel universes? Well, right before the turn of the century, science was able to confirm that they exist! Everyone started getting really excited \u2014 we could find some uninhabited parallel Earths and move people there! Earth's overpopulation problem was finally solved, and we wouldn't even have to spend decades or centuries on space ships to get people to their new homes!\n\nBut then came the bad news \u2014 they weren't like what we had imagined. Nothing was inhabitable. We sent an apple to one, and it just kind of fell apart into dust. We sent another apple to another, and that one melted. Turns out, physics plays a different game there.\n\nOne universe though, it was the jackpot. Still can't live there, but it's gonna make life a whole lot easier here \u2014 free energy. We've still got the saying \"there's no such thing as a free lunch\", but we're about to make that wrong. See, over here, energy's gotta be conserved. We've known about that for a long time now. Over there? Energy just pops into existence pretty much everywhere.\n\nAll we've gotta do is bring some of that energy over here. And that's what they're about to do. They're opening up the very first FEP (free energy plant) tomorrow. Flip that switch, and we've got free, clean energy flowing in for as long as we want it.\n\nThere's only one FEP for now, but they've already started building more. They're saying these new ones are going to give more energy too \u2014 they're learning more and more about how to bring the energy over, and they're putting that into the new plants to make them more efficient.\n\n1. Well, not for a really long time, at least.\n\n\u2022 That sounds interesting, but can you provided a method of doing so? \u2013\u00a0DohnJoe Oct 22 '15 at 22:56\n\u2022 @user2888833 The webcomic Schlock Mercenary uses this as the energy source for spaceships. The reactors are called \"annihilation plants\" with no attempt to explain how exactly they work. You're going to have to handwave it away at some point, so just pick a mechanism and don't worry about explaining the details. \u2013\u00a0Rob Watts Oct 23 '15 at 1:48\n\u2022 This idea was the basis of the classic Issac Asimov story, The Gods Themselves. \u2013\u00a0user243 Oct 24 '15 at 20:40\n\u2022 @JonofAllTrades I still haven't read that one, but based on the summary I found on Wikipedia I agree - my second idea is basically the same as the one in that story. \u2013\u00a0Rob Watts Oct 25 '15 at 2:44\n\nCouldn't you just use the old tried and true antimatter approach?\n\n\u2022 Meets the destructiveness criterion if you get enough of it together\n\u2022 But the amount of energy is directly proportional to the amount of antimatter you annihilate, so it could still work to power cars, airplanes, trains, etc.\n\u2022 While it could theoretically power phones and such, if it's already being generated at a power plant, you don't really need to change electricity. Just make it be antimatter explosions that generate the electricity rather than nuclear energy.\n\u2022 We already can make small amounts of antimatter, it's just that the cost to create it vastly outweighs the gain we would get from annihilating it, so that gives you a paragraph of technobabble to explain how we've learned to efficiently blah blah blah.\n\u2022 Could you give more information on antimatter? \u2013\u00a0DohnJoe Oct 22 '15 at 22:56\n\u2022 Antimatter is basically normal matter but with various properties flipped - protons with negative charge, electrons with positive charge, etc. When a particle and its antiparticle (a proton and an antiproton, an electron and a positron, etc) collide they release energy in the form of gamma rays - but it's almost pure energy conversion, not lost to heat or friction or anything like that. However, it's currently the most expensive stuff on Earth when it comes to producing it, so it would take a big breakthrough to get it cheaply in large quantities. \u2013\u00a0John Robinson Oct 22 '15 at 23:05\n\u2022 @user2888833 You could also do an Internet search for \"what is antimatter\"... \u2013\u00a0Frostfyre Oct 22 '15 at 23:16\n\nAntimatter is the one that comes quickly to mind (yes, like in Star Trek). As opposed to the puny amount of matter converted to energy in nuclear, in antimatter annihilations 100% of the matter is converted to energy, hence much more energy produced per gram of matter. When harnessed this would mean high yield for generators, but also immense destructive ability for weapons.\n\nIf you're unsure of what it is, the energy derived from antimatter comes from the collision and subsequent annihilation of a particle with its anti-particle (such an electron and a positron).\n\nSince we already know the ultimate answer is antimatter the real question is how do we get it? Making it in particle accelerators is amazingly inefficient (output isn't measured in kilograms or ounces but individual antiprotons).\n\nThe best answer so far seems to be to \"harvest\" antiprotons in the magnetosphere around Earth, or perhaps better yet, in deep space. The energy cost of getting to Jupiter can be easily recovered by the energy released by the antiprotons being harvested. NextBigFuture has posted on this idea:\n\nand Centauri Dreams:\n\nhttp:\/\/www.centauri-dreams.org\/?p=1569\n\nGiven the amazing energy release of antimatter-matter reactions (and the fact that antimatter reactions unfold even faster than nuclear reactions), the best place to utilize the antimatter harvested in space is in space. Harvesting antimatter and then trying to bring it down from orbit is just asking for trouble.\n\nAfter factoring the fact that a proton - antiproton reaction results in many different types of particles (mostly pions and kaons) and about 1\/3 of these hold a neutral charge - it means that the reaction loses at least 1\/3 of the reaction energy to waste heat. Meaning matter - antimatter may convert up to 2\/3 of its fuel mass into energy.\n\nWhile it is true that many fusion reactions lose substantial portions of their energy to liberated neutrons, there are a few aneutonic reactions that limit the amount of this energy loss. Fusion reactions may convert as much as 0.8% of its fuel mass into useful energy.\n\nFission reactions are less efficient and can only turn less than 0.2% of its rest mass into useful energy - much of it as heat.\n\nThere is a method of generating energy far more efficiently than any of these - extracting rotational energy from a black hole:\n\nhttps:\/\/physics.stackexchange.com\/questions\/20813\/how-would-a-black-hole-power-plant-work\n\nIn the example provided above, they're using the black hole essentially like a giant battery. It's continued use would eventually use up all the available rotational energy.\n\nAlternatively, dropping mass into a black hole releases almost as much potential energy as the rest mass of the object.\n\nAnti-matter is pretty wonderful. It converts to 100% energy when it meets with normal matter -- e=mcc. I have no idea how you'd manufacture it, store it safely (magnetism?), or release its power gradually. Maybe that's the scientific breakthrough!\n\n\u2022 Could you give more information on antimatter? \u2013\u00a0DohnJoe Oct 22 '15 at 22:56\n\u2022 I only know the basics of it: When it touches ordinary matter they both convert into pure energy. The Wikipedia page en.wikipedia.org\/wiki\/Antimatter will tell you a lot more than I could! \u2013\u00a0BrettFromLA Oct 23 '15 at 17:19","date":"2021-06-20 22:57:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 4, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.467462956905365, \"perplexity\": 745.456058477512}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623488257796.77\/warc\/CC-MAIN-20210620205203-20210620235203-00634.warc.gz\"}"} | null | null |
Enith Salón Marcuello (Montcada, 24 de setembre de 2001) és una futbolista valenciana, que juga com a portera al València CF. Forma part del planter del València des dels 11 anys, debutant al primer equip el 2019.
Referències
Futbolistes del València CF femení de la dècada de 2010
Futbolistes del València CF femení de la dècada de 2020
Montcadins
Futbolistes de l'Horta Nord
Futbolistes valencianes | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/datascience.stackexchange.com\/questions\/66640\/trying-to-return-more-than-just-the-top-result-from-sklearn-nearestneighbors\/66653","text":"# Trying to return more than just the top result from sklearn NearestNeighbors\n\nI'm trying to compare a list of names (duplicated into a clean file and a messy file). I then compare the files against each other. My problem is that it returns only the top 1 result for each, which is itself (the identical record in each file). What I am trying to capture is the second result, which would be the closest match, not being itself.\n\nnames = pd.read_csv('C:\/Temp\/messynames.txt', sep='\\t')\norg_names = names['VariationName'].unique()\nvectorizer = TfidfVectorizer(min_df=1, analyzer=ngrams)\ntf_idf_matrix = vectorizer.fit_transform(org_names)\n\norg_name_clean = clean_org_names['StandardName'].unique()\nvectorizer = TfidfVectorizer(min_df=1, analyzer=ngrams, lowercase=False)\ntfidf = vectorizer.fit_transform(org_name_clean)\n\nnbrs = NearestNeighbors(n_neighbors=3, n_jobs=-1).fit(tfidf)\nunique_org = set(names['VariationName'].values)\n\ndef getNearestN(query):\nqueryTFIDF_ = vectorizer.transform(query)\ndistances, indices = nbrs.kneighbors(queryTFIDF_)\nreturn distances, indices\ndistances, indices = getNearestN(unique_org)\n\nunique_org = list(unique_org) #need to convert back to a list\n\nmatches = []\nfor i,j in enumerate(indices):\ntemp = [round(distances[i][0],2), clean_org_names.values[j][0][0],unique_org[i]]\nmatches.append(temp)\n\nmatches = pd.DataFrame(matches, columns=['Match confidence (lower is better)','Matched name','Original name'])\nmatches.to_csv('C:\/Temp\/matchednames.txt', sep='\\t', encoding='utf-8', index=False, quoting=3)\n\n\nFor a file with the following four names:\n\nNOKIA\nNOKIAA\nNOKIA LMD\nNOKIA LTD\n\n\nThe results looks like this:\n\nMatch confidence Matched name Original name\n0 0.0 NOKIA LMD NOKIA LMD\n1 0.0 NOKIAA NOKIAA\n2 0.0 NOKIA NOKIA\n3 0.0 NOKIA LTD NOKIA LTD\n\n\nI'm trying to get to something more like:\n\nMatch confidence Matched name Original name\n0 0.1 NOKIA LTD NOKIA LMD\n1 0.1 NOKIA NOKIAA\n\n\nMy guess would be that in the following piece of code:\n\nfor i,j in enumerate(indices):\ntemp = [round(distances[i][0],2), clean_org_names.values[j][0][0],unique_org[i]]\nmatches.append(temp)\n\n\nThe variable (array element):\n\ndistances[i][0]\n\n\ncontains the top element match for the $$i^{th}$$ line. Replace the second array index so it becomes:\n\ndistances[i][1]\n\n\nYou may want to increase one of the two zero indices in the following line as well to have a distance like value:\n\nclean_org_names.values[j][0][0]","date":"2020-02-28 13:07:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.18630532920360565, \"perplexity\": 11112.668109894348}, \"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-2020-10\/segments\/1581875147154.70\/warc\/CC-MAIN-20200228104413-20200228134413-00160.warc.gz\"}"} | null | null |
/**
* @fileoverview Definitions for node's util module. Depends on the stream module.
* @see http://nodejs.org/api/util.html
* @see https://github.com/joyent/node/blob/master/lib/util.js
* @externs
* @author Daniel Wirtz <dcode@dcode.io>
*/
/**
BEGIN_NODE_INCLUDE
var util = require('util');
END_NODE_INCLUDE
*/
/**
* @type {Object.<string,*>}
*/
var util = {};
/**
* @param {string} format
* @param {...*} var_args
* @return {string}
* @nosideeffects
*/
util.format = function(format, var_args) {};
/**
* @param {string} string
*/
util.debug = function(string) {};
/**
* @param {...*} var_args
*/
util.error = function(var_args) {};
/**
* @param {...*} var_args
*/
util.puts = function(var_args) {};
/**
* @param {...*} var_args
*/
util.print = function(var_args) {};
/**
* @param {string} string
*/
util.log = function(string) {};
/**
* @param {*} object
* @param {boolean=} showHidden
* @param {number=} depth
* @param {boolean=} colors
* @return {string}
* @nosideeffects
*/
util.inspect = function(object, showHidden, depth, colors) {};
/**
* @param {*} object
* @return {boolean}
* @nosideeffects
*/
util.isArray = function(object) {};
/**
* @param {*} object
* @return {boolean}
* @nosideeffects
*/
util.isRegExp = function(object) {};
/**
* @param {*} object
* @return {boolean}
* @nosideeffects
*/
util.isDate = function(object) {};
/**
* @param {*} object
* @return {boolean}
* @nosideeffects
*/
util.isError = function(object) {};
/**
* @param {stream.ReadableStream} readableStream
* @param {stream.WritableStream} writableStream
* @param {function(...)=} callback
* @deprecated
*/
util.pump = function(readableStream, writableStream, callback) {};
/**
* @param {Function} constructor
* @param {Function} superConstructor
*/
util.inherits = function(constructor, superConstructor) {};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,183 |
{"url":"https:\/\/nigerianscholars.com\/past-questions\/general-paper\/question\/204011\/","text":"Home \u00bb \u00bb A man took a loan of NP at the rate of 4% per annum simple interest. If at the e...\n\n# A man took a loan of NP at the rate of 4% per annum simple interest. If at the e...\n\n### Question\n\nA man took a loan of NP at the rate of 4% per annum simple interest. If at the end of 5 year she paid back \u20a6720, find the value of P?\n\nA) \u20a6456\n\nB) \u20a6500\n\nC) \u20a6556\n\nD) \u20a6600\n\n### Explanation:\n\nAmount = Principal (P) + Interest (I).\n720 = P + I\n$$720 = P + \\cfrac{PRT}{100}$$\n72000 = 100P + PRT\n72000 = 100P + P $$\\times$$ 4 $$\\times$$ 5\n72000 = 100P + 20P\n72000 = 120P\nP = $$\\cfrac{72000}{120}$$ = 600\nP = \u20a6600\n\n## Dicussion (1)\n\n\u2022 Amount = Principal (P) + Interest (I).\n720 = P + I\n$$720 = P + \\cfrac{PRT}{100}$$\n72000 = 100P + PRT\n72000 = 100P + P $$\\times$$ 4 $$\\times$$ 5\n72000 = 100P + 20P\n72000 = 120P\nP = $$\\cfrac{72000}{120}$$ = 600\nP = \u20a6600","date":"2021-12-01 08:48:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.6222061514854431, \"perplexity\": 2888.671853914567}, \"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-2021-49\/segments\/1637964359976.94\/warc\/CC-MAIN-20211201083001-20211201113001-00271.warc.gz\"}"} | null | null |
<?xml version="1.0" encoding="UTF-8"?>
<project version="4">
<component name="ProjectModuleManager">
<modules>
<module fileurl="file://$PROJECT_DIR$/.idea/conjoint.iml" filepath="$PROJECT_DIR$/.idea/conjoint.iml" />
</modules>
</component>
</project> | {
"redpajama_set_name": "RedPajamaGithub"
} | 1,112 |
/**
* Adds an ellipsis or clips the text
* Options are "ellipsis" or "clip"
* @see http://www.css3.info/preview/text-overflow/
*/
=text-overflow(!type = "ellipsis")
{
text-overflow:!type;
-o-text-overflow:!type;
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,190 |
Q: Assigning values from Firestore to variables in swift I've been pulling my hair out for hours during the switch from Real-time Database to Firestore, I'm trying to figure out a way to assign fields from multiple documents (each document contains a photo and a caption) for display in-app, my code as of now looks like a 3-year-old just threw a tantrum on the keyboard but any help on how to do this would be greatly appreciated.
I've been through the documentation hundreds of times, and read basically all similar questions here on StackOverflow but nothings working.
P.S. I've not slept in over 36 hours over this.
db.collection("posts").addSnapshotListener { (querySnapshot, error) in
// get the data of all the documents into an array
var data = querySnapshot.docs.map(function (documentSnapshot) {
return documentSnapshot.data();
});
}
A: Here is a very simplified version of what you are trying to do. Once you have your data in hand, you can unwrap them individually, like in the example below, or map them to custom Swift objects. But it appears that your question is only about getting data from Firestore.
featuredAttractionsQuery.addSnapshotListener { (snapshot, error) in
guard let snapshot = snapshot else { // unable to get snapshot
if let error = error {
print(error)
}
return // terminate query
}
guard !snapshot.isEmpty else { // snapshot is empty
print("snapshot is empty")
return // terminate query
}
for doc in snapshot.documents { // iterate through documents
guard let caption = doc.get("caption") as? String,
let imagePath = doc.get("imagePath") as? String else {
continue // if we can't get these two values,
// move to next iteration and continue loop
// calling return here would exit the function
}
// do something with this document's data
// most likely you'll parse it into a native object
// and add it to an array
}
// the loop is complete, reload the table view or collection view
}
There is a lot more that goes into this, such as using dispatch queues to parse your data in the background and dispatch groups to handle image downloads which return asynchronously. But this is the basic starting point that virtually all parsing is founded on.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,223 |
<?php
# 1 - home, 2 - author ,3 - topics,4 - quotes of the day,5 - pictures,6
namespace AppBundle\Controller;
use Sensio\Bundle\FrameworkExtraBundle\Configuration\Route;
use Symfony\Bundle\FrameworkBundle\Controller\Controller;
use Symfony\Component\HttpFoundation\Request;
use Symfony\Component\Validator\Constraints\DateTime;
use AppBundle\Entity\Topicshits;
use AppBundle\Entity\Authorshits;
use AppBundle\Entity\Nationalityhits;
class DefaultController extends Controller
{
/**
* @Route("/", name="homepage")
*/
public function indexAction(Request $request)
{
$em = $this->getDoctrine()->getManager();
$topics = $this->getDoctrine()
->getRepository('AppBundle:Topics')
->findByTopEighteen();
$authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByTopEighteen();
$homepic = $this->getDoctrine()
->getRepository('AppBundle:Quotes')
->findByPicHome(2);
$date = new \DateTime();
# todo: rank-aar ne shvvj xaryylax
$query = $em->createQuery(
"SELECT p
FROM AppBundle:Authors p
WHERE p.born LIKE :date"
)->setParameter('date', '%'.$date->format('-m-d') );
$birthdays = $query->setMaxResults(5)->getResult();
// replace this example code with whatever you need
return $this->render('default/index.html.twig', array(
'topics' => $topics,
'authors' => $authors,
'birthdays' => $birthdays,
'homepic' => $homepic,
'menu' => '1'
));
}
/**
* @Route("/quotes/topics.{_format}", name="topicspage")
*/
public function topicsAction(Request $request)
{
$topics = $this->getDoctrine()
->getRepository('AppBundle:Topics')
->findAll();
if (!$topics) {
throw $this->createNotFoundException('The product does not exist');
}
// replace this example code with whatever you need
return $this->render('default/topics.html.twig', array(
'topics' => $topics,
'menu' => '3'
));
}
/**
* @Route("/quotes/topics/{slug}{page}.{_format}",
* name="onetopicpage",
* defaults={"page": "1"},
* requirements={"page": "\d+"}
* )
*/
public function onetopicAction(Request $request, $slug, $page)
{
$topics = $this->getDoctrine()
->getRepository('AppBundle:Topics')
->findAll();
$topic = $this->getDoctrine()
->getRepository('AppBundle:Topics')
->findOneBy(array('slug' => $slug, ));
if (!$topic) {
throw $this->createNotFoundException('The topic does not exist');
}
if (!$topics) {
throw $this->createNotFoundException('The topics does not exist');
}
# get ip address
$ip = $request->getClientIp();
$mycount = $this->getDoctrine()
->getRepository('AppBundle:Topicshits')
->findOneBy(array('topic' => $topic, 'ip' => $ip));
$topicshits = new Topicshits();
if($mycount){
# valid record;
} else {
$topicshits->setIp($ip);
$topicshits->setCreateAt();
$topicshits->setTopic($topic);
$em = $this->getDoctrine()->getManager();
$em->persist($topicshits);
$topic->setHits($topic->getHits() + 1);
$em->persist($topic);
$em->flush();
}
$slidequotes = $this->getDoctrine()
->getRepository('AppBundle:Quotes')
->findBySlideTopicsHome($topic->getName());
$quotes = $this->getDoctrine()
->getRepository('AppBundle:Quotes')
->findByTopic($topic->getName());
$paginator = $this->get('knp_paginator');
$pagination = $paginator->paginate(
$quotes, /* query NOT result */
$request->query->getInt('page', 1)/*page number*/,
27/*limit per page*/
);
$rpic_num = rand( 1, 88);
// replace this example code with whatever you need
return $this->render('default/onetopic.html.twig', array(
'topics' => $topics,
'topic' => $topic,
'pagination' => $pagination,
'slidequotes'=> $slidequotes,
'rpic_num' => $rpic_num,
'menu' => '3'
));
}
/**
* @Route("/quotes/keywords/{slug}{page}.{_format}",
* name="onekeywordpage",
* defaults={"page": "1"},
* requirements={"page": "\d+", "slug": "[a-z]+"}
* )
*/
public function onekeywordAction(Request $request, $slug, $page)
{
$topics = $this->getDoctrine()
->getRepository('AppBundle:Topics')
->findAll();
if (!$topics) {
throw $this->createNotFoundException('The topics does not exist');
}
$pageshow = 27;
$quotes = $this->getDoctrine()
->getRepository('AppBundle:Quotes')
->findByKeyword($slug);
$slidequotes = $this->getDoctrine()
->getRepository('AppBundle:Quotes')
->findBySlideTopicsHome($slug);
// to get just one result:
// $product = $query->setMaxResults(1)->getOneOrNullResult();
$quotes = $this->getDoctrine()
->getRepository('AppBundle:Quotes')
->findByKeyword($slug);
$paginator = $this->get('knp_paginator');
$pagination = $paginator->paginate(
$quotes, /* query NOT result */
$request->query->getInt('page', 1)/*page number*/,
27/*limit per page*/
);
$rpic_num = rand( 1, 88);
// replace this example code with whatever you need
return $this->render('default/onekeyword.html.twig', array(
'topics' => $topics,
'keyword' => $slug,
'pagination' => $pagination,
'slidequotes'=> $slidequotes,
'rpic_num' => $rpic_num,
'menu' => ''
));
}
/**
* @Route("/profession.{_format}", name="professionpage")
*/
public function professionAction(Request $request)
{
$profession = $this->getDoctrine()
->getRepository('AppBundle:Profession')
->findAll();
if (!$profession) {
throw $this->createNotFoundException('The product does not exist');
}
// replace this example code with whatever you need
return $this->render('default/profession.html.twig', array(
'profession' => $profession,
'menu' => ''
));
}
/**
* @Route("/profession/{slug}_quotes.{_format}", name="oneprofessionpage")
*/
public function oneprofessionAction(Request $request, $slug)
{
$profession = $this->getDoctrine()
->getRepository('AppBundle:Profession')
->findOneBy(array('slug' => $slug, ));
if (!$profession) {
throw $this->createNotFoundException('The product does not exist');
}
$authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findBy(array('profession' => $profession, ));
// replace this example code with whatever you need
return $this->render('default/oneprofession.html.twig', array(
'profession' => $profession,
'authors' => $authors,
'menu' => ''
));
}
/**
* @Route("/birthdays.{_format}", name="birthdayspage")
*/
public function birthdaysAction(Request $request)
{
$cal_days = array(
array('month' => 'January',
'lday' => cal_days_in_month(CAL_GREGORIAN, 1, date('Y')) ),
array('month' => 'February',
'lday' => cal_days_in_month(CAL_GREGORIAN, 2, date('Y')) ),
array('month' => 'March' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 3, date('Y')) ),
array('month' => 'April' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 4, date('Y')) ),
array('month' => 'May' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 5, date('Y')) ),
array('month' => 'June' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 6, date('Y')) ),
array('month' => 'July' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 7, date('Y')) ),
array('month' => 'August' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 8, date('Y')) ),
array('month' => 'September' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 9, date('Y')) ),
array('month' => 'October' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 10, date('Y')) ),
array('month' => 'November' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 11, date('Y')) ),
array('month' => 'December' ,
'lday' => cal_days_in_month(CAL_GREGORIAN, 12, date('Y')) )
);
// replace this example code with whatever you need
return $this->render('default/birthdays.html.twig', array(
'cal_days' => $cal_days,
'menu' => ''
));
}
/**
* @Route("/birthdays/{date}.{_format}", name="birthdaysonpage")
*/
public function birthdaysonAction(Request $request, $date)
{
$em = $this->getDoctrine()->getManager();
# declare Months list
$mons = array(
"January" => "01",
"February" => "02",
"March" => "03",
"April" => "04",
"May" => "05",
"June" => "06",
"July" => "07",
"August" => "08",
"September" => "09",
"October" => "10",
"November" => "11",
"December" => "12"
);
// split get month and day
list($month, $day) = split('[_.-]', $date);
$searchbirthday = "%-".$mons[$month]."-".str_pad($day,2,"0",STR_PAD_LEFT);
$query = $em->createQuery(
"SELECT p
FROM AppBundle:Authors p
WHERE p.born LIKE :date"
)->setParameter('date', $searchbirthday );
$authors = $query->getResult();
// to get just one result:
// $product = $query->setMaxResults(1)->getOneOrNullResult();
// replace this example code with whatever you need
return $this->render('default/birthdayson.html.twig', array(
'date' => $date,
'authors' => $authors,
'menu' => ''
));
}
/**
* @Route("/quotes/authors/{slug}{page}.{_format}",
* name="quotesbyauthor",
* defaults={"page": "1"},
* requirements={"page": "\d+"})
*/
public function quotesbyauthorAction(Request $request, $slug, $page)
{
$author = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findOneBy(array('slug' => $slug));
$topics = $this->getDoctrine()
->getRepository('AppBundle:Topics')
->findAll();
if (!$author) {
throw $this->createNotFoundException('The author does not exist');
}
if (!$topics) {
throw $this->createNotFoundException('The topics does not exist');
}
# get ip address
$ip = $request->getClientIp();
$mycount = $this->getDoctrine()
->getRepository('AppBundle:Authorshits')
->findOneBy(array('author' => $author, 'ip' => $ip));
$authorshits = new Authorshits();
if($mycount){
# valid record;
} else {
$authorshits->setIp($ip);
$authorshits->setCreateAt();
$authorshits->setAuthor($author);
$em = $this->getDoctrine()->getManager();
$em->persist($authorshits);
$author->setHits($author->getHits() + 1);
$em->persist($author);
$em->flush();
}
$slidequotes = $this->getDoctrine()
->getRepository('AppBundle:Quotes')
->findBySlideAuthorHome($author);
# start pagenation
$authors = $this->getDoctrine()
->getRepository('AppBundle:Quotes')
->findBy(array('author' => $author));;
$paginator = $this->get('knp_paginator');
$pagination = $paginator->paginate(
$authors, /* query NOT result */
$request->query->getInt('page', 1)/*page number*/,
27/*limit per page*/
);
# end pagenation
$rpic_num = rand( 1, 88);
// replace this example code with whatever you need
return $this->render('default/quotesbyauthor.html.twig', array(
'topics' => $topics,
'author' => $author,
'pagination' => $pagination,
'slidequotes'=> $slidequotes,
'rpic_num' => $rpic_num,
'menu' => '2'
));
}
/**
* @Route("/nationality.{_format}",
* name="nationality")
*/
public function nationalityAction(Request $request)
{
$nationality = $this->getDoctrine()
->getRepository('AppBundle:Nationality')
->findBy(array(),array('name' => 'ASC' ));
return $this->render('default/nationality.html.twig', array(
'nationality' => $nationality,
'menu' => ''
));
}
/**
* @Route("/nationality/{slug}_quotes.{_format}",
* name="quotesbynationality")
*/
public function quotesbynationalityAction(Request $request, $slug)
{
$nationality = $this->getDoctrine()
->getRepository('AppBundle:Nationality')
->findOneBy(array('slug' => $slug, ));
if (!$nationality) {
throw $this->createNotFoundException('The nationality does not exist');
}
# get ip address
$ip = $request->getClientIp();
$mycount = $this->getDoctrine()
->getRepository('AppBundle:Nationalityhits')
->findOneBy(array('nationality' => $nationality, 'ip' => $ip));
$nationalityhits = new Nationalityhits();
if($mycount){
# valid record;
} else {
$nationalityhits->setIp($ip);
$nationalityhits->setCreatedAt();
$nationalityhits->setNationality($nationality);
$em = $this->getDoctrine()->getManager();
$em->persist($nationalityhits);
$nationality->setHits($nationality->getHits() + 1);
$em->persist($nationality);
$em->flush();
}
// get authors by nationality
$authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findBy(array('nationality' => $nationality, ));
return $this->render('default/quotesbynationality.html.twig', array(
'nationality' => $nationality,
'authors' => $authors,
'menu' => ''
));
}
/**
* @Route("/quotes_of_the_day_{page}.{_format}",
* name="quotesofday",
* defaults={"page": "1"},
* requirements={"page": "\d+"})
*/
public function quotesofdayAction(Request $request)
{
$nationality = $this->getDoctrine()
->getRepository('AppBundle:Nationality')
->findBy(array(),array('name' => 'ASC' ));
//lkkjhgfasdfghjkl
return $this->render('default/quotesofday.html.twig', array(
'nationality' => $nationality,
'menu' => '4'
));
}
/**
* @Route("/quotes/quote_pictures{page}.{_format}",
* name="quotespictures",
* defaults={"page": "1"},
* requirements={"page": "\d+"})
*/
public function quotespictureAction(Request $request)
{
$authors = $this->getDoctrine()
->getRepository('AppBundle:Quotes')
->findByPictures();
$paginator = $this->get('knp_paginator');
$pagination = $paginator->paginate(
$authors, /* query NOT result */
$request->query->getInt('page', 1)/*page number*/,
30/*limit per page*/
);
return $this->render('default/quotespicture.html.twig', array(
'pagination' => $pagination,
'menu' => '5'
));
}
/**
* @Route("/quotes/favorites.{_format}",
* name="favorites")
*/
public function favoritesAction(Request $request)
{
# todo: rank-aar ne shvvj xaryylax
$a_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('a', 26);
$b_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('b', 39);
$c_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('c', 31);
$d_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('d', 24);
$e_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('e', 14);
$f_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('f', 18);
$g_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('g', 18);
$h_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('h', 25);
$i_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('i', 2);
$j_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('j', 13);
$k_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('k', 19);
$l_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('l', 18);
$m_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('m', 40);
$n_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('n', 9);
$o_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('o', 7);
$p_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('p', 22);
$q_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('q', 1);
$r_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('r', 24);
$s_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('s', 37);
$t_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('t', 24);
$v_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('v', 6);
$w_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('w', 26);
$y_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('y', 3);
$z_authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findByLetter('z', 2);
return $this->render('default/favorites.html.twig', array(
'a_authors' => $a_authors,
'b_authors' => $b_authors,
'c_authors' => $c_authors,
'd_authors' => $d_authors,
'e_authors' => $e_authors,
'f_authors' => $f_authors,
'g_authors' => $g_authors,
'h_authors' => $h_authors,
'i_authors' => $i_authors,
'j_authors' => $j_authors,
'k_authors' => $k_authors,
'l_authors' => $l_authors,
'm_authors' => $m_authors,
'n_authors' => $n_authors,
'o_authors' => $o_authors,
'p_authors' => $p_authors,
'q_authors' => $q_authors,
'r_authors' => $r_authors,
's_authors' => $s_authors,
't_authors' => $t_authors,
'v_authors' => $v_authors,
'w_authors' => $w_authors,
'y_authors' => $y_authors,
'z_authors' => $z_authors,
'menu' => '2'
));
}
/**
* @Route("/authors/{char}",
* name="authors")
*/
public function authorsAction(Request $request, $char)
{
# todo: rank-aar ne shvvj xaryylax
$authors = $this->getDoctrine()
->getRepository('AppBundle:Authors')
->findBy(array('tick' => $char),array('name' => 'ASC' ));
$paginator = $this->get('knp_paginator');
$pagination = $paginator->paginate(
$authors, /* query NOT result */
$request->query->getInt('page', 1)/*page number*/,
26/*limit per page*/
);
return $this->render('default/authors.html.twig', array(
'char' => $char,
'pagination' => $pagination,
'menu' => ''
));
}
/**
* @Route("/about/{slug}",
* name="about")
*/
public function aboutAction(Request $request, $slug)
{
$currSlug = $this->getDoctrine()
->getRepository('AppBundle:Slug')
->findOneBy(array('slug' => $slug));
$abouts = $this->getDoctrine()
->getRepository('AppBundle:About')
->findBy(array('slug' => $currSlug));
#var_dump($about);die();
$slugs = $this->getDoctrine()
->getRepository('AppBundle:Slug')
->findAll();
return $this->render('default/about.html.twig', array(
'slugs' => $slugs,
'currSlug' => $currSlug,
'abouts' => $abouts,
'menu' => ''
));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,443 |
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN">
<html>
<head>
<title>STUI:Instruments:APOGEE Window</title>
</head>
<body>
<h2><a href="../index.html">STUI</a>:<a href="index.html">Instruments</a>:APOGEE Window</h2>
<p>The APOGEE window offers status and basic control of the APOGEE instrument. This window is primarily intended for status monitoring and engineering; for normal observing APOGEE should be controlled via the <a href="SOPWindow.html">SOP window</a>
<h3>Telemetry</h3>
<p>Shows the state of vacuum, liquid nitrogen and temperatures. Also shows array power if Off (typically it is ? or On). A summary is printed to the right of the Telemetry button. Press the Telemetry button to show or hide the details.
<h3>Shutter</h3>
<p>Status and controls for APOGEE's cold shutter and associated LEDs. Click the "Shutter" button to show/hide details.
<h3>Cal Box</h3>
<p>Status and controls for APOGEE's calibration box. Click "Cal Box" button to show/hide details, including the controls. Note that the summary string is always blue (as a warning) unless the calibration box is fully off (shutter closed and all lamps off).
<h3>Collimator</h3>
<p>The controls are usually hidden because the collimator is typically only moved during engineering. Press the Collimator button to show or hide the controls.
<p>You may apply relative offsets to the following (absolute moves are not supported by the APOGEE ICC):
<ul>
<li>Piston: positive values move the collimator closer to the instrument.
<li>Pitch: positive values tip the beam down.
<li>Yaw: positive values tip the beam to the right, as seen by the collimator.
</ul>
<p>To the right of the Collimator button is a summary field that shows one of the following:
<ul>
<li><b>OK</b>: the collimator controller is working and no limit switch has fired.
<li><b>Off</b> indicates that the APOGEE ICC cannot communicate with the collimator controller. This is a potentially serious condition.
<li><b>Limits rev1 fwd1 rev2 fwd2 rev3 fwd3</b> indicates at least one limit switch has fired or is in an unknown state. The values are the state of the reverse=home and forward limit switches for each actuator. The states are 0: not activated, 1: activated, ?: unknown. It is only shown if one or more switches is activated or unknown. If the reverse and forward switches are both activated for one actuator then the text is red because this indicates a serious failure.
</ul>
<h3>Dither</h3>
<p>Dither is typically toggled between A and B before each exposure. You may specify the dither as a named position or an explicit value in pixels (select Any in the pop-up menu). Note: if you specify a pixel value that matches A or B the pop-up menu will continue to say Any. This is intentional (it makes sure the right command is sent to the ICC).
<p>To the right of the current dither display (near the top of the window) is a summary field that shows one of the following:
<ul>
<li><b>OK</b>: the dither controller is working and no limit switch is pressed.
<li><b>Off</b> indicates that the APOGEE ICC cannot communicate with the dither controller. This is a very serious condition since one must change dither for most data collecting.
<li><b>Limits <i>rev</i> <i>fwd</i></b> indicates that one or both limit switches is pressed or is in an unknown state. Shows the state of the reverse=home and forward limit switches. The states are 0: not activated, 1: activated, ?: unknown. If both switches are activated the text is red because this indicates a serious failure.
</ul>
<h3>Exp Type</h3>
<p>Controls the exposure type. At the request of the APOGEE team you must reset it for each exposure.
<h3>Num Reads</h3>
<p>Controls the number of up-the-ramp reads. The associated exposure time is displayed to the right.
</body>
</html>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,683 |
Dr Boyce Watkins: How to build wealth by investing just $5 a day
Many people think that in order to build wealth, you have to already have it. They believe that the stock market is complicated, and that only the experts have what it takes to make it work in their favor.
But Dr Boyce Watkins has a plan that can help you and your family get ahead in ways you can't even imagine. In this video below, Dr Watkins explains that building wealth doesn't require more than $5 a day, and that there are numerous ways to build extraordinary amounts of capital, as long as you have a long-term plan. Watch the video below and tell us what you think. Is it possible for people to build wealth when they are poor, or is the situation hopeless?
Dr Boyce Watkins: How to build wealth by investing just $5 a day – Financial Juneteenth
Many people think that in order to build wealth, you have to already have it. They believe that the stock market is complicated, and… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,213 |
Turniej o Złoty Kask 1962 – rozegrany w sezonie 1962 turniej żużlowy z cyklu rozgrywek o "Złoty Kask". Wygrał Marian Kaiser, drugi był Florian Kapała i Joachim Maj stanął na najniższym stopniu.
Wyniki
Czołówka piątka
I turniej
5 maja 1962 r. (sobota), Rzeszów
II turniej
19 maja 1962 r. (sobota), Wrocław
III turniej
23 czerwca 1962 r. (sobota), Bydgoszcz
IV turniej
7 lipca 1962 r. (sobota), Gorzów Wielkopolski
V turniej
14 lipca 1962 r. (sobota), Rybnik
VI turniej
22 lipca 1962 r. (niedziela), Leszno
VII turniej
18 sierpnia 1962 r. (sobota), Częstochowa
VIII turniej
25 sierpnia 1962 r. (sobota), Gdańsk
Klasyfikacja końcowa
Uwaga!: Odliczono 2 najgorsze wyniki.
Bibliografia
Złoty Kask 1962
1962
1962 w sporcie żużlowym
1962 w polskim sporcie | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,297 |
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\section{Introduction}
\label{sec:intro}
The recent measurement of Single-Spin Asymmetries (SSA) in semi-inclusive $l
p^\uparrow \to l'\pi X$ Deep-Inelastic Scattering (SIDIS) on transversely
polarized hadronic
targets~\cite{Airapetian:2004tw,Diefenthaler:2005gx,Avakian:2005ps,Alexakhin:2005iw},
has renewed the interest in the problem of describing the spin structure of
hadrons within Quantum Chromo-Dynamics (QCD)~\cite{cerncourier}, and has
stimulated since then a large production of phenomenological and theoretical
papers. Experimental evidence of large SSA in hadron-hadron collisions was well
known since many years~\cite{Bunce:1976yb,Adams:1991cs}, but it has never been
consistently explained in the context of perturbative QCD in the collinear
massless approximation~\cite{Kane:1978nd}. The idea of going beyond the
collinear approximation opened new perspectives about the possibility of
explaining these SSA in terms of intrinsic transverse motion of partons inside
hadrons, and of correlations between such intrinsic transverse momenta and
transverse spin degrees of freedom. The most popular examples are the
Sivers~\cite{Sivers:1990cc} and the Collins~\cite{Collins:1993kk} effects. In
the former case, an asymmetric azimuthal distribution of detected hadrons (with
respect to the normal to the production plane) is obtained from the
nonperturbative correlation ${\bm p}_{\scriptscriptstyle T} \times {\bm P}\cdot {\bm S}_{\scriptscriptstyle T}$, where
${\bm p}_{\scriptscriptstyle T}$ is the intrinsic transverse momentum of an unpolarized parton
inside a target hadron with momentum ${\bm P}$ and transverse polarization
${\bm S}_{\scriptscriptstyle T}$. In the latter case, the asymmetry is obtained from the
correlation ${\bm k} \times {\bm P}_{h{\scriptscriptstyle T}} \cdot {\bm s}_{\scriptscriptstyle T}$, where a parton
with momentum ${\bm k}$ and transverse polarization ${\bm s}_{\scriptscriptstyle T}$ fragments into
an unpolarized hadron with transverse momentum ${\bm P}_{h{\scriptscriptstyle T}}$. In both cases,
the sizes of the effects are represented by new Transverse-Momentum Dependent
(TMD) partonic functions, the socalled Sivers and Collins functions,
respectively.
However, SSA data in hadronic collisions have been collected so far typically for
semi-inclusive $pp^{(\uparrow )}\to h^{(\uparrow )} X$ processes, where the
factorization proof is complicated by higher-twist correlators~\cite{Qiu:1991pp}
and the power-suppressed asymmetry can be produced by several (overlapping)
mechanisms. On the contrary, the theoretical situation of the SIDIS measurements
is more transparent. On the basis of a suitable factorization
theorem~\cite{Ji:2004wu,Collins:2004nx}, the cross section at leading twist
contains convolutions involving separately the Sivers and Collins functions with
different azimuthal dependences, $\sin (\phi - \phi_S)$ and $\sin (\phi +
\phi_S)$, respectively, where $\phi, \phi_S,$ are the azimuthal angles of the
produced hadron and of the target polarization with respect to the axis defined
by the virtual photon~\cite{Boer:1998nt}. According to the extracted azimuthal
dependence, the measured SSA can then be clearly related to one effect or the
other~\cite{Airapetian:2004tw,Diefenthaler:2005gx}.
Similarly, in the Drell-Yan process $H_1 H_2^\uparrow \to l^+ l^- X$ the cross
section displays at leading twist two terms weighted by $\sin (\phi - \phi_S)$
and $\sin (\phi + \phi_S)$, where now $\phi, \phi_S,$ are the azimuthal
orientations of the final lepton plane and of the hadron polarization with
respect to the reaction plane~\cite{Boer:1999mm}. Adopting the notations
recommended in Ref.~\cite{Bacchetta:2004jz}, the first one involves the
convolution of the Sivers function $f_{1{\scriptscriptstyle T}}^\perp$ with the standard
unpolarized parton distribution $f_1$. The second one involves the transversity
distribution $h_1$ and the Boer-Mulders function $h_1^\perp$, a TMD distribution
which is most likely responsible for the violation of the Lam-Tung sum rule in
the corresponding anomalous $\cos 2\phi$ asymmetry of the unpolarized Drell-Yan
cross section~\cite{Boer:1999mm}. Hence, a simultaneous measurement of
unpolarized and single-polarized Drell-Yan cross sections would allow to extract
all the unknowns from data~\cite{Bianconi:2004wu,Bianconi:2005px}. Both $h_1$
and $h_1^\perp$ describe the distribution of transversely polarized partons; but
the former applies to transversely polarized parent hadrons, while the latter to
unpolarized ones. On an equal footing, $f_{1{\scriptscriptstyle T}}^\perp$ and $f_1$ describe
distributions of unpolarized partons. The correlation between ${\bm p}_{\scriptscriptstyle T}$ and
${\bm S}_{\scriptscriptstyle T}$ inside $f_{1{\scriptscriptstyle T}}^\perp$ is possible only for a nonvanishing
orbital angular momentum of partons. Then, extraction of Sivers function from
SIDIS and Drell-Yan data would allow to study the orbital motion and the spatial
distribution of hidden confined partons~\cite{Burkardt:2003je}, as well as to
test its peculiar universality property~\cite{Collins:2002kn}.
In a series of previous papers, we performed numerical simulations of
single-polarized Drell-Yan SSA for the $p p^\uparrow \to \mu^+ \mu^-
X$~\cite{Bianconi:2005yj} and $\bar{p} p^\uparrow \to \mu^+ \mu^-
X$~\cite{Bianconi:2004wu} processes. With proton beams, we considered collisions
at $\sqrt{s}=200$ GeV in the kinematic conditions for the foreseen upgrade of
RHIC (RHIC II). Even if in $pp$ collisions the nonvalence partonic contribution
to the elementary annihilation is unavoidable (leading, in principle, to lower
counting rates), still the kinematics selects a portion of phase space that
emphasizes this contribution. The net result is that with a reasonable sample of
Drell-Yan events the statistical accuracy allows to unambiguously extract the
Sivers function from the corresponding $\sin (\phi -\phi_S)$ asymmetry, as well
as to clearly test its predicted sign change with respect to the SIDIS
asymmetry~\cite{Bianconi:2005yj}. In $\bar{p} p$ collisions, the cross section
is dominated by the valence contribution to the annihilation of a parton (from
$p$) and an antiparton (from $\bar{p}$); hence, in general it is not suppressed
as in the previous case (for a quantitative check in our Monte Carlo, see Sec.
IVB of Ref.~\cite{Bianconi:2005yj}). In Ref.~\cite{Bianconi:2004wu}, we selected
antiproton beams of 15 GeV, as they could be produced at the High Energy Storage
Ring (HESR) at GSI~\cite{Maggiora:2005cr,pax2}, and we simulated collisions at
$\sqrt{s}\sim 14$ GeV in the socalled asymmetric collider mode. The goal was to
explore the minimal conditions required for an unambiguous extraction of $h_1$
and $h_1^\perp$ from a combined analysis of the $\sin (\phi + \phi_S)$ and $\cos
2\phi$ asymmetries in the full (unpolarized + polarized) cross section.
Here, we will reconsider the same scenarios but for the $\pi^\pm p^\uparrow \to
\mu^+ \mu^- X$ process at the same $\sqrt{s}\sim 14$ GeV that can be reached at
COMPASS with pion beams of energy 100 GeV. As for $\bar{p}$ beams, the elementary
mechanism is dominated by the annihilation between valence partons (from $p$) and
valence antipartons (from $\pi$). Indeed, a large Sivers effect was predicted in
this context by usig the same Sivers function fitted to the measured $\sin (\phi
-\phi_S)$ asymmetry in SIDIS~\cite{Collins:2005rq}. Taking advantage on the high
statistics reachable with pions, in our Monte Carlo we simulate both $\sin (\phi
\pm \phi_S)$ SSA in the Drell-Yan cross section. For the Sivers effect we use
two parametrizations of $f_{1{\scriptscriptstyle T}}^\perp$: the one of
Ref.~\cite{Anselmino:2005ea}, which was deduced by fitting the recent HERMES
data for the $\sin (\phi - \phi_S)$ SSA~\cite{Diefenthaler:2005gx}; the one of
Ref.~\cite{Bianconi:2005yj}, which is constrained by the recent RHIC data for
the $pp^\uparrow \to \pi X$ process at higher energy~\cite{Adler:2005in}. For
the Boer-Mulders effect, since there is no such abundance of data and fits, we
follow Ref.~\cite{Boer:1999mm} to constrain $h_1^\perp$ by the azimuthal asymmetry of
the corresponding unpolarized Drell-Yan cross section (see also
Ref.~\cite{Sissakian:2005yp} for a similar analysis). Then, we insert, as we did in
Ref.~\cite{Bianconi:2004wu}, very different input test functions for $h_1$ in order to
explore the sensitivity of the simulated SSA within the statistical accuracy.
In Sec.~\ref{sec:mc}, we review the formalism and the details of the numerical
simulation. In Sec.~\ref{sec:out}, we present and discuss our results. Finally,
in Sec.~\ref{sec:end} we draw some conclusions.
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{dykin.eps}
\caption{The Collins-Soper frame.}
\label{fig:dyframe}
\end{figure}
\section{General framework for the numerical simulation}
\label{sec:mc}
In a Drell-Yan process, an antilepton-lepton pair with individual momenta $k_1$
and $k_2$ is produced from the collision of two hadrons with momentum $P_i$, mass
$M_i$, and spin $S_i$, with $i=1,2$. The center-of-mass (c.m.) square energy
available is $s=(P_1+P_2)^2$ and the invariant mass of the final lepton pair is
given by the time-like momentum transfer $q^2 \equiv M^2 = (k_1 + k_2)^2$. If
$M^2,s \rightarrow \infty$, while keeping the ratio $0\leq \tau = M^2/s \leq 1$
limited, a factorization theorem can be proven~\cite{Collins:1984kg} ensuring
that the elementary mechanism proceeds through the annihilation of a parton and
an antiparton with momenta $p_1$ and $p_2$, respectively, into a virtual photon
with time-like momentum $q^2$. If $P_1^+$ and $P_2^-$ are the dominant light-cone
components of hadron momenta in this regime, then the partons are approximately
collinear with the parent hadrons and carry the light-cone momentum fractions
$0\leq x_1 = p_1^+ / P_1^+ , \; x_2 = p_2^- / P_2^- \leq 1$, with $q^+ = p_1^+,
\; q^- = p_2^-$ by momentum conservation~\cite{Boer:1999mm}. The transverse
components ${\bm p}_{i{\scriptscriptstyle T}}$ of $p_i$ with respect to the direction defined by
${\bm P}_i (i=1,2)$, are constrained again by the momentum conservation
${\bm q}_{\scriptscriptstyle T} = {\bm p}_{1{\scriptscriptstyle T}} + {\bm p}_{2{\scriptscriptstyle T}}$, where ${\bm q}_{\scriptscriptstyle T}$ is the
transverse momentum of the final lepton pair. If ${\bm q}_{\scriptscriptstyle T} \neq 0$ the
annihilation direction is not known. Hence, it is convenient to select the
socalled Collins-Soper frame~\cite{Collins:1977iv} described in
Fig.~\ref{fig:dyframe}. The final lepton pair is detected in the solid angle
$(\theta, \phi )$, where, in particular, $\phi$ (and all other azimuthal angles)
is measured in a plane perpendicular to the indicated lepton plane but containing
$\hat{\bm h} = {\bm q}_{\scriptscriptstyle T} / |{\bm q}_{\scriptscriptstyle T}|$.
The full expression of the leading-twist differential cross section for the $H_1
H_2^\uparrow \to l^+ l^- X$ process can be written as~\cite{Boer:1999mm}
\begin{eqnarray}
\frac{d\sigma}{d\Omega dx_1 dx_2 d{\bm q}_{\scriptscriptstyle T}} &= &
\frac{d\sigma^o}{d\Omega dx_1 dx_2 d{\bm q}_{\scriptscriptstyle T}} +
\frac{d\Delta \sigma^\uparrow}{d\Omega dx_1 dx_2 d{\bm q}_{\scriptscriptstyle T}} \nonumber \\
&= &\frac{\alpha^2}{3Q^2}\,\sum_f\,e_f^2\,\Bigg\{ A(y) \,
{\cal F}\left[ f_1^f(H_1)\, f_1^f (H_2) \right] \nonumber \\
& &\mbox{\hspace{2cm}} + B(y) \, \cos 2\phi \,
{\cal F}\left[ \left( 2 \hat{\bm h}\cdot {\bm p}_{1{\scriptscriptstyle T}} \, \hat{\bm h} \cdot
{\bm p}_{2{\scriptscriptstyle T}} - {\bm p}_{1{\scriptscriptstyle T}} \cdot {\bm p}_{2{\scriptscriptstyle T}} \right) \,
\frac{h_1^{\perp\,f}(H_1)\,h_1^{\perp\,f}(H_2)}{M_1\,M_2}\,\right] \Bigg\} \nonumber \\
& &+ \frac{\alpha^2}{3Q^2}\,|{\bm S}_{2{\scriptscriptstyle T}}|\,\sum_f\,e_f^2\,\Bigg\{
A(y) \, \sin (\phi - \phi_{S_2})\, {\cal F}\left[ \hat{\bm h}\cdot
{\bm p}_{2{\scriptscriptstyle T}} \,\frac{f_1^f(H_1) \, f_{1{\scriptscriptstyle T}}^{\perp\,f}(H_2^\uparrow)}{M_2}\right]
\nonumber \\
& &\mbox{\hspace{3cm}} - B(y) \, \sin
(\phi + \phi_{S_2})\, {\cal F}\left[ \hat{\bm h}\cdot {\bm p}_{1{\scriptscriptstyle T}} \,
\frac{h_1^{\perp\,f}(H_1) \, h_1^f(H_2^\uparrow)}{M_1}\right] \nonumber \\
& &\mbox{\hspace{3cm}} - B(y) \, \sin (3\phi - \phi_{S_2})\, {\cal F}\left[
\left( 4 \hat{\bm h}\cdot {\bm p}_{1{\scriptscriptstyle T}} \, (\hat{\bm h} \cdot
{\bm p}_{2{\scriptscriptstyle T}})^2 - 2 \hat{\bm h} \cdot {\bm p}_{2{\scriptscriptstyle T}} \, {\bm p}_{1{\scriptscriptstyle T}} \cdot
{\bm p}_{2{\scriptscriptstyle T}} - \hat{\bm h}\cdot {\bm p}_{1{\scriptscriptstyle T}} \, {\bm p}_{2{\scriptscriptstyle T}}^2 \right)
\right. \nonumber \\
& &\mbox{\hspace{7cm}} \left. \times
\frac{h_1^{\perp\,f}(H_1) \, h_{1{\scriptscriptstyle T}}^{\perp\,f}(H_2^\uparrow)}{2 M_1\,M_2^2}\,\right]
\, \Bigg\} \; ,
\label{eq:xsect}
\end{eqnarray}
where $\alpha$ is the fine structure constant, $d\Omega = \sin \theta d\theta
d\phi$, $e_f$ is the charge of the parton with flavor $f$, $\phi_{S_i}$ is
the azimuthal angle of the transverse spin of hadron $i$, and
\begin{align}
A(y) = \left( \frac{1}{2} - y + y^2 \right) \, \stackrel{\mbox{cm}}{=}\,
\frac{1}{4}\left( 1 + \cos^2 \theta \right) &\mbox{\hspace{2cm}}
B(y) = y (1-y) \, \stackrel{\mbox{cm}}{=}\,\frac{1}{4}\, \sin^2 \theta \; .
\label{eq:lepton}
\end{align}
The TMD functions $f_1^f(H), \, h_1^{\perp\,f}(H)$, describe the distributions
of unpolarized and transversely polarized partons in an unpolarized hadron $H$,
respectively, while $f_{1{\scriptscriptstyle T}}^{\perp\,f}(H^\uparrow)$ and the pair $h_1^f (H^\uparrow),
h_{1{\scriptscriptstyle T}}^{\perp\,f}(H^\uparrow)$, have a similar interpretation but for transversely
polarized hadrons $H^\uparrow$. The convolutions are defined as
\begin{equation}
{\cal F} \left[ DF_1^f(H_1) \, DF_2^f(H_2^{(\uparrow )}) \right] \equiv \int
d{\bm p}_{1{\scriptscriptstyle T}} d{\bm p}_{2{\scriptscriptstyle T}}\, \delta \left( {\bm p}_{1{\scriptscriptstyle T}} + {\bm p}_{2{\scriptscriptstyle T}} -
{\bm q}_{\scriptscriptstyle T} \right) \, \left[ DF_1(x_1,{\bm p}_{1{\scriptscriptstyle T}}; \bar{f}/H_1)\,
DF_2(x_2,{\bm p}_{2{\scriptscriptstyle T}}; f/H_2^{(\uparrow )} ) + (f\leftrightarrow \bar{f}) \right]
\; .
\label{eq:convol}
\end{equation}
In previous papers, we made numerical simulations of the SSA generated in
Eq.~(\ref{eq:xsect}) by the azimuthal dependences $\cos 2\phi$ and
$\sin (\phi + \phi_{S_2})$ for antiproton beams
$H_1=\bar{p}$~\cite{Bianconi:2004wu}, by the $\sin (\phi -
\phi_{S_2})$ dependence for proton beams $H_1=p$~\cite{Bianconi:2005yj}, as well
as for double-polarized Drell-Yan processes with $H_1^\uparrow = H_2^\uparrow =
p^\uparrow$~\cite{Bianconi:2005bd}. A combined measurement of these SSA
allows to completely determine the intertwined unknown transversity $h_1$ and
Boer-Mulders function $h_1^\perp$, and the Sivers function $f_{1{\scriptscriptstyle T}}^\perp$. The
Monte Carlo simulation was performed for high-energy proton beams
($\sqrt{s}=200$ GeV) in the conditions of the foreseen upgrade of RHIC (RHIC
II), and for antiproton beams of 15 GeV as they could be produced at HESR-GSI. In
the latter case, several scenarios were explored for $5\lesssim \sqrt{s} \lesssim
14$ GeV and $1.5<M<2.5,\, 4<M<9$ GeV, in order to avoid overlaps with the
strange, charm, and bottom quarkonia [where the elementary annihilation does not
necessarily proceed through a simple intermediate virtual photon, as it is assumed
in Eq.~(\ref{eq:xsect})]. Here, we reconsider the $\sin (\phi - \phi_{S_2})$ and
$\sin (\phi + \phi_{S_2})$ asymmetries by using pion beams of 100 GeV as they can
be produced at COMPASS, in the fixed target mode such as to reach the
same maximum c.m. energy considered at HESR-GSI, namely $\sqrt{s}\sim 14$ GeV. Most of
the technical details of the simulation are mutuated from our previous works; hence,
we will heavily refer to Refs.~\cite{Bianconi:2004wu,Bianconi:2005yj} in the following.
The Monte Carlo events have been generated by the following cross
section~\cite{Bianconi:2004wu}:
\begin{equation}
\frac{d\sigma}{d\Omega dx_1 dx_2 d{\bm q}_{\scriptscriptstyle T}} = K \, \frac{1}{s}\,
|{\cal T}({\bm q}_{\scriptscriptstyle T}, x_1, x_2, M)|^2 \, \sum_{i=1}^4\, c_i ({\bm q}_{\scriptscriptstyle T},
x_1,x_2) \, S_i(\theta, \phi, \phi_{_{S_2}}) \; ,
\label{eq:mc-xsect}
\end{equation}
where the event distribution is driven by the elementary unpolarized annihilation,
whose transition amplitude ${\cal T}$ has been highlighted. In
Eq.~(\ref{eq:xsect}), we assume a factorized transverse-momentum dependence in
each parton distribution such as to break the convolution ${\cal F}$, leading to
\begin{equation}
|{\cal T}|^2 \approx A(q_{\scriptscriptstyle T},x_1,x_2,M) \, F(x_1,x_2) \; ,
\label{eq:factorized}
\end{equation}
where $q_{\scriptscriptstyle T} \equiv |{\bm q}_{\scriptscriptstyle T}|$. The function $A$ is parametrized and
normalized as in Ref.~\cite{Conway:1989fs}, where high-energy Drell-Yan $\pi - p$
collisions were considered. The average transverse momentum turns out to be
$\langle q_{\scriptscriptstyle T} \rangle > 1$ GeV/$c$ (see also the more recent
Ref.~\cite{Towell:2001nh}), which effectively reproduces the influence of sizable
QCD corrections beyond the parton model picture of Eq.~(\ref{eq:xsect}). It is
well known~\cite{Altarelli:1979ub} that such corrections induce also large $K$
factors and an $M$ scale dependence in parton distributions, determining their
evolution. As in our previous
works~\cite{Bianconi:2004wu,Bianconi:2005yj,Bianconi:2005bd}, we conventionally
assume in Eq.~(\ref{eq:mc-xsect}) that $K=2.5$, but we stress that in an
azimuthal asymmetry the corrections to the cross sections in the numerator and in
the denominator should compensate each other, as it turns out to actually happen
at RHIC c.m. square energies~\cite{Martin:1998rz}. Since the range of $M$ values
here explored is close to the one of Ref.~\cite{Conway:1989fs}, where the
parametrization of $A, F,$ and $c_i$ in Eq.~(\ref{eq:mc-xsect}) was deduced
assuming $M$-independent parton distributions, we keep our same previous
approach~\cite{Bianconi:2004wu,Bianconi:2005yj,Bianconi:2005bd} and use
\begin{equation}
F(x_1,x_2) = \frac{\alpha^2}{12 Q^2}\,\sum_f\,e_f^2\,
f_1^f(x_1; \bar{f}/H_1) \, f_1^f (x_2; f/H_2) + (\bar{f} \leftrightarrow f) \; ,
\label{eq:mcF}
\end{equation}
where the unpolarized distribution $f_1^f (x)$ for various flavors $f=u,d,s,$ is
taken again from Ref.~\cite{Conway:1989fs}.
The whole solid angle $(\theta, \phi)$ of the final lepton pair in the
Collins-Soper frame is randomly distributed in each variable. The explicit form
for sorting it in the Monte-Carlo is~\cite{Bianconi:2004wu,Bianconi:2005yj}
\begin{eqnarray}
\sum_{i=1}^4\, c_i (q_{\scriptscriptstyle T},x_1,x_2) \, S_i(\theta, \phi, \phi_{S_2}) &=
&1 + \cos^2 \theta + \frac{\nu (x_1,x_2,q_{\scriptscriptstyle T})}{2}\, \sin^2\theta \, \cos 2\phi
\nonumber \\
& &+ |{\bm S}_{2{\scriptscriptstyle T}}|\, c_4 (q_{\scriptscriptstyle T},x_1,x_2)\, S_4 (\theta, \phi, \phi_{S_2}) \; .
\label{eq:mcS}
\end{eqnarray}
If quarks were massless, the virtual photon would be only transversely polarized
and the angular dependence would be described by the functions $c_1 = S_1 = 1$
and $c_2 = 1, \, S_2 = \cos^2 \theta$. Violations of such azimuthal symmetry
induced by the function $c_3 \equiv \textstyle{\frac{\nu}{2}}$ are due to the
longitudinal polarization of the virtual photon and to the fact that quarks have
an intrinsic transverse momentum distribution, leading to the explicit violation
of the socalled Lam-Tung sum rule~\cite{Conway:1989fs}. QCD corrections influence
$\nu$, which in principle depends also on $M^2$~\cite{Conway:1989fs}. Azimuthal
$\cos 2\phi$ asymmetries induced by $\nu$ were simulated in
Ref.~\cite{Bianconi:2004wu} using the simple parametrization of
Ref.~\cite{Boer:1999mm} and testing it against the previous measurement of
Ref.~\cite{Conway:1989fs}.
If we consider the Sivers effect in Eq.~(\ref{eq:xsect}), the last term in
Eq.~(\ref{eq:mcS}) becomes
\begin{equation}
S_4 (\theta, \phi, \phi_{S_2}) = (1+\cos^2 \theta) \, \sin (\phi - \phi_{S_2})
\label{eq:mcS4-sivers}
\end{equation}
and the corresponding coefficient $c_4$ reads
\begin{equation}
c_4 (q_{\scriptscriptstyle T},x_1,x_2) = \frac{\sum_f\,e_f^2\,{\cal F}\left[ \hat{\bm h}\cdot
{\bm p}_{2{\scriptscriptstyle T}} \, \displaystyle{\frac{f_1^f(x_1,{\bm p}_{1{\scriptscriptstyle T}})\,
f_{1T}^{\perp\, f}(x_2,{\bm p}_{2{\scriptscriptstyle T}})}{M_2}} \right]}
{\sum_f\,e_f^2\,{\cal F}\left[ f_1^f(x_1,{\bm p}_{1{\scriptscriptstyle T}})\,
f_1^f(x_2,{\bm p}_{2{\scriptscriptstyle T}}) \right]} \; ,
\label{eq:mcc4-sivers}
\end{equation}
where the complete dependence of the involved TMD parton distributions has been
made explicit.
Viceversa, if we consider the Boer-Mulders effect in Eq.~(\ref{eq:xsect}) the
last term in Eq.~(\ref{eq:mcS}) becomes
\begin{equation}
S_4 (\theta, \phi, \phi_{S_2}) = \sin^2 \theta \, \sin (\phi + \phi_{S_2})
\label{eq:mcS4-boer}
\end{equation}
and the corresponding coefficient $c_4$ reads
\begin{equation}
c_4 (q_{\scriptscriptstyle T},x_1,x_2) = - \frac{\sum_f\,e_f^2\,{\cal F}\left[ \hat{\bm h}\cdot
{\bm p}_{1{\scriptscriptstyle T}} \, \displaystyle{\frac{h_1^{\perp\, f}(x_1,{\bm p}_{1{\scriptscriptstyle T}})\,
h_1^f(x_2,{\bm p}_{2{\scriptscriptstyle T}})}{M_1}} \right]}
{\sum_f\,e_f^2\,{\cal F}\left[ f_1^f(x_1,{\bm p}_{1{\scriptscriptstyle T}})\,
f_1^f(x_2,{\bm p}_{2{\scriptscriptstyle T}}) \right]} \; .
\label{eq:mcc4-boer}
\end{equation}
In the following, we will discuss different inputs for the $x$ and ${\bm p}_{\scriptscriptstyle T}$
dependence of these distributions which allow to calculate the convolutions and
determine $c_4$. In any case, following
Refs.~\cite{Bianconi:2004wu,Bianconi:2005yj,Bianconi:2005bd}, the general strategy
is to divide the event sample in two groups, one for positive values "$U$" of
$S_4$ in Eq.~(\ref{eq:mcS4-sivers}) or (\ref{eq:mcS4-boer}), and another one for
negative values "$D$", then taking the ratio $(U-D)/(U+D)$. Data are accumulated
only in the $x_2$ bin, i.e. they are summed upon $x_1, \theta,$ and
$q_{\scriptscriptstyle T}$. Statistical errors for the spin asymmetry $(U-D)/(U+D)$ are obtained by
making 10 independent repetitions of the simulation for each individual case, and
then calculating for each $x_2$ bin the average asymmetry value and the variance.
We checked that 10 repetitions are a reasonable threshold to have stable numbers,
since the results do not change significantly when increasing the number of
repetitions beyond 6.
\subsection{The Sivers effect}
\label{sec:sivers}
Recently, the HERMES collaboration released new SSA data for the SIDIS
process on transversely polarized protons~\cite{Diefenthaler:2005gx}, which
substantially increase the precision of the previous data
set~\cite{Airapetian:2004tw}. As a consequence, different parametrizations of the
Sivers function $f_{1T}^\perp$ have been extracted from this data set and found
compatible also with the recent COMPASS data~\cite{Alexakhin:2005iw} (for a useful
comparison among the various approaches see Ref.~\cite{Anselmino:2005an}).
Following Ref.~\cite{Bianconi:2005yj}, we first simulate the Sivers effect using
the parametrization of Ref.~\cite{Anselmino:2005ea},
\begin{eqnarray}
f_{1T}^{\perp\, f}(x,{\bm p}_{\scriptscriptstyle T}) &= &-2\, N_f\,
\frac{(a_f+b_f)^{a_f+b_f}}{a_f^{a_f}\,b_f^{b_f}}\,
x^{a_f}\,(1-x)^{b_f}\,\frac{M_2 M_0}{{\bm p}_{\scriptscriptstyle T}^2+M_0^2}\,
f_1^f(x,{\bm p}_{\scriptscriptstyle T}) \nonumber \\
&= &-2\, N_f\,\frac{1}{\pi \, \langle p_{\scriptscriptstyle T}^2 \rangle}\,
\frac{(a_f+b_f)^{a_f+b_f}}{a_f^{a_f}\, b_f^{b_f}} \, x^{a_f}\, (1-x)^{b_f}\,
\frac{M_2 M_0}{{\bm p}_{\scriptscriptstyle T}^2+M_0^2}\, e^{-p_{\scriptscriptstyle T}^2/\langle p_{\scriptscriptstyle T}^2 \rangle}\,
f_1^f(x) \; ,
\label{eq:pTanselm}
\end{eqnarray}
where $M_2$ is the mass of the polarized proton, $p_{\scriptscriptstyle T} \equiv |{\bm p}_{\scriptscriptstyle T}|$,
and $\langle p_{\scriptscriptstyle T}^2 \rangle = 0.25$ (GeV/$c$)$^2$ is deduced by assuming a
Gaussian ansatz for the ${\bm p}_{\scriptscriptstyle T}$ dependence of $f_1$ in order to reproduce the
azimuthal angular dependence of the SIDIS unpolarized cross section (Cahn effect).
Flavor-dependent normalization and parameters in the $x$ dependence are fitted to
SIDIS SSA data neglecting the (small) contribution of antiquarks. The resulting
parameters $M_0$ and $N_f,a_f,b_f,$ with $f=u,d$, are listed in
Tab.~\ref{tab:pTanselm}. The sometimes poor resolution of the fit forced us to select
only the central values in order to produce meaningful numerical simulations.
Following the steps described in Sec.~III-1 of Ref.~\cite{Bianconi:2005yj}, in
particular the predicted sign change of $f_{1{\scriptscriptstyle T}}^\perp$ when going from SIDIS to
Drell-Yan, we insert the opposite of Eq.~(\ref{eq:pTanselm}) into
Eq.~(\ref{eq:mcc4-sivers}) and simplify it down to
\begin{equation}
c_4 \approx \frac{4 M_0\,q_{\scriptscriptstyle T}}{q_{\scriptscriptstyle T}^2+4 M_0^2}\,
\frac{1}{9}\, \left[ 8\, N_u\, \frac{(a_u+b_u)^{a_u+b_u}}{a_u^{a_u}\,b_u^{b_u}}
\, x_2^{a_u}\,(1-x_2)^{b_u}\, + \, N_d\,
\frac{(a_d+b_d)^{a_d+b_d}}{a_d^{a_d}\,b_d^{b_d}} \, x_2^{a_d}\,(1-x_2)^{b_d}
\right] \; .
\label{eq:c4mc-anselm}
\end{equation}
\begin{table}[h]
\caption{\label{tab:pTanselm} Parameters for the Sivers distribution from
Ref.~\protect{\cite{Anselmino:2005ea}}}
\begin{ruledtabular}
\begin{tabular}{cccc}
quark up & {} & quark down & {} \\
\hline
$N_u$ & $0.32 \pm 0.11$ & $N_d$ & $-1.0 \pm 0.12$ \\
$a_u$ & $0.29 \pm 0.35$ & $a_d$ & $1.16 \pm 0.47$ \\
$b_u$ & $0.53 \pm 3.58$ & $b_d$ & $3.77 \pm 2.59$ \\
\hline
$M_0^2$ & $0.32 \pm 0.25$ (GeV/$c$)$^2$ & & \\
\end{tabular}
\end{ruledtabular}
\end{table}
As an alternative choice, we adopt the new parametrization described in
Ref.~\cite{Bianconi:2005yj}. It is inspired to the one of
Ref.~\cite{Vogelsang:2005cs}, where the transverse momentum of the detected pion in
the SIDIS process was assumed to come entirely from the ${\bm p}_{\scriptscriptstyle T}$ dependence of
the Sivers function, and was further integrated out building the fit in terms of
specific moments of the function itself. The $x$ dependence of that approach is
retained, but a different flavor-dependent normalization and an explicit
${\bm p}_{\scriptscriptstyle T}$ dependence are introduced that are bound to the shape of the recent
RHIC data on $pp^\uparrow \to \pi X$ at $\sqrt{s}=200$ GeV~\cite{Adler:2005in}, where
large persisting asymmetries are found that could be partly due to the leading-twist
Sivers mechanism. The expression adopted is
\begin{eqnarray}
f_{1T}^{\perp\, f}(x,{\bm p}_{\scriptscriptstyle T}) &=&N_f\,x\,(1-x)\,
\frac{M_2 p_0^2 p_{\scriptscriptstyle T}}{(p_{\scriptscriptstyle T}^2+\frac{p_0^2}{4})^2}\,f_1^f(x,{\bm p}_{\scriptscriptstyle T}) \nonumber \\
&= & N_f\,x\,(1-x)\,\frac{M_2 p_0^2 p_{\scriptscriptstyle T}}{(p_{\scriptscriptstyle T}^2+\frac{p_0^2}{4})^2}\,
\frac{1}{\pi \, \langle p_{\scriptscriptstyle T}^2 \rangle}\, e^{-p_{\scriptscriptstyle T}^2/\langle p_{\scriptscriptstyle T}^2 \rangle}\,
f_1^f(x) \; ,
\label{eq:pTnoi}
\end{eqnarray}
where $p_0 = 2$ GeV/$c$, and $N_u = - N_d = 0.7$. The sign, positive for $u$
quarks and negative for the $d$ ones, already takes into account the predicted sign
change of $f_{1{\scriptscriptstyle T}}^\perp$ from Drell-Yan to SIDIS.
Again, following the steps described in Sec.~III-2 of Ref.~\cite{Bianconi:2005yj}, we
can directly insert Eq.~(\ref{eq:pTnoi}) into Eq.~(\ref{eq:mcc4-sivers}) and get
\begin{equation}
c_4 \approx x_2 \, (1-x_2)\,
\left( \frac{2\, p_0 \, q_{\scriptscriptstyle T}}{q_{\scriptscriptstyle T}^2+p_0^2} \right)^2 \, \frac{8\, N_u + N_d}{9}
\; .
\label{eq:c4mc-noi}
\end{equation}
The $q_{\scriptscriptstyle T}$ shape is different from Eq.~(\ref{eq:c4mc-anselm}) and the peak position
is shifted at larger values. This is in agreement with a similar analysis of the
azimuthal asymmetry of the unpolarized Drell-Yan data (the violation of the Lam-Tung
sum rule~\cite{Boer:1999mm}). But, more specifically, it is induced by the observed
$x_{_F}-q_{\scriptscriptstyle T}$ correlation in the above mentioned RHIC data for
$pp^\uparrow \to \pi X$, when it is assumed that the SSA is entirely due to the
Sivers mechanism. This suggests that the maximum asymmetry is reached in
the upper valence region such that $x_{_F} \approx x_2 \sim \langle q_{\scriptscriptstyle T} \rangle /
5$~\cite{Adler:2005in}.
\subsection{The Boer-Mulders effect}
\label{sec:boer}
Contrary to the Sivers effect, the lack of data for the Boer-Mulders effect
does not allow to build reasonable parametrizations either of
$h_1^{\perp\,f}(x,{\bm p}_{\scriptscriptstyle T})$ or of $h_1^f(x,{\bm p}_{\scriptscriptstyle T})$. Therefore, similarly to
what was done in our previous papers~\cite{Bianconi:2004wu,Bianconi:2005bd}, the
strategy of the numerical simulation is based on making guesses for the input $x$ and
${\bm p}_{\scriptscriptstyle T}$ dependence of the parton distributions, and on trying to determine the
minimum number of events required to discriminate various SSA produced by very
different input guesses. In fact, this would be equivalent to state that in this
case some analytic information on the structure of these TMD parton distributions
could be extracted from the SSA measurement.
Following the steps in Sec.~IV~C of Ref.~\cite{Bianconi:2004wu} and in Sec.~VI of
Ref.~\cite{Boer:1999mm}, the ${\bm p}_{\scriptscriptstyle T}$ dependence of the parton distributions is
parametrized as
\begin{eqnarray}
f_1^f(x,{\bm p}_{\scriptscriptstyle T}) &= &\frac{\alpha_{\scriptscriptstyle T}}{\pi}\,e^{-\alpha_{\scriptscriptstyle T}\,{\bm p}_{\scriptscriptstyle T}^2}\,
f_1^f (x) \nonumber \\
h_1^{\perp\,f}(x,{\bm p}_{\scriptscriptstyle T}) &= &\frac{M_{_C}}{{\bm p}_{\scriptscriptstyle T}^2 +M_{_C}^2} \,
f_1^f (x,{\bm p}_{\scriptscriptstyle T}) \nonumber \\
h_1^f(x,{\bm p}_{\scriptscriptstyle T}) &= &\frac{\alpha_{\scriptscriptstyle T}}{\pi}\,e^{-\alpha_{\scriptscriptstyle T}\,{\bm p}_{\scriptscriptstyle T}^2}\,
h_1^f (x) \; ,
\label{eq:boerparams}
\end{eqnarray}
where $\alpha_{\scriptscriptstyle T} = 1$ GeV$^{-2}$ and $M_{_C}=2.3$ GeV. In particular, the
${\bm p}_{\scriptscriptstyle T}$ dependence of $h_1^{\perp}$ is fitted to the measured $\cos 2\phi$
asymmetry of the corresponding unpolarized Drell-Yan cross section, which is small for
$1\lesssim q_{\scriptscriptstyle T} \lesssim 3$ GeV/$c$ (see, for example, Fig.4 in
Ref.~\cite{Boer:1999mm}). Correspondingly, the
$\sin (\phi +\phi_S)$ SSA will turn out to be small for the considered statistically
relevant $q_{\scriptscriptstyle T}$ range (see Sec.~\ref{sec:out-boer}).
Inserting the expressions~(\ref{eq:boerparams}) into Eq.~(\ref{eq:mcc4-boer}), we get
\begin{eqnarray}
c_4 &= &-\frac{2 M_{_C}\,q_{\scriptscriptstyle T}}{q_{\scriptscriptstyle T}^2+4 M_{_C}^2}\,
\frac{\sum_f\,e_f^2\,f_1^f(x_1; \bar{f}/H_1)\,h_1^f(x_2; f/H_2^\uparrow)+
(\bar{f} \leftrightarrow f)}
{\sum_f\,e_f^2\,f_1^f(x_1; \bar{f}/H_1)\,f_1^f(x_2; f/H_2)+
(\bar{f} \leftrightarrow f)} \nonumber \\
&\approx &-\frac{2 M_{_C}\,q_{\scriptscriptstyle T}}{q_{\scriptscriptstyle T}^2+4 M_{_C}^2} \,
\frac{f(x_1; \langle \bar{f} \rangle/H_1) \, h_1(x_2;\langle f \rangle/H_2^\uparrow)}
{f(x_1; \langle \bar{f} \rangle/H_1) \, f_1(x_2;\langle f \rangle/H_2)} \equiv
- \frac{2 M_{_C}\,q_{\scriptscriptstyle T}}{q_{\scriptscriptstyle T}^2+4 M_{_C}^2} \,
\frac{h_1(x_2;\langle f \rangle/H_2^\uparrow)}{f_1(x_2;\langle f \rangle/H_2)} \; ,
\label{eq:mcc4-boer2}
\end{eqnarray}
where the second step is justified by assuming that the contribution of each flavor can
be approximated by a corresponding average function~\cite{Bianconi:2004wu}.
Two choices with opposite features will be selected for the ratio
$h_1(x_2;\langle f \rangle/H_2^\uparrow) / f_1(x_2;\langle f \rangle/H_2)$, namely the
ascending function $\sqrt{x_2}$ and the descending one $\sqrt{1-x_2}$, that both respect
the Soffer bound. The goal is to determine the minimum number of events (compatible
with the kinematical setup and cuts) required to produce azimuthal asymmetries that
can be clearly distinguished like the corresponding originating distributions. We
identify this as the criterion to establish when information on the analytical
structure of the involved parton distributions can be extracted from SSA data.
\section{Results of the Monte Carlo simulations}
\label{sec:out}
In this Section, we present results for Monte Carlo simulations of both the Sivers
and the Boer-Mulders effects in the Drell-Yan process $\pi^{\pm} p^\uparrow \to
\mu^+ \mu^- X$ using input from the previous Sec.~\ref{sec:sivers} and \ref{sec:boer},
respectively. The goal is twofold. On one side, to explore the sensitivity of the
simulated asymmetry to the different input parametrizations of
Eqs.~(\ref{eq:pTanselm}) and (\ref{eq:pTnoi}), as well as to directly verify, within
the reached statistical accuracy, the predicted sign change of the Sivers function
between SIDIS and Drell-Yan~\cite{Collins:2002kn}. On the other side, to make
realistic estimates of the minimum number of events required to extract as detailed
information as possible on the chiral-odd distributions $h_1^\perp$ and $h_1$.
We consider pion beams with energy of 100 GeV hitting a transversely polarized proton
target such that $\sqrt{s}\sim 14$ GeV, i.e. the same c.m. energy available at HESR at
GSI in the socalled asymmetric collider mode with antiprotons of 15 GeV and protons of
3.3 GeV~\cite{Bianconi:2004wu}. The transversely polarized proton target is obtained
from a $NH_3$ molecule where each $H$ nucleus is fully transversely polarized and the
number of "polarized" collisions is 25\% of the total number of
collisions~\cite{Bianconi:2004wu}. The muon pair invariant mass is constrained in the
range $4<M<9$ GeV, in order to avoid overlaps with the resonance regions of the
$\bar{c}c$ and $\bar{b}b$ quarkonium systems. At the same time, the theoretical
analysis based on the leading-twist cross section~(\ref{eq:xsect}) should be well
established, since higher-twist effects can be classified according to powers of
$M_p/M$, where $M_p$ is the proton mass.
In the Monte Carlo, the events are sorted according to the cross
section~(\ref{eq:mc-xsect}), supplemented by Eqs.~(\ref{eq:factorized}) and
(\ref{eq:mcF}). The asymmetry is simulated by Eq.~(\ref{eq:mcS}). In particular, for
the Sivers effect we use Eqs.~(\ref{eq:mcS4-sivers}) and (\ref{eq:c4mc-anselm}) or
(\ref{eq:c4mc-noi}), according to the input parametrization selected for the Sivers
function. For the Boer-Mulders effect, we use Eqs.~(\ref{eq:mcS4-boer}) and
(\ref{eq:mcc4-boer2}). The events are divided in two groups, one for positive values
($U$) of $\sin (\phi - \phi_{S_2})$ in Eq.~(\ref{eq:mcS4-sivers}) or of
$\sin (\phi + \phi_{S_2})$ in Eq.~(\ref{eq:mcS4-boer}), and another one for negative
values ($D$), and taking the ratio $(U-D)/(U+D)$. Data are accumulated only in the
$x_2$ bins of the polarized proton, i.e. they are summed over in the $x_1$ bins for
the pion, in the transverse momentum $q_{\scriptscriptstyle T}$ of the muon pair and in their zenithal
orientation $\theta$.
Proper cuts are applied to the $q_{\scriptscriptstyle T}$ distribution according to the different inputs.
As for the Sivers effect, the flavor-independent Lorentzian shape in the ${\bm p}_{\scriptscriptstyle T}$
dependence of Eq.~(\ref{eq:pTanselm}) produces a maximum asymmetry for $q_{\scriptscriptstyle T} \sim 1$
GeV/$c$ and a rapid decrease for larger values. Consequently, transverse momenta are
selected in the range $0.5< q_{\scriptscriptstyle T} < 2.5$ GeV/$c$, because for larger cutoffs the
asymmetry is diluted. For the case of Eq.~(\ref{eq:pTnoi}), the peak position in $q_{\scriptscriptstyle T}$
is shifted at higher values and the cut is modified as $1<q_{\scriptscriptstyle T} <3$ GeV/$c$. In this
way, the ratio between the absolute sizes of the asymmetry and the statistical errors
is optimized for each choice, while the resulting $\langle q_{\scriptscriptstyle T} \rangle \sim 1.8$
GeV/$c$ is in fair agreement with the one experimentally explored at
RHIC~\cite{Adler:2005in}. As for the Boer-Mulders effect, we keep the latter cut
$1<q_{\scriptscriptstyle T} <3$ GeV/$c$. The $\theta$ angular dependence for the Boer-Mulders effect is
constrained in the range $60^{\rm o}< \theta < 120^{\rm o}$ due to
Eq.~(\ref{eq:mcS4-boer}), because outside these limits the azimuthal asymmetry is too
small~\cite{Bianconi:2004wu}. On the contrary, for the Sivers effect there is no need
to introduce cuts because of the $(1+\cos^2 \theta)$ term in
Eq.~(\ref{eq:mcS4-sivers})~\cite{Bianconi:2005yj}.
We have considered different initial samples. The Sivers mechanism is explored
starting from $100\,000$ events with the $\pi^-$ beam and $25\,000$ with the
$\pi^+$ beam, because the Monte Carlo indicates that the cross section involving
$\pi^+$ is statistically disfavoured by approximately the factor
1/4~\cite{Bianconi:2005bv}; in such a way, the two samples can be collected in the same
time. As for the Boer-Mulders effect, the lacking of any parametrization makes it
impossible to perform an isospin analysis; hence, we used $50\,000$ events with the
$\pi^-$ beam. Statistical errors for $(U-D)/(U+D)$ are obtained by
making 10 independent repetitions of the simulation for each individual case, and
then calculating for each $x_2$ bin the average asymmetry value and the variance.
We checked that 10 repetitions are a reasonable threshold to have stable numbers,
since the results do not change significantly when increasing the number of
repetitions beyond 6.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig2.eps}
\caption{The samples of Drell-Yan events for the Sivers effect in the
$\pi^\pm p^\uparrow \to \mu^+ \mu^- X$ reaction at $\sqrt{s}\sim 14$ GeV,
$4<M<9$ GeV, and $0.5<q_{\scriptscriptstyle T} <2.5$ GeV/$c$, using the parametrization of
Eq.~(\protect{\ref{eq:pTanselm}}) (see text). a) left panel: $100\,000$ events with
the $\pi^-$ beam; the darker histogram collects events with positive
$\sin (\phi - \phi_{S_2})$, the superimposed lighter histogram collects the negative
ones. b) right panel: the same for $25\,000$ events with the $\pi^+$ beam.}
\label{fig:pi+-anselm_histo}
\end{figure}
\subsection{The Sivers effect}
\label{sec:out-sivers}
In Fig.~\ref{fig:pi+-anselm_histo}, the left panel a) displays the sample of $100\,000$
Drell-Yan events for the $\pi^- p^\uparrow \to \mu^+ \mu^- X$ reaction at
$\sqrt{s} \sim 14$ GeV as they are collected in $x_2$ bins for muon invariant mass in
the $4<M<9$ GeV range. The right panel b) contains $25\,000$ events for the
$\pi^+ p^\uparrow \to \mu^+ \mu^- X$ reaction in the same kinematic conditions. Both
samples can be accumulated approximately in the same time according to
Eq.~(\ref{eq:c4mc-anselm}) based on the parametrization~(\ref{eq:pTanselm}) of the
Sivers function~\cite{Anselmino:2005ea}; as already discussed, the transverse momentum
distribution is constrained in the range $0.5<q_{\scriptscriptstyle T} <2.5$ GeV/$c$. For each bin two
groups of events are stored, one corresponding to positive values of
$\sin (\phi - \phi_{S_2})$ in Eq.~(\ref{eq:mcS4-sivers}) (represented by the darker
histogram), and one for negative values (superimposed lighter histogram). Since the
$\bar{q} q \to \gamma^\ast$ mechanism tends to populate the phase space for the lowest
possible $\tau$ values~\cite{Bianconi:2004wu,Bianconi:2005bd,Bianconi:2005yj} compatible
with the explored range $0.08<\tau = x_1 \, x_2<0.4$, this reflects in a
$x_1-$integrated distribution which is peaked for $x_2$ values in the valence domain.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig3.eps}
\caption{The asymmetry $(U-D)/(U+D)$ corresponding to the histograms of
Fig.~\protect{\ref{fig:pi+-anselm_histo}}, where $U$ identifies the darker histograms
and $D$ the superimposed lighter ones (see text). Triangles for the
parametrization of Eq.~(\protect{\ref{eq:pTanselm}}) using the $\pi^-$ beam and with
$N_u>0$; squares for $N_u<0$. Open triangles using the $\pi^+$ beam and with
$N_u>0$; open squares for $N_u<0$.}
\label{fig:pi+-anselm_as}
\end{figure}
In Fig.~\ref{fig:pi+-anselm_as}, the asymmetry $(U-D)/(U+D)$ is shown for each bin $x_2$
between the events of the previous figure accumulated for the positive $(U)$ and
negative $(D)$ values of $\sin (\phi - \phi_{S_2})$ in Eq.~(\ref{eq:mcS4-sivers}).
Average asymmetries and (statistical) error bars are obtained by 10
independent repetitions of the simulation. Boundary values of $x_2$ beyond 0.7 are
excluded because of very low statistics. The triangles indicate the results with
the $\pi^-$ beam obtained by Eq.~(\ref{eq:c4mc-anselm}) assuming that
$f_{1{\scriptscriptstyle T}}^\perp$ changes sign from the parametrization~(\ref{eq:pTanselm}) of the SIDIS
data to the considered Drell-Yan~\cite{Collins:2002kn}. For sake of comparison, the
squares illustrate the opposite results that one would obtain by ignoring such
prediction. Finally, the open triangles and open squares refer to the same situation,
respectively, but for the $\pi^+$ beam. The sensitivity of the parameters in
Tab.~\ref{tab:pTanselm} to the HERMES results for the Sivers effect, reflects in a more
important relative weight of the $d$ quark over the $u$ one in the valence $x_2$ range,
with opposite signs for the corresponding normalization $N_f, \, f=u,d$. Consequently,
in the valence picture of the $(\pi^-)\pi^+-p$ collision where the $(\bar{u}u) \,
\bar{d}d$ annihilation dominates, the SSA for the Drell-Yan process induced by $\pi^+$
has opposite sign with respect to $\pi^-$. Moreover, it has an absolute bigger size
because the $\bar{d}d$ annihilations are weighted more than the $\bar{u}u$ ones. Apart
for very low $x_2$ values where the parton picture leading to Eq.~(\ref{eq:xsect})
becomes questionable, the error bars are very small and allow for a clean
reconstruction of the asymmetry shape and, more importantly, for a conclusive test of
the predicted sign change in $f_{1{\scriptscriptstyle T}}^\perp$.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig4.eps}
\caption{The same situation with the same notations as in
Fig.~\protect{\ref{fig:pi+-anselm_histo}}, but for the parametrization of
Eq.~(\protect{\ref{eq:pTnoi}}) with $1<q_{\scriptscriptstyle T} <3$ GeV/$c$ (see text).}
\label{fig:pi+-noi_histo}
\end{figure}
In Fig.~\ref{fig:pi+-noi_histo}, the Drell-Yan events are shown in the same conditions
and notations as in Fig.~\ref{fig:pi+-anselm_histo}, i.e. in the left panel a)
$100\,000$ events for the $\pi^- p^\uparrow \to \mu^+ \mu^- X$ reaction at $\sqrt{s}
\sim 14$ GeV and for $4<M<9$ GeV, and in the right panel b) $25\,000$ events for the
$\pi^+ p^\uparrow \to \mu^+ \mu^- X$ reaction in the same kinematic conditions. The
difference is that the events are now collected according to Eq.~(\ref{eq:c4mc-noi})
based on the parametrization~(\ref{eq:pTnoi}) of the Sivers
function~\cite{Bianconi:2005yj}; the cut in the transverse momentum distribution is
now $1<q_{\scriptscriptstyle T} <3$ GeV/$c$. Again, the darker histogram refers to events with positive
$\sin (\phi - \phi_{S_2})$ in Eq.~(\ref{eq:mcS4-sivers}), while the superimposed lighter
histogram to the negative ones. Similarly, the density of events is peaked for $x_2$
values in the valence domain because of the dominance of the low $\tau$ portion of the
phase space.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig5.eps}
\caption{The same situation with the same notations as in
Fig.~\protect{\ref{fig:pi+-anselm_as}}, but for the parametrization of
Eq.~(\protect{\ref{eq:pTnoi}}) (see text).}
\label{fig:pi+-noi_as}
\end{figure}
In Fig.~\ref{fig:pi+-noi_as}, the asymmetry $(U-D)/(U+D)$ is shown for each bin $x_2$
between the events of Fig.~\ref{fig:pi+-noi_histo} accumulated for the positive $(U)$
and negative $(D)$ values of $\sin (\phi - \phi_{S_2})$ in Eq.~(\ref{eq:mcS4-sivers}).
Notations are as in Fig.~\ref{fig:pi+-anselm_as}: the triangles indicate the
results with the $\pi^-$ beam obtained by Eq.~(\ref{eq:c4mc-noi}) using a positive
normalization $N_u$, which already accounts for the sign change of $f_{1{\scriptscriptstyle T}}^\perp$
from SIDIS to Drell-Yan; the squares illustrate the results obtained by ignoring such
prescription, while the open triangles and open squares refer to the same situation,
respectively, but for the $\pi^+$ beam. Again, the opposite normalizations of the two
flavors $u,d,$ determine the opposite SSA between the $\pi^-$ and the $\pi^+$ beams. But
now in Eq.~(\ref{eq:pTnoi}) the relative weight of $u$ and $d$ distributions is the
same, hence the absolute sizes of the SSA are approximately the same irrespectively of
the charge of the $\pi$ beam. As already anticipated in Sec.~\ref{sec:sivers}, the
$q_{\scriptscriptstyle T}$ distribution induced by the parametrization~(\ref{eq:pTnoi}) is also related to
the observed $x_{_F}-q_{\scriptscriptstyle T}$ correlation in the RHIC data for
$pp^\uparrow \to \pi X$~\cite{Adler:2005in}, when it is assumed that the SSA is
entirely due to the Sivers mechanism. This suggests that the maximum asymmetry is
reached in the upper valence region such that $x_{_F} \approx x_2 \sim \langle q_{\scriptscriptstyle T}
\rangle / 5 \sim 0.4$ for the considered cut in $q_{\scriptscriptstyle T}$, as it is confirmed in
Fig.~\ref{fig:pi+-noi_as}. Similarly to the case of the other parametrization, the
statistical error bars are very small and allow for a detailed analysis of the
(universality) properties of $f_{1{\scriptscriptstyle T}}^\perp$.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig6.eps}
\caption{The sample of $50\,000$ Drell-Yan events for the Boer-Mulders effect in the
$\pi^- p^\uparrow \to \mu^+ \mu^- X$ reaction at $\sqrt{s}\sim 14$ GeV,
$4<M<9$ GeV, and $1<q_{\scriptscriptstyle T} <3$ GeV/$c$ (see text). a) left panel for the choice
$h_1(x_2, \langle f \rangle / H_2^\uparrow) / f_1 (x_2, \langle f \rangle / H_2) =
\sqrt{1-x_2}$ ($\langle f \rangle$ represents a common average term that replaces each
contribution in the flavor sum, for further details see text); the darker histogram
collects events with positive $\sin (\phi + \phi_{S_2})$, the superimposed lighter
histogram collects the negative ones. b) right panel: the same for
$h_1(x_2, \langle f \rangle / H_2^\uparrow) / f_1 (x_2, \langle f \rangle / H_2) =
\sqrt{x_2}$.}
\label{fig:pi-boer_histo}
\end{figure}
\subsection{The Boer-Mulders effect}
\label{sec:out-boer}
In Fig.~\ref{fig:pi-boer_histo}, a sample of $50\,000$ Drell-Yan events for the
$\pi^- p^\uparrow \to \mu^+ \mu^- X$ reaction at $\sqrt{s} \sim 14$ GeV is displayed
in $x_2$ bins for muon invariant mass in the $4<M<9$ GeV range and for
$1<q_{\scriptscriptstyle T} <3$ GeV/$c$. Events are produced by the Boer-Mulders effect contained in
Eq.~(\ref{eq:mcc4-boer2}), where the left panel a) refers to the choice
$h_1(x_2, \langle f \rangle / H_2^\uparrow) / f_1 (x_2, \langle f \rangle / H_2) =
\sqrt{1-x_2}$ and the right panel b) to the
$h_1(x_2, \langle f \rangle / H_2^\uparrow) / f_1 (x_2, \langle f \rangle / H_2) =
\sqrt{x_2}$ one. Here, $\langle f \rangle$ means that each term contributing to the sum
upon flavors is replaced by a common flavor-averaged parton distribution. Following
previous notations, for each bin the darker histogram represents events with positive
values of $\sin (\phi + \phi_{S_2})$ in Eq.~(\ref{eq:mcS4-boer}) and the superimposed
lighter histogram indicates the ones with negative values. Similarly, the density of
events is peaked for $x_2$ values in the valence domain because of the dominance of the
low $\tau$ portion of the phase space.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig7.eps}
\caption{The asymmetry $(U-D)/(U+D)$ corresponding to the histograms of
Fig.~\protect{\ref{fig:pi-boer_histo}}, where $U$ identifies the darker histograms
and $D$ the superimposed lighter ones (see text). Triangles for
$h_1(x_2, \langle f \rangle / H_2^\uparrow) / f_1 (x_2, \langle f \rangle / H_2) =
\sqrt{1-x_2}$. Open triangles for
$h_1(x_2, \langle f \rangle / H_2^\uparrow) / f_1 (x_2, \langle f \rangle / H_2) =
\sqrt{x_2}$.}
\label{fig:pi-boer_as}
\end{figure}
In Fig.~\ref{fig:pi-boer_as}, the asymmetry $(U-D)/(U+D)$ is shown for each bin $x_2$
between the events of Fig.~\ref{fig:pi-boer_histo} accumulated for the positive $(U)$
and negative $(D)$ values of $\sin (\phi + \phi_{S_2})$ in Eq.~(\ref{eq:mcS4-boer}).
Triangles correspond to the $\sqrt{1-x_2}$ input function, open triangles to the
$\sqrt{x_2}$ one. Both choices respect the Soffer bound between $h_1$ and $f_1$ and have
an overall normalization 2/3, which seems a reasonable expectation on the basis of
lattice results and first SIDIS experimental data~\cite{Boer:1999mm}. The error bars
represent statistical errors only. As it is evident in the figure, the open triangles
describe a SSA which statistically reflects the ascending trend of the input function
$\sqrt{x_2}$, while it is not the case for the other choice. Despite the small error
bars, which allow to state that both SSA are nonvanishing and to distinguish the two
cases in the narrow range $0.1<x_2<0.3$, it is not possible to conclude that specific
information on the analytic dependence of $h_1(x)$ can be extracted from such
simulation, contrary to what is claimed in Ref.~\cite{Sissakian:2005yp}. The reached
statistical accuracy indicates that the size of the sample may
not be responsible for such failure. Rather, the ${\bm p}_{\scriptscriptstyle T}$ dependence of
$h_1^{\perp}$ in Eq.~(\ref{eq:boerparams}) induces the overall small size of the
displayed SSA. Moreover, the Soffer bound and the flavor independence of the analysis
further reduce the selectivity power of the final Monte
Carlo output. In particular, the latter issue calls for specific parametrizations
of $h_1^f(x,{\bm p}_{\scriptscriptstyle T})$ and $h_1^{\perp\,f}(x,{\bm p}_{\scriptscriptstyle T})$, whose unavailability
reflects the lacking of experimental data for $\sin (\phi + \phi_{S_2})$
asymmetries in single-polarized Drell-Yan processes.
\section{Conclusions}
\label{sec:end}
In a series of previous papers~\cite{Bianconi:2004wu,Bianconi:2005bd}, we investigated
the spin structure of the proton using numerical simulations of single- and
double-polarized Drell-Yan Single-Spin Asymmetries (SSA) for the
$\bar{p}^{(\uparrow )} p^\uparrow \to \mu^+ \mu^- X$ process as well as for the
$p p^\uparrow \to \mu^+ \mu^- X$ one~\cite{Bianconi:2005yj}. We selected muon pair
invariant masses in the range $4<M<9$ GeV (and also $12<M<40$ GeV for the case of
proton beams), where there is no overlap with the resonance regions of the $\bar{c}c$
and $\bar{b}b$ quarkonium systems and the elementary annihilation can be safely assumed
to proceed through the $\bar{q}q\to \gamma^\ast$ mechanism. In particular, the Monte Carlo
was based on the Drell-Yan leading-twist cross section, because higher twists may be
suppressed as $M_p/M$, where $M_p$ is the proton mass.
As for single-polarized reactions, two interesting contributions generate azimuthal
asymmetries of the kind $\sin (\phi + \phi_S)$ and $\sin (\phi - \phi_S)$, where $\phi$
and $\phi_S$ are the azimuthal orientations of the plane containing the final muon pair
and of the proton polarization, respectively, with respect to the reaction plane. The
first one involves the convolution of the transversity $h_1$, the missing piece
necessary to complete the knowledge of the nucleon spin structure at leading twist, and
the Boer-Mulders $h_1^\perp$, another chiral-odd parton density which is most likely
responsible for the violation of the Lam-Tung sum rule, the long-standing problem of an
anomalous $\cos 2\phi$ asymmetry of the corresponding unpolarized Drell-Yan cross
section~\cite{Boer:1999mm}. The second convolution involves the socalled Sivers
function $f_{1T}^\perp$~\cite{Sivers:1990cc}, a "naive T-odd" partonic density that
describes how the distribution of unpolarized quarks is distorted by the transverse
polarization of the parent hadron. As such, $f_{1T}^\perp$ contains unsuppressed
information on the orbital motion of hidden confined partons and on their spatial
distribution inside the proton~\cite{Burkardt:2003je}.
In this paper, we have reconsidered the same scenario but for the
$\pi^\pm p^\uparrow \to \mu^+ \mu^- X$ process at $\sqrt{s}\sim 14$ GeV, that
can be reached at COMPASS with pion beams of energy 100 GeV and transversely polarized
proton fixed targets. As with antiproton beams, the elementary mechanism is dominated
by the annihilation between valence partons (from $p$) and valence antipartons (from
$\pi$). Taking advantage on the high statistics reachable with pions, in our Monte
Carlo we have simulated both $\sin (\phi \pm \phi_S)$ SSA in the Drell-Yan cross
section. For the Sivers effect we have used two parametrizations of $f_{1{\scriptscriptstyle T}}^\perp$:
the one of Ref.~\cite{Anselmino:2005ea}, which was deduced by fitting the recent HERMES
data for the $\sin (\phi - \phi_S)$ SSA in SIDIS~\cite{Diefenthaler:2005gx}; the one of
Ref.~\cite{Bianconi:2005yj}, which is constrained by the recent RHIC data for
the $pp^\uparrow \to \pi X$ process at higher energy~\cite{Adler:2005in}, when it is
assumed that the SSA is driven by the Sivers mechanism only. The main difference is
that the former displays an emphasized relative importance of the unfavoured $d$ quark,
and it gives an average transverse momentum $\langle q_{\scriptscriptstyle T} \rangle$ of the lepton pair
lower than the latter. Consistently, we have built SSA by integrating the $q_{\scriptscriptstyle T}$
distribution with adequate cuts, namely $0.5<q_{\scriptscriptstyle T} < 2.5$ GeV/$c$ for the former
parametrization, and $1<q_{\scriptscriptstyle T}<3$ GeV/$c$ for the latter one. Results have been
presented as binned in the parton momenta $x_2$ of the polarized proton, i.e. by
integrating also upon the antiparton partner momenta $x_1$ and the zenithal muon pair
distribution $\theta$ with no further cuts. For the Boer-Mulders effect, since there is
no such abundance of data and fits, we have used, as we did in
Ref.~\cite{Bianconi:2004wu}, very different input test functions and we have explored
the sensitivity of the simulated $\sin (\phi + \phi_S)$ asymmetry within the reached
statistical accuracy, integrating $q_{\scriptscriptstyle T}$ in the range $1<q_{\scriptscriptstyle T}<3$ GeV/$c$. Again,
results have been presented as binned in $x_2$ by integrating also upon $x_1$ and
$\theta$, but with the further constraint $60^{\rm o}<\theta <120^{\rm o}$ induced by
the factor $\sin^2 \theta$ which drives the angular distribution of muon pairs.
Given the very different situations for the two analyses, also the goals are different.
For the Sivers effect, the numerical simulation aims to establish the necessary
statistical accuracy to distinguish different input parametrizations and to test the
(universality) properties of the Sivers function, in particular its predicted sign
change when going from SIDIS to the Drell-Yan process~\cite{Collins:2002kn}. As for the
Boer-Mulders effect, the goal is to make input guesses and to try to determine the
minimum number of events required to discriminate various SSA produced by very
different input guesses, that would allow to extract as detailed information as
possible on the chiral-odd distributions $h_1^\perp$ and $h_1$.
In all cases, sorted events have been divided in two groups, corresponding to opposite
azimuthal orientations of the muon pair with respect to the reaction plane
(conventionally indicated with $U$ and $D$), and the asymmetry $(U-D)/(U+D)$ has been
considered. Statistical errors have been obtained by making 10 independent repetitions
of the simulation for each individual case and, then, calculating for each $x_2$ bin
the average asymmetry and the variance. For the Sivers effect, a starting sample of
$100\,000$ events has been selected for the $\pi^-$ beam. Since, from the Monte Carlo,
the cross section with $\pi^+$ turns out statistically unfavoured by a factor
1/4~\cite{Bianconi:2005bv}, we have reduced the sample to $25\,000$ events for the
$\pi^+$ beam in order to compare situations with the same "effective luminosity".
As for the Boer-Mulders effect, because of the unavailability of fits and isospin
analyses, we have used $50\,000$ events with the $\pi^-$ beam. In all cases, the
$1/\tau$ behaviour of the cross section, induced by the $\gamma^\ast$ propagator, has a
twofold effect. It produces the highest density of events for bins in the valence
domain, typically for $x_2 \sim 0.3$. At the given $\sqrt{s}$, it also grants that the
considered invariant mass range allows to explore the most populated portion of phase
space, while avoiding overlaps with ranges where the elementary mechanism could be more
complicated and the leading-twist analysis more questionable. The direct consequence is
that, with a very large statistics of pions available, very small error bars are
reached, except for boundary $x_2$ values.
The availability of different parametrizations of the Sivers function, obtained from
independent sets of data, allows for a deep analysis of the flavor dependence of the
resulting Drell-Yan SSA, as well as for a test of the universal properties of this
parton density. It turns out that the asymmetry always changes sign when switching from
the $\pi^-$ to the $\pi^+$ beam, because in the valence picture of the
$(\pi^-) \pi^+-p$ collision the $(\bar{u}u) \, \bar{d}d$ annihilation dominates, and
both the parametrizations here considered have weights with opposite signs for the $u$
and $d$ valence quarks. The parametrization of
Ref.~\cite{Anselmino:2005ea}, being deduced by SIDIS data for the Sivers
effect~\cite{Diefenthaler:2005gx}, displays a more important relative weight of the
$d$ quark over the $u$, which reflects in a smaller absolute size of the SSA with the
$\pi^-$ beam with respect to the $\pi^+$ case. No such evidence is shown by the
parametrization of Ref.~\cite{Bianconi:2005yj}, constrained by data for the
$pp^\uparrow \to \pi X$ process at $\sqrt{s}=200$ GeV~\cite{Adler:2005in}, where also
the higher $\langle q_{\scriptscriptstyle T} \rangle$ induces a maximum of the asymmetry at higher $x_2$,
typically $x_2 \sim 0.4$. In both the considered cases, we have simulated the asymmetry
assuming or neglecting the predicted sign change of the Sivers function when replacing
the SIDIS with the Drell-Yan process~\cite{Collins:2002kn}. The corresponding results
have, of course, opposite signs, but, noticeably, the very small statistical error
bars allow to clearly distinguish between one choice or the other extreme. We conclude
that with the considered sample of events it is possible to perform such important test
of nonperturbative QCD using pion beams and transversely polarized proton targets in
the kinematic conditions that can be prepared at COMPASS.
Unfortunately, for the Boer-Mulders effect the lack of data and parametrizations of the
involved parton distributions forbids a thorough analysis. The ${\bm p}_{\scriptscriptstyle T}$
dependence of $h_1^{\perp}$ is inherited by fitting the measured $\cos 2\phi$
asymmetry of the corresponding unpolarized Drell-Yan cross section; for the
statistically relevant range $1\lesssim q_{\scriptscriptstyle T} \lesssim 3$ GeV/$c$, the
$\sin (\phi +\phi_S)$ asymmetry turns out to be small. We have further approximated
the transversity distribution by using a "flavor-averaged" ratio
between $h_1(x_2)$ itself and the unpolarized parton distribution $f_1(x_2)$,
and we have simulated it by integrating upon $x_1, q_{\scriptscriptstyle T}, \theta ,$ and inserting very
different input test functions of $x_2$, one ascending and one descending, but all
satisfying the general constraints (like the Soffer bound, that puts a strong upper
bound on the size of $h_1$). The small
statistical errors allow to conclude that the resulting $(U-D)/(U+D)$ asymmetries,
though small, are certainly nonvanishing. But the displayed trends in $x_2$ are very
similar and do not reflect the very different inputs. Hence, we conclude that in the
present stage a measurement of such an asymmetry would not help in extracting
information on the transversity $h_1$ and/or the Boer-Mulders function $h_1^\perp$.
\begin{acknowledgments}
This work is part of the European Integrated Infrastructure Initiative in Hadron
Physics project under the contract number RII3-CT-2004-506078.
\end{acknowledgments}
\bibliographystyle{apsrev}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,568 |
using System;
using System.Linq;
using System.Net;
using System.Threading.Tasks;
using Microsoft.AspNetCore.Authentication;
using Microsoft.AspNetCore.Http;
using Microsoft.AspNetCore.Http.Features;
using Microsoft.AspNetCore.Identity;
using Microsoft.Extensions.DependencyInjection;
using Microsoft.Extensions.Logging;
using Microsoft.Extensions.Options;
using Moq;
using NUnit.Framework;
using Umbraco.Cms.Core.Configuration.Models;
using Umbraco.Cms.Core.Events;
using Umbraco.Cms.Core.Net;
using Umbraco.Cms.Core.Security;
using Umbraco.Cms.Core.Services;
using Umbraco.Cms.Web.Common.Security;
namespace Umbraco.Cms.Tests.UnitTests.Umbraco.Web.Common.Security
{
[TestFixture]
public class MemberSignInManagerTests
{
private Mock<ILogger<SignInManager<MemberIdentityUser>>> _mockLogger;
private readonly Mock<MemberManager> _memberManager = MockMemberManager();
public UserClaimsPrincipalFactory<MemberIdentityUser> CreateClaimsFactory(MemberManager userMgr)
=> new UserClaimsPrincipalFactory<MemberIdentityUser>(userMgr, Options.Create(new IdentityOptions()));
public MemberSignInManager CreateSut()
{
// This all needs to be setup because internally aspnet resolves a bunch
// of services from the HttpContext.RequestServices.
var serviceProviderFactory = new DefaultServiceProviderFactory();
var serviceCollection = new ServiceCollection();
serviceCollection
.AddLogging()
.AddAuthentication()
.AddCookie(IdentityConstants.ApplicationScheme)
.AddCookie(IdentityConstants.ExternalScheme, o =>
{
o.Cookie.Name = IdentityConstants.ExternalScheme;
o.ExpireTimeSpan = TimeSpan.FromMinutes(5);
})
.AddCookie(IdentityConstants.TwoFactorUserIdScheme, o =>
{
o.Cookie.Name = IdentityConstants.TwoFactorUserIdScheme;
o.ExpireTimeSpan = TimeSpan.FromMinutes(5);
})
.AddCookie(IdentityConstants.TwoFactorRememberMeScheme, o =>
{
o.Cookie.Name = IdentityConstants.TwoFactorRememberMeScheme;
o.ExpireTimeSpan = TimeSpan.FromMinutes(5);
});
IServiceProvider serviceProvider = serviceProviderFactory.CreateServiceProvider(serviceCollection);
var httpContextFactory = new DefaultHttpContextFactory(serviceProvider);
IFeatureCollection features = new DefaultHttpContext().Features;
features.Set<IHttpConnectionFeature>(new HttpConnectionFeature
{
LocalIpAddress = IPAddress.Parse("127.0.0.1")
});
HttpContext httpContext = httpContextFactory.Create(features);
_mockLogger = new Mock<ILogger<SignInManager<MemberIdentityUser>>>();
return new MemberSignInManager(
_memberManager.Object,
Mock.Of<IHttpContextAccessor>(x => x.HttpContext == httpContext),
CreateClaimsFactory(_memberManager.Object),
Mock.Of<IOptions<IdentityOptions>>(),
_mockLogger.Object,
Mock.Of<IAuthenticationSchemeProvider>(),
Mock.Of<IUserConfirmation<MemberIdentityUser>>(),
Mock.Of<IMemberExternalLoginProviders>(),
Mock.Of<IEventAggregator>()
);
}
private static Mock<MemberManager> MockMemberManager()
=> new Mock<MemberManager>(
Mock.Of<IIpResolver>(),
Mock.Of<IMemberUserStore>(),
Options.Create(new IdentityOptions()),
Mock.Of<IPasswordHasher<MemberIdentityUser>>(),
Enumerable.Empty<IUserValidator<MemberIdentityUser>>(),
Enumerable.Empty<IPasswordValidator<MemberIdentityUser>>(),
new MembersErrorDescriber(Mock.Of<ILocalizedTextService>()),
Mock.Of<IServiceProvider>(),
Mock.Of<ILogger<UserManager<MemberIdentityUser>>>(),
Options.Create(new MemberPasswordConfigurationSettings()),
Mock.Of<IPublicAccessService>(),
Mock.Of<IHttpContextAccessor>());
[Test]
public async Task WhenPasswordSignInAsyncIsCalled_AndEverythingIsSetup_ThenASignInResultSucceededShouldBeReturnedAsync()
{
//arrange
var userId = "bo8w3d32q9b98";
MemberSignInManager sut = CreateSut();
var fakeUser = new MemberIdentityUser(777)
{
UserName = "TestUser",
};
var password = "testPassword";
var lockoutOnFailure = false;
var isPersistent = true;
_memberManager.Setup(x => x.GetUserIdAsync(It.IsAny<MemberIdentityUser>())).ReturnsAsync(userId);
_memberManager.Setup(x => x.GetUserNameAsync(It.IsAny<MemberIdentityUser>())).ReturnsAsync(fakeUser.UserName);
_memberManager.Setup(x => x.FindByNameAsync(It.IsAny<string>())).ReturnsAsync(fakeUser);
_memberManager.Setup(x => x.CheckPasswordAsync(fakeUser, password)).ReturnsAsync(true);
_memberManager.Setup(x => x.IsEmailConfirmedAsync(fakeUser)).ReturnsAsync(true);
_memberManager.Setup(x => x.IsLockedOutAsync(fakeUser)).ReturnsAsync(false);
//act
SignInResult actual = await sut.PasswordSignInAsync(fakeUser, password, isPersistent, lockoutOnFailure);
//assert
Assert.IsTrue(actual.Succeeded);
}
[Test]
public async Task WhenPasswordSignInAsyncIsCalled_AndTheResultFails_ThenASignInFailedResultShouldBeReturnedAsync()
{
//arrange
MemberSignInManager sut = CreateSut();
var fakeUser = new MemberIdentityUser(777)
{
UserName = "TestUser",
};
var password = "testPassword";
var lockoutOnFailure = false;
var isPersistent = true;
//act
SignInResult actual = await sut.PasswordSignInAsync(fakeUser, password, isPersistent, lockoutOnFailure);
//assert
Assert.IsFalse(actual.Succeeded);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,764 |
Hicham Zerouali (; born 17 January 1977 – 5 December 2004), nicknamed 'Zero' or the 'Moroccan Magician', was a Moroccan footballer. He played as a forward for clubs in Morocco, Scotland and the United Arab Emirates. He played internationally for Morocco with 7 caps.
Club career
Aberdeen
After playing in his native Morocco for Yaakoub El Mansouria and USP Police, Zerouali was signed by Aberdeen manager Ebbe Skovdahl from FUS de Rabat in November 1999 for a fee of £450,000, on the recommendation of the club's director of football, Keith Burkinshaw. He became the first player in Scotland to wear the shirt number '0' in 2000, which was outlawed the following season by the Scottish Premier League and the Premier League in England.
In January 2000, Zerouali scored a thirty-yard free kick in a fourth round Scottish Cup tie away to St Mirren to take the tie to a replay. Zerouali then scored in the 2–0 replay win at Pittodrie to help the club into the next round of the competition. Aberdeen reached both the 2000 Scottish League Cup Final and the 2000 Scottish Cup Final with Zerouali playing in both matches, although they ended in defeat to Celtic and Rangers respectively.
In August 2000, Zerouali was injured with a broken ankle during a match against Motherwell, and subsequently missed out on a place at the Sydney Olympic Games, and was eventually out of action for a full year. In one of his more memorable outings after his return, he scored a hat-trick against Dundee.
Al-Nassr and FAR Rabat
After his contract at Aberdeen expired, he then moved to play his football in the United Arab Emirates with the team Al-Nassr for a year, before returning to live in his home country of Morocco in 2003, signing for FAR Rabat, where he won the Coupe du Trône that year.
International career
Zerouali won 7 caps for the Morocco national football team and scored three goals. He featured for Morocco in the 2002 Africa Cup of Nations in Mali, and scored two goals in a 2–1 win over Burkina Faso. He also played in the 1997 FIFA World Youth Championship. He was in the international squad a month before his death.
Death
Zerouali was killed in a car accident in Rabat in December 2004 aged 27. Only the previous Saturday, he had scored two goals in a league game for his club. He left a daughter with his girlfriend in Aberdeen.
A firm fans' favourite at Aberdeen, he was affectionately known as 'Zero' to the supporters. A memorial and tribute was held at Pittodrie Stadium after he died, which was attended by thousands of fans, despite him not playing for the club at the time and only having spent a short period there.
References
External links
https://www.worldfootball.net/player_summary/hicham-zerouali/
1977 births
2004 deaths
Moroccan footballers
Footballers from Rabat
Morocco under-20 international footballers
Morocco international footballers
Association football forwards
Fath Union Sport players
Aberdeen F.C. players
Al Nassr FC players
AS FAR (football) players
Road incident deaths in Morocco
Moroccan expatriate footballers
Scottish Premier League players
Expatriate footballers in Scotland
Expatriate footballers in the United Arab Emirates
2002 African Cup of Nations players
Moroccan expatriate sportspeople in Scotland
Moroccan expatriate sportspeople in the United Arab Emirates
Saudi Professional League players | {
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Acid Reflux In Babies Videos
Acid reflux is a common ailment that affects many people. Here are six home remedies for acid reflux that can help provide relief.
But in recent years, when babies appear to be in real discomfort, there has been a tendency among doctors to prescribe the same kinds of medicines for them that adults use for acid reflux. Now, new research suggests that the drugs may.
For many years, pediatricians have wondered if colic might actually be a burning pain caused by acid reflux in babies (when stomach acid squirts back up the wrong way also known as gastroesophageal reflux, or GER). One book even trumpeted it as "the cause of all colic." But hundreds of millions of dollars spent on baby.
