text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
{"url":"http:\/\/mathoverflow.net\/questions\/80433\/is-it-possible-to-check-two-curves-on-birational-equivalence-by-some-computer-al?sort=votes","text":"# Is it possible to check two curves on birational equivalence by some computer algebra system?\n\nI have two curves, for example hyperelliptic:\n\n\\begin{align} &y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\\\\\\\ &y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1 \\end{align}\n\nIs it possible to check them on birational equivalence (is able one curve be birationally transformed to another?) via some computer algebra system (like GAP, Sage, Magma, Maple, Maxima or something)?\n\nIt would be great if such system be free, but it is almost OK if It isn't.\n\n-\n\nI suppose Magma's IsIsomorphic will do the job.\n\nFrom the documentation\n\nIsIsomorphic(C, D) : Crv, Crv -> BoolElt,MapSch Given irreducible curves C and D this function returns true is C and D are isomorphic over their common base field. If so, it also returns a scheme map giving an isomorphism between them. The curves C and D must be reduced. Currently the function requires that the curves are not both genus 0 nor both genus 1 unless the base field is finite.\n\nExample code in the web calculator\n\nK<x,y>:=AffineSpace(Rationals(),2);\nC1A:=Curve(K,x^10-1-y^2);\nC2A:=Curve(K,x^10-2^10-y^2);\nC1:=ProjectiveClosure(C1A);\nC2:=ProjectiveClosure(C2A);\nIsIsomorphic(C1,C2);\n\ntrue Mapping from: CrvPln: C1 to CrvPln: C2\nwith equations :\n-2*$.1 32*$.2\n$.3 Wish this is implemented in sage. - Your example is a bit of a red herring, as this is relatively easy for hyperelliptic curves. A hyperelliptic curve can be reconstructed uniquely from the data of the branch divisor of the degree$2$map to$\\mathbb{P}^1$. Furthermore, isomorphisms of hyperelliptic curves commute with the degree$2$map to$\\mathbb{P}^1$. Thus for two hyperelliptic curves, the only issue is whether or not the branch divisors are projectively equivalent, and this is quite straightforward to check. - So, how do you check whether the branch divisors are projectively equivalent? My gut says that this should be easy, but I don't actually know how. \u2013 David Speyer Nov 9 '11 at 2:59 I don't have a \"smart\" way, but if the curve is$y^2 = f(x)$then the branch points are the roots of$f$. Normalize the branch points of one of the curves by sending 3 points to$0,1,\\infty$. Given any 3 branch points of the second curve, map them to$0,1,\\infty$and check if the images of the other branch points coincide with the branch points of the original curve. \u2013 Jack Huizenga Nov 9 '11 at 3:47 The problem is just the classification of binary quadratic\/cubic\/quintic\/etc. forms, up to$GL(n)$action. (We view a function on$\\mathbb P^1$as a function on$\\mathbb A^2$, and see what kind of function it is). This is a fairly standard representation theory problem, but I don't know the answer. Probably, one can show that the ring of polynomial invariants completely classifies the orbit. For elliptic curves, the$j$-invariant, which is a rational function of the coefficients, fits the bill. But the formulas for these in higher dimensions might be even nastier. \u2013 Will Sawin Nov 9 '11 at 5:00 For genus two (as in the OP's example), there are the Igusa invariants. \u2013 Felipe Voloch Nov 9 '11 at 9:44 Re Jack Huizenga's first suggestion: You have to check all triples of roots of the second branch locus, you can't just choose$3$roots of$f$and$3$roots of$g\\$. \u2013\u00a0 David Speyer Nov 9 '11 at 12:52","date":"2014-10-25 17:13:51","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\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5112670660018921, \"perplexity\": 1187.651264282453}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1414119648706.40\/warc\/CC-MAIN-20141024030048-00257-ip-10-16-133-185.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-find-the-derivative-of-a-polynomial","text":"# How do you find the derivative of a polynomial?\n\nOct 17, 2014\n\nIf\n\n$f \\left(x\\right) = {a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + \\cdots + {a}_{1} x + {a}_{0}$,\n\nthen\n\n$f ' \\left(x\\right) = n {a}_{n} {x}^{n - 1} + \\left(n - 1\\right) {a}_{n - 1} {x}^{n - 2} + \\cdots + 2 {a}_{2} x + {a}_{1}$.\n\nI hope that this was helpful.","date":"2022-10-07 16:24:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 2, \"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\": 0.7027828097343445, \"perplexity\": 675.4678046767207}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030338213.55\/warc\/CC-MAIN-20221007143842-20221007173842-00480.warc.gz\"}"} | null | null |
Q: How to make directive work with text area using AngularJS ? I have angularJS directive that is working good according to below code but here i have problem its not letting me to increase textarea rows i am not sure why its happening i spent hours but could not figure out the