If babies are having fever, it's more likely due to an infection. We sort of have to just manage it. "I have acid reflux. I cough a lot when I go to bed. Is that normal?" Dr. Riley: Actually, acid reflux is one of the common causes of chronic.
More baby videos. How can I prevent. Hi cristy2012,where can i buy the babies magic tea?my baby cannot sleep well,i suspected its an acid reflux.is your baby got.
One out of four babies in the world suffers from gastroesophageal reflux (GER) or , to put it simply, acid reflux. For Acid reflux in babies, it is caused by the regurgitating of acids and contents of the stomach back into the esophagus. Naturally, there is a. To learn more about treating infant acid reflux, watch the video below:.
Reflux & Reflux disease/ allergy / Intolerance & Gi issues in babies and young children. Information and support for parents and caregiver of babies and young children with related Reflux issues. All information provided has been from my own personal experience as a mother to a child with reflux disease. Watch videos on.
Do you suspect your child has stomach acid problems, do you have a silent reflux baby? It's not. It is fairly common for babies to experience multiple degrees of reflux, and all due to the muscular valves strength at the end of their food pipe. This is yet to fully. In Babies. Here's a short video with some interesting points too.
Many children respond to natural cures for acid reflux such as apple cider vinegar, dietary changes, and coconut oil.
Gerd History Gastro-esophageal reflux disease (GERD) is one of the most common disorders of the digestive tract. The two most typical symptoms are heartburn and regurgitation of. Gerd-Rainer Horn. Chercheur permanent. Professor of political History at Sciences Po Paris. Academic Work Experience. Professeur d'Histoire Politique, Institut d'Études Politiques de Paris (Sciences Po), 2013; Professor, Department of History, University of Warwick, 2010 -2012; Senior Lecturer, Department of History, a history of narrowing of the esophagus. The test might also be appropriate if you are a man 50 or older with GERD symptoms for more than five years plus other risk factors such as
Manage Frequent Spit-Up with Help from Enfamil A.R.™. Try it Today! Learn More.
Like adults, babies can also suffer from frequent acid refluxes. But there are several home remedies that can help them get relief.
Spit up and sleep deprivation come with newborns, but a growing number of parents are seeking acid reflux medications to ease their babies' blues. The U.S. Food and Drug Administration has not approved some of the most common acid.
What are the causes of acid reflux in infants? Find out what the symptoms are of acid reflux in children.
According to paediatric gastroenterologist Dr Bryan Vartabedian, from Texas Children's Hospital, author of Colic Solved and the father of two babies with acid reflux, babies at extreme ends of this spectrum – that is, the happy chuckers.
Acid Reflux: Newborn Enemy #1 – Mommy Shorts – I think this is my favorite travel video so far. YouTube link in bio!. My middle child had acid reflux when he was a baby and also spit up everywhere.
but the trend is neither based in science nor beneficial to babies, a commentary suggests. Advertising for acid reflux medications, misinterpretations of infants' normal responses and an over- medicalized culture are to blame,
Babies spit up, but when do you need to worry about it? GastroKids explains the difference between normal spitting up and reflux and GERD
I am worried I may have reflux. Could you explain what it is? Heartburn and acid reflux can be very common symptoms. It can be prevented by vaccination with the routine baby immunisations given as part of the childhood vaccination.
What are the causes of acid reflux in infants?. it is easier for the acid to come up, thus causing reflux. Videos You May Like.
Baby videos; Toddler videos. A baby may have reflux when her esophageal sphincter. Some babies respond right away to antacids or acid blockers.
Oct 20, 2011 · Are Too Many Babies Getting Acid Reflux Drugs? Pediatrician Suggests Many Healthy Infants Are Prescribed GERD Drugs Unnecessarily
Treating Acid Reflux and GERD Naturally. If your gastro-intestinal tract is like plumbing, then low stomach acid is like a drain clog that is still draining a.
We love posting cute videos of newborn and infant babies doing all sorts of sweet, fun and crazy things because we love to celebrate life. But we love pregnant moms too. They are heroes in more ways than one because of the.
In a video that's now been watched more than six million. Folding the arms echoes the effect of swaddling, which is known to reduce fussiness; holding a baby at an upright angle can ease acid reflux; and gentle motion (never shake a.
In a few children, particularly those who have reflux complications or GORD, doctors might recommend pH monitoring (checking the level of acid) to see whether there's too much acid from the stomach in the oesophagus. This is done by placing a tiny tube with a sensor in the baby's oesophagus for 24 hours. Sometimes the.
A look at gastroesophageal reflux disease — also called GERD — in infants and children.
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However, experts warn that tens of thousands of babies will be left with painful symptoms of acid reflux as a result of the new guidelines. Four in ten babies will 'spit-up' or posset a small amount of their milk feed back up soon after.
Acid Reflux In Babies Videos – Metacafe – One of the world's largest video sites, serving the best videos, funniest movies and clips.
Indeed, research suggests that proton pump inhibitors do very little to help reflux in babies and children. Some experts believe this is because the real problem is the baby's anatomy, or an allergy (and not stomach acid). Reflux is often.
"Some pregnant women have such severe acid reflux they can't eat because they are in so much pain," Hait says. "This is obviously not good for the mother or the baby. So each pregnant woman suffering from acid reflux, with the.
The signs of silent reflux in babies aren't always obvious. Videos; Newsletter; Baby. Regular reflux occurs when food and digestive acid leaks up from the.
GERD: Acid Reflux Symptoms, Treatment & Heartburn. – Learn about gastroesophageal reflux disease (GERD, acid reflux, heartburn) symptoms like heartburn, chest pain, regurgitation, and nausea. Diet, causes, diagnosis.
Acid reflux in babies, also called Gastroesophageal Reflux, (GER) occurs when the muscle at the entrance of the stomach, is not strong enough to keep the acids inside. This then leads to pain, and results in a. nerve controls the digestive system. Our page on massage advantages, techniques and step by step videos.
Is it a Colic Cry or Reflux Cry? Whether your baby has colic or. Dr. Harvey Karp's Happiest Baby book and video. a patient expert for Acid Reflux.
Videos & Tools; Español. Acid Reflux (GER & GERD) in Infants. Get Reflux in Infants updates by email What's this? GO. National Institutes of Health.
The American Academy of Pediatrics, in fact, has advised against using acid-blockers to treat infant reflux that doesn't affect growth or breathing, and physicians seem to be taking note and avoiding such medications. But plenty still.
When you think of reflux. a feeding specialist because that baby has developed oral aversion," he said. "They are so used to being in pain and then they feed, which relieves the pain temporarily, then that causes acid to reproduce and.
Despite this recent spike in prescriptions, Hassall argues that the vast majority of infants don't have gastroesophageal reflux disease (GERD), which is what these drugs are meant to treat. "There was no good medical reason" for this.
Acid Reflux in Babies A video of my son who was 1 month at the time of video suffering from a reflux "episode". If you have a baby that has been having.
Acid-suppressing medications such as proton pump inhibitors are often prescribed to infants who frequently spit up, are irritable and cry. But gastroesophageal reflux disease or GERD is not usually the cause of the symptoms in otherwise.
Learn from WebMD about acid reflux in infants and children, including causes, diagnosis, and treatment.
I've taken countless videos of my daughter since. which could signal anything from acid reflux to a heart attack. Likewise, there's probably not one single cause for nonstop crying in babies. Beyond that, the blessing and the curse.
After a follow-up visit to her pediatrician, Roach says the doctor explained it was caused by acid reflux. "The stomach acid often causes babies to hold their breath and have what is called a vagal event," says Dr. Garnet Maharajh, a.
The signs of silent reflux in babies aren't. Symptoms and remedies for silent. Regular reflux occurs when food and digestive acid leaks up from.
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It is common for babies to spit up a few times a day. But if a child often vomits, it can lead to other problems or symptoms. Frequent spitting up or vomiting in children is often due to acid reflux, also called gastroesophageal.
Infant reflux — Comprehensive overview covers symptoms, treatment of this messy, but common, condition.
Nov 11, 2011. Babies will be babies, said Dr. Eric Hassall, staff pediatric gastroenterologist at Sutter Pacific Medical Foundation in San Francisco. They'll cry and spit up. Hassall reiterates that point in a commentary published in the Journal of Pediatrics in which he expresses worry over the number of parents who give.
Last Updated on: January 30th, 2018 at 8:38 pm, by | {
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{"url":"https:\/\/www.zora.uzh.ch\/id\/eprint\/160183\/","text":"# Electroweak production of two jets in association with a Z boson in proton\u2013proton collisions at $\\sqrt s$ = 13 TeV\n\n## Abstract\n\nA measurement of the electroweak (EW) production of two jets in association with a Z boson in proton-proton collisions at s\u221a=13TeV is presented, based on data recorded in 2016 by the CMS experiment at the LHC corresponding to an integrated luminosity of 35.9fb--1. The measurement is performed in the \u2113\u2113jj final state with \u2113 including electrons and muons, and the jets j corresponding to the quarks produced in the hard interaction. The measured cross section in a kinematic region defined by invariant masses m\u2113\u2113>50GeV, mjj>120GeV, and transverse momenta pTj>25GeV is \u03c3EW(\u2113\u2113jj)=534\u00b120(stat)\u00b157(syst) fb, in agreement with leading-order standard model predictions. The final state is also used to perform a search for anomalous trilinear gauge couplings. No evidence is found and limits on anomalous trilinear gauge couplings associated with dimension-six operators are given in the framework of an effective field theory. The corresponding 95% confidence level intervals are \u22122.6<cWWW\/\u039b2<2.6TeV\u22122 and \u22128.4<cW\/\u039b2<10.1TeV\u22122. The additional jet activity of events in a signal-enriched region is also studied, and the measurements are in agreement with predictions.\n\n## Abstract\n\nA measurement of the electroweak (EW) production of two jets in association with a Z boson in proton-proton collisions at s\u221a=13TeV is presented, based on data recorded in 2016 by the CMS experiment at the LHC corresponding to an integrated luminosity of 35.9fb--1. The measurement is performed in the \u2113\u2113jj final state with \u2113 including electrons and muons, and the jets j corresponding to the quarks produced in the hard interaction. The measured cross section in a kinematic region defined by invariant masses m\u2113\u2113>50GeV, mjj>120GeV, and transverse momenta pTj>25GeV is \u03c3EW(\u2113\u2113jj)=534\u00b120(stat)\u00b157(syst) fb, in agreement with leading-order standard model predictions. The final state is also used to perform a search for anomalous trilinear gauge couplings. No evidence is found and limits on anomalous trilinear gauge couplings associated with dimension-six operators are given in the framework of an effective field theory. The corresponding 95% confidence level intervals are \u22122.6<cWWW\/\u039b2<2.6TeV\u22122 and \u22128.4<cW\/\u039b2<10.1TeV\u22122. The additional jet activity of events in a signal-enriched region is also studied, and the measurements are in agreement with predictions.\n\n## Statistics\n\n### Citations\n\nDimensions.ai Metrics\n3 citations in Web of Science\u00ae\n1 citation in Scopus\u00ae\n\n### Altmetrics\n\nDetailed statistics","date":"2019-04-22 01:12:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.8634430170059204, \"perplexity\": 1386.2015956193486}, \"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-2019-18\/segments\/1555578532948.2\/warc\/CC-MAIN-20190421235818-20190422021818-00175.warc.gz\"}"} | null | null |
{"url":"http:\/\/jafeb.org\/journal\/article.php?code=66686","text":"Journal Search Engine\nISSN : 2288-4637(Print)\nISSN : 2288-4645(Online)\nThe Journal of Asian Finance, Economics and Business Vol.6 No.2 pp.257-267\nDOI : https:\/\/doi.org\/10.13106\/jafeb.2019.vol6.no2.257\n\n# Long-run and Short-run Causality from Exchange Rates to the Korea Composite Stock Price Index\n\nJung Wan LEE1,Tantatape BRAHMASRENE2\n* The paper with the title [\u201cAssessing Dynamic Relationship between Stock Prices and Exchange Rates in Korea\u201d] was presented initially at 2014 International Conference on Business and Information (BAI2014) that was held in Osaka, Japan, July 03\u201005, 2014. The paper has been recognized as one of Best Paper Awards at BAI2014 conference. This paper is a substantially revised and expanded version of the paper presented at BAI2014 conference. The authors have taken into account all the comments of Editors, Session Chairs and Reviewers in the revised manuscript. The authors greatly appreciate Editors, Session Chairs and Reviewers for their valuable comments, interest in and support of this research.\n1 First Author. Assistant Professor, Administrative Sciences Department, Metropolitan College, Boston University, Boston, United States. E-mail: jwlee119@bu.edu\n2 Corresponding Author. Professor of Finance & International Business, College of Business, Purdue University Northwest [Postal Address: 1401 S. US. Hwy. 421, Westville, IN 46391, United States] E-mail: tapeb@pnw.edu\nApril 7, 2019 April 14, 2019 April 20, 2019\n\n## 2. Literature Review and Hypotheses\n\n### 2.1. Long-run Relationship between Exchange Rates and Stock Prices\n\nPrevious studies have addressed different findings regarding the causal relations between stock prices and exchange rates in various countries. For example, Nieh and Lee (2001) found no long-run relationship between stock prices and exchange rates in the Group-7 countries. Yau and Nieh (2006) suggested no clear long-run relation between the new Taiwan dollar and the Japanese yen exchange rates and the stock prices of Taiwan and Japan. Zhao (2010) did not find a stable long-run relationship between the Chinese yuan real effective exchange rates and stock prices. Ramasamy and Yeung (2002) indicated inconsistent results for bidirectional causality between stock prices and exchange rates for six Asian countries over the period of 1995 and 2001. Kutty (2010) found that stock prices Granger caused exchange rates in the short run but there was not a long-run significant relationship in Mexico between January 1989 and December 2006. Other studies from Griffin and Stulz (2001), Fernandez (2006), Fernandes (2009), Hartmann and Pierdzioch (2007), and Zhao (2010) suggest no relationship between exchange rates and stock prices.\n\nOn the other hand, the long-run relationship has been confirmed in some studies. For example, Lee and Zhao (2014) report that there exists negative long-run causality from foreign exchange rates to the Shanghai Stock Exchange Composite Index using monthly data from January 2002 to December 2012 of the People\u2019s Republic of China. Yau and Nieh (2009) found a long-run and asymmetric causal relationship between the exchange rates of the new Taiwan dollar and the Japanese yen and their stock prices in Japan and Taiwan. Whether empirically or theoretically, the above studies have suggested a significant relationship between exchange rates and stock prices. However, the results have been mixed for the sign and causal direction between exchange rates and stock prices. Thus, the following hypothesis is advanced:\n\nHypothesis 1: Exchange rates lead stock price dynamics in the long-run in the Korean stock market.\n\n### 2.2. Short-run Relationship between Exchange Rates and Stock Prices\n\nPhylaktis and Ravazzolo (2005) employed cointegration and multivariate Granger causality tests that resulted in positive short-run causality between stock prices and exchange rates in some Pacific Basin countries. Aloui (2007) indicated that movements of stock prices affect the exchange rate dynamics for the two periods pre- and post-Euro in the United States and Western European markets. Pan, Fok and Liu (2007) reported a causal relation from exchange rates to stock prices for East Asian countries. Yang and Doong (2004) suggested exchange rate changes directly impacted future changes of stock prices for the Group-7 countries from 1979 to 1999. Nandha and Hammoudeh (2007) argued that stock prices were affected by changes in the exchange rate for nine Asia-Pacific countries while Wu (2001) showed Singapore dollar exchange rates Granger caused stock prices. Further research emphasizing a positive relationship between exchange rates and stock prices can be found in Chiang and Yang (2003), Ratanapakorn and Sharma (2007), Kolari, Moorman, and Sorescu (2008), Aydemir and Demirhan (2009), Ning (2010), and Eichler (2011). Mun (2007) specified that higher exchange rate variability mostly increases local stock market volatility, but decreases volatility for the United States stock markets. Exchange rate exposure has negative and significant impact on emerging market stock returns in a study by Chue and Cook (2008) while the S&P 500 stock price is negatively related to the real exchange rates in Kim (2003)\u2019s research. Lee and Zhao (2014) report that there exists negative short-run causality from foreign exchange rates i.e. the Chinese yuan\/the Korean won and the Chinese yuan\/the Japanese yen, to the Shanghai Stock Exchange Composite Index using monthly data from January 2002 to December 2012 of the People\u2019s Republic of China.\u00a0 Thus far, the relationship between stock prices and exchange rates is still inconclusive. East Asian countries have strong economic relationships with each other within the intraregion as well as the United States and European countries (Ogawa & Yang, 2008). Therefore, further study is needed to elucidate the causality between stock prices and exchange rates. Hence:\n\nHypothesis 2: In the short-run, the stock price dynamics in the Korean stock market (KOSPI) are affected by the nominal exchange rates of the Korean won per the U.S. dollar (USD), the Korean won per the Euro (EUR), the Korean won per the Japanese yen (JPY) and the Korean won per the Chinese yuan (CNY).\n\n### 2.3. The Impact of Global Financial Crises on Stock Market Returns\n\nThe linkage between exchange rates and stock prices vary across economies with respect to exchange rate regimes, the trade size, the degree of capital control and the size of equity market (Pan et al., 2007). Mercereau (2006) suggested that the financial structure of an equity market influenced its real exchange rate, as well as the volatility of this exchange rate, whereas Walid, Chaker, Masood, and Fry (2011) asserted that the stock price volatility responded asymmetrically to events in the foreign exchange market. Diamandis and Drakos (2011) found that there is a significant long-run relationship between the local stock market and the exchange rate market, but that the stability of the relationship is affected by financial and currency shocks such as the Mexican currency crisis in 1994 and the global financial crisis from 2007 to 2009. In addition, the process of creating the Mercosur between Argentina, Brazil, Paraguay and Uruguay in Latin American countries led to the local currency devaluation. These exchange rate movements have substantial negative impact on the respective stock prices (Allegret & Sand-Zantman, 2009; Alvarez-Plata & Schrooten, 2004; Camarero, Flores, & Tamarit, 2006).\n\nIn spite of existing coupling-decoupling effect, some researchers find markets integrate immediately after financial crisis. For example, Kim, Lee and Park (2011) claimed that after the Asian financial crisis of 1997-1998, the degree of both regional and global dependence have increased drastically. Latin America also experience financial crisis shortly after the end of Asian financial crisis. Frijns, Tourani-Rad, and Indriawan (2012) suggested that Latin American countries integrate largely after the dot-com bubble. Their results are line up with empirical results of Liu and Wan (2012)\u2019s research, which shows there are neither linear nor nonlinear causality between Shanghai Composite Index and exchange rates before current financial crisis, but exchange rate can either linearly or nonlinearly impact Shanghai Composite Index after that financial crisis. Empirical evidence from these three research might imply that if there are negative or positive correlations between exchange rate and stock prices, the strength of these correlations may be enhanced after financial crises.\n\nDue to geographic factors, the relation between stock prices and exchange rate is observed regionally within a specific period of time. According to Pan et al. (2007), during the period of Asian financial crisis, causality from exchange rates to stock prices is founded in Hong Kong, Japan, Korea, Singapore, Taiwan and Thailand, except Malaysia; while no causality from stock prices to exchange rate has been observed in countries mentioned above. Nonetheless, Hatemi-J, and Roca (2005) reported that the two variables are significantly linked in the non-crisis period, but not at all during the 1997 Asian financial crisis period for the Association of Southeast Asian Nations (ASEAN) countries. Choi and Park (2008) found no evidence that interest rate differentials caused spot exchange rates in Indonesia, Korea and Thailand during the 1997 Asian financial crisis. Furthermore, Japanese foreign direct investment declined with a depreciation of the Japanese yen against host country currencies in Asia but with little effect from the 1997 Asian financial crisis (Takagi & Shi, 2011). On the other hand, Liao, Chu, and You (2011) examined 30 Taiwan categorical stock indexes and 13 historic data of foreign exchange index, and found a significantly inverse correlation between them. Moreover, Liao et al. (2011) also suggested that the foreign exchange correlation rules should be examined and discussed into two categories: appreciation term and depreciation term. As a complement to the previous points, Fung and Liu (2009) discovered that a real depreciation of the New Taiwanese dollar led to an increase in exports, domestic sales, total sales and productivity during the 1997 Asian financial crisis.\n\nGlobal financial shocks impacted exchange rate and stock prices simultaneously as well. Dooley and Hutchison (2009) evaluated the transmission effect of U.S. subprime crisis to emerging economies and found widely insulation and decoupling effect in emerging economies during early 2007 and mid 2008. But quickly after summer 2008, the linkages between exchange rate and stock prices re-appear dramatically. To be noticed, the timing of linkages re-emergence is uniform and remarkable within most of countries in which they observe. In addition, with much specification, Wen, Wei, and Huang (2012) found contagion effect between crude oil and U.S.\/Chinese stock market during most recent financial crisis. Hence, exchange rates and stock prices are correlated in a complicated manner (Kim, Yoon, & Kim, 2004; Tastan, 2006). Market interactions may destabilize stock markets, but may also play a stabilizing effect on the exchange rate market (Dieci & Westerhoff, 2010). In light of these findings, the following hypothesis is developed:\n\nHypothesis 3: The global financial crises of 1997-1998 and 2007-2009 have impacted the stock price dynamics in the Korean stock market.\n\n## 3. Empirical Specification\n\nEngle and Granger (1987) and Granger (1988) reported that if two or more variables are cointegrated, there always exists a corresponding error correction term in which the short-run dynamic of the variables in the system are influenced by the deviation from equilibrium. In this case, a vector error correction (VEC) model is formulated to reintroduce the information lost in the differencing process, thereby capturing long-run dynamics as well as short-run dynamics. The VEC model implies that changes in one variable are a function of the level of disequilibrium in the cointegrating relationship (captured by the error correction term), as well as changes in other explanatory variables. Thus, the VEC model is useful for capturing both the short-run and the long-run dynamics when the variables are cointegrated.\n\nThe following equation illustrates a multivariate vector autoregressive model with the error correction term. This equation will be used to test short-run and long-run Granger causation from exchange rates to the Korean stock market price.\n\nwhere:\n\n\u0394 indicates the difference operator;\n\n\u03b1 is the deterministic component;\n\n\u019f, \u03b2 and \u03b6 denote the parameters to be estimated;\n\net follows stationary random errors with mean zero, that is, white noise;\n\nj is the lag length;\n\nt represents 1, 2, 3, \u2026, n observation;\n\nECTt is the error correction term obtained from the cointegrating vectors deriving from the long-run cointegrating relationship via the Johansen maximum likelihood procedure.\n\nFC represents an exogenous variable of the global financial crises of 1997-1998 and 2007-2009.\n\nM2 is the broad definition of money supply.\n\nCPI refers to the inflation rate.\n\nINT represents the interest rate.\n\nThe long-run causality is indicated by the significance of t-statistics of the lagged error correction term (i.e. by testing null hypothesis: \u03b6 = 0). The asymptotic variance of the estimator is estimated so that the t-statistics have asymptotic standard normal distribution. Asymptotic t-statistics can be interpreted the same way as t-statistics. They are used to interpret the statistical significance of coefficients of the lagged error correction term.\n\n## 4.Data and Methodology\n\nThis section describes the data and outlines the methodology used in the selection of indicators and the normalization of data. The sample is restricted to the period in which monthly data are available from January 1986 to June 2018 (390 observations). All of the monthly time series data below has been collected and retrieved from the Economic Statistics System database sponsored by the Bank of Korea.\n\n### 4.1. Endogenous Variables\n\nKorea Stock Exchange. The Korea Composite Stock Price Index (symbol: KOSPI) is used as a proxy of the Korean stock market prices. This major stock market index tracks the general performance of all common shares listed on the Korean Stock Exchange. The KOSPI index is a capitalization-weighted index of all common shares on the Korean Stock Exchange. The index was developed with a base value of 100 as of January 4, 1980. Monthly average of this index is used in this study.\n\nForeign Exchange Rates. Based on the volume of international trade with South Korea, four major exchange rates are selected: the Korean won per the U.S. dollar (USD), the Korean won per the Euro (EUR), the Korean won per the Japanese yen (JPY) and the Korean won per the Chinese yuan (CNY). The time series data is a monthly adjusted average. In effect, USD, EUR, JPY and CNY are the logarithm of the nominal exchange rate of Korean won per U.S. dollar, Euro, Japanese yen and Chinese yuan, respectively.\n\nMacroeconomic Variables. Money supply (M2), inflation, and short-term interest rates are included in the model to address the omitted variable bias. In the empirical model, M2 is the logarithm of changes in the broad definition of money supply. Inflation is the logarithm of the consumer price index (CPI), a proxy of inflation rates. Interest rates (INT) are the logarithm of the short-term interest rate i.e. Certificate of Deposit 90 days.\n\n### 4.2. Exogenous Variable\n\nGlobal Financial Crises of 1997-1998 and 2007-2009. The global financial crises of 1997-1998 and 2007-2009 may cause structural breaks in the trend of stock prices in the Korean stock market, see Figure 1.\n\nBecause the global financial crisis would introduce some changes in the implementation of monetary policy, for example, from money targeting to interest rate targeting, this would introduce substantial instability in the system. An exogenous variable is used to examine the impact of the global financial crises of 1997-1998 and 2007-2009 on the Korean stock market. An exogenous variable (i.e. a financial crisis) is assumed to have systematically affected changes in the endogenous variables in the model (Brahmasrene, Huang, & Sissoko, 2014). A dummy variable with a value of zero will cause the variable\u2019s coefficient to disappear, and a dummy with a value one will cause the coefficient to act as a supplemental intercept in the regression model. Hence, the global financial crisis variable equals one if the period falls on either between January 1997 and June 2000 or between August 2006 and December 2010, and zero otherwise.\n\n### 4.3. Normalization and Transformation\n\nThe normalization of the data is necessary to transform the values to the same unit of measurement since KOSPI is presented as the base value of 100, while exchange rates are presented in the Korean won. Therefore, transformation into a natural log mitigates possible distortions of dynamic properties of the series. Log transformation is a preferred method since each resulting coefficient in the regression equation represents elasticity, which is the ratio of the incremental change of the logarithm of a function with respect to an incremental change of the logarithm of the argument.\n\nSelected tests on time series data as prescribed in the references in each section below are necessary for statistical accuracy and to avoid spurious results. To ascertain the order of integration of the variables, this study applied the augmented Dickey-Fuller (Dickey & Fuller, 1981) unit root test and the Phillips\u2013Perron (Phillips & Perron, 1988) test. The augmented Dickey-Fuller test assumes the errors to be independent and have constant variance while the Phillips\u2013Perron test allows for fairly mild assumptions about the distribution of errors. The two unit root tests are carried to test the null hypothesis of the unit root in the level and the first difference of the variables. All test equations were tested by the method of least squares, including an intercept and no time trend in the model. In the unit root tests, an optimal lag in the tests is automatically selected based on Schwarz information criterion and the lag length in the tests is automatically selected based on the Newey-West estimator using the Bartlett Kernel function.\n\nTable 1 reports the results of the two unit root tests. Table 1 indicates the null hypothesis of a unit root cannot be rejected in the level for KOSPI, USD, JPY, EUR and CNY. After first differencing, all null hypothesis of a unit root is rejected at the one percent significance level for these variables. The results in Table 1 unanimously confirm that all variables are integrated of order one or I(1).\n\nCheung and Ng (1998) reported that the Johansen approach is more efficient than the Engle-Granger single equation test method because the maximum likelihood procedure has large and finite sample properties. The Johansen\u2019s approach (1988) derives maximum likelihood estimators of the cointegrating vectors for an autoregressive process with independent errors. The Johansen cointegration test represents each variable as a function of all the lagged endogenous variables in the system. It uses two ratio tests, a trace test and a maximum eigenvalue test, to examine the number of cointegration relationships. Both tests could be used to determine the number of cointegrating vectors present, although they do not always indicate the same number of cointegrating vectors.\n\nTable 2 reports the results of the Johansen cointegration test. The model is tested by the method of least squares. The regression model allows for a linear deterministic trend in data and includes the intercept but no trend in the vector autoregressive model. Trace test statistics indicate that at least four cointegrating equations exist at the 0.05 level while maximum eigenvalue test statistic indicates that at least one cointegrating equation at the 0.01 level. The results of the Johansen cointegration test confirm that there exists cointegration between stock prices and foreign exchange rates in the Korean stock market. In this case, an unrestricted vector autoregressive model would not be an effective option for testing the short-run and long-run dynamics.\n\n## 5. Empirical Results\n\nStatistical inference is sensitive to parameter instability, serial correlation in residuals and residual skewness. Table 3 reports the results of VEC estimates, model diagnostic tests and residual diagnostic tests. Skewness of the series is not significantly different from a normal distribution. Histogram normality Jarque-Bera test (null hypothesis: residuals are multivariate normal) is not rejected. Breusch-Godfrey serial correlation Lagrange multiplier or LM test (null hypothesis: no serial correlation at lag order 2) is not rejected. Heteroskedasticity test (null hypothesis: no autoregressive conditional heteroskedasticity or ARCH effect at lag order 1) is not rejected. Thus, this model yields acceptable results. There are considerably fewer outliers and the fluctuation bands are smaller (Figure 2).\n\nThe long-run causality is determined by the error correction term. If the coefficient of the error correction term is significant, then it indicates the evidence of the long-run causality from the explanatory variables to the dependent variable (Toda & Phillips, 1993, 1994). This contains the long-run causal relationship information because it is derived from the cointegrating equation. Table 3 reports the results of VECM estimates. The significance of the coefficient of ECT(t-1) indicates that the lagged structure of ECT(t-1) is unstable, that is, the ECT has a negative effect on the Korean stock market in the long-run dynamics (see Figure 3).\n\nTable 3 reports the results of VECM estimates. Recall hypothesis 1 that exchange rates lead to stock price dynamics in the long-run. For the long-run causality, the results of VEC estimates indicate that the coefficient of the error correction term is significant at the 0.01 level. The results indicate that exchange rates have caused a negative effect to KOSPI in the long-run. The observation confirms that the long-run causality from exchange rates to the Korean stock market exists.\n\nThe short-run causality in the VEC model can be tested by the block exogeneity Wald test. The block exogeneity Wald test in the VEC system provides Chi-square statistics of coefficients on the lagged endogenous variables. These are used to interpret the statistical significance of coefficients of the endogenous variables. The hypothesis in this test is that the lagged endogenous variable does not Granger cause the dependent variable. Hypothesis 2 states that in the short-run, the stock price dynamics in the Korean stock market (KOSPI) are affected by the nominal exchange rates of the Korean won per the U.S. dollar (USD), the Korean won per the Euro (EUR), the Korean won per the Japanese yen (JPY) and the Korean won per the Chinese yuan (CNY). For the short-run causality, the results support hypothesis 2 for the Japanese yen only. The null of hypothesis can be rejected at the 0.05 level, which means the nominal exchange rate of the Korean won per the Japanese yen leads positively stock price dynamics in the short\u2013run in the Korean stock market.\n\nFor hypothesis 3 that the global financial crises of 1997-1998 and 2007-2009 have impacted the stock price dynamics and the Korean economy, the result of VEC estimates indicates that the coefficient of the exogenous variable is significant at the 0.05 level. The observation confirms that the global financial crises and\/or external shocks have caused instability in the VEC model, that is, the financial crises have affected changes in the endogenous variables in the VEC model, especially a significant negative effect on money supply. Thus, the significance of the coefficient of the exogenous variable in VEC model 1 indicates that the endogenous variables are affected by both the exogenous shock and their own dynamics in the VEC model. When compared to the results of VEC model 2, the results of two VEC models are very similar in terms of their signs and weights, with only exception of money supply(M2(t-2)). The observation suggests that the impact of the global financial crises on the endogenous variables in the tested VEC model has been marginalized, with reservation on money supply.\n\nIn addition to the findings corresponding to the above hypotheses, the impulse responses and variance decomposition may be noteworthy in their impact. A shock to the j-th variable not only directly affects the j-th variable but is also transmitted to all of the other endogenous variables through the dynamic (lag) structure of the vector autoregressive. The effects of the shocks on the endogenous variables can be assessed by estimating impulse responses and variance decomposition functions. An impulse response function traces the effect of a one-time shock to one of the innovations on current and future values of the endogenous variables. Since innovations are usually correlated, it may be viewed as having a common component that cannot be associated with a specific variable. In order to interpret the impulses, it is common to apply a transformation to the innovations so that they become uncorrelated. The Cholesky transforming method uses the inverse of the Cholesky factor of the residual covariance matrix to orthogonalize the impulses. This method imposes an ordering of the variables in the vector autoregressive and attributes all of the effect of any common component to the variable that comes first in the vector autoregressive system. For stationary vector autoregressive models, the impulse responses should die out to zero and the accumulated responses should be asymptote to some constant.\n\nFigure 4 presents the results of the impulse responses of stock prices to Cholesky one standard deviation innovations of three significant endogenous variables: the Korean stock price (KOSPI) and the Korean won per the Japanese yen exchange rate. The graphs of impulse responses confirm that the stock price dynamics in the Korean stock market are affected largely by their own stock market (KOSPI) dynamics and, to some degree, by the nominal exchange rates of the Korean won per the Japanese yen. The response of KOSPI itself to each shock shows a positive impact in the first few months, then declines and levels off after 24 months. Hence, the impulse response of KOSPI is largely determined by its own shock. In addition, the response of KOSPI to the Japanese yen exchange rate shows a positive impact in the short-run while widening but leveling off after 24 months eventually. One might expect that the nominal exchange rate of the Korean won per the Japanese yen has a positive impact on the Korean stock market prices as the inflow in portfolio investment from Japan increases when the Korean won has depreciated against the Japanese yen.\n\n## 6. Discussion\n\nThis research contributes to the existing literature in three main aspects. First, this research uncovers the fact that there exists long-run causality from exchange rates to stock prices in the Korean stock market. To be noticed, our results do not say anything about causality from stock prices to exchange rates. This might imply that direction of causality is not considered in this research.\n\nSecond, the paper empirically explores short-term causality from exchange rates to stock prices in the Korean stock market. Short-run causality only from the Japanese yen exchange rate to the Korean stock prices is observed. This supports Ogawa and Yang (2008)\u2019s research that the monetary authorities of East Asian countries should care about not only the US dollar and the euro but also the Japanese yen, because Japan has a larger role in intra-regional economic relations.\n\nFinally, the global financial crises of 1997-1998 and 2007-2009 have some impact on the stock price dynamics and the Korean economy. In light of these findings, financial managers can enhance their understanding of the short-run movements of exchange rates and stock price dynamics. A better understanding of these short-run and long-run movements enables financial managers to make well informed investment and financial decisions. While sufficiently understand short-run and long-run movements of exchange rates and stock prices, financial managers and policymakers should as well be aware of that coupling and decoupling effect between specific currencies may exist. This paper suggests that the global financial crises have some impact on the Korean economy, especially on money supply, but it has its limitations in providing much reasoning and convincing points.\n\nRecommendations can be drawn from the findings presented in this paper. From a long-term perspective, policymakers should consider the exchange rates as a policy tool aiming at the Korean stock market since the results of VEC estimates show evidence of the long-run causation from exchange rates such as the Korean won per the U.S. dollar (USD) and the Korean won per the Japanese yen (JPY) to stock prices in the Korean stock market. However, from a short-term perspective, policymakers may consider the exchange rates as a policy tool to manage and control the Korean stock market since the exchange rate of the Korean won per the Japanese yen, to some extent, would cause a positive effect on stock prices in Korea. This policy would be effective in the short-run and should be evaluated soundly.\n\n## 7.\u00a0Conclusions\n\nThe results of this study acknowledge that: first, there exists long-run causality from exchange rates to stock prices in the Korean stock market. Second, there exists short-run causality from the Japanese yen exchange rate to the Korean stock prices. Third, the global financial crises and\/or external shocks have caused instability, to some degree, in the Korean stock market. However, the Korean stock prices are determined largely by their own dynamics in the Korean stock market. It would appear that this inter-dependent behavior of the stock market with exchange rates could be a state of the market liberalization process for its economy to heal and propel itself toward long-run stability and a better future of the Korean stock market. Therefore, policymakers and investors should devote more time and effort acquiring not only the information on changes in monetary policy and temporary external shocks in the short-run but also the knowledge and information about the market itself.\n\nFurther research may be directed at some of the above issues. The empirical model may be estimated with alternative economic and financial factors or markets. Studies encompassing different economies should be conducted where data is available. Such research should contribute toward improving our understanding of the market mechanism and responses of each market to the frequently occurring phenomena of economic and financial crisis, whether regional or global.\n\n## Reference\n\n1. 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\section{Introduction}
Procedural Content Generation (PCG), represents a field of research into, and a set of techniques for, generating game content algorithmically.
PCG historically requires a significant amount of human-authored knowledge to generate content, such as rules, heuristics, and individual components, creating a time and design expertise burden. Procedural Content Generation via Machine Learning (PCGML) attempts to solve these issues by applying machine learning to extract this design knowledge from existing corpora of game content \cite{summerville2017procedural}. However, this approach has its own weaknesses; Applied naively, these models require machine learning literacy to understand and debug. Machine learning literacy is uncommon, especially among those designers who might most benefit from PCGML.
Explainable AI represents a field of research into opening up black box Artificial Intelligence and Machine Learning models to users \cite{biran2017explain}. The promise of explainable AI is not just that it will help users understand such models, but also tweak these models to their needs \cite{olah2018building}. If we could include some representation of an individual game designer's knowledge into a model, we could help designers without ML expertise better understand and alter these models to their needs.
Design patterns \cite{bjork2004pattern} represent one popular way to represent game design knowledge. A design pattern is a category of game structure that serves a general design purpose across similar games. Researchers tend to derive design patterns via subjective application of design expertise \cite{hullett2010fps}, which makes it difficult to broadly apply one set of patterns across different designers and games. The same subjective limitation also means that an individual set of design patterns can serve to clarify what elements of a game matter to an individual designer. Given a set of design patterns specialized to a particular designer one could leverage these design patterns in an Explainable PCGML system to help a designer understand and tweak a model to their needs. We note our usage of the term pattern differs from the literature. Typically, a design pattern generalizes across designers, whereas we apply it to indicate the unique structures across a game important to an individual designer.
\begin{figure*}[tb]
\centering
\includegraphics[height=1.25in]{autoencoder.pdf}
\caption{Network architecture for the Autoencoder.}
\label{fig:autoencoder}
\end{figure*}
We present an underlying system for a potential co-creative PCGML tool, intended for designers without ML expertise. This system takes user-defined design patterns for a target level, and outputs a PCGML model. The design patterns provided by designers and generated by our system can be understood as labels on level structure, which allow our PCGML model to better represent and reflect the design values of an individual designer. This system has two major components: (1) a classification system that learns to classify level structures with the user-specified design pattern labels. This system ensures a user does not have to label all existing content to train (2) a level generation system that incorporates the user's level design patterns, and can use these patterns as a vocabulary with which to interact with the user. For example, generating labels on level structure to represent the model's interpretation of that structure to the user.
The rest of this paper is organized as follows. First, we relate our work to prior, related work. Second, we describe our Explainable PCGML (XPCGML) system in terms of the two major components. Third, we discuss the three evaluations we ran with five expert designers. We end with a discussion of the systems limitations, future work, and conclusions. Our major contributions are the first application of explainable AI to PCGML, the use of a random forest classifier to minimize user effort, and the results of our evaluations. Our results demonstrate both the promise of these pattern labels in improving user interaction and a positive impact on the underlying model's performance.
\section{Related Work}
There exist many prior approaches to co-creative or mixed-initiative design agents and editors \cite{yannakakis2014mixed,deterding2017mixed}. However, the majority of existing approaches have relied upon search or grammar-based approaches instead of machine learning, making it difficult to adapt to the needs of a particular designer over time \cite{liapis2013sentient,shaker2013ropossum,baldwin2017mixed}. A final version of our system would focus on machine learning, adapting to the user, and \textit{explaining and visualizing its inner model/process}.
Procedural content generation via Machine Learning \cite{summerville2017procedural} is a relatively new field, focused on generating content through machine learning methods. The majority of PCGML approaches represent black box methods, without any prior approach focused on explainability or co-creativity. We note some discussion in the Summerville et al. survey paper on potential collaborative approaches. Summerville \shortcite{summerville2016learning} explored adapting levels to players, but no work to our knowledge looks at adapting models to individual designers.
Super Mario Bros. (SMB) represents a common area of research into PCGML \cite{dahlskog2012patterns,summerville2016super,jain2016autoencoders,snodgrass2017learning}.
Beyond explainability, our approach differs from prior SMB PCGML approaches in terms of representation quality and the size of generated content.
We focus on the generation of individual level sections instead of entire levels in order to better afford collaborative level building \cite{smith2011tanagara}.
Second, prior approaches have abstracted away the possible level components into higher order groups.
For example, treating all enemy types as equivalent and ignoring decorative elements.
We make use of a rich representation of all possible level components and an ordering that allows our approach to place decorative elements appropriately.
Explainable AI represents an emerging field of research \cite{biran2017explain}, focused on translating or rationalizing the behavior of black box models. To the best of our knowledge, this has not been previously applied to PCGML.
Codella et al. \shortcite{codella2018teaching} demonstrated how explanations could improve model accuracy on three tasks, but required that every sample be hand-labeled with an explanation and treated explanations from different authors as equivalent.
Ehsan et al. \shortcite{ehsan2017rationalization} made use of explainable AI for explainable agent behavior for automated game playing. Their approach relies on rationalization, which relies on a second machine learning interpretation of the original behavior, rather than visualizing or explaining the original model as our approach does.
Design patterns represent a well-researched approach to game design \cite{bjork2004pattern}. In theory, game design patterns describe general solutions to game design problems that occur across many different games. Game Design patterns have been used as heuristics in evolutionary PCG systems including in the domain of Super Mario Bros. \cite{dahlskog2012patterns}. Researchers tend to derive game design patterns through either rigorous, cross-domain analysis \cite{milam2010design} or based upon their subjective interpretation of game structure. We embrace this subjectivity in our work by having designers create a language of game design patterns unique to themselves with which to interact with a PCGML system.
\section{System Overview}
The approach presented in this paper builds an Explainable PCGML model based on existing level structure and an expert labeling design patterns upon that structure. We chose Super Mario Bros. as a domain given its familiarity to the game designers who took part in our evaluation. The general process for building a final model is as follows: First, users label existing game levels with the game design patterns they want to use for communicating with the system. For example, one might label both areas with large amounts of enemies and areas that require precise jumps as ``challenges''. The exact label can be anything as long as it is used consistently. Given this initial user labeling of level structure, we train a random forest classifier to classify additional level structure according to the labeled level chunks \cite{liaw2002classification}, which we then use to label all available levels with the user design pattern labels. Given this now larger training set of both level structure and labels, we train a convolutional neural network-based autoencoder on both levels structure and its associated labels \cite{lang1988time,lecun1989backpropagation}, which can then be used to generate new level structure and label its generated content with these design pattern labels \cite{jain2016autoencoders}.
We make use of Super Mario Bros. as our domain, and, in particular, we utilize those Super Mario Bros levels present in the Video Game Level Corpus \cite{summerville2016vglc}. We do not include underwater or boss/castle Super Mario Bros. levels. We made this choice as we perceived these two level types to be significantly different from all other level types. Further, while we make use of the VGLC levels, we do not make use of any of the VGLC Super Mario Bros. representations, which abstract away level components into higher order groups. Instead, we draw on the image parsing approach introduced in \cite{guzdial2016game}, using a spritesheet and OpenCV \cite{bradski2000opencv} to parse images of each level for a richer representation.
In total we identified thirty unique classes of level components, and make use of a matrix representation for each level section of size $8 \times 8 \times 30$. The first two dimensions determine the tiles in the $x$ and $y$ axes, while the last dimension represents a one-hot vector of length 30 expressing component class. This vector is all 0's for any empty tile of a Super Mario Bros. level, and otherwise has 1's at the index associated with that particular level component. Thus, we can represent all level components, including background decoration. We note that we treat palette swaps of the same component as equivalent in class.
We make use of the SciPy random forest classifier \cite{jones2014scipy} and tensorflow for the autoencoder \cite{abadi2016tensorflow}.
\subsection{Design Pattern Label Classifier}
Our goal for the design pattern label classifier is to minimize the amount of work and time costs for a potential user of the system. Users have to label level structure with the design patterns they would like to use, but the label classifier ensures they do not have to hand-label all available levels. The classifier for this task must be able to perform given access to whatever small amount of training data a designer is willing to label for it, along with being able to easily update its model given potential feedback from a user. We anticipate the exact amount of training data the system has access to will differ widely between users, but we do not wish to overburden authors with long data labeling tasks. Random forest classifiers are known to perform reasonably under these constraints \cite{michalski2013machine}.
The random forest model takes in an eight by eight level section and returns a level design pattern (either a user-defined design pattern or none). We train the random forest model based on the design pattern labels submitted by the user.
We use a forest of size 100 with a maximum depth of 100 in order to encourage generality.
In the case of an interactive, iterative system the random forest can be easily retrained. In the case where the random forest classifier correctly classifies any new design pattern there is no need for retraining. Otherwise, we can delete a subset of the trees of the random forest that incorrectly classified the design pattern, and retrain an appropriate number of trees to return to the maximum forest size on the existing labels and any additional new information.
Even with the design pattern level classifier this system requires the somewhat unusual step of labeling existing level structure with design patterns a user finds important. However, this is a necessary step for the benefit of a shared vocabulary, and labeling content is much easier than designing new content. Further, we note that when two humans collaborate they must negotiate a shared vocabulary.
\subsection{Generator}
The existing level generation system is based on an autoencoder, and we visualize its architecture in Figure \ref{fig:autoencoder}. The input comes in the form of a chunk of level content and the associated design patterns label, such as ``intro'' in the figure. This chunk is represented as an eight by eight by thirty input tensor plus a tensor of size $n$ where $n$ indicates the total number of design pattern labels given by the user. This last vector of size $n$ is a one-hot encoded vector of level design pattern labels.
After input, the level structure and design pattern label vector are separated. The level structure passes through a two layer convolutional neural network (CNN). We note that we placed a dropout layer in between the two CNN layers to allow better generalization. After the CNN layers the output of this section and the design patterns vector recombine and pass through a fully connected layer with relu activation to an embedded vector of size 512. We note that, while large, this is much smaller than the 1920+$n$ features of the input layer. The decoder section is an inverse of the encoder section of the architecture, starting with a relu fully connected layer, then deconvolutional neural network layers with upsampling handling the level structure. We implemented this model with an adam optimizer and mean square loss. Note that for the purposes of evaluation this is a standard autoencoder. We intend to make use of a variational autoencoder in future work \cite{kingma2013auto}.
\section{Evaluation}
Our system has two major parts: (1) a random forest classifier that attempts to label additional content with user-provided design patterns to learn the designer's vocabulary and (2) an autoencoder over level structure and associated patterns for generation. In this section we present three evaluations of our system. The first addresses the random forest classifier of labels, the second the entirety of the system, and the third addresses the limiting factor of time in human computer interactions. For all three evaluations we make use of a dataset of levels from Super Mario Bros. labeled by five expert designers.
\subsection{Dataset Collection}
We reached out to ten design experts to label three or more Super Mario Bros. levels of their choice to serve as a dataset for this evaluation. We do not include prior, published academic patterns of Super Mario Bros. levels (e.g. \cite{dahlskog2012patterns}) as these patterns were designed for general automated design instead of explainable co-creation. Our goals for choosing these ten designers were to get as diverse a pool of labels as possible. Of these ten, five responded and took part in this study.
\begin{itemize}
\item \textbf{Adam Le Doux}: Le Doux is a game developer and designer best known for his Bitsy game engine. He is currently a Narrative Tool Developer at Bungie.
\item \textbf{Dee Del Rosario}: Del Rosario is an events and community organizer in games with organizations such as Different Games Collective and Seattle Indies, along with being a gamedev hobbyist. They currently work as a web developer and educator.
\item \textbf{Kartik Kini}: Kini is an indie game developer through his studio Finite Reflection, and an associate producer at Cartoon Network Games.
\item \textbf{Gillian Smith}: Smith is an Assistant Professor at WPI. She focuses on game design, AI, craft, and generative design.
\item \textbf{Kelly Snyder}: Snyder is an Art Producer at Bethesda and previously a Technical Producer at Bungie.
\end{itemize}
\begin{table}[tb]
\begin{center}
\begin{tabular}{ |p{1cm}|c|c|p{4.2cm}| }
\hline
set & labels & total & top three \\
\hline
Le Doux & 47 & 259 & platform (29), jump (22), pipe-flower (22)\\
\hline
Del Rosario & 26 & 38 & concept introduction (3), objective (3), completionist reward (2)\\
\hline
Smith & 21 & 95 & enemy pair (17), staircase (11), ditch (9)\\
\hline
Smith-Naive & 25 & 118 & multi level (18), enemy pair (17), staircase (13)\\
\hline
Kini & 23 & 28 & hazard introduction (3), hidden powerup (2), pipe trap (2)\\
\hline
Snyder & 34 & 155 & walking enemy (18), power up block (17), coins (13)\\
\hline
\end{tabular}
\caption{A table comparing the characteristics of our six sets of design pattern labels.}
\end{center}
\label{tab:labels}
\end{table}
All five of these experts were asked to label their choice of three levels with labels that established ``a common language/vocabulary that you'd use if you were designing levels like this with another human". Of these experts only Smith had any knowledge of the underlying system. She produced two sets of design patterns for the levels she labeled, one including only those patterns she felt the system could understand and the second including all patterns that matched the above criteria. We refer to these labels as Smith and Smith-Naive through the rest of this section, respectively.
These experts labeled static images of non-boss and non-underwater Super Mario Bros. levels present in the Video Game Level Corpus (VGLC) \cite{summerville2016vglc}. The experts labeled these images by drawing a rectangle over the level structure in which the design pattern occurred with some string to define the pattern. These rectangles could be of arbitrary size, but we translated each into either a single training example centered on the eight by eight chunk our system requires, or multiple training examples if it was larger than eight by eight.
\begin{figure}[tb]
\centering
\includegraphics[width=3in]{ExampleLabels.png}
\caption{The first example of the the top label in each set of design pattern labels.}
\label{fig:exampleLabels}
\end{figure}
\begin{table}[tb]
\begin{center}
\begin{tabular}{ |p{1cm}|c|c|c| }
\hline
set & approach & train & test \\
\hline
Le Doux & RF & 84.06\rpm0.76 & 28.73\rpm1.89\\
\hline
Le Doux & CNN & 49.06\rpm4.43 & 26.73\rpm3.70\\
\Xhline{4\arrayrulewidth}
Del Rosario & RF & 86.71\rpm0.78 & 0.77\rpm1.07\\
\hline
Del Rosario & CNN & 36.08\rpm19.01 & 1.04\rpm0.82\\
\Xhline{4\arrayrulewidth}
Smith & RF & 92.11\rpm0.89 & 33.28\rpm3.48\\
\hline
Smith & CNN & 59.76\rpm8.16 & 26.19\rpm5.02\\
\Xhline{4\arrayrulewidth}
Smith-Naive & RF & 88.49\rpm0.92 & 42.31\rpm2.96\\
\hline
Smith-Naive & CNN & 65.50\rpm19.68 & 36.82\rpm11.84\\
\Xhline{4\arrayrulewidth}
Kini & RF & 90.19\rpm0.69 & 41.93\rpm2.29\\
\hline
Kini & CNN & 44.32\rpm2.89 & 29.07\rpm2.13\\
\Xhline{4\arrayrulewidth}
Snyder & RF & 86.72\rpm1.62 & 1.85\rpm5.56\\
\hline
Snyder & CNN & 49.40\rpm0.77 & 0.00\rpm0.00\\
\hline
\end{tabular}
\caption{A table comparing the accuracy of our random forest label classifier compared to a CNN baseline.}
\end{center}
\label{tab:autoResults}
\end{table}
We include some summarizing information about these six sets of design pattern labels in Table 1. Specifically, we include the total number of labels and the top three labels, sorted by frequency and alphabetically, of each set. Each expert produced very distinct labels, with less than one percent of labels shared between different experts. We include the first example for the top label for each set of design patterns in Figure \ref{fig:exampleLabels}. Even in the case of Kini and Del Rosario, where there is a similar area and design pattern label, the focus differs. We train six separate models, one for each set of design pattern labels (Smith has two).
\subsection{Label Classifier Evaluation}
In this section we seek to understand how well our random forest classifier is able to identify design patterns in level structure. For the purposes of this evaluation we made use of AlexNet as a baseline \cite{szegedy2016rethinking}, given that a convolutional neural network would be the naive way one might anticipate solving this problem. We chose AlexNet given its popularity and success at similar image recognition tasks. In all instances we trained the AlexNet until its error converged. We make use of a three-fold cross validation on the labels for this and the remaining evaluations. We make use of a three-fold validation to address the variance across even a single expert's labels and due to the small set of labels available for some experts.
\begin{table}[tb]
\begin{center}
\begin{tabular}{ |p{1cm}|c|c|c| }
\hline
set & No labels & No Auto Tag & Full \\
\hline
Le Doux & 12.3\rpm6.3 & 152.8\rpm4.8 & \textbf{10.6\rpm5.6}\\
\hline
Del Rosario & 10.4\rpm5.0 & 135.7\rpm3.8 & \textbf{9.0\rpm4.4}\\
\hline
Smith & 11.5\rpm6.1 & 157.2\rpm3.5 & \textbf{10.4\rpm5.6}\\
\hline
Smith-Naive & 12.7\rpm4.8 & 167.6\rpm4.0 & \textbf{11.5\rpm4.4}\\
\hline
Kini & \textbf{9.4\rpm4.4} & 129.6\rpm3.6 & 10.6\rpm3.3\\
\hline
Snyder & 28.6\rpm9.9 & 144.2\rpm5.0 & \textbf{15.0\rpm9.4}\\
\hline
\end{tabular}
\caption{A table comparing the error in terms of incorrect sprites for our three generators. Smaller values represent fewer mistakes.}
\end{center}
\label{tab:generatorResults}
\end{table}
Our major focus is training and test accuracy across the folds. We summarize the results of this evaluation in Table 2, giving the average training and test accuracies across all folds along with the standard deviation. We note that in all instances our random forest (RF) approach outperformed AlexNet CNN in terms of training accuracy, and nearly always in terms of test accuracy. We note that given more training time AlexNet's training accuracy might improve, but at the cost of test accuracy. We further note that AlexNet was on average one and a half times slower than the random forest in terms of training time. These results indicate that our random forest produces a more general classifier compared to AlexNet.
We note that our random forest performed fairly consistently in terms of training accuracy, at around 85\%, but that the test accuracy varied significantly. Notably, the test accuracy did not vary according to the the number of training samples or number of labels per expert. This indicates that individual experts identify patterns that are more or less easy to classify automatically. Further we note that Snyder and Del Rosario had very low testing error across the board, which indicates a large amount of variance between tagged examples. Despite this, we demonstrate the utility of this approach in the next section.
\subsection{Autoencoder Structure Evaluation}
We hypothesize that the inclusion of design pattern labels into our autoencoder network will improve its overall representative quality. Further, that the use of an automatic label classifier will allow us to gather sufficient training data to train the autoencoder. This evaluation addresses both these hypotheses. We draw upon the same dataset and the same three folds from the prior evaluation and create three variations of our system. The first autoencoder variation has no design pattern labels and is trained on all $8\times 8$ chunks of level instead of only those chunks labeled or autolabeled with a design pattern. Given that this means fewer features and smaller input and output tensors, this model should outperform our full model unless the design pattern labels improve overall representative quality. The second autoencoder variation does not make use of the automatic design pattern label classifier, thus greatly reducing the training data. The last variation is simply our full system. For all approaches we trained till training error converged. We note that we trained a single 'no labels' variation and tested it on each expert, but trained models for the no automatic classifier and full versions of our approach for each expert.
Given these three variations, we chose to measure the difference in structure when the autoencoder was fed the test portions of each of the three folds. Specifically we capture the number of incorrect structure features predicted. This can be understood as a stand in for representation quality, given that the output of the autoencoder for the test sample will be the closest thing the autoencoder can represent to the test sample.
We give the average number and standard deviation of incorrect structural features/tiles over all three folds in Table 2. We note that the minimum value here would be 0 errors and the maximum value would be $8\times 8\times 30$ or 1920 incorrect structural feature values. For every expert except for Kini, who authored the smallest number of labels, our full system outperformed both variations. While some of these numbers are fairly close between the full and no labels variation, the values in bold were significantly lower according to the paired Wilcoxon Mann Whitney U test ($p<0.001$).
Given the results in Table 3. We argue that both our hypotheses were shown to be correct, granted that the expert gives sufficient labels, with the cut-off appearing to be between Kini's 28 and Del Rosario's 38. Specifically the representation quality is improved when labels are used, and the label classifier improves performance over not applying the label classifier.
\begin{table*}[tb]
\begin{center}
\begin{tabular}{ |l|c|c|c|c|c|c|c| }
\hline
& Le Doux & Del Rosario & Smith & Smith-Naive & Kini & Snyder \\
\hline
No labels & 12.3\rpm16.3 & 10.4\rpm5.0 & 11.5\rpm6.1 & 12.7\rpm4.8 & \textbf{9.3\rpm4.4} & 28.6\rpm9.9 \\
\hline
Transfer No Auto Tag & 11.1\rpm5.8 & 10.1\rpm5.0 & 11.2\rpm6.1 & 12.6\rpm4.8 & 9.3\rpm4.4 & 16.7\rpm9.8 \\
\hline
Transfer w/ Auto & 10.8\rpm5.8 & 9.8\rpm5.0 & 11.0\rpm6.0 & 11.8\rpm6.3 & 10.3\rpm4.2 & 16.1\rpm9.6 \\
\hline
Full & \textbf{10.6\rpm5.6} & \textbf{9.0\rpm4.4} & \textbf{10.4\rpm5.6} & \textbf{11.5\rpm4.4} & 10.6\rpm3.3 & \textbf{15.0\rpm9.4} \\
\hline
\end{tabular}
\caption{A table comparing the transfer learning approach structure errors to the full system structure errors.}
\end{center}
\label{tab:transferResults}
\end{table*}
\subsection{Transfer Evaluation}
A major concern for any co-creative tool based on Machine Learning is training time. In the prior autoencoder evaluation, both the no labels and full versions of our system took hours to train to convergence. This represents a major weakness, given that in some co-creative contexts designers may not want to wait for an offline training process, especially when we anticipate authors wanting to rapidly update their set of labels. Given these concerns, we evaluate a variation of our approach utilizing transfer learning. This drastically speeds up training time by adapting the weights of a pretrained network on one task to a new task.
We make use of student-teacher or born again neural networks, a transfer learning approach in which the weights of a pre-trained neural network are copied into another network of a different size. In this case we take the weights from our no labels autoencoder from the prior evaluation, copy them into our full architecture, and train from there. We construct two variations of this approach, once again depending on the use of the random forest label classifier or not. We compare both variations to the full and no labels system from the prior evaluation, using the same metric.
We present the results of this evaluation in Table 4. We note that, while the best performing variation did not change from the prior variation, in all cases except for the Kini models, the transfer approaches got closer to the full variation approach, sometimes off by as little as a fraction of one structure feature. Further, these approaches were significantly faster to train, with the no automatic labeling transfer approach training in an average of 4.48 seconds and the automatic labeler transfer approach training in an average of 144.92 seconds, compared to the average of roughly five hours of the full approach on the same computer. This points to a clear breakdown in when it makes sense to apply what variation of our approach, depending on time requirements and processing power. In addition, it continues to support our hypotheses concerning the use of automatic labeler and personal level design pattern labels.
\begin{figure}[tb]
\centering
\includegraphics[width=3in]{XPCGML.png}
\caption{Above: the two examples of the pattern ``completionist reward'' labeled by the expert Dee Del Rosario. Below: example of the system given the input on the left and that label and its output.}
\label{fig:example}
\end{figure}
\subsection{Qualitative Example}
We do not present a front-end or interaction paradigm for the use of this Explainable PCGML system, as we feel such implementation details will depend upon the intended audience.
However, it is illustrative to give an example of how the system could be used.
In Figure \ref{fig:example} we present an example of the two training examples of the pattern ``completionist reward'' labeled by the expert Dee Del Rosario.
The full system, including the random forest classifier, trains on these examples (and the other labels from Del Rosario), and is then given as input the eight by eight chunk with only the floating bar within it on the left of the image along with the desired label ``completionist reward''. One can imagine that Del Rosario as a user wants to add a reward to this section, but doesn't have any strong ideas. Given this input the system outputs the image on the right.
We asked Del Rosario what they thought of the performance of the system and whether they considered this output matched their definition of completionist reward. They replied ``Yes -- I think? I would because I'm focusing on the position of the coins.'' We note that Del Rosario did not see the most decisive patch when making this statement, which we extracted as in \cite{olah2018building}. This clearly demonstrates some harmony between the learned model and the design intent. However, Del Rosario went on to say ``I think if I were to go... more strict with the definition/phrase, I'd think of some other configuration that would make you think, `oooooh, what a tricky design!!' ''. This indicates a desire to further clarify the model. Thus, we imagine an iterative model is necessary for a tool utilizing this system and a user to reach a state of harmonious interaction.
\section{Conclusions}
In this paper, we present an approach to explainable PCGML (XPCGML) through user-authored design pattern labels over existing level structure.
We evaluate our autoencoder and random forest labeler components on levels labeled by game design experts.
These labels serve as a shared language between the user and level design agent, which allows for the possibility of explainability and meaningful collaborative interaction.
We intend to take our system and incorporate it into a co-creative tool for novice and expert level designers. To the best of our knowledge this represents the first approach to explainable PCGML.
\section{Acknowledgements}
We want to thank our five expert volunteers. This material is based upon work supported by the National Science Foundation under Grant No. IIS-1525967. We would also like to thank the organizers and attendees of Dagstuhl Seminar 17471 on Artificial and Computational Intelligence in Games: AI-Driven Game Design, where the discussion that lead to this research began.
\bibliographystyle{aaai}
| {
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{"url":"http:\/\/aas.org\/archives\/BAAS\/v26n4\/aas185\/abs\/S4514.html","text":"1-5 micron Arcturus Atlas\nSession 45 -- Stellar Activity I\nDisplay presentation, Tuesday, 10, 1995, 9:20am - 6:30pm\n\n## [45.14] 1-5 micron Arcturus Atlas\n\nK.H.Hinkle, L. Wallace, W. Livingston (NOAO)\n\nAn atlas of the infrared spectrum of the bright K 2 giant Arcturus recently has been completed using the 4 meter Mayall telescope and FTS. The 0.9-5.4 $\\mu$m spectrum of Arcturus was observed at high signal-to-noise with a resolution of 100,000. Telluric lines were removed by using telluric transmission spectra generated from McMath-Pierce solar spectra and lunar spectra. The spectrum of Arcturus was observed on two different dates selected to give large opposite heliocentric shifts. The spectra observed on the different dates have been corrected for telluric absorption and then overplotted, with the result that the telluric spectrum has been effectively removed from all but the most obscured wavelengths of the Arcturus spectrum. The overplotted spectra also illustrate the signal-to-noise and the believability of spectral features.\n\nAn effort has been undertaken to identify lines with central depths stronger than a few percent. Identifications are marked on the atlas pages. While the majority of the lines are identified, a few strong unidentified lines are present. Identifications of the molecular features seem well in hand and the unidentified lines appear to be atomic. Frequencies for unidentified lines will be listed. Information concerning the availability of the atlas will be given.","date":"2014-08-29 01:49:02","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\": 0.5889369249343872, \"perplexity\": 4450.010836789117}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-35\/segments\/1408500831174.98\/warc\/CC-MAIN-20140820021351-00119-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
Gaoua är en stad och kommun i södra Burkina Faso och är administrativ huvudort för provinsen Poni. Staden hade 25 104 invånare vid folkräkningen 2006, med totalt 52 733 invånare i hela kommunen.
Källor
Orter i Sud-Ouest, Burkina Faso | {
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{"url":"https:\/\/www.semanticscholar.org\/paper\/Evolution-of-atmospheric-escape-in-close-in-giant-Allan-Vidotto\/80495819284b9f2e780ebd33e0a5fa911eadf8a4","text":"Evolution of atmospheric escape in close-in giant planets and their associated Ly$\\alpha$ and H$\\alpha$ transit predictions\n\n@article{Allan2019EvolutionOA,\ntitle={Evolution of atmospheric escape in close-in giant planets and their associated Ly\\$\\alpha\\$ and H\\$\\alpha\\$ transit predictions},\nauthor={Andrew Allan and A. Vidotto},\njournal={arXiv: Earth and Planetary Astrophysics},\nyear={2019}\n}\n\u2022 Published 2019\n\u2022 Physics\n\u2022 arXiv: Earth and Planetary Astrophysics\n\u2022 Strong atmospheric escape has been detected in several close-in exoplanets. As these planets are mostly made of hydrogen, observations in hydrogen lines, such as Ly-alpha and H-alpha lines, are powerful diagnostics of escape. Here, we simulate the evolution of atmospheric escape of close-in giant planets and calculate their associated Ly-alpha and H-alpha transits. For that, we use a 1D hydrodynamic escape model to compute the physical properties of the escaping atmosphere and a ray tracing\u2026\u00a0CONTINUE READING\n\nReferences\n\nPublications referenced by this paper.\nSHOWING 1-10 OF 36 REFERENCES","date":"2020-09-30 10:21:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2518577575683594, \"perplexity\": 7501.0262130713745}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600402123173.74\/warc\/CC-MAIN-20200930075754-20200930105754-00141.warc.gz\"}"} | null | null |
\section{Introduction} \label{sec:intro}
Time-domain astronomy has uncovered
a new class of stellar `superflares' with bolometric energies higher than $10^{33}$ erg \citep{maehara2012}, events that significantly influence planetary habitability \citep{armstrong2016,lingam2017} and may have been imprinted on meteoritic chemical abundances in our own solar system \citep{mishra2019}.
The most surprising superflare discoveries come from the low-mass end of the main sequence, the ultracool dwarfs (UCDs) of spectral type L. Four white-light superflares from L dwarfs have been found in All Sky Automated Survey for Supernovae \citep{schmidt2016}, {\em Kepler} \citep{gizis2017,paudel2018} and Next Generation Transient Survey data
\citep{jackman2019}, each
extremely bright (up to $\Delta V=-11$), releasing up to $4\times10^{34}$ erg of energy.
These bursts are comparable to the strongest flares in FGKM stars, and are
remarkable given that most tracers of quiescent magnetic emission (H$\alpha$ and X-ray) decline in the L dwarf sequence \citep{schmidt2015,stelzer2006}. This decline is the result of increasingly neutral photospheres \citep{mohanty2002} and possibly of changes in magnetic reconnection \citep{mullan2010} or dynamo processes \citep{cook2014}. Furthermore, while young stars are typically the most magnetically active, three of the superflare L dwarfs appear to be older field stars.
What enables these dim stars to undergo one of the most dramatic stellar outbursts, and the frequency of such events, remains a mystery.
All of the L dwarf superflares reported to date have been detected in white-light observations.
In the standard flare scenario \citep[e.g.][]{cargill83}
white-light flares are produced in the lower atmosphere following the bombardment with particles that are accelerated in magnetic reconnection events. The ensuing
heating of the chromosphere leads to upwards plasma motions and, subsequently, to an X-ray flare from the hot matter confined in coronal magnetic loops.
Even for the Sun, observationally associating X-ray and optical flares has remained difficult \citep{hao17}.
For L dwarfs the lack of simultaneous multi-band observations has so far prevented tests of the solar flare scenario.
Here we report on the first detection of an X-ray superflare emitted by a distant ($\sim240$ pc)
early L-type dwarf.
Previously, only one other L dwarf had been detected in (quiescent) X-rays, the binary system Kelu 1AB at 19 pc \citep{audard2007}.
Our results were obtained within the context of {\em Exploring the X-ray Transient and variable Sky\footnote{\url{www.extras-fp7.eu}}} (EXTraS), an EU-funded project aimed at extracting and characterising all temporal information stored in serendipitous {\em XMM-Newton} data \citep[][De Luca et al., in preparation]{deluca17}.
This letter is organised as follows: in Sect.~\ref{sec:selection} we describe the discovery of the flare; Sects.~\ref{sec:flare} -- \ref{sec:star} give temporal and spectral properties of the flare, set constraints on the flaring rate and on the quiescent X-ray emission of the source, and give results on the spectral classification of the dwarf and on its properties.
All results are discussed in Sect.~\ref{sec:disc}.
\section{The X-ray flare of J0331-27} \label{sec:selection}
A large catalogue listing 11945 candidate ultracool dwarfs of the L and T spectral classes was recently published by \citet{carnerorosell2019}. This catalogue is based on photometric classification of data from the Dark Energy Survey Year 3 release \citep{des2016} covering $\sim2400$ square degrees down to $i_{AB}=22$, matched to photometry from the Vista Hemisphere Survey \citep{mcmahon2013} and Wide-field Infrared Survey Explorer \citep{wright2010,mainzer2011}. Of the sources of this catalogue, 515 are located within the field of view of at least one {\em XMM-Newton} observation. Cross-correlation of the catalogue
with the EXTraS database singles out 3XMM\,J033158.9-273925 (hereafter J0331-27) as a very interesting case. J0331-27 matches within 1$\farcs$1 the position of the L0 candidate Obj.\
ID\ 366318917 ($\rm RA=03^h31^m59\fs07$, $\rm Dec=-27^{\circ}39'25\farcs7$; J2000) in \citet{carnerorosell2019}, and is listed as a variable X-ray source in the EXTraS database. Inspection of light curves in the EXTraS Public Archive\footnote{\url{https://www88.lamp.le.ac.uk/extras/archive}} clearly shows an X-ray flare.
Moreover, the source is located within the Extended {\em Chandra} Deep Field South \citep[ECDFS,][]{lehmer2005}, one of the most scrutinised portions of the sky, with a large amount of deep multi-wavelength data available.
\section{Temporal and spectral properties of the flare} \label{sec:flare}
The X-ray flare from J0331-27 is clearly seen in data collected on 2008 July 5 by all detectors of the European Photon Imaging Camera (EPIC) instrument on board {\em XMM-Newton}, namely the pn camera \citep{Strueder2001} and the two MOS cameras \citep{Turner2001}. The observation (Obs. ID 0555780101) lasted 130 ks and was performed as a part of the {\em XMM-Newton} Ultra-deep Survey of the {\em Chandra} Deep Field South \citep[XMM-CDFS;][]{comastri11,ranalli2013}.
In Fig.~\ref{epiclc} we show a background-subtracted light curve combining pn and MOS data, generated using an updated version of the EXTraS software \citep{marelli17}. The flare profile is nicely described (reduced $\chi^2=0.88$, 35 d.o.f.) by a linear rise (rise time of $1600\pm300$ s), peaking at MJD=$54652.8939\pm0.0032$ UTC (in Barycentric Dynamical Time), followed by an exponential decay of the form $e^{-t/\tau_{decay}}$ with $\tau_{decay}=2400\pm400$ s.
The peak flux is $(9.0\pm1.5)\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$ and the integrated flare energy flux ({\em fluence}), evaluated by integrating the model of the flare time profile, is $\sim2.7\times10^{-10}$ erg cm$^{-2}$ (all values refer to the 0.5-2 keV energy range).
The best fit model has a continuum level consistent with zero; no significant emission is seen apart from the flare.
To perform spectroscopy of the flare, we extracted source counts from a circle with radius of $15^{\prime\prime}$ and background counts from a nearby, source-free region. We only considered observing times starting 3000 s before the peak of the flare and extending up to 12000 s after the peak. Time intervals with high backgrounds were excluded following the prescription by \citet{delucamolendi2004}.
The resulting flare spectrum contains $100\pm12$ background-subtracted counts from the three EPIC detectors. We generated a response matrix and effective area file using the SAS tasks \texttt{rmfgen} and \texttt{arfgen}. Spectral modelling was performed using the XSPEC v12.10 Software. A good description of the data is given by an optically thin thermal plasma model (apec) with kT$=1.39^{+0.27}_{-0.11}$ keV and fixed abundance assumed to be 0.3 solar. The absorbing column is consistent with zero (N$_H<1.7\times10^{20}$ cm$^{-2}$ at 1$\sigma$); total Galactic absorption in the direction of the target is indeed very low \citep[N$_{H,Gal}=6\times10^{19}$ cm$^{-2}$,][]{benbekhti2016}. The goodness of the fit, evaluated as the percentage of Monte Carlo realisations that had Kendall's W-statistic values lower than that of the best fit, is 84\% (based on $10^4$ simulations).
\begin{figure}
\centering
\includegraphics[width=6cm,angle=-90]{lc_flare.pdf}
\caption{Background-subtracted light curve of J0331-27 in the 0.2-2 keV energy range combining pn and MOS data. A 30 ks portion of the observation is shown. Count rate has been converted to flux in the 0.5-2 keV energy range using the best fit spectral model (see text). The flare from the source is apparent, and is nicely described by a linear rise and exponential decay model (see text).}
\label{epiclc}
\end{figure}
The field of J0331-27 was observed 33 times by {\em XMM-Newton}.
We took advantage of this large dataset (total nominal exposure of $3.45$\,Ms) to search for other flares from J0331-27.
No flares were detected within the blind search for transient sources performed by EXTraS,
with duration $<10000$ s.
To set an upper limit to the integrated energy flux of unseen flares,
we evaluated the average background level close to the position of the source in individual observations.
We followed \citet{ranalli2013} in their steps for data reduction, event filtering (including soft proton flare screening), and astrometry correction. We excluded observation 0555780101, in which the flare was detected. Images and exposure maps were generated for each camera and for each observation in the 0.5-2 keV energy range. As a sanity check, we performed aperture photometry by extracting source counts from a circle of $10^{\prime\prime}$ radius and background counts from a contiguous circle with the same radius, combining results from all EPIC cameras for each observation; in no case was significant emission detected from J0331-27. Based on the observed average background rate (assuming a Poissonian regime) and adopting the spectral model best fitting the observed flare of J0331-27, we estimated that an integrated energy flux of $\sim4\times10^{-11}$ erg cm$^{-2}$ in 5 ks would have produced a $3\sigma$ excess of counts, which would have triggered the detection algorithm in EXTraS.
We note that the 3XMM catalogue lists a detection of J0331-27 in dataset 0108060701. However, the source is detected at the $3\sigma$ level only in the MOS1 hard band (4.5-12 keV), with upper limits at softer energies not consistent with a thermal spectrum.
We disregard this detection as an artefact.
\section{Search for X-ray quiescent emission} \label{sec:qemiss}
We searched for quiescent emission from J0331-27 by stacking all data from the XMM-CDFS survey, excluding Obs. ID 0555780101 ($\sim2.5$ Ms good observing time after screening for soft proton flares). Considering only PN data and the 0.5–2 keV band, we extracted 362 counts for J0331-27 from a circular region with a 10$''$ radius and 349 background counts from an adjacent circular region with the same radius. Assuming a Poisson background distribution, the observed counts can be safely attributed to a background fluctuation.
An upper limit to the quiescent flux of J0331-27 can be calculated by requiring that a detection with the PN camera had at least 50 more counts than in the background region. Formally, such an excess of counts would only have a 0.4\% probability of being due to a background fluctuation. In practice, requiring an excess of 50 counts over the background is consistent with what is observed for other off-axis sources in the XMM-CDFS catalogue \citep{ranalli2013}.
To convert counts to flux, we generated response matrices and effective area files for all XMM-CDFS observations (again excluding Obs. ID 0555780101) and averaged them following \citet{georgantopoulos2013}. Assuming the same spectral model as observed during the flare, we calculate the upper limit to the quiescent emission as $1\times 10^{-16}$ erg~s$^{-1}$~cm$^{-2}$ in the 0.5-2 keV energy range.
The field of J0331-27 was observed several times by {\em Chandra}.
It is included in the Extended CDFS, for a total of $\sim250$\,ks (Obs. ID 5017 and 5018, PI Brandt). The source was not detected by \cite{lehmer2005}. We retrieved {\em Chandra} data and analysed them with the CIAO v4.11 software and CALDB v4.8.3. We reprocessed level 2 data using the \texttt{chandra\_repro} script and merged the resulting event files using the \texttt{reproject\_obs} script. Using the \texttt{srcflux} script and adopting the spectral shape of the flare emission, we found a $3 \sigma $ upper limit of $5\times10^{-16}$ erg cm$^{-2}$ s$^{-1}$ to the flux of J0331-27 in the 0.5-2 keV energy range.
The position of J0331-27 lies also very close to the edge of the field of view in the CDFS 7Ms survey, at an off-axis angle of $\sim11$ arcmin, covered with an exposure time (corrected for vignetting) of $\sim400$ ks. The source is not detected in these data \citep{luo2017}. We retrieved the image and exposure map of the survey in the 0.5-2 keV energy range made available by the CDFS team\footnote{\url{http://personal.psu.edu/wnb3/cdfs/cdfs-chandra.html}}. Following \citet{lima1983}, we computed the count rate needed to produce a $3\sigma$ excess in the image at the position of J0331-27 within the PSF area and converted it into flux assuming the observed spectral shape of the flare. The resulting upper limit to the source flux is also $\sim5\times10^{-16}$ erg cm$^{-2}$ s$^{-1}$.
\section{Classification and properties of the UCD} \label{sec:star}
\vspace{0.5cm}
\subsection{Spectral classification}
An optical spectrum of J0331-27 was observed with the Visible Multi Object Spectrograph (VIMOS) instrument \citep{2003SPIE.4841.1670L} as part of the VIMOS/VLT Deep Survey of the CDFS \citep{2005A&A...439..845L}, where the source is listed with Object ID 38073.
Data were reduced using the automatic pipeline developed by the survey team. The spectrum (Figure~\ref{spectrum}) shows the characteristic steep rise in continuum across the optical band, with strong absorption features of K~I, TiO, VO, FeH, CrH apparent, typical of late-M and L dwarfs \citep{1999ApJ...519..802K}. This spectrum was compared to red optical templates of M and L dwarfs compiled from Sloan Digital Sky Survey (SDSS) optical spectroscopy by \citet{2007AJ....133..531B} and \citet{2014PASP..126..642S} in the 6500-8000~{\AA} range using a $\chi^2$ statistic. The best fit templates were L1 and (as a close second) L0, with a weighted mean of L0.7$\pm$1.3, consistent with the photometric classification of \citet{carnerorosell2019}\footnote{ The source had been previously classified as an M6 dwarf, or as an unresolved elliptical galaxy at $z\sim1.9$, by \citet{groenewegen2002} based on five-band (UBVRI) photometry and morphology (source ID J033159.06-273925.5 in their catalogues). It was also classified as a galaxy by \citet{wolf2008}, based on multi-band photometry (Obj. 51627 in their catalogue).}. There is no evidence of H$\alpha$ emission, and low signal-to-noise ratio of the data made it impossible to determine the presence of Li~I absorption (a diagnostics of substellarity) or features indicative of low surface gravity \citep{cruz2009}.
\begin{figure}
\centering
\includegraphics[width=9cm]{spectral_plot.pdf}
\caption{{\em Top panel}: VIMOS spectrum of J0331-27 (black line) compared to an L1 SDSS spectral template from
\citet[magenta dot-dashed line]{2014PASP..126..642S}. The spectral data uncertainty is shown as the dashed line.
{\em Bottom panel}: Difference between the observed data and template ($\Delta$) normalised to the uncertainty ($\sigma$). }
\label{spectrum}
\end{figure}
\vspace{0.5cm}
\subsection{Photometric distance}\label{sect:dist}
J0331-27{} is not listed in Gaia Data Release 2, thus we have to rely on photometric distance estimates.
\cite{carnerorosell2019}
found $\sim280$ pc
from the comparison
between the observed near-infrared (NIR) magnitudes and the absolute magnitudes for L0 subtype.
Absolute magnitudes are anchored to $M_{\rm W1}$ and $M_{\rm W2}$ from Table 14 of \cite{dupuyliu2012}. The other absolute magnitudes are obtained from the colours in Table 3 of \cite{carnerorosell2019}. Applying the same procedure under the hypothesis of spectral type L1 and averaging over all available bands we obtain a distance of
$240_{-20}^{+40}\,{\rm pc}$.
This is the value we adopt throughout this paper.
\subsection{Age characterisation}
The age of J0331-27 is of interest as younger stars tend to be more active, whereas the white light superflare UCDs identified to date appear to be older field objects \citep{schmidt2016}. Age determinations for individual UCDs are challenging, however, as traditional age metrics such as
(quiescent) magnetic activity level and rotation rates appear to decouple from age at the lowest stellar masses and coolest temperatures (e.g. \citealt{2011ApJ...727...56I,schmidt2015}). Surface gravity-sensitive features in the optical spectra of L dwarfs, such as enhanced VO band and weakened alkali line absorption \citep{2008ApJ...689.1295K,cruz2009} provide approximate age constraints for sources younger than $\sim$300~Myr; unfortunately, the VIMOS spectrum of J0331-27 is too noisy to discern these features. Similarly, kinematics cannot be used as the VIMOS data cannot provide radial velocity information, and no proper motion has been reported for this source.
Colour provides an additional age diagnostic.
\cite{faherty2016} show that NIR colours can be used to distinguish between low-gravity and field-gravity dwarfs, with the latter appearing systematically redder than the mean of the field dwarfs at a given spectral type. Table~\ref{tab:colors} compares the colours of J0331-27 to the average colours of low surface gravity and field L1 dwarfs reported in \cite{faherty2016}. Rather than being systematically redder, J0331-27 is bluer than field L1 dwarfs, particularly in $J-K_s$ and $J-W1$ colours, consistent with older (high surface gravity) and slightly metal-poor L dwarfs \citep{2018MNRAS.480.5447Z}.
Thus, it is most likely that J0331-27 is an older field L dwarf.
\begin{table}[!tb]
\centering
\caption{Infrared colours of J0331-27 compared to mean colours of the low-gravity and field-gravity L1 dwarfs from \citep{faherty2016}.}
\label{tab:colors}
\begin{tabular}{lccc} \hline
Colour & J0331-27 & <Low-g> & <Field>\\ \hline
${\rm J-H}$ &$0.61\pm0.15$ &$0.94\pm 0.06$& $0.81\pm0.14$\\
${\rm J-K_s}$ &$1.10\pm0.18$ &$1.61\pm0.13$ & $1.35\pm0.19$\\
${\rm J-W1}$ & $1.27\pm0.19$&$2.17\pm0.20$ & $1.71\pm0.21$\\
${\rm J-W2}$ &$1.80\pm0.33$ & $2.55\pm0.25$& $1.97\pm0.23$\\
${\rm H-K_s}$ &$0.49\pm0.19$ &$0.67\pm0.11$ &$0.54\pm0.13$\\
${\rm H-W1}$ &$0.66\pm0.20$ &$1.21\pm0.17$ & $0.91\pm0.15$\\
${\rm H-W2}$ &$1.20\pm0.34$ &$1.60\pm0.22$ & $1.17\pm0.18$\\
${\rm K_s-W1}$ &$0.16\pm0.23$ &$0.54\pm0.11$ &$0.37\pm0.11$\\
${\rm K_s-W2}$ &$0.70\pm0.37$ &$0.92\pm0.14$ & $0.63\pm0.14$\\
${\rm W1-W2 }$&$0.54\pm0.38$ &$0.38\pm0.05$ & $0.26\pm0.06$\\ \hline
\end{tabular}
\end{table}
\begin{figure}[!htb]
\centering
\includegraphics[width=9.1cm]{ucd_xray_bis.png}
\caption{$L_X$ and $L_X/L_{bol}$ vs. spectral type for J0331-27{} (flare peak luminosity and upper limit on the quiescent luminosity), compared with the other UCDs for which X-ray data are available (from \citealt{stelzer2006}, \citealt{williams2014}, \citealt{cook2014} and references therein; \citealt{robrade2010} and \citealt{gupta2011}).
}
\label{fig:ucds_xray}
\end{figure}
\section{Discussion} \label{sec:disc}
The association of the flaring source J0331-27 with the L1 dwarf is robust.
First, the chance alignment probability is
low ($<2\times10^{-4}$). Second,
archival {\em HST} images show that the dwarf is the only point-like source within the error circle of the X-ray source.
Assuming the two sources to be unrelated would require that the flaring source counterpart have m$_V>27$ and m$_{z}>26$. This requires a Galactic flaring X-ray source {extremely dim} at optical--NIR wavelengths, or an extragalactic transient. Both cases would require peculiar and unlikely scenarios to explain the X-ray emission and temporal behaviour.
The available data do not allow us to precisely constrain the age of J0331-27{}, although its IR colours indicate that it is likely an older source. Assuming an age $\gtrsim1$\,Gyr, evolutionary models \citep{2003A&A...402..701B,2001RvMP...73..719B} predict the mass of J0331-27{} to be above the hydrogen burning minimum mass limit (M $>$ 0.072~M$_{\odot}$).
In Fig. \ref{fig:ucds_xray} we put the X-ray emission of J0331-27 in context to that of other UCDs.
J0331-27 is the first L dwarf to be detected in an X-ray flare. We measure
a flare peak luminosity of
$\log{L_{\rm x,peak}}=29.8\,{\rm (erg/s)}$ and $\log{(L_{\rm x,peak}/L_{\rm bol})}=-0.1$,
similar to X-ray flares observed on late-M dwarfs (e.g. \citealt{stelzer2006b}, \citealt{hambaryan2004}, \citealt{gupta2011}) and actually somewhat larger than the majority of them.
The quiescent luminosity is only weakly constrained with an upper limit of
$\log{L_{\rm x,qui}}<27.0\,{\rm (erg/s)}$
and ${\log{(L_{\rm x,qui}/L_{\rm bol})}<-2.9}$, a result of the large distance of J0331-27{} compared with the other UCDs studied in X-rays so far.
Similarly, an upper limit on the radio flux of J0331-27 of $ 7.4\,\mu$Jy at 1.4GHz \citep{miller2013}, translates into a very weak constraint on the radio luminosity, $ \log{\rm L_{\nu,R}}<14.7$ (erg s$ ^{-1}$ Hz$ ^{-1}$), preventing analysis of the well-established deviation of UCDs from the G\"udel-Benz radio/X-ray relation \citep{guedelbenz93,berger2002}.
The short observed flare duration (decay timescale $\tau \simeq 2400\,{\rm s}$) is also consistent with those of the
X-ray flares on late-M dwarfs. This indicates a compact size for the flaring region on the UCD surface (see \citealt{stelzer2006b}).
Finally,
the observed X-ray emitting plasma temperature of $\sim 16\,{\rm MK}$
is within the range of values
reported for late-M dwarf flares. In summary, our observation shows that no qualitative change takes place in the properties of X-ray flares at the bottom of the main sequence down to
$T_{\rm eff} \sim 2100$\,K.
The remarkable presence of a single superflare in
$\sim2.5$\,Ms of {\em XMM-Newton} data
gives rise to the question on the frequency of such events and of X-ray flares in general on L-type dwarfs. The integrated X-ray flare energy of the event on J0331-27{} is
$\log{E_{\rm X,F}}=33.3\,{\rm(erg)}$,
and the flare frequency is $\nu (\log{E_{\rm X,F}} \gtrsim 33.3) \sim 1 / 30\,{\rm d^{-1}}$.
Based on the sensitivity of the available {\em XMM-Newton} observations (assuming a typical flare timescale of 5,000 s, see Sect.~\ref{sec:flare}) we estimate that $\sim$eight flares above $\log{E_{\rm X,F}} \sim 32.4\,{\rm (erg)}$ would have been detected if J0331-27{} obeyed the canonical power law for the flare energy number distribution, $\frac{dN}{dE_{\rm F}} \sim E_{\rm F}^{\alpha}$ with $\alpha \approx -2$ \citep[see references in][]{Argiroffi19.0}. The fact that these flares are not seen suggests a non-standard flare energy distribution for J0331-27{}.
Contrary to the large flare on the M8 dwarf LP412-32, which was observed simultaneously with the {\em XMM-Newton}{} X-ray instruments and its Optical Monitor \citep{stelzer2006b}, no contemporaneous optical data is available for the event on J0331-27{}. However, in view of the similarities of the J0331-27{} X-ray flare and the X-ray flares on late-M dwarfs described above, we can use the observed optical-to-X-ray energy ratio of LP412-32
($E_{\rm opt,F} \approx E_{\rm X,F}$)
to estimate an optical counterpart of $E_{\rm opt,F} \gtrsim 10^{33}$\,erg for the X-ray superflare on J0331-27{}.
Remarkably, other white-light superflares observed on L dwarfs (without simultaneous X-ray data) show flare energies of the same order (\citealt{jackman2019}, \citealt{gizis2017}).
We note that for a flare on an early M dwarf observed simultaneously with {\em XMM-Newton} and {\em Kepler} \citep[KIC\,8454353; see][]{Pizzocaro19.0} we also find that the X-ray and optical flare energy are within a factor of two of each other. On the other hand, \cite{Guarcello19.0} find significantly higher emission in the optical {\em Kepler} band with respect to X-rays for some flares in the Pleiades.
Simultaneous optical--X-ray studies of
larger samples are required to nail down the relative radiative output of chromosphere and corona, and whether this depends on other stellar parameters.
Systematic searches are also likely to yield better constraints on the frequency of X-ray flares on L dwarfs. Our
cross-correlation of the catalogue of UCD candidates by \citet{carnerorosell2019} with the {\em XMM-Newton}{} serendipitous source catalogue \citep[3XMM-DR8\footnote{\url{http://xmmssc.irap.omp.eu/Catalogue/3XMM-DR8/3XMM_DR8.html}},][]{rosen16} yielded two additional close matches between an L dwarf candidate and an X-ray source\footnote{3XMM J205205.8-610355, within $\sim1.6''$ of
Obj. ID 185442097 in \citet{carnerorosell2019} and 3XMM J232604.0-543340, within $\sim1.0''$
Obj.\ ID 133082583 in \citet{carnerorosell2019}}.
Interestingly, both X-ray sources display possible flaring activity, as seen in EPIC light curves produced by the EXTraS software. Optical--NIR spectroscopy
is
needed to confirm the UCD classification
of these objects.
Finally, we estimate that the upcoming All-Sky Survey of {\em eROSITA} \citep{Merloni12.0} will be able to detect superflares of the size of the event on J0331-27{} described here within a volume of $\sim 100$\,pc.
\begin{acknowledgements}
This work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. This research has made use of data produced by the EXTraS project, funded by the European Union's Seventh Framework Programme under grant agreement no 607452. We also used observations made by the Chandra X-ray Observatory, and obtained from the Chandra Data Archive. We thank Gian Luca Israel for useful discussions. We acknowledge the computing centre of INAF Osservatorio Astrofisico di Catania for the availability of computing resources and support under the coordination of the CHIPP project. We acknowledge financial support from ASI under ASI/INAF agreement N.2017-14.H.0.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,877 |
Coca Cola drops huge Christmas truck tour hints for 2021
It is one of the highlights of the festive season for many people
Neil Shaw
Holidays are coming as Coca Cola has dropped huge hints that its festive truck will be on tour this Christmas, reports The Daily Star.
Thousands of people flock to see the lorry when it visits around 40 destinations across the UK each Christmas - although the tour has been disrupted by Covid restrictions.
Coca Cola has so far not released details of dates or locations for this year's appearance.
The truck is an opportunity to meet Santa and pick up free Coke, and people queue to get their picture taken with the lorry.
According to the Star, Coca Cola has confirmed the appearance this year in a series of cryptic social media messages - advising people to: "Keep your eyes on our socials..." and saying: "We're just recruiting the last of our reindeer..."
Coca-Cola also tweeted: "Holidays are coming..." with a Santa Claus emoji.
Coca-Cola said on social media: "We update our route every year to make sure we include more and more locations. We hope we get to see you!"
During last year's cancellation they tweeted: "We know it's disappointing, but we'll continue to share special Christmas moments throughout the festive season. We look forward to seeing you next year!"
RSPCA warning to parents over Minecraft salamander trend
Price of a pint has gone up by up to almost 20% since start of pandemic | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,029 |
Q: Code after for loop does not execute (Java) I have a small snippet of code that is supposed to check the inputted codename and password against what is stored in a text file. If there is a match, it will start the game and everything is fine. But if there is no match, a dialog is supposed to pop up asking the user if they want to try logging in again.
int input=0; //yes
do {
codename=JOptionPane.showInputDialog(null,"Enter Codename: ");
String password=JOptionPane.showInputDialog(null, "Enter Password: ");
for(int i=0;i<users.length;i++){
if((codename.equals(users[i].getCodeName())) && (password.equals(users[i].getPassword()))){
System.out.println("\n\nCorrect");
new Game();
} else {
System.out.println("\n\nIncorrect");
}
}
input = JOptionPane.showConfirmDialog(null, "Incorrect User name/Password\nWould you like to try again?");
} while(input==0); //run while input is yes
The problem: the code after the for loop does not execute. If I check the variables against users[i] the code after the for loop does not run, but if I check against users[2] for example, then it works fine.
Not sure if this matters but I always get this error:
Exception in thread "main" java.lang.NullPointerException
at com.ramtin.Game.logOn(Game.java:505)
at com.ramtin.Game.main(Game.java:397)
I get it even when the password and codename match and the program runs perfectly.
FULL CODE for the above code:
public static void logOn(){
//ASK FOR CODENAME & PASSWORD FROM TEXTFILE BEFORE GAME BEGINS
//read from text file
UserData[]users=new UserData[20];
int countU=0;
try{
BufferedReader readU = new BufferedReader(new FileReader("userdata.txt"));
String line;
while((line=readU.readLine())!=null){
String []parts=line.split("#");
String codeName = parts[0];
String password=parts[1];
// System.out.println(parts[0]);
// System.out.println(parts[1]);
users[countU]=new UserData(codeName, password);
countU++;
}
readU.close();
}
catch(FileNotFoundException e){
}
catch(IOException e){
}
//PASSWORD & CODENAME
int input=0; //yes
do{
codename=JOptionPane.showInputDialog(null,"Enter Codename: ");
String password=JOptionPane.showInputDialog(null, "Enter Password: ");
for(int i=0;i<users.length;i++){
if((codename.equals(users[i].getCodeName()))&&(password.equals(users[i].getPassword()))){
System.out.println("\n\nCorrect");
new Game();
}
else{
System.out.println("\n\nIncorrect");
}
}
input = JOptionPane.showConfirmDialog(null, "Incorrect Username/Password\nWould you like to try again?");
}
while(input==0); //run while input is yes
}
}
FULL CODE for UserData:
public class UserData {
private String codename;
private String password;
UserData (String codeName, String password)
{
this.codename = codeName;
this.password= password;
}
String getCodeName()
{
return codename;
}
String getPassword()
{
return password;
}
public String toString ()
{
String temp = "\nCode name: "+codename+"\nPassword: " + password;
return temp;
}
}
A: Don't do for(int i=0;i<users.length;i++){
Do do for(int i=0;i<countU;i++){
Every array element after countU will cause NPE when you call a method on a null element like users[i].getCodeName()
A: Instead of using an array with the length 20, you can use an ArrayList as it will expand dynamically:
List<UserData> users = new ArrayList<>();
Then add each userdata to the list
users.add(new UserData(codeName, password));
and iterate with
for(int i=0 ;i<users.size(); i++) {
This will prevent the NullPointer as you only have as many entries as you have users (and will also dynamically grow/shrink with the number of users you have).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,217 |
{"url":"https:\/\/www.physicsforums.com\/threads\/ftl-communication-might-qm-entanglement-trump-relativity.281580\/","text":"# FTL communication: might QM entanglement trump relativity?\n\n#### Untangled\n\nFTL communication: might QM entanglement \"trump\" relativity?\n\nAs we presently understand entanglement, it can't be used for FTL communication, since some speed-of-light exchage of information between Bob and Alice is required. So c apparently still remains the communication speed limit.\n\nBut is it crystal clear that entanglement does not somehow permit FTL communication? My thinking is as follows:The speed of light limit on signaling times is predicted by a \"classical\" physics theory, i.e., relativity. But the apparent instantaneity (>10,000xc according to a recent experiment) of \"effects\" between entangled particles is a quantum mechanical phenomenon. At this point, the two theories are of course incompatible and thus incomplete, since no testable unified theory has yet been developed. But when experiments examined Bell's inequality, and thus whether (classical) locality\/separability or QM entanglement is correct, QM won out. This type of result implies to me that the current QM theory may represent a significantly more basic description of the universe than does the current relativity theory. If so, perhaps the final unified theory will in fact permit FTL signaling, utilizing some aspect of entanglement not yet envisaged.\n\nRelated Quantum Physics News on Phys.org\n\nStaff Emeritus\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nBefore I answer this, I have to know - do you really think you were the first person to think of this? And that somehow Einstein, Pauli, Dirac, Fermi and everybody else working in the past century somehow missed this?\n\nThere is no way to signal faster than light using entanglement. Every measurement to date - and there have been billions - is consistent with both SR and QM. If you want to argue SR is wrong, you will have to explain why it agrees with measurement after measurement.\n\n#### turbo\n\nGold Member\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nIt's not a bad question, Vanadium 50. Einstein himself worried about how quantum theory might invalidate field theory. Please read chapter 1 of \"The Philosophy of Vacuum\" by Saunders and Brown. That chapter is a translation of Einstein's \"On the Ether\" (1924) by Simon Saunders and it is under copyright so I will not transcribe it here.\n\n#### mgb_phys\n\nHomework Helper\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nEPR doesn't violate causality because you can't choose which state the particle will collapse into - so you can't use it to send data. You are allowed to transmit random numbers faster than light.\n\nSome instantaneous symmetry breaking functions can be used to send data faster than light.\nThe famous philosopher Pratchett suggested that since when a king dies the oldest prince instantly becomes king - you can use this to send information faster than light if your are able to kill enough kings and detect the resulting state of kinglyness in the heir.\n\nStaff Emeritus\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nBut it's not a new question either. Hence my point.\n\n#### turbo\n\nGold Member\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nBut it's not a new question either. Hence my point.\nI realize that, and it is a valid complaint. If you will refer to Einstein's memoriam on the death of Ernst Mach, you will see that he placed an extremely high value on epistemology - the questioning of previously-held beliefs and opinions to re-affirm or falsify current theory. Einstein bemoaned the fact that epistemology was ignored by modern science ( in his life-time) as a critical tool. He was ignored then, and he is being ignored today. Without epistemology, science heads down blind alleys for decades (or longer) without a clue.\n\n#### Phrak\n\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nThe famous philosopher Pratchett...\n?? Is this the renowned T. Pratchett of infinite (turtle) regression fame?\n\n#### turbo\n\nGold Member\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\n?? Is this the renowned T. Pratchett of infinite (turtle) regression fame?\nIt's turtles all the way down????\n\n#### Hurkyl\n\nStaff Emeritus\nGold Member\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nBut is it crystal clear that entanglement does not somehow permit FTL communication?\nWhen presented in a form like this, it is crystal clear: local time evolution is completely determined by local 'components' of quantum state. (Disclaimer: I believe QFT has not actually been shown to satisfy such axioms)\n\nAnother strike against is that an experiment involving only one half of an entangled pair only has access to the relative state of that particle -- thus losing all information about the other half -- and so such an experiment cannot possibly be influenced by interactions involving the other half.\n\nIf so, perhaps the final unified theory will in fact permit FTL signaling, utilizing some aspect of entanglement not yet envisaged.\nThen it's no longer the phenomenon of entanglement that QM describes, but instead some phenomenon described by some other scientific theory. If you want to speculate about the latter, it's quite misleading to speak as if you're talking about the former. (Even if you expect them to approximate each other)\n\nLast edited:\n\n#### Phrak\n\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nIt's turtles all the way down????\nDon't be silly. The world rests upon the backs of three great Elephants astride a Tortise (no turtles here), on his way to Andromida. But this could be erroniously construed as off topic.\n\n#### ZapperZ\n\nStaff Emeritus\n2018 Award\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nI don't see any incompatibility here between QM and SR. Here's the reason.\n\nDid anyone detect ANY signal that traveled between the entangled pair when a measurement is made? Has this ever been published in any of the EPR\/Bell-type experiments at all? I'll answer that for you : NO.\n\nIn other words, there is nothing that \"traveled\" between one location in space to another. What QM does say is that there is an intrinsic connection between the entangled pair that does not depend on things moving between the two. This is what is meant by \"non-local\", something that is very common in physics, believe it or not (look at the wavefunction of a free particle).\n\nThis is not what SR describes, where a signal or information has to traveled through both space and time. Even when we have the apparent FTL group velocity in the NEC experiment from several years ago, when something has to make such a travel, all the postulates of SR are still obeyed.\n\nAs http:\/\/sciencenow.sciencemag.org\/cgi\/content\/full\/2008\/813\/3\",\n\n\"yet another experiment that tells us quantum mechanics is right\" and that there \"really is an intrinsic connection between entangled particles, not that some signal passes quickly between them when an observation is performed.\"\nWe don't know yet how they connect with each other, or even if there is a way to know. What we DO know is that they don't pass information between each other via sending a signal that requires motion from one location to another over time. This is the part that does not violate Relativity.\n\nZz.\n\nLast edited by a moderator:\n\n#### Marcaias\n\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nEntanglement is not some mathematically mysterious process that can't be analyzed. In fact, it's fairly easy to convince yourself that FTL communication via entanglement is not possible in the simplest case, when two qubits are entangled.\n\nIf (as you say) Alice and Bob share an entangled pair $$|\\phi\\rangle = |00\\rangle + |11\\rangle$$, you might hope that Bob could selectively increase the amplitude only the \"11\" part of the qubits through only local operations on his qubit, but (assuming the Schroedinger equation is correct and only unitary transformations are physically possible) any unitary operation U on his qubit will simply transform the system into $$|0\\hat 0\\rangle + |1\\hat 1\\rangle$$, where $$\\{ |\\hat 0\\rangle, |\\hat 1\\rangle\\}$$ is another orthonormal basis.\n\nTaking a partial trace reveals that the density operator of Alice's qubit by itself is the maximally-mixed $$I\/2 = 0.5|0\\rangle\\langle0| + 0.5|1\\rangle\\langle1|$$, exactly the same as if Bob did nothing first, and thus revealing nothing.\n\nOn the other hand, if we suppose that non-unitary operations ARE possible, then FTL communication IS possible. Then you would be able to \"pry apart\" an orthonormal basis and increase the amplitude of only the \"11\" or \"00\" part of the entangled system. But this would mean abandoning the Schroedinger equation, the cornerstone of quantum theory, when all the available evidence points to it being correct.\n\n#### colorSpace\n\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nIn other words, there is nothing that \"traveled\" between one location in space to another. What QM does say is that there is an intrinsic connection between the entangled pair that does not depend on things moving between the two. This is what is meant by \"non-local\", something that is very common in physics, believe it or not (look at the wavefunction of a free particle).\n\nThis is not what SR describes, where a signal or information has to traveled through both space and time. Even when we have the apparent FTL group velocity in the NEC experiment from several years ago, when something has to make such a travel, all the postulates of SR are still obeyed.\nIt seems to me a similar \"exception\" to SR takes place when photons are emitted (by a simple candle, for example).\n\nEven though SR says something with mass <> 0 can't be accelerated to the speed of light, in the \"creation\" of photons their constituting energy \"jumps\" to the speed of light.\n\nBTW, I've recently read that 'causality' is not really a requirement of SR, even though it is often presented as if it were. Comments on this are welcome :).\n\nStaff Emeritus\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nEven though SR says something with mass <> 0 can't be accelerated to the speed of light, in the \"creation\" of photons their constituting energy \"jumps\" to the speed of light.\nPhotons are massless, and they are not accelerated. It's not like the candle wick has a store of stopped photons and starts tossing them out when lit.\n\n#### ZapperZ\n\nStaff Emeritus\n2018 Award\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nEven though SR says something with mass <> 0 can't be accelerated to the speed of light, in the \"creation\" of photons their constituting energy \"jumps\" to the speed of light.\nThis statement makes very little sense to me. What does it mean that \"...their constituting energy \"jumps\" to the speed of light...\"? And what's so contradictory to SR is that?\n\nBTW, I've recently read that 'causality' is not really a requirement of SR, even though it is often presented as if it were. Comments on this are welcome :).\nIn this forum, whenever one says \"I've recently read...\", we require exact citation of the source. If not, we have no way to know whether you read some crackpottery, or if you've misinterpreted what you read. Till then, it is pointless to write any comments on something like this.\n\nZz.\n\n#### colorSpace\n\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nThis statement makes very little sense to me. What does it mean that \"...their constituting energy \"jumps\" to the speed of light...\"? And what's so contradictory to SR is that?\nSorry, I thought the meaning would be quite obvious: The photons are \"created\" from energy which the valence electrons release. The valence electrons, as far as I know, do not move with the speed of light, nor are they massless. The energy of the emitted photons contributed to the mass of the system before they were emitted. This mass has moved at less than the speed of light, and SR says (in general) that a mass cannot be accelerated to the speed of light. However, when the photon is emitted, a certain amount of mass in the valence electrons is gone, and instead we have a photon moving at the speed of light. Of course, this is not an \"acceleration\" in the usual sense, and that is why I see a similarity to the text which I responded to.\n\nIn this forum, whenever one says \"I've recently read...\", we require exact citation of the source.\nSure, if wikipedia may be quoted here: http:\/\/en.wikipedia.org\/wiki\/Faster-than-light#Faster_light_.28Casimir_vacuum_and_quantum_tunnelling.29\n\n\"Causality is not required by special or general relativity, but is nonetheless generally considered a basic property of the universe that should not be abandoned.\"\n\nLast edited:\n\n#### ZapperZ\n\nStaff Emeritus\n2018 Award\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nSorry, I thought the meaning would be quite obvious: The photons are \"created\" from energy which the valence electrons release. The valence electrons, as far as I know, do not move with the speed of light, nor are they massless. The energy of the emitted photons contributed to the mass of the system before they were emitted. This mass has moved at less than the speed of light, and SR says (in general) that a mass cannot be accelerated to the speed of light. However, when the photon is emitted, a certain amount of mass in the valence electrons is gone, and instead we have a photon moving at the speed of light. Of course, this is not an \"acceleration\" in the usual sense, and that is why I see a similarity to the text which I responded to.\nYou do know that this is not the only way a photon can be created, don't you?\n\nSure, if wikipedia may be quoted here: http:\/\/en.wikipedia.org\/wiki\/Faster-than-light#Faster_light_.28Casimir_vacuum_and_quantum_tunnelling.29\n\n\"Causality is not required by special or general relativity, but is nonetheless generally considered a basic property of the universe that should not be abandoned.\"\nI'm not the person you want to use a wikipedia as a reference. You shouldn't pay that much emphasis on exact quotes out of Wikipedia. I could easily go in and edit that away, and now what will you be left with?\n\nYou need a better source than wikipedia to back your claim. Please note that for the physics forum, our main preferences and what we highly recommend would be peer-reviewed publications as valid sources. The use of other sources is done at your own risk.\n\nZz.\n\n#### colorSpace\n\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nYou do know that this is not the only way a photon can be created, don't you?\nBy a candle? No, I thought in a candle that would be the only way.\n\nI'm not the person you want to use a wikipedia as a reference. You shouldn't pay that much emphasis on exact quotes out of Wikipedia. I could easily go in and edit that away, and now what will you be left with?\n\nYou need a better source than wikipedia to back your claim. Please note that for the physics forum, our main preferences and what we highly recommend would be peer-reviewed publications as valid sources. The use of other sources is done at your own risk.\n\nZz.\nIt's not a claim at all, just something I've read. Which is why I was wondering what the response on this forum would be, as I was expecting that this could be controversial. However I do have suspicions that FTL necessarily implies time-travel and is therefore a priori impossible, but that's just a personal opinion since I don't think that SR\/causality will be the first theory which won't be superseded by another one.\n\n#### ZapperZ\n\nStaff Emeritus\n2018 Award\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nBy a candle? No, I thought in a candle that would be the only way.\nPlease do a search on how EM radiation is created by your incandescent light bulb, and at all those synchrotron research centers.\n\nZz.\n\n#### colorSpace\n\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nPlease do a search on how EM radiation is created by your incandescent light bulb, and at all those synchrotron research centers.\n\nZz.\nIt seems that SR can't handle any of those cases either... :)\n\n#### ZapperZ\n\nStaff Emeritus\n2018 Award\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nIt seems that SR can't handle any of those cases either... :)\nProve it. SR doesn't say HOW light is created.\n\nAt some point, you have to stop making these type of outlandish claims without any solid basis other than what you read out of Wikipedia.\n\nZz.\n\n#### colorSpace\n\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nProve it. SR doesn't say HOW light is created.\n\nAt some point, you have to stop making these type of outlandish claims without any solid basis other than what you read out of Wikipedia.\n\nZz.\nA long time ago, I've actually studied physics for a couple of years, however what I have read on wikipedia and elsewhere does match my memories, so I am more than surprised that you call these claims \"outlandish\".\n\nBefore the emission of photons by valence electrons, the electrons have not only more energy, but also more mass. After the emission, instead of this energy and mass, there are new photons moving at the speed of light. Since this is not an \"acceleration\" in a strict sense, one might say it doesn't contradict SR, which says that acceleration of a mass to speed of light would require infinite energy; on the other hand, if it wasn't known that photons can be emitted in this way, then one might be inclined to believe that because of SR, it would be impossible.\n\nMy physics books have remained in another country, call that \"outlandish\" if you want.\n\nSo please give me an example of a proper argument and tell me what is wrong with my \"claim\".\n\n#### ZapperZ\n\nStaff Emeritus\n2018 Award\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nA long time ago, I've actually studied physics for a couple of years, however what I have read on wikipedia and elsewhere does match my memories, so I am more than surprised that you call these claims \"outlandish\".\n\nBefore the emission of photons by valence electrons, the electrons have not only more energy, but also more mass. After the emission, instead of this energy and mass, there are new photons moving at the speed of light. Since this is not an \"acceleration\" in a strict sense, one might say it doesn't contradict SR, which says that acceleration of a mass to speed of light would require infinite energy; on the other hand, if it wasn't known that photons can be emitted in this way, then one might be inclined to believe that because of SR, it would be impossible.\n\nMy physics books have remained in another country, call that \"outlandish\" if you want.\n\nSo please give me an example of a proper argument and tell me what is wrong with my \"claim\".\nYou need to look back at the thread and see what I was challenging you on.\n\n1. I asked you if you think that the ONLY way one can generate photons is via atomic transition. You obviously thought so based on your response. This is patently false. One only needs to look at a radio transmitter to know this. The radio waves that your analog radio receive did not come from some atomic transition. I can also easily wiggle a bunch of free electrons and they'll generate EM radiation. No atomic transition there! If you really had studied physics, then you have forgotten your classical E&M.\n\n2. What I called \"outlandish\" claim backed by weak sources is your \"quote\" about causality. Don't you think it is a bit of a stretch to quote some passage out of Wikipedia? I can see (even if I think it is also dubious) one referring the physics content out of Wikipedia. But to actually quote a passage out of it as if it is the definitive source is rather horrifying, considering that anyone can come in and edit such text.\n\nDo you now see what the problems are?\n\nZz.\n\n#### colorSpace\n\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nYou need to look back at the thread and see what I was challenging you on.\n\n1. I asked you if you think that the ONLY way one can generate photons is via atomic transition. You obviously thought so based on your response. This is patently false. One only needs to look at a radio transmitter to know this. The radio waves that your analog radio receive did not come from some atomic transition. I can also easily wiggle a bunch of free electrons and they'll generate EM radiation. No atomic transition there! If you really had studied physics, then you have forgotten your classical E&M.\nI'm not sure why you think it was obvious, as it isn't the case. I knew (or thought) that photons can be created when matter and antimatter meet, and that there are many other particle reactions in which energy is released as photons.\n\nHowever I did (and still do) believe that in a candle the only source are the valence electrons (following your request I did several searches but could not find any information specific to candles or fire).\n\n2. What I called \"outlandish\" claim backed by weak sources is your \"quote\" about causality. Don't you think it is a bit of a stretch to quote some passage out of Wikipedia? I can see (even if I think it is also dubious) one referring the physics content out of Wikipedia. But to actually quote a passage out of it as if it is the definitive source is rather horrifying, considering that anyone can come in and edit such text.\n\nDo you now see what the problems are?\n\nZz.\nAgain I'm not sure why you got the impression that wikipedia would be in any way thought of as a definitive source, if that where so, I would not have asked whether it can be quoted here. In past discussions, I have several times quoted wikipedia in cases where I thought that the content was wrong, in order to discuss that content. In this case, I quoted it without comment, since I had no definite view on the matter, but was interested how others think about it. The literal quote was made only because you asked for it, perhaps it was the language of the quote itself which gave the impression of claiming to be definitive source. I do think that the question of causality was relevant to the topic.\n\nBut the problems seems to be that I'd need to clarify those things in the first place.\n\n#### ZapperZ\n\nStaff Emeritus\n2018 Award\nRe: FTL communication: might QM entanglement \"trump\" relativity?\n\nI'm not sure why you think it was obvious, as it isn't the case. I knew (or thought) that photons can be created when matter and antimatter meet, and that there are many other particle reactions in which energy is released as photons.\n\nHowever I did (and still do) believe that in a candle the only source are the valence electrons (following your request I did several searches but could not find any information specific to candles or fire).\nThis has gone way off topic. The discussion on the light source for candles, incandescent light bulb, synchrotron light sources, etc. have already been discussed ad nauseum on here. I may even have to put up a FAQ on this sooner or later.\n\nIt is seriously wrong to assume that atomic transition (and matter-antimatter annihilation) are the only mechanism to generate EM radiation. It ignores THE most common terrestrial mechanism by human to generate EM radiation.\n\nAgain I'm not sure why you got the impression that wikipedia would be in any way thought of as a definitive source, if that where so, I would not have asked whether it can be quoted here. In past discussions, I have several times quoted wikipedia in cases where I thought that the content was wrong, in order to discuss that content. In this case, I quoted it without comment, since I had no definite view on the matter, but was interested how others think about it. The literal quote was made only because you asked for it, perhaps it was the language of the quote itself which gave the impression of claiming to be definitive source. I do think that the question of causality was relevant to the topic.\n\nBut the problems seems to be that I'd need to clarify those things in the first place.\nOK, then let's move on by saying that it would take us several lifetimes if we continue to discussion various passages out of Wikipedia. The very same way that we simply do not have the time or the patience to answer every single crackpottery that exists out there, I would also insist that, unless one has a more reputable sources, quoting passages out of Wikipedia should not be done in here.\n\nIs that clear enough?\n\nZz.\n\n### Physics Forums Values\n\nWe Value Quality\n\u2022 Topics based on mainstream science\n\u2022 Proper English grammar and spelling\nWe Value Civility\n\u2022 Positive and compassionate attitudes\n\u2022 Patience while debating\nWe Value Productivity\n\u2022 Disciplined to remain on-topic\n\u2022 Recognition of own weaknesses\n\u2022 Solo and co-op problem solving","date":"2019-10-15 04:08:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5953998565673828, \"perplexity\": 820.9465168695074}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986655864.19\/warc\/CC-MAIN-20191015032537-20191015060037-00457.warc.gz\"}"} | null | null |
Diabolism e бивша блек метъл група основана в Бургас през 1994 година, активна до 2003 година. Идеята за името "Диаболизъм" е вдъхновена от мистиката в природата и водещо началото си от древните езически ритуали на траки, славяни и прабългари.
История
Групата съществува от лятото на 1994 година, когато Deimoz и Agarvaen се събрат с Иван Ганчев-Чеха (ex. Biophobia). Тогава, под влиянието на Чеха и набиращата сила норвежка блек метъл вълна, е решено новата група да свири atmospheric black metal. Самата песен "Creeping into the Grave" е един остатък от предишния Death Metal проект на Agarvaen и Deimoz, наречен Infanticide. По това време съставът на Диаболизъм изглежда по следния начин: Чеха – вокали, Agarvaen – бас, Deimoz – китара, Пламен (Зомбата) – клавири, Михаела Стоева – цигулка, Радостин Димитров – барабани (негова е идеята за името на групата). Хармонията между членовете на групата благоприятства бързото и съзряване и поставянето на основата върху която е изграден по-късния период на групата.
Усилените репетиции от края на 1994 година дават своя резултат, и за около три месеца са създадени две нови песни – Freezing emotions under the dark clouds of my soul (включена в албума Endless Darkened Saga, издаден от музикална компания Wizard през 1997 г.) и ...The Hill of the Devil...The Hill of the Death... (нереализирана). Радостин, Пламен и Чеха влязоха в казармата, групата губи мястото си за репетиране в немската гимназия, Михаела се оттегля и от късната пролет на 1995, и до късната есен на същата година Deimoz и Agarvaen са единствените, които поддържат диаболистичния пламък.
Желанието за реализация на създадения материал подтиква Deimoz и Agarvaen да потърсят барабанист, който се появява в лицето на Angrist. Така в края на 1995 се оформи култовия състав на "Диаболизъм". Този период е характерен с усложняването на композициите и подобряването на аранжиментите на песните. Постепенно Диаболизъм премина към езическо-историческа тематика. За по-малко от шест месеца са създадени парчетата за албума Endless Darkened Saga. Репетициите, отново в немската гимназия, са безкрайни и вдъхновяващи. Когато пристигна предложението за участие в ежегодния тогава Дет Фест в Бургас, групата решава да покаже на живо най-доброто от себе си. Вариантът на Epos for the pagan times of winter, включен в този диск, е от този фестивал, на който групата е удостоен със специалната награда на Graffiti студио-запис на своя песен. Така през септември 1996, за само 47 студийни часа е записан целия албум Endless Darkened Saga. Беше проявен интерес за издаване от страна на Wizard, стигна се до договор и през ранната пролет на 1997-а, албумът видя бял свят. През това време Диаболизъм провежда редица концерти в култовия клуб "Сатан" в Бургас, заедно с водещите имена на бургаската метъл сцена. Най-якият концерт е в началото на лятото, заедно с Демонизъм и Иудициум, който се провежда пред близо шестстотин души, отново в "Сатан". По това време бяха измислени песните "Мрачна прелюдия", "Страната, гдето се раждат ветровете" и почти изцяло Nebelwerfersflucht, вдъхновени от old-school pagan идеята. Когато вече всичко сочеше, че "Диаболизъм" ще са от основните фигури на наближаващия Дет Фест, организаторите не им отреждат подобаващото им се място на фестивала. Това довежда до търкания в групата и тя почти се разпада, имайки предвид започналото следване на Agarvaen и влизането в казарма на Deimoz. Angrist замина за Германия и всичко изглежда приключило.
Към края на 1998 година, след като Deimoz се завръща от казармата, отново започва да се говори за Диаболизъм. Старите неразбирателства со отхвърлени и двамата с Agarvaen отново творят заедно. През 1999 година са измислени последните три песни на Диаболизъм: Pamirian call for holocaust, Celestial beauty disgraced и Tristia. Липсата на репетиционна, постоянната ангажираност на Agarvaen със следването му в друг град и отсъствието на Angrist от страната са непреодолима пречка за съществуването на Диаболизъм като група. В този момент се появява Firebliss (основната фигура в REUNION), който предлага на Agarvaen и Deimoz да запишат при него напълно безплатно една песен с програмирани барабани. Това вдъхва живот на групата и в началото на 1999 година е записана "Мрачна прелюдия". По-късно Firebliss написа компютърни барабани и клавири за Nebelwerfersflucht, "Страната, гдето се раждат ветровете", Epos for the pagan times of winter, Sunset in heroic shades / "Легенда за края на българското царство" и Pamirian call for holocaust. Така става възможноу макар и без жив изпълнител на барабани "Диаболизъм" да имат изяви на живо. Участват в Hysteria'99, после на рок фестивал в Разград през септември, отново в "Сатан" през октомври (в състав с Angrist) и в края на 1999.
През 2000 година започна работа по записването на всички неиздадени песни при Firebliss. През пролетта работата по записите е прекъсната поради ангажименти на Deimoz и Agarvaen. През есента на същата година с Диаболизъм се свърза Кирил Ненов (ex-Саракт) от Пловдив с предложение музикантите да запишат нереализираните си песни в студио на негови разноски. И така през декември 2000 година в Пловдив са записани барабаните на всички песни създадени след "Endless Darkened Saga". Впоследствие работата със студиото в Пловдив е прекъсната, но може да се каже, че Кирил Ненов има съществен принос за последвалата реализация на "Страната, гдето се раждат ветровете", Nebelwerfersflucht и Tristia. След 1997 година групата съществува най-вече поради желанието на фенове като Firebliss и Киро Ненов, които оказват голяма помощ на групата. През лятото на 2001 Диаболизъм се свързват с Astaroth и в края на годината започва работа по три от песните записани в Пловдив. Записите са завършени към началото на лятото на 2002 година.
Последна изява на живо на членовете Deimoz и Agarvaen е на 25 януари 2003 година в "Роял Клуб" Бургас.
Дискография
Издадени са:
1997 – "Endless Darkened Saga"
2004 – "Concealed Craft", Компилация
2006 – "Concealed Craft", EP
Състав
Ивайло Братованов-Agarvaen – вокал (1994 – 2003)
Златин Атанасов-Deimoz – китара (1994 – 2003)
Радостин Димитров-Radogeddon – барабани (1994)
Пламен-Zombata – клавири (1994)
Михаела Стоева – цигулка (1994 – 1995)
Иван Ганчев – вокал (1994)
Калин Георгиев-Angrist – барабани (1995 – 1997)
Източници
Външни препратки
Diabolism в Myspace
Diabolism в Български рок архиви
Diabolism в Encyclopaedia Metallum
Diabolism в Last.fm
Diabolism в Spirit of metal
Diabolism в Rockdetector.com
Български блекметъл групи
Музикални групи в Бургас
Основани в България през 1994 година | {
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Q: Deriving Bernoulli's equation via conservation of E So I'm not OK with how some people derive this equation.
These people consider a pipe whose endings have cross-sectional areas and heights which are different. They then use the conservation of energy principle by saying $dW = dK + dU$ (Where $W$ is work, $K$ is kinetic energy, and $U$ is potential energy).
For this they consider that the work done on the system would be due to external pressure forces exerted on the whole system of water along the pipe. And here comes the part where I disagree: they use this Work to calculate the change in Potential and Kinetic energy for just a small slab of water within the whole system. This is completely invalid isn't it? I mean you would have to consider the entire system, I think.
My way of interpreting the derivation is if you consider just one slab the whole time. Is this a valid way of thinking?
Thanks!
edit: In fact, in one video I saw, the person just says "the middle chunk of water stays the same the whole time, so we can just ignore it".
A: You have to consider the assumptions that go into deriving Bernoulli's equation in that manner; that the fluid is incompressible, non-viscous, and experiences non-turbulent flow.
If these are the assumptions under which you've built your model for fluid flow, when you apply some sort of pressure to one end that results in a work on the fluid, because the fluid can't compress at all to absorb the energy nor can it interact with other parts of the fluid to have some resistive drag that causes some of the fluid to behave different then other sections of the fluid, every part of the fluid responds similarly.
Treating one small part of the fluid and its behavior should therefore be the same as any other part of the fluid. Indeed, it's as if the whole fluid is reacting to the force applied to it at once time. This is why Bernoulli's Equation tells us that energy is conserved per unit volume of the fluid, regardless of where it is.
In general, a more rigorous derivation is needed for more complicated fluid models, but that one suffices for the basic dynamics of fluid flow.
A: You cannot derive the classic Bernoulli Equation from conservation of energy, because, contrary to popular opinion, it is actually not an expression of conservation of energy at all. It is more accurately construed as an integrated expression of the conservation of linear momentum, $F=ma$.
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\section{Introduction}
\label{section: introduction}
The dusty debris disks around main sequence stars are hallmarks of mature planetary systems. The small quantity ($<<1$\,M$_\earth$) of dust in debris disks is generated by destructive processes involving grinding collisions of planetesimals \citep[][and references therein]{2018Hughes}. Recent observations have revealed that the dust is commonly accompanied by detectable amounts of molecular gas \citep[e.g.,][]{1995zuc,moor2011,2017moor}. The only { molecular} species so far detected is CO, which is detected most commonly around A- and B-type stars \citep{2016lieman} despite their harsh UV radiation fields that ought to photodissociate the gas on timescales much shorter than the age of the star \citep[e.g.,][]{2009visser,2013ApJkospal}. Surprisingly, some of these systems exhibit CO emission comparable to that from the protoplanetary disks around pre-main sequence stars.
Work to understand the origin and implications of the molecular gas in debris disks has centered around the question of whether the gas is effectively primordial, i.e., surviving from the protoplanetary disk phase and therefore H$_2$-dominated, or secondary, i.e., produced in destructive processes like the dust in debris disks and therefore H$_2$-poor. The former category is sometimes referred to as ``hybrid" disks \citep[e.g.,][]{2013ApJkospal}, since it would imply the coexistence of primordial gas and secondary dust.
Observationally, it is difficult to distinguish the origin of the gas on the basis of CO observations alone, although the weight of the evidence leans towards it being second-generation origin. The low excitation temperatures are suggestive of a low H$_2$ gas mass, unlike the LTE conditions in protoplanetary disks \citep{2013ApJkospal}. A secondary origin is also supported by the unresolved scale height in the 49 Ceti disk at a resolution of 0\farcs4 that suggests a high mean molecular weight \citep{Hughes2017}. The presence of large amounts of CI gas in at least two systems has also been interpreted as evidence of second-generation origin \citep{2017higatomic}. Certainly the asymmetries associated with individual systems like $\beta$ Pic and Fomalhaut are consistent with a second-generation origin \citep{dent2014,2017fomalhaut}. On the other hand, the gas release rates required to sustain the large mass CO for the brightest systems, which can require vaporizing a small comet every few seconds if the CO is unshielded or a Hale-Bopp-size comet roughly once an hour \citep{2012Zuckerman,2013ApJkospal}, are difficult to reconcile with the putative mass reservoirs in debris disks and the high frequency with which gas is observed.
Even if the origin of the gas is secondary, multiple mechanisms have been proposed to generate secondary gas. {These mechanisms include} photodesorption \citep{gri07}, collisional vaporization of icy dust grains \citep{cze07}, and low-velocity collisions, either at the site of a resonant point with an unseen planet \citep{dent2014} or as the result of a major collision between planetary embryos \citep{jac14}, collisional desorption from comets \citep{2012Zuckerman}, or sublimation of comets in the inner regions of the system to explain absorption features \citep{beu90}. Notably, \citet{kra17} developed a model based on the destruction of volatile-rich planetesimals that can explain the CO, C, and O content of most of the debris disks observed to date. In that work they suggest that some of the higher-mass sources may be hybrid disks. However, an extension of this basic model posits that even young debris disks with protoplanetary-level CO masses may in fact be composed of second-generation gas shielded by CI \citep{2018kral}.
The chemical composition of the molecular gas is a relatively unexplored dimension of the properties of gas-bearing debris disks that can provide insight into the nature and origin of the gas. Here we present the deepest search to date for five molecules other than CO in the debris disk around 49 Ceti. 49 Ceti is a 40 Myr \citep{2012Zuckerman} A-type star located 57.1\,pc from Earth \citep{2018gaia} that hosts one of the two brightest (in CO) and best-studied of the gas-bearing debris disks. The other system is $\beta$ Pictoris, which has an almost identical CO(3-2) line flux at a factor of $\sim$3 closer distance \citep{2017luca}. Despite their similar CO fluxes, 49 Ceti is a better candidate for a molecular line survey because its larger gas densities are more readily capable of exciting emission from higher critical density molecules, and its higher disk-averaged excitation temperature (32\,K vs. 12\,K in $\beta$ Pic) is better for detecting lines with higher upper-level energies, thus implying that we will get the strongest constraints on the molecular abundances. Its gas disk is also symmetric, unlike the $\beta$ Pic disk, and covers a smaller solid angle on the sky, making it easier to spatially average the emission to maximize sensitivity from faint lines. Previous studies of the 49 Ceti system have spatially resolved the CO emission from the disk, revealing it to be axisymmetric and mostly consistent with Keplerian rotation about the central star between radii of $\sim$20 and $\sim$200\,au \citep{2008Hughes,Hughes2017}. There are hints of an anomalous { (supersolar)} C/O ratio from {\it Herschel} spectroscopy \citep{rob14}, as well as evidence of active CO photodissociation with low H$_2$ abundances from the detection of strong CI emission \citep{hig17}.
The five molecules selected for the search are CN, HCN, HCO$^+$, SiO, and CH$_3$OH. The low cross-section of CN to photodissociation makes it more likely to survive in the harsh radiation field of an A star, and in fact models of second-generation gas based on comets with Solar System abundances indicate that CN should be about as bright as, if not brighter than, CO \citep{2018mat}. It is also an excellent tracer of regions affected by UV radiation \citep{rod98}, and seems to be enhanced in evolved protoplanetary disks \citep{2014kast}. CN is the daughter product of HCN photodissociation. HCN tends to be bright in both protoplanetary disks and comets \citep{2002comet,2010oberkar}, although intriguingly it was not detected in previous, less-sensitive ALMA observations of the 49 Ceti system \citep{Hughes2017}. HCO$^+$ is similarly bright in both disks and comets \citep{2016guillo,1997comet}, and tends to be produced in regions where water has been photodissociated \citep{mil04}. It is also a sensitive tracer of ionization conditions in protoplanetary disks \citep{cle15}. While SiO has never been detected at millimeter wavelengths in protoplanetary disks or comets, it would provide unequivocal evidence for grain-grain collisions as the source of the gas. Methanol (CH$_3$OH) can be as abundant as CO in comets \citep{boc04,mum11}, but has only been detected once at a very low level in the nearest protoplanetary disk \citep{2016diskch3oh} and in an outbursting young star \citep{hoff18}.
In Section~\ref{section: observations} we present the ALMA observations, and in Section~\ref{section: results} we describe the stringent upper limits we are able to place on the flux of all five molecules. In Section~\ref{section: analysis} we apply a spectral shifting technique pioneered by \citet{2017luca} and similar to that of \citep{yen16} that takes advantage of the known CO velocity field to maximize any signal present in the noisy data. We then discuss our results in the context of known and predicted molecular abundances for protoplanetary disks, comets, and second-generation gas in Section~\ref{section:discussion}, and summarize our conclusions in Section~\ref{section: conclusion}.
\section{Observations}
\label{section: observations}
The observations (2017.1.00941.S, PI Hughes) utilized 48 of the 12-meter ALMA antennas in the most compact configuration available during Cycle 5. We observed using two spectral setups in Band 7, each executed on two separate dates. On 2018 June 2 and June 6, the spectral setup targeted HCN(4-3) at a rest frequency of {354.5} GHz, CH$_3$OH 1(1,1)-0(0,0) at a rest frequency of 350.9 GHz and several hyperfine transitions of CN(3-2) with the strongest at a rest frequency of 340.2 GHz. The other spectral setup observed on 2018 June 9 and 21 targeted SiO(8-7) at a rest frequency of 347.3\,GHz, CH$_3$OH 4(1,3)-3(0,3) at a rest frequency of 358.6 GHz and HCO$^{+}$(4-3) at a rest frequency of 356.7 GHz. The three spectral windows targeting line emission used a spectral resolution of 976\,kHz (0.8\,km\,s$^{-1}$) and bandwidth 1.875\,GHz to maximize { continuum} sensitivity. One spectral window in each setup was configured for continuum observations (at frequencies of 341.5 and 345.3\,GHz), with a bandwidth of 1.875 GHz and channel widths of 31.250 MHz { (27.5\,km\,s$^{-1}$)}. The total on-source time was 1.6 hours per setup.
Table~\ref{table:observational_params} gives the basic observational parameters for both spectral setups and all observing dates, including the number of antennas, time on source, naturally weighted beam size ($"$), the coordinates of the phase center in RA and Dec, bandpass calibrators, flux calibrators, and gain calibrators. Table~\ref{table:results} lists the rest frequency of each line (note that some were offset from the central frequency of the spectral window to satisfy constraints on the independently tunable intermediate frequencies (IFs).
\begin{comment}
\begin{table}[ht]
\centering
\caption{49 Ceti Observational Parameters}
\label{table:observational_params}
\begin{tabular}{lcc}
\hline
\hline
$\#$ Antennas & \multicolumn{2}{c}{48} \\
Baseline distance & \multicolumn{2}{c}{12 - 250 m}\\
Time on Source &
\multicolumn{2}{c}{3.2 h$^a$} \\
Naturally Weighted Beam Size & \multicolumn{2}{c}{1\farcs1 x 0\farcs9} \\
Phase center $\alpha$ & \multicolumn{2}{c}{01:34:37.900573}\\
Phase center $\delta$ & \multicolumn{2}{c}{-15:40:34.94661}\\
\bottomrule
\hline
Dates & 02-Jun-2018 & 09-Jun-2018 \\
\hline
Bandpass Calibrator & J0006-0623 &J0006-0623\\
Flux Calibrator & J0006-0623 & J0141-0928 \\
Gain Calibrator & J0116-1136 & J0423-0120 \\
\hline
& 06-Jun-2018 & 21-Jun-2018 \\
\hline
Bandpass Calibrator &J0006-0623 &J0423-0120 \\
Flux Calibrator & J0006-0623 & J0423-0120 \\
Gain Calibrator & J0116-1136 & J0141-0928 \\
\bottomrule
\end{tabular}
\end{table}
\end{comment}
\begin{table}[ht]
\centering
\caption{49 Ceti Observational Parameters}
\label{table:observational_params}
\begin{tabular}{cccc}
\hline
\hline
\multicolumn{2}{l} {$\#$ Antennas} & \multicolumn{2}{c}{48} \\
\multicolumn{2}{l}{ Baseline distance} & \multicolumn{2}{c}{12 - 250 m}\\
\multicolumn{2}{l}{Time on Source} &
\multicolumn{2}{c}{3.2 h$^{a}$} \\
\multicolumn{2}{l}{Naturally Weighted Beam Size} & \multicolumn{2}{c}{1\farcs1 x 0\farcs9} \\
\multicolumn{2}{l}{ Phase center $\alpha$} & \multicolumn{2}{c}{01:34:37.900573}\\
\multicolumn{2}{l}{Phase center $\delta$} & \multicolumn{2}{c}{-15:40:34.94661}\\
\bottomrule
Dates & Bandpass Calibrator & Flux Calibrator & Gain Calibrator\\
\bottomrule
02-Jun-2018 & J0006-0623 & J0006-0623 &J0116-1136 \\
06-Jun-2018 & J0006-0623 & J0006-0623 &J0116-1136 \\
09-Jun-2018 & J0006-0623 & J0141-0928 &J0423-0120 \\
21-Jun-2018 & J0423-0120 & J0423-0120 &J0141-0928 \\
\bottomrule
\end{tabular}
\tablenotetext{a}{Divided into 1.6\,h per spectral setup}
\end{table}
\begin{comment}
\begin{table}[ht]
\centering
\caption{Central Frequency of Spectral Windows (GHz)}
\label{table:observations_freq2}
\begin{tabular}{*2c}
\toprule
Species & Transition \\
\midrule
\centering
\textnormal{C}H$_{3}$OH * & 350.9051\\
\textnormal{C}H$_{3}$OH ** & 358.6058\\
HCN(4-3) & 354.5054 \\
CN***(3-2) & 340.2477 \\
HCO$^{+}$(4-3) & 356.7342 \\
SiO(8-7) & 347.3305 \\
\bottomrule
&*1(1,1)-0(0,0)\\
&**4(1,3)-3(0,3) \\
& *** J=
($\frac{7}{2}$ \textnormal{-} $\frac{5}{2}$) \textnormal{F} = ($\frac{9}{2}$ \textnormal{-} $\frac{7}{2}$)\\
\bottomrule
\end{tabular}
\tablenotetext{a}{https://www.cv.nrao.edu/php/splat/}
\end{table}
\end{comment}
The raw data were calibrated via the ALMA pipeline using the Common Astronomy Software Applications Package \citep[CASA;][]{2007mcmull} { version 5.5.0-149}. We used the CASA task {\tt cvel} to convert the velocity axis from topocentric rest frame to the Kinematic Local Standard of Rest (LSRK). We used {\tt UVCONTSUB} to fit a constant (a zeroth order polynomial) to channels more than 50 km s$^{-1}$ away from the known systemic velocity so that we could subtract the continuum. We used {\tt TCLEAN} with natural weighting to produce channel maps for each molecule centered at the 2.78\,km\,s$^{-1}$ LSR systemic velocity of the 49 Ceti system \citep{Hughes2017} with pixel size 0\farcs15 covering an area 22\farcs5$\times$22\farcs5 on the sky.
\section{Results}
\label{section: results}
We begin to search for emission from the five molecules by { using the} channel maps and zeroth-moment (velocity-integrated intensity) maps { generated} from the visibilities. We use the CASA task {\tt immoments} to generate the moment 0 map, integrating channels within $\pm5$\,km\,s$^{-1}$ of the systemic velocity to ensure that all emission within the 8\,km\,s$^{-1}$ FWHM of the CO line \citep{2008Hughes} is incorporated. The moment 0 maps are displayed in Figure~\ref{fig:Moment0Maps} and show no significant gas emission.
CH$_{3}$OH and CN have multiple transitions in their spectral windows. CH$_{3}$OH has two transitions, 1(1,1)-0(0,0) and 4(1,3)-3(0,3), in two different spectral windows. CN exhibits many different hyperfine transitions within the same spectral window, all of which are listed in Table \ref{table:CN}. For our initial reconnaissance, we shifted all of the hyperfine lines to a common velocity channel and added them together in both the { disk-integrated, shifted} spectra produced from the channel maps { (see Section~\ref{section: analysis} for details)}, and in the moment 0 map, but did not detect any emission. For all subsequent analyses, we generate channel maps, spectra, and moment 0 maps for each methanol transition separately, and only for the strongest (highest intensity) of the CN transitions.
\begin{figure}[htb]
\centering
\includegraphics[width=1.0\textwidth,keepaspectratio]{5kmsmoment0_v7.pdf}
\caption{\label{fig:Moment0Maps} Moment 0 maps for all six molecular lines in the survey, integrated within $\pm5$\,km\,s$^{-1}$ of the 2.78\,km\,s$^{-1}$ systemic velocity of 49 Ceti. The naturally weighted beam is indicated by the hatched ellipse located on the bottom left, below a 50\,au scale bar. The red contour is an overlay of the CO(3-2) moment 0 map, indicating the spatial extent of the molecular line emission from the disk. The white contours are $\pm2\sigma$, where $\sigma$ is the rms noise in the image: 4.6, 5.5, 6.4, 7.2, 3.7, and 3.4\,mJy\,km\,s$^{-1}$\,beam$^{-1}$, for HCN(4-3), SiO(8-7), HCO$^+$(4-3), CH$_{3}$OH 4(1,3)-3(0,3), CH$_3$OH 1(1,1)-0(0,0), and CN(3-2), respectively. There is no statistically significant emission evident in any of the moment 0 maps.}
\end{figure}
\begin{table}[ht!]
\centering
\caption{CN Hyperfine Transitions}
\label{table:CN}
\begin{tabular}{*3c}
\toprule
Rest Frequency$^a$ (GHz)& Transition & CDMS/Lovas Intensity$^a$ \\
\midrule
339.44677 & N= 3-2, J=5/2-5/2, F=3/2-3/2 & -3.674 \\
339.45999 & N= 3-2, J=5/2-5/2, F=3/2-5/2 & -4.3924 \\
339.46263 & N= 3-2, J=5/2-5/2, F=5/2-3/2 & -4.3831 \\
339.47590 & N= 3-2, J=5/2-5/2, F=5/2-5/2 & -3.5256 \\
339.49321 & N= 3-2, J=5/2-5/2, F=5/2-7/2 & -4.3777 \\
339.49928 & N= 3-2, J=5/2-5/2, F=7/2-5/2 & -4.3598 \\
339.51663 & N= 3-2, J=5/2-5/2, F=7/2-7/2 & -3.3239 \\
339.99225 & N= 3-2, J=5/2-3/2, F=3/2-5/2 & -4.4392 \\
340.00809 & N= 3-2, J=5/2-3/2, F=5/2-5/2 & -3.0613 \\
340.01960 & N= 3-2, J=5/2-3/2, F=3/2-3/2 & -3.0625 \\
340.03154 & N= 3-2, J=5/2-3/2, F=7/2-5/2 & -2.1437 \\
340.03540 & N= 3-2, J=5/2-3/2, F=3/2-1/2 & -2.5692 \\
340.03540 & N= 3-2, J=5/2-3/2, F=5/2-3/2 & -2.3442 \\
340.03540 & N= 3-2, J=5/2-3/2, F=3/2-1/2 & -2.5692 \\
340.03540 & N= 3-2, J=5/2-3/2, F=5/2-3/2 & -2.3442 \\
340.24777 & N= 3-2, J=7/2-5/2, F=7/2-5/2 & -2.1495 \\
340.24777$^{*}$& N= 3-2, J=7/2-5/2, F=9/2-7/2 & -2.0159 \\
340.24854 & N= 3-2, J=7/2-5/2, F=5/2-3/2 & -2.2887 \\
340.26177 & N= 3-2, J=7/2-5/2, F=5/2-5/2 & -3.2027 \\
340.26494 & N= 3-2, J=7/2-5/2, F=7/2-7/2 & -3.2039 \\
340.27912 & N= 3-2, J=7/2-5/2, F=5/2-7/2 & -4.8867 \\
\bottomrule
$^{*}$ strongest transition \\
\end{tabular}
\tablenotetext{a}{https://www.cv.nrao.edu/php/splat/}
\end{table}
\section{Analysis}
\label{section: analysis}
To obtain the most sensitive upper limits on the flux density of the molecules, we utilize a spatial-spectral shifting technique pioneered by \citet{2017fomalhaut}. The original technique uses a model velocity field to identify the expected location of flux within the 3-D position-position-velocity cube and shift the expected peak of the spectrum of each spatial pixel to a common velocity, thereby concentrating signal in a few channels. This technique substantially increases the signal-to-noise ratio (by concentrating signal rather than changing the noise level), as evidenced by the detection of a small quantity of CO in the disk around Fomalhaut that was not evident in the unprocessed channel or moment 0 maps \citep{2017fomalhaut}.
We modify the technique slightly to utilize an observed velocity field rather than a model of a flat Keplerian disk. Previous observations of the 49 Ceti disk have demonstrated that 80\% of the CO(3-2) flux is well described by an axisymmetric power-law disk in Keplerian rotation \citep{Hughes2017}, although the remaining 20\% of the flux shows intriguing deviations from axisymmetry. We therefore use the moment 1 map of the CO(3-2) emission as the ``model" velocity field, and shift each pixel according to the intensity-weighted velocity of CO at that location. Applying the method to the CO(3-2) data cube boosts the signal-to-noise ratio of the line by a factor of 2-3 over a flat Keplerian model for 49 Ceti \citep{2018lambros}, although { extending that result to the current data} relies on the assumption that the targeted molecules share a common spatial distribution and velocity field with CO. Figure~\ref{Moment1co} shows the CO(3-2) moment 1 map generated from the data in \citet{Hughes2017}, using the CASA task {\tt immoments} and excluding pixels below three times the rms noise.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.6\textwidth,height=0.5\textheight,keepaspectratio]{co_moment1_v1.pdf}
\caption{A moment 1 map of CO(3-2) in the 49 Ceti debris disk generated from data presented in \citet{Hughes2017}. We use this observed intensity-weighted velocity field as the input for the spectral shifting technique drawn from \citet{2017fomalhaut}.}
\label{Moment1co}
\end{figure}
We applied this technique to our five molecules, subtracting the observed CO(3-2) gas velocity from the spectrum of each molecule in each pixel to shift any flux to a common zero-velocity channel. We then use the MIRIAD\footnote{Multichannel Image Reconstruction, Image Analysis and Display; \citep{sault95}} task {\tt cgcurs} to integrate the flux within the mask defined by the 3$\sigma$ contours of the CO(3-2) moment 0 map. For molecules with multiple transitions (i.e., the CN hyperfine transitions listed in Table~\ref{table:CN} and the two methanol transitions), we initially combined all the transitions by independently shifting and then subsequently adding together the 3-D position-position velocity cube for each transition before integrating the spectrum; however since no emission was detected we ultimately reported the upper limit for only the strongest transition in each spectral window. The resulting shifted spectra for all molecules are displayed in Figure~\ref{fig:shiftspec}. None of the spectra exhibit statistically significant { ($>3\sigma$)} emission { near the expected velocity}.
{ Table~\ref{table:results} lists the 3$\sigma$ upper limit on the integrated flux for each line, calculated within the area enclosed by the 3$\sigma$ contours of the shifted CO moment { 0} map and integrated over $\pm$2.0\,km\,s$^{-1}$ along the velocity axis. We restrict the velocity to this range because \citet{2018lambros} demonstrated that applying the spectral shifting technique to the CO data from 49 Ceti narrowed the line considerably from its original 8\,km\,s$^{-1}$ width, but that a range of $\pm2$\,km\,s$^{-1}$ was still required to detect all of the shifted flux. The upper limit is calculated by measuring the rms noise in the moment 0 map, multiplying by the square root of the number of synthesized beams enclosed within the 3$\sigma$ contours of the shifted CO moment map (the area within which the integrated flux is measured), and then multiplying by three to obtain the 3$\sigma$ upper limit.}
\begin{table}[H]
\centering
\caption{Upper limits (3$\sigma$) on molecular gas emission from the 49 Ceti disk}
\label{table:results}
\begin{tabular}{*4c}
\toprule
Species& Rest Frequency$^a$ & Upper Limit & Gas Mass \\
& GHz &\ mJy\,km\,s$^{-1}$ & $M_\oplus$\\
\midrule
CN(3-2)&340.248*& {25}& {$<$2.0 $\times$ 10$^{-8}$ }\\
SiO(8-7)&347.3305 & {34}&{$<$8.3 $\times$ 10$^{-9}$} \\
HCN(4-3)&354.5054 & {31} &{$<$1.7 $\times$ 10$^{-9}$} \\
HCO$^{+}$(4-3)&356.7342 & {37} &{$<$1.3 $\times$ 10$^{-9}$ } \\
\textnormal{C}H$_{3}$OH **&350.9051 & {22} &{$<$1.5 $\times$ 10$^{-7}$} \\
\textnormal{C}H$_{3}$OH ***&358.6058 & {56} & {$<$7.6 $\times$ 10$^{-7}$} \\
\bottomrule
& *N=3-2, J=7/2-5/2, F=9/2-7/2 **1(1,1)-0(0,0) & ***4(1,3)-3(0,3) \\
\bottomrule
\end{tabular}
\tablenotetext{a}{https://www.cv.nrao.edu/php/splat/}
\end{table}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=1.0\textwidth,keepaspectratio]{specplots_3.pdf}
\end{center}
\caption{\label{fig:shiftspec}The shifted spectra for all five targeted molecules in the 49 Ceti system. { The x-axis shows the velocity relative to the LSR velocity of 49 Ceti (so that 0\,km\,s$^{-1}$ is the expected velocity of the emission).} Each spectrum was created utilizing the shifted channel maps, which were then integrated over the area of the disk enclosed by the 3$\sigma$ contours of the CO(3-2) moment 0 map. The CN spectrum shows only the strongest (highest intensity) of the hyperfine transitions. The red horizontal line denotes the 3$\sigma$ flux level for each spectrum (three times the rms noise for the spectrum, calculated across at least 100 channels around the line within each spectral window). There is no significant ($>3\sigma$) emission near a velocity of 0\,km\,s$^{-1}$ for any of the observed lines.}
\end{figure*}
Even with the signal boost from the spatial-spectral shifting technique, we do not detect any gas emission from the five molecules. We convert the flux upper limits (calculated based on the noise level {in the shifted moment 0 maps with the velocity range restricted to $\pm2$\,km\,s$^{-1}$}) to mass upper limits, assuming optically thin emission and local thermodynamic equilibrium (LTE) and adopting the disk-averaged excitation temperature of T$_\mathrm{ex}$ = 32\,K from the CO line ratio in \citet{Hughes2017}, using the following relationships:
\begin{align}
M = \frac{4\pi}{h\nu_{0}}\frac{F_{J_{u} - J_{l}} \textnormal{m}_{mol} \textnormal{d}^{2}}{A_{J_{u}- J_{l}
}X_{u}}
\end{align}
\begin{align}
X_{u} = \frac{g_u}{Q(T)}e^{-E_u/k T_{ex}}
\end{align}
where $m_\mathrm{mol}$ is the molecular mass, $d$ is the distance to the source, $\nu_{0}$ is the rest frequency for each transition taken from the Splatalogue database, $A_{J_{u}-J_{l}}$ is the Einstein coefficient { (also taken from Splatalogue)}, X$_{u}$ represents the fraction of molecules in the upper energy state, and $F_{J_{u}-J_{l}}$ is the upper limit on the flux density for a given transition from Table~\ref{table:results}. The mass of each molecule, $m_\mathrm{mol}$, the degeneracy of the upper state $g_u$, the energy of the upper state $E_u$, and the tabulated partition function at a temperature of 37.5K (the closest to our assumed excitation temperature) $Q(T)$, were taken from the Cologne Database for Molecular Spectroscopy (CDMS).
Table~\ref{table:molecular_params} summarizes the parameters for each molecule, and Table~\ref{table:results} lists the corresponding upper limit on the gas mass of each molecule.
\begin{comment}
\begin{table}[ht!]
\center
\caption{Molecular Parameters}
\label{table:molecular_params}
\scalebox{0.8}{%
\begin{tabular}{*4c}
\toprule
{Parameters} && Molecular Species \\
\midrule
&HCO+ & CN & \\
\midrule
$\nu_{0}$(GHz){$^{a}$} &356.73424 & 340.24777 & \\
& & *Table 4.2 & \\
mass(Da)$^{b}$ & 29.00274$^{d}$ & 26.003622 & \\
Flux (mJy km s$^{-1}$) &21 &10 & \\
A (s$^{-1}$)$^{a}$ & $3.57215 \times 10^{-3}$ & $9.64051 \times 10^{-3}$ & \\
B$_{0}$ (cm$^{-1}$)$^{c}$ &1.475 & 1.900 & \\
J$_{u}$ &4 & 3 & \\
\midrule
& HCN & SiO & CH$_{3}$OH \\
\midrule
$\nu_{0}$(GHz)$^{a}$ &354.505473 & 347.3305786 & 350.90507 \\
& & & 358.60587 \\
mass(g)$^{b}$ & 27.010899 & 43.97184 & 32.026215 \\
Flux (mJy km s$^{-1}$) & 24 & 17 & 11 \\
A (s$^{-1}$)$^{a}$ & $2.053761 \times 10^{-3}$ & $2.20303 \times 10^{-3}$ & $3.31520 \times 10^{-4}$ \\
B$_{0}$ (cm$^{-1}$)$^{c}$& 1.478 & 0.7268 & 4.254 \\
J$_{u}$ &4 &8 & 4 \\
\bottomrule
\end{tabular}}
\tablenotetext{a}{www.cv.nrao.edu/php/splat}
\tablenotetext{b}{www.chemspider.com}
\tablenotetext{c}{~~~~cdms.astro.uni-koeln.de \citep{2016JMoSpCDMS}}
\tablenotetext{d}{Kim, S., Chen, J., Cheng, T., Gindulyte, A., He, J., He, S., … Bolton, E. E. (2019). PubChem 2019 update: improved access to chemical data. Nucleic acids research, 47(D1), D1102–D1109. doi:10.1093/nar/gky1033}
\end{table}\vfill
\end{comment}
\begin{table}[ht!]
\center
\caption{Molecular Parameters}
\label{table:molecular_params}
\scalebox{0.8}{%
\begin{tabular}{*6c}
\toprule
Species & Mass & A (s$^{-1}$)$^{a}$ & $E_u$ (cm$^{-1}$)$^b$ & g$_{u}^b$ & Q(T)$^b$ \\
&(Da) & (mJy km s$^{-1}$) & & & \\
\midrule
CN & 26.00362$^{b}$& 4.1313 $\times$ 10$^{-4}$& 22.7028 & 10 & 84.7308 \\
SiO & 43.97184$^{b}$& 2.2030 $\times$ 10$^{-3}$& 52.1397 & 17 & 36.3268\\
HCN & 27.01090$^{a}$ & 2.0605 $\times$ 10$^{-3}$& 29.5633 & 27 & 53.9106\\
HCO$^{+}$ & 29.00274$^{c}$& 3.5722 $\times$ 10$^{-3}$ & 29.7491 & 9 & {17.8601} \\
CH$_{3}$OH & 32.02622$^{b}$& 3.3148 $\times$ 10$^{-4}$ & 11.7049 & 12 & 920.963739\\
CH$_{3}$OH & 32.02622$^{b}$& 1.3193 $\times$ 10$^{-4}$& 30.7644 & 36 & 920.963739\\
\bottomrule
\end{tabular}}
\tablenotetext{b}{www.chemspider.com}
\tablenotetext{c}{~~~~cdms.astro.uni-koeln.de \citep{2016JMoSpCDMS}}
\tablenotetext{d}{\citet{kim2019}}
\end{table}\vfill
\section{Discussion}
\label{section:discussion}
To date, CO remains the only firmly detected gas-phase molecule in debris disks \citep{2018Hughes}. The upper limit on HCN(4-3) from our line survey is deeper than a previous search for HCN in the 49 Ceti disk \citep{Hughes2017}, by a factor of about 2.5 in rms noise, which is enhanced to { $\sim5\times$} by our use of the spectral shifting technique. \citet{2018mat} conducted a Submillimeter Array (SMA) and ALMA survey for CN, HCN, HCO$^+$, N$_2$H$^+$, H$_2$CO, H$_2$S, CH$_3$OH, SiO, and DCN in the $\beta$ Pic disk that also returned upper limits. When applying a simple scaling relationship for the differences in distance, frequency, { and Einstein coefficient} to the flux upper limits, our survey is roughly { 7--13}$\times$ more sensitive to CN, HCN, HCO$^+$, { and methanol}. The emerging picture seems to be that the abundance of molecules other than CO is surprisingly low in debris disks, even for those molecules that exhibit relatively high abundances in (1) protoplanetary disks, (2) solar system comets, and (3) models of secondary gas emission. Here we explore how the gas composition implied by our upper limits compares with the abundance ratios of those three categories of objects.
\subsection{Comparison with Observed Molecular Abundances of Protoplanetary Disks and Comets}
Protoplanetary disks exhibit a rich chemistry that has been studied both observationally and theoretically, with the field undergoing a particularly rapid expansion recently as ALMA has reached maturity \citep{hen13,dut14,ber18}. All of the molecules in our survey, with the exception of SiO, have been detected in one or more protoplanetary disks.
Models of protoplanetary disk chemistry predict a rich abundance of molecular species, as a result of stellar and interstellar radiation, cosmic rays, and grain surface chemistry \citep[e.g.,][]{1999Aikawa,1999yuri}. HCO$^+$ and HCN are particularly abundant, both in the models and subsequent observations. In a survey of eight T Tauri disks and four Herbig Ae disks with the SMA, HCO$^+$(3-2) emission was detected towards all of the surveyed objects and HCN(3-2) was at least tentatively detected towards 10 of the 12 sources \citep{2010oberkar,2011chemoberg}. Typical rms noise levels in the survey were of order $\sim0.1$\,Jy\,km\,s$^{-1}$ at distances of $\sim$140\,pc { (although V4046 Sgr was closer)}. If we scale those thresholds by $\nu^2$ (assuming optically thick emission) and by $d^2$ to account for the difference in distance to the 49 Ceti system, then our upper limits on the integrated HCO$^+$(4-3) and HCN(4-3) flux from 49 Ceti are about a factor of 50 below the sensitivity of the \citet{2010oberkar,2011chemoberg} survey. The CO(2-1) emission from 49 Ceti \citep[$3.87\pm0.41$\,Jy\,km\,s$^{-1}$,][]{moor2019} is 26 times lower than the average for the disks in the \citet{2011chemoberg} survey, and 10 times lower than the faintest disk in the survey. { Looking at it another way, the ratio of { HCO$^+$/CO} flux within a given band varies in the \citet{2010oberkar,2011chemoberg} sample between { 0.11 and 0.59}, while the ratio for 49 Ceti at Band 7 is { $<0.006$}. HCN is less conclusive, since there are two upper limits within the sample of 12, but for sources in which it is detected the HCN/CO ratio ranges from 2.0 to 18, while the ratio for 49 Ceti is $<0.005$.} A protoplanetary composition for the 49 Ceti disk is not supported by the low limits on both HCO$^+$ and HCN compared to CO emission.
The nondetections of CH$_3$OH and CN are less relevant for comparison with a protoplanetary disk composition. While \citet{2014protopredic} predicted detectable CH$_3$OH emission from protoplanetary disks based on models of grain-surface chemistry, nonthermal desorption, and irradiation of grain mantles, subsequent observations yielded lower abundances than expected by two orders of magnitude \citep{2016diskch3oh}. Observations of the nearest and brightest Herbig Ae disk (around HD 163296) have demonstrated an even lower CH$_3$OH/H$_2$CO ratio than in the TW Hya system (the closest protoplanetary disk and the only non-outbursting source in which methanol has been detected to date), suggesting that abundances may be suppressed even further in the radiation field of an A star. We therefore would not expect to detect CH$_3$OH in the 49 Ceti system { even} if its abundance relative to CO were protoplanetary. Similarly, a recent ALMA survey of molecular lines from protoplanetary disks in Lupus returned detections of CN(3-2) in only 38\% of surveyed sources \citep{2019cnlupus}. Typical upper limits were of order $\sim100$\,mJy\,km\,s$^{-1}$, however, which is about a factor of { 30} above our upper limits when scaled to the appropriate distance. This result is similar to previous surveys showing that $\sim$50\% of bright protoplanetary disks exhibit detectable CN at modest sensitivity \citep[e.g.,][]{gui13}. Our nondetection of CN is therefore inconclusive as a probe of protoplanetary composition.
Solar system comets serve as a foundation for predicting the likely composition of second-generation gas in extrasolar systems, although they are imperfect comparison objects for many reasons -- foremost that most Solar System comets are observed while sublimating at distances of a few au from the Sun, whereas the gas in extrasolar debris disks originates from tens to hundreds of au from the central star. The radiation field of an A star differs markedly from that of the Sun, and the mechanism behind the gas release may differ as well (collisional vs. radiative).
In a millimeter-wavelength survey of the composition of 24 comets with the Institut de Radioastronomie Millimetrique 30m, James Clerk Maxwell Telecope, Caltech Submillimeter Observatory, and Swedish-ESO Submillimeter Telescope, HCN was detected in all 24 comets with an abundance relative to CO of order $\sim$0.1-1 \citep{2002comet}. HCN lines in this survey were typically as bright or brighter than the CO lines observed in the same band. These compositional relationships are also borne out across larger and more modern samples of comets \citep{mum11,boc17}. In the 49 Ceti system, the HCN(4-3) upper limit of {31}\,mJy\,km\,s$^{-1}$ from this work implies an HCN(4-3)/CO(3-2) line ratio of {$< 0.0052$}, using the CO(3-2) flux of $6.0\pm 0.1$\,Jy\,km\,s$^{-1}$ from \citet{Hughes2017}. This limit is well below the observed range of HCN(4-3)/CO(3-2) line ratios for Solar System comets, which span values of { $\sim$1-20, although for most systems CO was not detected at all with lower limits of order $\gtrsim 10$} \citep[][and references therein]{2002comet}. HCO$^+$ has also been detected in comets and may be particularly prevalent in regions where water has been photodissociated \citep{mil04}. If the 49 Ceti disk exhibited the same abundance patters as Solar System comets, we would therefore strongly expect to detect HCN, and would likely detect methanol and HCO$^+$. The lack of detection of any of these molecules suggests that we need to dig deeper into modeling of second-generation gas to understand why the abundance ratios indicated by our observations differ so markedly from {\it both} { protoplanetary} disks and comets.
\subsection{Models of Second-Generation Gas Production}
In this section we explore models of second-generation gas production, beginning with an estimate of the CO photodissociation and survival timescale assuming only shielding by H$_2$ and self-shielding by CO (Section~\ref{COphd}). We then extend the analysis to CN, which is the molecule in our survey that has the best combination of a long survival timescale against photodissociation and intrinsically strong molecular transitions (Section~\ref{CNphd}). Finally, we derive constraints on the HCN/(CO+CO$_2$) outgassing rate ratios (Section~\ref{HCN_rate}). The latter two sections follow the methodology derived in \citet{2018mat}, applying it to these new and more stringent constraints for the specific case of the 49 Ceti system.
\subsubsection{CO photodissociation and survival timescale}
\label{COphd}
As ALMA has matured and detections of gas in debris disks have started piling up \citep[e.g.,][]{1995zuc,moor2011,moo15,2017moor,dent2014,gre16,2016lieman,2016diskwcomets,mar17,2017fomalhaut}, there have been a number of advances in models of second-generation gas production in debris disks. There are several proposed mechanisms, including collisional gas release \citep{2012Zuckerman,dent2014}, thermal desorption \citep{cze07}, and photodesorption \citep{gri07}. Recent modeling efforts have primarily focused on collisional gas release from icy planetesimals as the origin of the outgassing responsible for the observed molecular lines \citep{2017luca,kra17}. In this section we apply the model presented in \citet{2017luca} to our observations of the 49 Ceti disk to investigate the origin of the gas.
We first estimate the photodissociation timescale for CO using models of the temperature and density structure of the CO disk presented in \citet{Hughes2017}. We calculate { half} the vertical column density through the disk. We perform this calculation for both scenarios investigated in the paper: a primordial scenario in which the CO/H$_2$ abundance is assumed to equal 10$^{-4}$, and a second-generation scenario in which the CO/H$_2$ abundance is assumed to equal 1. We also scale up the mass of the \citet{Hughes2017} best-fit models to match the total CO mass of (1.11 $\pm 0.13) \times 10^{-2}$ M$_{\oplus}$ derived by \citet{moor2019}. In the primordial scenario (where the gas is in LTE) the CO column density is { 2.0} $\times$ 10$^{16}$ cm$^{-2}$ and the H$_2$ column density is { 2.0} $\times$ 10$^{20}$ cm$^{-2}$, whereas in the second-generation scenario (where the conditions are non-LTE), both CO and H$_{2}$ have column densities of { 3.6} $\times$ 10$^{16}$ cm$^{-2}$.
Since the dust in the 49 Ceti system is optically thin, we assume negligible dust attenuation. We then estimate the photodissociation timescale using the following relationships, following \citet{fla19}:
\begin{align}
k&=\chi k_{0} \Theta e^{-\gamma A_{v}} \\
t&=1/k
\end{align}
In these equations, $\chi$ is the radiation field in units of the Draine field \citep[one Draine field is $\sim$1.7 Habings, where a Habing is $1.6\times10^{-3}$\,erg\,cm$^{-2}$\,s$^{-1}$;][]{dra78}. We consider only the interstellar radiation field (ISRF), and ignore contributions from the stellar radiation field (which would only decrease the photodissociation timescale). $k_{0}$ is the unattenuated photodissociation rate in units of photodissociations per second for CO molecules. We used $k_{0}$ = 2.590 $\times$ 10$^{-10}$\,s$^{-1}$ from Table 6 of \cite{2009visser}, since Table 6 assumes an excitation temperature of 50\,K which is closest to the disk-averaged excitation temperature of 32\,K derived by \citet{Hughes2017}. $\Theta$ is the attenuation factor for self-shielding and H$_{2}$ shielding, tabulated in \citep{2009visser} for different values of CO and H$_2$ column density. $\gamma$ is the UV extinction relative to visual extinction ($\approx$2 for small dust grains). The photodissociation timescale is then $1/k$. We consider the simple case with no dust attenuation throughout the disk ($A_V = 0$), using the value of $\Theta$ corresponding to the column density, which amounts to an assumption that the photodissociation timescale is set by the most heavily shielded molecules in the disk \citep{2009visser}. We calculate a photodissociation timescale of { $\sim$6,000\,yr} for the primordial scenario and { $\sim$3,300\,yr} for the second-generation case (due to the differences in column density, which correspond to different values of $\Theta$). These timescales should be considered upper limits, since we have neglected stellar radiation, which could contribute up to the same amount of radiation as the ISRF depending on the optical thickness of the radial gas column.
These results present a familiar dilemma: the photodissociation timescale is so short compared to the age of the star that a primordial origin for the gas is unlikely, but the combination of large CO mass and short photodissociation timescale requires an uncomfortably high gas release rate from large bodies to sustain the observed gas \citep[see, e.g.,][]{2013ApJkospal}. Recent work by \citet{2018kral} suggests that the solution to the dilemma may lie in shielding of CO by neutral carbon in the disk, which is not included in the standard calculation of the photodissociation timescale that we performed, but which can extend the lifetime of CO by orders of magnitude. Observations by \citet{2017higatomic} demonstrated substantial reservoirs of neutral carbon in the disk, as expected due to rapid initial photodissociation of CO, and a detailed analysis by \citet{moor2019} demonstrated that the amount of CO is consistent with the \citet{2018kral} model given the observed quantity of CI in the 49 Ceti system. These results provide evidence for a secondary origin for CO in the disk. They are supported by the unresolved vertical height of the CO disk, which is smaller than expected for a primordial composition and implies a mean molecular weight more consistent with CO (rather than H$_2$) as the dominant constituent \citep{Hughes2017}. However, it is worth noting that the long photodissociation timescale of CO implied by CI shielding would apply equally to CO of primordial or secondary origin.
\subsubsection{CN photodissociation, mass, and survival timescale}
\label{CNphd}
{ While the LTE estimates provide a ballpark value for the conversions between flux and mass, if the H$_2$ density is low, then it is likely that at least some species are not in LTE.} Moving on to molecules other than CO, we can convert the upper limits in our survey to gas masses via the non-LTE calculation described in \citet{2018mat}. We focus on CN, since for Solar System comet-like compositions it is expected to be the most favourable for mm-wave detection \citep{2018mat}, compared to other molecules expected in a cometary environment (like the observed CH$_3$OH and HCN). This is largely due to CN having the best combination of a long survival timescale against photodissociation, and intrinsically strong molecular transitions. We do not consider SiO gas as we are not aware of its detection in Solar System comets, and HCO$^+$ because connecting its gas mass to parent species in exocometary ice would depend on a complex network involving chemistry and ionization balance, which is beyond the scope of this paper.
We begin by converting the upper limit on the CN flux of {25}\,mJy\,km\,s$^{-1}$ to a mass upper limit using Eq. 2 from \citet{2015matranogasfo}. This calculation assumes that any CN emission at millimeter wavelengths is optically thin, which we will verify {\it a posteriori}.
To calculate the excitation of the molecule and hence the upper level fractional population, $x_j$, we use the NLTE code developed in the same paper, where $x_j$ will depend on the density of collisional partners (e.g., electrons) and the gas kinetic temperature, $T_k$. Fig.~\ref{fig:luca} {(top panel) shows the derived total CN masses} from the measured upper limit as a function of collider density and temperature. The excitation calculation includes fluorescence due to absorption of stellar and interstellar optical/UV photons and later re-emission at lower-energy transitions \citep{2018mat}. For the stellar radiation field, we assume a PHOENIX stellar model with $T_\mathrm{eff}$=8800\,K, $R_*$=1.7\,R$_\odot$, $\log(g) = 4.0$, $[M/H]=0.06$ (G. Kennedy, private communication), and scaled its flux to what would be received by a CN molecule at 140\,au. This analysis leads to mass upper limits ranging between {(6.2-23)}$\times 10^{-9}$\,M$_\earth$.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.7\textwidth,keepaspectratio]{49Ceti_CNmass_vs_ncoll_Tk_v210621.pdf}
\includegraphics[width=0.7\textwidth,keepaspectratio]{49Ceti_tau_vs_ncoll_Tk_v210621.pdf}
\caption{(Top) Total CN mass as a function of collider density (with primary collider assumed to be electrons) and kinetic temperature. The total mass of CN from these models is $(6.2-23) \times 10^{-9}$ M$_{\oplus}$. (Bottom) Optical depth as a function of collider density and kinetic temperature. In order to check the assumption that CN was optically thin in the debris disk, the calculated CO column densities through the disk midplane were re-scaled by the CN/CO total mass ratio {to obtain a CN column density. This was then turned into an optical depth using Eq.~3 in \cite{2017luca}.} }\label{fig:luca}
\end{figure}
We then check whether the resulting emission is optically thin along the line of sight to Earth. {We rescale the CO radial column density from the secondary gas model of \citet{Hughes2017} (measured from the star to the disk's critical radius, and multiplied by two to account for the entire radial extent of the disk) by the CN/CO total abundance ratio to get the CN radial column density,} which we assume is comparable to the column density of CN along the line of sight to Earth. These assumptions result in CN column densities of {(1.0-3.9)}$\times 10^{12}$\,cm$^{-2}$, depending on excitation, which (via Eq.~3 in \cite{2017luca}, assuming line widths produced by thermal broadening by $T_K$) lead to optical depths of {0.001-0.36} (bottom panel of Fig.~\ref{fig:luca}). Therefore, CN emission, even if just below our upper limit, would be {largely} optically thin. {Note that this calculation} does not account for uncertainties in the amount of stellar radiation, or the radius at which it is received (140\,au is assumed), or the distance to the star from Earth.
From there, we can investigate CN photodissociation. By rescaling the previously calculated CO column density along the {vertical direction}, we obtain CN column densities of {(2.2-8.0)$\times 10^{10}$\,cm$^{-2}$}, for which self-shielding is negligible\footnote{https://home.strw.leidenuniv.nl/$\sim$ewine/photo/data/photo\_data/line\_shielding\_functions/CN/photodissociation\_ISRF}. The shielding factors due to CO, H$_2$, and CI column densities \citep[taking CI/CO = 54 from][]{2017higatomic} are {$\sim$0.91, $\sim$0.98 and $\sim$0.43, giving a combined shielding factor of {$\sim$0.38} \citep{2017Heays}}. {Note that} the combined shielding factor is approximated as the product of the shielding factors of the individual species, which assumes that the wavelength regimes over which the shielding takes place do not overlap, but we consider this a reasonable approximation especially given the minor contributions from CO and H$_2$. Considering only the {ISRF UV field impinging on the disk from the vertical direction}, we obtain a photodissociation timescale for CN of order {$\sim$161} years.
\subsubsection{Constraints on the HCN/(CO+CO$_2$) outgassing rate ratios}
\label{HCN_rate}
{If we assume that 1) all CN gas is produced by photodissociation of exocometary HCN gas, which is itself produced by cometary release of HCN ice; 2) all CO gas is produced by either cometary release of CO ice, or photodissociation of CO$_2$ released from CO$_2$ ice; and 3) that the production and destruction rates of each species are balanced (i.e., we are in steady state, although this is not necessarily the case in young systems, see \citet{marino2020}), we can relate the observed CN and CO gas masses to the HCN/(CO+CO$_2$) outgassing rate ratio. From Eq. 3 in \citet{2018mat}, this can be expressed as }
\begin{equation}
\label{HCNoverCOoutgassing}
\frac{M_{\rm CN}}{M_{\rm CO}}=\frac{\tau_{\rm CN, phd}^-m_{\rm CN}}{\tau_{\rm CO, phd}^-m_{\rm CO}}\frac{\dot{N}_{\rm HCN}^+}{\dot{N}_{\rm CO+CO_2}^+},
\end{equation}
{where $m_{\rm i}$ is the molecular mass in kg, $\dot{N_{\rm i}^+}$ its production (outgassing) rate in number of molecules per second, $M_{\rm i}$ the observed gas mass, and $\tau_{\rm i}^-$ its photodissociation timescale of the gas species.}
{The observed gas masses $M_{\rm CO}$ and $M_{\rm CN}$ are well constrained observationally from \citet{moor2019} and our data, and the CN photodissociation timescale $\tau_{\rm CN, phd}^-$ is also relatively well constrained, as shielding only increases it by a factor of a few (Sect. \ref{CNphd}). On the other hand, the main uncertainty in the calculation remains the CO photodissociation timescale $\tau_{\rm CO, phd}^-$, which can range from a few hundred years { if shielding by CI is neglected} to much longer than the system age of 40 Myr.}
{Taking the CO photodissociation timescale $\tau_{\rm CO, phd}^-\sim${3,300} yr derived in our second-generation, CI-free model} would imply a HCN/(CO+CO$_2$) outgassing rate ratio of {(1.2-4.6)} $\times 10^{-5}$. This is at least a factor of $\sim${65} below the observed values for Solar system comets, ranging from about $3\times10^{-3}-2$ \citep[see Fig.~5 in][]{2018mat}. Therefore, in the absence of CI shielding the constraint on the flux density of CN and HCN implies that either CO may be preferentially released from the exocometary ice relative to HCN, or there may be a true HCN depletion compared to CO ice in 49 Ceti's exocomets.
Perhaps more likely, however, given the work of \citet{2018kral} and the observed presence of CI, is that CI is preferentially shielding CO from UV photodissociation. For example, if we assume the CI/CO ratio of 54 derived from \citet{2017higatomic} (which does not account for optical depth and excitation effects) we obtain a CO shielding factor {of $\sim10^{-15}$. This is} so small that it would imply that CO is hardly photodissociating at all, allowing it to survive for much longer than the age of the system. Shielding by CI has a much weaker effect on other molecules, since the CO photodissociation bands are all at wavelengths shorter than the CI ionization threshold (110 nm), whereas some photodissociation bands of CN (and those of most other cometary molecules) extend beyond 110\,nm, where the stellar radiation field is orders of magnitude stronger.
It therefore seems likely that the only reason CO appears so anomalously abundant is due to shielding by CI, and that CN is in fact photodissociated about as rapidly as we would expect given the harsh UV radiation field from both the ISRF and the central A star. {Whether CI shielding is the cause of the overabundance of CO compared to other molecules could be confirmed with an accurate measurement of the CI column density. This can be achieved with future high resolution, optically thin, multi-transition CI observations, which are best suited to solving the degeneracy between excitation, spatial structure, mass, and optical depth which affects single line measurements. If CI column densities are confirmed to be very high,} then the CO photodissociation timescale $\tau_{\rm CO, phd}^-$ in Eq. \ref{HCNoverCOoutgassing} could be orders of magnitude higher than derived in the absence of CI shielding, and allow us to reconcile our extreme CN/CO flux ratios around 49 Ceti with Solar System comet -like HCN/(CO+CO$_2$) ice abundances.
\section{Conclusions}
\label{section: conclusion}
In order to constrain the composition of the gas in nearby debris disks, we performed a deep survey for five molecular species in the disk around 49 Ceti. After 3.2 hours of integration with ALMA at low angular and spectral resolution to maximize sensitivity, and after applying a spatial-spectral shifting technique adapted from \citet{2017fomalhaut}, we obtain stringent upper limits { from the shifted and velocity-range-restricted moment 0 maps} that represent the most sensitive search to date for molecules other than CO in a debris disk: { HCN(4-3) at 31\,mJy\,km\,s$^{-1}$, HCO$^{+}$(4-3) at 37\,mJy\,km\,s$^{-1}$, CN(3-2) at 25\,mJy\,km\,s$^{-1}$, SiO(8-7) at 34\,mJy\,km\,s$^{-1}$, CH$_3$OH 1(1,1)-0(0,0) at 22\,mJy\,km\,s$^{-1}$, and CH$_3$OH 4(1,3)-3(0,3) at 56\,mJy\,km\,s$^{-1}$}.
We use our upper limits on the flux density to calculate flux ratios that we can compare with the composition of protoplanetary disks and solar system comets. The upper limits on the HCN/CO and HCO$^+$/CO line ratios are inconsistent with a protoplanetary composition for the disk material. The upper limits on the HCN/CO ratio and the nondetection of HCO$^+$ and methanol are also inconsistent with the range of cometary compositions observed within the Solar System. Since the temperatures and mechanisms involved in the release of exocometary gas at tens to hundreds of au from the central star may differ markedly from those involved in the au-scale sublimation of Solar System comets, we turn to models of second-generation gas production to resolve this apparent conundrum. While the CO photodissociation timescale is apparently much shorter than the age of the system, it can be extended by orders of magnitude by taking into account shielding by CI \citep{2018kral}, which is observed to be present in abundance in the 49 Ceti system, likely as a byproduct of CO photodissociation \citep{2017higatomic}. Using a NLTE code from \citet{2018mat}, we demonstrate that if CI shielding is neglected, then the implied HCN/CO outgassing ratio for 49 Ceti would be orders of magnitude lower than the range observed for Solar System comets.
Our results imply that either there is a marked difference in the chemical abundance of the 49 Ceti system relative to all known Solar System comets and protoplanetary disks, an unspecified mechanism causing preferential release of CO relative to HCN, or more plausibly, that our results provide support for a picture where CI shields CO, but cannot shield CN and other (exo-)cometary molecules. The apparently high abundance of CO relative to the other molecules therefore may not reflect the true chemical abundances of the icy bodies in the system, but rather reflects the fact that only CO is shielded from photodissociation and therefore accumulates in substantial amounts while the other molecules are photodissociated so rapidly as to be effectively invisible. Our deep limits on molecular species readily detected in protoplanetary disks and comets provide support for the shielding of CO by CI in the 49 Ceti system.
\section*{Acknowledgments}
We thank Stew Novick for help with deciphering the CDMS database, and Grant Kennedy for providing the stellar photospheric model fits. Support for this work was provided by the NSF through award SOSPA5-009 from the NRAO. A.M.H. is supported by a Cottrell Scholar Award from the Research Corporation for Science Advancement. A.M. acknowledges the support of the Hungarian National Research, Development and Innovation Office NKFIH Grant KH-130526.
LM acknowledges support from the Smithsonian Institution as a Submillimeter Array (SMA) Fellow. The work of JK is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.
This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2017.1.00941.S
ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}
Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}).
Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement.
\software{
\texttt{CASA} \citep{2007mcmull},
\texttt{MIRIAD} \citep{sault95},
\texttt{NumPy} \citep{van2011numpy},
\texttt{Astropy} \citep{astropy},
\texttt{Pandas} \citep{mckinney},
\texttt{Uncertainties}, \url{http://pythonhosted.org/uncertainties/}
}
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,807 |
Q: Reusable javascript object with public and private methods I have tried so many things, each and everyone either doesnt work, or breaks another part.
How can I get the following code to work, the requirements are:
*
*just one instance of the object
*i want to be able to pass in a parameter every time i use the object
*i dont want to pass it/use sub methods, such as getter or setter. not for this specific scenario.
*i want to be able to have private methods
*i want to be able to have public methods
Regarding #5, i have tried with this.func() and prototype.func() but cant get it to work.
I have tried but failed, but think perhaps i need to have a return value? or perhaps self-invoking functions?
below is example/pseudo code of what i have:
var myobj = new function(param){
//code here, able to use input param and private function 'privatefunc' below
var privatefunc=function(){
console.log("i am a private function");
}
this.publicfunc=function(){
console.log("i am a public function");
}
}
myobj("batman");
myobj("superman");
myobj("catwoman");
myobj().publfunc(); // should print out: i am a public function
thank you for your time!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,620 |
package client // import "github.com/docker/docker/client"
import (
"bytes"
"encoding/json"
"fmt"
"io/ioutil"
"net/http"
"strings"
"testing"
"github.com/docker/docker/api/types"
"golang.org/x/net/context"
)
func TestCheckpointListError(t *testing.T) {
client := &Client{
client: newMockClient(errorMock(http.StatusInternalServerError, "Server error")),
}
_, err := client.CheckpointList(context.Background(), "container_id", types.CheckpointListOptions{})
if err == nil || err.Error() != "Error response from daemon: Server error" {
t.Fatalf("expected a Server Error, got %v", err)
}
}
func TestCheckpointList(t *testing.T) {
expectedURL := "/containers/container_id/checkpoints"
client := &Client{
client: newMockClient(func(req *http.Request) (*http.Response, error) {
if !strings.HasPrefix(req.URL.Path, expectedURL) {
return nil, fmt.Errorf("Expected URL '%s', got '%s'", expectedURL, req.URL)
}
content, err := json.Marshal([]types.Checkpoint{
{
Name: "checkpoint",
},
})
if err != nil {
return nil, err
}
return &http.Response{
StatusCode: http.StatusOK,
Body: ioutil.NopCloser(bytes.NewReader(content)),
}, nil
}),
}
checkpoints, err := client.CheckpointList(context.Background(), "container_id", types.CheckpointListOptions{})
if err != nil {
t.Fatal(err)
}
if len(checkpoints) != 1 {
t.Fatalf("expected 1 checkpoint, got %v", checkpoints)
}
}
func TestCheckpointListContainerNotFound(t *testing.T) {
client := &Client{
client: newMockClient(errorMock(http.StatusNotFound, "Server error")),
}
_, err := client.CheckpointList(context.Background(), "unknown", types.CheckpointListOptions{})
if err == nil || !IsErrNotFound(err) {
t.Fatalf("expected a containerNotFound error, got %v", err)
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,716 |
\section{Introduction}
Generalised polynomial sequences are expressions built up from the usual polynomials with the use of addition,
multiplication, and the floor function, such as
\[
g(n) = 2n \ip{ \sqrt{2}n \ipnormal{\sqrt{3}n }^2+\sqrt{5}n^3} - 7 n^2\,.
\]
They have been investigated by many authors, with particular emphasis placed on problems involving uniform distribution and taking their source in
Weyl's equidistribution theorem for classical polynomials.
See, for instance, \cite{Haland-1993,Haland-1994, BergelsonHaland-1993, BergelsonLeibman-2007,Leibman-2012,BergelsonHalandSon-2020} and the references therein.
An important feature that distinguishes generalised polynomials from ordinary polynomials is that they can be bounded without being constant.
As a basic example, for any polynomial $p(x)\in\mathbb R[x]$, the sequence of fractional parts $\bra{\fp{p(n)}}_{n=0}^\infty$ is a generalised polynomial
sequence with values in $[0,1)$, which is non-constant for most choices of $p$.
In fact, it is also possible for non-constant generalised polynomials sequences to take only finitely many values.
A notable example of a class of such finitely-valued generalised polynomial sequences is provided by Sturmian words.
Indeed, every Sturmian word $\bb a = (a_n)_{n=0}^\infty$ over $\{0,1\}$ can be defined by a generalised polynomial formula
\begin{equation}\label{eq:intro-1}
a_n = {\ip{\a n+\b} - \ip{\a (n-1) + \b}}
\end{equation}
for some $\a \in [0,1) \setminus \QQ$ and $\beta \in [0,1)$ (possibly with the floor function $\ip{\cdot}$ replaced by the ceiling function $\ceil{\cdot}$).
Motivated by this fundamental example, we endeavour to investigate finitely-valued generalised polynomials from the perspective of combinatorics on words.
More precisely, we study letter-to-letter codings of finitely-valued generalised polynomial sequences, which we dub \emph{bracket words}
(cf.{} Definition \ref{def:def:bra-word}). Thus, a {\gpword} $\bb a = (a_n)_{n=0}^\infty$ over an alphabet $\Sigma$ takes the form $a_n = c(g(n))$, were
$g\colon \NN_0 \to \RR$ is a finitely-valued generalised polynomial sequence and $c$ is a map from the finite set $g(\NN_0)$ to $\Sigma$. Throughout the paper, we let $\NN = \{1,2,\dots\}$ denote the set of positive integers and put $\NN_0 = \NN \cup \{0\}$.
Bracket words are thus first thought of as a broad arithmetical generalisation of Sturmian words.
Among the several equivalent definitions of Sturmian words, one involves codings of circle rotations.
Indeed, the word $\bb a$ defined by \eqref{eq:intro-1} can be described with the help of the rotation $R_\alpha \colon \RR/\ZZ \to \RR/\ZZ$, $x \mapsto x+\alpha$, as
\begin{equation}\label{eq:intro-2}
a_n =
\begin{cases}
1 &\text{if } R_\alpha^n(\beta) \in [0,\alpha)\,,\\
0 &\text{otherwise.}
\end{cases}
\end{equation}
A seminal paper of Bergelson and Leibman \cite{BergelsonLeibman-2007} provides a dynamical representation of any bounded generalised polynomial sequence
in terms of nilrotations, which are translations on nilmanifolds, hence linking the theory of generalised polynomials with nilpotent dynamics.
As a consequence, each bracket word can be obtained as a coding of a nilrotation with respect to a semialgebraic partition.
Conversely, each coding of a nilrotation which involves a semialgebraic partition gives rise to a bracket word.
Nilsystems have received a considerable amount of attention in the past two decades. A major source of interest stems from their significance in the study
of multiple ergodic averages, as demonstrated by Host and Kra \cite{HostKra-2005} and Ziegler \cite{Ziegler-2007}.
Nilsystems also play a key role in additive combinatorics, specifically in the relatively modern branch of higher order Fourier analysis,
initiated by Gowers in his work on an alternative proof of {S}zemer\'{e}di's theorem \cite{Gowers-2001}.
The importance of nilsystems was revealed by Green, Tao, and Ziegler, who characterised Gowers uniformity in terms of correlations
with nilsequences \cite{GreenTaoZiegler-2012}, see also \cite{GreenTao-2010}. For an introduction to higher order Fourier analysis, we refer to \cite{Tao-book}.
Thus, the characterisation of bracket words mentioned above, which we discuss in more detail in Section \ref{sec:nil},
allows us to take advantage of deep pre-existing results.
A great number of combinatorial results concerning Sturmian words can be derived from
the continued fraction expansion of the irrational parameter $\alpha$ and some related expansion, the so-called Ostrowski expansion, of the real parameter
$\beta$ (these parameters being defined as in \eqref{eq:intro-1} or \eqref{eq:intro-2}).
In contrast, the study of bracket words involves much more general Diophantine problems related to
simultaneous rational approximations of real numbers.
For this reason, one cannot expect to always obtain results as accurate as for Sturmian words.
For instance, Sturmian words can be charaterised in terms of their subword complexity. Recall that for an infinite word $\bb a$,
the subword complexity $p_{\bb a}(N)$ is the count of distinct length-$N$ subwords which appear in $\bb a$, and that $\bb a$ is Sturmian
if and only if $p_{\bb a}(N) = N+1$ for each $N \geq 1$, which is the slowest rate of growth possible for a word which is not eventually periodic.
Contrary to Sturmian words, bracket words cannot be characterised in terms of their subword complexity, but we still prove the following
polynomial upper bound, which is our main new result. It is also shown in Section \ref{sec:subword} that this result is essentially the best possible
(c.f. Propositions \ref{prop: comp1} and \ref{prop: comp2}).
\begin{restatable}{alphatheorem}{thmsubword}
\label{thm:A}Let $\bb a$ be a {\gpword}. Then there exists a constant $C > 0$ such that $p_{\bb a}(N) = O(N^C)$ for all $N \in \NN$.
\end{restatable}
In another direction, Sturmian words can also be characterised in terms of balance: a word $\bb a$ over $\{0,1\}$ is Sturmian if and only if it is not eventually periodic and for each $N \geq 1$ and each pair $u,v$ of length-$N$ subwords of $\bb a$, the number occurrences of the symbol $1$ in $u$ and in $v$ is either the same or differs by $1$.
For bracket words, we have weaker estimates on frequencies of symbols, which we explore in Sections \ref{sec:facts} and \ref{sec:growth}.
Since Sturmian words are {\gpwords}, all of our results apply in particular to Sturmian words. In all instances, the special case involving Sturmian words was shown earlier than the general case, or follows quickly from earlier results. However, there are several facts which we suspect might not be very widely known. These facts include existence of frequencies along the primes (Theorem \ref{thm:facts:prime-freq-exists}) and the $\mathrm{IP}^*$ recurrence (Theorem \ref{thm:nil:IP*-rec}).
A different source from which we draw inspiration is the theory of automatic sequences, and computation theory in general. Recall that a word $\bb a= (a_n)_{n=0}^\infty$ is $k$-automatic if there exists a deterministic finite automaton which, given the base-$k$ expansion of $n$ as input, produces $a_n$ as output (for extensive introduction, see \cite{AlloucheShallit-book}).
Although $k$-automatic words are generally not {\gpwords} (cf.\ Remark \ref{rmk:NOT-auto} below), we find this comparison helpful because the classes of $k$-automatic sequences and the {\gpwords} enjoy similar closure properties with respect to Cartesian products, codings, finite modifications and certain types of rearrangements, as discussed at length in Section \ref{sec:cl}.
For instance, bracket words taking values in a ring, equipped with coordinatewise addition and multiplication, form a ring. In contrast, the class of Sturmian sequences is considerably too small for analogous closure properties to be true.
Access to the closure properties mentioned above allows us to look at bracket words from a computational perspective. For example, given three bracket words
$\bb a^{(0)},\bb a^{(1)} \in \Sigma^\infty$, and $\bb b \in \{0,1\}^\infty$, we can construct a new bracket word $\bb a$ given by
\[
a_n =
\begin{cases}
a^{(0)}_n &\text{if } b_n = 0\,,\\
a^{(1)}_n &\text{if } b_n = 1\,.
\end{cases}
\]
As a consequence, bracket words can encode any finite computation including real constants, the basic arithmetic operations $+,\times$, the operation of taking the integer part $\ip{\cdot}$ and conditional statements involving $=$ and $<$ (c.f.{} Proposition \ref{prop:cl:g-in-I}).
\begin{remark}\label{rmk:NOT-auto}
We stress that, even though we use automatic sequences as a motivating example, automatic words in general are not bracket words. In fact, there are no automatic words that are also bracket words, except for eventually periodic ones (see Theorem \ref{thm:BK}). Rather, our interest in automatic sequences stems from the fact that they share with bracket words certain properties, as discussed above. Other classes of sequences which we discuss in this context are morphic sequences and regular sequences (in the sense of Allouche and Shallit, \cite{AlloucheShallit-1992}).
\end{remark}
In light of the discussion above, it is natural to ask which properties of bracket words can be tested algorithmically. We will say that a problem is \emph{decidable} if there exists an algorithm which solves it on all inputs, and \emph{undecidable} otherwise. Many properties are known to be decidable for $k$-automatic sequences, see e.g., \cite{CharlierRampersadShallit-2012} and references therein; a practical implementation is discussed in \cite{GocHenshallShallit-2013} and \cite{Mousavi-2016}. Similarly, there are many results on decidability for $k$-regular sequences (see e.g.\ \cite{AlloucheShallit-1992}, \cite{KrennShallit-2022}) and for morphic sequences (see e.g.\ \cite{Durand-2013,Durand-2013b}).
For problems involving generalised polynomial sequences, we assume that the algorithm is provided with a formula involving only polynomials, addition, multiplication and the floor function which represents the sequence. In \cite{Leibman-2012}, Leibman constructed a ``canonical'' representation of a bounded generalised polynomial, which is essentially unique. As a consequence, the problem of determining if a given generalised polynomial is zero almost everywhere is decidable. Likewise, it is decidable whether two given bracket words are equal almost everywhere. Here, a statement $\varphi(n)$, involving a parameter $n \in \NN_0$, is said to hold almost everywhere if the set $\set{n \in \NN_0}{\neg \varphi(n)}$ of positions where it is false has asymptotic density zero. Somewhat surprisingly, the problem of verifying equality everywhere turns out to be undecidable, as shown in Section \ref{sec:comp}. (For terminology used, see Section \ref{ssec:GP-def} and Remark \ref{rmk:alg-coeff}.)
\begin{restatable}{alphatheorem}{thmdecide}
\label{thm:comp}
It is undecidable if two given bracket words $\bb a$ and $\bb b$ with algebraic coefficients
defined over a finite alphabet $\Sigma$ are equal.
\end{restatable}
Finally, we discuss examples of ``naturally occurring'' words for which we can show that they are, or that they are not, bracket words. In the positive direction, we note that the characteristic word $\bb 1_{F} = 1111010010010001\cdots$ of the Fibonacci numbers is a bracket word, which can be traced back to the observation that the golden ratio
$(1+\sqrt{5})/2$ is a Pisot unit and the group of units of the field it generates has rank $1$. In Section \ref{sec:pisot} we discuss generalisations of this example, corresponding to other Pisot and Salem numbers. Similar but slightly weaker results were obtained in \cite{ByszewskiKonieczny-2018-TAMS}.
In the negative direction, several criteria for proving that an infinite word is not a {\gpword} follow from results discussed in the reminder of the paper. Several other techniques were developed in a series of papers by Byszewski and the second-named author \cite{ByszewskiKonieczny-2018-TAMS,ByszewskiKonieczny-2020-CJM,Konieczny-2021-JLM}, leading to a proof that automatic sequences which are not eventually periodic are not bracket words. As explicit applications of the aforementioned methods, we mention the characteristic words of primes and of squares are not bracket words. The same applies to many words coming from number theory, such as $(\varphi(n) \bmod q)_{n=0}^\infty$, where $\varphi$ denotes the totient function and $q \geq 3$.
We point out that a significant part of the paper is devoted to a survey of known results concerning generalised polynomial sequences and their interpretation in terms of {\gpwords}. Beyond that, we prove several new results, including Theorems \ref{thm:A} and \ref{thm:comp}. The paper is organised as follows.
\subsection*{Organisation of the paper}
Sections \ref{sec:def} and \ref{sec:ex} are concerned with setting up the terminology and providing examples of {\gpwords}.
In Section \ref{sec:nil}, we discuss the connection between dynamics on nilmanifolds and generalised polynomials, which is one of the key tools used in subsequent sections.
For the sake of readability, we delegate some related material to Appendix \ref{app:nil}.
In Sections \ref{sec:constr} and \ref{sec:cl}, we discuss closure properties of {\gpwords} and other ways in which {\gpwords} can be constructed. These results allow us to perform many basic operations on {\gpwords} later in the paper.
Sections \ref{sec:facts}--\ref{sec:comp} each concern a different facet of {\gpwords} and can mostly be read independently from one another. In Sections \ref{sec:facts} and \ref{sec:growth}, we discuss frequencies of symbols and subwords in {\gpwords}; qualitative results are included in \ref{sec:facts} and quantitative --- in \ref{sec:growth}.
In Section \ref{sec:comp}, we discuss the canonical representation of {\gpwords}, based on \cite{Leibman-2012}, and its consequences in terms of decidability; next, we prove several undecidability results, including Theorem \ref{thm:comp}.
In Section \ref{sec:pisot}, we consider a class of {\gpwords}, consisting of characteristic words of certain sets of integers described by a linear recurrence. In Section \ref{sec:neg}, we consider the problem of proving that a given word is not a {\gpword}. We collect several criteria from previous sections and from \cite{ByszewskiKonieczny-2018-TAMS,Konieczny-2021-JLM}, and give a number of new examples and applications.
In Section \ref{sec:subword}, we discuss subword complexity of {\gpwords} and lay the lay the groundwork for Theorem \ref{thm:A}, which we prove in Sections \ref{sec:sc-induction}--\ref{sec:sc-proof}.
\subsection*{Acknowledgements}
The second-named author works within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Universit\'e de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR). We are also grateful to Val\'{e}rie Berth\'{e}, Jakub Byszewski, and Sam Chow for helpful comments, and to the anonymous referee for careful reading of our paper and valuable corrections.
\section{Definitions and notation}\label{sec:def}
\subsection{Combinatorics on words}
An alphabet $\Sigma$ is a finite set of symbols, also called letters. A finite word over $\Sigma$ is a
finite sequence of letters in $\Sigma$ or, equivalently, an element of $\Sigma^*= \bigcup_{\ell = 0}^\infty \Sigma^{\ell}$,
the free monoid generated by $\Sigma$
with respect to the concatenation of finite words.
The length of a finite word $w$, that is, the number
of symbols in $w$, is denoted by $\vert w\vert$.
We let $\epsilon$ denote the empty word, the neutral element of $\Sigma^*$.
An infinite word $\bb a= (a_n)_{n = 0}^\infty$ over $\Sigma$ is an element of $\Sigma^{\infty}$, or, equivalently, an infinite sequence with values in $\Sigma$,
({\it i.e.}, a map from $\NN_0$ to $\Sigma$). It is sometimes represented as $\bb a=a_0a_1\cdots$.
Throughout, we use bold symbols $\bb a, \bb b, \dots$ to denote infinite words.
Let $\Sigma$ and $\Pi$ be two alphabets. A \emph{morphism}
is a map $\sigma \colon \Sigma^* \to \Pi^*$ that obeys the identity $\sigma(\fword{u} \fword{v}) = \sigma(\fword{u})\sigma(\fword{v})$ for
all words $\fword{u},\fword{v}\in \Sigma^*$. Note that a morphism $\sigma$ is uniquely determined by the knowledge of $\sigma(x)$
for all $x \in \Sigma$.
A map from $\Sigma$ to $\Pi^*$ naturally (and uniquely) extends as a morphism from $\Sigma^*$ to $\Pi^*$.
A morphism is said to be \emph{non-erasing} if $\sigma(x)\not= \epsilon$ for all $x\in \Sigma$.
A morphism $\sigma$ over $\Sigma^*$ is said to be $k$-\emph{uniform} if
$\vert \sigma(a)\vert =k$ for every letter $a$ in~$\Sigma$, and just
\emph{uniform} if it is $k$-uniform for some~$k$. A $1$-uniform morphism is called a $\emph{coding}$.
Furthermore, there is a natural way to extend the action of a non-erasing morphism to infinite words, that is, as a map
from $\Sigma^\infty$ to $\Pi^\infty$ defined by $\sigma(\bb a) = \sigma(a_0)\sigma(a_1) \cdots$ for $\bb a \in \Sigma^\infty$.
Such a map is still called a morphism and denoted by $\sigma$.
\subsection{Generalised polynomials and bracket words}\label{ssec:GP-def}
For $x \in \RR$, we let $\ip{x} \in \ZZ$ denote the integer part of $x$ (also known as the floor), which is the unique integer with $\ip{x} \leq x < \ip{x}+1$. Similarly, we let $\fp{x} = x - \ip{x} \in [0,1)$ denote the fractional part, $\ceil{x} = -\ip{-x} \in \ZZ$ --- the ceiling, $\nint{x} = \ip{x+1/2} \in \ZZ$ --- the nearest integer, and $\fpa{x} = \abs{ x - \nint{x}} \in [0,1/2]$ --- the distance to the nearest integer.
We extend the notions introduced above to $x = \bra{x_i}_{i=1}^d \in \RR^d$ for $d \geq 2$ coordinate-wise, meaning that $\ip{x} = \bra{\ip{x_i}}_{i=1}^d \in \ZZ^d$, etc. Similarly, for a map $f \colon X \to \RR$ (where $X$ is any set) we define $\ip{f} \colon X \to \ZZ$ by $\ip{f}(x) = \ip{f(x)}$.
Let $d \in \NN$. We define \emph{{\gp} maps} (or {\genpoly} maps) $\RR^d \to \RR$ as the smallest family such that
\begin{enumerate}
\item\label{it:def:i} each polynomial map $\RR^d \to \RR$ is a {\gp} map;
\item\label{it:def:ii} if $g,h \colon \RR^d \to \RR$ are {\gp} maps then $g+h$ and $g \cdot h$ are {\gp} maps;
\item\label{it:def:iii} if $g \colon \RR^d \to \RR$ is a {\gp} map then $\ip{g}$ is a {\gp} map.
\end{enumerate}
Note that $x \mapsto \fp{x},\ceil{x},\nint{x}$ are {\gp} maps, and the definition of a {\gp} map does not change if in \ref{it:def:iii} we replace $\ip{\cdot}$ with $\fp{\cdot}$, $\ceil{\cdot}$ or $\nint{\cdot}$. The distance for the nearest integer can also be expressed by a generalised polynomial formula, such as $\fpa{x} = \bra{x-\nint{x}}\cdot\bra{2\ceil{x-\nint{x}}-1}$. (Since all maps $\RR^d \to \RR$ obtained from polynomials using addition, multiplication and $\fpa{\cdot}$ is continuous, replacing $\ip{\cdot}$ with $\fpa{\cdot}$ in \ref{it:def:iii} yields a strictly smaller family of maps.)
\begin{remark}\label{rem:composition}
If $g:\RR \to \RR$ and $h: \RR^d \to \RR$ are {\gp} maps, then $g\circ h$ is also a {\gp} map.
\end{remark}
A {\gp} map on a domain $\Omega \subset \RR^d$ (e.g.{} $\Omega = \ZZ^d$ or $\NN_0^d$) is simply the restriction of a {\gp} map on $\RR^d$. In particular, each {\gp} map on $\Omega$ can be extended to $\RR^d$, but the extensions is usually not unique. In this paper, we are particularly interested in finitely-valued {\gp} sequences (i.e.{}, {\gp} maps $\NN_0 \to \RR$), as seen from the perspective of combinatorics on words. This motivates us to pose the following definition.\footnote{Several authors have expressed the sentiment that a better name for ``generalised polynomials'' would have been ``bracket polynomials'', and the main reason to not adopt the latter name is that it is already used in knot theory (see e.g.{} \cite{Leibman-2012}). Fortunately, similar considerations do not apply to the term ``{\gpword}''.}
\begin{definition}\label{def:def:bra-word}
A \emph{\gpword} over a (finite) alphabet $\Sigma$ is an infinite word $\bb a = (a_n)_{n=0}^\infty \in \Sigma^\infty$ of the form $a_n = c(g(n))$ for all $n \in \NN_0$, where $g \colon \NN_0 \to \RR$ is a finitely-valued {\gp} map and $c \colon g(\NN_0) \to \Sigma$ is an arbitrary map. \end{definition}
\begin{remark}\label{rmk:coding}
The inclusion of the coding in Definition \ref{def:def:bra-word} does not significantly increase the level of generality. Indeed, if $\bb a = (a_n)_{n=0}^\infty$ is a bracket word over an alphabet $\Sigma$ and $\phi \colon \Sigma \to \RR$ is an arbitrary map then the map $g \colon \NN_0 \to \RR$ given by $g(n) = \phi(a_n)$ is a {\gp} map as a consequence of Corollary \ref{cor:constr:gpword-nice}. However, from the perspective of combinatorics on words, it would be unnatural to restrict our attention to words over alphabets contained in $\RR$. We also point out that in Section \ref{sec:cl} it is frequently more natural to consider words over more general alphabets, especially when it comes to closure under Cartesian products in Proposition \ref{prop:cl:prod}.
\end{remark}
\begin{remark}\label{rmk:bw-multidim}
In analogy with Definition \ref{def:def:bra-word}, one can define \emph{$d$-dimensional {\gpwords}} over a finite alphabet $\Sigma$ to be $d$-dimensional infinite words \[ \bb a = (a_{n_1,n_2,\dots,n_d})_{n_1,n_2,\dots,n_d=0}^\infty \in \Sigma^\infty \times \Sigma^\infty \times \cdots \times\Sigma^\infty\]
of the form $a_{n_1,n_2,\dots,n_d} = c(g(n_1,n_2,\dots,n_d))$ where $g \colon \NN_0^d \to \RR$ is a finitely-valued {\gp} map and $c \colon g(\NN_0^d) \to \Sigma$ is an arbitrary map. We limit the discussion to the $1$-dimensional case for the sake of clarity, but many of the results have their multidimensional analogues. In particular, we point out that multidimensional Sturmian sequences (as defined in \cite[Def.\ 4]{Fernique-2006}), and more generally rotation sequences (as defined in \cite[Def.\ 5]{Fernique-2006}) are multidimensional {\gpwords} (cf.\ \cite[Prop.\ 2]{Fernique-2006}).
\end{remark}
{
\begin{remark}\label{rmk:alg-coeff}
For a ring $\ZZ \subset A \subset \RR$ we define \emph{{\gp} maps with coefficients in $A$} in a fully analogous way, except that in \ref{it:def:i} we only include polynomial maps with coefficients in $A$. We will be especially interested in {\gp} maps with algebraic coefficients, i.e., $A = \bar\QQ$. One can also define the set of coefficients of a {\genpoly} (see e.g.{} \cite{Haland-1994}), but we avoid using this notion since it depends on the choice of a representation, which is usually not unique (see Section \ref{sec:comp} for further discussion). Slightly informally, we will say that an infinite word $\bb a$ over $\Sigma$ is a \emph{{\gpword} arising from a {\gp} map with algebraic coefficients}, or simply a \emph{{\gpword} with algebraic coefficients}, if $a_n = c(g(n))$ for every $n \in \NN_0$, where $g \colon \NN_0 \to \RR$ is a finitely-valued {\gp} map with coefficients in $\bar{\QQ}$ and $c \colon g(\NN_0) \to \Sigma$ is a map.
\end{remark}
}
For $k \geq 2$, a map $ g = (g_i)_{i=1}^k \colon \RR^{d} \to \RR^k$ is {\gp} if for each $i$, $1 \leq i \leq k$, the coordinate map $g_i$ is {\gp}. In most cases, we find it simpler to speak of $k$-tuples of {\gp} maps instead.
A \emph{{\gp} subset of $\Omega \subset \RR^d$} (or simply \emph{a {\gp} set}, if $\Omega$ is clear from the context) is the zero locus of a {\gp} map, that is, a set $E \subset \Omega$ which takes the form
\(E = \set{\vec x \in \Omega}{ g(\vec x) = 0}\)
for some {\gp} map $g \colon \Omega \to \RR$. As we will see in Section \ref{sec:constr}, $E \subset \NN_0$ is a {\gp} set if and only if $\bb 1_E$ is a {\gpword}, cf.\ Proposition \ref{prop:cl:g-in-I}. (Here and elsewhere, $\bb{1}_E \in \{0,1\}^\infty$ is given by $(\bb{1}_E)_n = 1$ if $n \in E$ and $(\bb{1}_E)_n = 0$ otherwise). We stress that this notion depends on $\Omega$; in particular, a {\gp} subset of $\NN_0$ will usually not be a {\gp} subset of $\ZZ$.
\subsection{Other notation}\label{sec:Notation}
We briefly summarise some other pieces of notation we use.
For $N \in \NN_0$, we let $[N] = \{0,1,\dots,N-1\}$. For a quantity $X$, we let $O(X)$ denote any quantity bounded in absolute value by $CX$, where $C$ is a constant. When the $C$ is additionally allowed to depend on a parameter $Y$, we write $O_Y(X)$ instead. Similarly, assuming that $X > 0$, we let $\Omega(X)$ denote any quantity bounded from below by $cX$, where $c > 0$ is a constant. If $X = O(Y)$ and $Y = \Omega(X)$, we write $Y = \Theta(X)$. We occasionally also use the notation $Y \ll X$ when $Y = O(Y)$.
Lastly, we write $o_{n\to\infty}(X)$ for any quantity $Y$ with $\lim_{n \to \infty} Y/X = 0$; if the parameter $n$ is clear from the context, we write $o(X)$ instead.
In what follows, we will use the Iverson bracket notation.
For a sentence $\varphi$ we put $\braif{\varphi} = 1$ if $\varphi$ is a true and $\braif{\varphi} = 0$ otherwise.
By a slight abuse of notation, we also use the Iverson bracket to define infinite words over the alphabet $\{0,1\}$.
For instance, if $X,Y$ are sets with $Y \subset X$ and $f \colon \NN_0 \to X$ is a map then
$\braif{f \in Y} \in \{0,1\}^\infty$ is given by $\braif{f \in Y}_n = \braif{f(n) \in Y}$ for all $n \in \NN_0$.
\color{black}
\section{Examples}\label{sec:ex}
Let us now present several examples of {\gpwords}. In some cases, the fact that the sequence under consideration
indeed is a {\gpword} will follow directly from the definition, while in other cases it may be more surprising.
\begin{example}\label{ex:ex:1_0}
Let $\alpha \in \RR \setminus \QQ$ and let $\bb a$ be the {\gpword} defined by $a_n = \ip{1-\fp{n\a}}$ for $n \in \NN_0$.
Then $\bb a = \bb{1}_{\{0\}} = 100\cdots$. More generally, all eventually constant sequences are {\gpwords}.
Explicitly, if $\bb a$ is a word over $\Sigma$ with $a_n = b$ for all $n \geq N$ then $a_n = c(g(n))$,
where $g \colon \ZZ \to \{-1,0,1,\dots,N-1\}$ is the {\gp} map given by
\[
g(n) = -1+ \sum_{m = 0}^{N-1} (m+1) \ip{1-\fp{(n-m)\a}},
\]
and $c \colon \{-1,0,1,\dots,N-1\} \to \Sigma$ is given by $c(n) = a_n$ if $n \neq -1$ and $c(-1) = b$.
\end{example}
\begin{example}\label{ex:ex:per}
Let $Q \in \NN$. Then $\bb a \in [Q]^\infty$ defined by $a_n = n \bmod Q = Q\fp{n/Q}$ is a {\gpword}.
More generally, all eventually periodic words are {\gpwords}. For details, see Section \ref{sec:cl}.
\end{example}
\begin{example}\label{ex:ex:sturm}
One way to define Sturmian words is by an explicit formula. Namely, an infinite word $\bb a$ over $\{0,1\}$ is Sturmian if it is the discrete derivative of a Beatty sequence, meaning that it takes one of the following forms:
\begin{align}
\label{eq:802:1} a_n &= \ip{n\a + \b} - \ip{\a (n-1) + \b}, \quad \text{or}\\
\label{eq:802:2} a_n &= \ceil{n\a + \b} - \ceil{\a (n-1) + \b},
\end{align}
for some $\a,\b \in [0,1)$ with $\a$ irrational. (Note that \eqref{eq:802:1} and \eqref{eq:802:2} differ for at most one value of $n$.)
Thus, Sturmian words are {\gpwords} (while Beatty sequences are unbounded {\gp} sequences). We also point out that Sturmian words
arise from codings of rotations, which gives another way to see that they are {\gpwords}; we explore this point of view further in Section \ref{sec:nil}.
More concretely, setting $\a = \frac{\sqrt{5}-1}{2}$ and $\b = 0$ we obtain the Fibonacci word, whose initial values are:
\begin{align*}
1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 \cdots
\end{align*}
\end{example}
\begin{example}\label{ex:ex:poly}
As a generalisation of Example \ref{ex:ex:sturm}, let $p \colon \RR \to \RR$ be a polynomial and let $I \subset [0,1)$
be an interval (or a finite union thereof). Let $\bb a$ be the infinite word over $\{0,1\}$ defined as
\[
a_n =
\begin{cases}
1 & \text{ if } \fp{p(n)} \in I,\\
0 & \text{ otherwise}.
\end{cases}
\]
Then $\bb a$ is a {\gpword}. As a concrete illustration, let $p(x) = \varphi x^2$, where $\varphi = \frac{\sqrt{5}+1}{2}$ is the golden ratio,
and $I = [0,1/4) \cup (3/4,1)$. Then
\[
a_n =
\begin{cases}
1 & \text{ if } \fpa{ \varphi n^2} < 1/4,\\
0 & \text{ if } \fpa{ \varphi n^2} \geq 1/4 .
\end{cases}
\]
The initial values of $\bb a$ are:
\begin{align*}
&1000101001111011101110111010011111011101111000111110111\cdots
\end{align*}
\end{example}
\begin{example}\label{ex:ex:rec}
Let $F = \{0,1,2,3,5,8,13,\dots\}$ be the set of all Fibonacci numbers. Then $\bb{1}_F$ is a {\gpword}.
Similarly, let $(t_i)_{i=0}^\infty$ be the sequence given by $t_0 = 0$, $t_1= t_2 =1$ and
$t_{i+3} = t_{i+2} + t_{i+1} + t_{i}$ for all $i \in \NN_0$, sometimes called the Tribonacci numbers,
and let $T = \set{t_i}{i \in \NN_0}$. Then $\bb{1}_T$ is a {\gpword}. These are special cases of Proposition \ref{prop:pisot:rec}.
In an upcoming preprint by the second-named author and Byszewski \cite{ByszewskiKonieczny-upcoming}, it is shown that, more generally, for each $E \subset F$, $\bb{1}_E$ is a {\gpword}.
\end{example}
\begin{example}\label{ex: littlewood}
Let $\a,\b \in \RR$ and $\e > 0$. Then the infinite word $\bb a$ over $\{0,1\}$ defined by
\[
a_n =
\begin{cases}
1 & \text{ if } n \neq 0 \text{ and } \fpa{\a n} \cdot \fpa{\b n} < \e/n,\\
0 & \text{ otherwise},
\end{cases}
\]
is a {\gpword}, as follows from Proposition \ref{prop:cl:g-in-I}.
We point out that a famous conjecture in Diophantine approximation, the Littlewood conjecture,
is equivalent to the statement that, for each choice of $\a,\b,\e$, the {\gpword} $\bb a$ defined above is not identically zero.
Indeed, in its usual formulation, Littlewood's conjecture asserts that
\begin{equation}\label{eq:232:1}
\liminf_{n \to \infty} n \cdot \fpa{\a n} \cdot \fpa{\b n} = 0
\end{equation}
for all $\a,\b \in [0,1)$. A landmark result toward its resolution is due to Einsiedler, Katok and Lindenstrauss \cite{EinsiedlerKatokLindenstrauss-2006}:
the set of possible exceptions ({\it i.e.}, the set of pairs $(\a,\b)$ for which \eqref{eq:232:1} is false) has Hausdorff dimension zero.
\end{example}
\begin{example}
Let $(n_i)_{i=0}^\infty$ be a sequence of positive integers with $n_{i+1} \geq n_{i}^2$ for all $i$, such as $n_i = 2^{2^i}$.
Put $E = \set{n_i}{i \in \NN_0}$. Then $\bb{1}_E$ is a {\gpword}. This is a special case of Proposition \ref{prop:constr:super-sparse}.
\end{example}
\section{Dynamical representation}\label{sec:nil}
In this section, we discuss a dynamical description of bounded {\gp} sequences and {\gpwords}.
Specifically, we briefly introduce basic facts about nilmanifolds and nilsystems and explain their relation to generalised polynomials,
as established by Bergelson and Leibman \cite{BergelsonLeibman-2007}.
\subsection{Nilsystems and generalized polynomials}
Classical theory of nilpotent dynamics can be found in \cite{AuslanderGreenHahn-book}.
In order to maintain the introductory nature of this section, we delegate some of the more technical results to Appendix \ref{app:nil}.
We also refer to cited references, such as \cite{BergelsonLeibman-2007}, for precise definitions and a more detailed discussion.
\subsubsection{Nilpotent Lie groups}
Let $G$ be a group. The \emph{lower central series} $(G_i)_{i\geq 0}$ is the chain of subgroups of $G$ inductively defined by
$G_0= G_1 = G$ and $G_{i+1} = [G,G_i]$ for $i\geq 1$. Here, we let $[G,H]$ denote the group generated by the commutators $[g,h] = ghg^{-1}h^{-1}$ for $g \in G, h \in H$.
The group $G$ is nilpotent if there exists $s$ such that $G_{s+1}=\{\id_G\}$.
The smallest such $s$ is called the nilpotency class of $G$ and $G$ is said to be nilpotent of class $s$ or a $s$-step nilpotent group.
We recall that a Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable.
A nilpotent Lie group is a Lie group that is nilpotent.
\subsubsection{Nilrotations}
{
A \emph{nilmanifold} is a quotient space $G/\Gamma$ where $G$ is a nilpotent Lie group and $\Gamma$ is a discrete cocompact subgroup.
A \emph{nilsystem} is a dynamical system of the form $(G/\Gamma, T_g),$ where $G/\Gamma$ is a nilmanifold and $T_g$ is a \emph{nilrotation}.
That is, there exists some $g \in G$ such that $T_g(h\Gamma) = gh\Gamma$ for all $h \in G$.
In general, there is no guarantee that $G$ is connected, and we let $G^{\circ}$ denote the connected component of $\id_G$.
We may assume without loss of generality that $G^{\circ}$ is simply connected.
The simplest example of a nilsystem is the aforementioned rotation on the torus, where we take $G = \RR$, $\Gamma = \ZZ$ (see Section \ref{ssec:nil:sturm}).
\subsubsection{Mal'cev basis}
Let $G$ be a connected and simply connected $s$-step nilpotent Lie group and let $\Gamma < G$ be a discrete cocompact subgroup.
In this case, for $g \in G$ and $t \in \RR$, one can use the Lie algebra of $G$ to define $g^t \in G$.
We let $\dim G$ denote the dimension of $G$ as a Lie group.
A \emph{Mal'cev basis} of $G$ is a sequence $h_1,h_2, \dots, h_d \in \Gamma$ satisfying the following conditions.
\begin{enumerate}
\item Every $g \in G$ has a unique representation $h_1^{t_1} h_2^{t_2} \cdots h_d^{t_d}$ with $t_i \in \RR$.
\item There exists an increasing sequence of natural numbers $$1 = k_1 < k_2 < \dots < k_{s} = d+1$$ such that for each $1 \leq j \leq s$,
the quotient $G_{j}/G_{j+1}$ is spanned by $h_{k_j}, \dots, h_{k_{j+1}-1}$.
\end{enumerate}
Existence of such bases was established by Mal'cev \cite{Malcev-1949,Malcev-1949-tran}.
Given a Mal'cev basis, we let $\tilde \tau \colon G \to \RR^d$ denote the coordinate map, characterised by the property that
\begin{equation}\label{eq:Malcev-G}
\tilde\tau\bra{ h_1^{t_1} h_2^{t_2} \cdots h_d^{t_d} } = (t_1,t_2,\dots, t_d)\,, \qquad t_i \in \RR\,.
\end{equation}
This induces also the coordinate map $\tau \colon G/\Gamma \to [0,1)^d$, similarly characterised by
\begin{equation}\label{eq:Malcev-X}
\tau\bra{ h_1^{t_1} h_2^{t_2} \cdots h_d^{t_d} \Gamma} = (t_1,t_2,\dots, t_d)\,, \qquad t_i \in [0,1)\,.
\end{equation}
Thus, the nilmanifold $G/\Gamma$ can be identified with a cube $[0,1)^{\dim G}$ via
\emph{Mal'cev coordinates} $\tau \colon G/\Gamma \to [0,1)^{\dim G}$. The coordinate map $\tau$ is a bijection; $\tau^{-1}$ is continuous,
its restriction to $ (0,1)^{\dim G}$ is a diffeomorphism. The nilmanifold $G/\Gamma$ carries a natural probability measure, the Haar measure,
which we denote by $\mu_{G/\Gamma}$.
\subsubsection{Semialgebraic sets}\label{ssec:semialgebraic}
An algebraic variety in $\RR^d$ is a set defined by a finite number of polynomial equations. More generally, a semialgebraic set is a set defined by a finite number of polynomial equations and inequalities, or a finite union of sets of this form.
A map $f \colon \RR^d \to \RR$ is \emph{piecewise polynomial} if there exists a partition $\RR^d = S_1 \cup S_2 \cup \cdots \cup S_r$
into semialgebraic pieces such that, for every $i$, $1 \leq i \leq r$, the restriction $f|_{S_i}$ is a polynomial map.
A map $f \colon G/\Gamma \to \RR$ is piecewise polynomial if it takes the form $f = \tilde f \circ \tau$ for a piecewise polynomial
map $\tilde f \colon \RR^{\dim G} \to \RR$.
The notion of a piecewise polynomial map is independent of the choice of Mal'cev basis (see \cite[Sec.{} 0.18]{BergelsonLeibman-2007}).
\subsubsection{Connectedness}
Let us now return to the general case, where $G$ may be disconnected.
If $G/\Gamma$ is connected, then it remains true that $G/\Gamma = G^{\circ}/\Gamma \cap G^{\circ}$,
and the previous discussion applies verbatim (note, however, that not every translation $T_g$, with $g \in G$,
can be represented as $T_h$, with $h \in G^{\circ}$). If $G/\Gamma$ is disconnected, then it can be decomposed as a finite union
of connected components which again are nilmanifolds, and we can apply the discussion above to each component separately.
A map $f \colon G/\Gamma \to \RR$ is piecewise polynomial if its restriction to each connected component of $G/\Gamma$ is piecewise polynomial.
Finally, we recall that a topological dynamical system $(X,T)$ is \emph{minimal} if there is no closed subset $Y \subset X$ with $T(Y) \subset Y$.
We have now introduced all the terminology which is needed to state the $1$-dimensional case of the main result of \cite{BergelsonLeibman-2007}.
\begin{theorem}[{\cite[Thm.\ A]{BergelsonLeibman-2007}}]\label{thm:BL-mini}
Any bounded {\gp} map $g \colon \ZZ \to \RR$ admits a representation $g(n) = f(T^n(x))$, where $(X,T)$ is a minimal nilsystem,
$f \colon X \to \RR$ is piecewise polynomial, and $x \in X$. Conversely, for any nilsystem $(X,T)$, any piecewise polynomial map
$f \colon X \to \RR$, and any $x \in X$, the map $n \mapsto f(T^n(x))$ from $\ZZ$ to $\RR$ is a bounded {\gp} map.
\end{theorem}
For future reference, we record the following special case of Theorem \ref{thm:BL-mini} applicable to {\gpwords}.
\begin{theorem}\label{thm:BL-word}
For any {\gpword} $\bb a$ over an alphabet $\Sigma$, there exists a minimal nilsystem $(X,T)$, a point $x \in X$, and a partition
$X = \bigcup_{i \in \Sigma} S_i$ into pairwise disjoint semialgebraic pieces, such that for each $n \in \NN$ and $i \in \Sigma$,
\begin{align}\label{eq:418:1}
a_n = i && \text{ if and only if }&& T^n(x) \in S_i \,.
\end{align}
Conversely, for any nilsystem $(X,T)$, any point $x \in X$ and any partition $X = \bigcup_{i \in \Sigma} S_i$ into pairwise disjoint semialgebraic pieces,
\eqref{eq:418:1} defines a {\gpword}.
\end{theorem}
\begin{proof}
Pick a representation $a_n = c(g(n))$ ($n \in \NN_0$), where $g$ is a finitely-valued {\gp} map and $c$ is a coding. Let $g(n) = f(T^n(x))$ be the representation of $g$ as in Theorem \ref{thm:BL-mini}. Then, for each $i \in \Sigma$, $S_i = f^{-1}(c^{-1}(i))$ is a semialgebraic set such that \eqref{eq:418:1} holds.
The converse implication follows along similar lines.
\end{proof}
Now, we give two emblematic examples of nilsystems and the corresponding {\gp} maps. While we do not use them elsewhere in the paper, we believe they help to illustrate Theorem \ref{thm:BL-mini}.
\subsection{One dimensional torus and Sturmian words}\label{ssec:nil:sturm}
As already mentioned in the introduction, Sturmian words can be dynamically represented as codings of irrational translations on the one-dimensional torus.
In this case, we simply take $G=\mathbb R$, $\Gamma=\mathbb Z$, and the nilrotation $T=T_\a$ is just an irrational translation on $\RR/\ZZ$. More precisely,
let $\bb a$ be the Sturmian word defined by
\eqref{eq:802:1} (or, respectively, by \eqref{eq:802:2}).
Let $T_\a \colon \RR/\ZZ \to \RR/\ZZ$ denote the translation by $\a$ on $\RR/\ZZ$,
meaning that $T_\a(x) = x+\a$ ($x \in \RR/\ZZ$). Let $I \subset \RR/\ZZ$ be the interval $[0,\a)$ (resp.{} $(0,\a]$).
Then, \eqref{eq:802:1} (resp. \eqref{eq:802:2}) is equivalent to:
\[
a_n = 1_I(T_\a^n(\beta))
= \begin{cases}
1 &\text{ if } T_\a^n(\beta) \in I\, ,\\
0 &\text{ otherwise.}
\end{cases}
\]
\subsection{Heisenberg group}
Another helpful example to keep in mind is the Heisenberg nilsystem. It is a standard example, and appears e.g.\ in
\cite[Sec.\ 0.14]{BergelsonLeibman-2007},
\cite[Sec.\ 5]{GreenTao-2012},
\cite[Sec.\ 1]{GreenTao-2010},
\cite[p.\ 555]{GreenTao-2012-Mobius},
\cite{GreenTaoZiegler-2012} (as a running example).
Pick any $\a,\b,\c \in \RR$, and set
\[
G =
\begin{bmatrix}
1 & \RR & \RR \\
0 & 1 & \RR \\
0 & 0 & 1
\end{bmatrix}, \;\;\;
\Gamma =
\begin{bmatrix}
1 & \ZZ & \ZZ \\
0 & 1 & \ZZ \\
0 & 0 & 1
\end{bmatrix}, \;\;\mbox{ and }\;\;
h = \begin{bmatrix}
1 & \b & \c + \a\b/2\\
0 & 1 & \a \\
0 & 0 & 1
\end{bmatrix} \,.
\]
One possible choice for Mal'cev coordinates is given by
\[
\tau\bra{
\begin{bmatrix}
1 & y & z \\
0 & 1 & x \\
0 & 0 & 1
\end{bmatrix}
} = (x,y,z)\, .
\]
Then we can compute that
\begin{align*}
h^n \Gamma &=
\begin{bmatrix}
1 & n\b & n\c + n^2 \a\b/2 \\
0 & 1 & n\a \\
0 & 0 & 1
\end{bmatrix}\Gamma\\& =
\begin{bmatrix}
1 & \fp{n\b} & \fp{ n\b \fp{n\a} - n^2 \a\b/2 + \c n} \\
0 & 1 & \fp{n\a} \\
0 & 0 & 1
\end{bmatrix}\Gamma \,.
\end{align*}
Hence, using the nilsystem $(G/\Gamma,T_h)$ and taking $f = \tau_3$ (\emph{i.e.}, the third entry of the coordinate map $\tau$),
we obtain a representation of the bounded {\gp} map
\[ g(n) = \fp{ n\b \fp{n\a} - n^2 \a\b/2 + \c n}.\]
A slightly more complicated but similar construction involving matrices in dimension $4$ discussed in \cite[Sec.{} 0.14]{BergelsonLeibman-2007}
provides a dynamical representation of the sequence $\fp{ n\b \fp{n\a}}$.
\section{Representations and constructions}\label{sec:constr}
In this section we discuss methods by which {\gpwords} can be represented and constructed.
\subsection{Representations} It is clear from the definition that {\gp} maps $\colon \NN_0 \to \RR$ are precisely those maps which can be expressed using (classical) polynomials, addition, multiplication and the floor function. For finitely-valued {\gp} maps and {\gpwords}, the situation becomes more complicated. As we will see in Section \ref{sec:comp}, it is not always possible to decide if a given {\gp} map $g \colon \NN_0 \to \RR$ is finitely-valued
(and hence relevant to the study of {\gpwords}) or not. Thus, it is not always possible to see if a given formula represents a {\gpword}. On the other hand, for suitably constructed {\gp} maps, it is easy to see that they must be finitely-valued.
For instance, if $g(n) = \ip{2\fp{h(n)}}$ for some {\gp} map $h \colon \NN_0 \to \RR$, then evidently $g(\NN_0) \subset \{0,1\}$. In general, one can always represent a finitely-valued {\gp} map in a form which makes it easy to estimate the cardinality of the image. We stress that Proposition \ref{lem:constr:reg-rep} provides a concrete way to generate
all finitely valued \gp{} maps from $\NN_0$ to $\RR$, and hence all bracket words.
\begin{proposition}\label{lem:constr:reg-rep}
Let $g \colon \NN_0 \to \RR$ be a finitely-valued {\gp} map. Then $g$ can be written in the form $g(n) = f\bra{ \ip{N { \fp{h(n)}}}}$, where
$f$ and $h$ are {\gp} maps from $\NN_0$ to $\RR$ and $N = \abs{g(\NN_0)}$. Conversely, any map $g \colon \NN_0 \to \RR$ of the form
$g(n) = f\bra{ \ip{N { \fp{h(n)}}}}$, where $f$ and $h$ are {\gp} maps from $\NN_0$ to $\RR$ and $N \in \NN$, is a {\gp} map which
takes at most $N$ distinct values.
\end{proposition}
In the course of the proof of Proposition \ref{lem:constr:reg-rep}, we will need the following simple fact, which clarifies the relation between {\gpwords} and finitely-valued {\gp} maps.
\begin{lemma}\label{lem:constr:gp-word-vs-fun}
Let $f \colon \NN_0 \to \RR$ be a map taking finitely many values. Then the two following properties are equivalent.
\begin{enumerate}
\item\label{it:685:a} The word $\bb f=\bra{f(n)}_{n=0}^\infty$ is a {\gpword}.
\item\label{it:685:b} The map $f \colon \NN_0 \to \RR$ is a {\gp} map.
\end{enumerate}
\end{lemma}
\begin{proof}
It is clear that \ref{it:685:b} implies \ref{it:685:a}: in Definition \ref{def:def:bra-word}, we can take $g=f$ and $c = \mathrm{id}$.
For the converse implication, suppose that $f(n) = c(g(n))$, where $g \colon \NN_0 \to \RR$ has finite image and
$c \colon g(\NN_0) \to \RR$. There exists a polynomial map $p \colon \RR \to \RR$ such that $p(x) = c(x)$ for all $x \in g(\NN_0)$.
Hence, $f = p \circ g$ is a {\gp} map.
\end{proof}
\begin{proof}[Proof of Proposition \ref{lem:constr:reg-rep}]
Let $c \colon g(\NN_0) \to [N]$ be any bijective map. Then the word $\bb a = (c(g(n))_{n=0}^\infty$ is a {\gpword} taking values in $[N]$. Hence, by Lemma \ref{lem:constr:gp-word-vs-fun}, $c \circ g$ is a {\gp} map. Let $h$ be the {\gp} map defined by $h(n) = c(g(n))/N$, and let $f \colon \NN_0 \to \RR$ be a polynomial map such that $f(c(x)) = x$ for each $x \in g(\NN_0)$ (such $f$ exists by polynomial interpolation). It is straightforward to check that
\[ f(\ip{N\fp{h(n)}}) = f(c(g(n))) = g(n) \]
for all $n \in \NN_0$, as needed.
The converse direction holds trivially.
\end{proof}
As a consequence of Proposition \ref{lem:constr:reg-rep}, for each {\gpword} we can construct a particularly convenient representation.
\begin{corollary}\label{cor:constr:gpword-nice}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$ and let $N = \abs{\Sigma}$.
Then there exist a {\gp} map $g \colon \NN_0 \to [N]$ and a map $c \colon [N] \to \Sigma$ such that $a_n = c(g(n))$ for all $n \in \NN_0$.
\end{corollary}
\begin{proof}
Let $a_n = c'(g'(n))$, where $g' \colon \NN_0 \to \RR$ is a finitely-valued {\gp} map and $c' \colon g'(\NN_0) \to \Sigma$. Let $g'(n) = f(\ip{N\fp{h(n)}})$ be the representation of $g'$ from Proposition \ref{lem:constr:reg-rep}. It remains to define $g$ and $c$ by $g(n) = \ip{N\fp{h(n)}}$ and $c(x) = c'(f(x))$.
\end{proof}
\subsection{Constructions}
Next, we discuss a basic tool, which can be used to construct potentially interesting examples of {\gpwords}. In what follows, we will use the Iverson bracket notation, where $\braif{g \in I}$ denotes the word over the alphabet $\{0,1\}$ which records the positions $n$ such that $g(n) \in I$ (see Section \ref{sec:Notation} for exact definition).
\begin{proposition}\label{prop:cl:g-in-I}
Let $g \colon \NN_0 \to \RR$ be a {\gp} map and $I \subset \RR$ be an interval (possibly infinite or degenerate). Then $\braif{g \in I}$ is a {\gpword}.
\end{proposition}
\begin{proof}
The case where $I$ is bounded is covered by {\cite[Lemma 1.2]{ByszewskiKonieczny-2018-TAMS}}, while the case where $I$ is unbounded follows from \cite[Lemma B.3]{Konieczny-2021-JLM}.
\end{proof}
\begin{remark}
We point out that the analogous result with $\NN_0$ replaced by $\ZZ$ is true
for bounded $I$ and false for unbounded $I$. For instance, $1_{\NN}(n) = \braif{n \in (0,\infty)}$ is not a {\gp} map on $\ZZ$ (c.f.\ \cite[Ex.\ B.2]{Konieczny-2021-JLM}). This is one of the reasons why we focus on one-sided {\gpwords} $(a_n)_{n=0}^\infty$,
rather than on their two-sided analogues $(a_n)_{n \in \ZZ}$.
\end{remark}
We will use later the following slight refinement of Proposition \ref{prop:cl:g-in-I} in the case $I = \{0\}$.
\begin{lemma}\label{lem:cl:[g=0]}
Let $g \colon \NN_0 \to \RR$ be a {\gp} map with coefficients in some field $K$ with $\QQ \subsetneq K \subseteq \RR$. Then $\braif{g = 0}$ is a {\gpword} with coefficients in $K$.
\end{lemma}
\begin{proof}
Pick any $\a \in K \setminus \QQ$. Note that the only solution to $\fpnormal{x} = \fpnormal{\a x} = 0$ in $\RR$ is $x = 0$. Hence,
\[
\ip{1 - \frac{1}{2} \fpnormal{x} - \frac{1}{2} \fpnormal{\a x} } = \braif{ x = 0}\,, \qquad (x \in \RR)\,.
\]
It follows that
\[
\braif{g(n) = 0} = \ip{1 - \frac{1}{2} \fpnormal{g(n)} - \frac{1}{2} \fpnormal{\a g(n)} }\,, \qquad n \in \NN_0\,. \qedhere
\]
\end{proof}
One of the main reasons for interest in Lemma \ref{lem:cl:[g=0]} is that it gives an alternative definition of {\gpwords},
phrased in terms of fibres.
\begin{corollary}\label{cor:constr:fibre}
Let $\bb a$ be an infinite word defined over a finite alphabet $\Sigma$. Then the two following properties are equivalent.
\begin{enumerate}
\item The word $\bb a$ is a {\gpword}.
\item For every $x \in \Sigma$, the fibre $\set{n \in \NN_0}{a_n = x}$ is a {\gp} subset of $\NN_0$.
\end{enumerate}
In particular, for a set $E \subset \NN_0$, $E$ is a {\gp} set if and only if $\bb{1}_E$ is a {\gpword}.
\end{corollary}
\begin{proof}
Let us assume that $\bb a$ is a {\gpword} defined over $\Sigma$ and let $x\in \Sigma$.
Let $a_n=c(g(n))$ be a representation of $\bb a$, where $g: \NN_0\to \RR$ is a finitely valued \gp{} map and $c$ is a map from $g(\NN_0)$ to $\Sigma$.
Then $h= \prod_{a\in c^{-1}(x)}(g-a)$ is a \gp{} map and it follows from Lemma \ref{lem:cl:[g=0]} that $\braif{h(n) = 0}=\{n\in \NN_0 \mid a_n=x\}$ is a \gp{} subset of $\NN_0$.
Conversely, let us assume that, for every $x \in \Sigma$, the fibre $F_x=\set{n \in \NN_0}{a_n = x}$ is a {\gp} subset of $\NN_0$. Hence $\bb{1}_{F_x}$ is a \gp{} map.
Let us assume that $\vert \Sigma\vert=N$ and let $x_1,\ldots,x_N$ denote an enumeration of the elements of $\Sigma$. Set $h= \sum_{i=1}^N i \bb{1}_{F_{x_i}}$ and let $c$ be the map defined by $c(i)=x_i$, $1\leq i\leq N$. Then $h$ is a \gp{} map and thus $a_n=c(h(n))$ is a {\gpword}.
\end{proof}
With some basic algebraic manipulations, one can extend Proposition \ref{prop:cl:g-in-I} to apparently more complicated conditions,
as shown by the following example.
\begin{example}
Let $g$ and $h$ be {\gp} maps from $\NN_0$ to $\RR$ and assume that $h(n) > 0$ for all $n \in \NN_0$. Then $\braif{g < 1/h} = \braif{gh < 1}$ is a {\gpword}. More generally, for each rational exponent $\lambda = p/q \in \QQ_{>0}$, also
$\braif{g < 1/h^{\lambda}} = \braif{g^q h^p < 1}$ is a {\gpword}.
As an explicit application, for every pair $(\a,c) \in \RR^2$ and $\lambda \in \QQ$, with $c$ and $\lambda$ positive,
the formula $a_n = \braif{ \fpa{\a n} < c/n^{\lambda} }$ defines a {\gpword} $\bb a$ which detects denominators of good rational approximations to $\a$.
\end{example}
\section{Closure properties}\label{sec:cl}
We will now discuss ways in which known instances of {\gpwords} can be used to construct new ones.
Compared to the earlier section, the results discussed here have a more computational flavour.
For instance, we point out that, with the sole exception of Proposition \ref{prop:cl:orb-cl}, all results in this section
are analogues of standard results about automatic sequences, in the sense that they remain true if the term ``{\gpword}'' is replaced with ``$k$-automatic sequence'' and the term ``{\gp} set'' is replaced with ``$k$-automatic set'' for fixed $k \geq 2$ (see \cite[Sec. 5]{AlloucheShallit-book}).
\subsection{Codings and products}
It is an almost immediate consequence of Definition \ref{def:def:bra-word} that {\gpwords} are preserved under coding.
\begin{lemma}\label{lem:cl:code}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$ and let $\varphi \colon \Sigma \to \Pi$ be a map to some other finite alphabet $\Pi$.
Then $\bra{\varphi(a_n)}_{n=0}^\infty$ is a {\gpword} over $\Pi$.
\end{lemma}
\begin{proof}
If $a_n = c( g(n))$ is the representation of $\bb a$ as in Definition \ref{def:def:bra-word},
then $\varphi(a_n)= c'( g(n))$ where $c' = \varphi \circ c$.
\end{proof}
Next, we note that the direct product of two {\gpwords} is again a {\gpword}.
\begin{proposition}\label{prop:cl:prod}
Let $\bb a$ and $\bb a'$ be {\gpwords} respectively defined over some finite alphabets $\Sigma$ and $\Sigma'$.
Then $\bb a \times \bb a' = \bra{(a_n,a_n')}_{n=0}^\infty$ is a {\gpword} over $\Sigma \times \Sigma'$.
\end{proposition}
\begin{proof}
Recall that, by definition of a {\gpword}, $\bb a$ has a representation $a_n = c( g(n))$, where $g \colon \NN_0 \to A$ is a
{\gp} map taking values in some finite set $A \subset \RR$, and $c \colon A \to \Sigma$ is an arbitrary map. Let $a_n' = c'( g'(n))$
be an analogous representation of $\bb a'$.
Replacing $g$ with $C g$ for a sufficiently large positive real number $C > 0$ (and modifying $A$ and $c$ accordingly),
we may assume that the only solutions to $x+x' = y+y'$ with $(x,y) \in A^2$ and $(x',y)' \in A'^2$ are the trivial ones: $x = x'$ and $y = y'$.
Let $B$ denote the sumset $A+A' = \set{x+x'}{x \in A,\ x' \in A'}$, let $h$ denote the {\gp} map $g+g' \colon \NN_0 \to B$, and let
$d \colon B \to \Sigma \times \Sigma'$ denote the unique map such that $d(x+x') = \bra{ c(x),c'(x') }$ for all $x \in A$ and $x' \in A'$.
Then $(a_n, a'_n) = d(h(n))$ for all $n \in \NN_0$.
\end{proof}
\begin{remark}
It follows that replacing, in Definition \ref{def:def:bra-word}, {\gp} maps $\NN_0 \to \RR$ with {\gp} maps $\NN_0 \to \RR^d$
for arbitrary $d \in \NN$, leaves unchanged the set of words so defined.
\end{remark}
In practice, Proposition \ref{prop:cl:prod} is mostly used via the following corollary.
\begin{corollary}\label{cor:cl:prod}
Let $\bb a$ and $\bb a'$ be {\gpwords} respectively defined over some finite alphabets $\Sigma$ and $\Sigma'$,
and let $f \colon \Sigma \times \Sigma' \to \Pi$ be a map to some other finite alphabet $\Pi$.
Then $\bra{f(a_n,a_n')}_{n=0}^\infty$ is a {\gpword} over $\Pi$.
\end{corollary}
\begin{proof}
This follows directly from Proposition \ref{prop:cl:prod} and Lemma \ref{lem:cl:code}.
\end{proof}
\begin{remark}
In particular, for any ring $R$, {\gpwords} taking values in $R$, equipped with coordinatewise addition and multiplication, form a ring.
\end{remark}
As a consequence of the two previous results, we see that {\gpwords} can be defined in a ``case-by-case'' manner.
\begin{proposition}\label{prop:cl:cases}
Let $\Sigma$ be a finite alphabet, $\NN_0 = \bigcup_{i=1}^r S_i$ be a partition of $\NN_0$ into pairwise disjoint {\gp} subsets,
and $\bb a^{(i)}$, $1 \leq i \leq r$, be {\gpwords} over $\Sigma$. Let $\bb a$ be defined by $a_n = a^{(i)}_n$ if $n \in S_i$.
Then $\bb a$ is a {\gpword}.
\end{proposition}
\begin{proof}
We first infer from a recursive use of Proposition \ref{prop:cl:prod} that the word $$(\bb{1}_{S_1},\ldots,\bb{1}_{S_r},\bb a^{(1)},\ldots,\bb a^{(r)})$$
defined over the alphabet
$\{0,1\}^r\times \Sigma^r$ is a bracket word. Now, letting $f$ be any function from $\{0,1\}^r\times \Sigma^r$ to $\Sigma$ such that
$$
f(\varepsilon_1,\ldots,\varepsilon_r,x_1,\ldots,x_r)=x_i \mbox{ if } \varepsilon_i=1 \mbox{ and } \varepsilon_j=0 \mbox { when } j\not=i \,,
$$
we infer from Corollary \ref{cor:cl:prod} that $f(\bb{1}_{S_1},\ldots,\bb{1}_{S_r},\bb a^{(1)},\ldots,\bb a^{(r)})$ is a bracket word.
It remains to see that $\bb a=f(\bb{1}_{S_1},\ldots,\bb{1}_{S_r},\bb a^{(1)},\ldots,\bb a^{(r)})$.
\end{proof}
\begin{example}
Let $\bb a$ be the infinite word over the alphabet $\{-2,-1,0,\dots,10\}$ given by
\[
a_n =
\begin{cases}
\ip{n\fp{\sqrt{2}n}} &\text{if } {n\fp{\sqrt{2}n} } \leq 10\,,\\
-1 &\text{if } {n\fp{\sqrt{2}n} } > 10 \text{ and } n^2 \fp{\sqrt{2}n\ip{\sqrt{3}n}} - n \fp{\sqrt{5}n} + 7 > 0\,,\\
-2 &\text{if } {n\fp{\sqrt{2}n} } > 10 \text{ and } n^2 \fp{\sqrt{2}n\ip{\sqrt{3}n}} - n \fp{\sqrt{5}n} + 7 \leq 0\,.
\end{cases}
\]
Then $\bb a$ is a {\gpword}.
\end{example}
Since each eventually constant sequence is a {\gpword}, it follows that {\gpwords} are also closed under finite modifications.
\begin{corollary}
Let $\bb a,\bb a'$ be infinite words over a finite alphabet. If $\bb a$ is a {\gpword} and $a'_n = a_n$ for all but finitely many $n \in \NN_0$ then $\bb a'$ is a {\gpword}.
\end{corollary}
\begin{proof}
The result follows directly from Proposition \ref{prop:cl:cases} and the fact that all finite subsets of $\NN_0$ are {\gp}.
\end{proof}
\subsection{Rearrangements and morphisms}
Many natural operations on infinite words can be described in terms of extracting or inserting entries in a regular manner. We record a simple observation, which can be used to find examples of operations of the aforementioned type which preserve {\gpwords}.
\begin{lemma}\label{lem:cl:subseq}
Let $\bb a$ be a {\gpword} and let $h \colon \NN_0 \to \NN_0$ be a {\gp} map. Then $\bra{ a_{h(n)}}_{n=0}^\infty$ is a {\gpword}.
\end{lemma}
\begin{proof}
If $a_n = c( g(n))$ is the representation of $\bb a$ as in Definition \ref{def:def:bra-word}, then $a_{h(n)} = c( g'(n))$,
where $g' = g \circ h$ is a {\gp} map (see Remark \ref{rem:composition}).
\end{proof}
Below, we list some applications of this result.
Recall that, for $A \in \NN$, the map $\NN_0 \to \NN_0$ given by $n \mapsto n \bmod A = A\fp{n/A}$ is a {\gp} map.
\begin{corollary}\label{cor:cl:rearrange}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$, let $A \in \NN$, $B \in \NN_0$, let $\diamondsuit$ be a symbol not belonging to $\Sigma$, and let $\pi \colon [A] \to [A]$ be a map.
Then the following infinite words are also {\gpwords}.
\begin{itemize}
\item[{\rm (i)}] $\bra{ a_{An+B}}_{n=0}^\infty$.
\item[{\rm (ii)}] $\bb a'$ defined over $\Sigma \cup \{\diamondsuit\}$ by $a'_n = a_{n/A}$ if $A \mid n$ and $a'_n = \diamondsuit$ otherwise.
\item[{\rm (iii)}] $\bra{ a_{ \floor{n/A} + \pi(n \bmod A) }}_{n=0}^\infty$.
\end{itemize}
\end{corollary}
\begin{proof}
The first item follows directly from Lemma \ref{lem:cl:subseq}. The second and third items follow from Lemma \ref{lem:cl:subseq}
and Proposition \ref{prop:cl:cases}.
\end{proof}
\begin{lemma}\label{lem:cl:subs}
Let $\Sigma$ and $\Pi$ be two alphabets. Let $\bb a$ be a {\gpword} over $\Sigma$ and $\sigma$ be a morphism of constant length from
$\Sigma^*$ to $\Pi^*$. Then $\sigma(\bb a)$ is a {\gpword}.
\end{lemma}
\begin{proof}
Let us assume that $\sigma$ has constant length $k$, and set $\bb a' = \sigma(\bb a)$.
For each $i \in [k]$ and $n \in \NN_0$, we have $a'_{kn+i} = \sigma(a_n)_i$, where we let $ \sigma(a_n)_i$ denote the $i$th letter occurring in
$\sigma(a_n)$.
It remains to apply Proposition \ref{prop:cl:cases} and Lemmas \ref{lem:cl:subseq} and \ref{lem:cl:code}.
\end{proof}
By similar techniques, we can show a result in the reverse direction: {\gpwords} are preserved under grouping blocks of constant length,
or, in other words, if the image of a word $\bb a$ by an injective morphism of constant length is a {\gpword}, then $\bb a$ itself is also a {\gpword}.
\begin{lemma}
Let $\bb a$ be an infinite word defined over a finite alphabet $\Sigma$ and let $k \in \NN$.
Consider the infinite word ${\bb a}'$ over $\Sigma^k$ given by $ a_n' = a_{kn} a_{kn+1} \cdots a_{kn+k-1}$.
Then ${\bb a}'$ is a {\gpword} if and only if $\bb a$ is a {\gpword}.
\end{lemma}
\begin{proof}
The proof is similar to the one of Lemma \ref{lem:cl:subs}.
\end{proof}
\subsection{Orbit closure}
Given an infinite word $\bb a$ defined over a finite alphabet $\Sigma$, we let $\cO(\bb a)$ denote the orbit of $\bb a$ under the shift,
that is, the set of all infinite words $ \bb a' $ given by $a'_n = a_{n+m}$ for some $m \in \NN_0$.
\begin{proposition}\label{prop:cl:orb-cl-0}
Let $\bb a$ be a {\gpword} and let $\bb a' \in \cO(\bb a)$. Then $\bb a'$ is a {\gpword}.
\end{proposition}
\begin{proof}
This follows from (i) of Corollary \ref{cor:cl:rearrange} in the special case where $A=1$.
\end{proof}
More generally, we can consider the orbit closure $\bar{\cO(\bb a)}$, that is, the closure of $\cO(\bb a)$ with respect to the product topology
on $\Sigma^{\infty}$. Explicitly, $\bb a'$ belongs to $\bar{\cO(\bb a)}$ if, for every $N \in \NN$, there exists $m$ such that $a'_n = a_{n+m}$ for all $n \in [N]$.
\begin{proposition}\label{prop:cl:orb-cl}
Let $\bb a$ be a {\gpword} and let $\bb a' \in \bar{\cO(\bb a)}$. Then $\bb a'$ is a {\gpword}.
\end{proposition}
\begin{proof}
This is a direct consequence of Lemmas \ref{lem:nil:shift} and \ref{lem:cl:limit} given in Appendix \ref{app:nil}.
\end{proof}
\begin{example}
Let $\bb a$ be the {\gpword} given by $a_n = \braif{ \fp{ \sqrt{2} n\fp{\sqrt{3} n}} < 1/4}$.
Then each $\bb a' \in \cO(\bb a)$ can be written in the form
\[ a'_n = \braif{ \fp{ (\sqrt{2} n+\a)\fp{\sqrt{3} n+\b} + \gamma n + \delta } < 1/4},\]
where $\a,\b,\gamma,\delta \in [0,1)$. Conversely, $\bar{\cO(\bb a)}$ contains all sequences $\bb a'$ of the aforementioned form,
as well as all sequences obtained from them by replacing any instances of $\fp{x}$ with $\fp{x}' = 1-\fp{-x}$,
or by replacing the strict inequality $<$ with $\leq$.
\end{example}
\subsection{{\gp} sets}\label{ssec:gp-sets}
The closure properties of {\gpwords} directly translate into closure properties of {\gp} subsets of $\NN_0$ thanks to Corollary \ref{cor:constr:fibre}.
In fact, we have already used this connection at several places. For ease of reference, we gather these properties here.
\begin{proposition}\label{prop:cl:sets}
The family of {\gp} subsets of $\NN_0$ is a field of sets. In other words, it contains the empty set and if $E,F \subset \NN_0$ are {\gp} sets
then so are $\NN_0 \setminus E$, $E \cup F$, and $E \cap F$.
\end{proposition}
\begin{proof}
The result follows from Proposition \ref{prop:cl:cases}.
\end{proof}
\begin{proposition}\label{prop:cl:sets-2}
Let $E \subset \NN_0$ be a {\gp} set. Then the following sets are also {\gp} sets.
\begin{itemize}
\item $m \cdot E = \set{nm}{n \in E}$, where $m \in \NN_0$.
\item $E / m = \set{n \in \NN_0}{ nm \in E}$, where $m \in \NN$.
\item $E - m = \set{n \in \NN_0}{ n+m \in E}$, where $m \in \ZZ$.
\end{itemize}
\end{proposition}
\begin{proof}
This is a consequence of Corollary \ref{cor:cl:rearrange}.
\end{proof}
\section{Frequencies and recurrence}\label{sec:facts}
In this section, we collect some facts concerning frequencies with which symbols and subwords appear in {\gpwords}, as well as additive structure of the set of positions at which a given symbol appears in a {\gpword}. Most of these facts are consequences of results concerning uniform
distribution and recurrence of orbits of point in nilsystems, and hence are related to the dynamical characterization of bracket
words discussed in Section \ref{sec:nil}. We point out that, similarly, many standard properties of Sturmian words can be inferred from dynamical properties of circle rotations.
\subsection{Uniform frequency}
Let $\Sigma$ be a finite alphabet. Given an infinite word $\bb a\in\Sigma^{\infty}$ and a letter $x \in \Sigma$, the (asymptotic)
\emph{frequency} of $x$ in $\bb a$ is defined as
\[ \freq(\bb a,x) = \lim_{N \to \infty} \abs{\set{n \in [N]}{a_n= x}}/N\,,\]
provided that the limit exists.
More generally, given a finite word $\fword{w} \in \Sigma^\ell$, the frequency of $\fword{w}$ in $\bb a$ is defined as
\[
\freq(\bb a, \fword{w}) = \lim_{N \to \infty} \abs{\set{n \in
[N]}{\bb a_{[n,n+\ell)} = \fword{w}}}/N\,,
\]
provided that the limit exists.
Here and elsewhere, $\bb a_{[n,n+\ell)}$ denotes the finite subword
$a_{n}a_{n+1}\cdots a_{n+\ell-1}$. We define also the \emph{recurrence function} of $\fword{w}$ in
$\bb a$ as the least length of a segment of $\bb a$ that is guaranteed to contain an occurrence of $\fword{w}$:
\[
\mathrm{rec}(\bb a, \fword{w}) = \inf \set{ r \in \NN }{
\parbox{17em}{
{for each } $m$ { there exists } $n \in [m,m+r)$ { such that } $\bb
a_{[n,n+\ell)} = \fword{w}$ }
} \in \NN \cup \{\infty\} \,.
\]
For {\gpwords}, we have a strong result about the existence of
frequencies, which is a rephrasing of \cite[Thm.{} B]{BergelsonLeibman-2007}.
\begin{theorem}\label{prop:facts:freq-exists}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$ and $\ell$
be a positive integer. Then for each $\fword{w} \in \Sigma^\ell$, the frequency $\freq(\bb
a, \fword{w})$ exists and, moreover, one has
\[
\abs{\set{n \in [M,M+N)}{\bb a_{[n,n+\ell)} = \fword{w}}}/N \to
\freq(\bb a, \fword{w})
\]
uniformly in $M$ as $N \to \infty$.
\end{theorem}
\begin{proof}
By definition, there exist a finitely valued \gp{} map $g:\NN_0 \to
\RR$ and a coding $c:g(\NN_0)\to \Sigma$ such that $a_n=c(g(n))$.
We deduce from \cite[Thm.{} B]{BergelsonLeibman-2007} that the desired
conclusion holds for the sequence $g(n)$.
The same conclusion remains true after applying a coding.
\end{proof}
\begin{corollary}\label{cor:facts:freq-exists}
For $\bb a$ and $\fword{w}$ as above, either $\freq(\bb a, \fword{w})
= 0$ or $\mathrm{rec}(\bb a, \fword{w}) < \infty$.
\end{corollary}
\begin{remark}
It can happen that $\freq(\bb a, \fword{w}) = 0$ but still
$\fword{w}$ appears in $\bb a$ infinitely often, see e.g.\ Example \ref{ex:ex:rec}. Hence, {\gpwords} are
not guaranteed to be uniformly recurrent\footnote{A sequence $\bb a$ is uniformly
recurrent if, for every finite word $\fword{w}$ which appears in $\bb a$,
there exists $\ell \in \NN$ such that $w$ appears in the
length-$\ell$ segment $\bb a|_{[n,n+\ell)}$ for all $n$.}.
\end{remark}
The same result can be stated in terms of densities of {\gp} sets.
Recall that the \emph{asymptotic density} of a set $E \subset \NN_0$
is defined by
\[ d(E) = \lim_{N\to\infty} \abs{E\cap [N]}/N\,,
\]
provided that the limit exists. In general, we let $\bar{d}(E)$ and
$\underline{d}(E)$ denote the upper and lower asymptotic densities,
obtained by replacing $\lim$ with $\limsup$ and $\liminf$
respectively. Additionally, we define the
\emph{upper and lower uniform (Banach) densities} by
$$d^*(E) = \limsup_{N\to\infty} \sup_{M} \abs{E\cap [M,M+N)}/N$$
and
$$
d_*(E) = \liminf_{N\to\infty} \inf_{M} \abs{E\cap [M,M+N)}/N \,.
$$
In general, we have the chain of inequalities:
\(
d_*(E) \leq \underline d(E) \leq \overline d(E) \leq d^*(E).
\)
\begin{theorem}\label{thm:facts:dens-exists}
Let $E$ be a {\gp} subset of $\NN_0$. Then $d(E)$ exists. Moreover,
$d^*(E) = d_*(E) = d(E)$.
\end{theorem}
\subsection{Distribution along subsequences}\label{ssec:facts:equidist}
\renewcommand{\t}{t}
Many of the desirable properties of bracket words are preserved under passing to subsequences. As a first instance of this principle, we
consider the simple case of polynomial subsequences.
\begin{proposition}\label{cor:facts:poly-freq-exists}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$, $p: \NN_0 \to \NN_0$ be a polynomial, and $\ell$ be a positive integer.
Then for each $\fword{w} \in \Sigma^\ell$,
the frequency $\freq \bra{ \bra{a_{p(n)}}_{n=0}^\infty, \fword{w}}$ exists.
\end{proposition}
\begin{proof}
Let $a_n=c(g(n))$ be a representation of $\bb a$, where $g: \NN_0 \to \RR$ is a \gp{} map. Then $g\cdot p : \NN_0 \to \NN_0$ is also a \gp{} map for \gp{} maps
are closed under composition, and $a_{p(n)}=c(g\circ p(n))$. The results follows now from Theorem \ref{prop:facts:freq-exists}.
\end{proof}
We point out that even in the simplest cases, passing to a subsequence
can alter frequencies with which symbols occur. For instance, if $\bb
a = 101010\cdots$ then $\freq(\bb a, 1) = 1/2$ but
$\freq((a_{2n})_{n=0}^\infty, 1) = 1$.
Next, let us consider distribution along the primes. In
\cite{GreenTao-2012-Mobius}, Green and Tao obtained quantitative
estimates on correlations between the M\"{o}bius function and
nilsequences, which have important consequences for the question at
hand, (cf.{} \cite[Sec.{} 5]{GreenTao-2012-Mobius}). Extending the
techniques developed by Green and Tao, and the connection between
{\gp} maps and nilsystems, Bergelson, H{\aa}land-Knutson and Son
\cite{BergelsonHalandSon-2020} showed that bounded {\gp} maps have
asymptotic distribution along the primes. The following result is
obtained by specialising \cite[Theorem 5.1]{BergelsonHalandSon-2020}
to finitely-valued sequences. Below, we let $p_n$ denote the $n$-th
prime. We also mention related work of Eisner \cite{Eisner-2019} concerning
convergence of ergodic averages along primes in nilsystems. In the case of Sturmian words, this result is essentially due to Davenport \cite{Davenport-1937}.
\begin{theorem}\label{thm:facts:prime-freq-exists}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$.
Then for each $x \in \Sigma$, the frequency
$\freq \bra{ \bra{a_{p_n}}_{n=0}^\infty, x}$ exists.
\end{theorem}
\begin{proof}
This follows from {\cite[Thm.\ 5.1]{BergelsonHalandSon-2020}} using the same argument as in the proof of Theorem \ref{prop:facts:freq-exists}.
\end{proof}
Considering $\bb a = 101010\cdots$ again, we see that the frequencies $\freq \bra{ \bra{a_{p_n}}_{n=0}^\infty, x}$ need not be equal to $\freq \bra{ \bb a, x}$.
One can also inquire if it possible to generalise Theorem \ref{thm:facts:prime-freq-exists} from frequencies of symbols $x \in \Sigma$ to frequencies of
words $w \in \Sigma^\ell$, $\ell \geq 2$. While it seems plausible that this generalisation is true, it is out of reach of the current techniques.
Indeed, already in the case where $\bb a = (n \bmod q)_{n=0}^\infty$ is a periodic word with period $q \geq 3$, such a generalisation would require
us to understand the asymptotic behaviour of
\begin{equation}\label{eq:358:1}
\frac{1}{N} \abs{ \set{n < N}{ p_n \equiv w_1, p_{n+1} \equiv w_2, \dots, p_{n+\ell-1} = w_{n+\ell-1} \bmod{q}}}.
\end{equation}
It is conjectured that the expression in \eqref{eq:358:1} converges to $1/\varphi(q)^\ell$, where $\varphi$ denotes the totient function; in fact, the Main Conjecture in \cite{LemkeSoundararajan-2016} gives a more precise asymptotic expression. However, such estimate remains unknown for any $\ell \geq 2$, as discussed in \cite{LemkeSoundararajan-2016}.
Lastly, we consider a class of subsequences which preserve the frequencies of symbols.
As already alluded to earlier, Theorem \ref{thm:BL-mini} leads to a close connection between
equidistribution results for nilsystems and statements about frequencies of symbols in {\gpwords}. Recall
that each nilmanifold $X$ comes equipped with the Haar measure $\mu_{X}$. A sequence $(x_n)_{n=0}^\infty$ is
\emph{equidistributed} in $X$ if for each continuous map
$f \colon X \to \RR$ we have
\begin{equation}\label{eq: equidistributed}
\lim_{N \to \infty} \frac{1}{N} \sum_{n = 0}^{N-1} f(x_n) = \int_{X}
f d\mu_{X}\,.
\end{equation}
\begin{definition}\label{def:good-equidist}
Let $\t \colon \NN_0 \to \NN_0$. We say that $t$ is \emph{good for
equidistribution in nilsystems} if,
for all minimal nilsystems $(X,T)$ and all $x \in X$,
the sequence $(T^{\t(n)}(x))_{n=0}^\infty$ is equidistributed in $X$.
\end{definition}
\begin{proposition}\label{prop:nil:equidistribution}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$, and let
$t \colon \NN_0 \to \NN_0$ be good for equidistribution in nilsystems.
Then the frequencies of letters in $\bb a$ along the
subsequence $t(n)$ exist and remain the same as in $\bb a$:
\[
\freq\bra{ \bra{a_{t(n)}}_{n=0}^\infty, x} = \freq(\bb a,x) \;\;\quad
\forall x \in \Sigma\,.
\]
\end{proposition}
\begin{proof}
This is a consequence of the representation of $\bb a$ coming from Theorem \ref{thm:BL-word}.
Indeed, though the indicator functions ${\bb 1}_{S_i}$ of the
semialgebraic sets $S_i$ from Theorem \ref{thm:BL-word} are not
continuous, they can be efficiently approximated by continuous functions to which Equality \eqref{eq: equidistributed} can be applied.
\end{proof}
Hopefully, the following result of Frantzikinakis
\cite{Frantzikinakis-2009} provides a plentiful source of sequences
that satisfy Definition
\ref{def:good-equidist} and make Proposition
\ref{prop:nil:equidistribution} relevant.
\begin{theorem}[{\cite[Thm.\ 1.1]{Frantzikinakis-2009}}]\label{thm:Fran}
Let us assume that $f \colon \RR_{>0} \to \RR_{>0}$ satisfies the
following properties.
\begin{enumerate}
\item\label{it:574:1} The function $f$ belongs to some Hardy field.
\item\label{it:574:2} There exists $C > 0$ such that ${f(x)} = O(x^C)$ as $x \to \infty$.
\item\label{it:574:3} For every $p \in \ZZ[x]$ and $c \in \RR$, we have
\(\displaystyle \lim_{x \to \infty} \frac{\abs{ f(x) - c p(x)}}{\log
x} = \infty.\)
\end{enumerate}
Then $\ip{f}$ is good for equidistribution in nilsystems.
\end{theorem}
For a definition of a Hardy field, we refer the reader to
\cite{Frantzikinakis-2009}. Here, we just point out that one example of a Hardy field
is the logarithmic-exponential functions, that is, real-valued
functions defined on some interval $[c,\infty)$, $c \in \RR$, that can be constructed from polynomials using addition, multiplication, and the functions
$\exp$ and $\log$. Hence, the following formulae define sequences that
are good for equidistribution in nilsystems: $\ip{n^{3/2}}$, $\ip{n^{2}
\log^2 n}$, $\ip{n^{2} + \log^2 n}$, and $\ip{n^{2} + \sqrt{2}n}$.
Conversely, because of item \ref{it:574:3}, there is no guarantee that
the same should apply to $\ip{n^2 + \log n}$ or $\ip{\sqrt{2}n}$.
In fact, one can compute that
\[
\frac{1}{\sqrt{2}} \ip{\sqrt{2}n} = n + \frac{\fp{\sqrt{2}n} }{\sqrt{2}}
\equiv \frac{\fp{\sqrt{2}n} }{\sqrt{2}} \bmod{1},
\]
so $\ip{\sqrt{2}n}$ is not good for equidistribution for the circle
rotation by ${1}/{\sqrt{2}}$.
Extensions of some results in \cite{Frantzikinakis-2009} were
recently obtained by Richter \cite{Richter-2020}.
It is not possible to directly generalise Proposition \ref{prop:nil:equidistribution} to frequencies of words instead of symbols.
For instance, the sequence $t(n) = \ip{\sqrt{n}}$ is good for equidistribution in nilsystems and constant on each interval $[N^2,(N+1)^2)$, $N \in \NN$.
Hence, already for the periodic word $\bb a = 101010\cdots$ and $w \in \{0,1\}^2$ we have
\[
\freq\bra{ (a_{t(n)})_{n=0}^\infty, w } =
\begin{cases}
1/2 &\text{if } w \in \{00,11\},\\
0 & \text{if } w \in \{01,10\}.
\end{cases}
\]
Thus, the frequencies of words in $(a_{t(n)})_{n=0}^\infty$ bear little resemblance to the frequencies of symbols and words in $\bb a$.
On the other hand, under additional growth conditions, one can obtain positive results. The key technical component is the following theorem of Bergelson, Moreira and Richter. Below, we let $f^{(j)}$ denote the $j$-th derivative of $f$.
\begin{theorem}[{\cite[Thm. 5.6, special case]{BergelsonRichterMoreira-2020}}]\label{thm:BMR}
Let $(X,T)$ be a minimal nilsystem, let $z \in X$, and let $f_1,f_2,\dots,f_k$ be functions belonging to the same Hardy field. Suppose further that the following two conditions hold.
\begin{enumerate}
\item\label{it:BRM:1} For each $h \in \spanlin\set{ f_i^{(j)}}{1 \leq i \leq k, j \geq 0}$ and each $p(x) \in \RR[x]$, we have either $\abs{h(x) - p(x)} = O(1)$ or $\abs{h(x) - p(x)}/\log x \to \infty$ as $x \to \infty$.
\item\label{it:BRM:2} For each non-zero $h \in \spanlin\set{ f_i }{ 1 \leq i \leq k}$ and each $p(x) \in \RR[x]$, we have $\abs{h(x) - p(x)} \to \infty$ as $x \to \infty$.
\end{enumerate}
Then $\bra{\bra{T^{\ip{f_1(n}}(z),T^{\ip{f_2(n}}(z),\dots, T^{\ip{f_k(n}}(z) }}_{n=0}^\infty$ is equidistributed in $X^k$.
\end{theorem}
Let now $f$ be a function belonging to a Hardy field, and put $t = \ip{f}$ and $f_i(x) = f(x+i)$ for every $i$ such that $1 \leq i \leq k$. Using standard techniques, such as the Taylor expansion and estimates on derivatives of functions in Hardy fields \cite[Lem.\ 2.1]{Frantzikinakis-2009}, one can verify that conditions \ref{thm:BMR}.\ref{it:BRM:1} and \ref{thm:BMR}.\ref{it:BRM:2} are satisfied if
\begin{align*}
\frac{f(x)}{x^k \log x} &\to \infty & \text{and} && \frac{f(x)}{x^{k+1}} &\to 0 & \text{as }&& x \to \infty.
\end{align*}
Combining the conclusion of Theorem \ref{thm:BMR} with Theorem \ref{thm:BL-word}, one could show that for each {\gpword} $\bb a$ and each word $w \in \Sigma^k$ we have
\begin{equation}\label{eq:265:1}
\freq\bra{ \bra{a_{t(n)}}_{n=0}^\infty, w} = \prod_{i=0}^{k-1} \freq\bra{ \bb a, w_i}\,.
\end{equation}
We emphasise that the frequencies in \eqref{eq:265:1} are, in general, not equal to $\freq(\bb a,w)$; rather, they are the values which one would expect if $t$ exhibited random-like behaviour. We leave the details to the interested reader.
\subsection{IP recurrence}
Apart from asymptotic density, one can also inquire into combinatorial
richness of {\gp} subsets of $\NN$.
In order to state the relevant results, we introduce some terminology.
A set $E \subset \NN$ is an \emph{$\IP$-set} if it contains all finite
sums of a sequence of positive integers, that is, if there exists a
sequence of positive integers $(n_i)_{i=1}^\infty$ such that
$\sum_{i \in I} n_i \in E$ for all finite set $I \subset \NN$.
A set $F \subset \NN$ is $\IP^*$ if $E \cap F \neq \emptyset$ for
every $\IP$-set $E$, or, equivalently, if for every sequence of positive integers
$(n_i)_{i=1}^\infty$ there exists a finite set $I \subset \NN$ with
$\sum_{i \in I} n_i \in F$. We also define shifted variants of these
notions. A set $E \subset \NN$ is $\IP_+$ if $(E + n) \cap \NN$ is an $\IP$-set
for some $n \in \ZZ$.
Likewise, a set $F \subset \NN$ is $\IP^*_+$ if $(F + n) \cap \NN$ is
an $\IP^*$-set for some $n \in \ZZ$.
Each $\IP^*$-set is \emph{syndetic}, meaning that for every
$\IP^*$-set $F$ there exists an integer $N$ such that $F \cap [n,n+N)
\neq \emptyset$
for all $n \in \NN_0$. In particular, all $\IP^*$-sets have positive
lower uniform density.
Moreover, since for all $m \in \NN$, the multiples of $m$ form an
$\IP$-set, all $\IP^*$-sets contain a multiple of $m$.
The class of $\IP^*$-sets is closed under intersection, meaning that
for all pairs $F,F' \subset \NN$ of $\IP^*$-sets, $F \cap F'$ is also
an $\IP^*$-set.
In what follows, we say that a statement $\varphi(n)$ holds for
\emph{almost all} $n \in \NN_0$ if $d\bra{\set{n \in \NN_0}{\neg \varphi(n)}} = 0$.
\begin{theorem}[{\cite[Thm.{}D]{BergelsonLeibman-2007}}]\label{thm:nil:IP*-rec}
Let $\bb a$ be a {\gpword} over $\Sigma$. Then for almost all $n_0
\in \NN_0$ and all sequences of positive integers $(n_i)_{i=1}^\infty$,
there exists a finite set $I \subset \NN$ such that $a_{ n_0 +
\sum_{i \in I} n_i } = a_{n_0}$.
In other words, if $E \subset \NN$ is a {\gp} set with $d(E) > 0$,
then $E$ is $\IP^*_+$. Furthermore, $(E-n) \cap \NN$ is $\IP^*$ for
almost all $n \in E$.
\end{theorem}
\begin{remark}
For Sturmian words, a much simpler proof is possible, see e.g.\ a special case of \cite[Cor.\ 7.3]{Bergelson-2010}.
\end{remark}
\begin{remark}
One can obtain more precise versions of this result. Specifically, it
follows from \cite{BergelsonLeibman-2018} that one
can additionally require that $I \subset [r]$ for some integer $r$
that only depends on $\bb a$ and $n_0$.
In a different direction, it follows from \cite{Konieczny-2017-IJM}
that one can additionally require that the gaps between consecutive
elements of $I$ are bounded from above by some integer $d$ that only
depends on $\bb a$.
\end{remark}
\section{Counting function and discrepancy}\label{sec:growth}
\newcommand{\discr}{\Delta}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$.
In addition to the asymptotic frequency $\freq(\bb a, x)$ with which a symbol $x \in \Sigma$ appears in $\bb a$,
one can inquire into more quantitative estimates on the count of occurrences of $x$ in $\bb a$. This leads us to consider the
counting function
\begin{equation}\label{eq:growth:freq}
\cnt(\bb a, x; N) = \abs{\set{ n \in [N]}{ a_n = x}} \,,
\end{equation}
as well as the closely related notion of \emph{symbolic discrepancy} considered in \cite{Adamczewski-2004},
\begin{equation}\label{eq:growth:discr}
\discr(\bb a; N) = \max_{x \in \Sigma}\absBig{ \cnt(\bb a, x; N) -N \freq(\bb a,x)} \,.
\end{equation}
For example, if $\bb a$ is the Sturmian word given by $a_n = \ip{\a n + \b} - \ip{\a (n-1)+\b}$, then we have the estimates
\begin{align}\label{eq:growth:Sturm}
\cnt(\bb a, 1; N) &= \a N + O(1) && \text{and}& \discr(\bb a, N) = O(1)\,,
\end{align}
which are significantly stronger than $\freq(\bb a, 1) = \a$.
In this section, our aim is to study the possible rates of growth of $\cnt(\bb a,x;N)$ and $\discr(\bb a; N)$ when $\bb a$ is a bracket word. We note that, as a direct consequence of the definitions, for each $x \in \Sigma$ and $N \in \NN$ we have
\[ 0\leq \cnt(\bb a, x; N),\ \discr(\bb a;N) \leq N\,.\]
Since frequencies of all symbols exist, we also have
\[
\cnt(\bb a, x; N)/N \to \freq(\bb a,x) \text{ and } \discr(\bb a;N)/N \to 0 \ \text{as } N \to \infty\,.
\]
This quantitative approach is especially relevant in the case where $\freq(\bb a, x) = 0$. We point out that if there exists $z \in \Sigma$ with $\freq(\bb a,z) = 1$ then
\[ \discr(\bb a; N) = N - \cnt(\bb a,z ;N) = \sum_{x \in \Sigma \setminus \{z\}}\cnt(\bb a,x ;N) \,.\]
For instance, if we let $F=\{0,1,2,3,5,8,\ldots\}$ denote the set of Fibonacci numbers, $\varphi = (1+\sqrt 5)/2$ be the golden ratio, and
$E = \set{ n \in \NN_0}{ \fpa{\varphi n} < 1/\sqrt{n}}$,
then the frequencies $\freq(\bb{1}_{E}, 1)$ and $\freq(\bb{1}_F,1)$ are both zero, but
\[ \cnt(\bb{1}_F,1;N) = \discr(\bb{1}_F;N) = \Theta(\log N)\] grows much more
slowly (cf.{} Proposition \ref{prop:growth-N^lambda}) than
\[ \cnt(\bb{1}_E,1;N) = \discr(\bb{1}_E;N) =\Theta(\sqrt{N})\,.\]
For context, we mention that similar questions have been studied for other classes of sequences of combinatorial interest. For instance, it is well-known (see e.g.\ \cite[Sec.\ 1.2]{Rigo-book-2}) that for each $k$-automatic word $\bb a$ over a finite alphabet $\Sigma$ and each $x \in \Sigma$, the count of integers $n$ with $a_n = x$ whose base-$k$ expansion has length $L \geq 1$, $\cnt(\bb a, x; k^{L}) - \cnt(\bb a, x; k^{L-1})$, satisfies a linear recurrence in $L$. Thus, either $\cnt(\bb a,x;N) = \Theta(\log^d N)$ for some $d \in \NN_0$ or $\cnt(\bb a,x; N) = \Omega(N^c)$ for some $c \in (0,1]$ \cite[Thm.\ 1.48]{Rigo-book-2}. In particular, if $\cnt(\bb a,x; N) = N^{o(1)}$ then $\cnt(\bb a,x; N) = \log^{O(1)}N$. Another convenient consequence is that if $\freq(\bb a, x) = 0$ then $\cnt(\bb a, x;N) = O(N^{c})$ for some $c \in (0,1)$.
\subsection{Linear growth and low discrepancy}
As we have already observed, the rate of growth of $\cnt(\bb a,x;N)$ can be linear and $\discr(\bb a;N)$ can be bounded. In this context, it is helpful to recall the notion of a $C$-balanced word. An infinite word $\bb a$ over an alphabet $\Sigma$ is \emph{$C$-balanced} if the number of occurrences of a given symbol in any two subwords of $\bb a$ of equal length differs by at most $C$.
For instance, as we have already pointed out in the introduction, Sturmian words are $1$-balanced. It was shown in \cite[Prop.\ 7]{Adamczewski-2003} that an infinite word $\bb a$ is $C$-balanced for a finite value $C > 0$ if and only the discrepancy $\discr(\bb a; N)$ is bounded as $N \to \infty$.
\begin{lemma}\label{lem:growth:linear}
For every $\a \in (0,1]$, there exists a {\gpword} over $\{0,1\}$ with $\cnt(\bb a, 1; N) = \a N + O(1)$, and thus $\freq(\bb a, 1) = \a$ and $\discr(\bb a; N) = O(1)$.
\end{lemma}
\begin{proof}
Follows immediately from \eqref{eq:growth:Sturm}.
\end{proof}
The example above stems from the fact that $[0,\alpha)$ is a bounded remainder set for the rotation by $\a$ on the torus $\RR/\ZZ$. In general, given $d \in \NN$ and $\a = (\a_1,\a_2,\dots,\a_d)$ a measurable set $S \subset \RR^d/\ZZ^d$ is a \emph{bounded remainder set} for the rotation by $\a$ on $\RR^d/\ZZ^d$ if for almost all $x \in \RR^d/\ZZ^d$, the \emph{discrepancy}
\begin{equation*}
\Delta(S,x;N) = \abs{ \sum_{n = 0}^{N-1}1_S(x + n \a) - \lambda(S)N }
\end{equation*}
is bounded by a constant $C(S,\a)$ independent of $x$ and $N$. (Above, $\lambda$ denotes the Lebesgue measure.) A complete classification of Riemann measurable bounded remainder sets for rotations on tori was obtained in \cite{GrepstadLev-2015}. In particular, by \cite[Thm.\ 1]{GrepstadLev-2015}, if $v_1,v_2,\dots,v_d \in \ZZ \a + \ZZ^d$ are linearly independent vectors such that the parallelepiped
\[
P = \set{ \sum_{i=1}^d t_i v_i }{ t_1,t_2,\dots,t_d \in [0,1)}
\]
is contained in $[0,1)^d$ then $P$ (or, strictly speaking, its projection to $\RR^d/\ZZ^d$) is a bounded remainder set.
\begin{example}\label{ex:growth:BRS}
Let $d \in \NN$, let $\a = (\a_1,\a_2,\dots,\a_d) \in \RR^d$ be a $d$-tuple of real numbers such that $1,\a_1,\a_2,\dots,\a_d$ are linearly independent over $\QQ$, and let $S \subset \RR^d/\ZZ^d$ be a bounded remainder set for the rotation by $\a$ on $\RR^d/\ZZ^d$ which is also a semialgebraic set. Then $\bb a = \bra{1_S(n\a)}_{n=0}^\infty$ is a bracket word with $\freq(\bb a, 1) = \lambda(S)$ and $\discr(\bb a; N) = O(1)$ as $N \to \infty$.
\end{example}
A different way to generalise Lemma \ref{lem:growth:linear} is to consider iterated discrete derivatives of polynomials of higher degrees.
\begin{example}\label{ex:growth:diff-quad}
Let $\a \in (0,1)$ and let $\bb a$ be the {\gpword} given by
\begin{align*}
a_n &= \ip{\a (n+2)^2} - 2 \ip{\a(n+1)^2} + \ip{\a n^2} + 1
\\ &= - \fp{\a (n+2)^2} + 2 \fp{\a(n+1)^2} - \fp{\a n^2} + 1+2\a.
\end{align*}
It follows directly from the formula above that $a_n \in \ZZ$ and $-1 < a_n < 5$ for all $n \in \NN_0$, so $\bb a$ is a word over the alphabet $\{0,1,2,3,4\}$. Consider the word $\bb a'$ over the alphabet $\{0,1\}$ given by
\[
a_{4n+i}' =
\begin{cases}
1 & \text{if } a_n > i, \\
0 & \text{otherwise.}
\end{cases}
\qquad \text{for } n \in \NN_0,\ i \in \{0,1,2,3\} \,.
\]
Then we can compute that
\begin{align*}
\cnt(\bb a',1;N) &= \sum_{n=0}^{N-1} a_n' = \sum_{n=0}^{N/4-1} a_n + O(1)
\\ &= N/4 + \ip{\a(N/4+1)^2} - \ip{\a(N/4)^2} + O(1) = (1/4+\a/8)N + O(1).
\end{align*}
Hence, $\cnt(\bb a',1;N) = (1/4+\a/8)N + O(1)$ and $\discr(\bb a';N) = O(1)$.
\end{example}
We recall the words in Examples \ref{ex:growth:BRS} and \ref{ex:growth:diff-quad} are $C$-balanced for some finite constant $C$.
Thanks to Theorem \ref{thm:BL-word}, one could use Example \ref{ex:growth:diff-quad} to construct an explicit example of a bounded remainder set for a certain $2$-step nilsystem.
\subsection{Slow growth and small discrepancy}
At the other extreme in terms of the possible rates of growth of $\cnt(\bb a, x; N)$, it is possible for $\cnt(\bb a, x; N)$ to tend to $\infty$ arbitrarily slowly. In the case where $\abs{\Sigma} = 2$, $\discr(\bb a; N) = \cnt(\bb a, x; N)$ tends to $\infty$ at the same rate.
As a first example, we mention a result \cite[Theorem C]{ByszewskiKonieczny-2018-TAMS}
asserting that any ``sufficiently sparse'' subset of $\NN_0$ is a {\gp} set.
\begin{proposition}[{\cite[Thm. C]{ByszewskiKonieczny-2018-TAMS}}]\label{prop:constr:super-sparse}
Let $E = \set{n_i}{ i \in \NN_0} \subset \NN_{\geq 2}$ be a set with
\[
\liminf_{i \to \infty} \frac{\log n_{i+1}}{\log n_i} > 1.
\]
Then ${\bb 1}_E$ is a {\gpword}.
\end{proposition}
The proof of Proposition \ref{prop:constr:super-sparse} relies on an application of Proposition \ref{prop:cl:g-in-I} for a {\gp} map of the form
$g(n) = \braif{ \fpa{\a n} < 1/n^d}$ with large $d \in \NN$ and a carefully constructed $\a \in \RR$.
Of course, this result provides examples of bracket words
over $\{0,1\}$ for which the growth of $\cnt(\bb a, 1; N)$ tends to infinity as slowly as wanted.
Now, we prove the following general result that also covers growth order of type $\log N$.
\begin{proposition}\label{prop:growth:sub-log}
Let $f \colon \NN_0 \to \RR_{>0}$ be a non-decreasing map and assume that there exists a positive real number $C$ such that
$f(2n) \leq f(n) + C$ for all $n \in \NN$. Then there exists a {\gpword} $\bb a$ over $\{0,1\}$ such that $\cnt(\bb a,1;N) = f(N) + O(1)$.
\end{proposition}
\begin{proof}
Let $F$ be the set of all Fibonacci numbers, and let $F' = F+[C]$. Note that for all sufficiently large $n \in \NN$, we have $\abs{F' \cap [n,2n)} \geq C$.
In \cite{ByszewskiKonieczny-upcoming}, it is shown that any subset $E$ of $F$ is a {\gp} set, and hence the same also applies to each $E \subset F'$.
One can inductively construct a set $E \subset F'$ such that $\abs{E \cap [N]} = f(N)+ O(1)$ for all $N$: for each $n \in F'$, assuming that $E \cap [n]$
has already been constructed, we let $n \in E$ if and only if $f(n) > \abs{E \cap [n]}$. It remains to set $\bb a = \bb{1}_E$.
\end{proof}
\begin{example}
For every $\a \in \RR_{>0}$, there is a {\gpword} $\bb a$ over $\{0,1\}$ such that $\cnt(\bb a,1;N) = \a \log N + O(1)$.
\end{example}
For the sake of completeness, we mention that for even slower rates of growth, we can reformulate \cite[{Thm.{} C}]{ByszewskiKonieczny-2018-TAMS}
to obtain a slightly more precise result which does not involve an error term.
\begin{proposition}
Let $f \colon \RR_{>0} \to \RR_{>0}$ be a continuously differentiable increasing function with $\sup_x\abs{f'(x)} < \infty$.
Then there exists a {\gpword} $\bb a$ over $\{0,1\}$ with $\cnt(\bb a,1;N) = \floor{ f(\log \log N) }$ for all sufficiently large $N$.
\end{proposition}
\begin{proof}
Since $f(\log \log x)$ is increasing in $x$ and its derivative tends to $0$ as $x \to \infty$, we can construct a set $E = \{m_1 < m_2 < \cdots \}$ with $\abs{E \cap [N]} = \floor{ f(\log \log N) }$ for all sufficiently large $N$. Thus, for sufficiently large $n$ we have
\[ f(\log \log m_n) < n \leq f(\log \log (m_n+1))\,.\]
As a consequence, we find that
\[
f^{-1}(n- 1/2) \leq \log \log m_n \leq f^{-1}(n)\,.
\]
Applying this bound to $n$ and $n+1$, we obtain
\[
\frac{ \log m_{n+1} }{\log m_{n}} \geq \exp\bra{f^{-1}(n+1/2) - f^{-1}(n) }\,.
\]
Since $f'(x)$ is bounded as $x \to \infty$, we have
\[
\liminf_{n \to \infty} \bra{ f^{-1}(n+1/2) - f^{-1}(n) } > 0\,,
\]
and consequently
\[
\liminf_{n \to \infty} \frac{ \log m_{n+1} }{\log m_{n}} > 1\,.
\]
Thus, $\bb{1}_E$ is a {\gpword} by Proposition \ref{prop:constr:super-sparse}.
\end{proof}
\begin{example}
There are {\gpwords} $\bb a$ over $\{0,1\}$ such that $\cnt(\bb a,1;N)$ is any of the following: $\floor{ \a \log \log N}$
for any $\a \in \RR_{> 0}$, $\floor{ (\log \log N)^c}$ for any $c \in [0,1)$, $\floor{ \log \log \log N}$, and so on.
\end{example}
\subsection{Moderate behaviours}
Bearing in mind the examples mentioned above, we are left with the question of determining which rates of growth
between linear and logarithmic are possible for $\cnt(\bb a,x;N)$. Towards this end, we will make use of results concerning the asymptotic behaviour of Diophantine expressions of the form $\prod_{i=1}^d \fpa{\a_i n}$, which have received considerable attention in connection with the Littlewood conjecture (see Example \ref{ex: littlewood}). While the conjecture remains unsolved, one can obtain considerably more precise estimates for a generic choice of $(\a_1,\a_2,\dots,\a_d)$. For future reference, let us define the set
\[
E(\lambda,\a) = \set{n \in \NN}{ \prod_{i=1}^d \fpa{n\a_i} < n^{-1+\lambda} },
\]
where $d \in \NN$, $\lambda \in [0,1]$, and $\a = (\a_1,\a_2,\dots,\a_d) \in \RR^d$.
Note that $E(\lambda,\a)$ is a {\gp} set if $\lambda \in \QQ$.
\begin{proposition}\label{prop:growth-log^dN}
For every $\lambda \in [0,1) \cap \QQ$ and $c \in \NN_0$, there exists a {\gpword} $\bb a$ over $\{0,1\}$ with $\cnt(\bb a, 1; N) = \Theta(N^\lambda \log^c N)$.
\end{proposition}
\begin{proof}
It follows from a variant of the main result in \cite{WangYu-1981}, as cited in \cite[Thm.{} 1.6]{ChowTechnau-2022} and specialised to $\psi(n) = n^{-1+\lambda}$,
that for each $d \in \NN$ and almost all $\a \in \RR^d$, we have the asymptotic formula
\begin{equation}\label{eq:WY-001}
\abs{E(\lambda,\a) \cap [N]} =
\begin{cases}
\Theta(x^{\lambda} \log^{d-1} x) &\text{if } \lambda \neq 0,\\
\Theta(\log^{d} x) &\text{if } \lambda = 0.
\end{cases}
\end{equation}
If $\lambda = c = 0$ then one can simply take $\bb a = \bb 1_{\{0\}}$, so suppose this is not the case. Let $d = c+1$ if $\lambda \neq 0$ and $d = c$ if $\lambda = 0$. It follows from \eqref{eq:WY-001} that for almost all $\a \in \RR^d$ we may take $\bb a = \bb 1_{E(\lambda,\a)}$.
\end{proof}
Unfortunately, the main result in \cite{WangYu-1981} does not provide any explicit example of $\a \in \RR^d$ for which \eqref{eq:WY-001} holds, and as a consequence our proof of Proposition \ref{prop:growth-log^dN} does not provide any explicit example of {\gpwords} $\bb a$ such that $\cnt(\bb a, 1; N)$ has prescribed asymptotic behaviour. However, such examples can be constructed in the special case where (using notation from Proposition \ref{prop:growth-log^dN}) $d = 0$. Below, we let $\varphi$ denote the golden ratio.
\begin{proposition}\label{prop:growth-N^lambda}
Let $\lambda \in (0,1) \cap \QQ$, and let $\bb a = \bb 1_{E}$, where
\[
E = \set{ n \in \NN }{ \fpa{n\varphi} \leq n^{-1+\lambda}} \,.
\]
Then $\cnt(\bb a, 1; N) = \Theta(N^\lambda)$.
\end{proposition}
\begin{proof}
Recall that every positive integer $n$ has a unique expansion
\[
n = \sum_{i=2}^\infty \epsilon_i(n) f_i \,,
\]
where $f_i$ is the $i$-th Fibonacci number, $\epsilon_i(n) \in \{0,1\}$ and $(\epsilon_i(n),\epsilon_{i+1}(n)) \neq (1,1)$ for all $i \geq 2$.
Let $\nu(n)$ denote the least index $i \geq 1$ such that $\epsilon_{i+1}(n) = 1$, and let $\mu(n) = - \log_{\varphi} \fpa{n \varphi}$.
It follows from \cite[Lemma 1]{DrmotaMullerSpiegelhofer-2018} that the difference between $\nu(n)$ and $\mu(n)$ is bounded; in fact,
$\abs{\mu(n) - \nu(n) } \leq 3$ for all $n \in \NN$.
It is a standard exercise to show that for each $\ell \in \NN$, the number of sequences in $\{0,1\}^\ell$ with no pair of consecutive
$1$s is $f_{\ell+2} = \Theta(\varphi^\ell)$. As a consequence, for each sufficiently large $\ell \in \NN$ and each $c \in (-10,10)$, we have
\begin{equation}\label{eq:226:1}
\abs{ \set{ n \in [f_{\ell-1}, f_{\ell}) }{ \nu(n) \geq (1-\lambda) \ell + c} } = \Theta\bra{ \varphi^{\lambda \ell}}\,.
\end{equation}
Since $\abs{\mu(n) - \nu(n) } \leq 3$ and $\abs{\log_{\varphi} n - \ell} < 2$ for all $n \in [f_{\ell-1}, f_{\ell})$,
it follows that
\begin{equation}\label{eq:226:2}
\abs{ \set{ n \in [f_{\ell-1}, f_{\ell}) }{ \mu(n) \geq (1-\lambda) \log_{\varphi} n } } = \Theta\bra{ \varphi^{\lambda \ell}}\,.
\end{equation}
Let $N$ be a large integer, and let $i$ be the index with $f_{i-1} \leq N < f_{i}$. Summing \eqref{eq:226:2} over all $\ell \leq i$ (resp.{} $\ell \leq i-1$),
we conclude that
\[ \abs{E \cap [N]} = \abs{ \set{ n \in [N] }{ \mu(n) \geq (1-\lambda) \log_{\varphi} n } } = \Theta\bra{ \varphi^{\lambda i}}=\Theta\bra{ N^{\lambda}}\, . \qedhere
\]
\end{proof}
In general, it is not clear how to construct $\a \in \RR^d$ which satisfy \eqref{eq:WY-001} for $d \geq 2$. However, some progress in that direction follows from recent work of Chow and Technau. For instance, it follows as a special case of \cite[Theorem 1.9]{ChowTechnau-2022}, that for each $\lambda \in (0,1)$ and each $\alpha_1 \in \RR$ which is not a Liouville number (\textit{i.e.}, $\fpa{n\a_1}\gg 1/n^C$ for some constant $C$), for almost all $\alpha_2$ we have the we have the one-sided variant of \eqref{eq:WY-001},
\[
\abs{E(\lambda,(\a_1,\a_2)) \cap [N]} \gg N^{\lambda} \log N\,.
\]
\subsection{Almost linear growth}
It might come as a surprise that there exist {\gpwords} $\bb a$ over $\{0,1\}$ such that the growth rate $\cnt(\bb a,1; N)$ is slower than,
but arbitrarily close to, linear. This is closely related to a classical result of Khinchine asserting that there exist pairs $\a_1,\a_2 \in \RR$,
with $1,\a_1,\a_2$ linearly independent over $\QQ$, such that $\min_{\abs{n} < N} \fpa{n_1\a_1 + n_2\a_2}$ tends to $0$ arbitrarily fast
as $N \to \infty$, see e.g. \cite[Ch. V, Thm. XIV]{Cassels-book}. We will need the following elementary lemma.
\begin{lemma}\label{lem:exist-swell-approx}
Let $(N_i)_{i = 1}^\infty$ be a sequence of positive integers and $(\e_i)_{i = 1}^\infty$ be a sequence with values in $(0,1)$ such that
\[ N_{i+1} \geq 2 N_i/ \e_i\, .\]
Then there exists $\a \in \RR \setminus \QQ$ with $\fpa{N_i\a} \leq \e_i$ for all $i \in \NN$.
\end{lemma}
\begin{proof}
For every $j \in \NN$, we set
\[
A_j = \set{ \a \in \RR/\ZZ }{ \fpa{N_i\a} \leq \e_i \text{ for all } 1 \leq i \leq j}.
\]
It will suffice to show that the set $A= \bigcap_{j=1}^\infty A_j$ is uncountable.
Note that, for each $j \in \NN$, $A_{j+1}$ is the intersection of $A_{j}$ with the periodic set $\set{ \a \in \RR/\ZZ }{ \fpa{N_{j+1}\a} \leq \e_{j+1}}$, which is the union of the intervals
\[
I_{j+1}^{(m)} = \left[ \frac{m-\e_{j+1}}{N_{j+1}}, \frac{m+\e_{j+1}}{N_{j+1}} \right], \qquad m \in [N_{j+1}].
\]
This motivates us to recursively define a family of sets $A_j'$ by $A_1' = A_1$ and
\[
A_{j+1}' = \bigcup \set{I_{j+1}^{(m)} }{ m \in [N_{j+1}] \text{ and }I_{j+1}^{(m)} \subset A_{j}' }.
\]
A routine inductive argument shows that $A'_j \subset A_j$ for each $j \in \NN$, and it follows directly from the definition that $A'_j$ is a union of intervals with lengths ${2\e_{j}}/{N_j}$. Set $A'= \bigcap_{j=1}^\infty A_j' \subset A$.
Since ${2\e_{j}}/{N_i} \geq 4/N_{j+1}$, for each $m \in [N_j]$ there exists $m' \in [N_{j+1}]$ such that
\begin{equation}\label{eq:294:1}
I_{j+1}^{(m')}, I_{j+1}^{(m'+1)} \subset \left[ \frac{m'-1}{N_{j+1}}, \frac{m'+2}{N_{j+1}} \right] \subset I_{j}^{(m)}.
\end{equation}
Applying this observation inductively and using Cantor's intersection theorem, we conclude that $I_j^{(m)} \cap A' \neq \emptyset$ for each $j \in \NN$ and $m \in [N_j]$ such that $I_j^{(m)} \subset A_j$. In fact, since the left hand side of \eqref{eq:294:1} includes two disjoint intervals, we can use a similar argument to produce an injective map from $\{0,1\}^\NN$ to $A'$. Thus, $A'$ is uncountable, and so is $A$.
\end{proof}
\begin{proposition}\label{prop:growth-slow-o(N)}
Let $f \colon \NN \to (0,1]$ be any function with $f(N) \to 0$ as $N \to \infty$. There exists a {\gpword} $\bb a$ over $\{0,1\}$ such that $\freq(\bb a, 1) = 0$
but $\cnt(\bb a,1;N) = \discr(\bb a; N) \geq f(N)N$ for all $N \in \NN$.
\end{proposition}
\begin{proof}
We may assume without loss of generality that $f(n) \geq 1/n$ for all $n \in \NN$ and that $f$ is non-increasing; if this is not the case, we can freely replace $f(n)$ with $\max\bra{1/n,f(n),f(n+1),f(n+2),\dots}$.
For each $N$, let $h(N)$ denote the least positive integer with $f(h(N)) < 1/N$, and let $(N_i)_{i =1}^\infty$) be the sequence defined by
$N_1 = 1$ and
\[
N_{i+1} = 2 h(N_i)^2\, .
\]
Put also $\e_i = N_{i}/h(N_{i+1})^2 = 2N_{i}/N_{i+2}$. Note that $h(N) > N$ so $N_{i+1} > 2N_i^2$, and in particular the sequence $N_i$ is increasing.
By Lemma \ref{lem:exist-swell-approx}, there exist $\a_0, \a_1 \in \RR \setminus \QQ$ such that $\fpa{ \a_{j} N_{i} } \leq \e_{i}$
for all $j \in \{0,1\}$ and $i \in \NN$ with $i \equiv j \bmod{2}$.
For $j \in \{0,1\}$, set
\[
E_j = \set{ n \in \NN }{ \fpa{n \a_j} \leq 1/n}\, ,
\]
$E = E_0 \cup E_1 \cup [h(N_1)]$, and $\bb a = \bb{1}_E$. We claim that $\bb a$ satisfies the required conditions.
Since $\a_0$ and $\a_1$ are irrational, we have
\[ \freq(\bb a,1) = d(E) \leq d(E_0) + d(E_1) = 0\, .\]
It remains to show that $\cnt(\bb a,1;N) = \abs{E \cap [N]} \geq f(N)N$ for all $N$. If $N \leq h(N_1)$ then $[N] \subset E$,
so we may assume that $N > h(N_1)$. Let $i$ denote the unique index such that $h(N_i) \leq N < h(N_{i+1})$,
and put $\a = \a_{i \bmod 2}$. Since $\fpa{N_i \a} \leq \e_i$,
we have
\[
\fpa{mN_i \a} \leq m \fpa{N_i \a} \leq m \e_i \leq \frac{1}{mN_i}
\]
and thus $m N_i \in E$ for all integers $m$ with $1 \leq m \leq h(N_{i+1})/N_{i}$. It follows that
\[
\abs{ E \cap [N] } \geq N/N_i \geq f(h(N_{i}))N \geq f(N)N. \qedhere
\]
\end{proof}
\begin{example}
There exists a {\gpword} $\bb a$ over $\{0,1\}$ with $\freq(\bb a,1) = 0$ and $\cnt(\bb a,1;N) \geq N/\log\log\log N$ for all sufficiently large $N$.
Note, however, that Proposition \ref{prop:growth-slow-o(N)} does not ensure the existence of a bracket word $\bb a$ with
$\cnt(\bb a,1;N) = \Theta(N/\log\log\log N)$.
\end{example}
\subsection{Algebraic coefficients}
The construction in Proposition \ref{prop:growth-slow-o(N)} relies on finding a pair of real numbers $(\a_0,\a_1)$
with some rather unusual Diophantine properties. If we restrict our attention to {\gpwords} with algebraic coefficients (c.f. Remark \ref{rmk:alg-coeff}),
such constructions are no longer possible and some gap appears in the possible growth rates.
\begin{proposition}\label{prop:growth-alg-gap}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$ which {arises from a {\gp} map with algebraic coefficients}
and let $x \in \Sigma$. If $\freq(\bb a,x) = 0$, then there exists $c > 0$ such that $\cnt(\bb a, x; N) = O(N^{1-c})$.
\end{proposition}
\begin{proof}
In this proof, we will make extensive use of the material discussed in Appendix \ref{app:nil}. We also recall that definitions of algebraic varieties and semialgebraic sets are given in Section \ref{ssec:semialgebraic}.
By Theorem \ref{thm:BL-poly}, there exists a nilmanifold $X = G/\Gamma$ with $G$ connected and simply connected,
as well as a polynomial sequence $g \colon \ZZ \to G$ and a semialgebraic set $S \subset X$ such that for each $n \in \NN_0$
we have the equivalence: $a_n = x \Longleftrightarrow g(n)\Gamma \in S$.
The nilmanifold $X$ can be equipped with a Mal'cev coordinates $\tau \colon X \to [0,1)^d$, where we put $d = \dim G$. Recall also from \eqref{eq:Malcev-G} that we have a closely related coordinate map $\tilde\tau \colon G \to \RR^d$.
Furthermore, inspecting the construction in \cite{BergelsonLeibman-2007}, we see that $X$ can be equipped with a Mal'cev coordinates $\tau$ such that
$\tilde \tau \circ g$ is a polynomial with algebraic coefficients and $\tau(S)$ is defined by equations and inequalities with algebraic coefficients.
Since $\freq(\bb a,x) = 0$, the measure of $S$ is zero. As a consequence (see e.g.\ \cite[Sec.\ 2.8]{BochnakCosteRoy-book}), $S$ is contained in a proper algebraic variety $V \subsetneq \RR^d$, which is defined over $\QQ$.
Now, we pick a non-zero polynomial $p \colon \RR^d \to \RR$ that vanishes on $V \cup \partial [0,1]^d$, which exists because $V \cup \partial [0,1]^d$ is contained in a proper algebraic sub-variety of $\RR^d$ (specifically, in the union of the algebraic variety $V$ and $2d$ hyperplanes).
Throughout the argument, we allow all implicit constants to depend on $X$, $g$, and $S$.
There are now two cases to consider, depending on the distribution of $g(n)\Gamma$.
Suppose first that the sequence $(g(n)\Gamma)_{n=0}^\infty$ is equidistributed in $X$. Then by Lemma \ref{lem:quant-equi-algebraic},
there exists $c_1 > 0$ such that for each $N$ the sequence $\bra{ g(n)\Gamma }_{n=0}^{N-1}$ is $O(N^{-c_1})$-equidistributed for all $N \in \NN$.
For $\delta > 0$, consider the function $H_{\delta} \colon X \to [0,1]$ defined by
\[
H_\delta(x) = \max\braBig{ 1-\frac{\abs{p(\tau(x))}}{\delta}, 0 } \,.
\]
Then $\norm{H_{\delta}}_{\mathrm{Lip}} = O(1/\delta)$ and $\int H_{\delta} d\mu_{X} = O( \delta^{c_2})$ for some $c_2 > 0$.
We can now estimate
\[
\cnt(\bb a,x;N) \leq \sum_{n=0}^{N-1} H_\delta(g(n)\Gamma) \ll \frac{N^{1-c_1}}{\delta} + \frac{N}{\delta^{c_2} } \,\cdot
\]
Letting $\delta = N^{-c/c_2}$ for a sufficiently small $c > 0$, we obtain $\cnt(\bb a,x;N) = O(N^{1-c})$.
Let us assume now that the sequence $(g(n)\Gamma)_{n=0}^\infty$ is not equidistributed in $X$. In that case,
there exists a horizontal character $\eta \colon X \to \RR/\ZZ$ such that $\eta \circ g$ is constant.
Let $G'$ be the connected component of $\ker \eta$ and let $\Gamma' = \Gamma \cap G'$. Then, $g(n)$ can be decomposed as
$g(n) = g'(n)\gamma(n)$, where $g'$ takes values in $G'$ and $\gamma$ is periodic. This can be shown by adapting the proof of
\cite[Prop.{} 9.2]{GreenTao-2012}, or using this result as a black-box and passing to the limit $\delta \to 0$; periodicity of $\gamma$
follows from Lemma A.12 therein. Let $q$ denote a period of $\gamma$. Each of the sequences $\bb a^{(i)}$, $0 \leq i < q$, defined by
$a^{(i)}_n = a_{qn+i}$ can be represented using the nilmanifold $G'/\Gamma'$ rather than $X$.
Reasoning by induction with respect to the dimension $d$, we may assume that for each $i$, $0 \leq i < q$, there exists
$c_i$ such that $\cnt(\bb a^{(i)},x;N) = O( N^{1-c_i} )$. Letting $c = \min_{0 \leq i < q} c_i$, we see that $\cnt(\bb a,x;N) = O(N^{1-c})$.
\end{proof}
\subsection{Concluding remarks}
We close this section by mentioning several rates of growth about which we do not know if they can be realised.
It is also not clear if restriction to {\gpwords} with algebraic coefficients influences the answer.
\begin{question}
Does there exist a {\gpword} $\bb a$ over $\{0,1\}$ such that one of the following rate of growth holds?
\begin{enumerate}
\item $\cnt(\bb a, 1; N) = \Theta(N^{\lambda})$, where $\lambda \in (0,1) \setminus \QQ$\,.
\item $\cnt(\bb a, 1; N) = \Theta(\log^c N)$, where $c \in (1,\infty) \setminus \NN$\,.
\item $\cnt(\bb a, 1; N) = \Theta\bra{ e^ {\sqrt{\log N}}}$\,.
\end{enumerate}
\end{question}
\section{Computability}\label{sec:comp}
As we have already seen, a single {\gpword} can be represented by many different formulae.
This happens, for instance, in Lemma \ref{lem:cl:[g=0]}. An even simpler example is provided by the {\gpword} $\bb a = \bb 1_{0}$,
which can be represented as $a_n = \ip{1-\fp{\a n}}$ for any $\a \in \RR \setminus \QQ$.
A more surprising representation of the same {\word} is attributed to H{\aa}land Knutson in \cite[p.\ 2]{GrahamOBryant-2010}:
\[
a_n = \ip{ \ipnormal{\sqrt{2}n}2 \sqrt{2}n}- \ip{\sqrt{2}n}^2 - 2n^2 + 1\,.
\]
Existence of multiple representations of a single {\gpword} is not, in and of itself, surprising or worrying.
After all, the same phenomenon occurs for usual polynomials, say with real coefficients.
The reason why this ambiguity does not lead to problems in that case is that each polynomial $p(x) \in \RR[x]$ has
a canonical representation $p(x) = \sum_{i=0}^d \a_i x^d$, where $\a_i \in \RR$ and $\a_d \neq 0$.
Leibman \cite{Leibman-2012} constructed a similar canonical representation for {\gp} sequences.
The full statement of the main result, \cite[Thm.{} 0.1 \& 0.2]{Leibman-2012}, is rather technical, so we state only a simplified version.
Specifically, we restrict our attention to {\gpwords} (Leibman's result concerns arbitrary bounded {\gp} maps from $\ZZ^d$ to $\RR$) and do not
discuss the details of the construction (Leibman explicitly describes the families $\cF_M$ appearing below).
\begin{theorem}\label{thm:Leib}
Let $\bb a$ be a {\gpword} defined over a finite alphabet $\Sigma$. Then there exist families $\cF_M$, $M \in \NN$, of {\gp} maps from $\NN_0$ to $\RR$, depending only on polynomials which appear in a given representation of $\bb a$,
with the following properties.
\begin{enumerate}
\item\label{it:L:A} For each $M,d \in \NN$, and each $d$-tuple of different maps $v_1,v_2, \dots, v_d \in \cF_M$,
the sequence $(\fp{v_1(n)},\fp{v_2(n)},\dots,\fp{v_d(n)})_{n=0}^\infty$ is equidistributed in $[0,1)^d$.
\item\label{it:L:B} There exist integers $M,Q,d \in \NN$, maps $v_1,v_2,\dots,v_d \in \cF_M$, and a partition of
\[ \textstyle \{0,{1}/{Q},\dots,(Q-1)/{Q}\} \times [0,1)^d \]
into pairwise disjoint semialgebraic pieces $S_i$, $i\in \Sigma$, such that for each $n \in \NN$ and $i \in \Sigma$,
\begin{align}\label{eq:419:1}
a_n = i && \text{ if and only if }&& (\fp{n/Q},\fp{v_1(n)},\fp{v_2(n)},\dots,\fp{v_d(n)}) \in S_i\,.
\end{align}
\end{enumerate}
\end{theorem}
\begin{remark}
\begin{enumerate}
\item The families $\cF_M$ are explicitly constructed in \cite{Leibman-2012}.
We start with polynomial maps $p_1(x),p_2(x),\ldots \in \RR[x]$ that span the $\QQ$-algebra generated by the polynomials which
appear in some fixed representation of $\bb a$. As the first step, $p_1/M,p_2/M, \ldots \in \cF_M$. In subsequent steps, we add to $\cF_M$
some elements of the form $u \fp{v}$ where $u,v \in \cF_M$ have already been constructed. For details on exactly which of such elements should be added,
we refer to the original paper.
\item The representation in \ref{thm:Leib}\ref{it:L:B} can be explicitly computed for a given representation of the {\gpword} $\bb a$.
For fixed $M$, $Q$, and $d$, the sets $S_i$ are determined uniquely up to a set of measure zero.
\end{enumerate}
\end{remark}
As a consequence of Theorem \ref{thm:Leib}, for a given bracket word $\bb a$ over an alphabet $\Sigma$ and $i \in \Sigma$, one can check if $a_n = i$
for almost all $n \in \NN_0$. Indeed, it is enough to verify if $S_i$ has full measure. Since, for any two {\gpwords} $\bb a$ and $\bb b$ over the same alphabet,
we can construct the {\gpword} $(\braif{a_n=b_n})_{n=0}^\infty$, one can determine whether $a_n = b_n$ for almost all $n \in \NN_0$.
The situation is totally different
when we insist on exact equality, even when restricting our attention to bracket words with algebraic coefficients.
\begin{theorem}\label{thm:compbis}
Given {\gp} map $g \colon \NN_0 \to \{0,1\}$ with algebraic coefficients, the following problems are undecidable.
\begin{enumerate}
\item\label{it:compbis-A} Determine if $g(n) = 0$ for at least one $n \in \NN_0$.
\item\label{it:compbis-B} Determine if $g(n) = 0$ infinitely many $n \in \NN_0$.
\item\label{it:compbis-B'}Determine if $g(n)=0$ for all but finitely many $n\in\NN_0$.
\item\label{it:compbis-A'}Determine if $g(n) = 0$ for all $n \in \NN_0$.
\end{enumerate}
\end{theorem}
\begin{corollary}
It is undecidable if a given {\gp} map $g \colon \NN_0 \to \RR$ with algebraic coefficients takes only finitely many values.
\end{corollary}
\begin{proof}
Take an arbitrary {\gp} map $g \colon \NN_0 \to \{0,1\}$ and consider the {\gp} map $h\colon \NN_0 \to \RR$ defined by $h(n)=g(n)n$. Then
$h$ takes finitely many values if and only if $g(n)=0$ for all but finitely many $n\in\NN_0$.
\end{proof}
\begin{remark}
\begin{enumerate}
\item In Theorem \ref{thm:compbis}, item \ref{it:compbis-A} is reminiscent of the undecidability of Hilbert's tenth problem, concerning the existence of integer solutions to polynomial equations. In fact, our argument proceeds by a reduction to this problem.
\item In the case of $k$-automatic sequences, the analogues of properties in \ref{thm:compbis}\ref{it:compbis-A}--\ref{it:compbis-A'} are easily seen to be decidable.
\item Another related result, due to Allouche and Shallit \cite[Thm.\ 5.2]{AlloucheShallit-1992}, asserts that it is undecidable if a given $k$-regular sequence with integer values has at least one vanishing term. We point out that, conversely to Theorem \ref{thm:compbis}\ref{it:compbis-A'}, it is decidable if a $k$-regular sequence is identically zero \cite[Thm.\ A]{KrennShallit-2022}.
\end{enumerate}
\end{remark}
A key component in the proof of Theorem \ref{thm:compbis} is the existence of a surjective {\gp} map from $\NN_0$ to $\ZZ^d$ for each $d \in \NN$.
\begin{proposition}\label{prop:comp:surjective-Z^d}
For each $d \in \NN$, there exists a {\gp} map $h_d \colon \NN_0 \to \ZZ^d$ with algebraic coefficients such that for each $x \in \ZZ^d$ there exist infinitely many $n \in \NN_0$ with $h_d(n) = x$. In particular, $h_d$ is surjective.
\end{proposition}
Before we proceed to construct the maps mentioned in Proposition \ref{prop:comp:surjective-Z^d}, let us show how their existence can be used to deduce Theorem \ref{thm:compbis} from classical undecidability results.
\begin{proof}[Proof of Theorem \ref{thm:compbis} assuming Proposition \ref{prop:comp:surjective-Z^d}]
Let $p$ be an arbitrary polynomial in $\mathbb Z[x_1,\ldots,x_d]$ and let $h \colon \NN_0 \to \ZZ^d$ be the map constructed in Proposition
\ref{prop:comp:surjective-Z^d}. Define $g_p$ by $g_p(n) = \braif{ p(h(n)) = 0}$ for $n\in \NN_0$. Then $g_p$ is a
{\gp} map from $\NN_0$ to $\{0,1\}$ with algebraic coefficients, and the following conditions are equivalent.
\begin{enumerate}
\item[{\rm (a)}] There exists $n \in \NN_0$ with $g_p(n) = 0$.
\item[{\rm (b)}] There exist infinitely many $n \in \NN_0$ with $g_p(n) = 0$.
\item[{\rm (c)}] There exist $x_1,x_2,\dots,x_d \in \ZZ$ with $p(x_1,x_2,\dots,x_d) = 0$.
\end{enumerate}
Hilbert's tenth problem is known to be undecidable (see, for instance, \cite{Matiyasevich-1993}). Hence, comparing (a) and (c), we conclude that there is no algorithm to determine whether an element of the set
$\{g_p \mid d\geq 1, \,p\in\mathbb Z[x_1,\ldots,x_d]\}$ has a zero. Similarly, comparing (b) and (c) we conclude that there is no algorithm to determine whether an element of the aforementioned set has infinitely many zeroes. This finishes the proof in cases \ref{it:compbis-A} and \ref{it:compbis-B}.
To derive case \ref{it:compbis-B'} from \ref{it:compbis-B}, it is enough to notice that, for any map $g \colon \NN_0 \to \{0,1\}$, the condition that $g(n) = 0$ for all but finitely many $n \in \NN_0$ is equivalent to the condition that there are finitely many $n \in \NN_0$ such that $1-g(n) = 0$. Likewise, \ref{it:compbis-A'} follows from \ref{it:compbis-A}
and the observation that, $g(n) = 0$ for all $n \in \NN_0$ is equivalent to the condition that there does not exist $n \in \NN_0$ such that $1-g(n) = 0$.
\end{proof}
We now turn to the proof of Proposition \ref{prop:comp:surjective-Z^d}. The main difficulty lies in the construction of a surjective {\gp} map $\NN_0 \to \NN_0^2$; once such a map is constructed, it will not be difficult to use it to construct surjective maps $\NN_0 \to \ZZ^d$ for all $d \in \NN$. As a source of motivation, let us consider a (random) map $\bb f \colon \NN_0 \to \NN_0^2$ given by $\bb f(n) = \bra{\ip{\bb X_n^2 n},\ip{\bb Y_n^2 n}}$, where $\bb X_n,\bb Y_n$ for $n \in \NN_0$ are jointly independent random variables, uniformly distributed in $[0,1)$. One can explicitly compute that, for fixed $k,l \in \NN$ and sufficiently large $N$, we have \[\PP(\bb f(n) = (k,l)) = \frac{\bra{\sqrt{k+1}-\sqrt{k}}\bra{\sqrt{l+1}-\sqrt{l}}}{n}\,\cdot\] Hence, by the second Borel–Cantelli lemma, almost surely there exists infinitely many $n \in \NN_0$ with $\bb f(n) = (k,l)$. This prompts us to consider generalised polynomials of the form $\bra{ \ip{ \fpnormal{\sqrt{2} n}^2 n }, \ip{ \fpnormal{\sqrt{3} n}^2 n }}$, or, more generally
\[
\bra{ \ip{ \fpnormal{\sqrt{2} n}^A n }, \ip{ \fpnormal{\sqrt{3} n}^A n }}
\]
for $A \geq 2$, and exploit equidistribution properties of the sequence $\bra{\fpnormal{\sqrt{2} n},\fpnormal{\sqrt{3} n}}$ in $[0,1)^2$. As a first step in that direction, we mention the following quantitative equidistribution estimate.
\begin{lemma}\label{lem:comp:equidist-linear}
There exist $N_0 > 0$ and $c > 0$ such that for each $M \in \ZZ$, $N \geq N_0$, and $(x,y) \in [0,1)^2$,
there exists $n \in [M,M+N)$ such that
$$\max\left\{\fpa{\sqrt{2}n - x}, \fpa{\sqrt{3}n-y}\right\} \leq 1/N^c\,.$$
\end{lemma}
\begin{proof}
A more precise version follows from \cite[Thm 1.(ii)]{Chen-2000} combined with standard estimates on the quality of approximate rational relations
between algebraic numbers. Specifically, we have the well-known estimate $$\min_{(n,m) \in [N]^2 \setminus \{(0,0)\}} \fpa{n\sqrt{2} + m \sqrt{3}} \gg N^{-3}\,,$$
which is easily obtained by noticing that the norm $\prod_{\sigma,\rho \in \{\pm 1\} }\bra{k +\sigma n\sqrt{2} + \rho m \sqrt{3}}$ is an integer for all $k,n,m \in \ZZ$. (Stronger estimates follow from Schmidt's subspace theorem \cite{Schmidt-1972}, but are not needed for our purposes.)
The theorem is applied with $n = 2$, $m=1$, $\a_1 = x$, $\a_2 = y$, $T_1 = M$, $T_1'=M+N$, $\delta_1=\delta_2=1$, and $M = N^{1/7}$;
then $\Delta = 2$, $\Lambda_2 \gg N^{-3}$, and $\norm{S(T)} = N$. The conclusion, after elementary manipulations, asserts that there exists
$n \in [M,M+N)$ such that
$$\max\left\{\fpa{\sqrt{2}n - x}, \fpa{\sqrt{3}n-y}\right\} \ll N^{-2/7}\,.$$ Alternatively, one can also derive this estimate as a special case of
Theorem \ref{thm:GT}.
\end{proof}
\begin{lemma}\label{lem:comp:surjective}
For $A \in \NN$, define the map $g_A \colon \NN_0 \to \NN_0^2$ by
\[
g_A(n) = \bra{ \ip{ \fpnormal{\sqrt{2} n}^A n }, \ip{ \fpnormal{\sqrt{3} n}^A n }}\,.
\]
There exists a constant $A_0$ such that for all $A > A_0$, for all $(k,l) \in \NN_0$ the set
\[
\set{ n \in \NN_0 }{ g_A(n) = (k,l) }
\]
is infinite. In particular, $g_A$ is surjective.
\end{lemma}
\begin{proof}
Let $c$ be the constant from Lemma \ref{lem:comp:equidist-linear} and let $A > 3/c$.
Fix $(k,l) \in \NN_0^2$, let $N$ be a sufficiently large integer (to be determined in the course of the argument), and put $M = N^2$.
It will suffice to show that there exists $n \in [M,M+N)$ with $g_A(n) = (k,l)$.
Pick $(x,y)\in [0,1)^2$ such that $x^A M = k$ and $y^A M = l$
(that is, $x = (k/M)^{1/A}$ and $y = (l/M)^{1/A}$). By Lemma \ref{lem:comp:equidist-linear}, we can find $n \in [M,M+N)$ such that
$x \leq \fp{\sqrt{2}n} < x+1/N^{c}$ and $y \leq \fp{\sqrt{3}n} < y+1/N^{c}$. It remains to show that
\begin{equation}\label{eq:785:1}
k \leq \fpnormal{\sqrt{2} n}^A n < k+1 \;\; \mbox{ and } \;\; l \leq \fpnormal{\sqrt{3} n}^A n < l+1\,.
\end{equation}
We only consider the first of these two conditions, the second one is fully analogous. The lower bound is immediate:
\[
\fpnormal{\sqrt{2} n}^A n \geq x^A n = kn/M \geq k\,.
\]
For the upper bound, we first obtain that
\begin{align*}
\fpnormal{\sqrt{2} n}^A n < (x+N^{-c})^A (M+N) \,.
\end{align*}
If $k = 0$, then $x = 0$ and thus $(x+N^{-c})^A (M+N) < 2 N^{2-Ac} < 1$ (recall that $A > 3/c$).
If $k \geq 1$, then $x > N^{-2c/3}$ and hence
\begin{align*}
(x+N^{-c})^A (N_0+N) &= x^A N_0 \bra{1+N^{-c}/x}^A(1+N/M)
\\ &\leq k \brabig{1+N^{-c/3}}^A(1+1/N) \leq k \exp\bra{ (A+1) N^{-c/3} } \,.
\end{align*}
Thus, we can find $N_0 = N_0(k) = O_{A}(k^{3/c})$ such that, for all $N \geq N_0$, we have
\[
\fpnormal{\sqrt{2} n}^A n\leq k \exp\bra{ (A+1) N^{-c/3} } < k+1 \,,
\]
as needed.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:comp:surjective-Z^d}]
Let $g_2 \colon \NN_0 \to \NN_0^2$ denote the map constructed in Lemma \ref{lem:comp:surjective}.
For $d \geq 2$, define $g_{d} \colon \NN_0 \to \NN_0^{d}$ inductively by
\[ g_{i+1}(n) = \bra{ g_2(n)_1, g_i(g_2(n)_2)}.\]
An inductive argument shows that for each $x \in \NN_0^d$, there are infinitely many $n \in \NN_0$ with $g_d(n) = x$. Next, define $h_d \colon \NN_0 \to \ZZ^d$ by
\[ h_d(n) = \bra{ g_{2d}(n)_1 - g_{2d}(n)_{d+1}, g_{2d}(n)_2 - g_{2d}(n)_{d+2}, \dots, g_{2d}(n)_{d} - g_{2d}(n)_{2d}}\,.\]
Since the map $\NN_0^2 \to \ZZ$, $(n,m) \mapsto n-m$ is surjective, for each $x \in \ZZ^d$, there are infinitely many $n \in \NN_0$ with $h_d(n) = x$.
\end{proof}
\subsection{Consequences for bracket words}
As a consequence of Theorem \ref{thm:compbis}, we deduce Theorem \ref{thm:comp}, which we restate below for the reader's convenience.
\thmdecide*
\begin{proof}
Given an arbitrary {\gp} map $g \colon \NN_0 \to \{0,1\}$, define $\bb a$ by $a_n = g(n)$ for all $n \in \NN_0$, and let $\bb b$ be defined by $b_n=0$
for all $n\in \NN_0$. Then $\bb a = \bb b$ if and only if $g(n) = 0$ for all $n \in \NN_0$.
Thus, the results follows from Theorem \ref{thm:comp}.
\end{proof}
\section{Linear recurrences}\label{sec:pisot}
In earlier sections, we have encountered examples of {\gpwords} related to linear recurrence sequences,
such as the Fibonacci numbers. Here, we discuss these results in more detail and provide some new arguments.
\subsection{Results}
We recall that a \emph{Pisot number} is a real algebraic integer $\beta > 1$ such that all Galois conjugates of $\beta$ have absolute value strictly less than $1$.
Similarly, a \emph{Salem number} is a real algebraic integer $\beta > 1$ whose Galois conjugates all have absolute value no greater than $1$, and at
least one of which has absolute value exactly $1$. The minimal polynomial for a Salem number must be reciprocal, which implies that $1/\beta$ is a Galois conjugate of $\beta$, and that all other roots have absolute value exactly one. As a consequence, a Salem number is a unit in the ring of algebraic integers.
\begin{theorem}\label{prop:pisot:rec}
Let $\beta > 1$ be an algebraic unit with minimal polynomial $p(x) = x^d - \sum_{i=1}^d x^{d-i} a_i$ for some $d \in \NN$ and $a_1,a_2, \dots, a_d \in \ZZ$.
Let $(n_i)_{i=0}^\infty$ be a sequence of non-negative integers satisfying the linear recurrence
\[ n_{i+d} = \sum_{i=1}^d a_i n_{i+d-1}\, , \qquad i \in \NN_0 \,,\]
and put $E = \set{n_i}{i \in \NN_0}$. Then $\bb{1}_E$ is a {\gpword} if one of the following holds.
\begin{enumerate}
\item\label{it:pisot:rec-1} $d = 2$ (in this case, $\beta$ must be a Pisot number).
\item\label{it:pisot:rec-2} $d = 3$ and $\beta$ is a Pisot number with no real Galois conjugate ({\it i.e.}, the discriminant of $p$ is negative).
\item\label{it:pisot:rec-3} $\beta$ is a Salem number.
\end{enumerate}
\end{theorem}
\begin{proof}
{ Case \ref{it:pisot:rec-1} is covered by \cite[Thm.{} B]{ByszewskiKonieczny-2018-TAMS}, using an argument which relies on best rational approximations to
quadratic irrationals. Case \ref{it:pisot:rec-2} likewise follows from \cite[Thm.{} B]{ByszewskiKonieczny-2018-TAMS} under mild additional assumptions.
Below in Theorem \ref{prop:pisot} we give a complete proof of Case \ref{it:pisot:rec-2} which has a distinctly algebraic flavour, in contrast to the argument in
\cite{ByszewskiKonieczny-2018-TAMS} relying on Diophantine approximation.
Finally, case \ref{it:pisot:rec-3} is covered in the upcoming preprint \cite{ByszewskiKonieczny-upcoming} using methods analogous to those used in Theorem \ref{prop:pisot}.
}
\end{proof}
In the remainder of this section, we will mostly speak of {\gp} subsets of $\NN_0$ rather than {\gpwords}. We recall that, as defined in Section \ref{sec:def}, these terms are closely related, and the connection between them if further elucidated in Corollary \ref{cor:constr:fibre}.
\subsection{Cubic Pisot units}
The main ingredient in the proof of Theorem \ref{prop:pisot:rec}\ref{it:pisot:rec-2} is the following result.
In fact, the two results are equivalent due to a reduction obtained in \cite[Prop.{} 5.1]{ByszewskiKonieczny-2018-TAMS}.
\begin{theorem}\label{prop:pisot}
Let $\beta > 1$ be a cubic Pisot unit with a pair of complex Galois conjugates $\a,\bar\a$ and let $E = \set{ \nint{\b^i} }{ i \in \NN_0}$. Then $\bb{1}_E$ is a {\gpword}.
\end{theorem}
\begin{proof}
We will devise a procedure that verifies if, for a given integer $n \in \ZZ$, we have
\begin{equation}\label{eq:n=b^i}
n = \nint{\b^i} \text{ for some } i \in \NN_0\,.
\end{equation}
Later, we will explain how this procedure can be encoded using a generalised polynomial formula.
Throughout, we assume that $\abs{n}$ is sufficiently large, which we may do because {\gp} sets are closed under finite modifications.
Set $L = \QQ(\b,\a,\bar\a)$ and $K = \QQ(\b)$, and let $p(x) = x^3 - ax^2 -bx-1$ denote the minimal polynomial of $\b$.
Suppose for a moment that \eqref{eq:n=b^i} holds, and hence in particular $n > 0$. Our first goal is to compute $\beta^i$ as a {\gp} function of $n$. Since $\b^i + \a^i + \bar\a^i$ is an integer and $\abs{\a^i} = \beta^{-i/2} \ll 1/\sqrt{n}$,
we see that
\begin{equation}\label{eq:374:00}
n = \b^i + \a^i + \bar\a^i \,.
\end{equation}
Similarly, for every $j \in \NN$, we have
\[
\b^j n = {\b^{i+j} + \a^{i+j} + \bar\a^{i+j}} + O(\b^j/\sqrt{n})\,,
\]
and, as a consequence, bearing in mind that $n$ is large enough, we obtain
\begin{align}
\label{eq:374:01}
\nint{\b n} &= \b^{i+1} + \a^{i+1} + \bar\a^{i+1}\\
\label{eq:374:02}
\nint{\b^2 n} &= \b^{i+2} + \a^{i+2} + \bar\a^{i+2}\,.
\end{align}
Thus, we have expressed $(n,\nint{\beta n},\nint{\beta^2 n})$ as a linear function of $(\beta^i,\alpha^i,\bar\alpha^i)$. As a consequence, we can compute $(\beta^i,\alpha^i,\bar\alpha^i)$ by solving a system of linear equations involving $(n,\nint{\beta n},\nint{\beta^2 n})$ and algebraic coefficients belonging to $L$.
For each $n \in \ZZ$, let $g(n)$, $h(n)$, and $h^*(n)$ be the solution to the system of equations
\begin{align}
\label{eq:374:10} g(n) + h(n) + h^*(n) &= n\\
\label{eq:374:11} g(n)\b + h(n)\a + h^*(n)\bar\a &= \nint{\b n}\\
\label{eq:374:12} g(n)\b^2 + h(n)\a^2 + h^*(n)\bar\a^2 &= \nint{\b^2 n}\, .
\end{align}
Note that \eqref{eq:374:10}--\eqref{eq:374:12} is non-singular, so $g(n)$, $h(n)$, and $h^*(n)$ are well-defined and unique.
In fact, one can explicitly compute that
\begin{align}
\label{eq:374:20} g(n) &= \frac{\nint{\b n}(\a+\bar\a)-\a\bar\a n-\nint{\b^2 n}}{(\b-\a)(\b-\bar\a)}
= \frac{\nint{\b n}(a-\b)-n/\b-\nint{\b^2 n}}{2\b^2 -a\b+1/\b} \\
\label{eq:374:21} h(n) &= \frac{\nint{\b n}(\b+\bar\a)-\b\bar\a n-\nint{\b^2 n}}{(\a-\bar\a)(\a-\b)}
= \frac{\nint{\b n}(a-\a)-n/\a-\nint{\b^2 n}}{2\a^2 -a\a+1/\a} \\
\label{eq:374:22} h^*(n) &= \frac{\nint{\b n}(\a+\b)-\a\b n-\nint{\b^2 n}}{(\bar\a-\a)(\bar\a-\b)}
= \frac{\nint{\b n}(a-\bar\a)-n/\bar\a-\nint{\b^2 n}}{2\bar\a^2 -a\bar\a+1/\bar\a}\,\cdot
\end{align}
The key reason for our interest in $g,h,h^*$ is that, if \eqref{eq:n=b^i} holds, then it follows from the discussion above that $g(n) = \beta^i$, $h(n) = \a^i$ and $h^*(n) = \bar\a^i$. Indeed, $(\beta^i,\alpha^i,\bar\alpha^i)$ is a solution to \eqref{eq:374:10}--\eqref{eq:374:12}, and the solution to \eqref{eq:374:10}--\eqref{eq:374:12} is unique.
It is apparent from formulae \eqref{eq:374:20}--\eqref{eq:374:22} (or from the symmetries of \eqref{eq:374:10}--\eqref{eq:374:12})
that $h^*(n) = \bar{h(n)}$ and $g(n) \in K = \QQ(\b)$, $h(n) \in \QQ(\a)$ for all $n \in \ZZ$. We also see that $g(n)$ is a linear combination
of $n$, $\nint{\b n}$, and $\nint{\b^2 n}$. Hence $g$ is a {\gp} map. For the same reason, $\Re(h)$ and $\Im(h)$ are {\gp} maps\footnote{Note that we have not introduced the notion of a {\gp} map $\NN_0 \to \CC$. However, if one identifies $\CC$ with $\RR^2$ in the standard way, then $h$ is a {\gp} map under this identification.}.
Our next goal is to express condition \eqref{eq:374:00} in terms of the maps $g,h,h^*$. By the Dirichlet unit theorem, we know that the group of units of $\cO_K$ has rank $1$. Since $\beta$ is a unit, it generates a group that has finite index in the group of all units of $\cO_K$. Assume for now that $\beta$ is a fundamental unit, meaning that all units in $\cO_K$ take the form $\pm \beta^i$ for $i \in \ZZ$. We address the general case at the end of the proof.
Recall that if \eqref{eq:n=b^i} holds for some $n \in \ZZ$, then $g(n) = \b^i$ and $h(n) = \a^i$. Thus, $g(n) \in \cO_K$ and $g(n)h(n)h^*(n) = 1$. If, additionally, $\abs{n}$ is sufficiently large, then $g(n) \in [n-1,n+1)$.
Suppose, conversely, that for some $n \in \ZZ$ we have $g(n) \in \cO_K$ and $g(n)h(n) h^*(n) = 1$.
Then $g(n)$ is a unit, since its norm $N_{L/\QQ}(g(n)) = g(n)h(n)h^*(n)$ is equal to $1$. As a consequence, we have $g(n) = \pm \b^i$ for some $i \in \ZZ$.
If additionally $g(n) \geq n-1$ then $g(n) = \b^i$ for some $i \in \NN$. Thus, for all but finitely many $n \in \ZZ$, \eqref{eq:n=b^i} is equivalent to
\begin{align}\label{cond:521:2}
g(n) \in \cO_K\,,\quad g(n)h(n)h^*(n) &= 1\,, &&\text{and}& g(n) &\in [n-1,n+1)\,.
\end{align}
Our final goal is to express the conditions in \eqref{cond:521:2} in terms of generalised polynomials. As a first step in this direction, we will need a more precise description of $g,h,h^*$.
Since $g(n) \in \QQ(\b)$, for each $n \in \ZZ$, there exists a decomposition
\begin{equation}\label{eq:374:30}
g(n) = u(n) + v(n)\b + w(n)\b^2\,,
\end{equation}
where $u(n)$, $v(n)$, and $w(n) \in \QQ$. If $\sigma \in \mathrm{Gal}(L/\QQ)$ is an automorphism with $\sigma(\b) = \a$, then
$\sigma(g(n)) = h(n)$, and as a consequence we also have
\begin{align}\label{eq:374:31}
h(n) &= u(n) + v(n)\a + w(n)\a^2\\
h^*(n) &= u(n) + v(n)\bar\a + w(n)\bar\a^2\,.
\end{align}
Arguing along similar lines as above, we can express $u(n)$, $v(n)$, and $w(n)$ as linear combinations of $g(n)$, $h(n)$, and $h^*(n)$,
and hence also as a linear combination of $n$, $\nint{\b n}$, and $\nint{\b^2 n}$. For instance,
\begin{align*}
u(n) = \frac{1}{\Delta}\Big(&
\bra{-2 a^3+a^2 b^2-10 a b+4 b^3-9}n
\\&+\bra{a^3 b-a^2+4 a b^2+6 b}\nint{\b n} +
\bra{-a^2 b +3 a-4 b^2}\nint{\b^2 n}
\Big)\,,
\end{align*}
where $\Delta = -4 a^3+a^2 b^2-18 a b+4 b^3-27$ is the discriminant of $p$.
In particular, $u$, $v$, and $w$ are {\gp} maps.
Since $g$, $\Re(h)$ and $\Im(h)$ are generalised polynomials,
it follows that
$$
E_1 = \set{n \in \ZZ}{g(n)h(n)h^*(n) = 1} \text{ and }
E_2 = \set{n \in \ZZ}{g(n)-n \in [-1,1)}
$$
are {\gp} sets. Let us consider the set
\[
\Lambda = \set{(x,y,z) \in \QQ^3}{ x + y\b + z\b^2 \in \cO_K} \,.
\]
Clearly, $\Lambda$ is a lattice. It follows that $\Lambda$ is a {\gp} subset of $\RR^3$. Indeed, if $\Lambda = A \ZZ^3$ for a matrix $A \in \mathrm{GL}(3;\RR)$,
then
\[
\Lambda = \set{x \in \RR^3}{ \fp{ \bra{A^{-1}x}_1} + \fp{ \bra{A^{-1}x}_2} + \fp{ \bra{A^{-1}x}_3} = 0}\, .
\]
Recalling that $u$, $v$, and $w$ are {\gp} maps, we conclude that
\[
E_3 = \set{n \in \ZZ}{ g(n) \in \cO_K} = \set{n \in \NN_0}{ (u(n),v(n),w(n)) \in \Lambda }
\]
is a {\gp} set. Hence, $E_1 \cap E_2 \cap E_3$ is a {\gp} set by Proposition \ref{prop:cl:sets}, and it is precisely the set of all $n \in \ZZ$
which satisfy \eqref{cond:521:2}, as needed.
Let us now consider the case where the fundamental unit of $K$ is some $\tilde\beta \neq \pm \beta$.
We may assume that $\tilde \beta > 0$. Since $\beta$ is a unit, we have $\beta = \tilde\beta^k$ for some $k \in \NN$.
It is straightforward to verify that $\tilde\beta$ is again a cubic Pisot unit with a pair of complex conjugates, say $(\gamma, \bar{\gamma})$,
so the above discussion applies with $\tilde\beta$ in place of $\beta$. Set $\tilde n_i = \tilde\b^i + \gamma^i + \bar{\gamma}^i$, for every $i \in \NN_0$.
Then $(\tilde n_i)_{i=0}^\infty$ satisfies a linear recurrence relation, and, for all sufficiently large $i$, we have
$\tilde{n_i} = \nintnormal{\tilde\b^i}$ and $\tilde n_{i+1} = \nintnormal{\tilde\beta \tilde n_i}$. Furthermore,
$\tilde E = \set{ \tilde n_i}{i \in \NN_0}$ is a {\gp} subset of $\NN_0$.
One can find an integer $m$ such that the sequence $(\tilde n_i \bmod{m})_{i=0}^\infty$ is periodic with minimal period $\ell$ that is a
multiple of $k$ \cite{Ward-1933}
\footnote{More precisely, by \cite[Thm.\ 1]{Ward-1933}, it is enough to consider the case where $k$ is a power of a prime $p$. By \cite[Thm.\ 10.2 \& Cor.\ 3]{Ward-1933}, the period of $(\tilde n_i \bmod{p^N})_i$ is a multiple of the order of $\tilde \beta$ in $\mathbb{F}_q[\tilde\beta]^{\times}$, where $q = p^{N-C}$ for a constant $C$. In particular, the period of $(\tilde n_i \bmod{p^{N}})_i$ is a multiple of $k$ for sufficiently large $N$.}
. Note that, for each sufficiently large $i$ and for each $j$, $0 \leq j < \ell$, we have
\[
\nint{\tilde \b^j \tilde n_i} \equiv \tilde n_{i+j\bmod \ell} \bmod m \,.
\]
This allows us to identify, for each residue class $r$, $0 \leq r < \ell$, the elements $n \in \tilde E$ which take the form
$n = \tilde n_i$ with $i \equiv r \bmod \ell$. With finitely many exceptions, these are precisely those $n \in \tilde E$ for which
$\nint{\tilde \b^j \tilde n} \equiv \tilde n_{r+j\bmod \ell} \bmod m$ for all $j$, $0 \leq j < \ell$. It follows that the set $\set{\nint{\b^i}}{i \in \NN_0}$
differs by finitely many elements from
\begin{equation}\label{eq:399:1}
\set{ n \in \tilde E}{ (\exists\, 0 \leq a < \ell/k) \ (\forall\, 0 \leq j < \ell)\ \nintnormal{\tilde\b^j n} \equiv \tilde n_{ak+j} \bmod{m} }\, .
\end{equation}
Since the set in \eqref{eq:399:1} is a {\gp} set, and since the property of being a {\gp} set is preserved under finite modifications,
we conclude that the set $\set{\nint{\b^i}}{i \in \NN_0}$ is also a {\gp} set.
\end{proof}
\begin{proof}[Proof of Theorem \ref{prop:pisot:rec}(ii)]
Follows immediately from Theorem \ref{prop:pisot} and \cite[Prop.{} 5.1]{ByszewskiKonieczny-2018-TAMS}.
\end{proof}
\subsection{Concluding remarks}
Let us now briefly discuss potential generalisations of Theorem \ref{prop:pisot:rec}. Let $p(x) = x^d - \sum_{i=1}^d a_ix^{d-i} \in \ZZ[x]$ be a monic irreducible polynomial of degree $d \in \NN$ with a root $\beta > 1$ which is either a Pisot unit or a Salem number, and let $\a_1,\a_2,\dots,\a_{d-1}$ denote the remaining roots.
Much of the reasoning in Theorem \ref{prop:pisot} carries through to this more general context, except that the group of units of $\cO_{K}$ now no longer has rank $1$ (here, $K = \QQ(\beta)$). Adapting the argument, we can hope to show that
\[
F = \set{ \operatorname{Tr}_{L/\QQ}(\mu) }{ \mu \text{ is a unit of } \cO_K}
\]
is a {\gp} subset of $\ZZ$, where $L = \QQ(\b,\a_1,\dots,\a_{d-1})$ is the splitting field of $p$.
Consider also the sequence given by $n_i = \mathrm{Tr}(\beta^i) = \beta^i + \sum_{k=1}^{d-1} \a_k^i \in \ZZ$, and note that $n_i$ obeys the linear recurrence
$$
n_{i+d} = \sum_{j=1}^d a_j n_{i+d-j}, \qquad i \in \NN_0\,.
$$
Set $E = \set{ n_i}{i \in \NN_0} \cap \NN$. Note that the assumption on $\beta$ ensures that $n_i = \nintnormal{\b^i} + O(1)$ for all $i \in \NN_0$.
In analogy with Theorem \ref{prop:pisot:rec}, one can ask if $E$ is a {\gp} subset of $\NN$.
In the case where the group of units of $\cO_{K}$ has rank $1$, the set $F$ is essentially equal to $E$, which is one of the key observations behind the
proof of Theorem \ref{prop:pisot}. In general, we have the inclusion $E \supset F$, but $E$ will usually be a proper subset of $F$ and there is no clear way of
describing $E$ inside of $F$ with a {\gp} formula.
In the case where $\beta$ is a Pisot number but not a unit, we have a negative result for $d = 1$: if $k \geq 2$ is and integer, then $E = \set{k^i}{i \in \NN_0}$
is not a {\gp} set. (We revisit this example in Section \ref{sec:neg}.) When $d \geq 2$, nothing is known, but, based on the previously mentioned result,
it seems reasonable to expect a negative result. In the case where $\beta$ is neither a Pisot nor a Salem number ({\it i.e.}, either $\beta$ is transcendental or
$\beta$ has a Galois conjugate with absolute value greater than $1$) nothing is known, and the techniques discussed in this section appear not to
be applicable. We end this section with the following two general problems.
\begin{problem}
\begin{enumerate}
\item
Classify all sequences of positive integers $(n_i)_{i=1}^\infty$ satisfying a linear recurrence such that the characteristic word $\bb{1}_E$ of the set $E = \set{n_i}{i \in \NN_0}$ is a {\gpword}.
\item
Classify all real numbers $\beta > 1$ such that the characteristic word $\bb{1}_E$ of the set $E = \set{ \nint{\b^i} }{ i \in \NN_0}$ is a {\gpword}.
\end{enumerate}
\end{problem}
\section{Negative results}\label{sec:neg}
In this section we discuss the problem of proving that a given infinite word cannot be described by a generalised polynomial formula.
We remark that, as a general principle, it is often harder to verify that a sequence does not admit a representation of a specified form than it is to find such a representation when it exists. This phenomenon is particularly prevalent from a computational point of view. For context, we also note that Allouche, Shallit, and Yassawi
\cite{AlloucheShallitYassawi-2021} recently survey ways in which one can show that a sequence is not automatic.
\subsection{General conditions}
Let us now review criteria that can be used to show that a given infinite word $\bb a$
is not a {\gpword}. In principle, each ``positive'' result about {\gpwords} gives rise to such a criterion, and \emph{vice versa}.
For each property $P$ which can apply to finitely-valued sequences, the implication ``\textit{If $\bb a$ is a {\gpword} then $P(\bb a)$}'' is
tantamount to ``\textit{If $\neg P(\bb a)$ then $\bb a$ is not a {\gpword}}''. In practice, depending on the aesthetic appeal of the property $P(\bb a)$ and
on the ease of verifying $\neg P(\bb a)$, one of these implications is more interesting than the other.
The following proposition gathers several such properties that arise from positive results discussed earlier.
\begin{proposition}\label{prop:neg:long-list}
Let $\bb a$ be an infinite word defined over a finite alphabet $\Sigma$. Then $\bb a$ is not a {\gpword}
if one of the following conditions hold.
\begin{enumerate}
\item\label{it:neg:long-list:0} There is $w \in \Sigma^*$ such that $\mathrm{freq}(\bb a,w)$ does not exist.
\item\label{it:neg:long-list:1} There is $\fword{w} \in \Sigma^*$ such that $\mathrm{freq}(\bb a,\fword{w}) > 0$ but $\mathrm{rec}(\bb a,\fword{w}) = \infty$.
\item\label{it:neg:long-list:2} The subword complexity of $\bb a$ satisfies $\limsup_{n\to \infty}\log(p_{\bb a}(N))/\log N = \infty$.
\item\label{it:neg:long-list:3} There exists $x \in \Sigma$ with $\freq(\bb a,x) < 1$ and $t \colon \NN_0 \to \NN_0$ that is good for equidistribution
on nilsystems (cf.{} Sec.{} \ref{ssec:facts:equidist}) such that $a_{t(n)} = x$ for all $n \in \NN_0$.
\end{enumerate}
\end{proposition}
\begin{proof}
These are immediate consequences of Theorem \ref{prop:facts:freq-exists}, Corollary \ref{cor:facts:freq-exists},
Theorem \ref{thm:A}, and Proposition \ref{prop:nil:equidistribution}.
\end{proof}
Another way to discriminate bracket words that we want to mention is related to periodicity.
Following \cite{ByszewskiKonieczny-2020-CJM}, we say that an infinite word
$\bb a$ is \emph{weakly periodic} if for every infinite arithmetic progression $\bra{kn+r}_{n =0}^\infty$ ($k \in \NN,\ r \in \NN_0$)
there exist two distinct sub-progressions with equal steps $\bra{k'n+r'}_{n =0}^\infty$ and $\bra{k'n+r''}_{n =0}^\infty$
($k' \in \NN$ and $k \mid k'$, $r'$ and $r''$ belong to $\NN_0$, $r'\equiv r'' \equiv r \bmod k$, and $r' \neq r''$)
such that the corresponding restrictions of $\bb a$ are the same: $ a_{k'n +r'} = a_{k'n+r''}$ for all $n \in \NN_0$.
For instance, all automatic sequences are weakly periodic, all Toeplitz sequences are weakly periodic, and the characteristic sequence of
the square-free integers is weakly periodic.\footnote{A sequence $\bb a$ is a Toeplitz sequence if for each $n$ there exists a period $q \in \NN$
such that $a_{n+qm} = a_{n}$ for all $m \in \NN$;
see, for instance, \cite{Downarowicz-2005} for background. An integer $n$ is square-free if, for every prime $p$, one has $p^2 \nmid n$.}
We will also say that $\bb a$ is \emph{almost everywhere periodic} if there exists an infinite periodic word $\bb a'$ such that $a_{n} = a'_{n}$ for almost all $n$, that is, $d\bra{\set{n \in \NN_0}{a_{n} \neq a'_{n}}} = 0$. The following result is a rephrasing of {\cite[{Thm.{} 2.6}]{ByszewskiKonieczny-2020-CJM}}.
\begin{proposition}[{\cite[{Thm.{} 2.6}]{ByszewskiKonieczny-2020-CJM}}]\label{prop:neg:weakly-periodic}
Let $\bb a$ be an infinite word that is weakly periodic but not almost everywhere periodic. Then $\bb a$ is not a {\gpword}.
\end{proposition}
The proof relies on Theorem \ref{thm:BL-mini} and the fact that each nilsystem $(X,T)$ can be partitioned into a finite number of components,
$X = X_1 \cup X_2 \cup \dots \cup X_d$, such that $(X_i,T^d)$ is a totally minimal\footnote{A topological dynamical system $(X,T)$ is said to be totally minimal if
$(X,T^d)$ is minimal for every $d\geq 1$.} dynamical system for all $i$, $1 \leq i \leq d$.
Next, we observe that, for a totally minimal dynamical system $(X,T)$, $x \in X$, open $S \subset X$, $k \in \NN$, $r,r' \in \NN_0$ with $r \neq r'$,
if for all $n \in \NN_0$ we have the implication
\[T^{kn+r}(x) \in S \Rightarrow T^{kn+r'}(x) \in S\,,\] then $S$ is either empty or dense in $X$ (see \cite[Lem.{} 2.4]{ByszewskiKonieczny-2020-CJM}).
\begin{example}
The indicator function of the square-free integers is not a {\gpword} (which can also be derived from item (iii) of Proposition \ref{prop:neg:long-list}).
A non-periodic Toeplitz sequence is not a {\gpword} (note that a Toeplitz sequence is periodic if and only if it is almost everywhere periodic).
It was shown in \cite{CassaigneKarhumaki-1997} that a large class of Toeplitz sequences have polynomial subword complexity, so item (iii) of Proposition \ref{prop:neg:long-list}
cannot be applied in this case, nor can item (ii) Proposition \ref{prop:neg:long-list}.
\end{example}
As a counterpart to Theorem \ref{thm:nil:IP*-rec},
it is shown in \cite{ByszewskiKonieczny-2018-TAMS} that any {\gp} subset $E \subset \NN_0$ with $d(E) = 0$ is very poor in terms of additive structure.
Here, we cite a slightly weaker (but more succinct) variant of {\cite[{Thm.{} A}]{ByszewskiKonieczny-2018-TAMS}}.
\begin{proposition}[{\cite[{Thm.{} A}]{ByszewskiKonieczny-2018-TAMS}}]\label{prop:neg:sparse-IP}
Let $E \subset \NN_0$ be a set. Suppose that $d(E) = 0$ and $E$ contains an $\IP$ set. Then $E$ is not a {\gp} set.
\end{proposition}
\begin{corollary}\label{cor:neg:sparse-IP}
Let $\bb a = (a_n)_{n=0}^\infty$ be an infinite word over a finite alphabet $\Sigma$. Suppose that there exists $x \in \Sigma$ and an $\IP$ set $E \subset \NN$ such that $\freq(\bb a, x) = 0$ and $a_n = 0$ for all $n \in E$. Then $\bb a$ is not a {\gpword}.
\end{corollary}
\begin{example}
The characteristic word of the set $ \set{\sum_{n \in I} n! }{ I \subset \NN,\ \abs{I} < \infty}$ is not a {\gpword}.
\end{example}
Finally, we cite a result recently obtained by the second-named author \cite[{Thm.{} A}]{Konieczny-2021-JLM}, as a final ingredient needed to finish the classification of automatic {\gpwords}, cf.\ Theorem \ref{thm:BK}.
Recall that a set $E \subset \NN_0$ is \emph{thick} if its complement is not syndetic, or equivalently if for each $\ell$ there exists $n$ with $n,n+1,\dots, n+\ell \in E$.
\begin{theorem}[{\cite[{Thm.{} A}]{Konieczny-2021-JLM}}]\label{thm:neg:powers}
Let $k \geq 2$ be an integer, let $E \subset \NN_0$, and put
\[
F = \set{ m \in \NN}{ m k^n \in E \text{ for infinitely many $n \in \NN_0$}}.
\]
If $\set{k^n}{n \in \NN_0} \subset E$ and $\NN \setminus F$ is thick then $E$ is not a {\gp} set.
\end{theorem}
\begin{example}
Let $k \geq 2$ be an integer. Then the characteristic word of the set $\set{k^n}{n \in \NN_0}$ is not a {\gpword}.
\end{example}
For a set $E \subset \NN_0$ and $k \in \NN$, we set $E/k = \set{n \in \NN_0}{kn \in E}$. Restricting our attention to sets $E \subset \NN_0$ with $E/k = E$, we obtain a cleaner statement, which can be phrased either in terms of {\gp} sets or {\gpwords}.
\begin{corollary}\label{cor:neg:powers}
Let $k \geq 2$ and $E \subset \NN$. If $E \neq \emptyset$, $d(E) = 0$, and $E/k = E$, then $E$ is not a {\gp} set.
\end{corollary}
\begin{proof}
Pick $n_0 \in E$ and let $E' = E/n_0$. Then $d(E') = 0$, $k^n \in E'$ for all $n$, and
\[
F' = \set{ m \in \NN}{ m k^n \in E' \text{ for infinitely many $n \in \NN_0$}} = E'.
\]
Since $d(F') = 0$, the set $\NN \setminus F'$ is thick and we can apply Theorem \ref{thm:neg:powers}.
\end{proof}
\begin{corollary}\label{cor:neg:powers-set}
Let $\bb a = (a_n)_{n=0}^\infty$ be an infinite word over a finite alphabet $\Sigma$, let $k \geq 2$ be an integer and let $x \in \Sigma$. Suppose that
\begin{inparaenum}[(i)]
\item $a_{kn} = a_{n}$ for all $n \in \NN$,
\item $\freq(\bb a; x) = 0$, and
\item $a_n = x$ for at least one $n \in \NN$.
\end{inparaenum}
Then $\bb a$ is not a {\gpword}.
\end{corollary}
\subsection{Primes and squares}
Let us consider two standard examples: the primes $\cP = \{2,3,5,7,\dots\}$ and the squares $\cS = \{0,1,4,9,\dots\}$.
In both cases, we can show that the corresponding characteristic word is not a {\gpword}.
The arguments are short and rely heavily on theorems from nilpotent dynamics.
\begin{proposition}\label{prop:neg:primes-not-gp}
The characteristic word ${\bb 1}_{\cP}$ of the set of primes is not a {\gpword}.
\end{proposition}
\begin{proof}
As a special case of \cite[Theorem 5.2]{GreenTao-2012-Mobius}, for each {\gp} map $g \colon \NN_0 \to [0,1]$ and for all sufficiently large integers $N$,
we have
\begin{equation}\label{eq:863:1}
\abs{\EEE_{n < N} \mu(n) g(n)} = O( \log^{-2} N)\,,
\end{equation}
where we let $\mu(n)$ denote the M\"{o}bius function ({\it i.e.}, $\mu(n) = 0$ if $n$ is divisible by a square and $\mu(n) = (-1)^k$ if $n$ is
the product of $k$ different primes), and the implicit constant is allowed to depend on $g$.
If $\bb{1}_{\cP}$ was a {\gpword}, then \eqref{eq:863:1} would imply that
\begin{equation}\label{eq:863:2}
{\abs{\cP \cap [N]}} =O( N/\log^{2} N)\, ,
\end{equation}
which would contradict the Prime Number Theorem.
\end{proof}
\begin{remark}
Note, in particular, that the cited theorem implies that $\mu$ is not a {\gpword}. The same applies, with virtually the same proof,
to the Liouville function, which is defined by $\lambda(n) = (-1)^k$ if $n$ is the product of $k$ primes (counting multiplicities).
Results from \cite{FrantzikinakisHost-2017} allow one to extend this observation to a wider class of multiplicative functions.
In a different direction, using the results in \cite{GreenTao-2012-Mobius}, one should be able to strengthen the result of
Proposition \ref{prop:neg:primes-not-gp} by showing that for each bounded {\gp} map $g \colon \NN_0 \to \RR$, the averages
$\EE_{p \in \cP \cap [N]} g(p)$ converge to the same limit as the averages $\EE_{n \in [N]} g(n)$ as $N \to \infty$.
\end{remark}
We next consider the set of squares.
\begin{proposition}\label{prop:neg:squares-not-gp}
The characteristic word ${\bb 1}_{\cS}$ of the set of squares is not a {\gpword}.
\end{proposition}
This follows immediately from the following more general result.
\begin{proposition}
Let $p \colon \ZZ \to \ZZ$ be a polynomial with $p(\NN_0) \subset \NN_0$ and $\deg p \geq 2$.
Then the set $ \set{p(n)}{ n \in \NN_0}$ is not a {\gp} subset of $\NN_0$.
\end{proposition}
\begin{proof}
We will prove marginally more, namely that if $p_1,p_2,\dots,p_s \colon \ZZ \to \ZZ$ are polynomials
with $p_j(\NN_0) \subset \NN_0$ and $\deg p_j \geq 2$ for all $j$, $1 \leq j \leq r$, then the set
\begin{equation}\label{eq:254:1}
E = \textstyle \bigcup_{j=1}^r p_j(\NN_0)
\end{equation}
is not a {\gp} subset of $\NN_0$. For the sake of contradiction, suppose that the converse is true.
Throughout the proof, we use terminology introduced in Appendix \ref{app:nil}.
Applying Theorem \ref{thm:BL-poly}, we can find a nilmanifold $X = G/\Gamma$ with $G$ connected and simply connected, a
semialgebraic subset $S \subset X$, and a polynomial sequence $g \colon \ZZ \to G$ such that $E = \set{n \in \NN_0}{g(n)\Gamma \in S}$.
Let us first consider the case where $(g(n)\Gamma)_{n=0}^\infty$ is equidistributed in $X$. Since $d(E) = 0$, we have $\mu_{X}(S) = 0$.
The polynomial sequence $\bra{ \pi_{\mathrm{ab}}(g(n)\Gamma) }_{n=0}^\infty$ is equidistributed in $X_{\mathrm{ab}}$.
Hence, we can conclude from Weyl's equidistribution theorem that $\bra{ \pi_{\mathrm{ab}}(g(p(n)\Gamma)) }_{n=0}^\infty$ is
equidistributed in $X_{\ab}$ for each non-constant polynomial map $p \colon \NN_0 \to \NN_0$.
It now follows from Theorem \ref{thm:Leib-nil} (that is, \cite[Thm.{} C]{Leibman-2005}) that $\bra{ g(p(n)\Gamma) }_{n=0}^\infty$ is
equidistributed in $X$. In particular $g(p(n))\Gamma \not \in S$ for almost all $n \in \NN_0$. Taking $p = p_1$ we reach a contradiction.
Suppose next that $(g(n)\Gamma)_{n=0}^\infty$ is not equidistributed in $X$. Then, it follows from \cite[Thm.{} B]{Leibman-2005}
that there exists $q \in \NN$ and sub-nilmanifolds $Y_0,Y_1,\dots,Y_{q-1} \subset X$ such that $\bra{g(qn+i)\Gamma}_{n=0}^\infty$ is
equidistributed in $Y_i$ for each $i \in [q]$. It remains to apply the previously considered special case to the sets $(E-i)/q$ for each $i \in [q]$
(note that these sets again take the form \eqref{eq:254:1}, possibly for some larger $r$).
\end{proof}
\subsection{Number-theoretical functions}
Number theory provides a plentiful source of examples of finitely-valued sequences, for which one can inquire into the existence of a
generalised polynomial representation. Multiplicative functions constitute a particularly interesting and well-studied class of sequences, with many applications to other problems.
Recall that a sequence $f \colon \NN \to \CC$ is \emph{multiplicative} if $f(nm) = f(n) f(m)$
for all $n,m \in \NN$ with $\gcd(n,m) = 1$. For instance, the M\"{o}bius function, mentioned in the previous sub-section, is often used in the study of the prime numbers. The Prime Number Theorem is equivalent to $\frac{1}{N} \sum_{n=1}^N \mu(n) \to 0$ as $N \to \infty$, while the more quantitative bound $\abs{\sum_{n=1}^N \mu(n)} = O(N^{1/2+\e})$ for each $\e > 0$ is equivalent to the Riemann hypothesis. The results from \cite{GreenTao-2012-Mobius} on correlations of the M\"{o}bius function were a crucial ingredient in the work of Green and Tao on linear patterns in the primes \cite{GreenTao-2010b}. As we pointed out earlier, the M\"{o}bius function is not a {\gpword}, and the same applies to the closely related Liouville function.
While the M\"{o}bius function and the Liouville function are bounded, there are many interesting examples of unbounded integer-valued multiplicative functions. In order to obtain finitely-valued sequences, we use reduction modulo a fixed integer.
Thus, in this section we investigate infinite words of the form $(\bra{f(n)\bmod{q}}_{n=1}^\infty$,
where $q \geq 2$ is an integer and $f \colon \NN \to \ZZ$ is multiplicative.
Analogous questions, with automatic sequences in place of {\gpwords}, were investigated in \cite{Yazdani-2001}.
As an illustrative example, we begin with the Euler totient function $\phi$, given by
\[
\phi(n) = \abs{ (\ZZ/n\ZZ)^*} = n\prod_{\mathcal P\ni p \mid n}\bra{1-\frac{1}{p}} \,, \qquad n \in \NN\,.
\]
\begin{proposition}
Let $q \geq 3$ be an integer. Then $\bra{\phi(n) \bmod q}_{n=1}^\infty$ is not a {\gpword}.
\end{proposition}
\begin{proof}
Let us assume by contradiction that $\bra{\phi(n) \bmod q}_{n=1}^\infty$ is a {\gpword}.
Replacing $q$ with a divisor, we may assume that $q = 4$ or $q$ is a prime.
We first assume that $q=4$. By direct inspection we have:
\[
\phi(n) \bmod 4 =
\begin{cases}
1 & \text{if } n \in \{1,2\};\\
2 & \text{if $n = 4$ or $n \in \{p,2p\}$ for a prime $p \equiv 3 \bmod 4$} ;\\
0 & \text{otherwise}.
\end{cases}
\]
Since $\bra{\phi(n) \bmod 4}_{n=1}^\infty$ is a {\gpword}, we deduce that $\cP \cap (4\ZZ+3)$ is a {\gp} set.
This is impossible by direct repetition of the argument in Proposition \ref{prop:neg:primes-not-gp}.
Suppose now that $q$ is a prime and set
\[
E = \set{ n \in \NN}{ \phi(n) \not \equiv 0 \bmod q}\, .
\]
Alternatively, $E$ is described by the condition: $n \in E$ if and only if $q^2 \nmid n$ and $n$ has no prime divisors congruent to $1$ modulo $q$.
Hence, applying the prime number theorem in arithmetic progressions, we conclude that $d(E) = 0$. By assumption, $E$ is a {\gp} set.
Pick a prime $p > q$ with $q \nmid p-1$. Then $E/p = \set{n \in \NN}{pn \in E} = E$, and hence we infer from Corollary \ref{cor:neg:powers}
that $E$ is not a {\gp} set. This provides a contradiction.
\end{proof}
Applying the same ideas, we can obtain the following result, which can be used to deal with most other ``naturally occurring'' multiplicative sequences.
\begin{proposition}\label{prop:neg:multi}
Let $f \colon \NN \to \ZZ$ be a multiplicative sequence and let $q \in \NN$. Suppose that the two following properties hold.
\begin{enumerate}
\item\label{it:138:2} There exists $p\in \cP$ such that the sequence $\bra{f(p^n) \bmod q}_{n=1}^\infty$ is eventually periodic but not eventually zero.
\item\label{it:138:3} There exist infinitely many $p \in \cP$ such that $f(p^n) \equiv 0 \bmod q$ for some $n \in \NN$.
\end{enumerate}
Then $\bra{f(n) \bmod q}_{n=1}^\infty$ is not a {\gpword}.
\end{proposition}
\begin{proof}
We argue by contradiction, assuming that $\bra{f(n) \bmod q}_{n=1}^\infty$ is a {\gpword}.
Using Lemma \ref{lem:cl:code}, we deduce that $\bra{f(n) \bmod p}_{n=1}^\infty$ is also a {\gpword} for every $p \mid q$.
Replacing $q$ with a prime divisor if necessary, we may thus assume that $q$ is prime.
Consider the {\gp} set
\[
E = \set{ n \in \NN}{ f(n) \not \equiv 0 \bmod q}\, .
\]
Let $p$ be a prime satisfying the conditions in \ref{it:138:2}. Let $c \in \NN_0$ and $d \in \NN$ be such that
$f(p^{n+d}) \equiv f(p^{n}) \bmod q$ for all $n \geq c$ and $f(p^c) \not \equiv 0 \bmod q$.
Let $E'$ be the {\gp} set defined by $E' = E/p^c = \set{n \in \NN}{p^c n \in E}$. Then $E'/p^d = E'$.
Next, we show that $d(E) = 0$. Suppose, conversely, that $d(E) > 0$. Then, by Theorem \ref{thm:facts:dens-exists},
$E$ is syndetic, meaning that we can find $N$ such that $E$ intersects any interval $[n,n+N)$ ($n \in \NN$).
By \ref{it:138:3}, we can pick pairwise coprime integers $r_0,r_1,\dots,r_{N-1}$ with $f(r_i) \equiv 0 \bmod q$ for all $i \in [N]$.
By the Chinese remainder theorem, we can find $n \in \NN$ such that $n+i \equiv r_i \bmod r_i^{2}$ for all $i \in [N]$.
Then we have $f(n+i) \equiv 0 \bmod q$ and $n+i \not \in E$ for all $i \in [N]$, contradicting the defining condition of $N$.
Hence $d(E) = 0$.
Since $d(E) = 0$, we also have $d(E') = 0$. Thus, we infer from Corollary \ref{cor:neg:powers} that $E'$ is not a {\gp} set, which contradicts
previous observations.
\end{proof}
\begin{example}
Given an integer $k \geq 0$, we let $\sigma_k \colon \NN \to \NN$ denote the $k$-th power divisor-sum function defined by
\[
\sigma_k(n) = \sum_{d \mid n} d^k = \prod_{p\mid n} \frac{p^{k(\nu_p(n)+1)}-1}{p^k-1}\,, \qquad n \in \NN\,,
\]
where the sum runs over all positive integers $d$ that divide $n$. Note that $\sigma_0(n) = d(n)$ is the number of divisors of $n$
and $\sigma_1(n) = \sigma(n)$ is the sum of divisors. For each integer $q \geq 2$, the assumptions of Proposition \ref{prop:neg:multi} are satisfied:
if $k \geq 1$ then in \ref{it:138:2} we can take $p=q$, and in \ref{it:138:3} we can take any $p \equiv 1 \bmod{q}$ and $n = 1$
(infinitude of such primes follows from Dirichlet's theorem). If $k = 0$, both \ref{it:138:2} and \ref{it:138:3} hold for all primes $p$.
Hence, $\bra{\sigma_k(n) \bmod q}_{n=1}^\infty$ is not a {\gpword}.
\end{example}
\begin{example}
Given an integer $k \geq 1$, we let $\tau_k \colon \NN \to \NN$ denote the number of representations as the product of $k$ positive integers, that is
\[
\tau_k(n) = \abs{ \set{(d_1,d_2,\dots,d_k) \in \NN^k}{ \prod_{i=1}^k d_i = n} } = \prod_{p\mid n} \binom{k+\nu_p(n)-1}{k-1}\,, \qquad n \in \NN\,.
\]
For each integer $q \geq 2$, the assumptions of Proposition \ref{prop:neg:multi} are satisfied: both \ref{it:138:2} and \ref{it:138:3} hold for all primes $p$.
Hence, $\bra{\tau_k(n) \bmod q}_{n=1}^\infty$ is not a {\gpword}.
\end{example}
\begin{example}
Let $\tau \colon \NN \to \ZZ$ denote the Ramanujan function, and let $q$ be an integer with $\gcd(q,6) = 1$. For each prime $p$, the sequence $\bra{\tau(p^n)}_{n=1}^\infty$ satisfies the recurrence
\[
\tau(p^{n+2}) = \tau(p) \tau(p^{n+1}) - p^{11}\tau(p^n)\,,
\]
and hence the sequence $\bra{\tau(p^n) \bmod q}_{n=1}^\infty$ is eventually periodic. In particular, since $\tau(2) = -24 \not\equiv 0 \bmod q$, in Proposition \ref{prop:neg:multi} assumption \ref{it:138:2} is satisfied for $p = 2$. It is known that the set of $n \in \NN$ with $\tau(n) \not\equiv 0 \bmod q$ has asymptotic density $0$ \cite{Ramanujan}. As a consequence, assumption \ref{prop:neg:multi}\ref{it:138:3} is also satisfied (otherwise, there would exist $P \in \NN$ such that $\tau(n) \not \equiv 0 \bmod q$ for all $n \in P\NN + 1$). Hence, $\bra{\tau(n) \bmod q}_{n=1}^\infty$ is not a {\gpword}.
\end{example}
We close this section on multiplicative functions with another result, Corollary \ref{cor:prop:neg:fin-primes-multi}, which applies to multiplicative sequences that exhibit
non-trivial behaviour only for a finite set of primes. In particular, these sequences do not satisfy condition (ii) of Proposition \ref{prop:neg:multi}. In fact, we can state a somewhat more general result. For a prime $p$ and an integer $n$, we let $\nu_p(n)$ denote the $p$-adic valuation of $n$.
\begin{proposition}\label{prop:neg:fin-primes}
Let $\ell \in \NN$, let $p_1,p_2,\dots,p_{\ell}$ be primes, let $\Sigma$ be a finite alphabet, and let $F \colon \NN_0^\ell \to \Sigma$. Let $\bb a$ be the infinite word defined
over $\Sigma$ by
\[ a_n = F\bra{\nu_{p_1}(n),\nu_{p_2}(n), \dots,\nu_{p_{\ell}}(n)} , \qquad n \in \NN. \]
Suppose that $\bb a$ is not almost everywhere periodic. Then $\bb a$ is not a {\gpword}.
\end{proposition}
\begin{proof}
By Proposition \ref{prop:neg:weakly-periodic}, it will suffice to check that $\bb a$ is weakly periodic. Pick any $k \in \NN,r \in \NN_0$. We can find $k'$ with $k \mid k'$ such that $\nu_{p_i}(k') > \nu_{p_i}(r)$ for all $1 \leq i \leq \ell$. Thus, $a_{k'n + r} = a_{k'n + k'+r}$ for all $n \in \NN_0$, so, with notation as in the definition of weak periodicity,
we may take $r' = r$ and $r'' = r+k'$.
\end{proof}
\begin{corollary}\label{cor:prop:neg:fin-primes-multi}
Let $f \colon \NN \to \ZZ$ be a multiplicative sequence and let $q \in \cP$.
Suppose that $(f(n) \bmod q)_{n=1}^\infty$ is not periodic and that there exists $p_0 \in \cP$ such that $f(p^n) \equiv 1 \bmod q$
for all primes $p \geq p_0$ and all $n \in \NN$. Then $\bra{f(n) \bmod q}_{n=1}^\infty$ is not a {\gpword}.
\end{corollary}
\begin{proof}
Note that the sequence $(f(n) \bmod q)_{n=1}^\infty$ is Toeplitz, since, for every positive integer $n$, one has $f(n+dm) = f(n)$ for all $m \in \NN$, for any choice of $d \in \NN$
such that $\nu_p(d) > \nu_p(n)$ for all $p < p_0$. Note also that $(f(n) \bmod q)_{n=1}^\infty$ is not periodic, and thus it is also not almost everywhere periodic. It remains to apply Proposition \ref{prop:neg:fin-primes}.
\end{proof}
\begin{example}
The sequence $\bra{ (-1)^{\nu_2(n) + \nu_3(n) }}_{n=1}^\infty$ is not a {\gpword}.
\end{example}
\subsection{Automatic sequences}\label{ssec:Negative-Automatic}
Let us now consider the problem of classifying automatic sequences which are bracket words.
To begin with, we very briefly recall the definition of an automatic sequence; for extensive background, see \cite{AlloucheShallit-book}.
Let $k \in \NN$, let $\Sigma_k = \{0,1,\dots,k-1\}$ denote the base-$k$ alphabet, and let $\Omega$ be a finite set.
A sequence $\bb a$ over $\Omega$ is $k$-automatic if there exists a finite $k$-automaton which computes $\bb a$. More explicitly,
this means that there exist a finite set of states $Q$, a distinguished state $q_0 \in Q$, a transition function
$\delta \colon Q \times \Sigma_k\to Q$, and an output map $\tau \colon Q \to \Omega$
such that for each integer $n \in \NN_0$ with base-$k$ expansion $n = n_0 + kn_1 + k^2 n_2 + \dots + k^\ell n_{\ell}$
we have
\[
a_n =
\tau\left(\delta\Big(
\delta \big(
\dots
\delta(
\delta(q_0, n_0)
, n_1)
\dots
, n_{\ell-1} \big)
, n_{\ell} \Big)\right)\,.
\]
It is relatively easy to show that if an automatic sequence $\bb a = (a_n)_{n=0}^\infty$ coincides with a bracket word $\bb b = (b_n)_{n=0}^\infty$ almost everywhere, that is,
if $$d\bra{\set{n \in \NN_0}{a_n \neq b_n}} = 0 \,,$$
then $\bb a$ must also be periodic almost everywhere ({\it i.e.}, there exists a periodic word $\bb c$
such that $d\bra{\set{n \in \NN_0}{a_n \neq c_n}} = 0$). Indeed, this result follows from Proposition \ref{prop:neg:weakly-periodic} combined with the fact that
a $k$-automatic sequence has a finite $k$-kernel; see \cite{ByszewskiKonieczny-2020-CJM} for details.
Here, the $k$-kernel of a sequence $\bb a = (a_n)_{n=0}^\infty$ is defined as
the set of subsequences $\set{ \bra{a_{k^i n+r}}_{n=0}^\infty}{r,i \in \NN_0,\ r < k^i}$.
As a consequence of this ``density $1$'' result, in order to classify all automatic sequences which are bracket words,
it is enough to consider sequences $\bb a$ over $\{0,1\}$, with $\mathrm{\freq}(\bb a,1) = 0$. Equivalently, we let $\bb a = \bb 1_{E}$,
where $E \subset \NN_0$ and $d(E) = 0$. In \cite{ByszewskiKonieczny-2018-TAMS}, we investigated such ``sparse'' bracket words and showed
a slightly stronger variant of Proposition \ref{prop:neg:sparse-IP}. Combining it with facts concerning additive richness of automatic sets,
a full classification was obtained in \cite{ByszewskiKonieczny-2020-CJM}, conditional on the conjecture that for $k \geq 2$
the set $\set{k^i}{i \in \NN_0}$ of powers of $k$ is not a {\gp} subset of $\NN_0$. This conjecture was finally proved in \cite{Konieczny-2021-JLM},
leading to the following result.
\begin{theorem}[{\cite[Thm.{} B]{Konieczny-2021-JLM}}]\label{thm:BK}
Let $\bb a$ be an automatic sequence that is not eventually periodic. Then $\bb a$ is not a {\gpword}.
\end{theorem}
Note that this theorem extends the classical result claiming that a Sturmian word cannot be automatic.
\section{Subword complexity}\label{sec:subword}
Since generalised polynomials are defined by relatively simple formulae, it is natural to inquire if they also have low complexity when viewed
from different perspectives. In particular, we investigate \emph{subword complexity}, that is, the number $p_{\bb a}(N)$ of different length-$N$ factors
(or subwords) of an infinite word $\bb a$. An overview of various results concerning subword complexity can be found in a number of surveys,
such as \cite{CassaigneNicolas-2010} or \cite{Allouche-1994}, and connections with the theory of dynamical systems are discussed in \cite{Ferenczi-1999}.
\subsection{Background}
For an infinite word $\bb a=(a_n)_{n=0}^\infty$ defined over an alphabet $\Sigma$, the complexity function of $\bb a$ is the function that associates
with each positive integer $n$ the positive integer
\[ p_{\bb a}(N) = \abs{\set{ a_{n}a_{n+1}\cdots a_{n+N-1}}{n \in \NN_0}} \,.\]
The simplest sequences, in terms of subword complexity, are the eventually periodic ones: If $\bb a$ is eventually periodic then $p_{\bb a}(N)$ is bounded.
Conversely, it was shown by Morse and Hedlund that if there exists $N$ with $p_{\bb a}(N) \leq N$ then $\bb a$ must be eventually periodic
(see, for instance, \cite{MorseHedlund-1938} or \cite[Prop.{} 2]{Ferenczi-1999}). Thus, for any sequence $\bb a$, which is not eventually periodic, and every $N$
we have $p_{\bb a}(N) \geq N+1$. The Sturmian words, discussed in Section \ref{ssec:nil:sturm}, are characterised by the property that $p_{\bb a}(N) = N+1$ for all $N$.
Recall that Sturmian words are, in particular, {\gpwords}.
At the other extreme, if $\vert \Sigma\vert=k$, we have the upper bound $p_{\bb a}(N) \leq k^N$ for all $N$, and there exist sequences for which equality holds. (In fact, this is the case for almost all sequences with respect to the natural probability measure on $\Sigma^{\NN_0}$).
Because of the elementary inequality $p_{\bb a}(N+M) \leq p_{\bb a}(N) p_{\bb a}(M)$, the limit
\[
h(\bb a) = \lim_{N \to \infty} \frac{\log p_{\bb a}(N)}{N}
\]
exists for any finitely-valued sequence $\bb a$; its value is called the \emph{entropy} of $\bb a$, and is closely connected to the notion of
topological entropy coming from the theory of dynamical systems.
\subsection{New result}
Because {\gpwords} can be represented using nilsystems as in Theorem \ref{thm:BL-word} and because nilsystems have zero entropy,
one could show\footnote{Since we are about to prove a more precise estimate, we do not go into the details of the argument.
If in Theorem \ref{thm:BL-word} the sets $S_i$ were open, then the conclusion would be an immediate consequence of standard facts about topological entropy.
In general, the sets $S_i$ are not open, but their boundaries $\partial S_i$ have zero $\mu_{G/\Gamma}$-measure, which is sufficient for this application.}
that $h(\bb a) = 0$ for each {\gpword} $\bb a$, meaning that $p_{\bb a}(N) = \exp({o(N)})$ as $N \to \infty$. Our main new result concerning the complexity
of bracket words is a polynomial bound, given in Theorem \ref{thm:A} and restated below for the reader's convenience.
\thmsubword*
\begin{remark}
We note that related results concerning topological complexity of nilsystems were obtained in
\cite{HostKraMaass-2014}, but are not directly applicable in our context (the crucial obstacle is the fact that the representation in
Theorem \ref{thm:BL-word} involves a partition into semialgebraic sets $S_i$ which are neither open nor closed, while the results
in \cite[Sec.{} 3]{HostKraMaass-2014} are applicable to open covers).
\end{remark}
The proof of this result is carried out in Sections \ref{sec:sc-induction}, \ref{sec:sc-auxiliary}, and \ref{sec:sc-proof}.
\subsection{Complementary results}
\newcommand{\C}{\lambda}
In view of Theorem \ref{thm:A}, it is natural to inquire into more precise asymptotics for
$p_{\bb a}(N)$, where $\bb a$ is a {\gpword}. Let
\[
\C(\bb a) = \limsup_{N \to \infty} \frac{\log p_{\bb a}(N)}{\log N}
\]
be the smallest exponent such that $p_{\bb a}(N) \leq N^{\C(\bb a)+o(1)}$ as $N \to \infty$.
Theorem \ref{thm:A} asserts that $\C(\bb a)$ is finite for all {\gpwords} $\bb a$.
The exact value of $\C(\bb a)$ is known only in the simplest examples, such as the Sturmian words.
Already in relatively simple cases, such as $a_n = \braif{ \fp{ \sqrt{2}n \ip{\sqrt{3} n}} > 1/4}$, it would be interesting to compute $\C(\bb a)$ exactly.
Theorem 3.4 of \cite{HostKraMaass-2014} asserts that the topological complexity $S(\e,N)$ of a minimal nilsystem $(X,T)$ obeys the bounds
$C(\e) N^c \leq S(\e,N) \leq C'(\e) N^c$, where $c$ is the total commutator dimension of $X$ (see the original paper for the relevant definitions).
Let $\bb a$ be a {\gpword} represented on the nilsystem $(X,T)$ as in Theorem \ref{thm:BL-word}. It seems plausible that, under mild additional assumptions,
similar estimates might hold for $p_{\bb a}(N)$, implying in particular that $\C(\bb a) = c$.
We prove now that $\lambda(\bb a)$ can be arbitrarily large. In fact, for each $d \in \NN$, we can construct a {\gpword} $\bb a$ with $\lambda(\bb a) = d$.
\begin{proposition}\label{prop: comp1}
Fix $d \in \NN$, and $\a_1,\a_2, \dots, \a_d \in \RR$ such that $1,\a_1,\a_2,\dots,\a_d$ are $\QQ$-linearly independent.
For every $i$, $1 \leq i \leq d$, let $\bb a^{(i)}$ be the Sturmian word defined by $a^{(i)}_n = \ip{n\a_i} - \ip{(n-1) \a_i}$ for $n \in \NN_0$, and set
$\bb a = \prod_{i=1}^d \bb a^{(i)}$. Then $\bb a$ is a bracket word with $\lambda(\bb a) = d$.
\end{proposition}
\begin{proof} We first observe that, according to Corollary \ref{cor:cl:prod}, and since Sturmian words are bracket words, the word $\bb a$ is a bracket word.
For each $i$, $1 \leq i \leq d$, we have $p_{\bb a^{(i)}}(N) = N+1$. As a general fact about products, we have
$p_{\bb a}(N) \leq \prod_{i=1}^d p_{\bb a^{(i)}}(N)$. It follows that $\lambda(\bb a) \leq \sum_{i=1}^d \lambda\bra{ \bb a^{(i)}} = d$.
Conversely, for each $i$, $1 \leq i \leq d$, and each length-$N$ subword $w^{(i)}$ of $\bb a^{(i)}$, there exists an interval (not degenerate to a point)
$I \subset \RR/\ZZ$ such that, for $n \in \NN_0$, $\bb a^{(i)}|_{[n,n+N)} = w^{(i)}$ if and only if $n \a_{i} \bmod \ZZ \in I$. Since by assumption the numbers
$1,\a_1,\a_2,\dots,\a_d$ are $\QQ$-linearly independent, the sequence $(n\a_1,n\a_2,\dots,n\a_d) \bmod \ZZ^d$ is equidistributed in $\RR^d/\ZZ^d$.
It follows that for any $d$-tuple $w^{(1)},w^{(2)},\dots,w^{(d)}$ of factors of $\bb a^{(1)}, \bb a^{(2)},\dots, \bb a^{(d)}$ respectively,
$\prod_{i=1}^d w^{(i)}$ is a factor of $\bb a$. Hence,
$p_{\bb a}(N) \geq \prod_{i=1}^d p_{\bb a^{(i)}}(N)$, and consequently $\lambda(\bb a) \geq d$. This ends the proof.
\end{proof}
We end this section with an explicit example showing that $\C(\bb a)$ takes on arbitrarily large values for {\gpwords} $\bb a$,
even when the size of the alphabet of $\bb a$ remains bounded.
\begin{proposition}\label{prop: comp2}
Fix $d \in \NN$. Let $\a \in \RR \setminus \QQ$ and let $\bb a$ be the {\gpword} over $\{0,1,\dots,9\}$ given by
$a_n = \ip{ 10 \fp{\a n^d} }$. Then $p_{\bb a}(N) \gg_d N^{d(d-1)/2}$ for all $N \in \NN$.
\end{proposition}
\begin{proof}
For each pair $(m,n) \in \ZZ^2$, we can compute that $a_{n+m} = \ip{10\fp{h_m(n)}}$, where
\[
h_m(n) = \sum_{i=0}^d \a_i(m) n^{d-i} \qquad \text{ and } \qquad \a_i(m) = \fp{ \a m^i \binom{d}{i} }\,, \qquad 0 \leq i \leq d\,.
\]
Note that $\a_0(m) = \fp{\a}$ for all $m$. Let $\vec \a(m) = \bra{\vec \a_i(m)}_{i=1}^{d} \in [0,1]^{d}$, and let $A = \set{ \vec\a(m)}{ m \in \ZZ}$.
By Weyl's equidistribution theorem, $A$ is dense in $[0,1]^{d}$.
Let $N$ be a sufficiently large integer. Suppose that for some $m,m' \in \NN_0$, $\bb a|_{[m,m+N)} = \bb a|_{[m',m'+N)}$.
Then
\begin{equation}\label{eq:245:1}
\fpa{h_m(n) - h_{m'}(n)} \leq 1/5\,, \;\;\;\; \forall n \in [N]\, .
\end{equation}
In general, if $\beta = (\beta_0,\beta_1,\dots,\beta_{d-1}) \in [0,1)^{d}$ is such that $\fpa{\sum_{i=0}^{d-1} \b_i n^i} \leq 1/5$ for all $n \in [N]$, then it follows from the quantitative version of the Weyl equidistribution
theorem that there exists a non-zero integer $q = O_d(1)$ such that $\fpa{q\beta_i} = O_d (N^{-(d-i)})$ for all $i$, $0 \leq i \leq d$.
(This can be derived using \cite[Lemma 2.4]{HardyLittlewood-book}, or as a special case of a much more general statement in Theorem \ref{thm:GT}.) Applying this observation with $\b = \a(m) - \a(m')$, we see that there exists a set $R \subset \RR^d/\ZZ^d$, which is a union of $O_d(1)$
rectangles with side lengths $O_d (N^{-(d-i)})$, $1 \leq i \leq d$, such that \eqref{eq:245:1} implies $\a(m)-\a(m') \bmod \ZZ^d \in R$. Note that the measure of $R$ is
\[ 0< \lambda\bra{R} = O_d\bra{N^{-(d-1)} N^{-(d-2)} \cdots N^0} = O_d\bra{ N^{-d(d-1)/2}}\,.\]
Put $r = \ip{1/\lambda(R)}$. We can inductively construct an increasing sequence of integers $m_1,m_2,\dots,m_r$ with the property that $\a(m_j)-\a(m_i) \bmod \ZZ^d \not \in R$ for all $1 \leq i < j \leq r$. Indeed, if $m_1,m_2,\dots,m_{j-1}$ have already been constructed ($1 \leq j \leq r$) then it is enough to pick any $m_{j}$ with
\[ \a(m_{j}) \bmod \ZZ^d \not \in \textstyle\bigcup_{i=1}^{j-1} \bra{R + \a(m_i)},\]
which is possible because the measure of the union above is strictly less than $1$ and $A$ is dense in $[0,1]^d$. By construction, all factors $\bb a|_{[m_i,m_i+N)}$ with $1 \leq i \leq r$ are distinct. It follows that \( p_{\bb a}(N) \geq r \gg_{d} N^{d(d-1)/2}.\)
\end{proof}
\subsection{Outline of the proof of Theorem \ref{thm:A}}
We illustrate our strategy of the proof of Theorem \ref{thm:A} with a specific example, which already employs most of the tools used in the proof of the general case. Since the subsequent discussion serves only as illustration and motivation, we skip some of the technical details.
Let $\bb a$ be the {\gpword} over the alphabet $\{0,1\}$ given by
\[
a_n = \braif{\fp{\sqrt{2}n \ip{\sqrt{3}n}} < \frac{1}{2}}.
\]
(Above, we use Iverson bracket, introduced in Section \ref{sec:Notation}.) Our goal is to obtain a polynomial bound for $p_{\bb a}(N)$. We may write $a_n = c(g(n))$, where
\[
g(n) = \ip{2\fp{\sqrt{2}n \ip{\sqrt{3}n}}},
\]
and $c \colon \{0,1\} \to \{0,1\}$ is given by $c(0) = 1$, $c(1) = 0$. Of course, the subword complexity of $\bb a$ is the same as the subword complexity of $\bra{g(n)}_{n=0}^\infty$.
Our first step (cf.\ Example \ref{ex:par-gp}) is to note that for each $m \in \NN_0$ there exist $\a,\b,\gamma,\delta \in [0,1)$ such that for all $n \in \NN_0$ we have
\begin{align*}
g(n+m) &= \ip{ 2\fp{ (\sqrt{2}n+\a) { \ip{\sqrt{3} n + \b}} + \gamma n + \delta } } = \tilde g_{\a,\b,\gamma,\delta}(n).
\end{align*}
Thus, instead of estimating the number of subwords of $\bra{g(n)}_{n=0}^\infty$ of a given length $N$, it will suffice to estimate the number of sequences $\bra{\tilde g_{\a,\b,\gamma,\delta}(n)}_{n=0}^{N-1}$, where $(\a,\b,\gamma,\delta)$ varies over $[0,1)^4$. This step is the key reason why we introduce the notion of a parametric {\gp} map in Section \ref{ssec:sc-para-gp}.
For the purpose of counting, it will be more convenient to work with the operation $\ip{\cdot}$ rather than $\fp{\cdot}$. We can remove $\fp{\cdot}$ from the definition of $\tilde g$ by writing
\begin{align*}
\tilde g_{\a,\b,\gamma,\delta}(n) &=
2\ip{ (\sqrt{2}n+\a) { \ip{\sqrt{3} n + \b}} + \gamma n + \delta } \\&+
\ip{ 2\bra{ (\sqrt{2}n+\a) { \ip{\sqrt{3} n + \b}} + \gamma n + \delta }}
= 2 g'_{\a,\b,\gamma,\delta}(n) - g''_{\a,\b,\gamma,\delta}(n).
\end{align*}
Thus, to obtain a polynomial bound for $p_{\bb a}(N)$, it will suffice to obtain polynomial bounds for the number of sequences $\bra{ g'_{\a,\b,\gamma,\delta}(n)}_{n=0}^{N-1}$ and $\bra{ g''_{\a,\b,\gamma,\delta}(n)}_{n=0}^{N-1}$, where $(\a,\b,\gamma,\delta)$ varies over $[0,1)^4$. We only consider the first of these two bounds, the second one being analogous. (Actually, the two classes of sequences are related by $g'_{\a,\b,\gamma,\delta}(n) = \ip{g''_{\a,\b,\gamma,\delta}(n)/2}$, but we do not need this fact.)
Expanding the brackets inside $\ip{\cdot}$, we may bring $g'_{\a,\b,\gamma,\delta}(n)$ into a more convenient form
\begin{align*}
g'_{\a,\b,\gamma,\delta}(n)
&= \ip{
\sqrt{2} n \ip{\sqrt{3} n + \b} + \a \ip{\sqrt{3} n + \b} + \gamma n + \delta
}
\\ &= \ip{
\sqrt{2} h^{(1)}_{\b}(n) + \a h^{(2)}_{\b}(n) + \gamma h^{(3)}(n) + \delta h^{(4)}(n)
},
\end{align*}
where $ h^{(1)}_{\b}(n) = n \ip{\sqrt{3} n + \b}$, $h^{(2)}_{\b}(n) = \ip{\sqrt{3} n + \b}$, $h^{(3)}(n) = n$ and $h^{(4)}(n) = 1$ are integer-valued {\gp} maps, which are strictly simpler than $g'_{\a,\b,\gamma,\delta}(n)$ in a sense that is made precise in Section \ref{ssec:sc-height}. Note also that for each of the maps $h^{(i)}$ and each $n \in [N]$ we have $0 \leq h^{(i)}(n) < 10 N^2$.
Suppose that we have already proved a polynomial bound on the number of sequences $\brabig{h^{(1)}_{\b}(n)}_{n=0}^{N-1}$ and $\brabig{h^{(2)}_{\b}(n)}_{n=0}^{N-1}$ as $\beta$ varies over $[0,1)$. (Our proof of Theorem \ref{thm:A} proceeds by induction, cf.\ Section \ref{ssec:sc-induction}. Here, we omit the discussion of relatively simple and uninteresting cases.) Since the only dependence of $g'_{\a,\b,\gamma,\delta}(n)$ on $d\b$ is through $h^{(1)}_{\b}$ and $h^{(2)}_{\b}$, it will suffice to obtain for each $\b \in [0,1)$ a polynomial bound for the number of sequences $\bra{ g'_{\a,\b,\gamma,\delta}(n)}_{n=0}^{N-1}$, where $(\a,\gamma,\delta)$ varies over $[0,1)^3$. Thus, have reduced the number of parameters from $4$ to $3$. Let us fix the choice of $\b \in [0,1)$, and let $h^{(1)}(n) = h^{(1)}_{\b}(n)$, $h^{(2)}(n) = h^{(2)}_{\b}(n)$. We point out that from this point we will no longer need any information about the maps $h^{(i)}$ ($1 \leq i \leq 4$) other than that they map $[N]$ to $[10 N^2]$; in particular, we will not use the fact that they are {\gp} (cf.\ Proposition \ref{prop:inductive}).
We are thus left with the task of obtaining a polynomial bound on the number of sequences
\[
\bra{ g'_{\a,\b,\gamma,\delta}(n)}_{n=0}^{N-1} =
\bra{
\ip{
\sqrt{2} h^{(1)}(n) + \a h^{(2)}(n) + \gamma h^{(3)}(n) + \delta h^{(4)}(n)
}}_{n=0}^{N-1}
\]
as $\a,\gamma,\delta$ varies over $[0,1)^3$. A natural approach at this point is to approximate $\a,\gamma,\delta$ by rational numbers $\a^*$, $\gamma^*$, $\delta^*$ with denominators $Q$ (to be optimised in the course of the argument) and (say) $0 \leq \a - \a^*,\ \gamma - \gamma^*, \delta-\delta^* \leq 1/Q$. Indeed, as long as we have a polynomial bound $Q = N^{O(1)}$, the choice of $\a^*$, $\gamma^*$, $\delta^*$ will only contribute a polynomial factor to our bound on the subword complexity of $\bb a$. At the same time, for $n \in [N]$ we expect (cf.\ eq.\ \eqref{eq:521:1}), at least heuristically, that
\begin{align*}
g'_{\a,\b,\gamma,\delta}(n) &= \ip{
\sqrt{2} h^{(1)}(n) + \a h^{(2)}(n) + \gamma h^{(3)}(n) + \delta h^{(4)}(n)}
\\
&=
\ip{
\sqrt{2} h^{(1)}(n) + \a^* h^{(2)}(n) + \gamma^* h^{(3)}(n) + \delta^* h^{(4)}(n)
} = g'_{\a^*,\b,\gamma^*,\delta^*}(n)
.
\end{align*}
Let us make that last point somewhat more precise. We have
\[
\abs{ (\a-\a^*) h^{(2)}(n) + (\gamma-\gamma^*) h^{(3)}(n) + (\delta-\delta^*) h^{(4)}(n) } \leq 30 N^2/Q.
\]
Thus, (cf.\ eq.\ \eqref{eq:521:2})
\(
g'_{\a,\b,\gamma,\delta}(n) = g'_{\a^*,\b,\gamma^*,\delta^*}(n)
\)
as long as we have
\begin{equation}\label{eq:sc:ex:001}
\fpa{
\sqrt{2} h^{(1)}(n) + \a h^{(2)}(n) + \gamma h^{(3)}(n) + \delta h^{(4)}(n)} > 30N^2/Q.
\end{equation}
Next, we will need to better understand the situation where \eqref{eq:sc:ex:001} does not hold. It will be convenient to define (cf.\ eq.\ \eqref{eq:521:15})
\[
h^{(0)}(n) =
-\nint{
\sqrt{2} h^{(1)}(n) + \a^* h^{(2)}(n) + \gamma^* h^{(3)}(n) + \delta^* h^{(4)}(n).
}
\]
Then we have the bound $0 \geq h^{0}(n) > -50 N^2$, and we may express \eqref{eq:sc:ex:001} in a slightly more convenient form
\begin{align*}
\abs{ h^{(0)}(n) + \sqrt{2} h^{(1)}(n) + \a h^{(2)}(n) + \gamma h^{(3)}(n) + \delta h^{(4)}(n) } > 30N^2/Q.
\end{align*}
Let $B \subset \ZZ^5$ denote the set of vectors $m = (m_0,m_1,m_2,m_3,m_4)$ with $0 \geq m_0 > -50N^2$ and $0 \leq m_1,m_2,m_3,m_4 < 10N^2$ (thus, $\bra{ h^{(0)}(n), h^{(1)}(n),\dots, h^{(4)}(n) } \in B$ for $n \in [N]$).
The set of vectors $m = (m_0,m_1,m_2,m_3,m_4)$ such that
\begin{align}\label{eq:sc:ex:002}
\abs{ m_0 + \sqrt{2} m_1 + \a m_2 + \gamma m_3 + \delta m_4 } \leq 30N^2/Q
\end{align}
can be thought of as a discrete approximation of a hyperplane, and thus we expect it to be additively structured. Indeed, we show in Section \ref{sec:sc-auxiliary} (Proposition \ref{prop:lattice-approx}) that there exists a lattice $\Lambda$ with the following properties:
\begin{itemize}
\item If $m \in B$ and \eqref{eq:sc:ex:002} holds then $m \in \Lambda$.
\item If $m \in B \cap \Lambda$ then
\begin{align}\label{eq:sc:ex:003}
\abs{ m_0 + \sqrt{2} m_1 + \a m_2 + \gamma m_3 + \delta m_4 } \leq C N^{12}/Q
\end{align}
for some absolute constant $C > 0$.
\end{itemize}
Consider $n \in [N]$ such that $\bra{ h^{(0)}(n), h^{(1)}(n),\dots, h^{(4)}(n) } \in \Lambda$. Then (cf.\ eq.\ \eqref{eq:521:3})
\begin{align*}
g'_{\a,\b,\gamma,\delta}(n) &= - h^{(0)}(n) + \ip{
h^{(0)}(n) + \sqrt{2} h^{(1)}(n) + \a h^{(2)}(n) + \gamma h^{(3)}(n) + \delta h^{(4)}(n)},
\end{align*}
where the expression under the $\ip{\cdot}$ is bounded in absolute value by $C N^{12}/Q$. We will take $Q > CN^{12}$, meaning that
\begin{align}\label{eq:sc:ex:004}
g'_{\a,\b,\gamma,\delta}(n) &= -h^{(0)}(n),\ &&\text{or}& g'_{\a,\b,\gamma,\delta}(n) = -h^{(0)}(n)-1.
\end{align}
Next, we need to develop a better understanding of when each of the two possibilities mentioned above occurs. Put
\begin{align*}
\Lambda^+ &= \set{ m \in \Lambda }{m_0 + \sqrt{2} m_1 + \a m_2 + \gamma m_3 + \delta m_4 \geq 0},\\
\Lambda^- &= \set{ m \in \Lambda }{m_0 + \sqrt{2} m_1 + \a m_2 + \gamma m_3 + \delta m_4 < 0}.
\end{align*}
Then $\Lambda = \Lambda^+ \cup \Lambda^-$ is a partition obtained by cutting $\Lambda$ with a hyperplane. We can now make \eqref{eq:sc:ex:004} more precise (cf.\ eq.\ \eqref{eq:521:4}):
\begin{equation}\label{eq:sc:ex:005}
g'_{\a,\b,\gamma,\delta}(n) =
\begin{cases}
-h^{(0)}(n) & \text{if } \bra{ h^{(0)}(n), h^{(1)}(n),\dots, h^{(4)}(n) } \in \Lambda^+,\\
-h^{(0)}(n)-1
& \text{if } \bra{ h^{(0)}(n), h^{(1)}(n),\dots, h^{(4)}(n) } \in \Lambda^-.\\
\end{cases}
\end{equation}
Recall from \eqref{eq:sc:ex:001} that if $ \bra{ h^{(0)}(n), h^{(1)}(n),\dots, h^{(4)}(n) } \not \in \Lambda$ then $g'_{\a,\b,\gamma,\delta}(n) = g'_{\a^*,\b,\gamma^*,\delta^*}(n)$. Combining this observation with \eqref{eq:sc:ex:005}, we see that the sequence $\bra{ g'_{\a,\b,\gamma,\delta}(n)}_{n=0}^{N-1}$ is completely determined by the following data:
\begin{itemize}
\item the rational approximations $\a^*,\gamma^*,\delta^*$;
\item the restrictions of the sequences $h^{(1)}_{\b}$ and $h^{(2)}_{\b}$ to $[N]$;
\item the intersection of the lattice $\Lambda$ with the box $B$;
\item the hyperplane partition $\Lambda \cap B = (\Lambda^+ \cap B) \cup (\Lambda^- \cap B)$.
\end{itemize}
As we have seen, we may take $Q$ of the form $Q = C'N^{12}$ for a large constant $C'$. Then the number of choices of $\a^*,\gamma^*,\delta^*$ is $O(N^{36})$, and hence polynomial in $N$. The contribution from $h^{(1)}_{\b}$ and $h^{(2)}_{\b}$ is polynomial by the inductive assumption. Thus, it remains to estimate the number of ways in which the set $B$ can be partitioned into the components $B = (B \setminus \Lambda) \cup (\Lambda^+ \cap B) \cup (\Lambda^- \cap B)$, as described above. A polynomial estimate on the number of such partitions is obtained in Section \ref{sec:sc-auxiliary} (Proposition \ref{prop:half-lattice-count}) using the techniques of additive geometry.
\section{Proof of Theorem \ref{thm:A}: notation and induction}\label{sec:sc-induction}
In this section we set up the inductive scheme and introduce notation which will be used in the proof of Theorem \ref{thm:A}.
\subsection{Height}\label{ssec:sc-height}
One of several measures of complexity of a generalised polynomial is the \emph{height}, that is, the number of nested instances of the floor function.
Let $d \in \NN$, and let $\mathrm{GP}_0$ denote the polynomial maps from $\RR^d$ to $\RR$.
Inductively, for each $i \in \NN$, let $\mathrm{GP}_{i}$ be the smallest class of maps from $\RR^d$ to $\RR$ that is closed under sums and products,
and which contains $g$ and $\ip{g}$ for each $g \in \mathrm{GP}_{i-1}$. By definition, for each {\gp} map $g \colon \RR^d \to \RR$,
there exists an integer $i \in \NN_0$ such that $g \in \mathrm{GP}_i$.
The height of a {\gp} map $g \colon \RR^d \to \RR$,
denoted by $\cmp(g)$, is the least of such integers $i \in \NN_0$.
More generally, if $\Omega \subset \RR^d$ and $g \colon \Omega \to \RR$ is a {\gp} map then $\cmp(g)$ is defined as the least possible value
of $\cmp(\tilde g)$ where $\tilde g \colon \RR^d \to \RR$ is a {\gp} map and $\tilde g|_{\Omega} = g$. For instance, if $g$ is the {\gp} map
$\ZZ \to \ZZ \subset \RR$ given by
\( g(n) = \ip{ \sqrt{2} n \ipnormal{\sqrt{3} n}+\sqrt{5}n^2}\)
then $\cmp(g) \leq 2$. For the following result, we recall that by convention we treat the empty product $\prod_{j=1}^{0}(\cdots)$ as being identically equal to $1$.
\begin{lemma}\label{lem:gp-sum-rep}
Let $d \in \NN$ and let $g \colon \RR^d \to \RR$ be a {\gp} map. Then $g$ can be written as
\begin{equation}\label{eq:gp-sum-rep}
g = \sum_{i=1}^s p_i \prod_{j=1}^{r_i} \ip{ h_{i,j} },
\end{equation}
where $s \in \NN$, $p_i \colon \RR^d \to \RR$ ($1 \leq i \leq s$) are polynomials, and for each $1 \leq i \leq s$, $r_i \in \NN_0$ and $h_{i,j} \colon \RR^d \to \RR$ ($1 \leq j \leq r_i$) are a GP maps with $\cmp(h_{i,j}) < \cmp(g)$.
\end{lemma}
\begin{proof}
We proceed by induction on $\cmp(g)$. If $g$ is a polynomial, there is nothing to prove, since we can take $s = 1$ and $p_1 = g$.
If $g = \ip{h}$ for some $h$ with $\cmp(h) < \cmp(g)$ then we can take $s = 1$, $r_1 = 1$, $p_1 = 1$, and $h_{1,1} = h$.
If $g = h + h'$ or $g = h \cdot h'$ for some $h$ and $h'$ with $\max\{\cmp(h),\cmp(h')\} < \cmp(g)$, then a representation of $g$ of the form
\eqref{eq:gp-sum-rep} can be obtained from the analogous representations of $h$ and $h'$.
\end{proof}
\subsection{Parametric generalised polynomials}\label{ssec:sc-para-gp}
It will be convenient to state some of our results in terms of \emph{parametric {\gp} maps}, by which we mean families of {\gp}
maps which include real-valued parameters as coefficients. For instance, the formula
\[ g_{\a,\b}(n) = \ip{ \a n \ip{\b n}+\sqrt{2}n^2}\]
defines, for each $\a,\b \in \RR$, a {\gp} map from $\ZZ$ to $\RR$, and we will refer to $g_{\bullet}$ as a parametric {\gp}
map\footnote{We use ``$\bullet$'' as a placeholder for a variable. Thus, in the discussion above, we let $g_{\bullet}$ denote
the map $\RR^2 \to \RR^{\ZZ}$, $(\a,\b) \mapsto g_{\a,\b}$, which can be identified with a map $\RR^2 \times \ZZ \to \RR$ in a natural way.}
$\ZZ \to \RR$. We make this notion precise in the following definition. Below and elsewhere,
if $I \subset J$ are finite sets and $\vec\beta \in \RR^J$, we let $\vec\beta|I \in \RR^I$ denote the restriction of $\beta$ to $I$,
that is, $(\beta|I)_i = \beta_i$ for all $i \in I$.
\begin{definition}
Let $I$ be a finite set. A \emph{{\pgp} map} $g_{\bullet} \colon \ZZ \to \RR$ \emph{with index set I} is a map $\RR^I \to \RR^{\ZZ}$, $\a \mapsto g_{\a}$,
such that the combined map $\RR^I \times \ZZ \to \RR$, $(\a,n) \mapsto g_{\a}(n)$ is a {\gp} map.
The height of the {\pgp} map $g_\bullet$, denoted by $\cmp(g_\bullet)$, is the height of the corresponding {\gp} map $(\vec\a,n) \mapsto g_{\vec\a}(n)$.
\end{definition}
\begin{definition}
Let $g_\bullet,h_\bullet \colon \ZZ \to \RR$ be two {\pgp} maps with index sets ${I}$ and ${J}$ respectively. We define the sum
$ g_\bullet + h_\bullet $ to be the {\pgp} map with index set $I \cup J$ given by $(g+h)_{\vec\a}(n) = g_{\vec\a|I}(n) + h_{\vec\a|J}(n)$, $\vec\a \in \RR^{I\cup J}$, $n \in \ZZ$.
The product $g_\bullet \cdot h_\bullet$ is defined accordingly by $(g \cdot h)_{\vec\a}(n) = g_{\vec\a|I}(n) \cdot h_{\vec\a|J}(n)$, $\vec\a \in \RR^{I\cup J}$, $n \in \ZZ$.
\end{definition}
Our interest in {\pgp} maps stems largely from the following lemma, which allows us to replace the study of subwords in {\gp}
sequences by the study of prefixes in {\pgp} sequences.
\begin{lemma}\label{lem:subword->prefix}
Let $g \colon \ZZ \to \RR$ be a bounded {\gp} map. Then there exists a {\pgp} map
$\tilde g_{\bullet}$ with index set $I$, such that, for each $m \in \ZZ$, there exists $\vec\a \in [0,1)^I$
such that $g(n+m) = \tilde g_{\vec\a}(n)$ for all $n \in \ZZ$.
\end{lemma}
\begin{proof}
This is just a rephrasing of Lemma \ref{lem:nil:shift}.
\end{proof}
In the examples below, we use, as in Section \ref{sec:constr}, the Iverson bracket convention:
$\braif{\varphi} = 1$ if $\varphi$ is a true sentence and $\braif{\varphi} = 0$ otherwise.
\begin{example}
Let $d \in \NN$ and $g(n) = \braif{ \fp{ \sqrt{2}n^d } < \frac{1}{2} }$. Pick any $m \in \ZZ$.
Then $g(n+m) = \tilde g_{\a}(n)$ for all $n \in \ZZ$, where $\a = (\a_i)_{i=0}^d$ is given by
$\alpha_i = \fp{ \sqrt{2} m^i \binom{d}{i} }$ ($0 \leq i \leq d$) and $\tilde g_{\bullet}$ is given by
\[
\tilde g_{\a}(n) = \braif{ \fp{ \sum_{i=0}^d \alpha_i n^{d-i}} < \frac{1}{2} }\,, \qquad \a \in \RR^{d+1},\ n \in \ZZ\,.
\]
\end{example}
\begin{example}\label{ex:par-gp} Let $g(n) = \braif{ \fp{ \sqrt{2}n \ip{\sqrt{3} n} } < \frac{1}{2} }$. Pick any $m \in \ZZ$ and
let $a= \ip{\sqrt{2}m}$, $\a ={\sqrt{2}m}$, $b = \ip{\sqrt{3}m}$, $\b = \fp{\sqrt{3}m}$, $\gamma = \fp{\sqrt{2}b}$, and $\delta = \fp{\a b}$. Then
\begin{align*}
g(n+m) &= \braif{ \fp{ (\sqrt{2}n +a+\a) \bra{ \ip{\sqrt{3} n + \b} +b} } < \frac{1}{2} } \\
& = \braif{ \fp{ (\sqrt{2}n+\a) { \ip{\sqrt{3} n + \b}} + \sqrt{2}b n + \a b } < \frac{1}{2} } \\
& = \braif{ \fp{ (\sqrt{2}n+\a) { \ip{\sqrt{3} n + \b}} + \gamma n + \delta } < \frac{1}{2} } \\ &= \tilde g_{\a,\b,\gamma,\delta}(n)\,,
\end{align*}
where $\tilde g_{\bullet}$ is the parametric {\gp} map with index set $\{1,2,3,4\}$ given by
\[
\tilde g_{\a_1,\a_2,\a_3,\a_4}(n) = \braif{ \fp{ (\sqrt{2}n+\a_1) { \ip{\sqrt{3} n + \a_2}} + \a_3 n + \a_4 } < \frac{1}{2} }\, .
\]
\end{example}
When the dependence of a {\pgp} map on the parameters becomes too complicated, it is often more convenient to instead work
with a {\pgp} map which has more parameters but depends on them in a simpler way.
For instance, if $g_{\bullet}$ is a {\pgp} map of the form
\begin{equation}\label{eq:295:1}
g_{\vec \a}(n) = f_1(\vec \a) h_{\vec \a}^{(1)}(n) + f_2(\vec \a) h_{\vec \a}^{(2)}(n) + f_3(\vec \a) h_{\vec \a}^{(3)}(n)\, ,
\end{equation}
where $f_1$, $f_2$, and $f_3$ are {\gp} maps and $h^{(1)}_\bullet$, $h^{(2)}_\bullet$, and $h^{(3)}_\bullet$
are {\pgp} maps,
then it might be preferable to instead work with the {\pgp} map
\begin{equation}\label{eq:295:2}
g_{\vec \a,\vec \b}'(n) = \b_1 h_{\vec \a}^{(1)}(n) + \b_2 h_{\vec \a}^{(2)}(n) + \b_3 h_{\vec \a}^{(3)}(n)\,.
\end{equation}
We make this idea precise in the following definition.
\begin{definition}\label{def:ind:succ}
Let $g_\bullet$ and $h_\bullet$ be two {\pgp} maps with index sets $I$ and $J$ respectively.
Then we say that $h_\bullet$ \emph{extends} $g_\bullet$, denoted $h_\bullet \succeq g_\bullet$, if there exists a {\gp} map
$\varphi \colon \RR^I \to \RR^J$ such that $g_{\vec \a} = h_{\varphi(\vec\a)}$ for all $\vec\a \in \RR^I$.
\end{definition}
It is routine to check that the relation $\succeq$ defined above is a partial order.
\begin{example}
If $g_\bullet$ and $g'_\bullet$ are respectively given by \eqref{eq:295:1} and \eqref{eq:295:2},
then $g'_\bullet \succeq g_\bullet$. One can take $\varphi(\vec\a) = (\vec\a, f_1(\vec\a),f_2(\vec\a),f_3(\vec\a))$.
\end{example}
\subsection{Induction scheme}\label{ssec:sc-induction}
Using the terminology introduced above, we are ready to explain the induction scheme that will be used in the proof of Theorem \ref{thm:A}.
It can be construed as an analogue of the inductive definition of generalised polynomials, but restricted to $\ZZ$-valued sequences.
Note that {\gp} maps from $\ZZ$ to $\RR$ can be identified with {\pgp} maps with an empty index set.
\begin{proposition}\label{prop:gen-poly-induction}
Let $\cG$ be a family of {\pgp} maps from $\ZZ$ to $\ZZ$ with index sets contained in $\NN_0$. Suppose that $\cG$ has the following closure properties.
\begin{enumerate}
\item\label{it:A-1} All {\gp} maps $\ZZ \to \ZZ$ belong to $\cG$.
\item\label{it:A-2} For every $g_{\bullet}$ and $h_{\bullet} \in \cG$, it holds that $g_{\bullet}+h_{\bullet} \in \cG$ and $g_{\bullet} \cdot h_{\bullet} \in \cG$.
\item\label{it:A-4} For every $g_\bullet \in \cG$, $\cG$ contains all the {\pgp} maps $g'_\bullet \colon \ZZ \to \ZZ$ satisfying $g_\bullet \succeq g'_\bullet$.
\item\label{it:A-3} For every pair of disjoint finite sets $I \subset \NN$, $J \subset \NN$, and every sequence of {\pgp} maps $h^{(i)}_{\bullet} \in \cG$, $i \in I$,
with index set $J$, $\cG$ contains the {\pgp} map $g_\bullet$ defined by
\[ g_{\vec \a,\vec\b}(n) = \ip{\sum_{i\in I} \a_i h^{(i)}_{\vec\b}(n)}\,, \qquad n \in \ZZ\,,\ \vec\a \in \RR^{I}\,,\ \vec\b \in \RR^J\,.\]
\end{enumerate}
Then $\cG$ contains all {\pgp} maps $\ZZ \to \ZZ$ with index sets contained in $\NN_0$.
\end{proposition}
\begin{proof}
Since each {\pgp} map from $\ZZ$ to $\ZZ$ takes the form $\ip{g_\bullet}$ for some {\pgp} map $g_\bullet \colon \ZZ \to \RR$,
it suffices to show that $\ip{g_\bullet} \in \cG$ for each {\pgp} $g_\bullet$ with index set $I \subset \NN_0$.
We proceed by induction on $\cmp(g_\bullet)$.
Suppose first that $\cmp(g_\bullet) = 0$, that is, $g_\bullet$ is a polynomial. Expanding, we can write
\[
g_{\vec\a}(n) = \sum_{i=0}^d q_i(\vec \a) n^i\,, \qquad n \in \ZZ\,,\ \vec\a \in \RR^I\,,
\]
where $d \in \NN_0$ and $q_i \colon \RR^I \to \RR$ are polynomials.
It follows from \ref{it:A-1} and \ref{it:A-3} that $\cG$ also contains the {\pgp} map with index set $\{0,1,\dots,d\}$ given by
\[
g'_{\vec\a}(n) = \ip{\sum_{i=0}^d \a_i n^i}\,, \qquad n \in \ZZ,\ \vec\a \in \RR^{d+1}\, .
\]
Since $g'_\bullet \succeq \ip{g_\bullet}$, it follows from $\ref{it:A-4}$ that $g_\bullet \in \cG$.
Suppose next that $\cmp(g_\bullet) \geq 1$. Using Lemma \ref{lem:gp-sum-rep} and expanding out the polynomial contributions,
we can represent $g_{\bullet}$ in the form
\[
g_{\vec\b}(n) = \sum_{i=1}^s q_i(\vec \b) \tilde h^{(i)}_{\vec\b}(n)\,, \qquad n \in \ZZ\,,\ \vec\b \in \RR^J\,,
\]
where for every $i$, $1 \leq i \leq s$, $q_i$ is a polynomial, and $\tilde h^{(i)}_{\bullet}$ is a {\pgp} map of the form
\[ \tilde h^{(i)}_{\vec\b}(n) = n^{d_i} \prod_{j=1}^{r_i} \ip{h^{(i,j)}_{\vec\b}(n)} \qquad n \in \ZZ\,,\ \vec\b \in \RR^J\,, \]
where $d_i \in \NN_0$, $r_i \in \NN_0$, and $\cmp( h^{(i,j)}_\bullet) < \cmp(g_\bullet)$ for all $j$, $1 \leq j \leq r_i$.
By the inductive assumption, for each pair $(i,j)$, with $1 \leq i \leq s$ and $1 \leq j \leq r_i$, we have $\ip{h^{(i,j)}_\bullet }\in \cG$.
Consequently, applying \ref{it:A-1} and \ref{it:A-2}, we conclude that also $\tilde h^{(i)}_{\bullet} \in \cG$.
For notational convenience, assume that $\min J > s$ and put $I = \{1,\dots,s\}$
(since the ordering of the parameters does not play any role, this does not decrease the level of generality).
It follows from \ref{it:A-3} that $\cG$ also contains the {\pgp} map with index set ${I \cup J}$ that is defined by
\[
g'_{\vec\a,\vec\b}(n) = \ip{\sum_{i=1}^s \a_i \tilde h^{(i)}_{\vec\b}(n) }\,, \qquad n \in \ZZ\,,\ \vec\a \in \RR^{I}\,,\ \vec\b \in \RR^{J}\,.
\]
Since $g'_\bullet \succeq \ip{g_\bullet}$, it follows from \ref{it:A-4} that $\ip{g_\bullet} \in \cG$.
This ends the proof.
\end{proof}
\section{Proof of Theorem \ref{thm:A}: auxiliary results}\label{sec:sc-auxiliary}
In this section we discuss some results in additive combinatorics and Diophantine approximation which will be used
in the course of the proof of Theorem \ref{thm:A}.
\subsection{Additive geometry}
Let $r\in \NN$. A \emph{symmetric generalised arithmetic progression} of rank $r$ in an abelian group $Z$ with steps $x_1,x_2,\dots,x_r \in Z$ and side lengths
$\ell_1,\ell_2,\dots, \ell_r \in \NN_0$ is defined as the set
\[
\fS(x_1, x_2,\dots x_r; \ell_1,\ell_2,\dots, \ell_r) =
\set{ \sum_{i=1}^r n_i x_i}{ - \ell_i < n_i < \ell_i \text{ for all } i,\,1 \leq i \leq r }\,.
\]
Let $d \in \NN$. By a \emph{lattice} in $\RR^d$ we mean a discrete subgroup of $\RR^d$.
In particular, we do not require $\Lambda$ to have full rank, which is slightly non-standard but consistent (in particular with \cite{TaoVu-book}).
If $\Lambda < \RR^d$ is a lattice of full rank, we let $\covol \Lambda = \vol(\RR^d/\Lambda)$ denote its covolume
(that is, the measure of a fundamental domain of $\Lambda$).
By a \emph{convex body} in $\RR^d$, we mean a convex, open, non-empty, and bounded set, and by a \emph{half-space},
we always mean a closed half-space, that is, a set of the form
\[
\fH(\vec x, t) = \set{\vec y \in \RR^d}{ \vec x \cdot \vec y \geq t}
\]
for some $\vec x \in \RR^d$ and $t \in \RR$.
The classical John's theorem asserts that each symmetric convex body $B$ in $\RR^d$ can be efficiently approximated by an ellipsoid $E$,
in the sense that $E \subset B \subset \sqrt{d} E$. We will use a discrete analogue of this result.
\begin{theorem}[Discrete John's Theorem, {\cite[Thm.{} 3.36]{TaoVu-book}}]\label{thm:John}
Let $d \in \NN$, $B \subset \RR^d$ be a symmetric convex body, and $\Lambda < \RR^d$ be a lattice of rank $r$.
Then there exist vectors $\vec v_1,\vec v_2,\dots, \vec v_r \in \RR^d$ and integers $\ell_1,\ell_2,\dots,\ell_r \in \NN$ such that
\begin{align*}
\fS(\vec v_1,\vec v_2,\dots, \vec v_r; \ell_1,\ell_2,\dots,\ell_r)
& \subset B \cap \Lambda \\
&\subset \fS(\vec v_1,\vec v_2,\dots, \vec v_r; r^{2r}\ell_1,r^{2r}\ell_2,\dots,r^{2r}\ell_r)\,.
\end{align*}
\end{theorem}
We also mention an estimate on the number of partitions of a finite set using a hyperplane.
\begin{theorem}[{\cite[Thm.\ 1]{Harding-1966}}]\label{thm:partition-cell}
Let $d$ and $n$ be two positive integers, and let $S \subset \RR^d$ be a set with $\abs{S} = n$.
Then the number of sets of the form $S \cap H$, where $H$ is a half-space, is at most $2\sum_{i=0}^d \binom{n-1}{i}$.
\end{theorem}
In the course of the argument, we will need estimates on the number of sets that can be obtained from a given convex body in $\RR^d$
by intersecting it with a sub-lattice of $\ZZ^d$ and a half-space.
\begin{proposition}\label{prop:half-lattice-count}
Let $d \in \NN$, $\Gamma < \RR^d$ be a lattice, and $B \subset \RR^d$ be a symmetric convex body.
Then the number of pairs of sets of the form $\bra{ \Lambda \cap B, \Lambda \cap B \cap H }$,
where $\Lambda < \Gamma$ is a lattice and $H$ is a half-space, belongs to $O_d(\abs{B \cap \Gamma}^{2d})$.
\end{proposition}
\begin{proof}
Put $M = \abs{B \cap \Gamma}$. Let $\Lambda < \Gamma$ be a sub-lattice.
It follows form Theorem \ref{thm:John} that there exist vectors $\vec v_1,\dots,\vec v_d \in \Lambda \cap B$ such that
\(
\Lambda \cap B \subset \spanZ\{\vec v_1, \dots, \vec v_d\}.
\)
As a consequence,
\[
\Lambda \cap B = \spanZ\{\vec v_1, \dots, \vec v_d\} \cap B\,.
\]
In particular, the set $\Lambda \cap B$ is completely determined by the vectors $\vec v_1, \dots, \vec v_d$ which belong to $B \cap \Gamma$, and
hence can be chosen in at most $M^d$ ways.
For a fixed choice of $\Lambda$, we infer from Theorem \ref{thm:partition-cell} that the number of sets of the form $\Lambda \cap B \cap H$,
where $H$ is a half-space, belongs to $O_d(M^d)$. Combining these two estimates yields the claim.
\end{proof}
\newcommand{\Q}{q}
\renewcommand{\P}{p}
\subsection{Diophantine approximation}
In this subsection, we briefly discuss approximate linear relations with integer coefficients.
Let $d \in \NN$. Given a vector $\a = (\a_i)_{i=1}^d \in \RR^d$, $\e > 0$, and $N \in \NN$, we set
\begin{equation}\label{eq:def-of-R}
\cR_N(\vec \a, \e) = \set{ \vec n \in \ZZ^d }{\norm{n}_{\infty} < N, \ \abs{\sum_{i=1}^d n_i \a_i } < \e}\,.
\end{equation}
These sets naturally appear in the theory of Diophantine approximation. For instance, a classical theorem of Dirichlet asserts that for all
$\alpha \in [0,1)^d$, the set $\cR_{N}(\a,N^{-1/(d-1)})$ is non-empty, and the exponent ${-1/(d-1)}$ cannot be improved in general \cite{Cassels-1955}.
In our application, we will be interested in the case where $\e$ is considerably smaller. In this regime,
the sets $\cR_N(\vec\a, \e)$ can be approximated by lattices in a sense that is made precise by the following proposition.
\begin{proposition}\label{prop:lattice-approx}
Let $d \in \NN$. There exists a positive real number $C_d$ such that for each $\vec\a = (\a_i)_{i=1}^d \in \RR^d$, $\e > 0$, and $N \in \NN$,
there exists a lattice $\Lambda = \Lambda(\vec\a, \e, N)$ satisfying
\begin{equation}\label{eq:def-of-Lambda}
\cR_N(\vec\a,\e) \subseteq \Lambda \cap (-N,N)^d \subseteq \cR_N(\vec\a, C_d N^d \e)\,.
\end{equation}
\end{proposition}
For later reference, we let $\Lambda(\vec\a, \e, N)$ denote a lattice satisfying \eqref{eq:def-of-Lambda}.
The remaining part of this section is devoted to the proof of Proposition \ref{prop:lattice-approx}.
We begin with a lemma concerning finite groups, where the situation is simpler.
Recall that, for an abelian group $G$ and $g \in G$, the order $\ord(g)$ is the least $q \in \NN$ with $qg = e_G$, the order of $G$ is the cardinality of $G$, and the exponent of $G$ is the least common multiple of $\ord(g)$ for $g \in G$. For a set $S \subset G$ an and $k \in \NN$, we let $kS$ denote the $k$-fold sumset
$S+S+ \dots + S = \set{a_1+a_2+\dots+a_k}{ a_i \in S}$.
\begin{lemma}\label{lem:fill-group}
Let $G$ be a finite abelian group with order $M \geq 2$ and exponent $\Q$, and let $S \subset G$ be a generating set with $e_G \in S$.
Then there exists $k \leq {\Q \log M}/{\log \Q}$ such that $kS = G$.
\end{lemma}
\begin{proof}
We proceed by induction on $M$. Pick any $g \in S \setminus \{e_G\}$ and let $\ell = \ord(g)$.
Since $G$ is non-trivial, we may assume that $\ell \geq 2$. Put $f(x) = {x}/{\log x}$ ($x > 1$) and note that $f(n) \leq f(m)$ for all pairs of integers
$(n,m)$ with $m\geq n\geq 2$ and $n \mid m$. Our goal is to find $k \leq f(q)\log M$ with $kS = G$.
Let $\bar G = G/\left< g \right>$, and let $\pi \colon G \to \bar G$ be the quotient map.
Then $G$ has order $\bar M = M/\ell$ and exponent $\bar \Q$ which divides $\Q$.
The set $\bar S = \pi(S)$ generates $\bar G$. If $\bar G$ is trivial then $G$ is the cyclic group generated by $g$,
and hence $\Q = M$ and $k S = G$ for $k = f(\Q) \log M = M$, as needed. (Note that in this step we use the assumption that $e_G \in S$.)
Suppose next that $\bar G$ is non-trivial, meaning that $\bar M \geq 2$. By the inductive assumption, there exists $\bar k \leq f(\bar \Q) \log \bar M$
with $\bar k \bar S = \bar G$. It follows that $k S = G$, where $k = \bar k + \ell$. We can estimate
\[ k \leq f(\bar \Q) \log \bar M + \ell = f(\Q) \log M - \bra{f(\Q) - f(\bar \Q)}\log M + \bra{ f(\ell) - f(\bar \Q)} \log \ell.\]
Thus, it is enough to show that
\[ \bra{f(\Q) - f(\bar \Q)}\log M \geq \bra{ f(\ell) - f(\bar \Q)} \log \ell \,\]
which follows from the observations that $M \geq \ell$, $\ell\mid \Q$, and $\bar \Q \mid \Q$.
\end{proof}
\begin{lemma}\label{lem:span-Z-quant}
Let $d \in \NN$, $B \subset \RR^d$ be symmetric convex body, and $S \subset B$ be a symmetric set with $0 \in S$.
Put $\Lambda = \spanZ S$ and assume that $\Lambda$ is a lattice.
Then there exists $k = O_d(\abs{\Lambda \cap B})$ such that
\( kS \cap B = \Lambda \cap B.\)
\end{lemma}
\begin{proof}
Put $M= \abs{\Lambda \cap B}$.
We proceed by induction on $d$, including the degenerate case $d = 0$ for which one can take $k = 1$.
Assume that $d \geq 1$ and that the claim is proved for $d-1$. We may also assume without loss of generality that $\Lambda$ has full rank,
since otherwise we could replace $\RR^d$ with the subspace spanned by $\Lambda$.
Applying Theorem \ref{thm:John}, we conclude that there exist vectors $\vec w_1,\vec w_2, \dots, \vec w_d \in \Lambda$
and side lengths $\ell_1, \ell_2,\dots, \ell_d \in \NN$ such that
\[
\fS(\vec w_1, \vec w_2, \dots, \vec w_d; \ell_1,\ell_2,\dots,\ell_d) \subset B \cap \Lambda
\subset \fS(\vec w_1, \vec w_2, \dots, \vec w_d; D\ell_1,D\ell_2,\dots,D\ell_d)\,,
\]
where $D = d^{2d} = O_d(1)$. Since $\Lambda$ has rank $d$, the vectors $w_1,w_2,\dots,w_d$ form a basis of $\RR^d$.
Note that $\prod_{i=1}^d \ell_i \leq M$. Applying a change of basis, we may assume that $\vec w_i = \vec e_i$,
the $i$-th standard basis vector, for all $i$, $1 \leq i \leq d$. In particular, $\Lambda = \ZZ^d$.
Pick any $d$-tuple of linearly independent vectors $\vec v_1,\vec v_2,\dots,\vec v_d \in S$.
Let $A \in \RR^{d \times d}$ be the matrix satisfying $A \vec e_i = \vec v_i$, and put $\Q = \det A = \covol\bra{ \spanZ(\vec v_1, \vec v_2,\dots, \vec v_d)}$.
Then $\Q \neq 0$ and the Leibniz formula gives the estimate $\abs{\Q} \leq d! D^d \prod_{i=1}^d \ell_i = O_d(M)$.
Using Cramer's rule to find the inverse of $A$, we observe that the matrix $\Q A^{-1}$ has integer entries and satisfies
$\abs{ \Q ( A^{-1})_{i,j} } = O_d(M/\ell_i)$ for each pair $(i,j)$ with $1 \leq i,j \leq d$.
Since $\Q e_i = \sum_{j=1}^d \Q (A^{-1})_{i,j} v_j$ and since $0 \in S$, we conclude that there exists $\P = O_d(M)$
such that $\Q \vec e_i \in \floor{\P/\ell_i} S$ for every $i$, $1 \leq i \leq d$.
Pick any $\vec u \in \ZZ^d \cap B$. Since $\vec u \in \ZZ^d$, by Lemma \ref{lem:fill-group} applied to the group $G = \ZZ^d/\Q\ZZ^d$
and the set\footnote{Note that $\spanZ S=\Lambda=\mathbb Z^d$. Hence it makes sense to consider the set $S\bmod q$. Furthermore, since by assumption
$0\in S$, it follows that $0\in S\bmod q$ as needed.} $\bra{ S \bmod \Q} \subset G$, there exists $\pvec u' = (u_i')_{i=1}^d \in \ZZ^d$ such that $\vec u \in \Q\pvec u' + n S$ for some $n \leq d\Q$. Then for every $i$, $1 \leq i \leq d$, we have $\abs{u_i'} \leq D \ell_i$, so $\Q \pvec u' \in m S$, where
\[
m = \sum_{i=1}^d \abs{u_i'} \floor{\P/\ell_i} \leq d D \P = O_{d}(M)\,.
\]
It follows that we can take $k = n + m = O_d(M)$.
\end{proof}
\begin{example}
Pick any $d,N \in \NN$. Let $S = \{\vec 0, \pm \vec v_1, \pm \vec v_2, \dots, \pm \vec v_d\}$, where $\vec v_1 = \vec e_1$ and
$\vec v_i = \vec e_i - N \vec e_{i-1}$ for every $i$, $2 \leq i \leq d$. Thus, $\spanZ(S) = \ZZ^d$. We can compute that
\[
\vec e_d = \vec v_{d} + N \vec v_{d-1} + \dots + N^{d-1} \vec v_1\,.
\]
Taking $B = (-2N,2N)^d$, we see that $\abs{B \cap \ZZ^d} \leq (4N)^d$ and if $k S \supset B \cap \ZZ^d$ then $k \geq N^d$. Hence, the bound in Lemma \ref{lem:span-Z-quant} is tight up to a constant factor.
\end{example}
\begin{proof}[Proof of Proposition \ref{prop:lattice-approx}]
Apply Lemma \ref{lem:span-Z-quant} to $B = (-N,N)^d$ and $S = \cR_{N}(\vec\a,\e)$.
It is immediate from the definition that $\cR_N(\a,\e) \subset \Lambda \cap B$, where $\Lambda = \spanZ S$ is a sub-lattice of $\ZZ^d$.
For the other inclusion, we note that $\Lambda \cap B = kS = k \cR_N(\a,\e) \subset \cR_N(\a,k\e)$, where $k = O_d(\abs{\Lambda \cap B}) = O_d(N^d)$.
\end{proof}
\section{Proof of Theorem \ref{thm:A}: combining the ingredients}\label{sec:sc-proof}
We are now ready to prove a proposition which serves as the inductive step in the proof Theorem \ref{thm:A}.
In the argument, we will use the following elementary fact.
\begin{lemma}\label{lem:sc-pr:tech}
Let $x$ and $x^*$ be two real numbers and assume that $\abs{x-x^*} < \min\bra{ \abs{x-\nint{x^*}}, 1/2}$. Then $\ip{x} = \ip{x^*}$.
\end{lemma}
\begin{proof}
If $\fp{x^*} \in [0,1/2)$, then $\abs{x-\nint{x^*}} > \abs{x-x^*}$ implies that $x > \nint{x^*} = \ip{x^*}$, and $\abs{x-x^*} < 1/2$
implies that $x < \ip{x^*} + 1$. Thus, $\ip{x} = \ip{x^*}$. The case where $\fp{x^*} \in [1/2,1)$ is analogous.
\end{proof}
\begin{proposition}\label{prop:inductive}
Let $d \in \NN$, let $h_1,h_2,\dots,h_d \colon [N] \to \ZZ$ be sequences, with $\norm{h_i}_{\infty} \leq H$ for every $i$, $1 \leq i \leq d$.
For $\vec\a \in \RR^d$, we let $g_{\vec\a} \colon [N] \to \ZZ$ denote the sequence defined by
\[ g_{\vec\a}(n) = \ip{\sum_{i=1}^d \alpha_i h_i(n)}\,,\quad n \in [N]\,.\]
Then $\abs{\set{ g_{\vec\a}}{ \vec\a \in [-R,R)^d}}=O_d(R^dH^{3d^2})$.
\end{proposition}
\begin{proof}
Note that $g_{\vec\a}(n) = g_{\fp{\vec\a}}(n) + g_{\ip{\vec\a}}(n)$, where $\fp{\vec\a} \in [0,1)^d$ and
$\ip{\vec\a} \in \{-R,-R+1,\dots,R-2,R-1\}^d$, which is a set whose cardinality is in $O_d(R^d)$.
Thus, it remains to show that the cardinality of the set $\set{ g_{\vec\a}}{ \vec\a \in [0,1)^d}$ is in $O_d(H^{3d^2})$.
Pick any $\vec\a \in [0,1)^d$. Our plan is to construct an alternative description of $g_{\vec\a}$ that, instead of $\vec\a$, includes a finite number
of parameters chosen from sets whose cardinality can be estimated in an easier way.
Put $\e = 1/\bra{100C_d(dH)^d}$, where $C_d$ is the constant from Proposition \ref{prop:lattice-approx}.
Let $\pvec\a^* = (\a_i^*)_{i=1}^d \in [0,1)^d$ be any vector such that $\abs{\a_i - \a^*_i} < \e/dH$ and $\a_i^* \in (\e/d H)\ZZ$ for every $i$,
$1 \leq i \leq d$. This ensures that
\begin{equation}\label{eq:521:1}
\abs{\sum_{i=1}^d \a_i h_i(n) - \sum_{i=1}^d \a_i^* h_i(n) } < \e\,, \qquad \forall\ n \in [N]\,.
\end{equation}
Let $h_0 \colon [N] \to \ZZ$ be the map given by
\begin{equation}\label{eq:521:15}
h_0(n) = - \nint{\sum_{i=1}^d \a_i^* h_i(n) }
\end{equation}
and let $\vec h = (h_0,h_1,h_2,\dots,h_d) \colon [N] \to \ZZ^{d+1}$.
Note that $\norm{h_0}_\infty \leq dH$ and, since $\e < 1/2$, that $h_0(n) + g_{\vec\a}(n) \in \{0,-1\}$ for all $n \in [N]$.
Put $B = [-dH,dH] \times [-H,H]^d$. Let also $\Lambda = \Lambda(1\vec\a, \e, d H)$ be the lattice constructed in
Proposition \ref{prop:lattice-approx} and set
\[\Lambda^+ = \set{\vec m \in \Lambda}{ 1\vec\a \cdot \vec m \geq 0}\,.\]
Here and elsewhere, we use the shorthand $1\vec\a = (1,\a_1,\a_2,\dots,\a_d)$. Hence $(1\a)_0 = 1$ and $(1\a)_i = \a_i$ for every $i \geq 1$.
We have now introduced all the objects needed to obtain a more well-behaved formula for $g_{\vec\a}(n)$.
We consider two cases, depending on whether $\vec h(n) \in \Lambda$ or not. If $\vec h(n) \in B \setminus \Lambda$,
then $\abs{1\vec\a \cdot \vec h(n)} > \e$ and hence, by Lemma \ref{lem:sc-pr:tech}, we find
\begin{align}\label{eq:521:2}
g_{\vec\a}(n) = \ip{\sum_{i=1}^d \a_i h_i(n)} = \ip{\sum_{i=1}^d \a_i^* h_i(n)}\,.
\end{align}
Next, if $\vec h(n) \in B \cap \Lambda$, then $\abs{1\vec\a \cdot \vec h(n)} < C_d(dH)^{d} \e < 1/2$. Since $h_0(n) \in \ZZ$, we have
\begin{align}\label{eq:521:3}
g_{\vec\a}(n) = -h_0(n) + \ip{1\vec\a \cdot \vec h(n)} \,.
\end{align}
If $n \in \Lambda^+$, then $1\vec\a \cdot \vec h(n) \in [0,\frac{1}{2})$ and hence $\ip{1\vec\a \cdot \vec h(n)} = 0$.
Similarly, if $\vec h(n) \in \Lambda \setminus \Lambda^+$, then $1\vec\a \cdot \vec h(n) \in (-\frac{1}{2},0)$ and
hence $\ip{1\vec\a \cdot \vec h(n)} = -1$.
Combining \eqref{eq:521:2} and \eqref{eq:521:3}, and expanding out the definition of $h_0$, we conclude that
\begin{align}\label{eq:521:4}
g_{\vec\a}(n) =
\begin{dcases}
\ip{\sum_{i=1}^d \a_i^* h_i(n)} & \text{ if } \vec h(n) \in B \setminus \Lambda\,, \\
\ip{\sum_{i=1}^d \a_i^* h_i(n)+\frac{1}{2}} & \text{ if } \vec h(n) \in B \cap \Lambda^+ \,,\\
\ip{\sum_{i=1}^d \a_i^* h_i(n)-\frac{1}{2}} & \text{ if } \vec h(n) \in B \cap \Lambda \setminus \Lambda^+.
\end{dcases}
\end{align}
It follows from \eqref{eq:521:4} that, in order to determine $g_{\vec\a}|_{[N]}$, it is sufficient to know the following data:
$\pvec\a^*$, $B \cap \Lambda$, and $B \cap \Lambda^+$. The number of possible choices of $\pvec\a^*$ is in $O_d(H^{d^2})$ directly by the definition.
Furthermore, it follows from Proposition \ref{prop:half-lattice-count}, applied with $\Gamma = \ZZ^d$, that the number of possible choices for $B \cap \Lambda$
and $B \cap \Lambda^+$ are both in $O_{d}\bra{ H^{2d^2} }$. Consequently, the number of distinct sequences of the form $g_{\vec\a}|_{[N]}$ is in
$O_{d}\bra{H^{3d^2}}$. This ends the proof.
\end{proof}
We have now collected all the components needed to prove our main result.
\begin{proposition}\label{thm:main}
Let $g_\bullet:\ZZ \to \ZZ$ be a {\pgp} map with index set $I \subset \NN_0$.
Then there exist two positive real numbers $B = B(g_\bullet)$ and $C = C(g_\bullet)$ such that
\begin{equation}\label{eq:469:1}
\abs{ \set{g_{\vec\a}|_{[N]}}{\a \in \RR^I,\ \norm{\vec\a}_{\infty} \leq R }} = O_g\bra{R^B N^{C}}\,.
\end{equation}
\end{proposition}
\begin{proof}
We proceed by structural induction with respect to $g_\bullet$, using the scheme introduced by Proposition \ref{prop:gen-poly-induction}.
Let $\cG'$ denote the set of all {\pgp} maps $g_\bullet$ such that \eqref{eq:469:1} holds.
\begin{enumerate}
\item If $g \colon \ZZ \to \ZZ$ is a {\gp} map (viewed as a {\pgp} map with empty index set), then the set in \eqref{eq:469:1} has only one element
and thus $g \in \cG$ holds trivially.
\item For each $g_\bullet,h_\bullet \in \cG'$ with index sets $I,J \subset \NN_0$ respectively, and for each $ N \in \NN$, we note that
$(g + h)_{\gamma}|_{[N]}$, $\gamma \in \RR^{I\cup J}$, is uniquely determined by $g_{\a}|_{[N]}$ and $h_{\b}|_{[N]}$,
where $\a = \gamma|I$ and $\b = \gamma|J$. As a consequence, $g_\bullet + h_{\bullet} \in \cG'$ and we can take
\begin{equation}\label{eq:469:2}
B(g_\bullet + h_{\bullet}) \leq B(g_\bullet) + B(h_{\bullet}) \;\;\mbox{ and }\;\;
C(g_\bullet + h_{\bullet}) \leq C(g_\bullet) + C(h_{\bullet})\,.
\end{equation}
The same reasoning also applies to the product $g_\bullet \cdot h_{\bullet} $, with the same bounds.
\item Next, let $g_\bullet$ and $g_\bullet'$ be two {\pgp} maps with index sets $I,J \subset \NN_0$ respectively,
and suppose that $g_\bullet \in \cG'$ and $g_\bullet \succeq g'_\bullet$. By Definition \ref{def:ind:succ},
there exists a {\gp} map $\varphi \colon \RR^J \to \RR^I$ such that $g_{\vec\b}' = g_{\varphi(\vec\b)}$ for all $\b \in \RR^J$.
Since $\varphi$ is a {\gp} map, there exists a constant $D = D(\varphi)$ such that $\norm{\varphi(\b)}_\infty \leq D R^D$
for all $\b \in \RR^J$ with $\norm{\b}_\infty \leq R$. Hence, for every pair of positive integers $(N,R)$, we have
\begin{align*}
\abs{\set{ g'_{\vec\b}|_{[N]} }{\b \in \RR^J,\ \normbig{\vec\b}_{\infty} \leq R}}
& \leq
\abs{ \set{ g_{\vec\a}|_{[N]} }{\a \in \RR^I,\ \norm{\vec\a}_{\infty} \leq D R^D } },
\end{align*}
which implies that $g'_\bullet \in \cG'$ and that we can take
\begin{equation}\label{eq:469:3}
B(g_\bullet') \leq D B(g_\bullet) \;\;\mbox{ and }\;\;
C(g'_\bullet) \leq C(g_\bullet)\,.
\end{equation}
\item Finally, let $I \subset \NN_0$ be a finite set, and, for every $i\in I$, let $h^{(i)}_\bullet \in \cG'$ be a {\pgp} map with index sets $J^{(i)} \subset \NN_0 \setminus I$.
Set $J = \bigcup_{i\in I} J^{(i)}$. Our aim is to show that $\cG'$ also contains the {\pgp} map $g_\bullet$ with index set $I \cup J$ defined by
\[
g_{\vec\alpha, \vec\b}(n) = \ip{ \sum_{i \in I} \alpha_i h^{(i)}_{\vec\b} (n)}\,,\qquad n \in \ZZ\,,\ \vec \alpha \in \RR^I\,,\ \vec\b \in \RR^J\,.
\]
Given any $N,R \in \NN$, we set
\[ H = \max_{i \in I} \max_{n \in [N]} \sup_{\normsml{ \vec\b }_{\infty} \leq R } \abs{h^{(i)}_{\vec\b}(n)}\,.\]
Since the $h^{(i)}_\bullet$'s are {\pgp} maps, we have $H = O_g(R^D N^D)$ for some constant $D = D(g_\bullet)$.
For every $i \in I$, set $B_i = B(h^{(i)}_\bullet)$, $C_i = C(h^{(i)}_\bullet)$, and
\[
\cH_i = \set{h^{(i)}_{\vec\b}|_{[N]}}{\b \in \RR^J,\ \norm{\vec\b}_{\infty} \leq R }\,.
\]
By Proposition \ref{prop:inductive}, for each $h_1 \in \cH_1, h_2 \in \cH_2, \dots, h_d \in \cH_d$,
the number of sequences of the form $\ip{\sum_{i=1}^d \alpha_i h_i}$, where $\vec\a \in \RR^d$ and $\norm{\vec\a}_\infty \leq R$,
belongs to $O_d(R^d H^{3d^2})$.
It follows that the number of distinct sequences of the form $g_{\vec\a,\vec\b}|_{[N]}$, where
$\a \in \RR^I$, $\b \in \RR^J$, and $\max\{\norm{\a}_\infty,\norm{\b}_\infty\} \leq R$, is at most
\[
\prod_{i=1}^d \abs{\cH_i} \cdot O_d\bra{R^d H^{3d^2}} = O_g(R^B N^C)\,,
\]
where
\begin{equation*}\label{eq:469:4}
B = \sum_{i \in I} B_i + d + 3d^2 D \;\;\mbox{ and }\;\;
C = \sum_{i \in I} C_i + 3d^2 D\,.
\end{equation*}
In particular, $g_{\bullet} \in \cG'$ and we can take $B(g_{\bullet})\leq B$ and $C(g_{\bullet})\leq C$.
\end{enumerate}
It now follows from Proposition \ref{prop:gen-poly-induction} that $\cG'$ contains all {\pgp} maps $\ZZ \to \ZZ$
with index sets contained in $\NN_0$. This ends the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:A}]
Let $\bb a$ be a bracket word defined over an alphabet $\Sigma$. We infer from Corollary \ref{cor:constr:gpword-nice} that
there exist a finitely-valued {\gp} map $g \colon \NN_0 \to \NN$ and
$c \colon g(\NN_0) \to \Sigma$ such that $\bb a = \brabig{ c(g(n)) }_{n=0}^\infty$.
Let $\bb g = (g(n))_{n=0}^\infty$. It follows from Proposition \ref{thm:main} and Lemma \ref{lem:subword->prefix} that $p_{\bb g}(N) = O_g(N^C)$ for some $C > 0$.
Since $p_{\bb a}(N) \leq p_{\bb g}(N)$ for all $N \in \NN$, this ends the proof.
\end{proof}
\color{black} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,154 |
Q: Center a Background Jumbotron Image and Make it Mobile Responsive I have been fighting with a jumbotron background image which I managed to properly line up on the website but I can't seem to make it be mobile responsive. The link for the site is puremistdiffusers.com
Here is the HTML:
<div class="jumbotron" style="background-image: url(https://puremistdiffusers.com/wp-content/uploads/2017/02/slider-candle-jpeg-dark.jpg); background-repeat: no-repeat;">
<div class="container">
<h1 class="text-uppercase">YOUR AROMATHERAPY</h1>
<p>UP TO 40% OFF ON ALL PRODUCTS</p>
<a class="btn btn-default btn-inverse btn-lg text-uppercase topbutton" href="https://puremistdiffusers.com/product/" role="button">Shop Now</a>
</div>
</div>
and here is the CSS:
}
.jumbotron {
background-color: #fff !important;
}
.jumbotron {
padding-top: 48px;
padding-bottom: 48px;
height: 397px !important;
background-position: center;
width: 100%;
}
.jumbotron .container {
max-width: 100%;
}
.jumbotron h1, .jumbotron .h1 {
color: #fff;
font-size: 50px !important;
text-align: center;
}
.jumbotron h1, .jumbotron .h1 {
color: #fff;
font-family: "Adobe Garamond Pro", "Adobe Caslon Pro", Helvetica, Arial, sans-serif;
font-size: 1.5rem;
}
.jumbotron p {
margin-bottom: 15px;
font-size: 30px;
font-weight: 400;
color: #fff;
text-align: center;
}
.btn-default.btn-inverse {
color: #FFFFFF;
background-color: #162e00;
border-color: #FFFFFF;
}
.topbutton {
padding: 1% 6%;
font-size: 20px;
line-height: 2.375rem;
border-radius: 0.1875rem;
margin-left: 38%;
margin-top: 3%;
}
.btn:hover{
background:transparent;
color:#fff;
}
@media (max-width:768px){
.jumbotron h1, .jumbotron .h1 {
color: #fff;
font-size: 30px !important;
}
.jumbotron p {
font-size: 17px;
}
.jumbotron {
height: 259px !important;
}
.btn {
margin-left: 30%;
}
}
@media (max-width:640px){
.jumbotron h1, .jumbotron .h1 {
color: #fff;
font-size: 20px !important;
}
.jumbotron p {
font-size: 12px;
}
.jumbotron {
height: 259px !important;
}
.btn {
margin-left: 30%;
}
}
@media (max-width:480px){
.jumbotron h1, .jumbotron .h1 {
color: #fff;
font-size: 17px !important;
}
.jumbotron p {
font-size: 12px;
}
.jumbotron {
height: 259px !important;
}
.btn {
margin-left: 30%;
}
}
I tried several different techniques that I found here on stackoverflow but I would either get everything to line up on the left and not be centred or centre everything but loose the mobile responsiveness. Thanks in advance.
A: Ok , to add this lines :
.jumbotron {
background-image: url('your image background');
height: 100%;
width: 100%;
background-size: cover;
}
.jumbotron .container {
position: relative;
top:100px;
}
and in your head tag in html add this :
<meta name="viewport" content="width=device-width, initial-scale=1">
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,181 |
Q: Error Connecting QODBC to MySQL through migration wizard I am currently attempting to migrate my QB Data through QODBC to my MySQL server, however I have run into the following error:
Starting...
Connect to source DBMS...
- Connecting to source...
Connecting to ...
Opening ODBC connection to DSN=QuickBooks Data 64-Bit QRemote...
Connected
Connect to source DBMS done
Check target DBMS connection...
- Connecting to target...
Check target DBMS connection done
Retrieve schema list from source....
- Checking connection...
Connection to apparently lost, reconnecting...
Connecting to ...
Opening ODBC connection to DSN=QuickBooks Data 64-Bit QRemote...
Connected
- Fetching catalog names...
Traceback (most recent call last):
File "C:\Program Files\MySQL\MySQL Workbench 8.0 CE\modules\db_sql92_re_grt.py", line 368, in getCatalogNames
return Sql92ReverseEngineering.getCatalogNames(connection)
File "C:\Program Files\MySQL\MySQL Workbench 8.0 CE\modules\db_sql92_re_grt.py", line 43, in getCatalogNames
return sorted(list(set(row[0] for row in cls.execute_query(connection, 'SELECT TABLE_CATALOG FROM INFORMATION_SCHEMA.TABLES'))) )
File "C:\Program Files\MySQL\MySQL Workbench 8.0 CE\modules\db_generic_re_grt.py", line 80, in execute_query
return cls.get_connection(connection_object).cursor().execute(query, *args, **kwargs)
pyodbc.ProgrammingError: ('42000', '[42000] [QODBC] Unexpected extra token: . (11017) (SQLExecDirectW)')
Traceback (most recent call last):
File "C:\Program Files\MySQL\MySQL Workbench 8.0 CE\workbench\wizard_progress_page_widget.py", line 197, in thread_work
self.func()
File "C:\Program Files\MySQL\MySQL Workbench 8.0 CE\modules\migration_source_selection.py", line 469, in task_fetch_schemata
self.main.plan.migrationSource.doFetchSchemaNames(only_these_catalogs)
File "C:\Program Files\MySQL\MySQL Workbench 8.0 CE\modules\migration.py", line 245, in doFetchSchemaNames
catalog_names = self.getCatalogNames()
File "C:\Program Files\MySQL\MySQL Workbench 8.0 CE\modules\migration.py", line 209, in getCatalogNames
return self._rev_eng_module.getCatalogNames(self.connection)
SystemError: ProgrammingError("('42000', '[42000] [QODBC] Unexpected extra token: . (11017) (SQLExecDirectW)')"): error calling Python module function DbSql92RE.getCatalogNames
ERROR: Retrieve schema list from source: ProgrammingError("('42000', '[42000] [QODBC] Unexpected extra token: . (11017) (SQLExecDirectW)')"): error calling Python module function DbSql92RE.getCatalogNames
Failed
What should I do? I'm not quite sure how to diagnose this situation.
A: Instead of SELECT TABLE_CATALOG FROM INFORMATION_SCHEMA.TABLES use sp_tables
Please refer to http://qodbc.com/links/2342
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,910 |
package storm.lrb.tools;
import java.io.BufferedReader;
import java.io.FileNotFoundException;
import java.io.FileReader;
import java.io.IOException;
import java.util.HashMap;
import java.util.Map;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
public class CSVReader {
private final static Logger LOGGER = LoggerFactory.getLogger(CSVReader.class);
public CSVReader() {}
public Map<Integer, Map<String, Integer>> parseFile(String csvFile) {
Map<Integer, Map<String, Integer>> content = new HashMap<Integer, Map<String, Integer>>();
BufferedReader br = null;
try {
br = new BufferedReader(new FileReader(csvFile));
String line;
String cvsSplitBy = ",";
int cnt = 0;
while((line = br.readLine()) != null) {
if(cnt == 0) {
LOGGER.debug("Reading histfile line: %s", line);
}
cnt++;
// use comma as separator
String[] histdata = line.split(cvsSplitBy);
if(histdata.length != 4) {
return null;
}
Integer vid = new Integer(histdata[0]);
Integer day = new Integer(histdata[1]);
Integer xway = new Integer(histdata[2]);
String key = xway.toString() + "-" + day.toString();
Integer toll = new Integer(histdata[3]);
if(content.containsKey(vid)) {
content.get(vid).put(key, toll);
} else {
HashMap<String, Integer> tmp = new HashMap<String, Integer>();
tmp.put(key, toll);
content.put(vid, tmp);
}
}
cnt = 0;
for(Map.Entry<Integer, Map<String, Integer>> entry : content.entrySet()) {
if(cnt % 50 == 0) {
LOGGER.debug("Toll for vid: %d" + entry.getKey());
}
for(Map.Entry<String, Integer> toll : entry.getValue().entrySet()) {
if(cnt % 7 == 0) {
LOGGER.debug("[key=%d, toll=%d]", toll.getKey(), toll.getValue());
}
}
}
} catch(FileNotFoundException e) {
throw new RuntimeException(e);
} catch(IOException e) {
throw new RuntimeException(e);
} finally {
if(br != null) {
try {
br.close();
} catch(IOException e) {
LOGGER.error("IOException occured when trying to close input stream, ignoring", e);
return null;
}
}
}
return content;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,645 |
Azoxystrobin is the ISO common name for an organic compound that is used as a fungicide. It is a broad spectrum systemic active ingredient widely used in agriculture to protect crops from fungal diseases. It was first marketed in 1996 using the brand name Amistar and by 1999 it had been registered in 48 countries on more than 50 crops. In the year 2000 it was announced that it had been granted UK Millennium product status.
History
In 1977, academic research groups in Germany published details of two new antifungal antibiotics they had isolated from the basidiomycete fungus Strobilurus tenacellus. They named these strobilurin A and B but did not provide detailed structures, only data based on their high-resolution mass spectra, which showed that the simpler of the two had molecular formula C16H18O3. In the following year, further details including structures were published and a related fungicide, oudemansin A from the fungus Oudemansiella mucida, whose identity had been determined by X-ray crystallography, was disclosed.
When the fungicidal effects were shown to stem from what was then a novel mode of action, chemists at the Imperial Chemical Industries (ICI) research site at Jealott's Hill became interested to use them as leads to develop new fungicides suitable for use in agriculture. The first task was to synthesize a sample of strobilurin A for testing. In doing so, it was discovered that the structure that had been published was incorrect in the stereochemistry of one of the double bonds: the strobilurins, in fact, have the E,Z,E not E,E,E configuration. Once this was realised and the correct material was made and tested it was shown, as expected, to be active in vitro but insufficiently stable to light to be active in the glasshouse. A large programme of chemistry to make analogues was begun when it was discovered that a new stilbene structure containing the β-methoxyacrylate portion (shown in blue and believed to be the toxophore) had good activity in glasshouse tests but still lacked sufficient photostability. After more than 1400 analogues had been made and tested the team chose azoxystrobin for commercialisation and it was developed under the code number ICIA5504.
First sales were in 1996 using the brand name Amistar: it then gained fast-track registration in the United States, where it was marketed in 1997 as Heritage. By 1999 it had been registered in 48 countries on more than 50 crops. In the year 2000 it was announced that it had been granted Millennium product status by the UK Prime Minister, Tony Blair, as it had become in three years the world's best-selling fungicide. Meanwhile, BASF scientists who were collaborating with the German academic groups that had discovered strobilurin A had independently invented kresoxim-methyl, which was also launched in 1996.
Synthesis
The first synthesis of azoxystrobin was disclosed in patents filed by the ICI group. The sequence of the two substitution reactions allows for the synthesis of a diverse range of structural analogs via an Ullmann-type etherification between the aryl chloride of the first intermediate and a substituted phenol, thus introducing a range of structural diversity while maintaining the central strobin toxophore. The final choice of 2-cyano phenol in the second step of the synthesis was made after many other alternatives had been tested for their fungicidal properties.
The crystal structure was published in 2008.
Mechanism of action
Azoxystrobin and other strobilurins inhibit mitochondrial respiration by blocking electron transport. They bind at the quinol outer binding site of the cytochrome b-c1 complex, where ubiquinone (coenzyme Q10) would normally bind when carrying electrons to that protein. Thus production of ATP is prevented. The generic name for this mode of action is "Quinone Outside Inhibitors" QoI.
Formulations
Azoxystrobin is made available to end-users only in formulated products. Since the active ingredient has moderate solubility in water, formulations aid its use in water-based sprays by creating an emulsion when diluted. Modern products use non-powdery formulations with reduced or no use of hazardous solvents, for example suspension concentrates. The fungicide is compatible with many other pesticides and adjuvants when mixed by the farmer for spraying.
Usage
Azoxystrobin is a xylem-mobile systemic fungicide with translaminar, protectant and curative properties. In cereal crops, its main outlet, the length of disease control is generally about four to six weeks during the period of active stem elongation. All pesticides are required to seek registration from appropriate authorities in the country in which they will be used. In the United States, the Environmental Protection Agency (EPA) is responsible for regulating pesticides under the Federal Insecticide, Fungicide, and Rodenticide Act (FIFRA) and the Food Quality Protection Act (FQPA). A pesticide can only be used legally according to the directions on the label that is included at the time of the sale of the pesticide. The purpose of the label is "to provide clear directions for effective product performance while minimizing risks to human health and the environment". A label is a legally binding document that mandates how the pesticide can and must be used and failure to follow the label as written when using the pesticide is a federal offence.
Within the European Union, a 2-tiered approach is used for the approval and authorisation of pesticides. Firstly, before a formulated product can be developed for market, the active substance must be approved for the European Union. After this has been achieved, authorisation for the specific product must be sought from every Member State that the applicant wants to sell it to. Afterwards, there is a monitoring programme to make sure the pesticide residues in food are below the limits set by the European Food Safety Authority.
Agriculture and Horticulture
Azoxystrobin possesses a broad spectrum of activity, in common with other QoI inhibitors. Examples of the fungal groups on which it is effective are Ascomycota, Deuteromycota, and Basidiomycota, as well as the oomycetes. In addition, its properties mean that it can move systemically through plant tissue to protect parts of the crop that were not in contact with the spray. This combination of properties has meant that it achieved widespread use very quickly and has reached annual sales of more than $500 million. Important diseases which it controls include leaf spot, rusts, powdery mildew, downy mildew, net blotch and blight.
Worldwide, azoxystrobin is registered for use on all important crops. For example, in the European Union and United States, it is registered for use in wheat, barley, oats, rye, soya, cotton, rice, strawberry, peas, beans, onions and many other vegetables. The advantage to the farmer comes in the form of improved yield at harvest. Farmers can act in their best economic interest: the value of the additional yield can be estimated and the total cost of using the fungicide informs the decision to purchase. This cost-benefit analysis by the end user sets a maximum price which the supplier can demand and in practice pesticide prices fluctuate according to the current market value of the crops in which they are used. The estimated annual use of azoxystrobin in US agriculture is mapped by the US Geological Survey and shows an increasing trend from its introduction in 1997 to 2019, the latest date for which figures are available, and now reaching .
Home and garden
One of the earliest uses of azoxystrobin was to control fungal diseases of turf and it has been used on golf courses and lawns. It is now available for domestic markets under brand names such as Heritage and Azoxy 2SC.
Azoxystrobin is added to mold-resistant Purple wallboards (optiSHIELD AT, mixture of azoxystrobin and thiabendazole) and can leach into house dust, potentially providing a source of life-long exposure to children and adults.
Human safety
Azoxystrobin has little toxicity to mammals with an LD50 of over 5000 mg/kg (rats, oral). However, it can cause skin and eye irritation. First aid information is included with the label.
The World Health Organization (WHO) and Food and Agriculture Organization (FAO) joint meeting on pesticide residues has determined that the acceptable daily intake for azoxystrobin is 0-0.2 mg/kg bodyweight per day.
The Codex Alimentarius database maintained by the FAO lists the maximum residue limits for azoxystrobin in various food products.
Effects on the environment
Azoxystrobin is categorized as having a low potential for bioconcentration and of moderate risk to fish, earthworms and bees but of high risk to aquatic crustaceans, so care must be taken to avoid runoff into water bodies. Its main degradation product, the carboxylic acid resulting from hydrolysis of its methyl ester, is also potentially harmful to aquatic environments. The benefits and risks of use of QoI fungicides have been reviewed and there is extensive literature on azoxystrobin's environmental profile.
Ultimately it is the regulatory authorities in each country who must weigh up the benefits to end users and balance these against the compound's inherent hazards and consequent risks to consumers and the wider environment.
Resistance Management
Fungal populations have the ability to develop resistance to QoI inhibitors. This potential can be mitigated by careful management. Reports of individual pest species becoming resistant to azoxystrobin are monitored by manufacturers, regulatory bodies such as the EPA and the Fungicides Resistance Action Committee (FRAC), who assign fungicides into classes by mode of action. In some cases, the risks of resistance developing can be reduced by using a mixture of two or more fungicides which each have activity on relevant pests but with unrelated mechanisms of action. FRAC assigns fungicides into classes so as to facilitate this. On cereal crops in the US, for example, azoxystrobin may only be used in mixture, usually with an azole fungicide such as difenoconazole.
Brands
By international convention and in many countries the law, pesticide labels are required to include the common name of the active ingredients. These names are not the exclusive property of the holder of any patent or trademark and as such they are the easiest way for non-experts to refer to individual chemicals. Companies selling pesticides normally do so using a brand name or wordmark which allows them to distinguish their product from competitor products having the same active ingredient. In many cases, this branding is country and formulation-specific so after several years of sales there can be multiple brand names for a given active ingredient. The situation is made even more complicated when companies license their ingredients to others, as is often done. In addition, the product may be pre-mixed with other pesticides under a new brand name.
It is therefore difficult to provide a comprehensive list of brand names for products containing azoxystrobin. They include Amistar, Abound, Heritage, Olympus, Ortiva, Priori Xtra, Scotts DiseaseEx, Haedes and Quadris. Suppliers and brand names in the United States are listed in the National Pesticide Information Retrieval System.
References
Further reading
External links
National Pesticide Information Center
Quadris Fungicide Product - Syngenta US
Azoxystrobin on Pubchem
Fungicides
Pyrimidines
Phenol ethers
Benzonitriles
Methyl esters
Strobilurins
Diaryl ethers | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,204 |
Q: How do I continue a sentence that contains a quote mid-sentence The following question is quite similar, but it's not a duplicate because my example sentence does not follow up with a conjuction.
Which instance of the following sentence is right:
We cannot expect another person to say, "That is correct" if we appeal to circular reasoning.
We cannot expect another person to say, "That is correct," if we appeal to circular reasoning.
We cannot expect another person to say, "That is correct." if we appeal to circular reasoning.
I'm particularly interested in how this should be resolved according to Chicago Manual of Style.
I'm aware that I could reword the sentence in the ways presented below, but I would prefer to have an answer to my question:
We cannot expect another person to conclude that we are correct if we appeal to circular reasoning.
If we appeal to circular reasoning, we cannot expect another person to agree with us.
If we appeal to circular reasoning, we cannot expect another person to say, "That is correct."
A: I don't think The Chicago Manual of Style would say that any of your example sentences are correct. I believe it should be:
We cannot expect another person to say "that is correct" if we appeal to circular reasoning.
Here's why:
13.15: No comma to introduce a quotation
Many writers mistakenly use a comma to introduce any direct quotation,
regardless of its relationship to the surrounding text. But when a
quotation introduced midsentence forms a syntactical part of the
surrounding sentence, no comma or other mark of punctuation is needed
to introduce it, though punctuation may be required for other reasons.
Donovan made a slight bow and said he was "very glad."
One of the protesters scrawled "Long live opera!" in huge red letters.
According to one critic, Copland's style could be called "American
urban pastoral, with a touch of jazz and more than a hint of
Stravinsky."
She said she would "prefer not to comment."
So, there should be no initial comma. Nor should there be any other punctuation around the quote because there would not be any (syntactically) if it weren't a quote.
Further, this also applies (and was followed in several of the example sentences above):
13.19: Initial capital or lowercase—run-in quotations
When a quotation introduced midsentence forms a syntactical part of
the sentence (see also 13.15), it begins with a lowercase letter even
if the original begins with a capital.
Benjamin Franklin admonishes us to "plough deep while sluggards
sleep."
With another aphorism he reminded his readers that "experience keeps a
dear school, but fools will learn in no other"—an observation as true
today as then.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,260 |
Valley Forge Military College is pleased to announce that Nate Ware will return to Valley Forge as the Head Women's Basketball Coach. Nate began his coaching career at Devereux School in Berwyn, PA in 1970 as Head Boys Basketball Coach and Co-Head Baseball Coach. In 1972, he coached in the Sonny Hill Summer High School league through 1988.
Nate was hired as an assistant coach at Eastern University in 1977 and was named Head Coach in 1984, and continued in the position until 1995. He served as the Head Men's Basketball Coach at Valley Forge Military College from 1995-2001.
Nate left Valley Forge to further his career at the University of the Sciences in Philadelphia as their Head Women's Basketball Coach in 2001 through 2009. He continued his career at Wilmington University as the Assistant Men's Basketball Coach in 2010, the following season he served as the Women's Head Coach for the 2011 and 2012 seasons, and returned to the men's side in 2013 as the Men's Assistant Coach.
With a career record, 638 wins, over 30 championships including college tournaments, district and regional championships, coaching Eastern University's championship bid in 1987 in Chattanooga, TN, and the all-time current career winningest coach at both Eastern University and the University of the Sciences, Nate is a well-seasoned coach prepared to grow the Women's Basketball program at Valley Forge Military College.
Nate's awards include: Small College Hall of Fame, 1997; Eastern University Hall of Fame, 2009; Coach of the Year CACC Conference, 2004; NAIA Eastern Region Coach of the Year; and the NCCAA Eastern Region and National Coach of the Year. He is the current President of the Sam Cozen Small College Coaches Association in the Philadelphia Area, consisting of 27 member colleges within the tri-state area.
Please join us as we welcome Nate Ware back to the Valley Forge Military College community. We look forward to a great season of Women's Basketball, with Nate at the helm, as new members in the NJCAA! | {
"redpajama_set_name": "RedPajamaC4"
} | 6,857 |
helper = function () {
this.appPath = zvs.Processor.Utils.AppPath;
this.hostDetails = zvs.WPF.App.GetHostDetails;
this.userName = System.Environment.UserName;
this.applicationName = zvs.Processor.Utils.ApplicationName;
this.applicationVersion = zvs.Processor.Utils.ApplicationVersionLong;
this.applicationNameAndVersion = zvs.Processor.Utils.ApplicationNameAndVersion;
this.appDataPath = zvs.Processor.Utils.AppDataPath;
this.DBNamePlusFullPath = zvs.Processor.Utils.DBNamePlusFullPath;
this.app = System.Windows.Application.Current;
this.core = this.app.zvsCore;
this.pluginManager = this.core.PluginManager;
this.triggerManager = this.core.TriggerManager;
this.scheduledTaskManager = this.core.ScheduledTaskManager;
this.getPlugins = function () {
if (typeof this.pluginManager != 'undefined') return this.pluginManager.GetPlugins();
return undefined;
};
this.getPlugin = function (uniqueIdentifier) {
if (typeof this.pluginManager != 'undefined') return this.pluginManager.GetPlugin(uniqueIdentifier);
return undefined;
};
this.IsJavascriptCommand = function (cmd) {
return cmd.hasOwnProperty("Script");
};
this.SceneDetails = function () {
var sceneDetails = "";
if (HasScene) {
for (var cmd in Scene.Commands) {
if (!this.IsJavascriptCommand(cmd.Command)) {
sceneDetails += (cmd.Device.Name + ", Current Level: " + cmd.Device.CurrentLevelText + " (" + cmd.Device.CurrentLevelInt + ")") + "\n";
}
}
} else {
sceneDetails = "No scene information available";
}
return sceneDetails;
};
this.deviceByName = function (name) {
for (var dev in zvsContext.Devices) {
if (dev.Name == name) return dev;
}
return undefined;
};
this.deviceDetails = function (device) {
var details = "";
var name = device;
if (typeof name == 'string') {
device = this.deviceByName(device);
} else {
name = device.Name;
}
if (typeof device != 'undefined') {
details += "Name: " + device.Name + "\n";
details += "NodeNumber: " + device.NodeNumber + "\n";
details += "LastHeardFrom: " + device.LastHeardFrom + "\n";
details += "CurrentLevelText: " + device.CurrentLevelText + "\n";
details += "CurrentLevelInt: " + device.CurrentLevelInt + "\n";
details += "Type: " + device.Type.Name + "\n";
details += "Values: " + this.deviceValues(device) + "\n";
details += "Commands:{ \n" + this.deviceCommandLog(device) + " \n}";
} else {
details = "No device by that name: " + name;
}
return details;
};
this.sceneByName = function (name) {
for (var s in zvsContext.Scenes) {
if (s.Name == name) return s;
}
return undefined;
};
this.getDeviceValue = function (device, valueName) {
if (typeof device == 'string') device = this.deviceByName(device);
for (var v in device.Values) {
if (v.Name == valueName) {
return v.Value;
}
}
return "";
};
this.deviceValues = function (device) {
var details = "";
if (typeof device == 'string') device = this.deviceByName(device);
for (var value in device.Values) {
details += "("+value.Id+")" + value.Name + " = " + value.Value + "\n";
}
return details;
};
this.builtinCommands = function () {
return zvsContext.BuiltinCommands;
};
this.builtinCommand = function (name) {
var c = this.builtinCommands();
for (var cmd in c) {
if (cmd.Name == name || cmd.UniqueIdentifier == name) return cmd;
}
return undefined;
};
this.runBuiltinCommand = function (cmd, arg) {
if (typeof cmd == 'string') {
cmd = this.builtinCommand(cmd);
}
if (typeof cmd != 'undefined') {
cmd.Run(zvsContext, arg);
}
};
this.repollAll = function () {
return this.runBuiltinCommand("REPOLL_ALL");
};
this.repollDevice = function (dev) {
if (typeof dev != 'number') dev = dev.DeviceId;
return this.runBuiltinCommand("REPOLL_ME", dev);
};
this.groupOn = function (group) {
return this.runBuiltinCommand("GROUP_ON", group);
};
this.groupOff = function (group) {
return this.runBuiltinCommand("GROUP_OFF", group);
};
this.timeDelayScene = function (delay) {
return this.runBuiltinCommand("TIMEDELAY", delay);
};
this.deviceById = function (id) {
for (var dev in zvsContext.Devices) {
if (dev.Id == id) return dev;
}
return undefined;
};
this.deviceCommandByName = function(dev, commandName) {
var commands = this.deviceCommands(dev);
for(var c in commands) {
if(c.Name == commandName) return c;
}
return undefined;
}
this.deviceCommands = function(dev) {
if (typeof dev == 'number') {
dev = this.deviceById(dev);
}
return dev.Commands;
}
this.deviceCommandLog = function(dev) {
var c = "";
var commands = this.deviceCommands(dev);
for(var cmd in commands) {
c = c + "Id=" + cmd.Id + "\n";
c = c + "Name=" + cmd.Name + "\n";
c = c + "Value=" + cmd.Value + "\n";
c = c + "ArgumentType=" + cmd.ArgumentType + "\n";
c = c + "Description=" + cmd.Description + "\n";
c = c + "CustomData1=" + cmd.CustomData1 + "\n";
c = c + "CustomData2=" + cmd.CustomData2 + "\n";
c = c + "Help=" + cmd.Help + "\n";
c = c + "Options={\n";
for(var opt in cmd.Options) {
c = c + "Option.Id=" + opt.Id + "\n";
c = c + "Option.Name=" + opt.Name + "\n";
}
c = c + "Options=}\n\n\n";
}
return c;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,172 |
The Writing Program anticipates the following vacancies. Priority will be given to applications received by November 15th.
Director: Two- or three- year (renewable) position that may lead to tenure. AUC has a rapidly developing program in Rhetoric and Composition; the successful hire will help create and establish MA degree, add courses to the creative writing focus, and establish an already proposed writing minor. The position requires commitment to promoting effective teaching for freshmen and specialized upper level courses, experience with program administration and curriculum development, and a Ph.D. in Rhetoric and Composition. Duties include selecting and supervising a faculty of 30+, encouraging faculty development and program assessment activities, and serving as liaison to university departments. Rank open. Position # WP-1.
Writing Instructor/Assistant Professor: Several openings for experienced composition and rhetoric instructors to teach writing, rhetoric, and research in the Writing Program. Experience with information technology, Business/Technical Writing, and/or curriculum design would be a plus. Instructors must have MA in Composition, Literature, or related field within the humanities. Assistant Professor rank requires Ph.D., evidence of active research and publication agenda, and interest in curriculum development as well as teaching experience. Position # WP-2/5. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,851 |
{"url":"https:\/\/mersenneforum.org\/showthread.php?s=6483adc4d8f70622b61b679bc37690de&t=13656","text":"mersenneforum.org Best settings to factor Wagstaff p = (2^n +1) \/ 1\n Register FAQ Search Today's Posts Mark Forums Read\n\n 2010-07-26, 01:08 #2 kar_bon \u00a0 \u00a0 Mar 2006 Germany 23\u00b732\u00b741 Posts Try a smaller B1 like Code: echo {(2^^^^151 + 1) \/ 3} | ecm 100000 Even B1=10000 can do it! Last fiddled with by kar_bon on 2010-07-26 at 01:09\n2010-07-26, 01:14 \u00a0 #3\ndiep\n\nSep 2006\nThe Netherlands\n\n78210 Posts\n\nQuote:\n Originally Posted by kar_bon Try a smaller B1 like Code: echo {(2^^^^151 + 1) \/ 3} | ecm 100000 Even B1=10000 can do it!\nYou sure?\n\nC:\\factor\\ecm>echo {(2^^^^151 + 1) \/ 3} | ecm 10000\nInput number is {(2^151+1)\/3} (45 digits)\nUsing B1=10000, B2=1873422, polynomial x^1, sigma=878803321\nStep 1 took 31ms\nStep 2 took 47ms\n\nC:\\fail\\>\n\n2010-07-26, 01:18 \u00a0 #4\nkar_bon\n\nMar 2006\nGermany\n\n23\u00b732\u00b741 Posts\n\nQuote:\n Originally Posted by diep You sure? C:\\factor\\ecm>echo {(2^^^^151 + 1) \/ 3} | ecm 10000 GMP-ECM 6.2.3 [powered by GMP 4.2.1_MPIR_1.1.1] [ECM] Input number is {(2^151+1)\/3} (45 digits) Using B1=10000, B2=1873422, polynomial x^1, sigma=878803321 Step 1 took 31ms Step 2 took 47ms C:\\fail\\>\nYes!\n\nCode:\nE:\\ecm>echo {(2^^^^151 + 1) \/ 3} | ecm 10000\nInput number is {(2^151+1)\/3} (45 digits)\nUsing B1=10000, B2=1873422, polynomial x^1, sigma=184170618\nStep 1 took 47ms\nStep 2 took 31ms\n********** Factor found in step 2: 18717738334417\nFound probable prime factor of 14 digits: 18717738334417\nProbable prime cofactor ({(2^151+1)\/3})\/18717738334417 has 32 digits\nTry it more then once!\nDo you know something more about the ECM-algorithm? Read first more about it, like using the B1-param!\n\nLast fiddled with by kar_bon on 2010-07-26 at 01:19\n\n2010-07-26, 01:26 \u00a0 #5\ndiep\n\nSep 2006\nThe Netherlands\n\n2\u00d717\u00d723 Posts\n\nQuote:\n Originally Posted by kar_bon Yes! Code: E:\\ecm>echo {(2^^^^151 + 1) \/ 3} | ecm 10000 GMP-ECM 6.2.3 [powered by GMP 4.3.0] [ECM] Input number is {(2^151+1)\/3} (45 digits) Using B1=10000, B2=1873422, polynomial x^1, sigma=184170618 Step 1 took 47ms Step 2 took 31ms ********** Factor found in step 2: 18717738334417 Found probable prime factor of 14 digits: 18717738334417 Probable prime cofactor ({(2^151+1)\/3})\/18717738334417 has 32 digits Try it more then once! Do you know something more about the ECM-algorithm? Read first more about it, like using the B1-param!\nAh i see, ECM has a high degree of \"i feel so lucky\" push-button technology inside... ...after a few tries :)\n\nC:\\factor\\ecm>echo {(2^^^^151 + 1) \/ 3} | ecm 1000\nInput number is {(2^151+1)\/3} (45 digits)\nUsing B1=1000, B2=51606, polynomial x^1, sigma=42049832\nStep 1 took 0ms\nStep 2 took 0ms\n\nC:\\factor\\ecm>echo {(2^^^^151 + 1) \/ 3} | ecm 1000\nInput number is {(2^151+1)\/3} (45 digits)\nUsing B1=1000, B2=51606, polynomial x^1, sigma=3032908584\nStep 1 took 0ms\nStep 2 took 0ms\n\nC:\\factor\\ecm>echo {(2^^^^151 + 1) \/ 3} | ecm 1000\nInput number is {(2^151+1)\/3} (45 digits)\nUsing B1=1000, B2=51606, polynomial x^1, sigma=948727984\nStep 1 took 0ms\nStep 2 took 16ms\n\nC:\\factor\\ecm>echo {(2^^^^151 + 1) \/ 3} | ecm 1000\nInput number is {(2^151+1)\/3} (45 digits)\nUsing B1=1000, B2=51606, polynomial x^1, sigma=3156401853\nStep 1 took 0ms\nStep 2 took 15ms\n\nC:\\factor\\ecm>echo {(2^^^^151 + 1) \/ 3} | ecm 1000\nInput number is {(2^151+1)\/3} (45 digits)\nUsing B1=1000, B2=51606, polynomial x^1, sigma=2553158111\nStep 1 took 0ms\nStep 2 took 16ms\n********** Factor found in step 2: 18717738334417\nFound probable prime factor of 14 digits: 18717738334417\nProbable prime cofactor ({(2^151+1)\/3})\/18717738334417 has 32 digits\n\nC:\\factor\\ecm>\n\n 2010-07-26, 06:48 #6 10metreh \u00a0 \u00a0 Nov 2008 2\u00b733\u00b743 Posts Firstly, you used B1=1000 not 10000 for those last 5 runs. Secondly, you can tell ECM to run more than one curve with the -c switch. Last fiddled with by 10metreh on 2010-07-26 at 06:48\n2010-07-26, 13:47 \u00a0 #7\nR.D. Silverman\n\nNov 2003\n\n22\u00d75\u00d7373 Posts\n\nQuote:\n Originally Posted by kar_bon Do you know something more about the ECM-algorithm? Read first more about it, like using the B1-param!\nI know from long experience that most people never bother to\nread\/understand the algorithms and code that they use.\n\n2010-07-26, 14:24 \u00a0 #8\nkar_bon\n\nMar 2006\nGermany\n\n23\u00d732\u00d741 Posts\n\nQuote:\n Originally Posted by R.D. Silverman I know from long experience that most people never bother to read\/understand the algorithms and code that they use.\nI mean, before I use a program, I should know how to use it's parameters, read more about it from internet or this forum here, trying to learn about it, try others programs which do the same.\n\nAfter this is all done, I would ask for help.\nSure, I even don't know much about the exact alrogithm behind ECM, but I know how to use this program.\n\nPerhaps it's general people asking others first than searching by themselves!\n\n2010-07-26, 15:50 \u00a0 #9\nR.D. Silverman\n\nNov 2003\n\n11101001001002 Posts\n\nQuote:\n Originally Posted by kar_bon I mean, before I use a program, I should know how to use it's parameters, read more about it from internet or this forum here, trying to learn about it, try others programs which do the same. After this is all done, I would ask for help. Sure, I even don't know much about the exact alrogithm behind ECM, but I know how to use this program. Perhaps it's general people asking others first than searching by themselves!\nFor ECM one needs to know how the algorithm works before one can intelligently select its parameters.\n\n2010-07-26, 19:02 \u00a0 #10\nxilman\nBamboozled!\n\n\"\ud808\ude7a\ud808\udf0c\ud808\uddf7\ud808\uddb7\ud808\udc2d\"\nMay 2003\nDown not across\n\n101100001101112 Posts\n\nQuote:\n Originally Posted by R.D. Silverman For ECM one needs to know how the algorithm works before one can intelligently select its parameters.\nI beg to differ. Someone who does not know how the algorithm works may be quite capable of interpreting advice provided by someone else who does have an in-depth understanding.\n\nPaul\n\n2010-07-26, 21:33 \u00a0 #11\nR.D. Silverman\n\nNov 2003\n\n11101001001002 Posts\n\nQuote:\n Originally Posted by xilman I beg to differ. Someone who does not know how the algorithm works may be quite capable of interpreting advice provided by someone else who does have an in-depth understanding. Paul\nSo we should all go through life in ignorance and merely accept answers\nfrom others? And if someone (say) asks me and I give a deliberately\nwrong answer, how will they know that it is wrong?\n\nMath is an open subject. It can be read by anyone willing to take the\ntime to study it.\n\nIf you are interested in something you should take the time to learn how\nit works.\n\nExamples where people do not bother to learn for themselves readily\ncome to mind: The \"Anti-Vaccination\" kooks, the G-W deniers, and\nother similar flat earthers.......\n\n Similar Threads Thread Thread Starter Forum Replies Last Post T.Rex Wagstaff PRP Search 191 2021-06-30 17:22 ryanp Wagstaff PRP Search 26 2013-10-18 01:33 Batalov GMP-ECM 9 2012-08-24 10:26 davieddy Miscellaneous Math 209 2011-01-23 23:50 T.Rex Math 0 2007-09-04 07:10\n\nAll times are UTC. The time now is 20:46.\n\nThu May 26 20:46:55 UTC 2022 up 42 days, 18:48, 2 users, load averages: 1.78, 1.59, 1.64","date":"2022-05-26 20:46:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.3912040889263153, \"perplexity\": 11310.065406108579}, \"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-2022-21\/segments\/1652662625600.87\/warc\/CC-MAIN-20220526193923-20220526223923-00587.warc.gz\"}"} | null | null |
using System;
using System.Collections.Generic;
using System.Data;
using System.Data.SqlClient;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace DesignPatterns2
{
class NotasMusicais
{
private static IDictionary<string, INota> notas = new Dictionary<string, INota>(){
{"do", new Do()},
{"re", new Re()},
{"mi", new Mi()},
{"fa", new Fa()},
{"sol", new Sol()},
{"la", new La()},
{"si", new Si()},
};
public INota Pega(string nome)
{
return notas[nome];
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,843 |
setup.exe will have made mention about in-use files that should have gotten replaced but were only scheduled for replacement across the next reboot, and that you need to reboot in order to finish the upgrade.
Of course you can retry to shut down all processes and re-do the upgrade without a reboot, but nonetheless the reboot should happen soon, not after days or weeks with other intermitting setup.exe runs, else you might inadvertently be switched back to older installations on a reboot.
Be sure to carefully read the pop-up requesters and infobubbles from setup.exe. | {
"redpajama_set_name": "RedPajamaC4"
} | 416 |
# CARROTS LOVE TOMATOES
SECRETS OF COMPANION PLANTING FOR SUCCESSFUL GARDENING
Louise Riotte
_The mission of Storey Publishing is to serve our customers by
publishing practical information that encourages
personal independence in harmony with the environment_.
Edited by Julia Needham and Deborah Burns
Cover design by Meredith Maker
Cover illustration by Linda Devito Soltis
Text design by Cynthia McFarland
Text production by Eileen M. Clawson
Line drawings by the author
Indexed by Susan Olason, Indexes & Knowledge Maps
© 1975, 1998 by Storey Publishing, LLC
Cover illustration © Linda Devito Soltis
All rights reserved. No part of this book may be reproduced without written permission from the publisher, except by a reviewer who may quote brief passages or reproduce illustrations in a review with appropriate credits; nor may any part of this book be reproduced, stored in a retrieval system, or transmitted in any form or by any means — electronic, mechanical, photocopying, recording, or other — without written permission from the publisher.
The information in this book is true and complete to the best of our knowledge. All recommendations are made without guarantee on the part of the author or Storey Publishing. The author and publisher disclaim any liability in connection with the use of this information. For additional information please contact Storey Publishing, 210 MASS MoCA Way, North Adams, MA 01247.
Storey books are available for special premium and promotional uses and for customized editions. For further information, please call 1-800-793-9396.
Printed in the United States by Versa Press
30 29 28 27 26 25 24 23 22 21 20
**Library of Congress Cataloging-in-Publication Data**
Riotte, Louise.
Carrots love tomatoes : secrets of companion planting for successful gardening
/ Louise Riotte. — 2nd ed.
p. cm.
"A Storey Publishing book."
Rev. ed. of : Secrets of companion planting, © 1975.
Includes bibliographical references (p. ) and index.
ISBN 978-1-58017-027-7 (alk. paper)
1. Companion planting — Dictionaries. 2. Companion crops —
Dictionaries. 3. Gardening — Dictionaries. 4. Plants, Useful —
Dictionaries. I. Riotte, Louise. Secrets of companion planting. II. Title.
S603.5.R56 1998
635 — dc21
97-31914
CIP
## Table of Contents
**Introduction**
**Vegetables**
**Herbs**
**Wild Plants**
**Grasses, Grains, and Field Crops**
**First Steps for Home Fruit Growing**
**Nuts**
**Ornamental Trees and Shrubs**
**Garden Techniques**
**Soil Improvement**
**Pest Control**
**Poisonous Plants**
**Garden Plans**
**Sources**
**Suggested Reading**
**Index**
## Introduction
**The magic and mystery** of companion planting have intrigued and fascinated humans for centuries, yet it is a part of the gardening world that has never been fully explored. Even today we are just on the threshold. In years to come I hope that scientists, gardeners, and farmers everywhere will work together in making more discoveries that will prove of great value in augmenting the world's food supply.
Plants that assist each other to grow well, plants that repel insects, even plants that repel other plants — all are of great practical use. They always have been, but we are just beginning to find out why. Delving deeply into this fascinating aspect of gardening can provide for us both pleasure and very useful information. I hope that what I have written here will give you many of the tools to work with.
Vegetable growers find that companion planting provides many benefits, one of which is protection from pests. A major enemy of the carrot is the carrot fly, whereas the leek suffers from the onion fly and leek moth. Yet when leek and carrot live together in companionship, the strong and strangely different smell of the partner plant repels the insects so much that they do not even attempt to lay their eggs on the neighbor plant. They take off speedily to get away from the smell. This is why mixed plantings give better insect control than a monoculture, where many plants of the same type are planted together in row after row. Even when plants are affected by diseases, a mixed plant culture can usually alleviate the situation.
It is important to remember that not all "protective" botanicals act quickly. For example, marigolds, to be effective in nematode control, should be grown over at least one full season, and more is better, for their effect is cumulative. One should also realize that certain companion plants will diminish each other's natural repelling ability as they grow together. All through this book you will find "what to grow with" and "what _not_ to grow with." Both are equally important to gardening success.
The effects of plants on one another are important outside the vegetable garden, among trees and shrubs as well as grains, grasses, and field crops. These have chapters to themselves, as do herbs, the group of plants most widely used as protective companions.
Wild plants also play a vital part in the plant community. Some are accumulator plants — those that have the ability to collect trace minerals from the soil. They actually can store in their tissues up to several hundred times the amount contained in an equal amount of soil. These plants, many of which are considered weeds, are useful as compost, green manure, or mulch. Some are "deep diggers," sending their roots deep into the ground to penetrate hardpan and helping to condition the soil, and some have value as protectors of garden plants.
• • •
An entirely different type of community life is that of fruit and nut trees and the bush and bramble small fruits. For many of this group, the choice of good companions is not only helpful but also essential. Have you ever experienced the disappointment of having a beautiful fruit tree blossom, be visited by the bees, and yet fail to bear? There is a reason, of course, and it lies in pollinization. Pollen is the dust from blossoms that is needed to make the plant fruitful. If the tree is self-unfruitful and there is no pollenizer of the correct type growing near, it is doubtful that the tree will ever bear well. In the chapters on fruit and nut growing, I'll attempt to unravel some of this mystery, which seems particularly to plague new gardeners and orchardists.
• • •
A note on the chapter devoted to poisonous plants: This information is not meant to frighten but to warn, for most of the nursery catalogs do not tell us which plants are poisonous or to what degree. Even some of the gardening encyclopedias do not.
Cases of death resulting from poisonous plants are rare, but they do happen. In this book I refer to poisonous plants that are useful in the garden for various reasons. It is only fair to tell you that some of those most commonly used may be harmful to children, to livestock, or even to you.
Many of our loveliest and most decorative plants are poisonous — oleanders, daffodils, scillas, lily of the valley, hyacinths, and larkspurs. Other equally poisonous plants are of value for medicines or as insect repellents. To know is to be forewarned, and because we know, we may use them safely, for poisonous plants, unlike poisonous insects or animals, are never aggressive. You are in control of them at all times.
• • •
All of the suggestions given in this book for companion planting are only a beginning. I have included practical information on soil improvement and garden techniques, as well as some sample garden plans, to help you put companion plants to work for you. Your own experiments will lead you into many exciting pathways and discoveries.
## Vegetables
### Asparagus ( _Asparagus officinalis_ )
Parsley planted with asparagus gives added vigor to both. Asparagus also does well with basil, which itself is a good companion for tomatoes. Tomatoes will protect asparagus against asparagus beetles because they contain a substance called solanine. But if asparagus beetles are present in great numbers, they will attract and be controlled by their natural predators, making spraying unnecessary. A chemical derived from asparagus juice also has been found effective on tomato plants as a killer of nematodes, including root-knot, sting, stubby root, and meadow nematodes.
In my garden I grow asparagus in a long row at one side. After the spears are harvested in early spring I plant tomatoes on either side, and find that both plants prosper from the association. Cultivating the tomatoes also keeps down the weeds from the asparagus. The asparagus fronds should never be cut, if at all, until very late in the fall, as the roots need this top growth to enable them to make spears the following spring.
### Bean ( _Phaseolus_ and _Vicia_ )
Many different kinds of beans have been developed, each with its own lore of "good" and "bad" companions. Generally speaking, however, all will thrive when interplanted with carrots and cauliflower, the carrots especially helping the beans to grow. Beans grow well with beets, too, and are of aid to cucumbers and cabbages.
A moderate quantity of beans planted with leek and celeriac will help all, but planted too thickly they have an inhibiting effect, causing all three to make poor growth. Marigolds in bean rows help repel the Mexican bean beetle.
Summer savory with green beans improves their growth and flavor as well as deterring bean beetles. It is also very good to cook with beans.
Beans are inhibited by any member of the Onion family — garlic, shallots, or chives — and they also dislike being planted near gladiolus.
Broad beans are excellent companions with corn, climbing diligently up the cornstalks to reach the light. They not only anchor the corn more firmly, acting as a protection against the wind, but a heavy vine growth may also act as a deterrent to raccoons. Beans also increase the soil's nitrogen, which is needed by the corn.
### Bean, Bush ( _Phaseolus vulgaris_ )
Included with bush beans are those known as butter, green, snap, string, and wax beans. All will do well when planted with a moderate amount of celery, about one celery plant to every six or seven of beans.
Bush beans do well also when planted with cucumbers. They are mutually beneficial. Bush beans planted in strawberry rows are mutually helpful, both advancing more rapidly than if planted alone.
Bush beans will aid corn if planted in alternate rows. They grow well with summer savory but never should be planted near fennel. They also dislike onions, as do all beans.
### Bean, Lima ( _Phaseolus limensis_ )
Nearby locust trees have a good effect on the growth of lima beans. Other plants give them little or no assistance in repelling insects. Never cultivate lima beans when they are wet, because if anthracnose is present, this will cause it to spread. If the ground has sufficient lime and phosphorus, there will probably be little trouble from anthracnose and mildew.
### Bean, Pole
Like others of the family, pole beans do well with corn and summer savory. They also have some pronounced dislikes, such as kohlrabi and sunflower. Beets do not grow well with them but radishes and pole beans seem to derive mutual benefit.
### Beet ( _Beta vulgaris_ )
Beets grow well near bush beans, onions, and kohlrabi but are "turned off" by pole beans. Field mustard and charlock inhibit their growth. Lettuce and most members of the Cabbage family are "friendly" to them.
_Beets and kohlrabi make good companions. Both take the same kind of culture, and they take soil nourishment at different levels_.
### Broccoli ( _Brassica oleraceae_ )
Like all members of the Cabbage family, broccoli does well with such aromatic plants as dill, celery, camomile, sage, peppermint, and rosemary, and with other vegetables such as potatoes, beets, and onions. Do not plant it with tomatoes, pole beans, or strawberries. Use pyrethrum on broccoli against aphids, before the flower buds open. (See the Pest Control chapter.)
_If rabbits dig your cabbage patch, plant any member of the onion family among them. Or you can dust with ashes, powdered aloes, or cayenne pepper. Rabbits also shun dried blood and blood meal_.
### Cabbage (Brassicaceae)
The cabbage family includes not only cabbage but cauliflower, kale, kohlrabi, broccoli, collards, and Brussels sprouts — even rutabaga and turnip. While each plant of this group has been developed in a special way, they are all pretty much subject to the same likes and dislikes, insects and diseases. Hyssop, thyme, wormwood, and southernwood are helpful in repelling the white cabbage butterfly.
All members of the family are greatly helped by aromatic plants or those that have many blossoms. Good companions are celery, dill, camomile, sage, peppermint, rosemary, onions, and potatoes. Cabbages dislike strawberries, tomatoes, and pole beans.
All members of the family are heavy feeders and should have plenty of compost or well-decomposed cow manure worked into the ground previous to planting. Mulching will help if soil has a tendency to dry out in hot weather, and water should be given if necessary.
_Butterflies themselves do no harm and can help pollinate plants. It is their caterpillars that do much damage to the orchard and field crops. The white cabbage butterfly is perhaps the most destructive. Herbs will repel them: hyssop, peppermint, rosemary, sage, thyme, and southernwood_.
Cabbage and cauliflower are subject to clubroot, and if this occurs, try new soil in a different part of the garden. Dig to a depth of 12 inches and incorporate plenty of well-rotted manure into the soil. Rotate cabbage crops every two years.
If cabbage or broccoli plants do not head up well, it is a sign that lime, phosphorus, or potash is needed. Boron deficiency may cause the heart of cabbage to die out.
### Carrot ( _Daucus carota_ )
For sweet-tasting carrots your soil must have sufficient lime, humus, and potash. Too much nitrogen will cause poor flavor, as will a long period of hot weather.
Onions, leeks, and herbs such as rosemary, wormwood, and sage act as repellents to the carrot fly ( _Psila rosae_ ), whose maggot or larva often attacks the rootlets of young plants. Black salsify ( _Scorzonera hispanica_ ), sometimes called oyster plant, also is effective in repelling the carrot fly. Use as a mixed crop.
_Carrots are good to grow with tomatoes — also with leaf lettuce, chives, onions, leeks, radishes, rosemary, and sage. They have a pronounced dislike for dill. Carrot roots themselves contain an exudate beneficial to the growth of peas_.
Apples and carrots should be stored a distance from each other to prevent the carrots from taking on a bitter flavor.
### Cauliflower (Brassicaceae)
The white cabbage butterfly ( _Pieris rapae_ ) is repelled if celery plants are grown near the cauliflower, but cauliflower does not like tomatoes or strawberries. Extract from cauliflower seeds inactivates the bacteria causing black rot.
### Celeriac ( _Apium graveolens rapaceum_ )
A sowing of winter vetch before planting celeriac is helpful, for the plant needs a rich, loose soil with plenty of potassium. The leek, also a potassium lover, is a good companion in alternating rows, as are scarlet runner beans.
Celeriac does not need as much attention as celery since blanching is not necessary, but as the root starts to enlarge, the crown may be helped to better development and higher quality by removing the fine roots and the soil attached to them. Many lateral roots close to the top of the crown tend to make the fleshy part irregular and coarse.
### Celery ( _Apium graveolens_ )
Celery grows well with leeks, tomatoes, cauliflower, and cabbage, while bush beans and celery seem to give mutual assistance. One gardener believes that celery is particularly benefited if grown in a circle so that the lacy, loosely interwoven roots may make a more desirable home for earthworms and soil microbes. Celery and leeks both grow well when trenched.
Both celery and celeriac are reported to have a hormone that has an effect similar to insulin, making them an excellent seasoning for diabetics or for anyone on a salt-reduced diet.
Celery dinant or French celery dinant is a unique type that sends out a multitude of narrow stalks. I have found it easy to grow here in southern Oklahoma. It has a much fuller flavor than common celery and less should be used in cooking.
This celery is completely insect-free and grows well with all garden vegetables. Plants will freeze in winter but the root does not, and will put out new leaves from the center with the advent of warm spring weather. In a cold climate the leaves may be dried for winter use.
### Chayote ( _Sechium edule_ )
This is a perennial tropical vine, an annual in colder climates, which bears a delicious, light green, pear-shaped fruit in the fall. Two vines must be grown or it will not bear well. Chayotes in a cream sauce are a dish "fit for the gods." I grow them on my garden fence along with cucumbers, where both do exceptionally well. They apparently have no insect enemies and seem to be protective to the cucumbers.
### Chile Peppers ( _Capsicum_ spp.)
The Aztec "chilli" or the Spanish "chile" is known to have existed as early as 700 B.C., but its birthplace remains a mystery, says Anne Lindsay Greer writing in _Cuisine of the American Southwest_. She also claims that all chiles contain an element of unpredictability but insists that all Mexican food or all chiles are not hot. More chiles are produced and consumed than any other seasoning in the world. The hotness of each depends on the climate in which it is grown.
Early South and Central American cultures used chiles for medicinal purposes, currency, and as a discipline for disobedient children. Chiles are thought to be an aid to digestion, to give protection from colds, and to cure everything from toothache to colic to an indifference toward romance. But after separating fact from folklore, some of the mystique from chiles, certain facts remain undisputed:
• Chiles are high in vitamins A and C. One ounce of dried chile has two times the daily requirement of vitamin A.
• Chiles serve as a natural meat preservative by retarding the oxidation of fats.
• Chiles added personality and flavor to the otherwise bland diet of early southwesterners.
• Chiles can be used to reduce the amount of salt in many dishes, or eliminate salt altogether for those on a low-sodium diet.
Columbus is said to have discovered chiles in the West Indies (though he called them peppers). Thereafter they were enthusiastically adopted by the rest of the world. Indian food, Szechuan sauces, Cajun cooking, and southern African cuisines all favor chiles for their assertive flavors. Southwestern and Asian recipes are becoming increasingly popular and many call for chile peppers.
Chiles can be direct seeded after the last spring frost in long, warm-season areas or sown indoors 6 to 8 weeks before transplanting out. Chile peppers are easily transplanted, and most produce fruits with almost reckless abandon!
#### A CHILE LINEUP
**'Habanero'**. Hottest! Lantern-shaped, lime green to orange fruit. Aromatic and tasty in sauces or pickled. Wear gloves when preparing this chile.
**'Piquin'**. Very hot, round, or slightly pointed, perennial in Mexico and south Texas.
**'Thai Pepper'**. Tiny peppers (1 inch long), red when ripe, bred in Thailand, hot and the heat lingers. Use in Oriental dishes.
**'Anaheim "M" Chili'**. An 8-inch long, tapering chile, which can be used green for rellenos and green chile sauce, or dry in red enchilada sauce. A medium-hot, long-bearing pepper.
**'Aji'**. Hot, fruity Peruvian, a thin orange to red 3–5 inches long.
**'Centennial'**. Tiny ornamental purple to white to red simultaneously. Hot pods are edible at all stages. (Named for our flag's colors.)
**'De Arbol'**. Treelike plant; 3–4 feet tall; thin 3–4-inch chiles; hot and smoky. Good for drying.
**'Serrano'**. Hot, small, slender, rounded chile. Serve fresh, dry, or pickled. Good in salsas. May be frozen if blanched first.
**'Cayenne'**. Hot, short, slender. Best dried and ground for red-hot sauces. Used in Asian as well as Mexican and southwestern food.
**'Sandia Hot'**. Very hot, thin-walled, best dried and ground. A standard for hot chile lovers. May be used green or red, fresh or dried.
**'Chimayo'**. Pick green for your favorite stew or salsa. Dried and ground for red chile powder, it makes an excellent enchilada sauce.
**'Española' Improved**. A special short-season chile that will turn red and get hot even in northern climates. Narrow, pointed, and thin-walled. Use green or red.
**'Jalapeño'**. Everybody's favorite. Easily grown, prolific producer. A 3-inch, dark green, almost black chile with thick, meaty walls.
Good for pickling or fresh. Finely chopped, adds a new dimension to corn bread.
**'Santa Fe Grande'**. Short, thick, hot, yellow pepper, good in condiments or pickled. Very attractive as an ornamental.
**'Cascabel'**. Medium hot, tough, round, and dark red, the 1-inch pod makes a dark, smoky chile pepper.
**'NuMex Eclipse'**. Dark brown when ripe, medium hot. Adds color to vegetable dishes.
**'NuMex Sunrise'**. Bright yellow when ripe, medium hot. May be used as a green chile.
**'Big Jim'**. Five 'Big Jims' may weigh as much as one pound. This pepper is a perfect candidate for chile rellenos.
**'Vallero'**. Dark red, medium hot. Use dried for red chile sauces and soups.
**'Mulatto'**. Dark brown when dried, excellent for rellenos, very smoky flavor. 5 inches long, 2-inch pods are mild.
**'Guajillo'**. Used dry in soups and salsas.
**'Andho' ('Poblano')**. Large, heart-shaped, mild chile of excellent flavor. Use peeled 'Poblano' for rellenos, or cut into strips for soups and stews. Good dried.
**'Pasilla'**. Almost black when dried, use for mole. Pods are mild.
**'Pimento'**. Hungarians use for paprika. Heart-shaped, 4 inches long, red, mild, good stuffed or as a stuffing. Mild to hot.
### Chinese Celery Cabbage ( _Brassica chinensis_ )
This easy-to-grow vegetable, deserving to be better known in America, is one of the oldest vegetable crops in China. I grow both types: the tall, slender heads and the huge, deeply savoyed hybrid type, which makes a round head. I find that celery cabbage grows well, making enormous heads, if planted in my fall garden about two feet apart. I alternate the plants in the row with Brussels sprouts or cauliflower.
_Chinese celery cabbage does well in the fall garden, interplanted with Brussels sprouts_.
Celery cabbage has few insect enemies but should not be grown near corn, as the corn worms will infest it.
### Collard ( _Brassica oleracea_ , var. _acephala_ )
Cornell University's College of Agriculture has shown that collards benefit from interplanting with tomatoes since the flea beetle, the prime pests of collards, is significantly decreased.
Collards, widely grown in the southern states as a source of greens, are more nutritious than heading cabbage, and their taste is improved by freezing.
### Corn ( _Zea mays_ )
Sweet corn does well with potatoes, peas, beans, cucumbers, pumpkin, and squash. Peas and beans help corn by restoring to the soil the nitrogen used up by the corn. Is there anyone who hasn't heard the story of Indians putting a fish in every corn hill?
Melons, squash, pumpkins, and cucumbers like the shade provided by corn. In turn they benefit the corn, protecting it from the depredations of raccoons, which do not like to travel through the thick vines. Similarly, pole beans may be planted with corn to climb on the stalks. But don't plant tomatoes near corn because the tomato fruitworm and corn earworm are identical.
An experiment with odorless marigold showed that when it was planted next to corn, the Japanese beetle did not chew off the corn silks.
An experiment reported in the British _New Scientist_ in 1970 states that, "Reduced incidence of fall armyworm on maize [corn] and a correspondingly increased yield were obtained by growing the crop with sunflower in alternating strips... there were also large reductions in the numbers of _Carpophilus_ beetles in the sunflower strips, compared with unbroken areas of the crop. Some infestations were cut by more than half."
Research has shown that to remove corn suckers is a waste of time as well as being detrimental to the development of the ears.
### Cucumber ( _Cucumis sativus_ )
Cucumbers apparently are offensive to raccoons, so they are a good plant with corn. And corn seemingly protects the cucumber against the virus that causes wilt. Thin strips of cucumber will repel ants.
Cucumbers also like beans, peas, radishes, and sunflower, and, preferring some shade, they will grow well in young orchards.
Sow two or three radish seeds in cucumber hills to protect against cucumber beetles. Do not pull the radishes, but let them grow as long as they will, even blossoming and going to seed. Cucumber beetles also may be trapped by filling shallow containers about three-fourths full of water into which some cooking oil has been poured.
If cucumbers are attacked by nematodes, try a sugar spray. I boil a half cup of sugar in two cups of water, stirring until completely dissolved. Let cool and dilute with one gallon of water. Strange as it seems, sugar kills nematodes by drying them out. This will also attract honeybees, ensuring pollination and resulting in a bumper crop of cucumbers, so the spray is worth trying even if you don't suspect the presence of nematodes.
Beneficial fungi are another enemy of nematodes. If you suspect their presence, build up the humus content of your soil.
A chive spray is helpful for downy mildew on cucumbers, as is a spray made of horsetail ( _Equisetum arvense_ ). (See _Horsetail_ in the chapter on Herbs.)
Cucumbers dislike potatoes, while potatoes grown near cucumbers are more likely to be affected by phytophthora blight, so keep the two apart. Cucumbers also have a dislike for aromatic herbs.
### Disease-Resistant Vegetable Varieties
Many vegetable varieties have been specially bred to resist specific plant diseases, and more are constantly being developed.
It's important to know what diseases are most troublesome in your area. Your county's Cooperative Extension Service is a good source of information on this, and may have lists of varieties recommended for local conditions.
Seed catalogs normally indicate which of their offerings have inbred resistance to various problems, often using a code (for instance, "VFT" to show that a tomato cultivar is resistant to verticillium wilt, fusarium wilt, and tobacco mosaic). You can greatly increase your chances of success by heeding this information and matching your choices to the troubles you're most likely to run into.
### Eggplant ( _Solanum melongena_ )
Redroot pigweed makes eggplants more resistant to insect attack. During dry weather, mulching and irrigation will help to prevent wilt disease. Dry cayenne pepper sprinkled on plants while still wet with dew will repel caterpillars. Eggplants growing among green beans will be protected from the Colorado potato beetle. The beetles like eggplant even more than potatoes, but they find the beans repellent.
### High-Vitamin Vegetables
A new tomato, 'Doublerich', containing as much vitamin C as citrus fruit, was introduced in 1956. Prof. A. F. Yeager of the University of New Hampshire developed it using crosses of the tiny wild Peruvian tomato, which is four times richer in vitamin C than our ordinary garden types.
A few years later 'Caro-Red', containing about 10 times the amount of vitamin A found in standard varieties, was perfected at the Indiana Experiment Station. 'Caro-Red' owed its richness to its orange pigment, beta-carotene, and a single fruit could supply up to twice the minimum daily requirement of vitamin A for an adult. Perhaps best of all, this was a very delicious tomato to eat. Later came 'Caro-Rich', containing even more vitamin A.
These are just a few examples of the ongoing effort to improve the nutritional content of garden vegetables. Announcements of new vitamin-rich introductions appear in seed catalogs and gardening literature, and those who want the greatest possible food value from their growing space should take note of them.
### Horseradish ( _Ammoracia rusticana_ )
Horseradish and potatoes have a symbiotic effect on each other, causing the potatoes to be healthier and more resistant to disease. Plants should be set at the corners of the potato plot only and should be dug after each season to prevent spreading. Horseradish does not seem to protect against the potato beetle, but it is effective against the blister beetle. A tea made from horseradish is beneficial against monilia on apple trees.
### Jerusalem Artichoke ( _Helianthus tuberosus_ )
In Italy these are called _girasole_ , meaning "turn with the sun." They really are a type of sunflower and should not be confused with the globe artichoke, which is an entirely different plant.
Jerusalem artichokes, a native Amerian plant, were known to and used by the Indians. They are a good companion to corn. The tuber is the edible portion, for this sunflower has its surprise at the bottom, the flowers being attractive but not large.
The principal food content of the Jerusalem artichoke is inulin, a tasteless, white polysaccharide dissolved in the sap of the roots, which can be converted into levulose sugar. This is of special interest to diabetics, for levulose is highly nutritious and the sweetest of all known natural sugars. Levulose also occurs in most fruits, in the company of dextrose, which diabetics must avoid, but in the Jerusalem artichoke it is present alone. The artichokes are high in food value and rich in vitamins. They may be cooked or eaten raw in salads.
### Kale ( _Brassica oleracea acephala_ )
This cool-weather crop is fine to grow in the fall garden, and it will stand most average winters if given a little protection.
Kale does well in the same rows as late cabbage or potatoes. If planted about the first of August following late beans or peas, it will continue to grow until a hard freeze. A light freeze does not hurt it and even improves its flavor.
Wild mustard and kale sometimes are a problem in oat fields. Rolling is the best method of control. It should be practiced early in the morning while the plants are still wet with dew. The springy oats will pop back up again, but the mustard or kale will be broken.
### Kohlrabi (Brassicaceae)
Kohlrabi is mutually beneficial with onions or beets, with aromatic plants, and surprisingly with cucumbers, in part because they occupy different soil strata. It dislikes strawberries, tomatoes, and pole beans but helps protect the garden members of the mustard family.
It is a demanding plant, needing plenty of water but good drainage, as well as good supplies of compost. It grows best in filtered sunlight.
### Leek ( _Allium porrum_ )
Leek is one of the "heavy feeders" and should be planted in soil well fertilized with rotted manure. Leeks are usually sold in the grocery store (at least where I live) with the root still attached. I once bought several bunches and planted them; they grew well and propagated, and I've had leeks ever since.
Leeks are good plants to grow with celery and onions, and also are benefited by carrots. Returning the favor, leeks repel carrot flies.
### Lettuce ( _Lactuca sativa_ )
In spring I keep a supply of small lettuce plants growing in cold frames. When I pull every other green onion for table use, I pop in lettuce plants. They will aid the onions, and the compost in the onion row will still be in good supply for the lettuce to feed on, while the onion will repel any rabbits.
Lettuce grows well with strawberries, cucumbers, and carrots and it has long been considered good to team with carrots and radishes. Radishes grown with lettuce in summer are particularly succulent.
Lettuce needs cool weather and ample moisture to make its best growth, and I find that the seed will not germinate in very hot weather. Already-started lettuce should have some summer shade.
### Melon (Cucurbitaceae)
Crop rotation can be one of your best weapons against garden pests, but do not rotate melon, squash, and cucumber with each other, since all are cucurbits.
Timing is another weapon. Most cucurbits are not very susceptible to borers once they are past the seedling stage, so try either earlier or later plantings. I find that fall-planted cucumbers and squash are almost entirely insect-free.
Do not plant melons near potatoes, though they will grow well with corn and sunflowers. Morning glory is thought to stimulate the germination of melon seeds.
Heavy waxed paper placed under melons helps keep worms from entering, while sabadilla dust is effective, too. (See _Insecticides, Botanical_ , in the Pest Control chapter.) Melon leaves, rich in calcium, are good to place on the compost pile.
### Okra ( _Hibiscus esculentus_ )
This native of the Old World tropics is grown for its immature pods, which are called okra or gumbo. It's a warm-weather plant that will grow wherever melons or cucumbers thrive. I plant two rows, dig a trench between, and cover it with mulch. On the north side of my okra I plant a row of sweet bell peppers and on the south side a row of eggplant. All are well mulched as the season advances. When the weather becomes dry, in midsummer, I lay the hose in the trench and flood it so that all three companions grow well.
### Onion ( _Allium cepa_ )
Onions and all members of the Cabbage family get along well with each other. They also like beets, strawberries, tomatoes, lettuce, summer savory, and camomile (sparsely), but do _not_ like peas and beans. Ornamental relations of the onion are useful as protective companions for roses.
Since onion maggots travel from plant to plant when set in a row, scatter your onion plants throughout the garden.
### Parsley ( _Petroselinum hortense_ )
Parsley mixed with carrot seed helps to repel carrot flies by its masking aroma. It protects roses against rose beetles. Planted with tomatoes or with asparagus, it will give added vigor to both.
Poultry are sometimes turned loose at intervals in parsley patches where there are many parsley worms, which are the larvae of the black swallowtail butterfly.
A number of different strains of parsley, including the Hamburg ( _Petroselinum crispum latifolium_ ), are grown solely for the fleshy roots, which are cooked and eaten in the same way as parsnips.
_The Onion family is the gardener's best friend_.
### Parsnip ( _Pastinaca sativa_ )
The parsnip is of ancient culture, but remains a vegetable for the special palate. The parsnips have few insect enemies and suffer from few diseases, but both the foliage and roots make a safe insect spray. They are not injured by freezing and are often left in the ground over winter. The seeds germinate slowly and unevenly and should not be used if over a year old.
### Pea ( _Pisum sativum_ )
For large crops, treat pea and bean seed with inoculant, a natural bacterial agent available under various brand names from garden centers and seed catalogs. (See _Inoculants_ in the Soil Improvement chapter.)
Peas grow well with carrots, turnips, radishes, cucumbers, corn, beans, and potatoes, as well as many aromatic herbs. They do _not_ grow well with onions, garlic, and gladiolus.
Always plow pea vines under or return them to the compost pile. Wood ashes used around the base of pea vines help to control aphids.
### Peanut ( _Arachis hypogaea_ )
As members of the Legume family, peanuts are good soil builders. In many areas of the South and Southwest, they may be grown as a second crop after an earlier one, such as carrots or beets, has been harvested. They make a good ground cover in an orchard of young nut trees. (See _Legumes_ in the Soil Improvement chapter.)
### Pepper, Sweet ( _Capsicum frutescens_ var. _grossum_ )
The general requirements of sweet peppers are surprisingly like those of basil, so plant them together. Sweet peppers also grow well with okra, and since they are very brittle plants, the okra, growing taller, serves as a windbreak.
### Pumpkin ( _Cucurbita pepo_ )
Pumpkins grow well when jimsonweed ( _Datura_ ), sometimes called thorn apple, is in the vicinity. (See the Wild Plants chapter.)
Pumpkins grow well with corn, a practice followed by Native Americans, but pumpkins and potatoes have an inhibiting effect on each other.
Middle Eastern peoples consider the seeds an inexhaustible source of vigor offered by a bountiful nature. While we know today that there are no mysterious potions for tired lovers, we also know that some of the old formulas did perform seeming miracles — not through magic but through good nutrition — and pumpkin seeds are really vitamin-rich.
Pumpkin varieties have been developed with seeds that lack the shell of normal pumpkin and squash. The hull-less seeds may be removed and simply washed and dried; they are a delicious snack when roasted and lightly salted.
### Radish ( _Raphanus sativus_ )
Radish is aided by redroot pigweed, which loosens soil, by nasturtiums, and by mustard's protective oils. Do not rotate radish with cabbage, cauliflower, Brussels sprouts, kohlrabi, broccoli, or turnip, since all are members of the Cabbage family.
Early radishes are good to sow with beets, spinach, carrots, and parsnips to mark the row. Sow radishes with cucumbers, squash, and melons to repel the striped cucumber beetle, and with tomatoes to rout the two-spotted spider mite. Radishes grow well with kohlrabi, bush beans, and pole beans. The presence of leaf lettuce in summer will make radishes more tender. Tobacco dust protects them from flea beetles, and garlic juice from many diseases.
Radish and hyssop should never be sown near each other.
### Rhubarb ( _Rheum rhaponticum_ )
This very ornamental as well as useful plant is a good companion to columbines ( _Aquilegia_ ), helping to protect them against red spider.
Rhubarb leaves contain oxalic acid. They may be boiled in water and made into a spray that, watered into drillings before sowing plants of the Cabbage family, wallflowers, and other seeds, is helpful in preventing clubroot. It is also useful on roses against greenfly and black spot.
_Rhubarb stems make delicious pies, but the leaves are very toxic and sometimes also cause skin irritation. They are safe on the compost pile, however_.
Rhubarb, often called the pie plant, is technically a vegetable but is mostly used for dessert. It also has been long recognized as a laxative. This is one of our oldest garden plants, which Marco Polo found growing in China centuries ago.
### Salsify ( _Tragopogon porrifolius_ )
Sometimes this is called oyster plant. To achieve a delicate and different flavor, the milky, oyster-flavored roots need a moist, cool soil for at least four months before reaching maturity.
Salsify grows well with mustard greens, and try growing it with watermelons. Plant the warm-weather watermelon several weeks later than the cool-weather salsify. Let the melons fill the middles of the rows before hot, dry weather arrives. Acting as a living mulch and tending to rest on the ground, the melon vines still will leave the salsify foliage exposed to light and air.
Never use salsify seed over a year old.
### Shallot ( _Allium ascalonicum_ )
Shallots, more delicate in flavor than onions, are propagated by planting the sections or cloves that make up the large bulb. They are good to grow with most garden vegetables but, like onions and garlic, should not be located near peas or beans.
### Spinach ( _Spinacia oleracea_ )
Because of its saponin content, spinach is a useful pre-crop and does well planted with strawberries. (See _Saponin_ in the Soil Improvement chapter.) _Bacillus thuringiensis_ (see Pest Control) may be used as an insect control.
### Squash (Curcubitaceae)
Two or three icicle radishes planted in each hill will help prevent insects on squash as on cucumbers. Let the radishes grow and go to seed. Nasturtiums will repel squash bugs and so will cigarette ash and other tobacco residue if placed with the seed when it is planted. Squash planted either earlier or later than usual often will escape insect damage. Here in Oklahoma I find fall-planted squash almost entirely insect-free.
Early in the day, before the sun is strong, squash stinkbugs are sluggish, and in the small garden may be picked off. There also are insect-resistant strains of squash.
### Sweet Potato ( _Ipomea batalas_ )
Sweet potatoes generally have high energy value, only peas and beans yielding more. They have a common enemy, the fungus disease or wilt called stem rot, which can be controlled with diseasefree seed and by rotating the crop. White hellebore controls a number of leaf-eating insects. Nematodes can be a problem, and some varieties have been bred with built-in resistance.
If rabbits bother your sweet potato patch, spray with a diluted fish emulsion.
### Tomato ( _Lycopersicon esculentum_ )
Tomatoes and all members of the Brassica (Cabbage) family repel each other and should be kept apart. Tomatoes also dislike potatoes and fennel.
Tomatoes will protect asparagus against the asparagus beetle. Since they are tender plants, put tomatoes in during late spring after the early crop of asparagus spears has been harvested. Tomatoes protect gooseberries against insects.
Tomatoes are compatible with chives, onion, parsley, marigold, nasturtium, and carrot, and for several years I have planted garlic bulbs between my tomato plants to protect them from red spider mites. Stinging nettle growing nearby improves their keeping qualities, and redroot pigweed, in small quantities, is also beneficial.
Though not containing fungicidal elements, tomatoes will protect roses against black spot. The active principle of tomato leaves is solanine, a volatile alkaloid that at one time was used as an agricultural insecticide. To make a spray for roses: Make a solution of tomato leaves in your vegetable juicer, adding four or five pints of water and one tablespoon of cornstarch. Strain and spray on roses where it is not convenient to plant tomatoes as companions. Keep any unused spray refrigerated.
Root excretions of tomatoes have an inhibiting effect on young apricot trees, and don't plant tomatoes near corn, since the tomato fruitworm is identical with the corn earworm. Don't plant near potatoes, either, since tomatoes render them more susceptible to potato blight.
Unlike most other vegetables, tomatoes prefer to grow in the same place year after year. This is all right unless you have a disease problem, in which case plant your tomatoes in a new area. Since they are heavy feeders, give them ample quantities of compost or decomposed manure. Mulch and water in dry weather to maintain soil moisture and stave off wilt disease and blossom end rot. But never water tomatoes from the top. Water from below and water deeply.
If you smoke, be sure to wash your hands thoroughly before you work in your garden, for tomatoes are susceptible to diseases transmitted through tobacco.
### Turnip-Rutabaga ( _Brassica rapa_ and _Brassica napobrassica_ )
An accident revealed that hairy vetch and turnips are excellent companions. Turnip seeds became mixed with the vetch that a gardener planted, and they came up as volunteer plants. He found that the turnip greens were completely free of the aphids that usually infest them, apparently because the vetch provided shelter for ladybugs, which feast on aphids. Elsewhere it has been found that wood ashes around the base of turnip plants will control scab.
I find that peas planted near turnips are mutually benefited. Turnip and radish seed mixed with clover will bolster the nitrogen content of the soil. In your crop rotation it is good to follow the heavy feeders with light feeders such as turnips and rutabagas.
Turnips dislike hedge mustard and knotweed, and do not rotate them with other members of the Cabbage family such as broccoli and kohlrabi. A naturally occurring chemical compound in turnips when synthesized is deadly to aphids, spider mites, houseflies, German cockroaches, and bean beetles.
Rutabagas take much the same culture as turnips but require a longer growing season.
### Watermelon ( _Citrullus vulgaris_ )
Watermelons are good to interplant with potatoes, particularly if the potatoes are mulched with straw. The hybrid seedless watermelons, which set no pollen, will produce better if planted with a good pollinator such as 'Sugar Baby'. Watermelons need plenty of sunshine, so do not plant them with or near tall-growing vegetables.
## **Herbs**
### **Absinthium** ( _Artemisia absinthium_ )
Also called wormwood, this plant is grown as a border to keep animals from the garden. Ornamental species such as _A. pontica_ have leaves of great delicacy and are good to plant in flower beds and around choice evergreens. (See _Wormwood_.)
### Aloe Vera ( _Aloe barbadensis_ )
Nature's own medicine plant, known and used for centuries, is a vegetable belonging to the Lily family. The name "aloe vera" means "true aloe," and it is so named because, among 200 species of aloe, it has the best medicinal properties.
_Aloe vera is known as nature's medicine plant_.
The aloe genus belongs to a larger class of plants known as the xeroids, which possess the ability to close their stomata completely to avoid the loss of water. The plants are easy to grow outside in warm, frost-free climates, and equally easy to grow indoors in pots. Possibly because of the bitter taste of the gel, they appear to be completely disease- and insect-free. Almost all xeroids are laxative in nature and have a bitter flavor.
Aloin is the thick, mucilaginous yellow juice that occurs at the base of the aloe leaf, and is also present in the rind. The gel may be removed from the leaf as one would fillet a fish. It is a very slippery, clear, viscous juice useful for sunburns and for healing cutaneous ulcers of radioactive origin, as well as burns and scalds of various types. Many fishermen carry aloe vera aboard their boats to stop the pain of a sting from a Portuguese man-of-war. It also will stop the sting often experienced when gathering okra.
The gel may be used instead of tree wound dressing if it becomes necessary to cut a tree limb. The surface will heal over quickly and insects are repelled by the bitter taste. The juice may be mixed with water to make a spray for plants. Powdered aloes dusted on young plants will repel rabbits, but this must be repeated after rain. Aloe vera plants, thrown in the drinking water for chickens, are said to cure them of certain diseases.
### Anise ( _Pimpinella anisum_ )
The spicy seeds of this annual herb, related to caraway and dill, are used to flavor licorice as well as pastry, cookies, and certain kinds of cheese. The oil extracted from the seeds is used to make absinthe, and it is also used in medicine. The flower, powdered and infused with vermouth, is used for flavoring muscatel wine. Anise is antiseptic, and is useful as an ointment (when mixed with lard) for lice and itching from insect bites.
When sown with coriander, aniseed will germinate better, grow more vigorously, and form better heads.
### Basil ( _Ocimum basilicum_ )
Basil helps tomatoes to overcome both insects and disease, also improving growth and flavor. Since this is a small plant, one to two feet tall, grow it parallel to the tomatoes rather than among them. It repels mosquitoes and flies, and when laid over tomatoes in a serving bowl will deter fruit flies.
Sweet basil has inch-long, dark green leaves and a clovepepperish odor and taste. Pinch out the plant tops and they will grow into little bushes, the dwarf varieties especially becoming beautifully compact. As a kitchen herb, basil is used in vinegar, soup, stew, salad, chopped meat, and sausage, as well as in cottage cheese, egg, and tomato dishes, and may be sprinkled over vegetables. 'Dark Opal' makes a very handsome houseplant.
Though it is often said that herbs enhance everything except dessert, sweet basil is one that may be used to give a subtle, indefinable, but delicious flavor to pound cake. It is also one of the culinary herbs that may be used in certain dishes to replace black pepper. (See _Pepper Substitutes_ in this chapter.)
It has been known since ancient times that basil and rue dislike each other intensely. Perhaps this is because basil is sweet and rue is very bitter.
### Bay ( _Lauris nobilis_ )
Bay (or laurel) leaves put in stored grains such as wheat, rice, rye, beans, oats, and corn will eliminate weevils. The bay belongs to the same family as the cinnamon, camphor, avocado, and sassafras trees. I have tried sassafras leaves in grains and flours and find them also effective against insects and weevils.
### Bee Balm ( _Monarda_ )
This plant improves both the growth and flavor of tomatoes.
Borage
### Borage (Boraginaceae)
This is an excellent provider of organic potassium, calcium, and other natural minerals of benefit to plants. Grow this herb in orchards and as a border for strawberry beds. Honeybees like to feast on the blossoms.
### Camomile ( _Chamomile_ )
The real plant, the German or wild camomile ( _Matricaria chamomilla_ ), recognizable by the hollow bottom of the blossom and its highly aromatic odor, often is confused with Roman camomile ( _An themis nobilis_ ). This is an excellent companion to cabbages and onions, improving growth and flavor of both. But it should be grown sparingly, only one plant every 150 feet.
Camomile
Wheat grown with camomile in the proportion of 100 to 1 will grow heavier and with fuller ears, but too much will harm field crops.
Camomile contains the substance chamazulene, which has anti-allergic and anti-inflammatory properties when used in the form of tea. It is also used for diarrhea or scour in calves. A tea of one-third each camomile, lemon balm, and chervil applied as a warm compress will cure hoof rot in animals.
Camomile flowers may be used in the dog's bed against fleas. When using herbs in pet pillows, simply add the dried form to the stuffing, occasionally adding more to freshen up.
The blossoms soaked in cold water for a day or two can be used as a spray for treating many plant diseases and to control damping-off in greenhouses and cold frames.
A camomile rinse is excellent for blond hair. Use three or four tablespoons of dried flowers to a pint of water. Boil twenty to thirty minutes, straining when cool. Shampoo the hair before using, since it must be free of oil. Pour rinse over the hair several times and do not rinse with clear water after using. It will leave the hair smelling like sweet clover.
Camomile contains a hormone that stimulates the growth of yeast. Grown with peppermint in very small quantities, it increases the essential oil.
### Caraway ( _Carum carvi_ )
Since it is difficult to sprout caraway seed, sow it with a companion crop of peas. After harvesting the peas, harrow the area and the caraway will come up. It's good to plant on wet, heavy land, the long roots making an excellent substitute for subsoiling. Do not grow fennel near it.
Europeans like and use caraway seed more than we do. It is put in rye bread for its aromatic flavor and to make it more digestible. It is also used in cakes, cheeses, and apple and cabbage recipes.
### Catnip ( _Nepeta cataria_ )
Catnip contains an insect-repellent oil, nepetalactone, and fresh catnip steeped in water and sprinkled on plants will send flea beetles scurrying.
The catnip compound is chemically allied to those found in certain insects. Two of these occur in ants and another in the walkingstick insect, which ejects a spray against such predators as ants, beetles, spiders, birds, and even humans. Freshly picked catnip placed on infected shelves will drive away black ants. But my cats love catnip, and I like to eat it in salads.
### Chervil ( _Anthriscus cerefolium_ )
This is one of the few herbs that will grow better in partial shade, which can be provided by taller plants growing near it. It does not take kindly to transplanting. Chervil is a good companion to radishes, improving their growth and flavor.
### Chive ( _Allium schoenoprausum_ )
Chives are a good companion to carrots, improving both growth and flavor. Planted in apple orchards they are of benefit in preventing apple scab, or made into chive tea may be used as a spray for apple scab or against powdery mildew on gooseberries and cucumbers.
### Coffee Substitutes
The price of America's favorite hot brew fluctuates from one year to the next depending on weather conditions in the countries where it is grown. It is interesting to know that there are a number of herbal substitutes that can be made into an acceptable hot drink and are almost universally available. Like any food or beverage, however, the product should be weighed on its own merits. To taste parched and perked seeds, roots, or nuts with the thought of "coffee" in mind is unfair to the substitute.
_**Chicory** (Cichorium intybus_), which grows on roadsides and in waste clay soils from Canada southward, is identifiable when young by its leaves, which resemble those of the dandelion. But as the plants mature, a rigid, loosely branched 2-foot stem develops, and blue flowers bloom in midsummer. The tubular roots, which grow horizontally, should be dug in September after flowering is over. To use chicory as a substitute or as a coffee additive, the roots should be washed, coarsely ground, dried in a very slow oven for two or three hours, then roasted in a clean skillet. This should be done very slowly and the granulated chicory repeatedly stirred until the proper color and flavor are reached.
Chicory
Chicory does not taste like coffee, but when used as an additive, some people believe, it improves the color and flavor of South American coffee, as well as extending the number of cups. In France, and in our own South, subject to French influence, chicory is added to coffee, not as an adulterant but for its distinctive flavor.
According to Virginia Scully in _A Treasury of American Indian Herbs_ , the Native Americans roasted the roots and used them as they did the dandelion roots. Nelson Coon, author of _Using Plants for Healing_ , writes of chicory as "a plant of ancient usage," the name tracing back through Arab medicinal language to Greek and Egyptian, and mention of the use of chicory is found in Roman writings.
_**Cleavers** (Galium sparine_), after chicory, seems to be the most popular coffee substitute. Cleavers grows in Alaska, southward across Canada, and on down into Texas, and is found on seashores and in rocky woods. Although cleavers sprouts may be eaten in the spring, the tiny twin burr-seeds ripen later. Cleavers grows best in damp thickets. Cleavers, also known as goose grass, is eagerly sought by animals and poultry for its medicinal qualities. Cleavers also, according to all the old herbalists, has a reputation as a reducing diet par excellence, painlessly paring pounds from plump persons.
The usefulness of cleavers is not confined to its value as a reducing diet or as a palatable cooked vegetable. According to Euell Gibbons in _Stalking the Healthful Herbs_ , the little two-lobed seeds make the best coffee substitute of almost any plant growing in our range. Perhaps this is because cleavers belongs to the Rubiaceae, the natural order of plants to which the coffee tree also belongs.
The little hard fruits should be gathered during the summer when they are full-sized and roasted in an oven until they are very brown. They can then be made into a beverage that tastes much like coffee and also has a definite coffee aroma. Cleavers contains no caffeine.
Cleavers has long been considered a medicinal herb, its properties being listed as diuretic, tonic, alterative, and somewhat laxative. Gather the herb in May or June when in flower and dry in a warm room.
_**Nut grass** (Cyperus esculentus_) is an herb of almost universal distribution. If nut grass tubers are roasted until they are a very dark brown all through, then pulverized in a blender or coffee mill, they make a very palatable hot drink when brewed exactly as you do coffee. Nut grass coffee contains no harmful stimulants and can freely be given to children who insist on having "coffee" when the grown-ups do.
_**Sunflower** (Helianthus annus_) has long been used by Native Americans, who parched the highly nutritious seeds and made them into a meal for gruel and cakes. Frequently, they added water to the meal as a drink, and crushed roasted seeds to make a drink like coffee.
_**Sassafras** (Sassafras albidum_ and _Sassafras varifolium_ ) is also called wild cinnamon or mitten plant. It likes to grow in rich woods in humusy soil and both leaves and bark may be used, the leaves being gathered in the spring and the bark in the fall. The name sassafras is a corruption of the Spanish word _saxifrage_ , which in turn is derived from two Latin words, _saxum_ , meaning "a rock," and _frango_ , meaning "I break." It is interesting to note that herbalists recommend to those suffering with stones in the kidneys and other kidney diseases that sassafras be always included in a formula of kidney-stimulating herbs.
A hot drink of sassafras may be prepared by stirring a half teaspoonful of the ground bark in a cup of boiling water. Cover with a saucer for about five minutes. Stir, strain, and add a little honey for sweetness.
_**Dandelion roots** (Taraxacum officinale_), roasted, have long been used as a coffee substitute and taste much like chicory. The young leaves often appear in salads, and the Native Americans chewed the young stems like gum.
_**Dried persimmon seeds** (Diospyros), **wild senna seeds** (Cassia occidentalis_), and _**roasted acorn beans**_ have been ground and used as coffee. The black or red-oak ( _Quercus_ ) acorns were used.
Cereals, nuts, peas, soya beans, and even okra seeds have been roasted and used as coffee substitutes, dilutants (thinners), or additives. Again, when it comes to extending the family coffee supply, it is well to remember that some sacrifice of flavor must be made.
### Comfrey ( _Symphytum officinale_ )
Comfrey, also called knitbone or healing herb, is high in calcium, potassium, and phosphorus, and rich in vitamins A and C. It was an ancient belief that comfrey preparations taken internally or as a poultice bound to injured parts hastened the healing of broken bones.
It is possible that the nutrients present in comfrey actually do assist in the healing process, since we now know that the herb also contains a drug called allantoin, which promotes the strengthening of the lining of hollow internal organs. However, this herb also contains alkaloids that cause liver damage and is no longer considered safe for internal use.
The leaves of Russian comfrey are ideal for the compost heap, having a carbon-nitrogen ratio similar to that of barnyard manure.
### Coriander ( _Coriandrum sativum_ )
Coriander has a reputation for repelling aphids while being immune to them itself. It helps anise to germinate but hinders the seed formation of fennel. In blossom the herb is very attractive to bees.
Many people think the foliage and fresh seed of coriander have a disagreeable smell, but as the seeds ripen they gain a delicious fragrance that intensifies as they dry. The savory seeds, sometimes sugar-coated as a confection, are baked in breads or used to flavor meats.
Coriander has four times more carotene than parsley, three times as much calcium, more protein and minerals, more riboflavin, and more vitamin B1 and niacin. Oil of coriander is used medicinally to correct nausea.
### Dill ( _Anethum graveolens_ )
Dill is a good companion to cabbage, improving its growth and health. It does not do well by carrots and if allowed to mature will greatly reduce that crop, so pull it before it blooms.
Dill will do well if sowed in empty spaces where early beets have been harvested, and light sowings may be made with lettuce, onions, or cucumbers. Honeybees like to visit dill blossoms.
### Drying Herbs
Leaf herbs should be cut, washed, tied in loose bunches, and allowed to drip dry. Place upside down in large brown paper bags that have been labeled. Close the mouth of the bag about the stems; let the herbs hang free inside the bag. Hang the bags where they will have good air circulation. In this way none of the oil is absorbed by contact with the paper, as may be the case if dried herbs are stored in cardboard boxes.
With seed herbs, let the plants mature until the ripe seeds part from the dry umbels with a little pressure. This occurs after they lose their greenish color but before they will drop of their own accord. Cut the heads on a dry morning and spread them out on brown paper in the sun for the rest of the day, stirring occasionally. Do this for several days, taking them in at night, until they are thoroughly dry. Seeds may be stored in opaque glass bottles or clear ones away from the light.
_Dry herbs in paper bags hung in an airy place_.
### Elecampane ( _Inula helenium_ )
My German heritage bids me have great respect for elecampane, sometime called horseheal or horse alder. This herb was under the protection of the goddess Hulda, who first taught mortals the art of spinning and weaving flax. Candied elecampane, according to a 17th-century herbal, was thought to "cause mirth."
Elecampane came to America as a healer, being introduced into gardens as a home remedy. Finding the climate congenial, it went native and now grows wild in many places.
The substance most abundantly contained in elecampane root is inulin __, a sort of invert starch usable as a replacement for ordinary starch in the diet of diabetics. It also contains a volatile oil and several identifiable crystalline substances. The thick, yellow taproot has the odor of camphor.
Elecampane once was used in England for the heaves in horses. As far back as 1885 a Dr. Korab demonstrated that the active bitter principle of the plant, called helenin, was a powerful antiseptic and bactericide, particularly destructive to the tubercle bacillus.
Elecampane in the garden provides a six-foot-tall accent of bright yellow in midsummer when the large-rayed flowers stand above the enormous felty leaves. The plants are useful in providing a bit of shade for lower-growing mints.
### Fennel ( _Nigella sativa_ )
Most plants dislike fennel, and it is one herb that should be planted well away from the vegetable garden, since it has an inhibiting effect on bush beans, caraway, kohlrabi, and tomatoes. Fennel planted away from the garden is valuable for its masses of fringed foliage. At one time the fragrant seeds, which smell and taste like licorice, were made into a tea soothing to colicky babies. Mixed with peppermint leaves, it also makes a delicious tea for adults. Fennel is inhibited by the presence of coriander and will not form seed. It also dislikes wormwood.
### Garlic ( _Allium sativum_ )
The Babylonians and Hindus knew of garlic's medicinal power 3,000 years ago, and it was well known to the ancient Egyptians, who fed great quantities of garlic to the workers who built the pyramids. The Greek physicians, fathers of present-day medicine, used garlic in their precriptions, and it was rationed to the soldiers of the mighty Roman armies.
The 800-year-old medical school of Salerno included garlic in its _materia medica_ , and it always has been one of the standbys of folk medicine practitioners.
Garlic is an effective control for many insects. Try this recipe for garden use: Take three to four ounces of chopped garlic bulbs and soak in two tablespoons of mineral oil for a day. Add one pint of water in which one teaspoon of fish emulsion has been dissolved. Stir well. Strain the liquid and store in a glass or china container, as it reacts with metals. Dilute this, starting with one part to 20 parts of water, and use as a spray against your worst insect pests.
If sweet potatoes or other garden plants are attracting rabbits, try this spray. Rabbits dislike the smell of fish, too, so fish emulsion may help.
Garlic sprays are useful in controlling late blight on tomatoes and potatoes. Garlic is an effective destroyer of the diseases that damage stone fruits, cucumbers, radishes, spinach, beans, nuts, and tomatoes.
A garlic-based oil sprayed on breeding ponds showed a 100 percent kill of mosquito larvae in a University of California experiment. Garlic cloves stored in the grain are good against grain weevils.
Garlic grown in a circle around fruit trees is good against borers. It is beneficial to the growth of vetch and is protective planted with roses. All alliums, however, inhibit the growth of peas and beans. Plant garlic with tomatoes against red spider. I have done this for three successive years with good results.
### Herbs for Tea
Herb teas have both aided digestion and given pleasure to untold generations. Here is how to make tea:
Pick some of the leaves, using only those undamaged (or dried leaves may be used). You will need four or five leaves for a cup of tea and about a handful for a pot. Wash the leaves in cool water, put in a cup or pot, and pour boiling water over them. Let steep, covered, for three to five minutes. Add sugar or honey to sweeten.
Please note: Research on herbs is ongoing, and some that were formerly used internally are now thought to be potentially harmful. Make teas only from plants suggested for this purpose in up-to-date guides to the use of herbs.
### Horehound ( _Marrubian vulgare_ )
In centuries past, virtues attributed to horehound included the power to cure snakebite and merit as a fly repellent, vermifuge, and an ointment for wounds and itches. The Hebrew name for the plant, _marrob_ , means "a bitter juice." It was one of the five bitter herbs required to be eaten at the Passover feast. The Romans considered it a good and sometimes magical herb. Horehound's real value relates to pulmonary ailments and it is widely used as an ingredient in lozenges for coughs and colds.
For small stems and better quality, grow the plants intended for candy making close together. Either the fresh or dried herb can be used for this purpose.
Grasshoppers and other insects dislike the taste of horehound. The plant grows well with tomatoes, improving their quality, causing them to bear more abundantly and continue later in the season.
### Horsetail
See the Wild Plants chapter.
### Hyssop ( _Hyssopus officinalis_ )
It's hard to find anything more delightful than a hyssop hedge in full sun. The blue, white, and pink flowered hyssop makes an intriguing design grown with gray Roman wormwood.
Sow the hyssop seeds in late fall so they will germinate first thing in the spring. Planted near grapevines they will increase their yield, and near cabbages will lure away the cabbage butterfly. Bees are very fond of visiting hyssop blossoms, yet many other insects find the plant repellent. Radishes will not do well if hyssop is nearby.
**F RECKLE REMEDIES**
I do not dislike freckles — my mother always told me they were the scars left by the angels' kisses — but some lucky people seem to receive more kisses than they want. If freckles trouble you, elder flowers added to facial steam baths will clear and soften the skin, and are also good against freckles and faulty pigmentation, especially when they are used in conjunction with whey and yogurt. Such face packs are not only soothing but also tonic and stimulating. Parsley water externally used also is said to remove freckles or moles.
Freshly crushed leaves or freshly pressed juice of lady's-mantle ( _Alchemilla vulgaris_ ) is helpful against inflammation of the skin and acne, as well as freckles. Externally used, lime flowers ( _Tilia_ ) are a fine cosmetic against freckles, wrinkles, and impurities of the skin, and they also are stimulating to hair growth.
Hyssop leaves have a peculiar fragrance reminiscent of civet, yet some use them the same way as savory. A compress of hyssop leaves is good for removing black and blue spots from bruises.
This is another of the bitter herbs used in Jewish ceremonies but it is not the true hyssop of the Bible, which is believed to be a species of origanum.
### Lavender ( _Lavendula officinalis_ )
In a 2 percent emulsion spray for cotton pests, lavender kills somewhere between 50 to 80 percent within a period of 24 hours.
Few ticks are found in lavender plantations, although neighboring woods and shrubs may harbor many. It has been used effectively as a mouse repellent, and lavender sachets are often put in woolen clothing to repel moths, while leaves scattered under woolen carpets are helpful for the same purpose.
The plant grows from seed very slowly. Both plants and seeds are obtainable from Nichols Garden Nursery. (See Sources.)
### Lemon Balm ( _Melissa officinalis_ )
Lemon balm, often called the bee herb, has long been famous for its delightful, lemon-scented foliage and honeyed sweetness. _Melissa_ , the generic name, is Greek for "honeybee," and there is a very old belief that bees will not leave the hive area if melissa grows near it. Pliny wrote, "When bees have stayed away they do find their way back home by it."
Melissa tea calms the nervous system and stimulates the heart, it is very relaxing, and may even dispel headache or migraine. In pastures this plant increases the flow of cows' milk, and it is very good to give cows after calving in a tea with marjoram.
### Lovage ( _Levisticum officinale_ )
Lovage planted here and there will improve the health and flavor of other plants. It is one of the herbs that may be used to reduce the amount of salt used for seasoning and is delicious sprinkled on salads or used in cheese biscuits. In dishes that need strengthening, it can replace meat stock and is excellent in soups and casseroles.
Lovage will winter well, but in colder climates the roots should have some protection.
### Marjoram ( _Marjorana hortensis_ )
This small, easily grown plant is probably one of the oldest herbs in use. "Marjoram" really covers three very different kinds of marjoram, all of which belong to the Labiatae family.
The sweet marjoram, an annual, is the most popular for flavoring, especially in sausages. It was used extensively by the Greeks, who gave it the name that means "joy of the mountains." Its disinfectant and preserving qualities made it an invaluable culinary herb in the Middle Ages.
Pot marjoram is a perennial with a bit less flavor but more easily grown.
Wild marjoram ( _Oregano vulgare_ ) is a wild as well as a cultivated variety with a strong flavor, the pungency varying according to where it grows. This herb, also known as oregano, is used the world over in Italian, Mexican, and Spanish dishes, and is believed to have both stimulating and medicinal properties, since it contains thymol, a powerful antiseptic when used internally or externally. The whole plant of oregano is covered with hairy oil glands. The pleasant aromatic scent, reminiscent of thyme, is very lasting — even the dead leaves and stems retain it during the winter or when dried for culinary use.
In the garden all the marjorams have a beneficial effect on nearby plants, improving both growth and flavor.
Mint
### Mint ( _Mentha_ )
Mint is a good companion to cabbage and tomatoes, improving their health and flavor. Both mint and tomatoes are strengthened in the vicinity of stinging nettle. Mints such as apple, orange, and pineapple will thrive under English walnut trees, in part because of the filtered sunlight.
Mint deters the white cabbageworm by repelling the egg-laying butterflies. Spearmint repels ants and may help to keep aphids off nearby plants. (See also _Peppermint_ and _Spearmint_.)
Mint is a repellent against clothes moths when used indoors, and is useful against black flea beetles. The leaves strewn under rabbit cages will keep flies to a minimum, while dried leaves (or mint oils) will repel rats and mice.
### Mugwort ( _Artemisia vulgaris lactiflora_ )
Mugwort is one of the most useful members of the Artemisia family. Planted in chicken yards it will help repel lice, and since the chickens like it as a food, it is also thought to be helpful in ridding them of worms. Made into a weak tea, it may be used as a fruit tree spray.
Mugwort is not good too near other garden plants because it has a growth-retarding effect, particularly in years of heavy rainfall. The roots and leaves exude a toxic substance. This soluble toxic, absinthin, washed off by rain, soaks into the soil near the plant and remains active over an indefinite period of time.
### Orach ( _Atriplex hortensis_ )
This beautiful annual, sometimes called French spinach, has shield-shaped, wavy leaves of beet red. They have a mealy texture similar to its close relative, lamb's-quarters, and are also used as a potherb. While orach may be planted in the garden, it should never be placed near potatoes, since it has an inhibiting effect on their growth.
Old-time herbalists believe that orach had a cleansing quality either raw or cooked and if laid upon swollen glands of the throat would cure the condition.
### Oregano ( _Origanum vulgare_ )
Sow with broccoli to repel the cabbage butterfly. (See _Marjoram_.)
### Pennyroyal ( _Mentha pulegium_ )
Plant this with broccoli, Brussels sprouts, and cabbage against cabbage maggot. Like tansy, it may be grown at doorways to repel ants, and is also a good mosquito repellent if rubbed on the skin. Fresh or dried sprigs have long been used as a flea repellent.
### Pepper Substitutes
Basil, summer savory, thyme, marjoram, and nasturtium can help replace pepper in cooking for those who have digestive disturbances.
### Peppermint ( _Mentha piperita_ )
Of all herbs, this makes the greatest demand on the soil for humus and moisture. It will benefit from a small amount of chicken manure if well broken down.
Peppermint drives away red ants from shrubs, and planted among cabbage it will repel the white cabbage butterfly. When growing with camomile it will have less oil, but the camomile will benefit and have more. The oil of peppermint is increased when it is grown with stinging nettle.
Black mint is distinguished from other species by purple stems and dark green leaves. It grows about three feet tall and is crowned with spikes of lavender flowers in midsummer. It is widely used for medicinal and commercial purposes. (See also _Mint_ and _Spearmint_.)
### Rue ( _Ruta graveolens_ )
By now I'm sure everybody knows that rue doesn't like basil. But an authority as ancient as Pliny tells us that "rue and the fig tree are in great league and amitie together."
Rue planted near roses or raspberries will deter the Japanese beetle. It can be clipped and made into an attractive hedge, but first be sure you are not allergic to it, since the foliage can cause dermatitis as severe as that from poison ivy when the plant is coming into flower. The intensity of the eruption seems aggravated by the presence of sunlight. If you happen to be perspiring and working bare-handed with rue, you may get poisoned. If this happens, washing with brown soap or covering the exposed area with oil will help.
Rue may be grown among flowers as well as vegetables, where its good looks will add to the planting. It is protective to many trees and shrubs. It is good near manure piles and around barns for discouraging both house and stable flies.
The ancient Schola Salernitans wrote that "Rue putteth fleas to flight." However, it should be used only for dog pillows or beds, for cats do not like it. Anything rubbed with the leaves of rue will be free from cats' depredations — good to know if your house cats tend to claw the furniture.
### Sage ( _Salvia officinalis_ )
Sage is protective to cabbages and all their relatives against the white cabbage butterfly, and it also makes the cabbage plants more succulent and tasty.
The herb also is good to grow with carrots, protecting them against the carrot fly, and is mutually beneficial with rosemary. Do not plant sage with cucumber, which does not like aromatic herbs in general and sage in particular.
Originally sage was used medicinally in stuffing and meats to make them more digestible, but we have grown to like the flavor. It has often been made into a tea, but is now considered harmful if much is taken internally.
### Salt Substitutes
The clever use of herbs can replace salt in many dishes and reduce the amount needed in others. Those on a salt-free diet can flavor their food deliciously by using such herbs as celery, summer savory, thyme, lovage, and marjoram.
### Santolina ( _Chamaecyparis_ spp.)
This South European plant, sometimes called lavender cotton, is a good moth repellent. The name is from _sanctum linum_ , meaning "holy flax." The plants are improved by being pruned as soon as the blossoms fall.
### Savory, Summer ( _Satureia hortensis_ )
In Germany savory is called the bean herb because it's good to grow with beans and also to cook with them. It goes with onions as well, improving both growth and flavor.
### Savory, Winter ( _Satureia montana_ )
Winter savory is a subshrub about 15 inches tall. Its leaves, though not as delicate as summer savory, may be used in cooking. It is useful as an insect repellent, too.
### Sesame ( _Sesamum orientale_ )
Sesame is an herb grown in tropical countries, mainly for the oil obtained from its seeds. The oil is used in salad dressings and for cooking, while the delicious seeds are used to flavor bread, candy, biscuits, and other delicacies. Sesamin is derived from sesame oil and is used as a synergist to strengthen the effect of pyrethrum.
Sesame is very sensitive to root exudates from sorghum ( _Andropogon sorghum_ ) and will not ripen well when grown near it.
### Southernwood ( _Artemisia abrotanum_ )
Dry the leaves of southernwood, place them in nylon net bags, and hang them in the closet to prevent moths. Burned to ashes in the fireplace, they will remove any cooking odors from the house.
Southernwood has green, finely divided leaves with a lemonmixed-with-pine scent. Grown near cabbages, it will protect them from cabbageworm butterfly and near fruit trees will repel fruit tree moths. Among its names are old man, lad's love, and even maiden's ruin!
### Spearmint ( _Mentha spicata_ )
Spearmint also is called green mint, pea mint, and lamb mint. (See _Mint_.)
### Sugar Substitutes
Three herbs that can reduce the use of sugar in cooking and sweets are lemon balm, sweet cicely, and angelica. They are particularly good in tart fruit or fruit pies made of black currant, red currant, rhubarb, gooseberry, plums, and tart apples. Not only do these herbs make it possible to use sometimes half the usual quantity of sugar, but they also impart a delicious flavor. Chopped sweet cicely may be added to lightly sugared strawberries.
### Sweet Basil ( _Ocimum basilicum_ )
See _Basil_.
### Tarragon, French ( _Artemisia dracunculus_
Use potato fertilizer as a side dressing for tarragon in the spring and again right after the first cutting to increase the vitality of the plant. To reset tarragon successfully, the roots must be carefully untangled. Each section of root eased apart from the clump may be reset to form another plant. This is best done every third year in March or early April. As a cooking herb tarragon is something very special, and it is particularly good for flavoring vinegar.
### Thyme ( _Thymus vulgaris_ )
Thyme has an ancient history as a medicinal and culinary herb. The oil still is used as the basis of a patent cough medicine, while thymol has antibacterial powers of considerable importance. But thyme is of value mainly in cooking, being very good for poultry seasoning and dressing. Lemon thyme makes a delicious herbal tea.
The herb deters the cabbageworm and is good planted anywhere in the garden, accenting the aromatic qualities of other plants and herbs.
### Valerian ( _Valeriana officinalis_ )
This herb is good anywhere in the garden, particularly to give vegetables added vigor. It is rich in phosphorus and stimulates phosphorus activity where grown; it is attractive to earthworms and therefore particularly useful in the compost pile.
It is thought that the Pied Piper of Hamelin used valerian to clear the town of rats, yet many gardeners find it attracts cats, which love to nibble the leaves and roll aginst them. The _UniversalHerbal_ of 1820 states: "It is well known that cats are very fond of the roots of valerian; rats are equally partial to it — hence ratcatchers employ them to draw the vermin together."
Valerian enjoyed great prominence in colonial days as a drug plant, the strangely scented root being brewed into a sedative tea. Since the oil of valerian does have an effect on the central nervous system, the tea should not be drunk often.
Cold tea is made by this rule: Put one level teaspoon crushed dried valerian root to soak in one cup cold water. Cover and place in a cold place for 12 to 24 hours. Strain and drink about one hour before retiring.
### Woodruff, Sweet ( _Asperula odorata_ )
Sweet woodruff is an excellent ground cover, particularly under crab apple trees. While it will grow in the sun, the foliage is darker green and much more abundant if it receives shade at least half of the day.
### Wormwood ( _Artemisia absinthium_ )
This herb, particularly the variety _cineraria_ , will keep animals out of the garden when used as a border. It's a good repellent for moths, flea beetles, and cabbageworm butterfly. It discourages slugs if sprayed on the ground. Fleas on cats and dogs may be dislodged with a bath of wormwood tea.
Many artemisias are of value as ornamentals, their cool, silvery beauty providing a fine contrast for flowers, such as red geraniums, of brilliant color. They do not attract honeybees, but small wasps seem to be frequent visitors. Keep wormwood out of the garden since most plants growing near it do not do well, particularly anise, caraway, fennel, and sage.
Wormwood
## Wild Plants
### Algae
Here is a food source, not yet fully explored, that may prove of great value as populations increase, for the total amount of photosynthesis carried on by marine algae may be ten times greater than the total of such activity in all land plants. If this is true, then we may well look to the plants of the sea as a rich, abundant, and relatively untapped source of foodstuffs.
In some parts of the world, for example Japan and China, algae have for many years been important items in the human diet. Farmers in many coastal regions cultivate brown algae on bamboo stems pushed into the ocean bottom in shallow waters. Dried preparations of these brown algae, available in many food stores in the United States, are very rich in minerals and also have moderate quantities of carbohydrates and vitamins.
Brown algae, because of their rich mineral content, also are often used as soil fertilizers (spread on fields and plowed under), or they may be dried and burned and their ash used as fertilizer. Some species are a commercial source of iodine for medicinal use.
### Amaranth ( _Amaranthus retroflexus_ )
This plant, sometimes called rough pigweed and commonly found in disturbed ground everywhere, is one of the best weeds for pumping nutrients from the subsoil.
Amaranth
Amaranth loosens the soil for such crops as carrot, radish, and beet. It helps potatoes yield more abundantly and is good to grow with onions, corn, pepper, and eggplant — but keep it thinned. Tomatoes grown with the weed are more resistant to insect attack.
Euell Gibbons says in his _Stalking the Healthful Herbs_ that green amaranth has a higher iron content than any other green vegetable except parsley.
A type of amaranth grown widely in rural Mexico is known as "the sacred food of the Aztecs." The small seeds, easily threshed from the large heads, can be baked with bread. Sometimes they are even popped, mixed with honey, and eaten as a confection. In India certain species are eaten as a salad or cooked like spinach, and the seed is ground into flour.
Amaranth (which is distantly related to beets) has a higher protein content than the cultivated beet, is higher in vitamin C, and has approximately the same amount of vitamin A.
### Aster ( _Asteroides_ )
Many asters are soil indicators. Some like low, moist soil, so if the bushy type ( _Boltonia asteroides_ ) or the purple-stemmed aster ( _A. puniceus_ ) shows up in your pastures or fields, it indicates a need for drainage. The sea aster ( _A. tripolium_ ) grows on seasides and near salt mines and is a salt and soda collector. The poisonous woody aster ( _Xylorhiza parryi_ ) of the West indicates an alkaline soil.
### Baked-appleberry ( _Rubus chamaemorus_ )
Most of us are familiar with the blackberry-dewberry family, Rubus, but one member has been neglected. The baked-appleberry, or cloudberry, is found growing in wet areas in acid peats from the Arctic regions southwest through New England. It bears a single, soft, pinkish berry, which grows on a 12-inch stem bearing a few scalloped leaves. The berries make a delicious fresh dessert, and may be preserved in jams or prepared as juice.
### Broom Bush ( _Sarothamnus vulgaris_ )
This useful weed grows on the poorest, stoniest soils and those that are sandy and slightly to medium acid. Being rich in calcium carbonate, the plant improves the soil through decomposition of its leaves and stems. In a thin stand it will provide shelter for young tree seedlings, but it will choke them out if too many of the weeds are present.
### Burdock ( _Arctium_ )
Do not allow wild burdocks to grow, since they are robbers of the soil. Particularly do not allow them to go to seed, for the burrs will adhere to the hair of sheep, horses, and dogs, even clothing, and be spread far and wide.
Burdock roots have medicinal value and are said to alleviate gout and skin diseases. An edible burdock ('Takinogawa') has been developed in Japan, where the cooked roots are greatly relished for their refreshing, pungent flavor. The Japanese also value this burdock for its reputed blood-purifying qualities and the relief it is said to give to sufferers of arthritis. (It is available from Nichols Garden Nursery; see Sources.)
### **Calamus** ( _Araceae_ )
It is said that mosquitoes are never found in swamps or other standing water in which calamus, sometimes called sweet flag or sweet root, is growing.
### Charlock ( _Brassica arvensis_ )
Charlock or wild mustard is frequently found in grainfields. The seeds and those of wild radish can lie inert in the ground for 50 to 60 years, showing up again when the field is planted to grain, particularly oats.
### Cinquefoil ( _Potentilla monspeliensis_ )
The cinquefoils are considered a bad symptom when found on pastureland, for they indicate a very acid soil, and gradually they will choke out other grasses and clovers. They are very persistent and will last when other grasses are burned out by drought. It has been observed that the butternut or gray walnut tree ( _Juglans cinerea_ ) and the black walnut have an inhibitory effect on the growth of the creeping cinquefoil.
Cinquefoil
### Crowfoot (Ranunculaceae)
Like other members of its family, the common meadow buttercup secretes a substance in its roots that poisons the soil for clover by inhibiting the growth of nitrogen bacteria. So potent is this secretion that clover in time will disappear if buttercups begin to invade the field. Cattle will not eat the acrid, caustic plant, and children should be warned against biting the buttercup's stem and leaves, which are capable of raising blisters.
The garden monkshood ( _Aconitum_ ) is even more dangerous, being poisonous in all its parts, while other members of this family that are more or less poisonous are delphinium, columbine, and peony. They are beautiful, but grow them with care.
### Cypress Spurge ( _Euphorbia cyparissias_ )
This funny little plant escaped from eastern gardens where it was grown as an ornamental. The milky juice once was thought to be effective against warts, and it is used in France as a laxative. Here the plant has become a weed. Do not let it grow near grapes, since it may cause the vines to become sterile, and cattle eating hay containing the spurge are made ill.
### Dandelion ( _Taraxacum officinale_ )
Dandelion likes a good, deep soil, as do clover and alfalfa. Soil around dandelions is attractive to earthworms, for this plant is a natural humus producer.
Dandelion
Dandelions on your lawn may frustrate, but actually they are not in competition with the grasses because their three-footdeep roots take nutrients from a different level of the soil. They penetrate hardpan and bring up minerals, especially calcium, depositing them nearer the surface and thus restoring what the soil has lost by washing. When dandelions die, their root channels act like an elevator shaft for earthworms, permitting them to penetrate deeper into the soil than they might otherwise.
Dandelions exhale ethylene gas, which limits both the height and growth of neighboring plants. It also causes nearby flowers and fruits to mature early.
### Datura ( _Datura stramonium_ )
Datura
Other names for this weed are Jamestown weed, jimsonweed, apple of Peru, thorn apple, stinkweed, devil's trumpet, angel's trumpet, and dewtry. Although all parts of datura are poisonous, it is a source of valuable medicine. The weed is especially helpful to pumpkins and it protects other plants from Japanese beetles. The smoke from dried datura leaves is calming to honeybees when opening a hive, but use it sparingly.
### Dead Nettle ( _Lamium album_ )
Although classed as a nettle, this plant, sometimes called white archangel, has no relation to stinging nettle and does not sting. A similarity of the leaves may be the reason for its name. Dead nettle has a long season of showy white bloom and is one of the few herbs that will also grow in damp places in filtered sunlight. As such, it is a valuable companion for the garden and for grain crops.
### Devil's Shoestring ( _Tephrosia virginiana_ )
There are nineteen species of this native North American weed that have valuable insecticidal properties. It is low in toxicity to animals, but is reputed to contain a fish poison. Wild turkeys are fond of eating the plant.
### Dock ( _Rumex crispus_ )
Curled dock is both a food and a medicine. In the old days it was gathered to "thin and purify the blood" in the spring of the year. There is no evidence to support this claim, but the high vitamin C content of dock undoubtedly was beneficial, particularly after a winter diet deficient in greens. The greens also are richer in vitamin A than carrots.
Dock is good to calm the pain of stinging nettle. Crush the juicy leaves and rub on the affected area.
### Dyer's Greenweed ( _Genista tinctoria_ )
Once this was considered a very useful plant for dyeing. Bees like it for its honey and sheep and goats like to graze it, but it is thought that its bitter taste affects the milk of cows that eat it.
### Elgrass ( _Zostera marina_ )
Eelgrass has an edible grain and is a widely distributed sea plant along the coasts of North America and Eurasia. It is reportedly harvested in spring by the Seri Indians along the Gulf of California in Sonora, Mexico.
The upper stem fruits in the spring, then breaks off and floats on the surface of the water. The Seri harvest the grain when large quantities of the plant are found floating loose near the shore.
Eelgrass has a content of protein and starch similar to that of grains grown on land. Once it is dry, it is separated from the seaweed with which it grows, and the grains are toasted and ground into flour. Cooked into a gruel, it may be eaten with honey. It is the only ocean plant known that has a grain used as a human food resource.
The Seri find many other uses for the eelgrass, one of which is a cure for diarrhea. It is also piled over house frames for shade and made into toys for the children. Others find it makes an excellent mulch for garden or orchard.
Fresh-water eelgrass (Hydrocharitaceae, genus _Vallisneria_ , species _americana_ ), also known as tapegrass and wild celery, grows in the mud of shallow ponds, sending up ribbonlike leaves directly from the root. (See Sources.)
### Euphorbia (Euphorbiaceae)
A few well-placed plants of caper spurge ( _Euphorbia lathyrus_ ) will deter moles and mice and thus are good to plant near young fruit trees. They also are useful to repel rats and mice. Many of the spurge family like dry, light, sandy soils but will spread to cultivated land if given a chance.
Injury to stem or leaves of the euphorbias causes them to exude a milky-looking sap that is very acrid and poisonous. Great care should be taken that the sap does not touch a scratch, as it will even cause blisters on delicate skin. However, the juice of the leafy spurge ( _E. elsula_ ) and the cypress spurge have been used against warts.
Snow-on-the-mountain, one of the euphorbias, is an attractive annual plant that grows wild from Minnesota to Texas. Poinsettia, another lovely member of this family, was formerly considered highly toxic, but is no longer classed as a poisonous plant.
### Fern, Male ( _Dryopteris filix-mas_ )
My beloved old _Pharmacopoeia of the United States of America_ , Seventh Decennial Revision, 1890, speaks of this fern eloquently, referring to its taste as "sweetish, acrid, somewhat bitter, astringent and nauseous." It does not, however, say that fern seed will make you invisible, as the Doctrine of Signatures once stated.
That ferns have medicinal value has been recognized for centuries and they are still listed in the _Pharmacopoeia_ today, the useful species including also the evergreen wood fern ( _D. marginalis_ ). In the autumn the roots are carefully dug, cleaned, and dried and the substance oleoresin is extracted through the use of ether.
Perhaps it is because of the oleoresin that ferns and beeches ( _Fagus_ ) have an inhibiting effect on each other. However, a compost made of ferns assists tree seeding and is useful to tree nurseries to encourage germination.
### Fiddlehead Fern ( _Pteridium aquilinum_ )
Sometimes called bracken, fiddlehead fern is another sprout-type wild food. This coarse fern has a single thick base stalking into three distinct curled fronds growing several inches from the ground. This fern is found in low, moist land from Canada southward into Virginia. To gather, snap off the coiled fiddleheads. The coat of fuzz is easily removed by pulling each head through your fingers.
Cook the fiddleheads in boiling water for about an hour, season with pepper, salt, and melted butter, oil and vinegar, or sour cream. Some people like to nibble the fern heads raw. They may also be cooked like asparagus or fresh beans.
Mature fronds may be toxic to grazing animals but as long as the ferns are coiled and tender, we may eat them.
### Fleabane ( _Erigeron_ )
This is one of our native plants that have spread to Europe and taken possession of stony soils. It is used for medicinal purposes, and the acrid oil as a mosquito repellent. But some people are allergic to this plant, reacting to it as they would to poison ivy.
### Fungi
Fungi are plants without chlorophyll — some are very useful and even edible, others are very troublesome.
_**Mushrooms**_ have a natural affinity for plants in woods, fields, and meadows. The part of the mushroom we see is just a small portion of the entire plant. Most of it is underground, a tangled, twisted jungle of threads that form the vegetative part of the plant (mycelium). The mushroom itself is the reproductive part, the fruiting body, which grows easily under the right conditions of moisture and temperature.
My sister-in-law in Missouri once brought home some morels from her father's farm and took them outdoors to her backyard to trim. She threw the pan of trimmings under an apple tree in her small orchard and was surprised sometime later to find morels growing thriftily there. They continued to come up intermittently for several years.
The delicious morels are fairly common in the United States. The fruit body, like that of other mushrooms, grows above the ground and resembles a sponge. Because of this, they are easy to identify and safe for the collector to gather for the table. They are often found in apple orchards just about the time the trees are blossoming.
Morchella elata
Morchella semilbera
Many mushrooms are deadly poison and no one who is inexperienced should ever gather them for food. The deadly amanita, sometimes called the destroying angel, can cause death in less than six hours. Yet many of the poisonous mushrooms also serve a purpose in promoting healthy growth on other plants.
_**Truffles**_ , another type of fungus, grow several inches below the ground and are not visible. They are rare in the United States but are quite common in Spain, England, Italy, and France, where they grow under oak and chestnut trees. Here they are found by dogs and pigs specially trained to locate them by smell. Should you see a French farmer following a pig on a leash, this is simply a routine way of picking truffles, which sell for fantastic prices.
Because of their close association with oak and chestnut trees, scientists believe that the truffles help the tree roots assimilate chemicals from the soil. Truffles vary from 1 ⁄4 inch to 4 inches in diameter and resemble an acorn, a walnut, or a potato. The spores are borne within the tuberlike body of the fungus. They have a delicious taste and serve as a condiment. They are black and thus very attractive as a garnish for salads.
Fungus on tree roots first was reported in 1885 by the German botanist A. B. Frank. His belief that water and nutrients were entering the tree through the fungus was scoffed at, but we know today that the fungus acts as a link between the soil and the rootlets of the plant. The tree in turn helps the mycorrhizal fungus by providing root metabolites, substances that are vital to the fungus for the completion of its full life cycle. Harmless tree fungi called saprophytes help trees to resist such diseases as bark canker, decay fungi, and leaf rust fungus.
_**Mildews**_ are a type of fungi that can be extremely troublesome and difficult to control when they form on plants, usually due to combined moisture and humidity. They attack grapes, lettuce, tomatoes, roses, peas, tobacco, potatoes, cucumbers, and many other fruits and vegetables, usually forming a gray or white, powderlike coating on the surface of the leaves. I have found that it is possible to partly control this fungus with a dusting of wettable sulphur. Sunshine and good air circulation is the best remedy.
_**Smut**_ is a fungus that attacks such grains as wheat, barley, rye, corn, and oats. It looks like a large sac or tumor among the kernels when it appears on an ear of corn. The sac contains a large mass of black spores.
Smuts act on the host plant in a different way from most parasitic fungi. The mycelium that grows among the cells of the host stimulates them to produce a swelling, or gall. The spores develop as a black mass within the gall and are thrown into the air when the gall breaks.
Smuts are difficult to control as, unlike the spores of any other fungi that attach themselves to the seed, these lie dormant in the ground through the winter. In the spring new kinds of spores are germinated and reinfect the corn plants. Treating the seed is seldom effective in preventing corn smut. The best answer seems to be crop rotation, along with the development and use of strains of corn that are smut-resistant.
### Garlic, Meadow ( _Allium vineale_ )
Meadow garlic's very penetrating taste and odor give a bad taste to milk if eaten by cows, and the bulblets in wheat will spoil the flour. It's hard to eradicate because the little bulbs grow deeper and deeper into the soil with the passing years. If pastures or fields are badly infested, crop rotation is recommended.
But even wild garlic has a definite health value, and medicines derived from it are useful against high blood pressure and sclerosis.
### Ginger, Wild Ginger ( _Asarum canadensis_ )
Ginger grows in rich woods from Virginia to Minnesota on a low plant with root cords extending from a knotty but superficial root. The leathery leaves, mottled green, are heart-shaped, growing on tough, hairy stalks. A reddish brown, cup-shaped flower blooms during April and May at ground level, emerging between two-leaf stalks. Gingerroot is best dug in October when the roots are full and are better for candied ginger at this time.
### Ginseng ( _Panax quinquefolium_ )
Wild ginseng needs the companionship of trees to provide the filtered sunlight it requires to grow. When raised in beds for commercial use, it is covered with latticework to protect it from the heat of the sun, which would kill it.
The Chinese believe that ginseng will cure nearly every disease, yet even now Western science is not sure whether it has real value or not. Physicians regard its benefits as largely psychological, but tests in Russia and elsewhere indicate that infusions of the root actually may increase energy and resistance to infection.
### Goldenrod ( _Solidago_ )
There are more than 60 native species of goldenrods, some growing on dry soil low in humus, others on soil that is rich and moist. If you would eliminate the goldenrod, cut it before it goes to seed and improve your soil with organic matter and beneficial crops.
### Henbane ( _Hyoscyamusniger, L_.)
Henbane, like hemp, is an ancient narcotic that was once used to treat disease, but escaped to become naturalized in some parts of the country. The drug hyoscyamine, used to dilate the pupils of the eyes, is made from black henbane.
This poisonous herb is fatal to fowl, hence its name "henbane." All parts of the plant contain poisonous alkaloids, and even hogs sometimes are killed by eating its fleshy roots.
### Horsetail ( _Equisetum arvense_ )
The horsetails are the last remainder of the huge trees of the carboniferous forests. The most common is the field horsetail, which grows in sandy and gravelly soils on a high ground-water level.
Horsetail looks like a tiny Christmas tree and sometimes is called the meadow pine. The hollow-branched, jointed stems range from one to three feet tall. The plant does not produce flowers or seeds but sends up fertile spore-bearing stems resembling catkins. These are covered with powdery brown spores. After the spores drop, small green shoots emerge from the ground. The perennial horsetail has a rootstock that bears tuberous growths that store available carbohydrate against a future need.
The green shoots of horsetail contain a high percentage of silica, the controlling factor against fungus disease. If the green stems are burned in a hot but quiet flame, removing all organic parts, the white skeleton of silica that is left will show the original structure of the little stems.
Silicic acid can be found in many plants such as stinging nettle and quack or couch grass, but some, including horsetail and knotweed ( _Polygonum aviculare_ ), are particularly rich in it. It is this silica in horsetail that has made it valuable medicinally for centuries.
Silicic acid is traditionally believed to strengthen tissue, particularly that of the lungs, and at times to add to disease resistance. It has been reported to have a good influence on inflammation of the gums, the mouth, or the skin in general. Horsetail was specially recommended for diuretic purposes by Kneipp, the German priest who was so successful in using hydrotherapy in combination with herbs.
_The ancient, primitive horsetail (called equisetum) makes a fine plant spray. The plants, containing much silica, also are used to scour pots and pans_.
While horsetail itself will stop external bleeding, a horsetail brew also may be used as a healing agent for abscesses, burns, cuts, and scratches, for both animals and humans. Place a good handful of the dried leaves and stems in a stewpan with just enough vinegar (5 percent acidity) to cover. Simmer for no longer than 20 minutes, cool, and strain. Keep in the refrigerator. When needed for use, add 1 part brew to 2 parts cow or goat milk. Return any not needed to refrigerator. Horsetail brew is stingless, soothing, and gentle in its action. A plastic squeeze bottle may be used for convenient application.
A tea made of horsetail is very effective against mildew and other fungi found on fruit trees, grapevines, vegetables, and roses. It is gentle but swift in action and does not disturb soil life. Silicarich plants are valuable on the compost pile, too.
The parts used are the dried leaves and stems of the sterile form. Boil 2 or 3 teaspoons of the crumbled herb to a cup of water for 20 minutes. Or soak the leaves in water for several hours, boil for 10 minutes, and steep for another 10.
Made into a spray, it is particularly useful against powdery fungus and curly leaf on peach trees. It also controls mildew on roses, vegetables, grapes, and stone fruits, and has been found to have a cellstrengthening action on the plants sprayed with it.
According to Beatrice Trum Hunter's _Gardening Without Poisons_ , silica in the form of an aerogel is one of our most effective insecticides. It may be used against the flour beetle, rice weevil, granary weevil, and the larva of the Mediterranean flour moth.
Horsetails have still another use: Their high silica content makes them effective as "scouring brushes" for pots and pans, and campers still find them usually available and convenient for this purpose. In German-speaking countries horsetail was called _Zinnkraut_ , or "pewter plant," because the high silica content made it useful for cleaning and scrubbing copper, brass, pewter, and all fine metals.
### Indian Cucumber ( _Medeola virginiana_ )
Indian cucumbers are found from the Great Lakes eastward and south to Florida in rich woodlands. They are good eaten raw, cubed in salad, or pickled. "They grow in sociable groups," says Grace Firth in _A Natural Year_ , "and, like other sociables, are best when mildly pickled."
The single slender stems of Indian cucumbers grow out of a whorl of five or seven dark, elongate, visibly ribbed leaves. The spindly cream-colored flowers are followed by small black berries. The roots grow horizontally just below the ground surface, making the cucumbers easy to dig.
### Jewelweed ( _Impatiens bifora_ or _I. pallida_ )
Jewelweed is an excellent remedy for poison ivy, relieving the itching almost instantly. It does not have an adverse effect on the poison ivy plants, for the two often grow side by side. In fact, wherever poison ivy grows, jewelweed is likely in the vicinity. It blooms between July and September.
To make the remedy, boil a potful of jewelweed in water until the liquid is about half the original volume. The strained juice is effective both in preventing the rash after exposure and in treating the rash after it has developed. The best way to keep the extract is frozen and stored in cubes in the freezer. Pack them in small plastic bags and they will be at hand if needed.
Jewelweed
The young, tender sprouts of jewelweed are edible and are cooked like green beans.
### Knotweed ( _Polygonum aviculare_ )
These members of the Buckwheat family grow mostly on acid soils. The prostrate knotweed, frequently found in garden borders and along paths, doesn't mind being trampled on. All knotweeds are characterized by "knots" on their stems from which branches grow. Knotweeds in pastures are thought to be troublesome to sheep. They also inhibit the growth of turnips, and are very rich in silica.
### Lamb's-Quarters ( _Chenopodium album_ )
This plant, sometimes called smooth pigweed, is one of our most enduring annual weeds, producing a tremendous amount of seeds that are able to survive dormant in the soil for decades. It is among the weeds that follow human footsteps and cultivation, liking a soil with a well-fermented humus.
Lamb's-quarters is particularly stimulated when grown with potatoes, and it should be allowed to grow in the garden in moderate amounts, especially with corn. It also aids cucumber, muskmelon, pumpkin, and watermelon as well as giving additional vigor to zinnias, marigolds, peonies, and pansies.
This plant, a close relative of spinach, also is good to eat. The young shoots may be cooked and eaten like asparagus. It is richer in vitamin C than spinach, far richer in vitamin A, and, though not quite so rich in iron and potassium, is still a good source of these minerals. It is exceptionally rich in calcium.
Lamb's-quarters is a freebie that everyone should know about, for it is found in cultivated ground from north to south and east to west, and plants in the right stage for eating can usually be found from late spring until frost. It even grows in the Andes at a height of 12,000 feet, and here has become an important substitute for rye and barley, which cannot survive at such an altitude.
### Larkspur, Wild (Ranunculaceae)
Wild larkspur is detrimental to cattle and too much may cause poisoning. Barley as a crop, however, is a weed deterrent and will prevent wild larkspur or poppy from establishing itself. It is thought to promote vigor in winter wheat.
### Locoweed ( _Astragalus mollissimus_ and _A. diphysus_ )
Locoweed gets its name from the Spanish word for crazy, due to the strange actions of animals poisoned by it.
Strangely enough, the poisonous effect of locoweed depends on the soil in which the plants grow, because of their ability to absorb poisonous elements from the soil. Both the green and the dry plants are poisonous, the symptoms varying somewhat in horses, cattle, and sheep.
Horses become dull, drag their legs, seldom eat, lose muscle control, become thin, and then die. Cattle react similarly, but sometimes they run about wildly, or stagger and bump into objects in their path. Sheep are less apt to be injured by the poison.
Ranchers destroy locoweed by cutting the roots about two inches below the surface.
### Lupine ( _Lupinus perennis_ )
Sometimes called old maid's bonnets, wild pea, or sundial, the plant has vivid blue flowers, sometimes pink or white, with a butterfly shape that indicates its membership in the Legume family.
Farmers once thought that lupines preyed upon the fertility of their soil; hence the name derived from _lupus_ , "a wolf." In fact, they help the growth of corn, as well as most other cultivated crops.
Lupines grow best on steep, gravelly banks or exposed sunny hills, liking almost worthless land where their roots can penetrate to surprising depths, in time leaving behind them fine, friable soil. They are adventurous pioneers, spreading far and wide in thrifty colonies, and are among the first plants to grow on the barren pumice after a volcanic eruption.
The lupine is one of those interesting flowers that go to sleep at night. Some fold their leaflets not only at night but also during the day when there is movement in the leaves. Sundial, a popular name for the wild lupine, refers to this peculiarity. Among the nearly 100 kinds of lupines that grow in North America, some contain poisonous alkaloids, while the seeds of others can be eaten.
### Mayweed ( _Anthemis cotula_ )
Sometimes this is called dog fennel or fetid camomile because of its evil smell. Beekeepers used to rub it into their skin to repel bees. It also will repel fleas and may be rubbed into floors and walls of a granary to repel mice.
### Meadow Pink ( _Lychnis floscuculi_ )
The roots of all members of the Lychnis family contain saponin, which produces a soapy foam if stirred in water. Before the invention of soap these roots, together with those of the true saponaria, were used for washing. An interesting family member is the sleepy catchfly, so called because its flowers are closed most of the day, opening only in bright sunshine, while the gluey substance on its stems entangles flies.
### Mildew, Powdery (Erysiphaceae)
Mildew is a fungus of the type called an "obligate parasite" because it feeds on living plants. When moisture conditions are just right, wind-carried spore (little seeds) resting on a plant's leaves send out germ tubes, which grow into white threads (mycelia). These branch over the leaves in a white, soft, felty coating.
This type of fungus does not grow _inside_ the plant but sends its little suckers (haustoria) into the plant's sap. As chains are built up from the mycelium spore, the plant becomes covered in a few days. Eventually black fruiting bodies with the sexual or "overwintering" spores are formed.
Because it is on the surface, mildew is more easily controlled than many other fungi, and horsetail tea is an excellent spray to use (see _Horsetail_ in this chapter). During the season when green plants are available, it is also good to prepare an extract by covering freshly picked plants with water. Allow them to ferment for about 10 days, then dilute the liquid and use it as a spray in the same way as the tea. Mustard seed flour or sulphur dust also may be used, while polybutenes, oil derivatives, have been used successfully to control powdery mildew on cucurbits.
### Milkweed ( _Asclepias_ )
All of the many milkweeds exude milky juice when their leaves or stems are punctured. Roots are considered poisonous, but Native Americans have used them for various maladies, and some say that the juice cures warts or ringworm. Cows dislike the bitter, acrid plants but may eat them if hard-pressed for food.
### Mistletoe (Loranthaceae)
This parasite is the most legendary of plants. It was sacred to the ancient Druids, who cut it with a golden sickle (the symbol of the sun) and caught it in a cloth to prevent it from touching the ground.
Mistletoe grows on apple trees, oaks, and poplars, usually being sown by birds. Here in Oklahoma it is our state "flower" and grows profusely on hackberry trees.
People used to call the mistletoe "all heal" and thought there was no illness it could not cure. It is in fact poisonous, particularly the berries. If you find it growing on your trees, remove it, for it is a parasite and eventually will weaken the tree and possibly kill it.
### Nettles
See _Stinging Nettle_.
### Nightshade (Solanaceae)
The Nightshade family includes apple-of-Sodom, belladonna, bittersweet, capiscum, eggplant, jimsonweed ( _datura_ ), petunia, potato, snakeberry, tobacco, and tomato.
Where black nightshade ( _Solanum nigrum_ ) grows profusely, the soil is tired of growing root crops. This plant draws the Colorado potato beetle away from potatoes, since they prefer the weed though it is poisonous. The beetles eat it and die.
### Nut Grass ( _Cyperus esculentus_ )
The botanical name, _Cyperus esculentus_ , means "edible sedge." Nut grass is related to the tules and bulrushes and is almost as old as civilization. The ancient Egyptians developed cultivated strains more than five thousand years ago. The "nut", which is really a tuber, may be made into many tasty and unusual dishes. But consider well before growing this plant, for it can become a fearful weed. Having battled against it in my garden for many years, I have come to the conclusion that this is a plant only Euell Gibbons could love.
If the native species of nut grass plagues you, it can be discouraged by growing a heavy cover crop of cowpeas on the plot for several summers. Sow the cowpeas thickly to form a dense mat that shades the ground. Plow them under in late fall, in October or November, and they will add nitrogen to the soil. St. Augustine grass will choke out nut grass on a lawn.
### Opuntia (Cactaceae)
In the Southwest and Mexico, people eat the flat-leaved joints on opuntia cactus boiled or fried, and make the flowers into salads. The juicy fruits of opuntia are eaten raw or cooked; the seeds are ground up into a meal and made into cakes. The bushy cacti grow in hedges around houses, and where little else will survive they serve as windbreaks and ground cover.
In some regions in times of drought ranchers "burn off" the cactus spines so that cattle may eat the plants. Plant breeder Luther Burbank developed a spineless cactus that proved to be a useful source of food for both men and animals, and in some sections of California the opuntia type is a commercial crop. It is grown on sandy loam, fertilized with chicken manure, and needs no insecticides. The crop of young leaves, which measure about eight inches long at one to two months, are hand-picked to be sold diced, shredded, and spiced, or pickled.
Opuntia
Almost all of the many useful species of cactus will grow well with each other, requiring the same type of soil and cultural practices. Many have beautiful delicate blossoms and are grown as ornamentals, and there is a subzero cactus of the opuntia type that will grow in the North. (See Sources.)
### Ox-eye Daisy ( _Chrysanthemum leucanthemum_ )
Ox-eye daisy seeds are beneficial in small quantity mixed (1 to 100) with wheat grains, but in larger quantity the daisy will overwhelm the wheat.
### Pansy, Wild ( _Viola tricolor_ )
The wild form of the cultivated pansy is Shakespeare's "heartsease," and once was listed in the _U.S. Pharmacopoeia_ as a medicine. Many species of viola were candied as a sweet and were thought to be soothing and therapeutic to the heart.
Rye helps the wild pansy to germinate and is itself seemingly improved by a few pansies. But the pansy has an inhibiting effect on wheat.
### Pawpaw Trees ( _Asimina triloba_ )
These quick-growing, rangy trees rarely exceed 20 feet but bear large, outsized leaves. The fruit usually falls to the ground before it is good to eat but will ripen if held at room temperature. Mashed pawpaws may be made into a bread similar to banana bread.
### Pennycress ( _Thlaspi arvens_ )
Like shepherd's purse, this is often abundant where grain is grown. The seeds are 20 percent oil, and if accidentally ground with grain will spoil the flour. Mountain pennycress ( _Thlaspi alpestre_ var. _calaminare_ ) likes soils containing zinc.
### Peruvian Ground Cherry ( _Nicandra physalodes_ )
When planted in quantity near a barn or in a stableyard it will repel flies, and it also is effective against whitefly.
### Pigweed
See _Amaranth_ in this chapter.
Plantain
### Plantain ( _Plantago_ )
Of the many different plantain species, the largebracted plantain ( _P. aristata_ ) is the most prolific, one plant producing well over 3,000 seeds.
The narrow-leaved ( _P. lanceolata_ ) has been used as a home remedy for treating bruises and strained joints. It also has a cooling and astringent effect if a few leaves are squeezed over a bee sting.
Plantain and red clover frequently are found growing together, this because the plantains occur as impurities in grass and clover seeds. If plantains appear on the lawn, it is best to dig them out.
### Poison Ivy ( _Rhus radicans_ )
Poison ivy
Jewelweed will relieve the itching of poison ivy (see _Jewelweed_ ). If your land has poison ivy, the best thing you can do is eradicate it. Mow close to the ground in midsummer and follow this with plowing and harrowing, grubbing out small patches. Under trees or along a fence where mowing might be difficult, try smothering it with heavy cardboard or tar paper. A deep mulch of hay or straw may work as well.
Vines growing in trees may be cut near the ground and then pulled down a few days later. Be sure to wear gloves and protective clothing, washing well afterward, preferably with yellow soap.
### Pokeweed ( _Phytolacca americana_ )
I find that pokeweed grows well under my figs, Scotch pines, and other trees. Pokeberries and roots are poisonous, but the tiny, pinkish green, asparagus-like shoots are simply delicious. Poke is one of the first greens to come up in early spring, and these shoots should be cooked lightly in several changes of water. The berries and roots contain phytolaccin, a cathartic and slightly narcotic substance used for treating rheumatism.
Pokeweed
Poke should never be confused with the completely unrelated Indian poke, or white hellebore. This latter plant, also poisonous, grows in wet places and comes up very early in the spring before the edible poke starts growing.
### Potherbs
There is a wealth of wild greens available in April. Most of these belong to the Mustard family, Cruciferae, so called because their flower petals form a cross. The mustard cousins, Brassica, together with strong-growing young shoots of peppergrass ( _Lepidium_ ), horseradish ( _Armoracia lapathifolia_ ), pennycress ( _Thlaspi arvens_ ), searocket ( _Cakile edentula_ ), scurvy grass ( _Cochlearia_ ), and other cress greens are a gift of spring to those who love to search for wild greens. Cook them as you would turnip greens or spinach. If their turnipy-horseradishy flavor is a bit much for you, combine them with milder greens such as young amaranth ( _Amaranthus_ ), Japanese knotweed ( _Polygonum cuspidatum_ ), or dayflowers ( _Commelina communis_ ). Shepherd's purse ( _Capsella bursa-pastoris_ ) also combines well. Gather potherbs when the shoots are very young. You may wish to cook them through several waters.
### Puffball (Fungi)
Sometimes these "smoke balls" or "devil's snuffboxes" grow to be more than two feet across. A cut that is bleeding profusely may be covered with the powderlike spores from those that produce a puffor "smoke" when disturbed, and it will stop the bleeding. Puffball powder is very explosive, so if you store it, keep the container closed and away from fire.
### Purslane ( _Portulaca oleracea_ )
Purslane has a liking for good cultivated soil and is frequently found in gardens. But it is not altogether unwelcome, for though often considered a weed, it is cultivated in both England and Holland. It is a refreshing green with a slightly acid taste, and it may be cooked like spinach. One hundred grams of purslane contains 3.5 milligrams of iron, and this is all the more remarkable because the plant is 92.5 percent water.
### Pyrethrum ( _Chrysanthemum cinerariaefolium_ )
Pyrethrum is absolutely bugproof and will keep pests from plants close by. Few ticks are ever found where pyrethrum or sage forms a ground cover. Pyrethrum powder, generally considered a safe insecticide, is made from the dried flowers. It has a very short residual action, breaking down rapidly in sunlight. Because of this it can be used as a preharvest spray.
Records show that pyrethrum may have been used nearly two thousand years ago in China. As an insect repellent it became popular again in the nineteenth century, when it was the "secretingredient" in Persian insect powder. In 1828 this powder was produced on a commercial scale and introduced into Europe by an Armenian trader. By 1860 it was becoming well known in the United States.
The active principles in pyrethrum are the esters pyrethrin and cinerin. Certain nontoxic plant products such as asarinin (from the bark of southern prickly ash), sesamin (from sesame oil), and peperine (from black pepper) are added to pyrethrum to strengthen its effect.
### Rattlebox ( _Crotalaria_ )
This weed is valuable for its soil-improving qualities, but one variety, _C. sagittalis_ , found on bottomland in the Missouri and Mississippi basin, is very poisonous to cattle and horses and should be eradicated. Following its cutting or plowing under, plant a crop like cotton or corn that needs repeated cultivation.
### St.-John's-Wort ( _Hypericum perforatum_ )
This common pasture plant contains a red oil sometimes used as a home remedy for bronchitis and chest colds. It also is astringent and has been used against diarrhea and dysentery. The leaves have oily cells and a strong, peculiar smell. They look perforated if held against the light. It was once believed that if the plant was collected during St. John's Night (June 24), it would afford protection against witches and evil spirits.
### Shepherd's Purse ( _Capsella bursa-pastoris_ )
Shepherd's purse is very rich in minerals. Along with mustard, it absorbs excessive salts in the soil and returns them in organic form. If grown on a salty marsh and plowed under while still green, it will both sweeten the soil and discourage the weeds ordinarily growing on such soil. It has medicinal qualities and has been used as a styptic.
### Sow Thistle ( _Sonchus arvensis_ )
This plant has creeping, deep-growing roots, contains a milky, yellow-tinged juice, and grows on moist soil. It aids watermelon, muskmelon, pumpkin, and cucumber and in moderate amounts onions, tomatoes, and corn.
A cousin called blessed thistle ( _Cnicus benedictus_ ) has medicinal and industrial uses and is a basic ingredient of the Benedictine liqueur as well as certain bitter tonics.
### Spurge
See _Euphorbia_ in this chapter.
### Stinging Nettle ( _Urtica dioica_ )
Stinging nettle has many helpful qualities. It makes neighboring plants more insect-resistant. It also helps plants withstand lice, slugs, and snails during wet weather; strengthens growth of mint and tomatoes; and gives greater aromatic quality to herbs such as valerian, angelica, marjoram, sage, and peppermint. The nettle protects fruit from mold and thus enables it to keep better. Fruit packed in nettle hay ripens more quickly.
Stinging nettle
Stinging nettle is helpful to stimulate fermentation in compost or manure piles, according to British author M. E. Bruce, who advises making a crushed nettle solution. Good results can be had with less trouble by using the nettle in its original form, placing it in layers in the compost before the nettle seeds ripen. The plant is said to contain carbonic acid and ammonia, and these may be the factors that activate the compost. If you have the space, you might try raising a crop of nettle — somewhere away from the garden, for the plant spreads quickly.
Euell Gibbons reports that stinging nettles combined with young horseradish leaves are delicious as spring greens. They also combine well with lettuce or spinach. Since nettles are rich in vitamins and iron, they are a good remedy for anemia, while aiding blood circulation and acting as a stimulant.
The plant leaves, as rich in protein as cottonseed meal, are good for animals, too, though they will touch them only when the nettles are mowed and dried. Horses improve in health and cows will give more and richer milk. When powdered nettle leaves are added to their mash, hens will lay more eggs and the eggs will have a higher food value, chicks will grow faster, and turkeys will fatten. Even the manure from nettle-fed animals is better than that from others.
Be sure to wear gloves when picking young nettles, for the fine hairs on the leaves and stems contain formic acid, which irritates the skin when touched. Nettle rash can be relieved with the juice of the nettle plant itself, or by rubbing the skin with jewelweed, with rhubarb, or with any member of the Sorrel family.
### Tansy ( _Tanacetum vulgare_ )
Tansy, once used as a medicinal tea, is now considered dangerous if taken as an infusion (see the chapter on Poisonous Plants). It is also thought to be poisonous to cattle. But planted under fruit trees, particularly peach, it repels borers, and is a good companion to roses, raspberries, blackberries, grapes, and other cane fruits. It deters flying insects, Japanese beetles, striped cucumber beetles, and squash bugs, and helps repel flies and ants. The dried leaves are useful for storing woolens and furs. Because of its concentration of potassium, tansy is useful on the compost pile.
Tansy
Tansy also goes by the names of bitter buttons, ginger plant, hind heal, and scented fern. Its fernlike, attractive foliage is topped by composite heads of buttonlike flowers, and their scent is delightful. Though the roots have been used, it is the tops that are of primary importance.
Tansy once was used as a culinary herb in place of pepper, was widely used in churches in medieval times as a "strewing herb," and is one of the plants associated with the Virgin Mary.
### Thistle ( _Circium_ )
All thistles are rich in potassium and thus are useful in compost; but their prickly leaves make them unpopular in pasture, and certain types rob grainfields of food and moisture. To kill out thistles, be careful not to cut them before the blossoms are open or many more will grow from the rootstocks. If you cut just the blossom heads after the blossoms are pollinated, the plant will bleed to death.
Blessed thistle ( _Cnicus benedictus_ ) has medicinal and industrial uses, as well as being an insect repellent, while constituting a basic ingredient of the Benedictine liqueur and stomach bitters.
### Thornapple ( _Datura stramonium_ )
See _Datura_ in this chapter.
### Tillandsia ( _T. usneoides_ )
Tillandsia (also called Spanish moss, though not a true moss) occurs naturally from Virginia to Florida and Texas and southward to Argentina and Chile. Though it may be seen on other trees, its favorite host is the live oak. It is not a parasite, sapping the life of the tree, but a lodger that finds its own food supply through slender, grayish stems that look like hair hanging from the trees. It is related to the air plants of the Bromelia or Pineapple family, which largely draw their sustenance from the air itself. It may be used for mulching or for compost. The dried stems also are used industrially to stuff upholstery.
### Venus Flytrap ( _Dionaea muscipula_ )
This hungry little plant, which captures its own meals, must be grown in high humidity indoors or out. The insect traps on young plants, which develop in three to four weeks, consist of two leaves hinged in the center when open, and when closed forming a pouch in which the trapped insects are digested. When a leaf has caught several insects it withers and dies, but new ones take its place. Venus flytrap grows naturally in bogs where the soil lacks available nitrogen, and the insects supply this nutrient in the plant's diet.
### Water Hyacinth ( _Eichhornia crassipes_ )
This tropical American plant has "escaped" and is now growing profusely in the southern states, sometimes choking ponds and streams with its growth of floating leaves. The roots hang down in the water and receive the spawn of fish. The lovely violet flowers are large and showy.
If you live in an area where water hyacinth has gone wild, you will benefit both yourself and the waterway by dredging it out and using it for compost.
### Water Lily ( _Nymphaea odorata_ )
The American water lily, related to the lotus, sends long, stout leaf and flower stalks from the mud bottom of clear, shallow water. The beautiful flowers usually rise above the water on long stalks and may be as large as a foot across. In cultivation, water lilies grow well in a mixed planting of other water plants. In the wild state they, like water hyacinth, sometimes threaten to choke shallow pools of water and should be dredged out and composted.
### Weeds
Someone once said, "A weed is a plant out of place," but I am inclined to go along with Ralph Waldo Emerson, who believed that "a weed is a plant whose virtues have not yet been discovered." Weeds, wisely used, are some of our most important companion plants. Of course, they never should be allowed to overwhelm the food plants, but a few left here and there may surprise you by the influence they exert.
The extensive root growth of weeds penetrates the subsoil, breaking it up and making it easier for the roots of crop plants to go farther than usual as they search for water and nourishment. A few weeds are useful in shading the ground to keep seedling vegetables from drying out in the sun's heat. Meanwhile, moisture from the subsoil will travel by capillary action up the outside of the weed roots to a level where the young vegetables can use it.
Deep divers such as pigweed, lamb's-quarters, and thistles bring up minerals from the lower soil by way of their stalks and leaves. When these weeds are turned under, the minerals become available to shallower-rooting crops. Minerals, including trace minerals that may be leached away or become exhausted under a succession of crops, are thus retained.
Another interesting fact is that weeds seem to accumulate the nutrients in which a particular soil is deficient. Such weeds as sheep sorrel and plantain, which thrive best in acid soil, are rich in alkalinizing minerals such as calcium and magnesium. Bracken, which grows best in phosphorus-poor soil, is high in phosphorus. Turning these weeds under will release these minerals into the topsoil, again making them available to food plants.
Weeds also benefit the soil by conditioning it. Their extensive root sytems leave fibrous organic matter, which decays, adding humus to both topsoil and subsoil. Not only this, but they also leave channels for drainage and aeration. When decomposed, the root systems of dandelions provide subterranean channels for earthworms, which, in turn, enrich the soil with their castings. Soil texture is vastly improved and soil-inhabiting bacteria will multiply enormously.
Learning to read weeds can be very useful, for they are excellent indicators of the type of soil they select to grow on.
Weeds that delight in acid soil, and also indicate increasing acidity, are the docks, fingerleaf weeds, lady's thumb, and sorrels. Horsetail indicates slightly acid soil, as do hawkweed and knapweed.
Weeds that indicate a crust formation and hardpan are pennycress, morning glory, horse nettle, field mustard, camomiles, quack grass, and pineapple weed.
Weeds most likely to occur on cultivated land are chickweed, buttercup, dandelion, lamb's-quarters, plantain, nettle, prostrate knotweed, prickly lettuce, field speedwell, common horehound, celandine, mallows, rough pigweed, and carpetweed.
Sandy soils are favored by arrow-leaved wild lettuce, yellow toadflax, onions, partridge pea, broom bush, flowered aster, and most goldenrods.
On alkaline soils we are apt to find sagebrush and woody aster, while limestone soils grow field peppergrass, hare's ear mustard, wormseed, Canada bluegrass, cornelian cherry, pennycress, Barnaby's thistle, mountain bluet, yellow camomile, and field madder.
If a plot of land grows healthy weeds, it will probably grow good vegetable crops, too. Let the weeds reach full growth but cut them before they go to seed. Let them wilt a few days and then plow them under for green manure.
You may even find it helpful to your compost pile to bring in extra weeds such as those cut by the highway department along public roadways. This largesse often includes such items as nettles, sunflowers, yarrows, and sweet clover. These should be thoroughly composted to kill their seeds before being placed on the garden.
Weeds are not necessarily our enemies. With good management, they may well become friends and coworkers.
### Wild Carrot ( _Daucus carota_ )
Wild carrot does not always indicate bad soil, for its deep taproot implies a deep soil capable of good cultivation. A rich stand indicates a soil worth improving for crops. But it can become a pest, so prevent it from seeding by cutting the plant close to the ground shortly after pollination. Do not cut too early or many plants will spread out from the root.
### Wild Morning Glory (Convolvulaceae)
We have it from Native Americans that wild morning glory is beneficial to corn, but if allowed to go to seed it can become a great pest, coming up for years afterward. It may be killed out by spraying a little white vinegar into the center of each vine.
Wild morning glory
### Wild Parsnip ( _Pastinaca sativa_ )
Wild parsnip is a nourishing food plant that will give a good yield even on poor soils, but it soon becomes a weed and is hard to eradicate. The cow parsnip ( _Heracleum lanatum_ ) is poisonous.
### Wild Radish ( _Paphanus raphanistrum_ )
Wild radish spreads quickly in soils worn out from growing too many grain crops and depleted in nitrogen. It flourishes well, especially in wet years, where manure is scarce and potassium fertilizer abundant. Nevertheless, cattle are very fond of it, and it produces a good honey as well as an oil from the seed.
### Wild Rose (Eglantine)
When this pretty weed migrates from hedgerows to pastures, it indicates the pasture has not been grazed adequately and needs mowing and harrowing. The prickly canes are troublesome to sheep and cattle but do not particularly bother goats, which love all kinds of rosebushes. To eradicate, cut the canes while they are still soft.
### Wild Strawberry ( _Fragaria_ )
Wild strawberries are small, but have a delicious flavor quite unlike any other. Their presence in pastureland is an indicator of increasing acidity.
### Yarrow ( _Achillea millefolium_ )
Yarrow is a plant of both mystery and history. For centuries the Chinese mystic has cast yarrow stalks when consulting the _I Ching_.
Yarrow
According to the Bio-Dynamic book _Companion Plants_ , yarrow has a definite effect on the quality of neighboring plants, increasing not so much their size as their resistance to adverse conditions, and thereby improving their health. It is a good companion for medicinal herbs, enhancing their essential oils and increasing their vitality. It is also said to help cuts to heal.
Yarrow also gives nearby plants resistance to insects, perhaps because of its acrid, bitterly pungent odor.
Yarrow tea or yarrow hay is helpful to sheep, and I have given it to milk goats after kidding. It will grow almost anywhere and under any conditions and does not mind being walked upon. Where it grows in lawns and is cut by the mower, it simply spreads out in a low growth.
## Grasses, Grains, and Field Crops
**Sugarcane and bamboo** are both giant grasses, while the great cereals of the world, grasses too, are wheat, corn, oats, rye, barley, and rice. Wild grasses include bluegrass, esparto, reed, sandbur, Sudan grass, and wild barley, as well as many others.
Many grains and grasses grow well planted together, the growth of both being enhanced and increased.
### **Cotton** ( _Gossypium_ )
Alfalfa planted before cotton will put nitrogen in the soil, to the cotton's benefit, and alfalfa planted with it will discourage root rot.
Cotton growers try to keep the pink bollworm under control by isolating infected fields, sterilizing seeds and cotton, and by using machines that chop up leaves and other trash from the cotton. Shredding stalks in late summer and plowing them under helps control the worm and the boll weevil.
Farmers now protect their crops against the diseases of older cotton plants by growing varieties that are bred to resist such diseases as wilt and blight.
### Crabgrass ( _Digitaria sanguinalis_ )
Crabgrass is one of the most troublesome lawn pests. Handpulling is recommended on lawns (before the plants form a mat), and mulch and frequent cultivation in gardens. As with Bermuda grass, dry hot weather will wilt the roots if they are brought to the surface. (See also _Grass, Quack_.)
### Esparsette ( _Onobrychis viciaefolia_ )
This perennial forage legume of Eurasian origin also is called medick, sanfoin, lucifer, snail clover, and great trefoil. The plant was introduced into England by the Romans, but it is not much grown today, possibly because it takes several years to reach fruition. It produces three crops a year, however, and grows again from the same roots once it reaches maturity. Oddly enough, in England the spikelike flowers are violet but in China they are yellow.
Esparsette is a good food for cattle and is equally nourishing for humans. It is valuable for those suffering from weight loss yet reduces weight in those who are too heavy. It is considered a tonic for both the brain and spinal cord. The roots of a variety called black medick make a good tooth powder.
Esparsette is recommended as a border plant for small grains or vegetables, and in a thin stand it aids growth of corn. It also may be grown as a lightly scattered stand with small grains. The seed retains viability for up to three years, and grows well in limestone soils.
### Flax ( _Linum usitatissimum_ )
Flax is a good companion to both carrots and potatoes, improving both their growth and flavor. Flax planted near potatoes will protect against the Colorado potato beetle. (It is, however, poisonous; see the chapter on Poisonous Plants.)
Leaf extracts of the false flax ( _Camelina sativa_ and _C. micrcoarpa_ ), frequently found growing in flax fields, have an inhibiting effect on flax itself.
### Grass, Korean ( _Zoysia japonica_ )
This group of eastern Asiatic, perennial creeping grasses is widely used in the Southwest. Some are important horticulturally as lawn grasses, others as ornamentals. In hot, dry climates Korean grass is strikingly effective planted with such succulents as the porcelainlike aeonium, the texture contrast enhancing both plants. It grows well with _Cotyledon undulata_ and the delicately beautiful rosettes of _Graptopetalum paraguayense_.
### Grass, Molasses ( _Mellimus minutiflora_ )
See _Molasses Grass_ in the Pest Control chapter.
### Grass, Pampas ( _Cortaderia_ )
This ornamental grass, best grown in the South, produces beautiful flower plumes that, if cut when fully developed, are useful for decorative purposes indoors during the winter. It is increased by root division, and it grows well as a specimen plant in the lawn.
### Grass, Quack ( _Agropyron repens_ or _Triticum repens_ )
Quack grass indicates a crust formation and/or a hardpan in the soil. Choke it out by sowing millet, soybeans, or cowpeas, making sure that the land first is thoroughly cultivated and the weather hot and dry. Two successive crops of rye also will choke it out.
A concentrated brine of common salt (sodium chloride) will kill it out, too, if used after grass is freshly cut and applied in dry weather several times. Dry weather will wilt the roots of quack grass if they are brought to the surface. Hand-pulling is recommended if there are but a few plants.
Like so many other things, quack grass isn't all bad. It is a good cattle feed, and because of its persistence it makes a useful covering for gullies and road banks where live soil has been cut open and few other plants will grow. Though hard to get rid of once it is started, it does prepare the soil for better things. Oddly, it is wheat's nearest relative.
### Grasses, Lawn
_**Bermuda Grass** (Cynodon dactylon_) is an excellent lawn grass for the southern states. Bermuda withstands both heat and drought and will grow reasonably well even on poor soils. It may be started by seeding or sodding.
Never allow Bermuda to get started in the garden or flower beds, for it spreads quickly on cultivated soil, competing with flowers or vegetables for moisture. It may be killed out in the summer by hoeing and exposing the rhizomes to hot sunlight.
_**Kentucky Bluegrass** (Poa_) is an excellent grass for the North and East. It needs a quick-germinating and quick-growing grass, such as redtop, planted with it to provide a rapid ground cover that will help crowd out weeds during its early development. After it gets a good start,the bluegrass will crowd out the nurse grass.
_**St. Augustine** (Stenotaphrum secundatum_) also thrives in the South, being particularly good under trees or in other shady areas where Bermuda will not do well. It forms a thick mat and smothers weeds.
_**Zoysia**_ ( _Gramineae_ spp.) in cultivated species forms dense turf and is very valuable for planting on sandy soils, especially in the South. It is propagated vegetatively by means of small pieces of turf called plugs. Zoysia will choke out crabgrass and weeds.
### Grasses, Pasture
With the development of "beefalo" hybrid cattle by D. C. Basolo of Tracy, California, rich pasture grasses suitable for grazing livestock take on even greater importance. The Beefalo, which is three-eighths buffalo, three-eighths Charolais, and one-quarter Hereford, can be produced more economically than other breeds because it gains weight faster and can finish out on grass rather than grain.
Another oldtime breed is the Texas Longhorn, which, though not as tasty as the Beefalo, also will finish out on grass.
_**Bermuda** (Cynodon dactylon_), a very persistent and nutritious grass of the southern United States, is useful for both pasture and lawn.
_**Buffalo grass** (Buchloe dactyloides_), which grows on the western range where bison used to graze, still serves as food for herds of cattle. It is also useful for binding the soil, preventing erosion.
_**Grama grass**_ ( _Bouteloua_ spp.) __, occurring mostly in the Great Plains area, is excellent forage for livestock. It is widely used also in conservation to prevent soil erosion.
_**Johnson grass**_ ( _Sorghum halepense_ ) grows wild all over the Southwest and is often a great pest in gardens, yet in pastures it is very nutritious for cattle.
_**Ryegrass**_ ( _Lolium_ spp.) is often grown in nut orchards and serves as food for cattle that are allowed to graze and fertilize the land, from which they are removed at harvest time. Orchard grass growing under fruit trees can suppress the root growth of pears and apples.
_**Teosinte** (Zea mexicana_), an annual grass often used as fodder for livestock, is considered the nearest relative to maize or Indian corn of all the wild grasses. It grows wild in moist soil from Connecticut west to Kansas and south to Florida and Texas.
### Nurse Grass
A quick-germinating and quick-growing grass such as redtop frequently is used in lawn seed mixtures to provide a rapid ground cover that helps to crowd out weeds during the early development of the more permanent grasses, such as Kentucky bluegrass. The bluegrass may take two or three years to reach full development, but once it attains this under favorable conditions, it will crowd out the nurse grass.
### Oats ( _Avena sativa_ )
A cover crop of mixed clover and oats following sod and before corn is planted will lessen the white grubs that infest corn. Oats and vetch do well planted together.
Oats sometimes can be grown effectively as a trap crop to lure red-winged blackbirds away from other grains. The stand should be grown at some distance from the birds' roosting places.
### Pasture Weeds
Do not let sneezeweed ( _Helenium autumnale_ ), sometimes called swamp sunflower, grow in pastures. Most cows respect the bitter leaves, but many a pail of milk has been spoiled by a mouthful of helenium among the herbage. If you are wondering why this plant is called sneezeweed, take a whiff of snuff made from the dried and powdered leaves.
Wild larkspur is poisonous to cattle.
Other plants that should never be allowed to grow in pastures are the field and the meadow garlic ( _Allium vineale resp. canadense_ ). Just a few minutes after a cow has eaten some field garlic her entire body is penetrated, and after half an hour the milk is flavored with it and remains so for several hours. To avoid damage to the milk, it may be necessary to keep the cows off the pasture or let them graze for only a short time after milking, then remove them to another pasture.
### Poppy ( _Papaver_ spp.)
Poppy and wild larkspur like to grow with winter wheat but dislike barley. Wheat fields heavily infested with poppy yield a poor harvest of lightweight seeds.
Poppies are grown for both seed and oil but they rob the soil of nutrients, causing it to need rest and reinforcement afterward. This factor may be used to advantage, however, to choke out weeds that cannot be gotten rid of by any other means.
Poppy seeds may lie dormant in the ground for years and then show up again with a grain crop, particularly winter wheat.
_Poppies can become too much of a good thing, especially in plantings of barley, which they inhibit_.
### Rape ( _Brassica napus_ )
Rape is an annual plant cultivated for its leaves and used as temporary pasture crop for livestock. Because of its deep taproot, it loosens soil and improves drainage, leaving the land friable and ready to grow a more useful crop. It helps to heal soils injured by overdoses of mineral fertilizer.
The succulent rape grows fast, producing best under cool, moist conditions. It also resists rather severe frosts and is best seeded in the fall in the southern states and in the spring in the northern ones.
Do not grow rape near hedge or field mustard since both will inhibit its growth.
### Rice, Wild ( _Zizania aquatica_ )
This aquatic grass is not really rice at all, nor is it related to rice. It grows from four to eight feet tall in the shallow lakes of Minnesota, Wisconsin, and central Canada. There it has traditionally been harvested by Indians, who bend the heads of the plant over the edge of boats or canoes, beating the grains loose with two sticks.
Wild rice may be cultivated, growing best in quiet, pure water from one to six feet deep, along the margins of streams, ponds, and lakes or the floodplains of rivers with rich mud bottoms. It likes a slow current and will not grow in stagnant lakes or pools. A fairly shallow farm pond fed by streams can provide a good supply of this vitamin B–rich delicacy.
Do not try to plant the product you find in the grocery store, for only unhulled seed will sprout. Rice for planting must be sacked and kept wet. The seed may be planted by scattering over the surface of the water at the rate of a bushel per acre. The good seed will sink rapidly. If your area is small, use a large handful to a six-by-six-foot space. The best planting time is just before ice forms in late fall.
### Rye ( _Secale cereale_ )
Rye is an excellent crop to choke out chickweed and other low-growing weeds that survive the winter. Planted twice in succession it even will choke out quack grass. A cover crop following sod will reduce black spot on strawberries and pink root on onions.
Rye will be benefited by cornflowers in the ratio of 100 to 1. A few pansies in the field will aid it, and the wild pansy (viola) will germinate almost 100 percent if grown nearby. Rye has an inhibiting effect on field poppy, retarding both the germination of the seed and its growth.
Rye flour sprinkled over cabbage plants while they are wet with morning dew will dehydrate cabbageworms and moths. Refined diatomaceous earth is useful as an insecticide for stored rye, but it is not injurious to warm-blooded animals.
### Sorghum ( _Andropogon sorghum; Sorghum vulgare_ or _Holcus sorghum_ )
Several insect-resistant strains of sorghum have been developed: 'Atlas' is resistant to the chinch bug, while 'Milo' is susceptible; 'Sudan' is resistant to the corn leaf aphid, while 'White Martin' is susceptible.
The sweet sorghums or sorgos are grown especially for the production of sorghum syrup, which is made by pressing the juice out of the stems. For the gardener who would like to be self-sufficient, here is a source of sweetening for his other foods. To get the maximum amount of sugar in the juice, sorghum should be seeded on soils that are not too fertile. Large vigorous stalks usually are lower in sugar than those grown more slowly and not over a half-inch in diameter.
Root exudates of sorghum apparently are poisonous to sesame and wheat. Stored sorghum grain can be kept free of insects by refined diatomaceous earth used as a desiccant dust.
_**Sudan grass** (Sorghum vulgare sudanese_) is a tall annual sorghum whose thin stalks grow quickly and may reach a height of 10 feet. It serves as excellent summer pasturage and grows well with soybeans if sufficient moisture is present.
_**Johnson grass** (S. halepense_), a perennial sorghum, grows as a weed in the southern United States. It resembles Sudan grass but spreads by creeping rootstocks in gardens or on land needed for cotton or other row crops to become a pest, but it makes excellent hay for cattle feed.
### Soya Bean — Soybean ( _Glycine max_ and _G. soya_ )
Soya beans, native to China, are so rich in protein they have been called the "meat without a bone." They are perhaps the world's oldest food crop, and they have meant meat, milk, cheese, bread, and oil to the Asiatic peoples for centuries. Like all legumes, they loosen and enrich poor soil and are an excellent crop to grow preceding others that need nitrogen. They grow faster and thicker than weeds and will choke them out.
Soya bean
Soybeans planted near corn protect it against chinch bugs and Japanese beetles. They grow well with black-eyed peas and will choke out weeds because they grow so rapidly.
### Sugar Beet ( _Beta vulgaris_ )
Grain can be partially replaced as stock feed by sugar beets, which are liked by all animals and are good for increasing the milk flow of cows.
Cheat grass is often a despised weed but has the ability to quickly form a ground cover over denuded soil, preventing erosion. At the same time it replaces plants that are host to beet leafhopper, making it of considerable importance to sugar beet growers.
### Timothy ( _Phleum pratense_ )
Timothy, a valuable, cool-season grass perennial, sometimes called herd's grass and by the English cat's-tail, has slender stems bearing round spikes of tiny, tightly packed flowers. Farmers in both Canada and the United States often sow timothy in rotation with oats and other grains. It does not last long when cattle or other animals graze on it continually, and is not considered a satisfactory pasture grass unless mixed with hardier types.
Timothy and other small grains are benefited by planting them with legumes such as alfalfa and sweet clover as a protection against white grubs.
### Triticale ( _Triticale_ )
The International Wheat and Maize Center of Mexico produced a new grain, triticale, by crossing wheat and rye, gaining the high yield of wheat and the disease- and drought-resistance of rye. This was accomplished by the tedious process of cross-fertilization among different species. By nurturing the resulting embryo and chemically causing its chromosomes to duplicate themselves, the scientists succeeded in producing fertile plants bearing the characteristics of both parents.
Triticale
The name triticale (pronounced _trit-i-kay-lee_ ) derives from the scientific names for wheat and rye, _triticum_ and _secale_. The cross has a higher protein content and protein efficiency ratio than either wheat or corn — comparable to soy concentrate — and it is also higher than wheat in lysine and methionine, two of the life-sustaining amino acids.
A delicious bread made from triticale is now obtainable in many grocery stores throughout the Southwest. The flour's baking qualities are better than rye's.
Triticale has shown the ability to produce two or three times as much per acre as either wheat or rye and can be grown anywhere in the world where wheat is found. The grain is being improved constantly as new strains are developed at various experiment stations throughout the country.
### Vetch ( _Vicia_ )
Vetch, a relatively slow-growing perennial, is a good companion for oats and rye. Plant fast-growing rye or oats as a "nurse crop" to provide shade and check competitive growth. However, if this is done, the vetch should be planted more thinly than ordinarily or the annual nurse crop may be choked out by the sturdy perennial. Fall-planted vetch is one of our most valuable green manure crops. Being a legume, it enriches the soil with both nitrogen and humus.
### Wheat ( _Triticum vulgare_ )
There are two stories about the origin of wheat, both extremely interesting. The first is that bread wheat appeared around 8000 B.C. when wild wheat by accidental cross-pollination apparently formed a hybrid with a type of "goat grass," resulting in much plumper grains. This new plant, called emmer, again crossed with goat grass, forming an even more luxuriant hybrid. Because the husk of this grain was so tight that the whole grain would not scatter to the wind as other grass seeds do, the continued existence of "wheat" was dependent upon man, and thus bread wheat came into being.
The Theosophists, however, believe that mankind at a certain stage in his development was assisted by some high initiates coming from the planet Venus. They believe that these advanced beings not only gave moral and social guidance to man but also brought with them wheat grains to supply a better cereal, bees to produce honey and fertilize flowers, and ants. Rye, they think, was produced by man in imitation of wheat by selective breeding. Oats and barley are thought to be hybrids brought about by crossing with earthly grasses.
In some regions, poppies spring up and become a weed in wheat fields. They should not be allowed to spread, for they check the wheat's growth. On the other hand, camomile is beneficial when permitted to grow with wheat in a very small ratio (1 to 100), while in larger amounts it is harmful. Wheat will be increased by the presence of corn.
The growth of wheat is adversely affected by cherry, dogwood, pine, and tulip, as well as proximity to the roots of sorghum. Canada thistle and field bindweed are harmful to both wheat and flax.
I have grown a good stand of winter wheat by sowing it in the fall on my Bermuda grass lawn. In our mild climate it grows intermittently all winter, heading about the last of June. After it is harvested, the Bermuda grass takes over again and you would never know the wheat had been there. I don't know whether this would work with other lawn grasses.
If you do make a sowing of winter wheat, avoid the Hessian fly by planting it late, timed according to when this fly appears in your area.
## First Steps for Home Fruit Growing
**For the gardener** on a small lot, the site of the home orchard may be limited by necessity, the placement of trees being to a large extent dependent upon the overall landscape design. The homesteader, who has several acres, has at least a modest choice.
Since fruiting plants are more permanent than vegetables, their placement in the landscape design becomes most important. And often their usefulness may be doubled by considering also their ornamental and shade values.
Apples, plums, peaches, and pears are such beautiful flowering trees that they may be used for the same design scheme as crab apples, dogwoods, and redbuds. Pecans and walnuts (as well as apples and pears) make fine shade trees, too.
In areas where they grow well, blueberries will fit in nicely with other flowering shrubs such as forsythias, hydrangeas, and spiraeas. A trellis or arbor becomes both useful and beautiful if bunch grapes or muscadines are planted to grow on it. Unsightly fences may be covered or a patio comfortably shaded if a few grape plants are placed thereon.
Pollination, as applied to fruit and nut trees, vines, and bramble fruits, really is a matter of "companion planting," yet we seldom hear it called this.
Fruit and nut trees almost always do better if at least two of each kind are planted. For some varieties the need is imperative — they will bear scarcely at all without pollination help.
Few home gardens can accommodate more than two or three different kinds of fruit. To grow them successfully it is very important to consider varieties known to be self-fertile (also called self-fruitful), or known to be good pollinators for the other types you wish to grow. If your homesite is not large enough for many trees, check around the neighborhood and list the fruit and nut trees you find there. Some of them may be good pollinators for trees you would like to plant.
In this limited space I cannot possibly list every variety of each fruit that will ensure pollination, but there are a few general rules to follow for good results, and I have tried to include them in the chapters that follow. But remember that pollination, important as it is, is only one factor in success.
While I won't go into the details of cultural practices here, it should be said that trees in a healthy growing condition will naturally derive more benefit from correct companion plantings. Healthy trees produce more pollen. And this applies to all trees, whether standard or dwarf types.
## Getting Started
If you have room to set aside a definite orchard area, the first year you should do subsoiling, plowing, disking, and grading well in advance of planting. If possible, choose a gently sloping site with good air and good soil drainage. There is nothing a tree dislikes more than hardpan and wet feet.
Soil that absorbs water readily is the best, and you can test this by digging a 10-inch-deep hole and filling it with water. If the hole drains completely within about eight hours, drainage may be considered satisfactory. However, if the water remains much longer, drainage is poor. To prevent root rot, work crushed rock, gravel, or peat moss into the soil. Mixing compost with the soil will help in more ways than one.
Grow a nourishing cover crop such as rye, vetch, or soya beans, and disk this in after well-rotted manure or compost has been spread. Allow time for its decomposition, for the trees do not like raw manure or organic matter around their roots. In a natural forest setting, raw organic matter remains on top and only decomposed humus touches the roots.
For shrubs or bramble fruits, the materials should be worked into the soil at least one foot deeper than planting depth. For trees, mix the additions to the soil about two or three feet deeper than the intended planting hole.
The actual planting of the trees comes after the soil has settled. Planting trees in early spring is the generally accepted practice, but in the South or Southwest it is often possible to plant with good results in fall or early winter.
Shortly before planting, fill the hole with water and allow it to drain completely. This will prevent the surrounding soil from absorbing most of the water applied to the freshly planted shrub or tree.
Maintaining a layer of mulch around new plantings helps their growth, since it preserves moisture and in time becomes compost, providing plant nutrients.
## Culture
Dr. Ehrenfried E. Pfeiffer, author of _The Biodynamic Treatment of Fruit Trees, Berries and Shrubs_ , believed that a mixed culture in the orchard as well as in the garden helped to keep down insect pests. He advocated growing nasturtiums between fruit trees as a means of transmitting a "flavor" to the tree that made it disagreeable to insects. He considered it particularly effective when the flowers were grown under apple trees to repel woolly aphids. A washing down of the trees with nasturtium juice was recommended, if planting them was not possible.
Dr. Pfeiffer also suggested for orchard use stinging nettle, chives, garlic (against borers), tansy, horseradish, and southernwood. Permanent covers considered beneficial are clovers, alfalfa, and pasture grasses. Temporary crops to turn under for green manure are such biennial clovers as mammoth clover, red clover, and incarnate clover. He believed buckwheat useful on a light, sandy soil.
Though a mixture of red clover and mustards is considered ideal, Dr. Pfeiffer cautioned that mustard, while it sweetens the soil, can become a rapidly spreading weed and for this reason should not be allowed to go to seed. Alfalfa hay, particularly if shredded or chopped, was thought to have special benefit as a mulch.
Dr. Pfeiffer also recommended a paste for all fruit trees consisting of equal parts of cow manure, diatomaceous earth, and clay, to which horsetail tea is added. This mixture is applied with a whitewash brush or with spraying equipment in the larger orchard.
A number of excellent preparations for fruit trees are obtainable from the Bio-Dynamic Farming and Gardening Association (see Sources).
## More Hints for Fruit Growers
Here are some other helpful suggestions concerning fruiting plants:
• **Marigolds** planted near apple trees or between rows of nursery stock will benefit the trees used in grafting and budding.
• **Wild mustard** is beneficial to grapevines and fruit trees, but cut it before seeding.
• **Dandelions** in the area of fruits and flowers will stimulate them to ripen quickly.
• **Chives** improve the health of apple trees and will prevent apple scab. Use chive tea as a spray against apple scab and for powdery and downy mildew on gooseberries.
• **Pollination** is accomplished mainly by bees and other insects, so no sprays of any kind should be used at blossoming time.
• **Ripening apples** give off small amounts of ethylene gas, which sometimes limits the height of nearby plants but causes their flowers or fruit to mature earlier than normal.
• **Oats** may have an inhibitory effect on the growth of young apricot trees.
• If you must replace a **young fruit tree** on the same spot where an old one has been removed, choose a different variety.
• **Garlic juice** or the powdered extract contains a powerful antibacterial agent effective against diseases that damage stone fruits.
• Do not place **apples** near carrots in a root cellar, as they may cause the carrots to take on a bitter flavor. If apples and potatoes are stored near each other, both will develop an "off" flavor.
• **Nut trees** usually take a little longer to bear than fruit trees. While you are waiting for them to grow, interplant with peanuts (legumes). They will improve the soil and give you a crop as well.
• **Nut trees** are good to plant in pastures and near stables or manure and compost piles, to repel flies on cattle.
## The Fungus Connection
One gardener reported that her unthrifty young peach trees apparently were assisted by moldy oats from the cleanings of the oat bin when one bushel was applied to each tree. After several weeks all her slow-growing trees were putting out new, healthy leaves.
A possible reason for the good growth of the peach trees is in the mold, rather than the oats, for almost all trees have a symbiotic relationship with some fungi (including molds). The fungi grow around the plant roots and furnish vitamins and other natural compounds necessary for a fast-growing and healthy tree.
This brings us back to the soil again. Because of this relationship, it's a good idea to have some of the original soil packed around the roots when transplanting a shrub or tree. Quite likely there will be fungi in the soil beneficial to the plant.
You may even do this: If you have a tree that isn't doing well after being set the first time, take some soil from another tree of the same variety that _is_ growing well and dig it in around your problem tree. There's a good chance that your tree will perk up and grow.
This will work well not only with trees but also with other ornamentals and even with houseplants. If possible, investigate the original, invigorating habitat of such plants, remove some of the soil, and see if nature doesn't have a cure for the ailing plant far better than any commercial fertilizer you could buy.
## Small Fruits
### Blackberries ( _Rubus_ spp.)
Some self-unfruitful varieties of blackberries require cross-pollination. Others, even though self-fruitful, may benefit from the pollen-distributing visits of insects.
The flowers of blackberries are very attractive to their primary pollinators, honeybees. If a variety of blackberry is known to require cross-pollination, ensure a sufficient supply of pollinators in large acreages by placing colonies of bees in or near the field.
Do not grow blackberries near raspberries. Plant them in moderately acid soil, 5 to 5.7 pH.
Mulberries, chokecherries, and elderberries may be used to attract birds away from valued blackberry crops. Blackberries themselves are strong vital plants that help in preparing the soil to support the growth of trees.
### Blueberries ( _Vaccinium_ spp.)
Have at least two different varieties — any two — in a blueberry planting.
Blueberries like very acid (4 to 5 pH) and open, porous soils, such as a mixture of sand and peat with loam. The water table should be 14 to 30 inches below the surface.
### Boysenberry ( _Rubus ursinis_ 'Boysen')
Boysenberries are sometimes called trailing or semi-trailing blackberries.
### Grapes ( _Vitis_ spp.)
Bunch grapes such as 'Concord', 'Fredonia', and 'Niagara' are self-fertile, and one vine will give an abundance of grapes even if planted alone. Grapes like a moderately acid soil of 5 to 5.7 pH. Perhaps more than any other fruiting plant, they need good air circulation to prevent fungus disease such as mildew. This is particularly important in moist, humid climates.
Grapes
Grapes in their natural environment swing high in the trees, doing especially well if the tree happens to be an elm or a mulberry. Such grapes are seldom troubled by either brown rot or mildew. Since growing grapes in trees is impractical for most, the best solution is perhaps a terraced hillside unsuitable for other crops.
Try planting hyssop with your grapes for an increased yield, or use legumes as an intercrop. Cypress spurge is unfriendly, so do not let it grow nearby. To discourage the rose chafer, keep grass out of the vineyard, since its larvae feed on grass roots.
(See also _Muscadine_ below.)
### Muscadine
As natives of the southeastern United States, muscadines do well under the high temperature and humidity found in this area, but they also are resistant to drought conditions and disease. Under favorable conditions they will live many years, but are not hardy in the northern United States. Some varieties are self-pollinating, while others require a pollenizer.
### Raspberries ( _Rubus_ spp.)
Raspberries, which like a near-neutral soil (6.5 to 7 pH), are self-fertile.
Because of virus disease, black and purple raspberries should be planted no closer than 600 feet from red varieties. Do not grow raspberries and blackberries near each other, either. Do not plant any raspberries near potatoes, since they make the potatoes more susceptible to blight.
### Strawberry _(Fragaria_ x _ananassa)_
Almost all strawberries now sold, both June-bearing and everbearing, are self-fruitful. The "best" varieties vary from one area to another.
A cover crop of rye following sod will reduce the incidence of black rot on strawberries. They do well in combination with bush beans and spinach.
Strawberries will benefit if a few plants of borage, also a good attractant for honeybees, are grown near the bed. Lettuce is good used as a border. Pyrethrum, planted alongside, serves well as a pest preventative. A spruce hedge also is protective. White hellebore will control sawfly, and marigolds are useful, too, if you suspect the presence of nematodes.
Pine needles alone or mixed with straw make a fine mulch, said to make the berries taste more like the wild variety. Spruce needles also may be used as a mulch, but my personal preference is chopped alfalfa hay.
In some areas, growers plant strawberries as an intercrop in peach, apple, fig, orange, or other tree-fruit orchards. When the orchard is first planted, strawberries may be set out and grown for several years before the trees need all the ground. The strawberries furnish some income from the land, or at least pay the expense of caring for the orchard. The intensive cultivation given strawberries is especially good for young orchards. Also, because strawberries do not bear well unless moisture conditions are good, they will prove a good indicator of the orchard conditions.
## Tree Fruits
### Apples ( _Malus_ spp.)
Only a few apples will bear well if grown alone, producing a good crop from self-pollination. Most should not be planted alone or be depended upon for pollination in a combination; they are either low or lacking in viable pollen. Suppliers of fruit trees have information available on which combinations work best, and you should be sure you understand the needs of any tree you plan to buy.
If you have room only for one tree, there is still a way that you can have your favorite apple and pollinate it too. Graft a branch of a good pollinator somewhere on the host tree and this will serve your purpose. Apples like a near-neutral soil with a pH of 6.5 to 7.
### Apricots ( _Prunus_ spp.)
All apricots are self-fertile, but they will benefit from cross-pollination to bear more heavily.
### Cherries ( _Prunus_ spp.)
All sour pie cherries are self-fruitful and have no pollination problems. A single tree may be planted and expected to produce well from its own pollen.
Sweet cherries all are self-unfruitful and will require another variety nearby to enable them to set fruit. To further complicate things, there are even instances of pollen incompatibility among this group. A good nursery will give you information on pollination needs of the trees you are interested in.
Wheat is suppressed by the roots of cherry trees, and potatoes grown in the vicinity are less resistant to blight.
### Citrus ( _Citrus_ spp.)
Lime, lemon, orange, and grapefruit trees grow better in the area of guava, live oak, or rubber trees, which apparently exert a protective influence.
### Crab Apples ( _Malus_ spp.)
Crab apple trees are often planted simply for their beauty. But they are self-fertile, and a good variety such as 'Dolgo' will provide both beauty and fruit.
### Fig _(Ficus carica)_
Many people consider figs a tropical fruit, but there are varieties that will do well elsewhere. The fruit of the fig tree is peculiar in that the flowers form inside the fruit's skin. Pokeweed grows well as a fig's companion.
Most figs offered by general nurseries are self-fertile, but some varieties will not mature their fruits unless the tiny female flowers are fertilized by pollen from a special kind of fig tree called a caprifig. Other varieties bear larger fruit if they are subjected to this process, which is known as caprification. The pollen is transferred by a tiny wasp that spends part of its life in the fruits of the caprifig. In regions where Smyrna and other figs requiring caprification are grown, caprifigs are planted also.
Fig
### Mulberries ( _Morus_ spp.)
Mulberry trees have rather insipid-tasting fruits but can be very useful to lure birds away from cherries and berry plants. The birds seem actually to prefer mulberries.
The Russian mulberry ( _Morus alba_ 'Tatarica'), a rapid-growing tree, bears an abundant crop resembling blackberries, which may be made up into pies and jams.
### Nectarine _(Prunus persica_ var. _nucipersica)_
Nectarines are self-fruitful. They also will pollinate peaches, and peaches nearby will help the nectarine to set a larger crop.
### Peach _(Prunus persica)_
Most peaches are self-fruitful, but a few require a pollinator (which can be any other variety of peach). Peaches like a near-neutral soil with a pH from 6.5 to 7.
Never plant a young peach tree where an old one has been removed — plant a different fruit tree.
If peach leaf curl appears and only a few leaves are affected, pull them off by hand. Feeding the tree with well-rotted manure or compost high in nitrogen will help the tree back to health. Garlic planted close to the trunk will protect against borers.
### Pears ( _Pyrus_ spp.)
Almost all pears require other varieties nearby for a good fruit set, the exceptions under most conditions being 'Duchess' and 'Kieffer', which are self-fruitful.
'Bartlett' and 'Seckel' are not compatible, and 'Kieffer' is not always a good pollinator for 'Bartlett'.
If you live in an area where fire blight prevails, it will pay you to plant resistant varieties.
Some orchardists believe that pears are suppressed by the root excretions of grass, but a successful pear grower in California, believing the opposite, lets a variety of grasses and weeds grow in his orchard.
This same grower sprays against codling moth and leaf roller, using ryania because it is specific, killing only chewing insects. As a fertilizer he uses chicken manure to provide nitrogen, plus other animal manures, cottonseed meal, compost, and dried blood.
### Persimmons ( _Diospyros_ spp.)
There are two species of importance. The first, American persimmon _(Diospyros virginiana)_ , is native to a large part of the United States, and the second, the Oriental or Japanese persimmon _(Diospyros kaki)_ , is a native of China and Korea. American and Japanese trees are not interfruitful. Persimmons come in dozens of cultivated varieties, which are considered superior to the wild type.
The common persimmon is a small, low-growing tree perfectly adapted for the homeowner with limited space, since it ordinarily attains a height no greater than 40 or 50 feet. The incon spicuous, greenish yellow, urnshaped male and female flowers are borne on separate trees.
Persimmon
A number of excellent grafted persimmon varieties are offered by the Louis Gerardi Nursery. (See Sources.)
### Plums ( _Prunus_ spp.)
Almost all plums require pollination, though there are a few that will fruit alone. Plums like a moderately acid soil of 5 to 5.7 pH.
### Quince _(Cydonia oblonga)_
Quince trees are self-fruitful.
## Nuts
### Almond _(Prunus dulcis)_
The almond is not a true nut but belongs to the Rose family. All varieties produce better if pollen from another tree is available. Peaches and almonds, being of the same family, will pollinate each other.
### Butternut _(Juglans cinerea)_
Butternut has an inhibitory effect on plants within its immediate vicinity, but to a lesser degree than the black walnut. (See _Walnut_ entries.)
Butternuts
### Cashew _(Anacardium occidentale)_
This native of Brazil has become naturalized in many tropical countries and will grow on sandy soils in Florida. Cross-pollination is not necessary.
### Chestnut, American _(Castanea dentata)_
The chestnut blight has just about wiped out the native American species. However, much work has been done to develop disease-resistant varieties, and a revival of this tree is in progress.
### Chestnut, Chinese _(Castanea mollissima)_
Plant two or more varieties for cross-pollination.
### Filberts and Hazels ( _Corylus_ spp.)
In ancient times many believed that a forked hazel twig had supernatural powers. Such twigs are mentioned in the Bible, while the Romans also describe the magical quality of the branches and told of hazel divining rods being used to find water and precious minerals underground.
Hazelnuts
Hazels furnish valuable cover and food for wildlife. Homeowners also plant them as ornamentals or to shelter other plants. In some forests hazels form such dense thickets that tree seedlings cannot grow and heavy machinery is needed to uproot them so more valuable timber can be planted.
Hazel trees and bushes are beneficial in pastures and elsewhere against flies. Cows like to nibble on the leaves, which increase the butterfat in their milk, while the tannic acid also acts as a cleansing agent for their digestive systems.
It is recommended that two varieties be planted for cross-pollination and better crops.
### Hickories ( _Carya_ spp.)
The hickories, like the walnuts, have male and female flowers growing separately on the same shoot of the current season's growth. Many varieties appear to be self-unfruitful, so it is good practice to plant several varieties together to ensure cross-pollination.
### Pecan _(Carya illinoinensis)_
These trees in all their varieties give no evidence of crossincompatibility, and all will bear larger crops if two or more varieties are planted together.
Pecan trees like plenty of nitrogen. In the orchard, plant a winter and spring cover crop such as clover, which harbors nitrogen-fixing bacteria. For a lawn specimen, let a dense mat of grass grow near the trunk to conserve soil moisture and prevent sunscald of the roots. It is good to mulch with grass clippings, too.
Pecans
The casebearer and hickory shuckworm, the most serious pecan pests, are best foiled by releasing trichogramma wasps in the orchard. Do not store pecan meats near onions or oranges.
### Walnut, Black _(Juglans nigra)_
Grafted varieties of these self-fruitful trees usually produce each year, while wild trees generally produce well only in alternate years, some only every third year.
Black walnut trees are known to produce a substance called juglone, which is washed from the leaves to the soil, inhibiting the growth of many plants within the area where the trees grow. Cultivated plants not compatible with black walnuts are apples, alfalfa, potatoes, tomatoes, blackberries, azaleas, rhododendrons, and heathers. The butternut also seems to have this quality, but plants near it are less severely affected. (See also _Sycamore_ in the Trees and Shrubs chapter.)
Walnuts
Toxicity is contained in the roots of black walnuts as well as in the leaves, and because of this many plants will not grow near the tree. But not all are discouraged. Right at the drip line of a black walnut I have a bed of rainbow-colored iris, interplanted with daylilies, grape hyacinths, and daffodils, none of which appear to be in the least affected.
Drawbacks aside, the black walnut is prized for its valuable wood and delicious nuts. In addition, the tree's leaves scattered around the house or put in the dog kennel will repel fleas.
A Russian remedy to prevent sunburn is to rub freshly ground walnut leaves on the skin. The dark juice of walnut hulls applied to ringworm is said to heal the scalp. (See also _Walnuts, English_ , below.)
### Walnut, English _(Juglans regia)_
Unless you live in a favorable climate, you will probably be more successful with a tree of the Carpathian type, which will do well farther north.
Walnuts are monoecious — that is, the male and female blossoms are separate on the same tree. They are self-fertile but produce better in plantings of several nearby.
English walnuts do not have the level of detrimental leaf and root excretions found in black walnuts, but their shade makes it difficult to grow some plants nearby. Many of the fruit mints, such as apple, orange, pineapple, and spearmint, will do well, however, as will angelica, sweet anise, and other herbs that like filtered sunlight.
## Ornamental Trees and Shrubs
### Alder ( _Alnus tenufolia_ )
Closely related to the hornbeams and birches is this small, waterloving tree that grows very rapidly and serves definite, special uses. The genus _Alnus_ includes 20 species, nine of which grow in North America and six of which reach the height of trees. Alders may be planted in hedges along the borders of streams where their closely interlacing roots hold the banks from crumbling and keep the current clear in midstream. Like willows, alders are of assistance in draining wet soils.
In America the black alder ( _Ilex verticillata_ ) is often found in horticultural varieties. The daintiest are the cut-leaved forms, of which _imperialis_ , with leaves fingered like a white oak, is a good example. The root nodules add nitrogen to the soil, the black alder being the only nonleguminous plant that is able to perform this function. (See _Legumes_ in the Soil Improvement chapter.)
### Azalea ( _Rhododendron_ )
Azaleas, holly, pieris, and rhododendrons are good companions for a landscape planting because all like humusy, acid soil. Do not plant azaleas or rhododendrons near black walnut trees. The substance called juglone washed from the leaves of black walnuts is detrimental to them.
### Beech ( _Fagus_ )
Beech trees and ferns often grow together, and scilla bulbs do well under the trees. Beech trees in their infancy do well under the shade of other trees, so each fruiting tree is the mother of many young ones.
### Birch ( _Betula_ )
It is believed that birch roots excrete substances that encourage fermentation and make the trees useful to plant around manure and compost piles. Dr. Ehrenfried Pfeiffer, one of the early advocates of the biodynamic method of farming and gardening, observed that composts fermented in the vicinity of the gray birch derived benefit from it and suffered no losses of nutrients even if the roots actually penetrated the heap. It is considered best, though, to maintain a distance of at least six feet from the tree when building a compost pile.
### Conifers
Turpentine substances washing from the leaves of conifers such as pine trees will inhibit the fermentation process of compost piles. Interplanting onions with conifers will help prevent damage by squirrels, which eat the buds of Scotch, white, and red pines. Winter-hardy Egyptian onions are the best kind.
_Squirrel damage in pines and other trees can be limited by planting onions nearby_.
Pine needles make an attractive mulch and will increase the stem strength, flavor, and productiveness of strawberries. In general, conifers have an adverse effect on the growth of wheat, since rain washing over them picks up substances that inhibit the germination of seeds.
### Elderberry ( _Sambucus nigra_ and _Sambucus canadensis_ )
Elderberries, having a liking for moist soil, are helpful near compost yards that are difficult to drain, and will also assist in the fermentation of the compost. Elderberries are noted for their ability to produce very fine humus soil about their roots.
### Elm ( _Ulmus_ )
Grapevines that climb trees, swinging high in the air, are greatly benefited by good air circulation and sunlight. It is the sunlight on their leaves rather than on the grapes that causes them to ripen to perfection. Elm trees are particularly beneficial as supports for grapevines.
The slippery elm ( _U. fulva_ ) is also known as the red elm because its wood is red and the moose elm because moose are fond of browsing its young shoots. When the bark is stripped from this valuable tree, it is possible to scrape from its inner surface the thick, fragrant, mucilaginous cambium — a delectable substance that allays both hunger and thirst. The inner bark, dried, ground, and mixed with milk, is a valuable food for invalids. Fevers and acute inflammatory disorders have been treated with the bark, and poultices of the bark also relieve throat and chest ailments.
### Hedges
Hedges used as windbreaks are of particular value in dry, windy areas. Blueberries make a delightful hedge where they can be grown. Rosa rugosa makes an almost impenetrable hedge and also affords a harvest of vitamin-rich rose hips. 'Cardinal' autumn olive ( _Elaeagnus umbellata_ 'Cardinal') and dwarf burning bush ( _Euonymus alatus com pactus_ ) are beloved by birds, as is red-leaf barberry ( _Berberis thunbergii atropurpurea)_. For brilliant color there are 'Golden Prince' euonymus ( _Euonymus fortunei_ ) and 'Gold-flame' spiraea ( _Spiraea bumalda)_.
### Locust ( _Robinia_ spp.)
Sweet pea–type blossoms on a tree, or pods like the pea's swinging from the twigs, mean that it's a member of the pod-bearing Leguminosae family, to which both herbaceous and woody plants belong.
The black locusts ( _R. pseudacacia_ ) have nectar-laden, white flowers of "butterfly form," which honeybees (leading a host of other insects) swarm about as long as a flower remains to offer its sweet nectar. Cross-fertilization is the advantage the tree gains from all it gives.
Locust, good to plant as a border, has leaves, roots, and bark that are poisonous if eaten, but the pods of honey locust ( _Gleditsia triacanthos_ ) contain a sweetish pulp used as cattle feed and occa-sionally eaten by small boys, who brave the tree's thorns to get them.
_Being a leguminous tree, the black locust is a good companion to lima beans. There are toxins dangerous to humans in the leaves, bark, and roots, however_.
### Maple ( _Acer_ )
The single genus _Acer_ includes from 60 to 70 species widely distributed over North America. _Acer saccharum_ , the sugar maple, is the best known and economically the most important for both its beautiful wood and its sap, which yields maple syrup. The black maple ( _A. nigrum_ ) is the sugar maple of South Dakota and Iowa. Red maple ( _A. rubrum_ ), perhaps the most beautiful of all, is a swamp lover but will thrive on hillsides if the soil is moist. It is widely planted in parks and along streets.
Maples have shallow, spreading root systems and it is difficult to get other plants to grow near them. They may also excrete substances that inhibit the growth of some plants, particularly wheat. Maple leaves laid in layers between apples, carrots, potatoes, and other root vegetables have a preservative effect.
### Mulberry ( _Morus alba, M. rubra, M. nigra_ )
White mulberry is the chosen food of silkworms and no substitute has ever topped this tree's preeminence. The berries of the red mulberry ( _M. rubra_ ) do not compare with the cultivated type, but are of value in poultry yards and hog pastures, where they are eagerly devoured. The black mulberry ( _M. nigra_ ), believed a native of Persia, has large, dark red, juicy fruits but is hardy only in the southern and Pacific Coast states, where it is a desirable tree because it is so attractive to birds.
Mulberry trees are particularly good as a support for grapes. Tree-grown grapes are more difficult to pick than trellised grapes, but they will be relatively free of fungus diseases due to better circulation of air around them. Worms in horses may be repelled by mulberry leaves, and Russian mulberry is sometimes used as a trap crop to protect cherries and strawberries.
### Nurse Trees
As abandoned fields again become covered with vegetation, the brushland is gradually reforested. The first trees are quick-growing, short-lived types that provide conditions suitable for the slower-growing, longer-lived trees. Looking at the forest floor, you will see very few pine seedlings. Other seedlings — young oaks, black cherries, and hickories — do better. Gradually the pines will die off and the young hardwoods grow up and take their place. Should a forest fire occur, the whole process will start over again.
### Nitrogen-Fixing Trees
Pod-bearing (leguminous) trees have the power to take nitrogen out of the air and store it in their roots and stems. The decay of these parts restores to the soil the plant food that is most often lacking and most expensive to replace. These trees and shrubs include black locust ( _Robinia pseudacacia_ ); bristly locust ( _Robinia hispida_ ), sometimes called rose-acacia; clammy locust ( _Tobinia viscosa_ ); Scotch broom ( _Robinia scopariua_ ); honey locust ( _Gleditsia triacanthos_ ); Kentucky coffee tree ( _Gymnocladus dioicus_ ); redbud ( _Cercis canadensis_ ), sometimes called Judas tree; yellowwood ( _Cladrastis lutea_ ); woad waxes ( _Genista tinctoria_ ), sometimes called dyer's greenweed; indigo bush ( _Amorpha fruticosa_ ), sometimes called false indigo; mesquite ( _Prosopis juliflora_ ), screwbean ( _Prosopis pubescens_ ), a slender-trunked mesquite, sometimes called screwpod; Palo Verde acacia ( _Cercidium torreyanum_ ); Jamaica dogwood ( _Icthyomethia piscipula_ ); horse bean ( _Parkinsonia aculeata_ ); Texas ebony ( _Zigia flexicaulis_ ); and frijolito ( _Sophora secundiflora_ ).
Black alder ( _Ilex verticillata_ ) also adds nitrogen to the soil. It is the only known nonleguminous shrub with root nodules that can do this. (See _Alder_ in this chapter.)
### Oak ( _Quercus_ )
Oaks grown with American chestnuts seem to give them some resistance to chestnut blight. During their growth, oaks accumulate a large amount of calcium in their bark, yet amazingly the most calcium has been found in the ash of oak trees that grew in calcium-deficient soil.
A mulch of oak leaves serves to control radish and turnip maggots as well as repelling slugs, cutworms, and grubs of June bugs, but some gardeners believe the leaves have an inhibiting effect on certain vegetables. Therefore they should be fully composted before being spread on the garden.
In Germany it has long been a practice to control greenhouse pests such as ants, aphids, and small mites with the smoke from oak leaves. The smoke is not considered poisonous and will not kill bacteria in the soil, nor leave harmful residues.
Live oaks are believed to exert a protective influence on citrus trees.
The trichogramma wasp, whose larvae feed on moth eggs, helps keep oak trees green by controlling gypsy moths. _Bacillus thuringiensis_ (see the Pest Control chapter) will also control and kill various caterpillars on the trees.
### Osage Orange ( _Maclura pomifera_ )
This thorny tree is native from Arkansas to Texas and is hardy as far north as New England and central New York. It is valued for windbreaks or to grow in poor soils, and is an excellent hedge plant, being almost impenetrable when fully mature. It was widely planted by the pioneers as a living fence around their homes before barbed wire came into use. The name refers to the Osage Indians and to the yellow fruit, which looks like an orange but is inedible.
### Pine ( _Pinus_ )
Pine boughs are good to lay over peonies in winter for protection. Remove them in the spring before growth starts. Pine needles make a good mulch for azaleas, rhododendrons, and other acid-loving plants and will increase vigor and flavor in strawberries.
Pine needles contain terpene, which, washed down by rain, has an inhibiting effect on seed germination. It is not good to place a compost heap near pine trees.
### Poplar ( _Populus_ )
The quick-growing, short-lived poplar often fulfills the function of "nurse tree." When a fire sweeps through the forest, it is likely to be the first tree to grow again on the bare land. The poplar's abundant seed, much like willow's, is wind-sown far and wide. Lombardy poplars, which look like exclamation points, are often planted to shelter other plants from the wind.
In Canada, very good stock feed has been made by boiling poplar wood under pressure.
### Rosa Rugosa ( _Rosa_ )
This "hippy" rose has become so famous that it deserves to be mentioned all by itself. Grown in a mass, it makes a charming windbreak as well as an almost impenetrable barrier for animals. It grows better with purslane, parsley, and mignonette around it; is protected from rose bugs by alliums or onions nearby. Keep boxwood away. It blooms prolifically and is an excellent source for berries (hips) rich in vitamin C, containing more than oranges. The hips are used for making teas, jams, soups, and other dishes.
Rosa rugosa
### Rose ( _Rosa_ )
All the alliums — garlic, onions, chives, and shallots — are beneficial to roses, protecting them against black spot, mildew, and aphids. For a recipe to overcome black spot in roses, see _Tomatoes_ in the Vegetables chapter.
Garlic and onions are particularly beneficial to roses. In Bulgaria, where attar of roses is produced for perfumes, it is a common practice to interplant them with roses since they cause the roses to produce a stronger perfume in larger quantities.
Roses also are aided by parsley against rose beetles, by onions to repel rose chafers, by mignonette as a ground cover, and by lupines to increase soil nitrogen and attract earthworms. Marigolds are helpful against nematodes, and geraniums or milky spore disease against Japanese beetle. (See _Milky Spore Disease_ in the Pest Control chapter.)
A carpet of low-growing weeds from the Purslane family will improve the spongy soil around the roots of rosebushes. An infusion of elderberry leaves in lukewarm water sprinkled over roses is thought to control caterpillar damage and is also recommended for blight.
Do not plant roses with other plants that have woody, outspreading roots that will compete with the roses for soil nutrients.
### Sassafras ( _Sassafras albidum_ )
Sassafras is sometimes called the mitten tree from its peculiar leaves, which grow in three different shapes: the simple ovate leaf, a larger blade (oval in form but with one side extended and lobed to form a thumb), and third, a symmetrical three-lobed leaf, the pattern of a narrow mitten with a thumb on each side.
Sassafras will repel mosquitoes. The pungent oil has antiseptic properties, and the bark mixed with dried fruit wards off insects.
Sassafras
A tea made from the bark of young sassafras roots has been used for digestive disturbances. The dried leaves, called file, were formerly much used in the southern states as an ingredient in soups. However, sassafras is now regarded as unsafe for internal use.
### Spruce ( _Picea_ )
Three species of woodpeckers were credited with controlling a serious infestation of spruce beetles in Colorado in 1947. Naturally occurring _Bacillus thuringiensis_ (see the Pest Contol chapter) has been found to give good control of this beetle in some forests.
### Sycamore ( _Platanus occidentalis_ )
Studies conducted by American and Iraqi scientists show that sycamores inhibit the growth of other herbaceous plant species, and the decaying leaves cause significant reduction in seed germination and seedling growth. Organic compounds leached from the leaves often are allelopathic to plants, and virtually no herbaceous plants will grow under the trees.
Sycamore bark has value, however. Boiled in water and made into a poultice, it is good to use for poison ivy.
### Wild Cherry ( _Prunus pensylvanica_ )
The wild bird, pine, or red cherry grows from Newfoundland to Georgia and west to the Rocky Mountains in rocky woods, forming thickets that are valuable as nurse trees. Wild cherry often springs up in burned-over districts where its bird-sown pits take root, the young trees sheltering new pines and hardwoods. It provides berries for birds and nectar-laden flowers for bees, so it can scarcely be called worthless, even though it is a short-lived tree.
The wild black cherry ( _P. serotina_ ) is sometimes called the rum cherry. A tonic is derived from its bark, roots, and fruit, and brandies and cordials are made from its heavy-clustered fruits, which hang until late summer, turning black and losing their astringency when fully ripe. The wild black cherry makes an attractive shade and park tree, too.
The wild black cherry and the chokecherry ( _P. virginiana_ ) are both of value to attract birds. Unfortunately, the tent caterpillar favors them to lay its eggs, making the trees unpopular with farmers. The egg rings in the outer smaller branches are easily seen and removed.
### Willow ( _Salix_ )
The tough and fibrous roots of willow are useful in binding the banks of streams that may erode. Nature seems to have designed them specifically for this purpose, for wherever a twig lies upon the ground, it will strike root at every joint if the soil is sufficiently moist. The wind often breaks off twigs and the water carries them downstream where they lodge on banks and sandbars, which soon become covered with billows of green.
For thousands of years the bark and leaves of the willow have yielded resins and juices that eased the aches and pains of rheumatism and neuralgia or alleviated the distress of fevers. In the 1820s, salicin, the active principle of willow bark, was isolated, and in 1897 a synthetic derivative gave the world aspirin.
### Windbreaks
Before planting a windbreak, study your land carefully and plan to put it where it will do the most good. Consider prevailing wind directions and the location and relationship of your buildings to the area you want protected. Most often windbreaks are planted across the west and north sides of a property, but of course there are exceptions to this rule, depending on the configuration of the land and the winds.
Do not plant your screen too close to the garden, for if the windbreak is to consist of trees and shrubs, they will rob the soil of moisture and nutrients. If you have sufficient land, plant the windbreak at least 50 feet from field crops. Very possibly you do not have this much room, but be as generous as you can.
The protective factor of a windbreak is 20 times its height. Thus a 10-foot screen would give you protection up to 200 feet downwind from it. You will also receive protection for several feet in front of the tree belt because it causes the air to back up and act as an invisible wall before it hits the planting of trees. Not the least of its uses is to hold down soil against heavy winds and to keep snow from drifting over walks and driveways. It may even help you to reduce fuel bills.
In the prairie regions in particular, shelterbelt plantings have a marked influence on local climate, especially if they are placed at right angles to prevailing winds. A chain of such belts checks movement of the air, slowing down the wind velocity even before the windbreak is reached, and starts up a whole series of favorable climatic influences. These influences, such as a reduction of evaporation by increasing the humidity of the air, improve the yield of crops grown under their protection. (See _Hedges_ in this chapter, and _Vertical Gardening_ in Garden Techniques.)
### Witch Hazel ( _Hamamelis virginiana_ )
The witch hazel is a stout, many-stemmed shrub or small tree, characteristically an undergrowth of larger trees. Native Americans were the first to use the bark of the witch hazel for curing inflammations. An infusion of the twigs and roots is made by boiling them for 24 hours in water to which alcohol then is added. The extract distilled from this mixture is used for bruises and sprains and to allay the pain of burns.
Perhaps the alcohol is the effective agent, for chemists have failed to discover any medicinal properties in either bark or leaf — yet who knows, they may still find it.
The tree has the peculiar property of throwing its seeds, particularly in dry, frosty weather. This does for the parent tree what the winged seeds of other, taller trees accomplish.
Witch hazel gets its name from the fact that superstitious English miners once used the forked twigs as divining rods.
## Garden Techniques
### Border Plants
Castor beans planted around the perimeter of the garden will repel moles, while borders of daffodil, narcissus, scilla, and grape hyacinth around flower beds will discourage mice. If used in small amounts, dead nettle (henbit), sainfoin, esparsette, hyssop, lemon balm, and valerian are helpful to all vegetables. Yarrow is a good plant in paths, as well as borders, as it will grow well even if walked upon. Planted as a border to the herb garden, it enhances the growth of essential oils in the herbs.
### Catch Cropping
This simply means growing a quick-to-mature crop of some vegetable in ground you've reserved for a planting of a later or slower-growing crop such as tomatoes, or a member of the Cabbage family such as broccoli or cauliflower. While you are waiting, put in radishes, lettuce, or spinach as a catch crop.
### Climate
Since climates vary greatly throughout the world, where you live should always be taken into account when you plan your garden. Maximum summer and minimum winter temperature should be considered, as well as annual rainfall.
For best success, try plants recommended for your area, making these your garden basics. This determined, you can then have fun experimenting each year with a few borderline plants those that do best in either a warmer or colder climate. Often, by providing shelter or otherwise creating a "mini" climate you may grow these successfully. Winter protection will help in the North, shade or a windbreak in the South. Some natural feature of your land, such as a pond, may enable you to grow something that your neighbor a few miles away cannot. (See _Microclimate_ in this chapter.)
Mulching to keep the ground cool may be helpful for certain plants. Improving soil with humus often makes it possible to grow vegetables or plants that formerly were unsuccessful.
### Damping-Off
This is a disease caused by fungi, apparently present in the soil, that kills many young plants. It is characterized by collapse of the stems, or the seedlings falling over. It may occur before the seeds germinate or after the seedlings emerge. To avoid this, you can start seeds in a commercial soilless medium; but if you make your own potting soil in which seeds are to be planted, it should be treated to kill the fungi by steam-heating to 180°F for half an hour or more.
A simple method for the home gardener is dry-heating the soil in an oven. Place the soil 4 to 5 inches deep in a pan and bury a small potato about 1½ inches in diameter in it. Bake in 200°F oven until the potato is done and the soil is sterilized and ready to use.
### French Intensive Gardening
This type of gardening, which stresses maximum use of the soil and first became popular in the 1800s, is largely accomplished by using raised beds. These may be any length but narrow enough to permit easy handling from either side. Raised beds have the advantage of improved drainage and better aeration. The soil does not become waterlogged in winter and as a result it warms up faster in spring and produces earlier crops.
Prepare the soil by loosening to a depth of 12 inches and removing all weeds. Add compost or well-decomposed manure as well as any other organic amendments (agricultural lime, gypsum, bonemeal, phosphate rock, etc.) that a soil test may indicate. Double digging is then done. This means that where the first spade-depth of soil is removed, a second spade-depth of soil is loosened before soil from the top layer is replaced.
If you are working with extremely poor soil, the bottom spade-depth may need to have additional incorporations of sand, compost, and loamy soil. All this sounds like a lot of work, and it is, but as the soil is improved each year, the work gets easier.
_French intensive gardening is an ideal way to save space and to use companion planting effectively_.
The benefit derived from this intensive gardening method is the increased number of plants that may be grown in a very small area. Perhaps in no other form of gardening is companion planting so important, since herbs and vegetables are so closely crowded together.
In general, the smaller vegetables and salad greens are best suited to this type of culture, but there is no law that says you can't grow corn and pumpkins and sunflowers and cucumbers this way if you want to!
### Frost
Vegetables frequently are classified according to their ability to survive frosts. The U.S. Department of Agriculture defines the differences:
Hardy or cool-season crops will survive medium to heavy frosts. Seed from this group (peas, beets, kale, etc.) can be planted as soon as the soil can be prepared in the spring, or in midsummer for a late-fall crop.
Semi-hardy vegetables will survive a light frost. Seed will germinate at relatively low temperatures, and can be planted 2 to 3 weeks before the last frost date. This necessarily will vary in different sections of the country.
Tender or warm-season crops (tomatoes, eggplant, bell pepper, etc.) are injured or killed by frost, and their seeds seldom germinate in cold soil.
### Honeybee ( _Apis mellifera_ )
Both in the garden and in the orchard, honeybees are an important agent of pollination. They are particularly attracted to the often inconspicuous flowers of herb plants.
A hive of bees is a good weather indicator, for if drones are forced out of the hive during fair weather, it is a sign that cold, wet weather is imminent. When hiving a new swarm of bees, rub the hive's inside with lemon balm, which the bees like. A smoke from jimsonweed (datura) calms the bees when a hive is opened.
### Intercropping
This is really the heart of companion planting, for the idea is to have two or more different vegetables growing on the same piece of ground, or in the same row, providing diversification. And this idea need not be confined to vegetables. Flowers and herbs can happily bump shoulders with each other. In fact, they _should_.
If your garden is small and you don't want to have empty spaces between your peas and bean rows, intercrop with broccoli, Brussels sprouts, cabbages, cauliflower, kale, or even radishes or carrots. After the early peas and beans are out, the slower-growing vegetables have all the space to themselves, and you have room to walk again. This may make things a bit cramped at times, but if you have little space and a short growing season, it's well worth trying. Of course, to be successful you must keep up the fertility of your soil.
I like to keep a sort of "floating crop game" going in my own garden, making small plantings of quick-growing vegetables that sprout readily from seed, such as lettuce, radish, spinach, celery, cabbage, kale, chard, collards, and other greens. Staggered plantings mean fresh supplies coming on all through the season.
Plants that will help each other are put together as often as possible, either in the same row (for instance, marigolds with bush beans) or in adjacent rows. Lettuce and onions do well together, so I pop in a lettuce plant each time I pull a green onion for the table. I plant onions close together and pull every other one, letting the remaining onions mature for dry onions. In my climate even eggplant and green peppers benefit from a bit of shade, so I plant these together in a row next to okra.
Many vine crops, such as squash, cucumbers, and pumpkins, grow well with corn and may even protect it from raccoons. The corn is helpful in protecting vine crops from wilt. Many early crops do well following spinach, which is rich in saponin. (See _Saponin_ in the Soil Improvement chapter.) Early spinach also may be intercropped with strawberries.
Many vegetables are pretty enough to put in the flower beds. Parsley between bulbs provides an attractive background in the spring. Tomatoes can grow with roses and at the same time protect them against black spot.
Chive clumps are another attractive planting for the rose garden (see _Chive_ in the Herbs chapter for benefits). They grow larger each year, late spring bringing a pincushion of lavender blossoms that last for many days.
Hardy amaryllis, a member of the Lily family, sends out long strap leaves in early spring. After the leaves ripen and die, the ground is bare until August, when a sturdy stem emerges to grow quickly and bear fragrant pink lilies. A lettuce planting between the bulbs contrasts with the flower, making it far more beautiful.
Some vegetables have been especially hybridized for oramentation in a flower planting. Flowering cabbage and kale come in colorful shades of red, white, and green, yet have excellent flavor. Plant them with the same herbs (mints, thyme, rosemary, sage, hyssop) as the garden varieties.
In garden intercropping, try not to put a plant that needs light where other, taller-growing plants will shade it, nor a moistureloving plant with another that is greedy for water.
Just follow the general rules: asparagus with tomatoes; beans with carrots or summer savory; beet with onion or kohlrabi; members of the Cabbage family with aromatic plants or potatoes or celery; leeks with onions, celery, or carrots; turnips with peas.
Remember also the dislikes, and do not plant beans with onions, garlic, or gladiolus; beets with pole beans; the Cabbage family with strawberries, tomatoes, or pole beans; potatoes with pumpkin, squash, cucumber, sunflower, tomato, or raspberry.
### Microclimate
A microclimate — a small area with special growing conditions — may result from an unusual natural feature on your land, such as a pond that moderates air temperatures. Or you can create a microclimate yourself by varying your plantings, adding a hedge, or covering a fence with vines. A hedge makes a permanent windbreak, but rows of tall corn will grow quickly and serve the same purpose for a season: to shade, protect, and limit air circulation for tender plants. So will vine plantings, such as grape, and also cucumbers (though these must be kept well watered during the summer, particularly if they take the western sun).
### Mulch
Mulch can be almost anything that retards loss of moisture from the soil, but organic mulches, many of which also add nutrients, are considered the most helpful. These include chopped bark, buckwheat hulls, cocoa shells, coffee grounds, corncobs, cottonseed hulls, cranberry vines, evergreen boughs, grass clippings, hay, hops, leaves (particularly oak leaves, which repel slugs, cutworms, and grubs of June bugs), manure, peanut hulls, peat moss, pine needles (great to increase stem strength and flavor of strawberries), poultry litter, salt hay, sawdust, seaweed, stinging nettle, straw, sugarcane residue, tobacco stems, and wood chips and shavings. Particularly comprehensive information is found in _The Mulch Book_ by Stu Campbell. (See Suggested Reading.)
### Mulch, Sawdust
There is much to be said about a sawdust mulch, both for and against. Mulches like sawdust are particularly susceptible to spontaneous combustion, fresh sawdust can cause a depletion of soil nitrogen, and it is not good to use in summer because earthworms will avoid it.
On the good side, it is claimed by many authorities that blueberries mulched with sawdust will develop a larger, more fibrous root system and as an end result have a far higher yield. It is considered good mulch for raspberries and should be put on immediately after transplanting. Mixed with animal manures or poultry litter, it makes an acceptable mulch for many plants and shrubs where either one alone would not work well. Shavings or sawdust used for animal bedding makes an excellent mulch.
The type of tree from which the sawdust comes also has a bearing on the situation. Unweathered pine sawdust will decompose very slowly, so give it a bit of time to weather and turn gray before using. Sawdust from hardwood trees will rot much more rapidly than pine, spruce, or cedar, especially if weathered before using. Studies now show that the tannins and terpenes in sawdust that gardeners often fear really do little if any harm to the soil. (See _The Mulch Book_ in Suggested Reading.)
### pH
Experienced garden writers take it for granted that everybody knows what pH is all about. If you don't, relax; it isn't scary at all.
The pH of anything simply indicates its active acidity or alkalinity, expressed in units. The term is generally used in horticultural science to indicate a condition of the soil, and it's important to know, because many plants thrive only when the pH value of the soil closely approximates the optimum for their particular kind.
Soil acidity may be of two kinds, active and potential. It is a state in which the concentration of hydrogen ions (H+) exceeds that of hydroxyl ions (OH-). When you have an exact balance of H+ and OH- ions, you have neutrality. When the OH- ions exceed the H+ ions, you have alkalinity.
Active soil acidity represents the excess of H ions over the OH ions present in the soil solution. It is expressed in pH units on the pH scale. On this scale, 7 represents neutrality; higher readings indicate alkalinity and lower ones acidity. It is rare to find a soil with greater acidity than 3.5, or with greater alkalinity than 8.0. You should note, however, that the relationship between the figures is geometric. Acidity at pH 5 is ten times as great as at 6, and at pH 4, one hundred times.
What can you do about it if a soil test shows too much one way or the other? To neutralize acidity, the gardener adds lime, preferably the agricultural type. Gypsum or sulfur can be used to correct an alkaline condition. In my opinion all soils, but particularly alka-line ones, benefit from the use of compost or humus in the form of decomposed organic matter. A green manure crop plowed under also helps.
### Shade
Shade is sometimes the decisive factor in companion planting. Nature does not arrange plants in long, straight rows, as we often do in our gardens. Try radishes in a foot-wide bed with no thinning. Put fast-growing lettuce such as 'Buttercrunch', 'Simpson', or 'Oakleaf' between cabbages, broccoli, Brussels sprouts, or even tomatoes, which will shade the young plants while they are growing. The lettuce will be up and out of the way when the slower-maturing plants need the room. You'll have a double crop on half the ground and with half the work, and you will also find that the taller plants give the lettuce just enough shade to keep it coming on crisp and sweet right into hot weather.
If you interplant early beets with late potatoes, the shade of the growing potatoes will benefit the beets, keeping them tender and succulent right into warm weather.
Plant melons between your onion rows, and by the time the onions are harvested the melons will be taking over the ground. While the vines are growing, the onions will protect them from insects.
After you harvest your early corn, let the stalks remain a while to shade a planting of fall cabbage, beans, peas, and turnips. When the fall garden is well established and the sun less warm, remove the cornstalks and use them for mulch right on the ground where they grew.
Many of the mints take kindly to shade and may be grown under trees. Sweet woodruff also likes shade and makes an excellent ground cover, while retaining moisture for other plants that give it protection from the sun. Tarragon and chervil like partial shade, too.
### Succession Planting
This technique will enable you to make the most of a supply of compost or fertilizer. Heavy feeders such as broccoli, Brussels sprouts, cabbage, cauliflower, celeriac, celery, chard, cucumber, endive, kohlrabi, leek, lettuce, spinach, squash, sweet corn, and tomato should be planted in soil newly fertilized with well- decomposed manure.
Follow these heavy feeders with light feeders such as beet, carrot, radish, rutabaga, and turnip, which also like finely pulverized raw rocks and compost.
Legumes, the third group in succession planting, include broad and lima beans, bush and pole beans, peas, and soybeans. These soil improvers collect nitrogen on their roots and restore it to the soil.
### Suicide in Plants
Why do most annual plants die in the autumn? It is possible that seeds inside mature fruits such as soybean pods send out hormones that cause plants to yellow and die even before nights cold enough for freezing cut them down.
Gardeners for years have known that if faded flowers are picked before they form seeds, the plants will continue to produce more flowers. Pansies are a good example. Among the vegetables, okra will continue from early spring to frost if the pods are kept picked before they harden.
This idea has been tested on soybeans. Growing pods were plucked from one side of the plant only and allowed to remain on the other. The side with the mature pods and seeds turned yellow and died, while the other remained healthy.
### Two-Level Planting
Vegetables that occupy different soil strata often make good companions. Among these are asparagus with parsley and tomatoes, beets with kohlrabi, beets with onions, leeks with vine plants, garlic with tomatoes, carrots with peas, and also strawberries with bush beans.
Many combinations like this are possible, enabling the gardener with little space to virtually double the garden's yield, and at the same time improve the health and flavor of the vegetables planted together.
Do not put together plants that are competing for the same space and light, such as sunflowers and pole beans, or plants whose root excretions react unfavorably on each other, such as carrots and dill.
### Two-Season Planting
Gardeners in areas that have a long growing season may find both a spring and fall garden possible. In the fall here in Oklahoma I can grow cauliflower, broccoli, Brussels sprouts, cabbage, collards, lettuce, radishes, and English peas, and they are practically insect-free. Some of the vegetables that require a long growing season to head up are a complete failure for me if planted in the spring, because of the hot midsummer conditions. I cannot give you exact dates; this has to be worked out by area according to where you live, but is well worth trying experimentally.
In many places, early-planted squash is more likely to withstand borers, which lay their eggs in July. Where I live, we have a rule: "Plant squash and cucumbers the first day of May before the sun comes up and they will be free of beetles." I find that fall-planted squash often escapes insects as well.
Radishes and cabbage may escape root maggots by careful timing. The Hessian fly's attacks on winter wheat may be avoided if the wheat is sown after the first week in October, when the fly is no longer active.
South Texas cotton farmers have found they can control the pink bollworm without insecticides by carefully establishing deadline dates for both planting cotton and destroying the stalks after harvest.
Observe when insect infestations are worst on certain crops, and plant either earlier or later than you have in the past.
### Vertical Gardening
If you have a fenced garden, here is an opportunity for both beauty and increased productiveness. Many plants take kindly to climbing. Cucumbers (such as Burpee 'Burpless') grow longer and straighter when trained on a fence. Scarlet runner beans climb rapidly and make a beautiful as well as a tasty display. I plant these with my chayotes, which bear good-tasting and attractive fruit in September.
Morning glories and pole beans do well together, and rambling roses are happy with gourds. When the roses are gone, the gourds will bear attractive blossoms and fruits without damage to the roses. If you grow the birdhouse type of gourd, these will be a bonus for your garden, dried and hung the following season to attract birds.
You might follow Oriental practice to relieve the somber dark green of pines by allowing clematis to grow into the trees, particularly the white-blooming type that forms huge panicles of scented flowers in late fall.
If you don't have a fence, try a tepee or wigwam made of four or more poles fastened together near the top and with soft wire or twine tied from pole to pole. The growing plants are trained to the poles by tying loosely. When they reach the top, pinch out the growing point of each plant, causing them to produce side shoots. This system is very good for vining squash. Soon the wigwam will be covered with a mass of attractive flowers, bright green leaves, and squashes.
## Soil Improvement
### Alfalfa ( _Medicago sativa_ )
This is one of the most powerful nitrogen-fixers of all legumes. A good stand can take 250 pounds of nitrogen per acre from the air each year. Alfalfa needs a deep soil without hardpan or an underlying rock layer, because it sends its roots down deep. Research ers have traced them for well over 100 feet, and 20 to 30 feet is average.
Alfalfa's deep-rooting ability is the source of its great nutritional power, feeding as it does from mineral-rich subsoil that has not been worn out and depleted. Alfalfa is strong in iron and is also a good source of phosphorus, potassium, magnesium, and trace minerals.
You can easily sprout alfalfa seeds in the kitchen, and you may even want to grow some in your garden for highly nutritious alfalfa greens, or use its leaves for tea. Alfalfa used as a meal is a great compost stimulant and activator, particularly good for composting household garbage.
Alfalfa will make good growth wherever dandelions grow. Dandelions themselves are deep divers, their presence indicating that the subsoil is easy to penetrate. Alfalfa grown in pastures will give protective shelter for shallower-rooting grasses, keeping other plants alive longer during spells of dry weather. As a trap crop, alfalfa will draw lygus bugs away from cotton. Two percent alfalfa provides sufficient control, but it should be planted about a month before the cotton.
### Buckwheat ( _Fagopyrum esculentum_ )
Buckwheat is valuable as a soil builder and it will grow on very poor soils while collecting lots of calcium. Used in this manner it will take the light away from low-growing weeds, choking them out. If plowed under as green manure, it will sweeten the soil and make it more suitable for growing other crops. Buckwheat does not like winter wheat.
### Calcium
Peas, beans, cabbages, and turnips revel in soil containing lime, but a few plants — notably those belonging to the Heath family, such as erica, azalea, and rhododendron — actually dislike it. Potatoes and a few cereals are not at their best if lime is applied to the ground immediately before they are planted or sown.
Land in need of lime does not respond to cultivation and manuring as it should, and often coarse weeds such as sheep sorrel flourish. Sometimes a green scum grows over the surface. A soil test that reveals excessive acidity indicates the need for liming.
Buckwheat accumulates calcium, and when composted or plowed under as green manure enriches the soil. Lupine ( _Lupinus_ ) has roots that penetrate to surprising depths even on steep, gravelly banks or exposed sunny hills. It adds calcium to the soil, too, and is of value to grow on poor, sandy soils worthless for other purposes.
Scotch broom, a member of the Legume family, also accumulates calcium but may become a weed unless kept in check. Melon leaves are rich in calcium and should be added to the compost heap when the plants are spent.
### Clover (Leguminosae)
Planting clover between rows of grapes will add nitrogen to the soil. This also works well in orchards or with companion grasses. Clover dislikes henbane and also members of the Buttercup family, which secrete a substance in their roots that inhibits the growth of nitrogen bacteria and poisons the soil for clover. This poisoning is so effective that clover will disappear in a field if buttercups are increasing. Clover has a stimulating effect on the growth of black nightshade ( _Solanum nigrum_ ).
### Compost
Compost is largely composed of decayed organic matter that has heated sufficiently to kill weed seeds and then has thoroughly decomposed. Plant preparations may be used to influence or speed up the fermentation process, and these — even when added in small amounts — can influence the entire operation. Once conditions are right, earthworms enter the compost pile and assist the other microorganisms in the breaking-down process.
Certain plants such as stinging nettle may be used to speed up or assist fermentation in the compost heap or manure pile. This plant is an excellent soil builder and, like comfrey, has a carbon-nitrogen ratio similar to barnyard manure. Nettles also contain iron.
Several other herbs are particularly well endowed with minerals and can be of value when incorporated into compost. These include dandelion, which absorbs between two and three times as much iron from the soil as other weeds; salad burnet, with its rich magnesium content; sheep sorrel, which takes up phosphorus; and chicory, goosegrass, and bulbous buttercup, which accumulate potassium. Horsetail shares honors with ribwort and bush vetch for a capacity to store cobalt. Thistles contain copper as a trace element.
Compost is the best fertilizer for herbs as well as garden vegetables and is particularly rich if weeds are put in the pile instead of being destroyed. Use all herb refuse obtained in the garden, too, and naturally all the kitchen waste, particularly in a household where cooking with herbs is a frequent feature.
Any organic material can be added, but refuse of a woody nature will decompose more rapidly if it is crushed or chopped first and used as a base mixed with materials such as grass clippings. Leaves picked up in the fall make good compost, but since they are apt to pack, they should be mixed with other material. Bulky materials form a solid mass of rotting vegetation, lack oxygen, and quickly become sour and greasy.
When making a pile, the site first should be well dug to allow for a quick entry of earthworms. Incorporate sods into the heap. Build your pile up to a height of about six feet with straight sides and a slightly concave top. The contents must be kept moist and allowed to heat so that weed seeds are killed. The quicker the heating, the earlier the heap should be turned and restacked.
Turpentine substance, washing from the leaves of conifers, will retard fermentation. Birch trees in the vicinity assist fermentation, even if their roots penetrate the heap. It is best, however, to have them at least six feet away.
### Fertilizers, Nature's Own
We are all familiar with the ability of legumes to draw nitrogen from the air and "fix" it to their roots. Actually, there are many plants that get only about 5 percent of their nourishment directly from the soil.
Have you ever noticed how plants, particularly grass, look greener after a thunderstorm? This is not an optical illusion. They really _are_ greener as a result of the electrically charged air, which frees its 78 percent nitrogen content in a water-soluble form.
_**Rain and lightning**_ are fertilizing agents. Each time lightning strikes the earth, large amounts of nitrogen are charged into the ground. One authority states that 250,000 tons of natural nitrogen are produced every day in the 1,800 thunderstorms taking place somewhere on the earth. In some places this may amount to more than 100 pounds per acre per year. Rain also brings nitrogen — in some areas as much as 20 pounds per acre annually.
_**Sulfur**_ comes down with the rain, possibly producing as much as 40 pounds per acre per year. Rainwater also contains carbonic acid, forming carbon dioxide in the soil where it is needed for the plant-feeding process. Millions of tons fall yearly, and when we consider that nearly half the makeup of a plant is carbon we realize how important this is. Evidence also seems to show that rare minerals such as selenium and molybdenum are washed down in rain.
_**Snow**_ , which furnishes not only nitrogen but also phosphorus and other minerals, yields an extra bonus denied warm-climate areas. Snow contains 40 percent less heavy water, or deuterium oxide, than normal water. Deuterium is a heavy isotope, a form of hydrogen but a little different. Combined with water it does not form H2O, the water molecule, but D2O instead. Heavy water, according to the Russian scientists who observed this, slows down some chemical and biological processes of growing plants. When the heavy-water molecules are removed, plants seem to grow faster. Thus crops are aided in short-season, snowy climates such as that of Alaska. Even fog contributes to the soil's fertility, especially along the seacoast, where it brings in large quantities of iodine, nitrogen, and chlorine.
_**Dust**_ , though sometimes disagreeable, has its good points too, containing minerals, organic matter, and beneficial organisms often in substantial quantities essential to plant growth. Dust may be carried for thousands of miles, even being held suspended for long periods in the upper atmosphere to be washed down eventually by rain. Many believe that dust is one of the most significant factors in restoring minerals to the exhausted soil and that it also contains bacteria important to healthy soil life.
Here's a way you can obtain some of the benefits of electroculture without waiting for a thunderstorm. Tie your tomato plants to metal poles or trellises (ours are concrete-reinforcing wire bent in an inverted V-shape, with a row of tomato plants on each side), using nylon strips cut from discarded panty hose. These sturdy supports also attract static electricity. A friend who tried this reported that she harvested an abundant crop of extremely large tomatoes, and her plants continued producing right up to frost.
### Green Manures
Green manures are cover crops, usually achieved by planting lowpriced seed. If fall-planted, the cover protects the soil surfaces of fields or garden plots from erosion from winter winds, snowstorms, and quick thaws. It acts as an insulating blanket, keeping the soil warmer in winter and cooler in summer. This encourages soil life activity in general and earthworms in particular. The more earthworms in the soil, the more channels they'll burrow deep in the subsoil, bringing to the surface useful minerals and nutrients that will increase the health and insect-resistance of food plants.
Yellow sweet clover
Red clover
Alsike clover
White clover
_The clovers make fine green manure crops. Clover's growth is limited by Henbane and Buttercup family members_.
The roots of many green manure crops themselves reach deeply into the subsoil, where they absorb and bring up valuable nutrients. These revitalize the soil when the plants are plowed under to decompose.
Certain green manure crops, the legumes, have the ability to capture and fix large amounts of nitrogen from the air, also adding this important nutrient to the soil. Alfalfa is one of the best of these, and it is also high in terms of protein, which breaks down into usable nitrate fertilizer.
Other useful green manure cover crops are barley, bromegrass, buckwheat, cheat grass, alsike clover, cowpeas, lespedeza, millet, rape, spring rye, Italian ryegrass, winter rye, sorghum, soybeans, Sudan grass, sunflowers, common clovers, hairy vetch, and winter wheat.
For the home garden, kale makes a good, thick, nonleguminous green manure cover crop for the winter months. It is an easily grown, tasty food with a flavor that is improved by frost, and you can actually dig it out from under the snow in the dead of winter. When the kale starts growing again in the spring, it can be tilled under to add green manure to the soil for the spring garden.
### Inoculants
Sold under various names such as Legume Aid (Burpee) and Nitragen (Farmer), these preparations are aids to peas, beans, soybeans, sweet peas, etc., for better blooms and increased yields. Treating legume seeds with the proper inoculants helps them to develop root nodules, which convert free nitrogen into plant food.
### Legumes
Though nitrogen makes up 80 percent of the volume of the atmosphere, it is almost useless to most plants, for it must be changed into a compound before it can be used. Lightning combines or fixes small amounts of this nitrogen and oxygen in the air, thus forming oxides of nitrogen that are washed out of the atmosphere by rain or snow to reach the soil. (See _Fertilizers, Nature's Own_ in this chapter.)
Nitrogen-fixing bacteria living in nodules on the roots of legumes — alfalfa, beans, clover, esparsette, kudzu, lespedeza, peas, peanuts, soybeans, winter (hairy) vetch, and others — can change atmospheric nitrogen into nitrogen compounds useful to themselves and other plants. Farmers for centuries have rotated their crops to take advantage of this increased soil fertility produced by legumes.
Clover is particularly beneficial if used as a green manure crop and plowed under before planting a crop of wheat or corn the following season. The decaying legume, rich in fixed nitrogen, increases the nitrogen content of the soil without the need for commercial fertilizers. A green manure crop of alfalfa will benefit a crop of cotton. Red clover may be used on soils too acid and too poorly aerated for alfalfa. The optimum pH for red clover is between 5.8 and 6.8 but it can stand a pH below 6.0 and still do reasonably well.
The steps in the nitrogen cycle can be traced in the growth and use of clover:
1. Atmospheric nitrogen is changed into proteins by the action of nitrogen-fixing bacteria growing in nodules on the clover roots.
2. After plowing under, the clover proteins are changed into ammonia by ammonifying bacteria.
3. Ammonia is then changed into nitrates by nitrifying bacteria.
4. Both ammonia and nitrates are used by other plants to form plant proteins.
The nitrogen-fixing bacteria of legumes benefit not only themselves but also other plants nearby. Peas and beans, for instance, benefit potatoes, carrots, cucumbers, cauliflower, cabbage, summer savory, turnips, radishes, corn, and most other herbs and vegetables.
Peanuts, another legume, can be used as a second crop after an earlier one is out, provided that you have a long growing season.
On larger acreages, lespedeza, kudzu, and esparsette may be used to aerate the soil and put nitrogen into it, while winter vetch and soybeans make excellent cover crops. Legumes sown with a small amount of mustard are helpful to grapevines and fruit trees. Peanuts are excellent to grow in an orchard of newly set nut trees.
### Nitrogen-Fixing Plants
See _Legumes_ , above.
### Oats
See the chapter on Grasses, Grains, and Field Crops.
### Potassium
Certain weeds indicate a soil rich in potassium. These are marsh mallow, knapweed, wormwood, opium poppy, fumitory, Russian thistle, tansy, and sunflower. Red clover, however, is a good indicator of potassium deficiency.
Tobacco takes up potassium in its leaves and stalks and is thus a good plant on the compost pile if it has not been sprayed with chemicals. Vegetables that like potassium are celeriac and leek.
### Rhizobium
These are the bacteria that live in nodules on leguminous plants and turn atmospheric nitrogen into a useful form for building plant proteins. Unfortunately, they withhold their talents from such useful plants as corn and wheat. (See _Legumes_ in this chapter.)
### Saponin
Soapberry or chinaberry is the name of a group or genus of trees and shrubs that bear fruit containing a soapy substance called saponin. Bouncing Bet is probably our best-known saponin-rich plant. Other unrelated plants such as primroses and carnations also have this quality, as well as many legumes, cyclamen tubers, camellia, viola, mints, horse chestnut, orach, pokeweed, runner beans, tomatoes, mullein, potatoes, and spinach. These plants are important because their decomposing remains create a favorable environment for the plants that come after them.
Commercial saponin is used as a foam producer in beverages and fire extinguishers and also as a detergent useful in washing delicate fabrics.
### Seaweed
Kelp, a type of brown alga, provides fertilizer and is a source of the chemical element iodine, particularly the giant kelp of the Pacific. Chemists also extract from kelp large amounts of algin, which is useful in such commercial products as ice cream and salad dressings by virtue of its ability to hold several different liquids together.
Kelp is about 20 to 25 percent potassium chloride, and it also contains common salt, sodium carbonate, boron, iodine, and other trace elements. For gardeners who live on or near the seacoast, seaweeds are a natural and usually wasted resource that can be utilized for mulching and in compost piles. They are especially good materials to put around fruit trees. Another advantage is that decomposing seaweed is less attractive to mice than is straw.
Chopping seaweed may be advantageous if only for cosmetic reasons. It may be also advisable to rinse off the salt, but it is not necessary to be too fussy about this. The small amount left clinging will do no harm. Eelgrass, which dries into a light "hay" and doesn't pack down, is a perfect nonsmothering material for many plants. (See _Eelgrass_ in the Wild Plants chapter.)
Another use for "sea power" to promote fertility is liquefied seaweed, which may be applied as a foliar fertilizer directly to the leaves of plants. It is particularly helpful to trees.
It has been found that seaweed as a fertilizer helps to promote frost resistance in tomatoes and citrus fruits, as well as increasing the sweetness of some fruits and giving better resistance to pests and diseases. Beets and parsnips respond badly to boron shortages in the soil, so chopped kelp makes an excellent mulch for them.
Seaweed helps to break down certain insoluble elements in the soil, making them available to plants. This quickens seed germination and further aids development of blossoms and fruits, resulting in increased yields.
For those who cannot readily obtain seaweed, there are seaweed preparations and fish emulsion, available at most garden centers and through gardening catalogs.
_Eelgrass makes an enriching mulch and is fine for composting_.
Another idea for gardeners who, like myself, live far inland is to use water plants. Though not as rich in nutrients as seaweed, they make good mulch if chopped, and are good additions to the compost. Water hyacinth, a plant of tropical America, has escaped into the wild and has become a pest, choking many small streams. Dredged from waterways, it is valuable on the mulch pile. Here where I live, water lilies grow profusely in shallow pools and are easily obtainable. My husband, Carl, and I usually brought back a tubful of water weeds of some kind every time we went fishing.
### Sweet Clover ( _Melilotus alba_ and _M. indica_ )
This is not a true clover although it is a legume. White and yellow sweet clovers live for two years, and their large roots penetrate deeply into the soil. At the end of the second season, they decay to enrich the soil with nitrogen and decomposing vegetable material.
Sour clover, a kind of sweet clover used almost entirely to improve the soil, is often called melilot.
Spoiled sweet clover hay or poorly preserved silage never should be fed to animals. The hay contains coumarin, an anticoagulant, which develops toxicity as the clover decomposes and may cause both internal and external bleeding.
### Weeds as Soil Builders
Many weeds seem to have a mysterious capacity for enriching soils. Jimsonweed (datura) grown near pumpkins will promote their health and vigor; the best watermelons come from the weediest part of the patch; onions grown with weeds (but not allowed to be overwhelmed) are apt to be larger than those in clean-cultivated rows.
This is particularly true if the weeds are the so-called "deep divers" that break up the subsoil, allowing the roots of the vegetable plants to have a larger-than-usual feeding zone. Deep divers sometimes bring up from below the hardpan mineral elements that the roots of food plants cannot reach. The high mineral content of weeds is another reason for adding them to the compost pile. (See _Compost_ in this chapter.)
## Pest Control
### _Bacillus thuringiensis_ (BT)
This is a selective bacterial disease, highly effective against caterpillars and the larvae of moths. During spore production, _Bacillus thuringiensis_ produces crystals that act as a stomach poison on the insects eating the treated plants. This substance is not toxic to plants, man, or other animals, and can be applied even up to the day of harvest.
BT controls the fruit tree leaf roller, as well as tent caterpillars — the disease attacking the pest in the caterpillar stage after they have come out of the tent. The disease is widely used to protect commercial crops of celery, lettuce, cabbage, broccoli, cauliflower, mustard, kale, collards, and turnip greens. It is also effective against the tobacco budworm and the bollworm.
Special strains of BT have been developed to kill other pests such as mosquito larvae and Colorado potato beetles. BT products are widely available, under various brand names, from garden centers and mail-order catalogs.
### Bean and Potato Combination
Bush beans planted with potatoes protect them against the Colorado potato beetle. In return, the potatoes protect the bush beans from the Mexican bean beetle. It is considered best to plant the beans and potatoes in alternate rows.
### Birds
Birds around the garden are generally recognized as one of the best controls against insect pests. Particularly useful are the purple martins, which must catch flying insects almost constantly in order to live. Well-made martin houses set up away from large trees will attract martin colonies. Feeders and bird waterers will also encourage other birds to visit and nest near your garden.
Birds are particularly attracted to hackberry, chokecherry, elderberry, Tartarian honeysuckle, mulberry, dogwood, Japanese barberry, red and black raspberries, viburnum, Hansen's bush cherry, Russian olive, hawthorn, and sunflowers. Evergreen trees and thornbushes are attractive for nest-building. Certain forms of cactus are attractive for this purpose in the Southwest.
_The purple martins rid orchards and garden areas of incalculable numbers of injurious insects, devouring them in the winged stage_.
Birds, however, can be too much of a good thing at times and may become very destructive to the food plants we grow for ourselves. From the Chinese we have this suggestion: When fruits start to ripen, hang sliced onions in the trees — the birds dislike the scent and will avoid the fruit. Hanging empty milk cartons in fruit trees from a string so they twirl in the breeze will deter many birds. Change their position occasionally, and perhaps also use bright, fluttering ribbons or strips of cloth.
If crows are a problem for a corn or watermelon patch, put up several stakes and string white twine around the patch, crisscrossing it through the center. To the birds this will look like a trap and they will avoid the patch.
### Blackfly (Aphididae)
This insect is particularly detrimental to broad beans, and it is advisable to use a fermented extract of nettle to keep it under control. Its natural enemy, the lady beetle, will help. Intercropping with garlic, or placing an occasional plant of nasturtium, spear-mint, or southernwood in the rows of beans, also is a good plan.
### Borers
Nasturtiums planted around fruit trees repel borers, while garlic and other alliums such as onions and chives also are good.
### Castor Bean _(Ricinus communis)_
Experiments have shown that castor beans planted around a garden will repel moles, and they also are good to repel mosquitoes.
This is an agricultural crop in some areas where it is grown for the oil in the seed, yet all parts of the plants are poisonous to livestock and humans, particularly the seed. Two or three seeds eaten by a child can be fatal and as few as six can cause the death of an adult. The beans also carry an allergen that causes severe reactions in some people when handling castor pomace. This danger can be eliminated if the seed heads are clipped off or destroyed before they mature.
### Castor bean
To form an effective mole repellent, plant castor beans every five or six feet around the perimeter of the garden. Use them also as a companion crop: Plant several pole beans close to their bases and let them climb the tall-growing plants. The largest variety of castor bean, _zansibarensis_ , grows eight feet tall, and has large leaves and beautiful variegated seeds of various colors. _Sanguineus_ grows seven feet tall and 'Bronze King' five feet.
### Cockroach _(Blatella)_
Extracts of chinaberry have been found useful against cockroaches and termites. The little-known cockroach plant _(Haplophyton cimicidum)_ from Mexico also is valuable in controlling this pest.
### Cutworm (Noctuidae)
A three-inch cardboard collar around young plants extended one inch into the ground and two inches above will foil cutworms. Use cardboard cut from toilet paper or paper towel rolls, or cut off the top and bottom from a quart-size milk carton and cut the remainder into three collars.
A used matchstick, toothpick, small twig, or nail set against the plant stem will keep the cutworm from wrapping itself around the plant and cutting it. Oak leaf mulch will repel cutworms, too.
### Diatomaceous Earth
This effective remedy against many insects is made from the finely ground skeletons of small, fossilized, one-celled creatures called diatoms, which existed in the oceans and constructed tiny shells about themselves out of the silica they extracted from the waters. The microscopic shells, deposited on the floor of the ancient seas, collected into deposits sometimes thousands of feet deep.
This earth contains microscopic needles of silica, which do their work by puncturing the bodies of insects, allowing vital moisture to escape from them. The insects die from dehydration. This earth is so finely milled that it poses no threat to either humans or animals, but these particles, when taken internally by insects, interfere with breathing, digestion, and reproductive processes.
Diatomaceous earth will not harm earthworms, which are structurally different from the insects. The earthworm's outside mucus protection, coupled with its unique digestive system, enables it to move through soil treated with diatomaceous earth without harm.
Many gardeners use diatomaceous earth as a dusting agent to give effective control against gypsy moths, codling moths, pink boll weevils, lygus bugs, twig borers, thrips, mites, earwigs, cockroaches, slugs, adult mosquitoes, snails, nematodes, all species of flies, corn worms, tomato hornworms, mildew, and so on. For field crops and in orchards, the diatomite particles are best applied with an electrostatic charger, which gives the particles a negative charge, causing them to stick to plant surfaces.
### Eelworm
Eelworms or nematodes are tiny, sightless, eel-shaped organisms that pierce the roots of plants to feed or to lay their eggs inside, causing knots to form. When organic matter is incorporated into the soil, it encourages beneficial fungi and other nematodes that feed on the plant-parasitic variety of eelworms. These beneficial fungi grow in decomposing vegetable matter and kill the nematodes. It is possible to reduce wireworm and nematode attack, too, by placing a heavy dressing of barnyard manure on grassland before plowing.
These microscopic nuisances are discouraged by marigolds, scarlet sage _(Salvia)_ , or dahlias _(Dahlia)_. Asparagus is a natural nematocide. Tomatoes grown near asparagus thus are protected, while in turn the tomatoes protect the asparagus from the asparagus beetle.
### Flea Beetle _(Faltica_ and _Epitrix)_
Flying insects such as flea beetles are known to dislike moisture. Very often they can be discouraged by watering the garden in full sunlight. I find them annoying but not particularly destructive to eggplant, tomato, turnip, and radish plants. The damage is mostly cosmetic, and strong plants quickly grow out of it, the plant becoming less attractive to the beetles as the leaves enlarge and toughen a little. Light cultivation and the addition of organic matter to the soil both discourage the beetles and help the plants.
Bruised elderberry leaves laid over the rows of plants are a deterrent to the beetles, and they also are repelled by mint and wormwood. Beetles attracted to radish or kohlrabi may be controlled by interplantings of lettuce.
### Fumigation
Greenhouse gardeners, who frequently have difficulty controlling aphids, ants, and termites as well as that all-time pest, whitefly, find smoke from oak leaves effective. The leaves are not poisonous, do not kill soil bacteria, and leave no harmful residue. Smoke the leaves for about a half hour, keeping the greenhouse door tightly closed.
### Garlic
See the Herbs chapter.
### Gopher (Geomyidae)
These burrowing rodents may be repelled by plantings of scilla bulbs. The scillas, sometimes called squills, are flowering, bulblike ornamentals that have grasslike leaves and clusters of flowers at the top of long stems. They are easy to grow, and bloom in the early spring. They may be grown among vegetables as well as in flower beds. But be cautious — the bulbs should never be eaten.
### Grasshopper (Tettigoniidae and Locustidae)
Grasshoppers are very difficult to control, especially where they come in from surrounding fields, but this spray will help: Grind together two to four hot peppers, one mild green pepper, and one small onion, and add to one quart water. Let stand 24 hours and strain. This mixture also is good against aphids.
During times of grasshopper infestation, an after-harvest plowing will discourage egg-laying, while spring tilling before seeding will prevent grasshoppers from emerging from the eggs still present. Extracts of chinaberry have proved useful, while sabadilla dust or extract is effective against these and many other insects. (See _Insecticides, Botanical_ in this chapter.)
Grasshoppers can be attracted by this bait: Fill several twoquart mason jars with a 10 percent molasses solution and place around the area where the infestation is the worst.
Chickens are of great value in an orchard, for they eat grasshoppers and other insects and at the same time add their manure to aid the fertility of the soil. The coop may be moved every few days to a new location.
Wild birds attracted to the garden eat a great many grasshoppers and surprisingly cats will kill and eat them, too. I think this is partly for the fun they get out of the chase.
Grasshoppers will eat almost anything except horehound, but grasshopper-resistant varieties of corn and wheat have been developed.
### Ichneumonid Wasp _(Campoletis perdistinctus)_
This wasp has been found by the Brownsville Experiment Station in Texas to parasitize at least 27 destructive species of moths and butterflies, but it _prefers_ to deposit its eggs in bollworms and tobacco budworms.
### Insect Attractants, Botanical
Insects are largely attracted by scent and may be lured away from certain crops by other plants placed nearby. Plants such as nasturtium and mustard, both of which contain mustard oil, frequently are used for this purpose. These are called trap crops. Insects feed on and lay their eggs in trap crops, which should be destroyed before the eggs hatch.
### Insect Repellents, Botanical
Insect repellents may be prepared from crushed leaves, infusions, or essential oils of such botanicals as citronella, eucalyptus, pennyroyal, bergamot, cedarwood, clove, rose geranium, thyme, winter-green, lavender, cassia, anise, sassafras, bay laurel, and pine tar. Ginkgo, elder, pyrethrum, and lavender repel ticks or other insects. Other plants such as cedar, quassia, and teakwood themselves are immune to insects.
See also the Insect Control through Companion Planting chart, page 154, in this chapter.
### Insecticides, Botanical
Several plant derivatives are available commercially as insecticides. (See Sources.) These materials break down more quickly than man-made insecticides and (except for nicotine) are relatively nontoxic to humans when used as directed. They kill bees and other beneficial insects as well as pests, however, and can be toxic to other life-forms such as fish. As with any pest-control products, read the label and follow the directions exactly.
_**Pyrethrum**_ is prepared from the flowers of _Chrysanthemum cinerariafolium_ or _C. roseum_. It is used alone or in combination with other botanicals against a wide variety of pests, including aphids, leafhoppers, spider mites, harlequin bugs, and imported cabbageworms.
_**Rotenone**_ comes from the roots of tropical plants and is a powerful stomach poison for chewing insects.
_**Nicotine**._ Tobacco and its main alkaloid, nicotine, have been used as an insecticide since the late seventeenth century. Nicotine, a contact poison, is toxic to mammals as well as insects and should be handled very carefully.
_**Ryania** ,_ from a Latin American shrub native to Trinidad, does not always kill insects outright but rather makes them ill, causing them to stop feeding.
_**Sabadilla**._ When heated or treated with an alkali, the seeds of this plant become toxic to many insects. Oddly enough, this quality increases with age during storage. The extract is effective against a large group of insects such as grasshoppers, corn borers, codling moths, webworms, aphids, cabbage loopers, and squash bugs. Be cautious in handling the dust and do not breathe it in.
_**Neem oil**_ is made from the seeds of the neem tree, native to India. It is useful against Japanese beetles and the adults of many other chewing insects, acting as a repellent and as an appetite suppressant, and also stops the development of larvae.
### Insect-Resistant Vegetables and Grains
Every garden has both "good" and "bad" bugs, yet from a gardener's point of view only a few are relatively destructive. Some vegetables seem to have a natural, built-in resistance: carrots, beets, endive (including escarole and witloof chicory), chives, okra, Egyptian onions, parsley, peppers, and rhubarb. Under good growing conditions, lettuce might be added to this list, too.
Numerous vegetables and herbs listed in this book help other vegetables to resist insects when grown with or near them. Scientists have done a great deal of research, also, on why certain other plants are attractive to insects. They have come up with something that organic gardeners knew all along: Insects prefer to eat plants having high concentrations of free amino acids, such concentrations being enhanced if the plants are improperly nourished. Organically grown vegetables produced on balanced, healthy soils have significantly lower levels of free amino acids in their tissues than plants grown where chemical fertilizers have destroyed the balance. Such "organic" vegetables are less tasty to insects.
In addition, vegetable varieties have been bred for resistance to specific pests. Your county's Cooperative Extension Service may have recommendations on good choices for local conditions.
## INSECT CONTROL THROUGH COMPANION PLANTING
Legumes planted in a rotation will protect grain crops and grasses from white grubs and corn rootworm. Chinch bug on corn and flea beetles are controlled by growing soybeans to shade the bases of the plants. The herbs in this chart may be planted as specific controls.
### Ladybugs
These famous eaters of aphids may be purchased for garden release. (See Sources.) The problem, especially in small gardens, is to keep them there. If there is a real need for their services and there is plenty of ladybug food around, they are more likely to stay.
The way you release them will make a difference. Do not strew them about like grains of corn but rather place them by handfuls — carefully — around the base of infested plants. Their natural instinct is to climb the nearest plant and start hunting for food. Do your "seeding" gently, since rough handling, especially in warm weather, may excite them to flight. Early morning or evening is the best time.
Lamb's-quarters, which sometimes harbor the leaf miner, also may play host to the beneficial ladybug. Newly placed ladybugs are dependent on their hosts (which also may be Chinese celery, cabbage, or other plants), and they must find aphids in sufficient quantity to keep them in the vicinity and ensure reproduction. After one mating a female will produce from 200 to 1,000 offspring.
In the spring you can often find the eggs of ladybugs on the undersides of leaves, near their early food supply, aphids. You will see them standing on end in clusters of 5 to 50, generally yellow or orange in color. The alligator-shaped larvae are blueblack and orange-spotted.
### Mice
Mice and rats are repelled by fresh or dried leaves and the oils of mints, by camphor, and by pitch pine. Mothballs repel rabbits as well as mice but should not be used where food crops are grown or where children can pick them up.
Sea onions, white lavender, wormwood, and spurge repel mice, while everlasting pea is useful against field mice, and leaves of dwarf elder protect against mice in granaries.
_If mice like your garden too much, repel them with bulb plantings of daffodil, narcissus, scilla, or grape hyacinth_.
### Milk
Cows and goats give more and richer milk when fed on stinging nettle hay or members of the Umbelliferae family. We always saved discarded carrots and carrot tops when we kept milk goats, which also liked prunings from rosebushes if the thorns were not too prominent.
Skim milk may be used as a spray on tobacco and other plants subject to mosaic virus, and on peppers and tomatoes grown in the greenhouse. Pickers in commercial pepper and tomato plots find it useful to dip their hands in skim milk to avoid transmitting the mosaic virus, while whey proteins are effective, too.
A milk and blood spray has been used in orchards to control fungi, and a milk and coal tar mixture against chinch bug.
Sour milk or butttermilk may be sprinkled over cabbages against cabbageworms.
### Milky Spore Disease _(Bacillus popillae)_
This widely used bacterial organism gives protection against the Japanese beetle by producing a fatal disease in the grub. Because it brings about an abnormal white coloring in the insect, it is called milky.
It was developed in 1933 when a field survey in central New Jersey discoverd a few abnormally white Japanese beetle grubs, which examination showed were teeming with bacterial spore.
This disease was studied by the Bureau of Entomology and Plant Quarantine, where attempts were made to culture it for release. Treatment with the milky spore on experimental plots showed a more than 90 percent mortality in two months. In areas where the disease had been established naturally, equally high kills were found. This disease, occurring naturally in Japan, has kept the beetle from becoming a serious problem there.
The spore ordinarily needs only one application and then continues and spreads itself. It can be applied to the soil at any time except on a windy day or when the ground is frozen, but it is best to treat mowed or cropped areas. Apply a teaspoonful of the spore disease powder on grass or sod spots three to four feet apart and in rows the same distance apart. The beetle grub feeding in the soil will then take in the bacteria spores.
When a healthy grub becomes infected, the spores give rise to slender vegetative rods, which multiply in the blood by repeated divisions. In a short time the normally translucent blood of the grub will become milky in appearance, and eventually the grub dies. The spores stored in the body cavity then are released in the soil and are taken up by other grubs that in turn become infected. As the cycle continues, the spores increase in number and, since more grubs are killed, fewer and fewer adult beetles emerge to feed on crops. The disease is cumulative and self-perpetuating.
Milky spore disease is sold under several trade names. (See Sources.)
Neem oil is being increasingly used as a Japanese beetle control with considerable success. (See _Insecticides, Botanical_ in this chapter.)
Less effective but still useful methods of combating the Japanese beetle include companion plantings of geraniums among roses and grapes to drive the beetles away. Larkspur eaten by the beetles will kill them, while soybeans work as a trap crop. The beetles are rarely destructive to cabbage, carrots, cauliflower, eggplant, onions, lettuce, parsley, peas, potatoes, radishes, spinach, squash, sweet potatoes, tomatoes, and turnips.
### Molasses Grass _(Mellimus minutiflora)_
This is one of nature's very own insect traps. The covering of glandular hairs exudes a viscous oil capable of trapping small pests such as ticks. It does not kill them, but stops them from crawling upward to come in contact with an animal. Cattle pastured on this grass in Guatemala were found to be free of ticks within a year, and it has been planted in Florida with good results. At the same time it provides good forage for cattle, which like to graze on it. It also repels mosquitoes and the tsetse fly.
### Moles (Gryllidae)
Moles generally are considered a nuisance, and they do consume beneficial creatures such as earthworms, but they also feed on Japanese beetle grubs. They are deterred by a few plants of caper spurge _(Euphorbia lathyrus)_ strategically placed, by daffodil bulbs, and by castor bean plants. Thorny twigs of raspberry, rose, hawthorn, or mesquite pushed into burrows will scratch the moles and cause them to bleed to death.
### Mosquitoes
Garlic-based oil (see _Garlic_ in the Herbs chapter) is effective in killing mosquito larvae in ponds. Myristicin, a synthesized compound found in parsnips, also is effective as a selective insecticide against the larvae, as is _Bacillus thuringiensis_.
The leaves of molasses grass _(Mellimus minutiflora)_ and sassafras are mosquito repellents, and I find that castor beans planted around my garden make it more pleasant to work there in the long, cool western twilights when I do most of my garden work.
Euell Gibbons says that American pennyroyal _(Hedeoma pulegioides)_ , sometimes called squaw mint and a native American plant not to be confused with the European pennyroyal _(Mentha pulegium)_ , is a natural insect repellent. A handful crushed and rubbed on the skin will not only emit a pleasant smell but also repel mosquitoes and gnats.
### Moths and Millers
If moths and millers are troublesome in fruit trees, place one cup molasses in one and a half cups water and hang in small buckets or cans from the limbs. Remove the insects occasionally or make new solutions. This remedy is particularly helpful if used in peach trees.
### Nematodes
See _Eelworms_ in this chapter.
### Oil Sprays
Used properly and at the right time, dormant oil sprays are effective, particularly in orchards, against many chewing and sucking insects. The oils make a tight film over insect eggs, causing suffocation.
Apply heavy dormant oil sprays only in early spring over leafless trees. However, lighter, more refined oils have been developed in recent years and can be used at other stages of growth.
### Pepper, Hot _(Capsicum frutescens_ , var. _fasciculatum_ and var. _longum)_
Hot red peppers are among the most useful plants in the garden as well as the most flavorsome.
Grind hot peppers and mix with water and a little powdered real soap to make an infusion for spraying plants infested with aphids. Dry cayenne pepper may be dusted on tomato plants attacked by caterpillars.
However, if you note long green hornworms on tomatoes, do not be too quick to spray. Watch to see if any of the tiny parasitic wasps that build noticeable white cocoons all over tomato hornsworms are present. Do not harm these predators with sprays — they may be doing your work for you.
Ground red peppers placed around eggplants and rubbed on the leaves will help repel eggplant pests, and the dried pods, pulverized and sprinkled on corn silk, will give protection against raccoons.
Another all-purpose spray may be made of ground pepper pods, onions, and a bulb of garlic. Cover this mash with water, let stand 24 hours, and strain. Add enough water to make a gallon of spray. Use several times daily on roses, azaleas, chrysanthemums, or beans to hold down serious infestations. Do not throw away the mash, but bury the residue among the plants where insects occur.
Protect your hands with gloves when you work with hot peppers, and be very careful not to get the juice in your eyes.
### Praying Mantis (Mantidae)
Despite its ferocious appearance, this is one of our most beneficial insects, and it will not harm any vegetation in the garden, dining only on other insects. When young it eats the soft-bodied insects, the cutting and sucking aphids and leafhoppers; when fully mature it kills and eats chinch bugs, crickets, locusts, bees, wasps, beetles, flies, spiders, tent caterpillars, and many others.
Though mantids usually are found in warm countries, the common European mantid can live in the northern United States. A full-grown large mantid varies from two to five inches in length depending on the kind, yet it will easily escape notice because in form and color it closely resembles the plants on which it rests.
Female mantids lay their eggs in masses, gluing them to trees and shrubs with a sticky substance from their bodies. The eggs remain there all winter, and if you carefully examine thorny bushes, brush, hedges, and berry bushes in autumn after leaves have fallen, you may find the hardened froth mass egg cases. You can collect some from marshes and waste areas for your own garden, but never strip the area clean.
Walking stick
Praying mantis
To place them properly, allow one egg case for each major shrub or about four for each quarter acre (without shrubbery). Select a sheltered spot and tie or tape the cases securely about two to four feet above the ground.
Early in the spring the mantis will start to aid you, emerging in bright sunshine usually from early May to late June, just at the time when a large variety of insect fare is likely to be available.
Each creature is fastened to the egg by a tenuous cord that it must break. After doing so it will drop and then climb to surrounding foliage. Mantids are poor flyers and slow movers and usually remain in the area as long as they continue to find food. And once introduced, they are likely to multiply and extend control. Though many of the young may not live, still enough survive to perpetuate themselves. You can obtain mantid egg cases commercially through the mail, usually between November and early May. (See Sources.)
### Rabbit _(Oryctolagus cuniculus)_
Onions are repellent to rabbits and may be interplanted with cabbage and lettuce. An old garden hose cut in lengths of a few feet each and arranged to look like snakes will frighten rabbits away.
Animal fat can be painted on young fruit trees, and a thin line of dried blood or blood meal sprinkled around the garden often acts as a repellent. Try a dusting of aloes on young plants, or shake wood ashes, ground limestone, or cayenne peppers on plants while they are wet with dew.
### Raccoon _(Procyon lotor)_
Farmers have been planting pumpkins and corn together for centuries to discourage raccoons. Put the pumpkin seeds about four feet apart. As the corn approaches maturity, the big, wide pump-kin leaves will grow around the stalks. It is believed that the coons will not come into the corn rows because they like to be able to stand up and look around while they eat, and the big leaves make that impossible.
_Repellents for the corn-loving raccoon include nearby cucumber, melon, pumpkin, or squash vines. Red or black pepper on the corn silk also may help_.
Other methods to control raccoons include sprinkling black or red pepper on the corn silks. This does not affect the taste of the corn.
For a small planting, a wire "corn cage" or an electric fence is sure protection. If a six- or twelve-volt battery is used, the fence will be harmless. It need be turned on only at night, since raccoons sleep during the day. Another solution is to use a small transistor radio (enclosed in a plastic bag to protect it from rain or dew), placing it in the patch at ripening time. Turn it on at night.
### Root Knot
If the fresh tops of asparagus are crushed and steeped in water, the solution may offer protection to vegetables where root knot, stubby root, and meadow nematodes are suspected. Marigolds are also useful against meadow nematodes. (See _Eelworms_ in this chapter.)
### Root Maggot
Root maggots may be repelled by a mulch of oak leaves.
### Salt, Common (Sodium chloride)
Salt has a damaging effect on most plants and should not be used except where the soil is to be rested for a time afterward. It is useful to kill Canada thistle or quack grass and will give best results if put on after the weeds are freshly cut. Apply several times, taking care to do so only in dry weather. Salt put on slugs will dissolve them.
### Slugs and Snails ( _Agriolimax campestris_ and Helicidae)
These beasties cause most of their damage at night in places where the ground is damp. They are topers: They love beer and will drown in saucers of it. Even empty beer cans placed in the garden attract them by their odor. They also love honey and will drown in saucers of that. What a way to go!
_The skunk's protection, butyl mercaptan, can be neutralized with tomato juice. Skunks have some benefit in keeping large insects and mice in check_.
Since ordinary table salt will dissolve slugs, I find it handy to carry a small salt shaker in my pocket to use when I spot them. I feel no compunction, for these slimy creatures will destroy my choicest cabbage and lettuce heads if they get a chance.
Tobacco stem meal discourages slugs, and white hellebore controls them on grapes and cherries. They dislike tanbark or oak leaves, and wormwood will repel them by its bitterness. Snails are reluctant to cross lines of ashes or hydrated lime.
### Soap
In using most sprays made from botanicals it is a good idea to add a small amount of real soap to help the materials adhere. Washing plants down with just plain soapy water is an excellent practice in many instances, for soap appears to have antiseptic qualities useful against many plant diseases. Insecticidal soaps, which can be useful against a variety of insects, are available from garden centers and by mail order (see Sources).
### Spider _(Arachnida)_
Some mites and spiders are natural predators, our valuable allies that dine on many destructive insects. One type of predaceous mite is used to control plant-feeding insects on avocado and citrus crops.
_The garden spider is one of the best helpers to the vegetable grower_.
### Spider Mite _(Acarina)_
Spider mites, also called red spiders, are apt to show up suddenly in hot weather. They sometimes can be removed from plants by spraying forcibly with a stream of plain water, and once dislodged from a plant they seldom return. A 3 percent oil spray, pyrethrum dust, or a spray of onions and hot peppers may be used, and I have found that garlic will repel the mites on tomato plants, while ladybugs are their natural enemies.
### Tea Leaves
Mix tea leaves with radish and carrot seed to prevent maggots.
### Termite (Termitidae)
Since most termites cannot live without water, cutting off the source of moisture usually kills them. Silica aerogel is effective, and extracts of chinaberry will kill about 98 percent of them.
### Toad _(Bufo)_
Both toads and frogs consume many insects, one toad being able to eat up to 10,000 insects in three months' time — and many of these will be cutworms. Other pests eaten by toads include crickets, grubs, rose chafers, rose beetles, caterpillars, ants, squash bugs, sow (pill) bugs, potato beetles, moths, mosquitoes, flies, slugs, and even moles.
_Horned toads, like true toads, are great friends of the gardener, devouring great numbers of insects. Help them find shelter in your garden area_.
Toads do _not_ cause warts and are not poisonous to man, though they exude a slime distasteful to their enemies. If you want to secure some toads for your garden, look around the edges of swamps and ponds in spring. Once secured, they need shelter. A clay flowerpot with a small hole broken out of the side will serve for housing if buried a few inches in the ground in a shady place. They also need a shallow pan of water and, if your garden is not fenced, protection from dogs and other creatures.
In the Southwest, we use the horned toads (really lizards) in the same way. They can withstand more heat and thrive in my garden, apparently living on dew and insects. They bury themselves in the ground during winter, emerging when the weather warms up.
### Tobacco
See _Nicotine_ under _Insecticides, Botanical_ , in this chapter.
### Trap Crops
See _Insect Attractants, Botanical_ , in this chapter.
### Whitefly _(Trialeurodes vaporariorum)_
Whitefly is one of the insects known to thrive on certain shortages of minerals in the soil, and experiments have shown that greenhouse whiteflies attack tomatoes only when phosphorus or magnesium is deficient.
Botanical controls include nasturtium, particularly good to grow in greenhouses with tomatoes. Oak leaves burned in a greenhouse for a half-hour period are helpful, and nicotine may be used as a spray.
Whitefly can be controlled biologically by a small parasite called _Encorsia formosa_ (see Sources). Ladybugs also are a control, as are aphis lions (the larvae of lacewing flies).
### White Hellebore _(Veratrum album)_
White hellebore is a member of the Lily family whose roots and rhizomes contain insecticidal substances. It becomes less toxic upon exposure to light and air and has little residual effect, making it less poisonous to use than more persistent materials.
The use of white hellebore is centuries old. The Greeks mixed it with milk to kill flies, and it was a favorite remedy of the Romans against mice and rats. Today it is used to control many leaf-eating insects such as sawflies, which attack ripening fruit, and also for slugs and cabbageworms.
### Wood Ashes
Wood ashes sprinkled around the base of cauliflower and onion plants are a popular remedy to control maggots. They are also used against clubroot, red spider, bean beetles, and scab on beets and turnips, as well as aphids on peas and lettuce. They are good, too, around such plants as corn, which need to develop strong stalks.
A thin paste made of wood ashes and water and painted on the trunks is used to control tree borers. A handful each of wood ashes and hydrated lime, diluted with two gallons of water, makes a spray for the upper and lower sides of cucurbit leaves to control cucumber beetles.
### Woodchuck _(Marmota monax)_
If woodchucks are a problem, spray the plants they are nibbling with a solution of water and hot pepper.
## Poisonous Plants
**Humans perhaps** were given dominion over the birds and beasts and the lilies of the field (including thorn apple and castor bean) because they have (or should have) the ability to distinguish the good from the bad.
Since the beginning of time, people the world over have lived close to hundreds of plants that can cause irritation, illness, or even death. A few are seriously poisonous; a far greater number are moderately so, producing varying degrees of illness or irritation. Some cause dermatitis, hay fever, or other illnesses as a result of the allergic sensitivity rather than the direct toxicity of the plant.
It would be virtually impossible to ban every plant that is inedible, drug-producing, or even irritating to the skin. Moreover, many such plants are the source of valuable medicines or serve us well as natural insecticides when wisely and carefully used in our gardens.
Children should be taught which plants are harmful and which may be used or eaten or touched with safety. Caution the child old enough to understand about eating wild berries, fruits, and nuts or chewing on bark, branches, or stems. Watch younger children just as you keep them from running into the street or keep dangerous household preparations out of reach. For poisonous plants, whether present in your own garden or in fields nearby, _will_ be encountered.
_Apples, peaches, and castor beans contain highly poisonous parts_.
Dozens of these plants are attractive garden flowers or shrubs, greatly prized for their beauty or usefulness in landscaping, so many adults and most children do not realize their poisonous characteristics. Who would think, for instance, that daffodils are dangerous? Nevertheless, the bulbs can be fatal if eaten. The beautifully marked castor beans are a natural plaything for a small child, and little children often put things into their mouths. Yet for this poison there is no available antidote.
Even older children have been known to try the "nuts" inside a peach stone. How do I know? Because as a child I tried one of these seeds. They resemble almonds, which I dearly loved. I didn't know that they contain cyanic acid and that a few of them could kill me. Fortunately, the bitter flavor stopped me after the first bite. Parents can't think of everything to tell children not to do, so it is indeed fortunate that many dangerous plants do have an unpleasant taste.
But if my mother didn't know about the peach stones (and I kept my experiment a secret), there wasn't much else she didn't know about. She was a skilled herbalist, and I grew up with the salves and tonics she prepared and brewed from the plants she gathered.
She often told me of her adventures in the Indiana woods where she went to gather ginseng. Later, when she and my father were married, she transferred her herb-gathering activities to Kentucky, and still later, when the family came out to Oklahoma, then Indian territory, she found literally new fields to conquer. She learned much herbal lore from Native Americans, who gave her the name Hill-leatah-ha, meaning "one-who-thinks-good." In those days when there were few doctors and most people made their own medicines, her knowledge was greatly respected. And she was fortunate to be completely unaffected by poison ivy, an immunity that I too am happy to have.
To this day I still grow many of the plants my mother brought in from the woodland to use for their therapeutic or insecticidal properties. Until recently I did not think of them as "companion plants," though I was clearly aware of their effect when grown in proximity to flowers, fruits, and vegetables.
While I have known since childhood most of the plants likely to be encountered in garden or woodland, there are many things about plants that I am still learning — for instance, that some nonpoisonous plants have been contaminated by neighboring poisonous ones through a chemical chain. The poisonous plant absorbs selenium from the soil, something the "innocent" plants cannot do. In dying and decomposing, the poisonous plant leaves the selenium in a changed state, in a sense predigested, which causes the harmless plant during its next season's growth to draw the toxic substance into its own system.
I have no degree in botany, nor am I an expert on poisonous plants. Neither shall I try to cover _all_ of them in this brief chapter. Excellent books are available for those who are interested in pursuing the subject in greater detail. Here are only those plants that are most likely to be found in flower or vegetable gardens, or that may arrive, blown in by the wind or other means, as weeds. I have noted with a "D" those plants likely to cause dermatitis, a susceptibility that of course varies from person to person.
## POISONOUS PLANTS
_The handsome lily of the valley has poisons in both leaves and flowers_.
_The digitalis in foxglove is a powerful heart stimulant. It may be fatal if the leaves are eaten_.
_Mistletoe usually won't be touched by animals, but it is poisonous. However, birds eat the berries without harm_.
_All parts of the lovely rhododendron are very toxic if eaten_.
D = those plants likely to cause dermatitis (susceptibility varies from person to person)
## Garden Plans
**This chapter** contains an assortment of garden plans to suit the different needs of different gardeners.
### A Model Companion Garden
Planning your first organic garden may be a little tricky, but in time and with experience truly good gardeners get to know plants' likes and dislikes (which may sometimes vary in different localities), and they can spot just the place in their garden for any plant where it will do best.
Where it is at all possible, try to use double-purpose plants — those that are both decorative and sources of food, or those that are protective of other plants — in your garden.
Planning your garden on paper is one of the best ways to achieve success. Use large sheets ruled off into squares, using a scale of 2 feet to the inch. Draw large plants the size they will be at maturity. Locate existing features, such as trees, shrubs, hedges, buildings, and paths. This is important, as they may shade the garden during certain hours of the day. Mark off north and south on your plan. Then add each item you want in your garden, considering them in their order of importance to you and your family and the space and time you can give to them.
For your vegetable garden, plan on using and replacing early vegetables with those that do well in spring and summer. In turn these will be replaced with cool-season vegetables for the fall.
Keep records, for you will probably want to change your plan from season to season, including some new vegetables or different varieties and discarding others that may not have done well. Note how well certain plants did on certain sites, the effect of their form and color, and the rotation of your crops. Rotation is very important in garden plots that are planted year after year.
The heavy feeders (those that need generous fertilizing, including corn, tomatoes, cabbage, and other brassicas) should be followed by legumes to help the soil recover from the demands of the heavy feeders, especially in poorer soils. In a live humus soil legumes may go in third place if desired. The last group are the light feeders, which include root vegetables, bulbs, herbs, and protective flowers such as marigolds and nasturtiums.
For the serious gardener a complete record of space used and treated in a vegetable garden is an invaluable aid in planning the next year's garden. It is not enough to place the high vegetables to the north, the corn in blocks, and the asparagus where the tractor will not run it down. A map showing the pH factor in various garden areas should be made and kept up-to-date. Application dates of slowly available materials, such as rock phosphates, should be recorded. With such, you can make a long-range plan of your garden that will serve you well in future years.
Few garden projects are more fun to experiment with than companion planting. Good plant neighbors are those that occupy different soil levels, like beets and kohlrabi, or that find in each other's company the light requirements that best suit them. An example of this is the compatibility of celery and leek: The upright leek finds room near the bushy celery plant. Other compatible combinations are cabbage and beans, beets and onions, cucumbers and sweet corn, carrots and peas, cucumbers and beans, early potatoes and corn, early potatoes and horseradish, tomatoes and parsley. Asparagus beetles are repelled by tomatoes growing nearby. To repel the cabbage maggot, plant mint, tomato, rosemary, or sage in the next row. Radish will grow extra well and especially tender near lettuce.
On the other hand, avoid the incompatible combinations such as sunflowers and potatoes, tomatoes and fennel, tomatoes and kohlrabi, pole beans and beets, pole beans and kohlrabi. Note, however, that localities and seasons also seem to affect "companions." More than one gardener has gleefully told me she planted onions with peas or beans and had a wonderful crop of both.
We are sometimes surprised when we go out into our garden in the morning to find that the bugs have held a candlelight ceremony and arrived overnight. Timing can discourage these unwelcome guests. I find fall-planted squash is entirely free of squash bugs. Onions around spring-planted squash help to keep insects away. Aromatic herbs as border plants are very helpful to the garden.
In the drawing opposite you see a model companion garden very much like my own (though smaller). Walkways around the fence inside make it easy to use a tiller. Note, too, how a protective fence is used for such climbing plants as beans, cucumbers, peas, and grapes.
Flowers add beauty to the garden but are chosen as well for their beneficial influences on crop vegetables and fruits. Note vegetable interplantings, too, in most of the rows. Perennials such as asparagus and horseradish are placed at the outsides to make tilling of the full garden easier.
### The Weekend Garden
The primary object of the weekend garden is to get the "mostes' result with the leastes' effort." Remember that garden space may also be saved by doubling rows of narrow-growing crops. Thus, onions, which take only as much space as the diameter of the bulb, may be grown in double rows 6 inches apart. Mulch may be arranged between the two rows, when the plants are large enough, to cut down the amount of hand-weeding, if necessary. Carrots, beets, parsnips, and turnips may be handled in the same way. Cos or Romaine lettuce, which grows elegantly tall, is also a space saver and may be grown close together. Bush varieties of squash and cucumbers and beans also take less space.
Annual vegetables that like sole possession of their garden space for the entire summer include lima beans, Swiss chard, cucumbers, eggplant, okra, onions, parsley, parsnips, peppers, sweet potatoes, late white potatoes, salsify, squash — both summer and winter — New Zealand spinach (grows well in hot weather), and tomatoes.
Just because you are a weekend gardener doesn't mean that you cannot enjoy some of the perennial vegetables as well, if you are willing to plan for them and give them their own space. Asparagus, chives, horseradish, perennial onions, rhubarb, perennial herbs, and berries are possible, and most of these can stand a certain amount of benign neglect. They must, of course, be located where power tillage or cultivating tools will not be inconvenienced by them. Usually it's best to give them a plot of their own, removed from the rest of the garden.
Study your garden catalogs for small, space-saving varieties, even dwarf vegetables, as well as for early, midseason, and late varieties. If you have a gardening friend or neighbor, consult him as to the best varieties for growing in your area. Don't be shy — gardeners just love to give advice!
If your schedule allows you to garden only one or two days per week, here are some tips.
• Plan a small garden the first year.
• Prepare your garden soil in the previous fall, incorporating compost.
• Grow the vegetables your family likes best — choosing varieties that produce well in your area.
• Interplant some companion plants in each row.
• Plan for a watering system with a timer for use when rainfall is insufficient. Use a soaker hose in the garden.
• Harvest often to keep plants producing (peas, beans, tomatoes, peppers, okra, for example).
• Pull spent plants and replace with warm-weather plants either started indoors or purchased.
• As the season advances, start a fall garden in the shade of cornstalks, etc., and pull when young plants are well started.
• Mulch plants to cut down on weeding. Plow in mulch in the late fall as preparation for early-spring planting.
### The Weekend Garden
WARM-SEASON VEGETABLES PLANT
PERENNIAL VEGETABLES
## The Postage Stamp Garden
Whether the postage stamp garden is tucked away in the corner of a city backyard or blossoms in the rarefied atmosphere of a penthouse far above the traffic of the streets, it can bring the same rewards and satisfaction to its creator as a larger one of more impressive proportions. The small backyard garden on the ground may be cared for much the same as a larger garden would be and with good soil, water, and sufficient sunlight may be expected to have much the same degree of success.
But city gardening in high places is different from that in a city backyard. High above the street, dust and fumes may not be troublesome but there are other problems to deal with — excessive winds and beating sun. A certain amount of experimentation as to the placement of the garden will best determine the location most likely to be successful.
Because the roots of the plants are confined to limited quarters in their box or boxes, the soil must be prepared with care. It should be composed of sandy loam (topsoil), humus, and peat moss. Fertilizers such as decomposed manure and bonemeal should be mixed in the soil. Good drainage is essential.
Plants can be protected from high winds by windbreaks of various kinds, such as a fence of cedar saplings, a perpendicular board fence, or one of close basketweave. For a more decorative effect, use a screen of plate glass, glass brick, or other interesting material.
Plant as many started plants as possible, such as tomatoes, peppers (sweet and hot), onions, and herbs. Okra transplants well, so you might try growing a number of plants in a pot, setting them in when they are larger where you want them to grow. Yellow squash and zucchini can be grown in peat pots and handled the same way. All of these plants do well in sunshine if given wind protection. Avoid lettuce and other shade-loving plants. But you might like to tuck in here and there a few sun-loving flowers such as marigolds and nasturtiums, which can also do double duty as protection from insects. Yes, you may be surprised at the number and variety that will find their way to your garden! Some, serving as pollinators, are not all bad.
A roof garden may not be as easy as one on the ground but can be a lot of fun to plan, plant, and care for. To the confirmed gardener, having this miniature plot means not just having fresh vegetables for your table but also the joy and satisfaction of "growing your own." For the city dweller a "garden in the sky" may be just the stuff that dreams are made of.
**Tips for Roof Gardening**
**Soil**. Use a good, friable loam. Fertilize with dried manure. Be sure the roof is strong enough to support a box or boxes. It's advisable to check with a structural engineer.
**Mulch**. A mulch will help prevent sudden drying out of the soil from wind and sun. Use peat moss, decomposed manure, leafmold, or black plastic.
**Vegetables**. With full sun, good soil, and boxes about 8 inches deep, you can grow broccoli, parsley, radishes, bush beans, endive, onions (from sets), New Zealand spinach, Swiss chard, small-size tomatoes, and shortrooted carrots and beets. Cage peppers, standard-size tomatoes, and bush squash for wind protection.
**Herbs**. Herbs are a good choice for shallow boxes usually used on a roof. Try thyme, chives, parsley, mint, sage, and basil.
**Flowers**. Marigolds, zinnias, ageratum, petunias, calendulas, sweet alyssum, lobelia, portulaca, celosia, iberis, forget-me-not, salvia, coreopsis, aster, and scabiosa are all good choices. Many now come in dwarf varieties.
**Boxes**. Boxes should be deep enough to hold 8 to 12 inches of soil. Make them as wide as you wish or your space will accommodate. Provide drainage holes in the bottom of each box. Paint the insides with asphaltum, or other wood-preserving compound, and the outside with several coats of quality outdoor paint. Dark green is a good color. Cypress or redwood has good durability.
**Vines**. Vines are possible if the wind is not too strong. An arbor over part of the roof would make a good vine support and also supply some shade and shelter. Try ivy, honeysuckle, or morning glories.
## The Kitchen Herb Garden
Why have a culinary herb garden adjacent to your kitchen door? For the simple reason of convenience and quick accessibility. Whether you are a busy houseperson or a busy businessperson, you want to give your family the most inviting and taste-tempting meals you can prepare in the shortest possible time. Those of us who cook with herbs and delight in their flavor, fragrance, and freshness find this small kitchen garden a little luxury that is pleasurable, affordable, and easily obtainable, well worth the small amount of time its maintenance requires.
Most herbs originated in rather dry, often rocky soil and do best in soil that is lightly fertilized. Most herbs like sunlight, though some will do equally well in shade. And fresh herbs add that certain something hard to achieve with dried.
_**Basil**._ You haven't really lived until you have tasted fresh basil with tomatoes. This sweet, wondrous-smelling plant is at its very best when used fresh. Pesto is that delicious Italian sauce made with fresh basil pounded with olive oil, pine nuts, parmesan cheese, parsley, garlic, salt, and pepper. Use it on spaghetti or green noodles, spread on toasted bread, or on vegetables and in salads.
_**Fresh bay leaves**_ make up the ubiquitous bouquet garni that is used in many meat and vegetable dishes, noodle dishes, condiments and soups.
_**Thyme**._ Jeanne Rose, in _Herbal Guide to Food_ , describes the thyme as a little plant with a big flavor. It is available in an incredible range of scents and flavors from nutmeg, caraway, mint, pine, and pepper to lemon, citronella, and many more. Thyme is delicious in stuffings and salads, on vegetables such as beans, onions, and tomatoes, and in herb butter and breads. Try out several varieties and surprise family or guests with unusual flavors.
_**Mint**_ in its variations can liven up many things besides that timehonored southern favorite, mint julep, usually flavored with spearmint or peppermint. White mint is especially useful in jellies or with lamb. Apple mint can be used to scent and flavor salads. Pineapple mint, ginger mint, and peppermint can be used in teas. Mints are good in breads and stuffings, with vegetables, in soups, or with eggs, meats, or game. Many sauces can be given a delightfully tangy flavor with the use of various mints.
_**Marjoram**_ is indispensable in French cooking. It is used as a garnish, in soups, salads, eggs, cheese, vegetables, meat or poultry, stuffings, pasta, breads, and drinks.
_**Parsley**._ I need not dwell on the uses of parsley as it is so well known and universally used, so pretty and reliable in the herb garden. Just remember that it is an extremely rich source of minerals and vitamins as well, including vitamins A and C.
_**Chervil**_ is another herb essential to French cooking, and one that we need to show more respect. Try seasoning peeled avocados with olive oil, pepper, lemon juice, and a really generous dose of chopped chervil. Or use it generously sprinkled over cold cream soups, fresh green salads, carrots, or potatoes.
_**French tarragon**._ You can achieve herb happiness in your culinary art with French tarragon, often called the king of culinary herbs. Its tart, somewhat sweet licorice taste adds excitement to béarnaise or hollandaise sauce. Jeanne Rose suggests basting a roasting chicken with tarragon-mushroom butter. (See Suggested Reading.)
_**Sage**_ , an old kitchen favorite, is used to counteract the greasiness of fatty meats. It is used in stuffings of goose, duck, turkey, and chicken, and it is also good with rabbit, eggs, cheeses, beans, onions, and tomatoes.
_**Summer savory**_ is good chopped into salads or with baked fish or roasted pork.
_**Rosemary**_ is one of the most important herbs. Even the flowers are put into salads, and the herb is used for everything from garnishes to desserts and jellies.
To those accustomed to using only dried herbs, the truly fresh ones from your own kitchen garden will open up a whole new world of epicurean excitement.
## A Child's Garden
A little yard where children can play and have their own garden should be located where soil, sun, and drainage conditions are good — preferably in the service area where it can be watched from the house.
Water should be available nearby. Part of the playhouse may be used as a storage area.
Plant trees and shrubs that do not bear unwanted fruit for children to nibble on, fall on, or track into the house.
Plant a grass that can "take it," such as bluegrass, fescue, or Bermuda.
Use concrete or asphalt to build a "drag strip" for riding toys — making wide places for "turn-out stations."
Use chain-link fence or a similar type with clear visibility.
Floodlights, mounted on adjacent buildings or in trees or on a high pole, can add hours of evening playtime.
Buy started plants for the kids' garden, varieties that are quick and easy to grow, or prolific producers such as squash. If the child is old enough to help tend the garden, be lavish with praise when something of his or hers is brought to the table.
## The Able Disabled Gardener's Garden
A "no-stoop garden" is the nicest kind! Here are some suggestions to make a convenient garden for an older or disabled person.
• Box (or boxes) should be of a height easily reached by the individual.
• Boxes can be any length but should be just wide enough to be easily reached from either side.
• Box should be sturdy enough to support gardener, if using cane, walker, or wheelchair — also sturdy enough for soil when wet.
• Elevating an area improves drainage. Holes should be bored in the bottom of the box and in the pan underneath the box.
• Use a watering can with a comfortable handle and detachable sprinkler head. Try using a spray wand.
• For a larger garden you may want an overhead spray arrangement. For spot watering, use ice cubes; they'll melt slowly with little drip. Set a plant or two beneath the drip.
• Cover holes with copper screening to prevent plugging up.
• What will you grow? Just about anything you want to. Have planter boxes of different depths.
• Harvest often and keep a "floating crop game" going, quickly reseeding harvested plants.
Choose your tools carefully and buy good quality. You will not need a garden tractor but at the very least you should have a digging tool, a clipping tool, and a cultivator tool. Kathleen Yeomans, writing in _The Able Gardener_ , suggests that you "test-drive" before you buy. "Lift them," she says, "and manipulate the tool from the position in which you will be using it. Whether you stand, stoop, or sit to garden will determine the handle length you require. If you expect to be sitting while you garden, sit down and go through the motions of using the tool." Buy tools that fit your hand comfortably, especially if you have unusually small or extra-large hands. A tool that seems heavy in the store will feel twice as heavy when you are working with it in your garden.
Your little garden can be caught in a late freeze just like a big one. Have some type of hot cap handy to protect tender plants. In a pinch use grocery bags, weighted down with stones or clods. Remove them during the day if the weather is warm.
Keep a sharp eye out for insects that may suddenly discover your garden, often overnight, if they are in the immediate neighborhood. Seeing them almost eye to eye in the elevated garden makes them easy to spot and you can quickly take care of the problem. Just be sure they are "enemies" and not beneficial kinds.
Give careful thought to the location of your elevated garden. Will it receive full sunlight for at least six hours of the day? Will a large tree shade it morning or afternoon? Does it have protection from the wind? And is it located convenient to your degree of disability, if you garden from a wheelchair, standing with a cane, or using a walker?
And I urge you most sincerely not to let yourself get too tired. This is supposed to be fun, but sometimes we get carried away and just want to keep going. Take it from me, for I, too, am an "able disabled" gardener with weak hands, small strength, and painful arthritis.
### VEGETABLE GROWING GUIDE FOR MINIGARDENS
Group 1:Plant seeds of these vegetables in large containers (bushel baskets or 5-gallon buckets). They need full sunlight and warm weather. These vegetables should be planted outdoors when it is warm and the danger of frost is past. You can start tomato and pepper plants indoors six to ten weeks early, then move them outdoors at the proper time for earlier harvest.
Group 2: These vegetables can be grown in pots 6 to 10 inches. They can withstand a little shade and do well in cool weather. Plantings should be made when it is warm and all danger of frost is past.
Don't plant onion seed. Use dry sets or green plants, or use an onion from the kitchen that has sprouted; usually it will divide itself into several plants.
**The Tinest Garden of All**
Life can be very lonely for the elderly, housebound, or disabled. But how they can enjoy butterflies! Julia Percival and Pixie Burger give us a suggestion in their book _Household Ecology_. "Often you can persuade butterflies to settle and feed on a sunny windowsill if offered the right confection. This confection is simple. Half a teaspoon of honey and half a teaspoon of sugar mixed in a cup of warm water will make an adequate butterfly nectar.
"To make a feeder use a saucer and a wad of cotton. Butterflies taste through their feet, so they need an island of cotton in the middle of the nectar sea. The cotton will absorb the honey-water so that the butterfly, when it lands on the cotton, will taste the food through its feet and then it will stand on the island and lower its rolled tongue and suck the nectar up."
## The Spirit Garden
When our ancestors first happened on the formulas for making wine and beer, it was considered a tremendous boon from the gods. Alcohol was supposed to have an in-dwelling spirit adopted from the life of the fruit and plants from which it was made. Hence the origin of our modern term "spirits" for such drinks.
Beers, wines, brandies, cordials, bitters, and whiskeys can all be made from the herbs, fruits, grains, vegetables, and trees in your garden. (See the plan on pages 202–203.)
## The Spirit Garden
Trees
Persimmon, wine
Sassafras, root beer
Peach, wine, brandy; washed
peelings plus a few stones
can be used to make brandy
Pear, wine
Fir; Native Americans used
the young treetops to make
a beer said to prevent
scurvy
Plum, wine
Lime, wine additive
Lemon, wine additive
Orange, wine
Apple, wine, cider
Cherry, wine, cordial
Elderberry, wine
Birch, beer
Vines
Hop vine, beer
Honeysuckle, wine
Grape, wine
Vegetables (fully mature)
Parsnips
Potatoes
Beets
Carrots
Turnips
Rhubarb
Pumpkin
Hard-shelled winter squash
Herbs
Anise, used to flavor liqueurs
Wintergreen, wine
Ground ivy, bitters
Sage, wine
Maguey (Agave species), tequila
Nettle, beer
Flowers
Daisy, flower heads, wine
Dandelion, flower heads, wine
Roses, petals
Clover, flower heads, wine
Berries
Blackberry, wine, cordial
Raspberry, wine
Cranberry, wine
Strawberry, wine
Boysenberry, wine
Dewberry, wine
Various hybrids
Weeds
Barberry, wine
Burdock, beer, bitters
Red currant, wine
Black currant, wine
Gooseberry, wine
Wild grape, wine
Wild hops, beer, ale
Juniper, beer
Red mulberry, wine
Wild sarsaparilla, wine
Shadbush (Juneberry, serviceberry), wine
Wild strawberry, wine
Beverage Additives
Lemon, sangria blanca
Lime, margarita
Orange, tequila sunrise
Pineapple, piña colada
Tomato, sangria
Jalapeño chile, sangria
Onion, sangria
Coconut milk, piña colada
Mint — sprigs to mint julep
Whiskey
Corn, chief ingredient of
Bourbon whiskey
Rye whiskey, rye is chief
ingredient
Barley malt, Irish whiskey
Barley malt, Scotch whiskey
## The Aphrodisiac Herbal Window Box
Hundreds of effective aphrodisiacs have been discovered over the centuries. Many can achieve amazing results. Virility can be sustained, preserved, and even recaptured with specially selected herbs and foods.
Many herbs have proved effective as aphrodisiacs through the ages. They include basil, chervil, chives, mint, parsley, sage, tarragon, thyme, and garlic.
Soup, meat, fish, and desserts featuring the aphrodisiacs grown in your own garden can become powerful sexual stimulants. Suburbanites, with a small plot of land behind their house, can supply themselves with aphrodisiacs in abundance, but what of apartment dwellers who have no scope whatever for ordinary gardening? Must they be penalized because they have chosen (or been forced by circumstances) to occupy part of a building that is surrounded, not with the good earth, but with concrete, asphalt, or paving stones? Are they to be deprived of the pleasure and satisfaction of growing their own "contributory crops"?
Fortunately, the answer to these questions is "No," an emphatic "No!" Experiments have shown that a great deal can be achieved in that very small garden — the window box. While such small-scale gardeners cannot expect to produce the same quantity and variety as the owner of 600 square yards, neither do they need to. They can provide themselves with a surprisingly large number of herbs and vegetables of a stimulating nature, quite often sufficient to their needs.
Assuming that the apartment has a living room, bedroom, and kitchen, and that each has an average-sized window, each window box can accommodate a reasonable selection of plants. In fact, even one window box can grow a succession of herbal or vegetable plants, if you change them from time to time.
_**Chives**_ can be grown from seeds. The chopped leaves give a distinctive flavor to salads and a pleasing, tangy taste to soups and egg dishes. A bit milder than onions, they improve chicken and shrimp dishes. Chicken broth with chives, yolk of eggs, and powdered almonds makes an invigorating nightcap.
_**Garlic**_ as an aphrodisiac is second to none. The trouble with garlic is its powerful odor. If a garlic-flavored dish is prepared, it should be shared.
_**Mint**_ , on the other hand, has a delightful fragrance. It is a powerful amatory aid and can be said to stimulate the appetite. A meal of lamb, new potatoes, and peas can be deliciously enlivened with a mint sauce, doing something more than satisfying hunger. Even boiled cabbage sprinkled with this herb may leave one in "mint" condition.
_**Parsley**._ The aphrodisiac properties of parsley were appreciated by the ancient Greeks and Romans. Parsley is just chock-full of vitamin E. Parsley is more than just a pretty face; use it lavishly as a stimulating addition to boiled chicken and omelettes.
_**Thyme**._ That oldest and most attractive of kitchen herbs, thyme, can be grown either from seed or by root division. As an aphrodisiac it crops up with almost monotonous regularity in world literature. Far, far back in time it was found invigorating by the Egyptians. John Gerard in his _Herball_ , written in 1633, praises it; and Benedictine monks still use it as an ingredient in their liqueurs. Use discreetly, for a little goes a long way, and it tends to kill other and more subtle flavors.
_**Vegetables**._ While some apartment dwellers may choose to concentrate on herbs and fill their window boxes with only basil, tarragon, fennel, rosemary, lavender, and so on, others might prefer to add a few stimulating vegetables such as radishes, dwarf carrots, spinach, peas, and beans, all of which have been shown to be "contributory crops."
_**Cress**._ Another possibility is cress, the aphrodisiac properties of which have been recognized for thousands of years.
_**Oysters**._ If you like raw oysters, fine; if not, try them in a creamy stew with parsley, celery, and mushrooms.
_Three garden plans for window boxes_
**Construction of Box**
Measure window ledge. Box should be a few inches shorter in length than the window ledge of the same width and no more than 6 inches high. Use well-seasoned lumber, oak if possible. Spray interior with paraffin and apply match. Let surface oil burn off and then extinguish flame by turning box upside down. Paint outside of box. Color should harmonize with surroundings. Bore half-inch hole at five or six intervals in bottom of box for drainage and aeration. Nail small wedges of wood on base of box to keep box level and clear of the sill. Wedges should be high enough so shallow tray may be pushed under box to catch surplus water. In ends of box, screw metal hooks that will fit into wall staples and hold box securely in position.
In bottom of box put a layer of broken crocks or oyster shells to cover drainage holes. Second layer should consist of dead leaves, decayed turfs, and straw. Over this put good loam, filling box to within an inch of top. A bushel of soil is usually ample.
Invest in a small-spouted can for watering. Foil placed behind box will reflect sunlight most herbs need.
## Sources
Businesses relocate frequently. We try, but it is not always possible to keep our lists up-to-date. See your local extension agent for additional sources.
**Bio-Dynamic Farming and Gardening Association, Inc**.
888-516-7797
_www.biodynamics.com_
Sprays
**Burgess Seed & Plant Company**
309-662-7761
_www.eburgess.com_
General nursery plants
**Farmer Seed & Nursery**
507-334-1623
_www.farmerseed.com_
General nursery plants
**Henry Field Seed & Nursery**
513-354-1495
_henryfields.com_
General nursery plants
**Gurney's Seed & Nursery Co**.
513-354-1491 _<http://gurneys.com>_
General nursery plants
**Indiana Botanic Gardens**
800-644-8327
_www.botanicchoice.com_
Botanicals
**Nichols Garden Nursery**
800-422-3985
_www.nicholsgardennursery.com_
Rare herbs, vegetable seeds
**Park Seed**
800-213-0076
_www.parkseed.com_
Herb, vegetable seeds
**Richters**
905-640-6677
_www.richters.com_
Seeds and plants
**Spring Hill Nurseries**
513-354-1509
_<http://springhillnursery.com>_
General nursery plants
**Stark Brothers Nurseries & Orchards, Co**.
800-325-4180 _www.starkbros.com_
Fruit and nut trees
**Stokes Seeds, Inc**.
800-396-9238
_www.stokeseeds.com_
Untreated seeds
**Van Ness Water Gardens**
800-205-2425
_www.vnwg.com_
Horsetail ( _Equisetum_ ), eelgrass ( _Vallisneria_ ), water lilies
## Suggested Reading
Some of these books may be out of print but can be found at your local library, or purchased secondhand.
Ajilvsgi, Geyata. _Butterfly Gardening for the South_. Dallas, TX: Taylor Publishing, 1991.
Campbell, Stu. _The Mulch Book_. North Adams, MA: Storey Publishing, 1991.
Creasy, Rosalind. _The Complete Book of Edible Landscaping_. San Francisco, CA: Sierra Club Books, 1982.
Gibbons, Euell. _Stalking the Wild Asparagus_. Chambersburg, PA: Alan C. Hood & Company, 1988.
——————. _Stalking the Healthful Herbs_. Chambersburg, PA: Alan C. Hood & Company, 1989.
Philbrick, Helen, and Richard Gregg. _Companion Plants and How to Use Them_. Old Greenwich, CT: Devin-Adair Pub., 1990.
Riotte, Louise. _Catfish Ponds and Lily Pads_. Pownal, VT: Storey Publishing, 1997.
——————. _Successful Small Food Gardens_. North Adams, MA: Storey Publishing, 1993.
——————. _Roses Love Garlic_. North Adams, MA: Storey Publishing, 1998.
_Rodale's All-New Encyclopedia of Organic Gardening_. Emmaus, PA: Rodale Books, Inc., 1993.
Rose, Jeanne. _Herbs & Things, Jeanne Rose's Herbal_. New York: Berkley Publishing Group, 1972.
Telesco, Patricia. _A Kitchen Witch's Cookbook_. St. Paul, MN: Llewellyn, 1994.
Telesco, Patricia. _Folkways_. St. Paul, MN: Llewellyn Publications, 1995.
## Index
Page references in _italics_ indicate illustrations; **bold** indicates charts.
### A
_Able Gardner, The_ ,
Absinthium _(Artemisia absinthium)_ ,
_Acer_ , –13
_Achillea millefolium_ , –80
Acidity. _See_ Soil improvement
Acorns _(Quercus)_ ,
_Agropyron repens_ , , ,
Alder _(Alnus tenufolia)_ ,
Alfalfa _(Medicago sativa)_
companions, ,
cover crop,
soil improvement, –34,
Algae,
_Allium ascalonicum_ , –25,
_Allium cepa. See_ Onion
_Allium porrum_ , , , ,
_Allium sativum. See_ Garlic
_Allium schoenoprausum. See_ Chive
_Allium vineale_ , ,
Almond _(Prunus dulcis)_ ,
_Alnus tenufolia_ ,
Aloe vera _(Aloe barbadensis)_ , –29
Amaranth _(Amaranthus retroflexus)_ , –50,
_Ammoracia rusticana_ , ,
_Anacardium occidentale_ ,
_Andropogon sorghum_ , , ,
_Anethum graveolens_ , , , ,
Angelica, , ,
Anise _(Pimpinella anisum)_ , , ,
_Anthemis cotula_ ,
_Anthriscus cerefolium_ , ,
Ants
fumigation,
repellents, , , , ,
and toads,
Aphids
fumigation,
insecticides, , ,
predators, ,
repellents, , , , , , , ,
_Apium graveolens_ , –10, ,
_Apium graveolens rapaceum_ , –11
Apples _(Malus_ spp.), , , –101, –8
Apricots _(Prunus_ spp.), ,
_Araceae_ ,
_Arachis hypogaea_ , ,
_Arctium_ ,
_Artemisia abrotanum_ , , , ,
_Artemisia absinthium. See_ Wormwood
_Artemisia dracunculus_ , ,
_Artemisia vulgaris lactiflora_ ,
_Asarum canadensis_ , , –75
_Asclepias_ ,
_Asimina triloba_ ,
Asparagus _(Asparagus officinalis)_
companions, ,
and nematodes, –50,
overview,
_Asperula odorata_ ,
Aspirin,
Aster _(Asteroides)_ ,
_Astragalus mollissimus_ , –65
_Atriplex hortensis_ ,
_Avena sativa_ , –19, , ,
Azalea _(Rhododendron)_ , , –10, ,
### B
_Bacillus thuringiensis_ , , , , ,
Baked-appleberry _(Rubus chamaemorus)_ ,
Barley, ,
Basil _(Ocimum basilicum)_
companions, ,
in kitchen garden,
overview, –30
as pepper substitute,
Basolo, D. C.,
Bay _(Lauris nobilis)_ , ,
Bean _(Phaseolus_ and _Vicia)_
and herbs, ,
overview, –6
spray, pepper,
and vegetables, –18, , ,
Bean, bush _(Phaseolus vulgaris)_
companions, , , , ,
overview,
Bean, lima _(Phaseolus limensis)_ , –7,
Bean, pole, –8, , ,
Bee balm _(Monarda)_ ,
Beech _(Fagus)_ , ,
Beet _(Beta vulgaris)_
companions, –7, –20, , ,
overview,
Bermuda grass _(Cynodon dactylon)_ ,
_Beta vulgaris. See_ Beet
_Betula_ , ,
Bio-Dynamic Farming and Gardening Association, ,
Biodynamic gardening, –95,
_Biodynamic Treatment of Fruit Trees, Berries and Shrubs, The_ ,
Birch _(Betula)_ , ,
Blackberries _(Rubus_ spp.), , –99,
Blackfly,
Black spot, , , ,
Blessed thistle _(Cnicus benedictus)_ , ,
Blight, , –101,
Blueberries _(Vaccinium_ spp.),
Bollworms, , ,
Borage _(Boraginaceae)_ , ,
Borers
and diatomaceous earth,
insecticides,
overview,
planting, timing of,
repellents, , , , ,
Boysenberry _(Rubus ursinis_ 'Boysen'),
_Brassica arvensis_ , ,
_Brassicaceae_ , , , , . _See also_ Cabbage
Cauliflower _Brassica chinensis_ , –15
_Brassica napus_ ,
_Brassica oleracea acephala_ , , –19,
_Brassica oleraceae_ , –8, , –44
_Brassica oleracea_ , var. _acephala_ , ,
_Brassica rapa_ and _Brassica napobrassica_ , , –23, –27,
Broccoli _(Brassica oleraceae)_ , –8, , –44
Broom bush _(Sarothamnus vulgaris)_ ,
Bruce, M. E., –74
Brussels sprouts, , , ,
Buffalo grass _(Buchloe dactyloides)_ ,
Burbank, Luther,
Burdock _(Arctium)_ ,
Burger, Pixie,
Butternut _(Juglans cinerea)_ , ,
### C
Cabbage _(Brassicaceae)_
and grasses,
and herbs, , , –42, –46
overview,
and vegetables, , , ,
Cabbageworms insecticides,
repellents, , –48, , ,
_Cactaceae_ ,
Calamus _(Araceae)_ ,
Camomile _(Chamomile)_
companions, , , ,
overview,
Campbell, Stu,
Canada thistle, ,
Caper spurge _(Euphorbia lathyrus)_ , ,
_Capsella bursa-pastoris_ , ,
_Capsicum frutescens_ var. _fasciculatum_ and var. _longum_ , –61
_Capsicum frutescens_ var. _grossum_ ,
_Capsicum_ spp., –14
Caraway _(Carum carvi)_ , , ,
Carrot _(Daucus carota)_ and fruits,
and grasses,
and herbs, , ,
overview, –10
and vegetables, , , –23,
wild,
and wild plants,
_Carum carvi_ , , ,
_Carya illinoinensis_ , –7
_Carya_ spp.,
Cashew _(Anacardium occidentale)_ ,
_Castanea dentata_ , , ,
_Castanea mollissima_ , ,
Castor bean _(Ricinus communis)_ , –48,
Caterpillars
_Bacillus thuringiensis_ , ,
predators, ,
repellents, , ,
Catnip _(Nepeta cataria)_ ,
Cauliflower _(Brassicaceae)_ companions, , , , ,
overview,
Cayenne peppers, ,
Celeriac _(Apium graveolens rapaceum)_ , –11
Celery _(Apium graveolens)_ , –10, ,
_Chamaecyparis_ spp.,
_Chamomile. See_ Camomile
Charlock _(Brassica arvensis)_ , ,
Chayote _(Sechium edule)_ ,
Chenopodium album, ,
Cherries _(Prunus_ spp.), ,
Chervil _(Anthriscus cerefolium)_ , ,
Chestnut, American _(Castanea dentata)_ , , ,
Chestnut, Chinese _(Castanea mollissima)_ , ,
Chicory _(Cichorium intybus)_ , ,
Chile peppers _(Capsicum_ spp.), –14
Chinch bugs, , ,
Chinese celery cabbage _(Brassica chinensis)_ , –15
Chive _(Allium schoenoprausum)_
aphrodisiac,
and borers,
companions, , , ,
overview,
Chokecherry _(Prunus virginiana)_ , ,
_Chrysanthemum cinerariaefolium_ , , ,
_Chrysanthemum leucanthemum_ ,
_Cichorium intybus_ , ,
Cinquefoil _(Potentilla monspeliensis)_ ,
_Circium_ , , ,
_Citrullus vulgaris_ , , , ,
Citrus _(Citrus_ spp.), ,
Cleavers _(Galium sparine)_ , –34
Clover _(Leguminosae)_
companions, , ,
as cover crop,
soil improvement, –35,
Clubroot, , ,
_Cnicus benedictus_ , ,
Cockroach, ,
Codling moths, , ,
Collard _(Brassica oleracea_ , var. _acephala)_ , ,
Colorado potato beetles, , , , –46
Comfrey _(Symphytum officinale)_ , –36
Companion planting. _See also_ Mulching
Pest control
Soil improvement
_specific plants_
aphrodisiac herbal window box, –5, __
background, –4
biodynamic gardening, –96,
border plants,
business, sources, –8
butterfly garden,
catch cropping,
child's garden, _194–95_ and climate, –22
damping-off, ,
double digging,
French intensive gardening, –24
and frost,
fruits, –104
grasses and grains, –92
herbs, –48
intercropping, –26
kitchen herb garden, –92, _193_ microclimate, –27
minigardens,
model companion garden, –81, __
no-stoop garden, –97
nuts, –8
ornamental trees and shrubs, –20
poisonous plants, –79
pollination, –94, , –25
postage stamp garden, –89
raised beds, –24
roof garden,
rules for,
shade,
soil pH, –29
spirit garden, –201, _202–3_
succession planting, –30
suicide in,
two-level planting, –31
two-season planting,
vegetables, –27
vertical gardening, –32
weekend garden, –84, __ , **–88**
wild plants, –80
_Companion Plants_ ,
Compost
ingredients, , , , , –11, –44
overview, –36
Conifers, –11,
_Convolvulaceae_ , –79
Coon, Nelson,
Coriander _(Coriandrum sativum)_ , , ,
Corn _(Zea mays)_ and grasses, , , ,
overview,
and raccoons, –63
and vegetables, , , , , –23,
and wild plants, , –65, ,
_Corylus_ spp.,
Cotton _(Gossypium)_ , , ,
Crab apples _(Malus_ spp.),
Crabgrass _(Digitaria sanguinalis)_ ,
_Crotalaria_ ,
Crowfoot _(Ranunculaceae)_ ,
Cucumber _(Cucumis sativus)_ and herbs, ,
overview,
and vegetables, , , , –23
and wild plants, , , –74
and wood ashes,
_Cucurbitaceae_ , , , ,
_Cucurbita pepo. See_ Pumpkin _Cuisine of the American Southwest_ ,
_Curcubitaceae. See_ Squash Cutworms, , ,
_Cydonia oblonga_ ,
Cypress spurge _(Euphorbia cyparissias)_ , –53,
### D
Dandelion _(Taraxacum officinale)_
companions, ,
overview,
soil improvement, ,
Datura _(Datura stramonium)_ , , –54,
_Daucus carota. See_ Carrot
Dead nettle _(Lamium album)_ ,
Devil's shoestring _(Tephrosia virginiana)_ ,
_Digitaria sanguinalis_ ,
Dill _(Anethum graveolens)_ , , , ,
_Dionaea muscipula_ ,
Diseases. _See_ Pest control
_specific diseases_
Dock _(Rumex crispus)_ ,
Downy mildew, ,
_Dryopteris filix-mas_ ,
Dyer's greenweed _(Genista tinctoria)_ ,
### E
Eelgrass _(Zostera marina)_ , –55,
Eelworms. _See_ Nematodes
Eggplant _(Solanum melongena)_ , , ,
_Eglantine_ ,
_Eichhornia crassipes_ ,
Elderberry _(Sambucus nigra_ and _Sambucus canadensis)_ , , ,
Elecampane _(Inula helenium)_ , –38
Elm _(Ulmus)_ ,
Emerson, Ralph Waldo,
_Equisetum arvense_ , , –62, ,
_Erigeron_ ,
Esparsette _(Onobrychis viciaefolia)_ ,
Euphorbia _(Euphorbiaceae)_ ,
_Euphorbia cyparissias_ , –53,
_Euphorbia lathyrus_ , ,
### F
_Fagus_ , ,
Fennel _(Nigella sativa)_ companions, , , , ,
overview,
Fern, male _(Dryopteris filix-mas)_ ,
_Ficus carica_ , , –2
Fiddlehead fern _(Pteridium aquilinum)_ ,
Fig _(Ficus carica)_ , , –2
Filberts and hazels _(Corylus_ spp.),
Firth, Grace,
Flax _(Linum usitatissimum)_ ,
Fleabane _(Erigeron)_ ,
Flea beetles, , , ,
Fleas, –45, , ,
Flies
and diatomaceous earth,
predators, ,
repellents, , , , , , ,
_Fragaria_ ,
_Fragaria_ x _ananassa. See_ Strawberry Frank, A. B.,
Freckle remedies,
Fruits. _See also specific fruits_
culture, mixed, –96
and fungus, –98
hints for, –97
pollination, –94,
small, –100
soil preparation, –95
trees, , , –104
Fungi, –59, ,
### G
_Galium sparine_ , –34
Gardening. _See_ Companion planting
_Gardening Without Poisons_ ,
Garlic _(Allium sativum)_
aphrodisiac,
companions, , , –97,
overview, –39
as pest control, , ,
and roses,
Garlic, meadow _(Allium vineale)_ , ,
_Genista tinctoria_ ,
Geraniums, ,
Gerard, John,
Gibbons, Euell, , , ,
Ginger, wild ginger _(Asarum canadensis)_ , , –75
Ginseng _(Panax quinquefolium)_ , –60
Gladiolus, ,
_Glycine max_ and _G. soya_ , ,
Goldenrod _(Solidago)_ ,
Gooseberries, ,
Goose grass, –34,
Gophers,
_Gossypium_ , , ,
Gramma grass _(Bouteloua_ spp.),
Grapes _(Vitis_ spp.)
companions, –41, ,
overview,
and trees, ,
Grass, Korean _(Zoysia japonica)_ ,
Grass, quack _(Agropyron repens_ or _Triticum repens)_ , , ,
Grasses and grains, –92, . _See also specific plants_
Grasshoppers, , ,
Greer, Anne Lindsay,
Grubs, , ,
Gypsy moths, ,
### H
_Hamamelis virginiana_ , –20
_Helenium autumnale_ ,
_Helianthus annus_ , , , ,
_Helianthus tuberosus_ ,
Henbane _(Hyoscyamusniger, L.)_ , ,
_Herbal Guide to Food_ ,
_Herball_ ,
Herbs. _See also specific herbs_
aphrodisiac herbal window box, –5, _206_
coffee substitutes, –35
drying, –37
kitchen herb garden, –92, _193_
overview, –48
pepper substitutes,
salt substitutes,
sugar substitutes, –47
for tea,
_Hibiscus esculentus_ , ,
Hickories _(Carya_ spp.),
_Holcus sorghum_ , , ,
Honeybees, , –25
Horehound _(Marrubian vulgare)_ , –40
Horseradish _(Ammoracia rusticana)_ , ,
Horsetail _(Equisetum arvense)_ , , –62, ,
_Household Ecology_ ,
Hunter, Beatrice Trum,
_Hyoscyamusniger, L_., ,
_Hypericum perforatum_ ,
Hyssop _(Hyssopus officinalis)_ , , , –41,
### I
_Impatiens bifora_ or _I. pallida_ , ,
Indian cucumber _(Medeola virginiana)_ ,
Insects. _See_ Pest control
_Inula helenium_ , –38
Inulin, ,
_Ipomea batalas_ ,
### J
Japanese beetles insecticides,
and milky spore disease,
repellents, , , ,
and roses,
Jerusalem artichoke _(Helianthus tuberosus)_ ,
Jewelweed _(Impatiens bifora_ or _I. pallida)_ , ,
Jimsonweed _(Datura)_ , , –54,
Johnson grass _(Sorghum halepense)_ ,
_Juglans cincerea_ , ,
_Juglans nigra_ , –8
_Juglans regia_ ,
### K
Kale _(Brassica oleracea acephala)_ , , –19,
Kentucky Bluegrass _(Poa)_ ,
Knotweed _(Polygonum aviculare)_ , , , –64
Kohlrabi _(Brassicaceae)_ , , , ,
### L
_Lactuca sativa. See_ Lettuce
Lamb's-quarters _(Chenopodium album)_ , ,
_Lamium album_ ,
Larkspur, wild _(Ranunculaceae)_ , , ,
_Lauris nobilis_ , ,
Lavender _(Lavendula officinalis)_ ,
Leek _(Allium porrum)_ , , , ,
_Leguminosae. See_ Clover Lemon balm _(Melissa officinalis)_ , ,
Lettuce _(Lactuca sativa)_ and beetles,
and fruits,
and herbs,
overview,
and vegetables, , , ,
_Levisticum officinale_ , –42,
_Linum usitatissimum_ ,
Locoweed _(Astragalus mollissimus_ and _A. diphysus)_ , –65
Locust _(Robinia_ spp.), –7,
_Loranthaceae_ ,
Louis Gerardi Nursery, ,
Lovage _(Levisticum officinale)_ , –42,
Lupine _(Lupinus perennis)_ , ,
_Lychnis floscuculi_ ,
_Lycopersicon esculentum. See_ Tomato
### M
_Maclura pomifera_ ,
M.aggots, , , ,
_Malus_ spp., , , –101, –8
Maple _(Acer)_ , –13
Marigolds
companions, , , , ,
and nematodes, , –64
and roses,
Marjoram _(Marjorana hortensis)_ , , –45, ,
_Marrubian vulgare_ , –40
Mayweed _(Anthemis cotula)_ ,
Meadow pink _(Lychnis floscuculi)_ ,
_Medeola virginiana_ ,
_Medicago sativa. See_ Alfalfa
_Melilotus alba_ and _M. Indica_ , ,
_Melissa officinalis_ , ,
Melon _(Cucurbitaceae)_ , , , ,
_Mentha. See_ Mint
_Mentha piperita. See_ Peppermint
_Mentha pulegium_ ,
_Mentha spicata_ , , , , –92
Mexican bean beetles, ,
Mice
overview, –57
repellents, , , , ,
Mildew
and diatomaceous earth,
horsetail tea,
overview,
and plants, ,
and soil,
Mildew, powdery _(Erysiphaceae)_ , , ,
Milkweed _(Asclepias)_ ,
Milky spore disease _(Bacillus popillae)_ , , –59
Mint _(Mentha)_
aphrodisiac,
companions,
overview, –43, –92
as pest control, ,
Mistletoe _(Loranthaceae)_ ,
Moles, , –48, ,
_Monarda_ ,
Morning glory, wild _(Convolvulaceae)_ , –79
_Morus_ spp., , ,
Mosquitoes
and _Bacillus thuringiensis_ ,
and diatomaceous earth,
overview, –60
repellents, , , , , , ,
and toads,
Moths
and _Bacillus thuringiensis_ ,
overview,
predators, ,
repellents, , , , ,
Mugwort _(Artemisia vulgaris lactiflora)_ ,
Mulberries _(Morus_ spp.), , ,
_Mulch Book, The_ , –28
Mulching
and climate,
overview, –28
pine needles, ,
soil improvement,
Muscadine,
Mushrooms,
Muskmelons, ,
Mustard companions, ,
field,
greens,
insect attractants, –52
wild, –19, ,
### N
Nasturtiums
companions, , , ,
as pepper substitute,
pest control, , –52,
_Natural Year, A_ ,
Nectarine _(Prunus persica_ var. _nucipersica)_ ,
Nematodes overview, –50
repellents, , , ,
and roses,
_Nepeta cataria_ ,
_New Scientist_ ,
_Nicandra physalodes_ ,
Nichols Garden Nursery, , ,
_Nigella sativa. See_ Fennel
Nightshade _(Solanaceae)_ ,
Nut grass _(Cyperus esculentus)_ , , –68
Nuts, , , –8. _See also specific nuts_
_Nymphaea odorata_ ,
### O
Oak _(Quercus)_ , , –15
Oats _(Avena sativa)_ , –19, , ,
_Ocimum basilicum. See_ Basil
Okra _(Hibiscus esculentus)_ , ,
Onion _(Allium cepa)_ and conifers,
and herbs, , ,
overview,
as pest control, ,
and roses,
and rye,
and vegetables, –10, , ,
and wild plants, ,
_Onobrychis viciaefolia_ ,
Opuntia _(Cactaceae)_ ,
Orach _(Atriplex hortensis)_ ,
Oregano _(Origanum vulgare)_ , –44
Ornamental trees and shrubs, –20. _See also specific plants_
Osage orange _(Maclura pomifera)_ ,
Ox-eye daisy _(Chrysanthemum leucanthemum)_ ,
### P
_Panax quinquefolium_ , –60
Pansy, wild _(Viola tricolor)_ , , ,
_Papaver_ spp., , ,
_Paphanus raphanistrum_ ,
Parsley _(Petroselinum hortense)_
aphrodisiac,
companions, , ,
and freckles,
overview, ,
Parsnip _(Pastinaca sativa)_ , , ,
Pawpaw trees _(Asimina triloba)_ ,
Pea _(Pisum sativum)_
and herbs, ,
overview,
and vegetables, , –18, , ,
Peach _(Prunus persica)_ , ,
Peanut _(Arachis hypogaea)_ , ,
Pears _(Pyrus_ spp.),
Pecan _(Carya illinoinensis)_ , –7
Pennycress _(Thlaspi arvens)_ , ,
Pennyroyal _(Mentha pulegium)_ ,
Peonies, ,
Pepper, hot _(Capsicum frutescens_ var. _fasciculatum_ and var. _longum)_ , –61
Pepper, sweet _(Capsicum frutescens_ var. _grossum)_ ,
Peppermint _(Mentha piperita)_
companions, –9, ,
overview, , –92
Percival, Julia,
Persimmons _(Diospyros_ spp.), ,
Peruvian ground cherry _(Nicandra physalodes)_ ,
Pest control. _See also specific pests_
_Bacillus thuringiensis_ , , , , ,159
beans and potatoes, –46
birds, –47
castor beans, –48,
companion planting, , **–55**
diatomaceous earth, –49
_Encorsia formosa_ ,
fumigation,
ichneumonid wasp,
insect attractants, –52
insecticides, –53
ladybugs, ,
milky spore disease, , –59
molasses grass,
oak, smoke, ,
praying mantis, –62
pyrethrum, , , ,
repellents, , ,
salt, common, –65
sprays, , , , , , –61,
toads, –67
trichogramma wasps, ,
and two-season planting,
wood ashes, ,
_Petroselinum hortense. See_ Parsley
Pfeiffer, Ehrenfried E., –96,
_Pharmacopoeia of the United States of America_ , ,
_Phaseolus. See_ Bean
_Phaseolus limensis_ , –7,
_Phaseolus vulgaris. See_ Bean, bush
_Phleum pratense_ ,
_Phytolacca americana_ , ,
_Picea_ ,
Pigweed, , , , –50
_Pimpinella anisum_ , , ,
Pine _(Pinus)_ , ,
_Pisum sativum. See_ Pea
Plantain _(Plantago)_ , –70,
_Platanus occidentalis_ , –18
Plums _(Prunus_ spp.),
Poison ivy _(Rhus radicans)_ ,
Poisonous plants, –71, **–79**
Pokeweed _(Phytolacca americana)_ , ,
_Polygonum aviculare_ , , , –64
Poplar _(Populus)_ ,
Poppy _(Papaver_ spp.), , ,
_Portulaca oleracea_ , ,
Potatoes
and flax,
and fruits, , –101
and herbs, ,
and nuts,
and vegetables, –8, –16, , –23, –27
and wild plants,
_Potentilla monspeliensis_ ,
Potherbs,
_Prunus dulcis_ ,
_Prunus pensylvanica_ ,
_Prunus persica_ , ,
_Prunus_ spp., , , ,
_Prunus virginiana_ , ,
_Pteridium aquilinum_ ,
Puffball (Fungi),
Pumpkin _(Cucurbita pepo)_
companions, , , ,
overview, –23
and raccoons,
Purslane _(Portulaca oleracea)_ , ,
Pyrethrum _(Chrysanthemum cinerariaefolium)_ , , ,
_Pyrus_ spp.,
### Q
_Quercus_ , , , –15
Quince _(Cydonia oblonga)_ ,
### R
Rabbits, , , , ,
Raccoons, , –16, , –63
Radish _(Raphanus sativus)_
and herbs, , ,
overview,
and vegetables, , , , , ,
and wild plants,
Radish, wild _(Paphanus raphanistrum)_ ,
_Ranunculaceae_ , , , ,
Rape _(Brassica napus)_ ,
_Raphanus sativus. See_ Radish
Raspberries _(Rubus_ spp.), , , , –100
Rattlebox _(Crotalaria)_ ,
_Rheum rhaponticum_ , –24
Rhododendrons _(Rhododendron)_ , , –10, ,
Rhubarb _(Rheum rhaponticum)_ , –24
_Rhus radicans_ ,
Rice, wild _(Zizania aquatica)_ ,
_Robinia_ spp., –7,
Root knot, –64
Root maggot,
Rosa rugosa _(Rosa)_ ,
Rose _(Rosa)_ and herbs, ,
overview, –17
pepper spray,
and toads,
and vegetables, , ,
and wild plants,
Rose, Jeanne, ,
Rose, wild _(Eglantine)_ ,
Rosemary, –10, ,
_Rubus chamaemorus_ ,
_Rubus_ spp., , , –100,
_Rubus ursinis_ 'Boysen',
Rue _(Ruta graveolens)_ , , –45
_Rumex crispus_ ,
Rye _(Secale cereale)_ , , –88, ,
Ryegrass _(Lolium_ spp.),
### S
Sage _(Salvia officinalis)_
companions, –10, ,
and nematodes,
overview, ,
St. Augustine _(Stenotaphrum secundatum)_ ,
St.-John's-Wort _(Hypericum perforatum)_ ,
Salicin,
_Salix_ ,
Salsify _(Tragopogon porrifolius)_ ,
_Salvia officinalis. See_ Sage
_Sambucus nigra_ and _Sambucus canadensis_ , , ,
Santolina _(Chamaecyparis_ spp.),
Saponin, , –42
_Sarothamnus vulgaris_ ,
Sassafras _(Sassafras albidum_ and _S. varifolium)_ , –35, ,
Savory, summer _(Satureia Hortensis)_
companions, , ,
overview, ,
as pepper substitute,
as salt substitute,
Savory, winter _(Satureia montana)_ ,
Scab, , , ,
Scully, Virginia,
Seaweed, –44
_Secale cereale_ , , –88, ,
_Sechium edule_ ,
Senna seeds _(Cassia occidentalis)_ ,
Sesame _(Sesamum orientale)_ , ,
Shallot _(Allium ascalonicum)_ , –25,
Shepherd's purse _(Capsella bursa-pastoris)_ , ,
Silicic acid, –62
Slippery elm _(Ulmus fulva)_ ,
Slugs, , , , –68
Smut, –59
Snails, , , –65
Sneezeweed _(Helenium autumnale)_ ,
Soil improvement. _See also_ Compost
by alfalfa, , , , , –34,
by buckwheat,
calcium, , , , ,
by clover, , , , –35,
by earthworms, , , , –36,
fertilizers, –37,
fish emulsion, , ,
green manures, –39
indicator plants, ,
inoculants,
by legumes, , , –41,
magnesium, , ,
nitrogen, , –34, –41
pH, –29
phosphorus, , , ,
potassium, , , ,
rhizobium,
saponin, , –42
by seaweed, –44
by sweet clover, ,
by weeds, –78, ,
_Solanaceae_ ,
_Solanum melongena_ , , ,
_Solidago_ ,
_Sonchus arvensis_ ,
Sorghum _(Sorghum vulgare)_ , , ,
Southernwood _(Artemisia abrotanum)_ , , , ,
Sow thistle _(Sonchus arvensis)_ ,
Soya bean — soybean _(Glycine max_ and _G. soya)_ , ,
Spearmint _(Mentha spicata)_ , , , , –92
Spiders, , –27, , , –66
Spinach _(Spinacia oleracea)_ , , , ,
Spruce _(Picea)_ ,
Squash _(Curcubitaceae)_
companions, , , ,
overview,
and toads, –67
_Stalking the Healthful Herbs_ , ,
Stinging nettle _(Urtica dioica)_
companions, , , ,
and milk,
overview, –74
soil improvement,
Strawberry _(Fragaria_ x _ananassa)_
companions, –10, –20, , ,
and conifers,
overview,
Strawberry, wild _(Fragaria)_ ,
Sugar beet _(Beta vulgaris)_ ,
Sunflower _(Helianthus annus)_ , , , ,
Sweet clover _(Melilotus alba_ and _M. indica)_ , ,
Sweet potato _(Ipomea batalas)_ ,
Sycamore _(Platanus occidentalis)_ , –18
_Symphytum officinale_ , –36
### T
Tansy _(Tanacetum vulgare)_ , –75,
Tarragon, French _(Artemisia dracunculus)_ , ,
Teosinte _(Zea mexicana)_ ,
_Tephrosia virginiana_ ,
Termites, ,
Thistle _(Circium)_ , , ,
_Thlaspi arvens_ , ,
Thorn apple _(Datura stramonium)_ , , –54
Thyme _(Thymus vulgaris)_
aphrodisiac,
companions, , –45
overview, ,
Tillandsia _(Tillandsia usneoides)_ ,
Timothy _(Phleum pratense)_ ,
Tobacco, –26, , ,
Tomato _(Lycopersicon esculentum)_
and herbs, –30, –40,
and nuts,
overview, –26
and vegetables, –11, , , –21,
and wild plants, ,
_Tragopogon porrifolius_ ,
_Treasury of American Indian Herbs, A_ ,
Triticale _(Triticale)_ ,
_Triticum repens_ , , ,
_Triticum vulgare. See_ Wheat Truffles,
Turnip-rutabaga _(Brassica rapa_ and _B. napobrassica)_ , , –23, –27,
### U
_Ulmus_ ,
_Universal Herbal_ (of 1820), –48
_Urtica dioica. See_ Stinging nettle
_Using Plants for Healing_ ,
### V
_Vaccinium_ spp.,
Valerian _(Valeriana officinalis)_ , –48,
Vegetables. _See also specific vegetables_
as aphrodisiacs,
cool-season, , **** disease-resistant, –17
heavy feeders, –30,
high-vitamin,
insect-resistant,
legumes, , , –41,
light feeders, ,
for minigardens,
overview, –27
perennial, ****
semi-hardy,
warm-season, , ****
Venus flytrap _(Dionaea muscipula)_ ,
_Veratrum album_ , , , ,
Vetch _(Vicia)_ , , , –91
_Vicia. See_ Bean
_Viola tricolor_ , , ,
_Vitis_ spp. _See_ Grapes
### W
Walnut, black _(Juglans nigra)_ , –8
Walnut, English _(Juglans regia)_ ,
Water hyacinth _(Eichhornia crassipes)_ ,
Water lily _(Nymphaea odorata)_ ,
Watermelon _(Citrullus vulgaris)_ , , , ,
Weeds, –78, , . _See also specific weeds_
Weevils, , , ,
Wheat _(Triticum vulgare)_
companions, , –69,
and conifers, –11
and maples,
overview,
and sorghum,
winter, , ,
Whitefly _(Trialeurodes vaporariorum)_ , ,
White hellebore _(Veratrum album)_ , , , ,
Wild cherry _(Prunus pensylvanica)_ ,
Wild plants, –80. _See also specific plants_
Willow _(Salix)_ ,
Wilt, , ,
Witch hazel _(Hamamelis virginiana)_ , –20
Woodchucks,
Woodruff, sweet _(Asperula odorata)_ ,
Wormwood _(Artemisia absinthium)_
companions, ,
overview, ,
as pest control, , ,
### Y
Yarrow _(Achillea millefolium)_ , –80
Yeager, A. F.,
Yeomans, Kathleen,
### Z
_Zea mays. See_ Corn
_Zizania aquatica_ ,
_Zostera marina_ , –55,
Zoysia _(Gramineae_ spp.),
_Zoysia japonica_ ,
| {
"redpajama_set_name": "RedPajamaBook"
} | 6,237 |
La Vickers K, conosciuta tra i britannici anche come Vickers Gas Operated (VGO), era una mitragliatrice leggera ad alta cadenza di tiro sviluppata dalla Vickers-Armstrong per impiego aeronautico. L'alta cadenza era necessaria per consentire un efficace ingaggio nei brevi istanti nei quali il mitragliere poteva inquadrare un velivolo attaccante. Durante la seconda guerra mondiale l'arma venne adattata anche all'uso terrestre veicolare.
Storia
Sviluppo
La Vickers K era una evoluzione della mitragliatrice leggera da fanteria Vickers-Berthier (VB), adottata nel 1932 dal British Indian Army. La VB, come il Bren, usava un otturatore a blocco basculante, ma diversamente da questo bloccava l'otturatore in chiusura solo all'ultimo momento della sua corsa anterograda e permise così lo sviluppo della Vickers K a recupero di gas. Realizzata con componenti più leggere ma con lo stesso sistema di chiusura della VB, la Vickers K aveva una cadenza di tiro regolabile tra 950 e 1.200 colpi al minuto, maggiore della MG 34 tedesca. .
Impiego operativo
.
La VGO era anche l'arma standard del "mitragliere-telegrafista" sui velivoli imbarcati Fairey Swordfish, Fairey Albacore e Fairey Barracuda della Fleet Air Arm, l'aviazione della Royal Navy.
Man mano che le forniture di Browning .303 aumentavano, la VGO venne progressivamente dismessa dalla RAF. Queste armi vennero trasferite quindi alle forze di terra britanniche e del Commonwealth. L'arma rimase in servizio con l'aviazione di marina inglese. L'ultimo impiego documentato si ebbe da parte dei Barracuda dell'812 Naval Air Squadron a Hong Kong nell'ottobre 1945.
I Long Range Desert Group ricevettero grandi quantità di VGO per i loro veicoli. Le mitragliatrici vennero installate su supporti singoli o su affusti binati artigianali. Gli Special Air Service adottarono la VGO per le loro incursioni, montandole binate sulle loro Jeep. Successivamente si è ipotizzato che le due forze speciali avessero adottato l'arma perché non avevano potuto ottenere altre mitragliatrici, ma grazie all'alto rateo di fuoco ed alla resistenza all'inceppamento da sabbia essa si dimostrò alla prova sul campo nettamente superiore alla Vickers .303 raffreddata ad acqua ed al Bren. .
Dal 1942, le VGO imbarcate sulle motosiluranti e le motocannoniere delle Coastal Forces della Royal Navy iniziarono ad essere rimpiazzate dalle Lewis..
Varianti
VGO No. 1 Mk. I
La Gun, Machine, Vickers G.O. .303-inch, No. 1, Mk. I era la versione standard aeronautica, per installazione in torretta, "in caccia" o brandeggiabile "in fuga".
.
VGO No. 2 Mk. I
A partire dal 1943 la RAF acquistò anche una versione terrestre della mitragliatrice, denominata Gun, Machine, Vickers G.O. .303-inch, No. 2, Mk. I .
.
VGO No. 3 Mk. I
Nel gennaio 1945 venne realizzata per la RAF la versione Gun, Machine, Vickers G.O. .303-inch, No. 3, Mk. I, dichiarata obsoleta nel 1947. Era una versione aeronautica simile alla No. 1 Mk. I ma a controllo remoto.
VGO No. 4
La versione Gun, Machine, Vickers G.O. .303-inch, No. 4 era destinata ad equipaggiare le autoblindo, con blocco di presa dei gas modificato per diminuire la cadenza di tiro.
VGO No. 5 Mk. I
La versione Gun, Machine, Vickers G.O. .303-inch, No. 5 Mk. I venne realizzata per conversione di No. 1 e No. 3 su affusto quadrinato.
VGO Camera Gun
.
Velivoli armati
Impero britannico
Airspeed Oxford
Armstrong Whitworth AW.38 Whitley
Avro 652A Anson
Blackburn Shark
Blackburn Skua
Bristol Beaufort
Bristol Blenheim
Bristol Bombay
Curtiss P-36 Hawk
Douglas A-20 Havoc/Boston
Fairey Albacore
Fairey Barracuda
Fairey Battle
Fairey Fulmar
Fairey Swordfish
Handley Page Halifax
Handley Page Hampden
Miles Master
Saunders-Roe A.36 Lerwick
Short S.25 Sunderland
Supermarine Type 322
Supermarine Walrus
Vickers Wellesley
Westland Lysander
CAC Wirraway
Lublin R-XIII
RWD-14 Czapla
Martin 187 Baltimore
Martin 167 Maryland
Note
Bibliografia
Voci correlate
Vickers (mitragliatrice)
Altri progetti
Collegamenti esterni
Mitragliatrici leggere
Mitragliatrici aeronautiche
Armi della seconda guerra mondiale
Mitragliatrici per veicoli | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,515 |
As we continue to grow and flourish in a world connected by technology, River Valley United Way is working to make your volunteering life more productive, fun, and engaging! Recently, we partnered with Galaxy Digital, a web-based communications firm, to provide a robust and versatile volunteer engagement and management tool to communities across the River Valley.
Similar to social media platforms like Facebook or Google+, regional non-profit organizations create profiles on Get Connected that highlight their mission statements, contact information, needs, and events. Participating agencies can then post available advocacy, employment, and volunteer opportunities as well as any in-kind donations that they may need.
Likewise, interested volunteers create individual profiles specifically tailored to their schedule and interests, join volunteer groups, become fans of participating agencies, and track non-profit agencies' needs and events. Individuals in search of volunteer opportunities may also use Get Connected to conduct detailed searches on open opportunities.
Click here to access our Get Connected database. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,184 |
{"url":"https:\/\/datascience.stackexchange.com\/questions\/41044\/how-to-sort-list-by-parameter-in-python\/41047","text":"# How to sort list by parameter in python? [closed]\n\nI have a list of employee records. Each tuple of the list represent a person's record, which includes his name, ID, and age.\n\nFor example,\n\nemp_records = [('Karim',100, 45), ('Rahim',10, 30),('Salim', 300,60),('Abu',50,35)]\n\nNow, I want to sort emp_records based on ID.\n\nYou can use lambda function to pass argument by which you need to sort the records.\n\nFor example,\n\nsorted_list=sorted(emp_records, key=lambda emp : emp[1]) \/\/Here 1 means ID\nprint(sorted_list)\n\n\nYou can also use the operator module. The operator module has the itemgetter function, which takes item as argument and sort based on the argument\n\nFor example,\n\nfrom operator import itemgetter\nsorted_list=sorted(emp_records, key=itemgetter(1)) \/\/ Here 1 means ID\nprint(sorted_list)\n\n\nUse the sorted function. It accepts key as argument.\nYou can then use a lambda to sort on the relevant index of your tuple.\n\nsorted(emp_records, key=lambda x: x[1])","date":"2022-07-04 15:06:19","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\": 0.33776307106018066, \"perplexity\": 6268.782869576063}, \"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-2022-27\/segments\/1656104432674.76\/warc\/CC-MAIN-20220704141714-20220704171714-00083.warc.gz\"}"} | null | null |
<?php
namespace Atlas\Orm\Transaction;
use Atlas\Testing\DataSource\Employee\Employee;
class BeginOnWriteTest extends TransactionTest
{
public function testRead()
{
$this->assertFalse($this->connection->inTransaction());
$this->atlas->fetchRecord(Employee::CLASS, 1);
$this->assertFalse($this->connection->inTransaction());
}
public function testWrite()
{
$this->assertFalse($this->connection->inTransaction());
$employee = $this->atlas->fetchRecord(Employee::CLASS, 1);
$employee->name = 'changed';
$this->atlas->persist($employee);
$this->assertTrue($this->connection->inTransaction());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Q: Create distance matrix using a custom similarity function I have a data frame which looks like this:
data = pd.DataFrame({'id':[1,1,1,2,2,2,3,3,3],
'age':[20, 21,18,54,23,11, 19, 18,12],
'experience':[5,4,3,8,2,11,2,8,6]},columns=['id','age','experience'])
id age experience
0 1 20 5
1 1 21 4
2 1 18 3
3 2 54 8
4 2 23 2
5 2 11 11
6 3 19 2
7 3 18 8
8 3 12 6
I am using a custom distance function called dtw_path which calculates distance between tuples. I am not going into how exactly that function calculates distance as its a complex procedure, but it just outputs a scalar distance value between the tuples.
The tuple are being formed in the following fashion:
data['age_exp'] = data[['age', 'experience']].apply(tuple, axis=1)
id age experience age_exp
0 1 20 5 (20, 5)
1 1 21 4 (21, 4)
2 1 18 3 (18, 3)
3 2 54 8 (54, 8)
4 2 23 2 (23, 2)
5 2 11 11 (11, 11)
6 3 19 2 (19, 2)
7 3 18 8 (18, 8)
8 3 12 6 (12, 6)
So for the above data frame if i need to calculate distance between id 1 and id 2, I would calculate the distance as follows:
data1 = data[data['id']==1]
data1 = np.array(data1['age_exp'].tolist())
data1
array([[20, 5],
[21, 4],
[18, 3]])
data2 = data[data['id']==2]
data2 = np.array(data2['age_exp'].tolist())
data2
array([[54, 8],
[23, 2],
[11, 11]])
dtw_path(data1,data2)[1]
1.5
What i need help is how to loop through the data frame and create a distance matrix for id column i.e. something like this
1 2 3
1 0 1.5 2
2 1.5 0 2.3
3 2 2.3 0
A: It is not clear in your question what dtw_path is. I used here tslearn.metrics.dtw_path, which gives me different results. Yet, the rationale should be the same.
Let's first reshape a bit the original dataframe:
data2 = (data.groupby('id')
.apply(lambda x: np.array(list(zip(x['age'], x['experience']))))
).to_frame()
0
id
1 [[20, 5], [21, 4], [18, 3]]
2 [[54, 8], [23, 2], [11, 11]]
3 [[19, 2], [18, 8], [12, 6]]
NB. It requires to be two-dimensional (DataFrame) for the next step, thus the .to_frame()
Then, apply your dtw_path function using scipy.spatial.distance.pdist that can take an arbitrary distance function using the parameter metric and keep only the second element of the output. Finally, reshape the output as a square matrix using scipy.spatial.distance.squareform:
squareform(pdist(data2, metric=lambda x,y: dtw_path(x[0], y[0])[1]))
output:
array([[ 0. , 35.86084215, 8.94427191],
[35.86084215, 0. , 36.7151195 ],
[ 8.94427191, 36.7151195 , 0. ]])
| {
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{"url":"http:\/\/www.ck12.org\/analysis\/Degenerate-Conics\/lesson\/Degenerate-Conics-PCALC\/","text":"<img src=\"https:\/\/d5nxst8fruw4z.cloudfront.net\/atrk.gif?account=iA1Pi1a8Dy00ym\" style=\"display:none\" height=\"1\" width=\"1\" alt=\"\" \/>\n\nDegenerate Conics\n\nPoint, line, or pair of lines formed when some coefficients of a conic equal zero.\n\nEstimated16 minsto complete\n%\nProgress\nPractice Degenerate Conics\nProgress\nEstimated16 minsto complete\n%\nDegenerate Conics\n\nThe general equation of a conic is .\u00a0 This form is so general that it encompasses all regular lines, singular points and degenerate hyperbolas that look like an X.\u00a0 This is because there are a few special cases of how a plane can intersect a two sided cone.\u00a0 How are these degenerate shapes formed?\n\nGraphing Degenerate Conics\n\nA degenerate conic is a conic that does not have the usual properties of a conic. Degenerate conic equations simply cannot be written in graphing form.\u00a0 There are three types of degenerate conics:\n\n1. A singular point, which is of the form: . You can think of a singular point as a circle or an ellipse with an infinitely small radius.\n2. A line, which has coefficients\u00a0 in the general equation of a conic.\u00a0 The remaining portion of the equation is , which is a line.\n3. A degenerate hyperbola, which is of the form:\u00a0 .\u00a0 The result is two intersecting lines that make an \u201cX\u201d shape.\u00a0 The slopes of the intersecting lines forming the X are . This is because\u00a0 goes with the\u00a0 portion of the equation and is the rise, while\u00a0 goes with the\u00a0 portion of the equation and is the run.\n\nExamples\n\nExample 1\n\nEarlier, you were asked how degenerate conics are formed. When you intersect a plane with a two sided cone where the two cones touch, the intersection is a single point.\u00a0 When you intersect a plane with a two sided cone so that the plane touches the edge of one cone, passes through the central point and continues touching the edge of the other conic, this produces a line.\u00a0 When you intersect a plane with a two sided cone so that the plane passes vertically through the central point of the two cones, it produces a degenerate hyperbola.\n\nExample 2\n\nTransform the conic equation into standard form and sketch.\n\nThis is the line .\n\nExample 3\n\nTransform the conic equation into standard form and sketch.\n\nThe point (2, 1) is the result of this degenerate conic.\n\nExample 4\n\nTransform the conic equation into standard form and sketch.\n\nThis is a degenerate hyperbola.\n\nExample 5\n\n1. Create a conic that describes just the point (4, 7).\n\nReview\n\n1. What are the three degenerate conics?\n\nChange each equation into graphing form and state what type of conic or degenerate conic it is.\n\n2.\n\n3.\n\n4.\n\n5.\n\n6.\n\n7.\n\n8.\n\n9.\n\n10.\n\nSketch each conic or degenerate conic.\n\n11.\n\n12.\n\n13.\n\n14.\n\n15.\n\nTo see the Review\u00a0answers, open this PDF file and look for section 9.6.\n\nNotes\/Highlights Having trouble? Report an issue.\n\nColor Highlighted Text Notes\nPlease to create your own Highlights \/ Notes\n\nVocabulary Language: English\n\nConic\n\nConic sections are those curves that can be created by the intersection of a double cone and a plane. They include circles, ellipses, parabolas, and hyperbolas.\n\ndegenerate conic\n\nA degenerate conic is a conic that does not have the usual properties of a conic section. Since some of the coefficients of the general conic equation are zero, the basic shape of the conic is merely a point, a line or a pair of intersecting lines.\n\ndegenerate hyperbola\n\nA degenerate hyperbola is an example of a degenerate conic. Its equation takes the form $\\frac{(x-h)^2}{a}-\\frac{(y-k)^2}{b}=0$. It looks like two intersecting lines that make an \u201cX\u201d shape.","date":"2016-08-26 21:15:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 1, \"texerror\": 0, \"math_score\": 0.6058869957923889, \"perplexity\": 865.6579547822186}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-36\/segments\/1471982296571.15\/warc\/CC-MAIN-20160823195816-00142-ip-10-153-172-175.ec2.internal.warc.gz\"}"} | null | null |
Q: find object in list i have a list of items. i then have a function to find an item in there.
def find(self, runners):
for runners in self._runners:
if runners.name == name:
return runners
so when i type runnersworld.find(peter) it should bring back a print statement such as "record found or record not found. each record has a runnerid and a runner name. but when i run this function i get a global variable name it not defined. the two fines i have are:
RunnerClass
class Runner(object):
def __init__ (self, runnerid, name):
self._runnerid = runnerid
self._name = name
@property
def runnerid(self):
return self._runnerid
@property
def name(self):
return self._name
@name.setter
def name(self, name):
self._name = name
def get_fee(self, basicfee, moneyraised):
raise NotImplementedError("AbstractMethod")
def __str__(self):
rep = "Runner: " + self._runnerid + " " + self._name + "\n"
return rep
Club Class
class Club (object):
def __init__(self, clubid, name):
self._clubid = clubid
self._name = name
self._runners = []
@property
def clubid(self):
return self._clubid
@property
def name(self):
return self._name
@name.setter
def name(self, name):
self._name = name
@property
def runners (self):
return self._runners
@runners.setter
def runners(self, runners):
self._runners = runners
def add(self, runners):
if runners not in self._runners:
self._runners.append(runners)
def remove(self, runners):
if runners in self._runners:
self._runners.remove(runners)
def find(self, runners):
for runners in self._runners:
if runners.name == name:
return runners
def __str__(self):
rep = "Club: " + self._clubid + " " + self._name + "\n"
for runner in self._runners:
rep += str(runner)
return rep
i have edited my find function to:
def find(self, name):
for runners in self._runners:
if runners.name == name:
print("Runner found in Records")
else:
print("Runner not found in record")
it seems to work but when i try to run runnersworld.find(peter) and runners4life.find(peter) i get:
Runner not found in record
Runner not found in record
Runner not found in record
Runner not found in record
Runner not found in record
Runner not found in record
but there is defininatly a peter in runners4life
A: In the following piece of code :
def find(self, runners):
for runners in self._runners:
if runners.name == name:
return runners
there's no variable name defined. You probably mean
def find(self, name):
for runners in self._runners:
if runners.name == name:
return runners
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,822 |
package lfgen.platform;
import com.interpss.common.exp.InterpssException;
import lfgen.algo.impl.LoadFlowGenerator;
import lfgen.algo.impl.QChecker;
import lfgen.algo.impl.SpecialBusChecker;
import lfgen.algo.impl.VoltageGenerator;
import lfgen.datatype.AclfCase;
import lfgen.datatype.RefInfo;
import lfgen.datatype.VoltageGenCondition;
/**
* @author JeremyChenlk
* @version 2019Äê2ÔÂ13ÈÕ ÏÂÎç1:45:09
*
* Class description:
*
*/
public interface ILoadFlowCaseGenerator {
void init();
void boom() throws InterpssException;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,226 |
Chubby Checker, född Ernest Evans 3 oktober 1941 i Spring Gully, South Carolina, är en amerikansk sångare.
Han växte upp i Philadelphia, Pennsylvania.
Chubby Checker upptäcktes i slutet av 1950-talet då han ofta sjöng för och underhöll kunder i den butik han arbetade i. Det lokala skivbolaget fick upp ögonen för honom och 1960 släpptes "The Twist", en coverversion av en låt som Hank Ballard sjungit in två år tidigare. Låten gick direkt upp på förstaplatsen på den amerikanska Billboardlistan och gjorde den nya dansen "twist" populär.
Också uppföljaren "Pony Time" gick upp till förstaplatsen i USA, medan det var nästa singel, "Let's Twist Again", som blev hans stora genombrott i Europa. "The Twist" är den enda låt som gått upp till förstaplatsen på Billboardlistan två gånger - den hamnade där en gång till 1961. Sammanlagt hade Chubby Checker ett 20-tal hits fram till mitten av 1960-talet. Förutom de redan nämnda nådde "The Fly" (1961), "Slow Twistin' " (med anonym medverkan av Dee Dee Sharp) och "Limbo Rock" (båda 1962) topp 10-placering på amerikanska singellistan.
Chubby Checker uppträder än i dag runt om i USA och har dessutom intressen i en firma som tillverkar potatischips och andra tilltugg, "Chubby Checker's Snacks".
Diskografi (urval)
Singlar (topp 10 på Billboard Hot 100)
1960 – "The Twist" (#1)
1961 – "Pony Time" (#1)
1961 – "Let's Twist Again" (#8)
1961 – "The Fly" (#7)
1962 – "Slow Twistin' " (med Dee Dee Sharp) (#3)
1962 – "Limbo Rock" (#2)
Externa länkar
Officiell webbplats
Chubby Checker på Allmusic
Amerikanska musiker
Amerikanska sångare
Födda 1941
Levande personer
Män | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,888 |
Q: How to know which operating system is suitable for my PC between 32/64-bit? I'm using 32-bit operating system since I've my laptop. I've never used the 64-bit operating system so I'm much curious about this that if I upgrade to 64-bit still my pc will give me the same performance. However I've checked about my hardware from this question. I don't know about those result that what they are saying? So I'm here for little help to know that is there any performance issue after upgrading or not?
A: In common usage, the reason to go 64-bit is to make use of more RAM (memory) in your system.
In Windows, 32-bit is limited to a little over 2GB of RAM available to programs, while 64-bit is virtually limitless. To benefit from it, your processor must support 64-bit. The D.E.P. and virtualization are not related to performance in common desktop usage.
So, since Windows 32-bit already supports a little over 3GB of RAM:
*
*If you have 3GB of RAM or less, stay with 32-bit.
*If you have 4GB or more, go with 64-bit.
A: It is unlikely that you'll notice any difference in performance when moving from 32 to 64bit.
The note about no hardware virtualization can be ignored. If you didn't miss this feature while using your 32bit operating system, you won't miss it with 64bit either.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,834 |
\section{Introduction}
\noindent The exchanges between countries become closer with the progress of globalization. As countries began to communicate more politically, economically and academically, language understanding became a new challenge. Acronyms often appear in the scientific documents of different countries. Compared to English, acronyms are more challenging to understand in other languages. Acronyms will become a barrier for researchers to read scientific literature and affect exchanges and cooperation between countries.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{4.png}
\caption{Differences and challenges between English and other (such as French) phrases in acronym disambiguation. Red means wrong, green means right. Acronyms in English are often first letter acronyms, but not in other languages.}
\label{fig}
\end{figure}
Acronym disambiguation refers to when acronyms are used in a large number of scientific documents. For these acronyms, we need to find the correct one in the current context from the dictionary. For example, in ``The traditional Chinese sentences are transferred into SC'', ``SC'' means ``simplified Chinese'' rather than ``System Combination''. It is difficult for some people who are not familiar with a language to understand related acronyms. So we need to distinguish abbreviations, which is a challenging task.
In the datasets, 30,237 data in the four fields of English (science), English (legal), French and Spanish were given. Any data contains a sentence, and there will appear a word that is the first letter abbreviated. The task hopes to find the most suitable form of an extension for the first letter abbreviation.
In the past, researchers have tried to solve AD problems by means of character extraction \cite{li-etal-2018-guess}, word embedding \cite{charbonnier-wartena-2018-using}, and deep learning \cite{jin-etal-2019-deep}. Over the last few years, the BERT \cite{devlin-etal-2019-bert} model has emerged, which adopts a method of pre-training in a large language library. Many studies have shown that these pre-training models (PTMs) have gained a wealth of generic characteristics. Recently, They \cite{Pan2021BERTbasedAD,LeveragingDomain} have achieved remarkable effects using the BERT model in AD tasks.
However, these methods do not work well in other languages. So we used the following methods to further enhance the model's out-of-data test performance to help better researchers understand and communicate multilingual multi-domain scientific documents.
\begin{itemize}
\item A simple ADBCMM approach was proposed to use other language data as counterfacts datasets in AD tasks, solving the model bias.
\item We tried to use the Multiple-Choice Model framework to make the model more focused on word-to-word comparisons to help the model better understand the first letter abbreviation.
\item Our results achieved SOTA effects in both the French and Spanish of the AD dataset, showing outstanding performance, surpassing all other baselines methods.
\end{itemize}
\section{Related Work}
\noindent In this section, we will introduce AD datasets and how to solve AD tasks in English scenarios in the past, while introducing the difficulties of AD tasks in other languages.
\subsection{AD dataset}
\begin{table}[h]
\centering
\renewcommand\arraystretch{2.2}
\begin{tabular}{c|cccc}
\noalign{\hrule height 1pt}
\textbf{Data} & \textbf{En(Lagel)} & \textbf{En(Sci)} & \textbf{French} & \textbf{Spanish}\\
\noalign{\hrule height 0.5pt}
\textbf{Train} & 2949 & 7532 & 7851 & 6267 \\
\textbf{Dev} & 385 & 894 & 909 & 818 \\
\textbf{Test} & 383 & 574 & 813 & 862 \\ \noalign{\hrule height 0.75pt}
\textbf{Total} & 3717 & 9000 & 9573 & 7947 \\ \noalign{\hrule height 1pt}
\end{tabular}
\caption{Specific number of AD datasets, including AD tasks for 4 different fields. The total number of data sets is not more than 10,000.}
\label{table1}
\vspace{-0.2cm}
\end{table}
\noindent In this AD task, the abbreviation appears in scientific documents in English and other languages. AD datasets provide datasets in French and Spanish in addition to English. Each data gives a dictionary, and each language split has its test set with acronyms not appearing in their training set.
\subsection{Previous work}
In the AD of SDU@AAAI-21, the teams presented their methodologies and submitted a total of 10 papers. Those papers included some excellent projects.
Pan \cite{Pan2021BERTbasedAD} trained a Binary Classification Model incorporating BERT and several training strategies. His program includes dynamic adverse sample selection, task adaptive pretraining, adversarial training \cite{Goodfellow2015ExplainingAH} and pseudo labelling in his paper. This model achieved its first achievement.
Zhong \cite{LeveragingDomain} took into account the field unknowledge and specific knowledge often encountered in AD tasks. He proposed a Hierarchical Dual-path BERT method to capture general and professional field language, while using RoBERTa and SciBERT to perceive and predict text. He eventually reached a 93.73\% F1 value in the SciAD datasets.
\subsection{Difficulty in multilingual}
In the AD of SDU@AAAI-22, the organizers released AD datasets covering French and Spanish, which have the following difficulties compared to the English environment:
\begin{itemize}
\item In Figure 1, we can find that the extension of other languages does not necessarily contain an acronym of the first letter, and it isn't easy to match directly through the rules.
\item Other languages lack PLMs trained in scientific language.
\item In Table 1, the number of datasets in French and Spanish is small. Training models are prone to bias and over-adaptation.
\end{itemize}
\section{Methods}
In this section, we will describe the framework for the overall model, as well as a range of methods for AD datasets for other languages, including ADBCMM, In-Trust-loss \cite{huang-etal-2021-named}, Child-Tuning \cite{xu-etal-2021-raise} and R-Drop\cite{liang2021rdrop}.
\subsection{The model framework}
We use the Multiple-Choice model framework, which is different from the Binary Classification Model used by Pan \cite{Pan2021BERTbasedAD}.
The Multiple-Choice model \cite{wolf-etal-2020-transformers} refers to adding a classifier to the end output of the BERT model. Each sentence has only a single output value to represent the probability of this option.
In Figure 2, when we use the Multiple-Choice model, each batch will enter all the possible options in the same set during the training. If the word in the dictionary is insufficient, we use ``Padding'' for filling, eventually at the output end for softmax classification and calculation of losses.
Thus, we can more accurately derive the probability that each option should be by comparing methods. Compared with Binary Classification Model, Multiple-Choice model capturing more semantic characteristics and make the model more comprehensively trained and predicted on differences, rather than the error interference model caused by the dynamic construction of negative samples.
\begin{figure*}[htb]
\centering
\includegraphics[width=1\linewidth]{9.png}
\caption{Multiple-Choice Model}
\label{fig}
\end{figure*}
\subsection{ADBCMM}
PLM has achieved excellent results in many NLP tasks, but the potential bias in training data can harm out-of-data testing performance. Counterfactually augmented datasets is a recent solution \cite{kaushik2021explaining}. But if man-built counterfactual samples, it would be expensive and time-consuming.
We found many word-like but meaning-different samples by analyzing erroneous samples on dev datasets. We think these samples errors are mainly due to model bias, over-training leads to over-adaptation seriously, and data set performance is poor. That's why we used different language markup information to use other language samples as new counterfactual samples after being modified.
In Figure 3, the training process is like a pyramid. We first train using data in multiple languages, and then we do secondary training in a single language based on pre-training.
Why continue training with single-language materials after multilingual mixed training instead of testing directly after multilingual Counterfacts datasets training? Because in our experiment, with the addition of more language samples, the models may become overwhelming. Even though French, English and Spanish belong to the Indo-European language family, they all have unique language properties, syntax and vocabulary. This would be a noise interference for different languages. Models may ignore semantic characteristics that are unique to a particular language and prefer to learn more common ones.
In addition, to address the noise problem of multilingual mixing caused by ADBCMM. We replaced the original CE loss with In-Trust-Loss. This incomplete trust loss function avoids model over-adaptation noise (other languages data) samples while trusting label information and model output. Combined with our ADBCMM method, it has achieved practical results in multilingual hybrid training scenarios.
Our ADBCMM approach can also be further extended to translation, Ner, conversation generation and other tasks. The ADBCMM approach helps address biases caused by insufficient data in small-language environments.
\begin{figure}[htb]
\centering
\includegraphics[width=1\linewidth]{10.png}
\caption{Training Process}
\label{fig}
\end{figure}
\subsection{Child-Tuning}
Because AD data sets are smaller and can easily be learned, resulting in the model's poor centralized generalization capacity during testing. We used the Child-Tuning method proposed to address this discrepancy. The Child-Tuning strategy only updates the corresponding Child Network when the parameters are updated backwards, without adjusting all the parameters. This approach like the reverse Dropout \cite{JMLR:v15:srivastava14a}, it can bring performance improvements to our models.
\subsection{R-Drop}
In the R-Drop work, the authors used the model to open Dropout during the training and then made two inputs, so the results of the two inputs would not be the same because the model opened Dropout. In addition to calculating the loss of label information, the Kullback-Leibler divergence was also calculated between the same two inputs but different outputs. This R-Drop method can play the role of normalizing and increasing robustness. In our experiment, R-Drop improved greater performance.
\begin{table*}[t]
\centering
\renewcommand\arraystretch{1.5} \setlength{\tabcolsep}{4mm}
\begin{tabular}{c|ccc|ccc}
\noalign{\hrule height 1pt}
& & French & & & Spanish & \\
Model/Method & Precision & Recall & Macro F1 & Precision & Recall & Macro F1 \\
\noalign{\hrule height 1pt}
BETO & N/A & N/A & N/A & 0.8063 & 0.7510 & 0.7777 \\
Flaubert-base-cased & 0.7796 & 0.6786 & 0.7256 & N/A & N/A & N/A\\
mDeberta-v3-base & 0.7244 & 0.6001 & 0.6564 & 0.7176 & 0.6491 & 0.6816\\
\noalign{\hrule height 0.5pt}
+ ADBCMM & 0.8087 & 0.7213 & 0.7625 & 0.8558 & 0.8236 & 0.8394 \\
+ Child-Tuning & 0.7438 & 0.6232 & 0.6782 & 0.7512 & 0.6834 & 0.7157 \\
+ R-Drop & 0.7467 & 0.6337 & 0.6856 & 0.7492 & 0.7019 & 0.7248 \\
\textbf{ALLs} & \textbf{0.8423} & \textbf{0.7712} & \textbf{0.8052} & \textbf{0.8859} & \textbf{0.8352} & \textbf{0.8598} \\
\noalign{\hrule height 1pt}
\textbf{Finally in Test} & \textbf{0.8942} & \textbf{0.7934} & \textbf{0.8408} & \textbf{0.9107} & \textbf{0.8514} & \textbf{0.8801} \\
\noalign{\hrule height 1pt}
\end{tabular}
\caption{Experimental results in French and Spanish AD datasets. BETO is a Spanish pre-training model, tested only on Spanish data in AD; Flaubert-base-cased is a French pre-training model, tested only on French data in AD; mDeberta is a multi-language pre-training model, we test in both French and Spanish. Additionally, methods including ``ADBCMM'', ``Child-Tuning'', ``R-Drop'' and ``Alls'' are fine-tuned on mDeberta models, ``Alls'' refers to using all of the above methods. In addition to ``Finally in Test'', we test the results of the Dev series. ``Finally in Test'' also uses model fusion to improve our performance.}
\label{sci}
\end{table*}
\section{Experimental Setting}
This section will subsequently present our Baseline, experimental models, experimental settings, control of variables experiment.
\subsection{Baseline}
For both French and Spanish languages, we used Flaubert-base-cased \cite{le2020flaubert} models and BETO \cite{CaneteCFP2020} cased models respectively. These models are Bidirectional Encoder Representations from Transformers \cite{devlin-etal-2019-bert}, and the size is both bases. These models have a lot of MLM training in the related large single-language repository and have SOTA results in the related languages. These pre-trained models can better capture the semantic information of words.
But there is no additional training, so the two models still need to fine-tune AD data centralization to solve AD tasks. We will add a classification layer behind these models, and then the models become Multiple-Choice Models. We trained the models in a single language. Their results will be used as our Baseline, and the results of other models will be compared with them.
\subsection{Model}
To better adapt to the ADBCMM method, we used the DeBERTa model \cite{he2021debertav3} for pre-training in the multilingual repository CC100 \cite{conneau-etal-2020-unsupervised} . The authors of DeBERTa replaced the MLM objective with the RTD (Replaced Token Detection) intent introduced by ELECTRA for pre-training.
Specifically, we used the mdeberta-v3-base model in the experiment, with a total of 280M and containing 250,000 tokens. MDeberta supports 100 languages in 100 countries, including English, French and Spanish.
Of course, to ensure that the ADBCMM method rather than the mDeberta model brought us practical performance enhancements, we also used mDeberta only in French or Spanish as a contrast experiment.
\subsection{Parameters Setup}
We used three pre-training models, including Flaubert, BETO and mDeberta, for a total of 15 training sessions. We use argmax to choose the maximum of all values as the final result for the word to be selected.
In all the experiments, we set 16 epochs and decided to use the 1e-5 learning rate (we used warmup simultaneously). We put gradient decrease 1e-5 and batch size 1 (each batch contains 14 different options). We select AdamW Optimizer. We only use the first 300 tokens for each sample. On a 10900K server with 128G memory, we used a 24G NVIDIA 3090 GPU to train our model.
\subsection{Assessment of indicators}
In AD tasks, Macro F1 was used as an assessment indicator by calculating the accuracy and recall rate of the final result.
$$Precision = \frac{TP}{TP+FP}$$
$$Recall = \frac{TP}{TP+FN}$$
$$F1 = \frac{2PrecisionRecall}{Precision+Recall}$$
$$Macro F1 = \frac{\sum_{i=1}^{n} F1_i}{n}$$
$n$ means that the higher the total number of categories, accuracy, recall rate, and MacroF1. The higher the F1 method, the better the performance.\footnote{Below is the specific meaning of the formula.
TP: The prediction is correct and the sample is correct.
FP: The prediction is wrong and the sample is correct.
FN: The predicting is correct and the sample is wrong.}
\begin{table*}[t]
\centering
\renewcommand\arraystretch{1.2} \setlength{\tabcolsep}{4mm}
\begin{tabular}{c|ccc|ccc}
\noalign{\hrule height 1pt}
& & French & & & Spanish & \\
Ranked & Precision & Recall & Macro F1 & Precision & Recall & Macro F1 \\
\noalign{\hrule height 1pt}
\textbf{Rank1(Ours)} & \textbf{0.89} & \textbf{0.79} & \textbf{0.84} & \textbf{0.91} & \textbf{0.85} & \textbf{0.88} \\
Rank2 & 0.85 & 0.73 & 0.78 & 0.88 & 0.79 & 0.83 \\
Rank3 & 0.81 & 0.72 & 0.76 & 0.86 & 0.80 & 0.83\\
Rank4 & 0.76 & 0.70 & 0.73 & 0.83 & 0.80 & 0.81\\
Rank5 & 0.73 & 0.64 & 0.68 & 0.86 & 0.77 & 0.81 \\
\noalign{\hrule height 1pt}
\end{tabular}
\caption{SDU@AAAI ranks AD tasks in French and Spanish}
\label{sci}
\end{table*}
\section{Results}
In Table 2, we can find that under the same conditions, mDeberta performs less well in French than in Flaubert-base-cased, and less well in Spanish than in BETO. We speculate that because mDeberta uses a large number of data in different languages during the pre-training phase. Still, after spinning into other languages, due to the further side focus, it may not necessarily accurately record the semantic characteristics of a single language so that the actual performance will be slightly worse compared to BETO and Flaubert. They have been pre-trained only in a single language.
Both Child-Tuning and R-Drop showed excellent performance in English and Spanish, bringing a 3-5\% F1 boost to our model. But compared to the ADBCMM method, they were still slightly underperforming. Our ADBCMM method brought more than 10\% performance boost directly to our mDeberta model. This is indeed incredible. To ensure the repetitiveness of the experiment, we repeated three experiments. The mDeberta models using the ADBCMM method were compared to their mDeberta model F1 performance over 10\% in these three experiments.
We think that ADBCMM can significantly boost our models because of the reliable Counterfacts datasets. First, they can match upstream and downstream training data; second, counterfacts datasets can reduce the model's bias, learning from more text data to more relevant information with AD tasks; third, even if the datasets are collected from different languages or fields, but they are scientific documents, so the general language training mDeberta model can learn the syntax characteristics of scientific documents in more scientific documents and further improve performance.
Finally, we followed ADBCMM-based methods and achieved SOTA scores in both SDU@AAAI's French and Spanish. In AD tasks, our methods of Precision, Recall and Macro F1 are SOTA. Remarkably, our approach leads us to the second F1 score of 5\% - 6\%.
\section{Conclusion}
In this article, we mainly talk about how to use ADBCMM in AD tasks at SDU@AAAI-22 and compare it with other Models or Methods to ultimately SOTA. We used a straightforward method to build counterfacts datasets in ADBCMM. We directly use other language datasets for training and secondary Fine-Tune in their language, which gives our models a remarkable effect. After combining the Multiple-Choice Model, Child-Tuning, R-Drop and other methods, our approach leads ahead of all different systems. Apparently, in multilingual data aggregation, simply using other languages as counterfacts datasets can improve performance. At the same time, our work provides practical help for researchers to understand scientific documentation better.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,694 |
Сеидли () — село в Агдамском районе Азербайджана.
История
Во времена царской России территория села входила в состав Елизаветпольской губернии, Шушинского уезда, магала Кабирли. Территория села в 1930 года году входила в состав Агдамского района Азербайджанской ССР.
Контроль над селом перешло к армянским вооруженным силам летом 1993 года.
После окончания вооружённого конфликта в Нагорном Карабахе в 2020 году было подписано трехстороннее соглашение и с 20 ноября 2020 года вся территория Агдамского района перешла под контроль азербайджанских ВС.
23 декабря 2020 года Минобороны Азербайджана опубликовало видеокадры из села Сеидли, где остались только руины.
География
Высота села над уровнем моря — 252 м. Село находится в 1 км к северо-востоку от районного центра, в 5 км от железнодорожной станции Агдам.
Население
По данным 1983 года население села Сеидли составляло 3478 человек.
Климат
В селе холодный семиаридный климат.
Примечания
Внешние ссылки
Эмин Мусеви нашел разрушенный отчий дом в селе Сеидли Агдамского района
Населённые пункты Агдамского района | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,150 |
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The Chennai Super Kings (CSK) are a franchise cricket team based in Chennai, Tamil Nadu, which plays in the Indian Premier League (IPL). Founded in 2008, the team plays its home matches at the M. A. Chidambaram Stadium in Chennai. After serving a two-year suspension from the IPL starting July 2015 for the alleged involvement of their owners in the 2013 IPL betting case (along with Rajasthan Royals), the Super Kings returned to the league in 2018, winning the championship in the comeback season, its third overall to tie Mumbai Indians for the record for most IPL title wins. The team is captained by Mahendra Singh Dhoni who has led the team to three victories since 2008 and coached by Stephen Fleming.
The Super Kings have lifted the IPL title thrice (in 2010, 2011 and 2018), and have the best win percentage among all teams in the IPL (61.56).They hold the records of most appearances in the IPL playoffs (nine) and in the final (seven). In addition, they have also won the Champions League Twenty20 in 2010 and 2014. The brand value of the Super Kings in 2018 was estimated at $65 million, making them the most valuable franchise in the IPL.
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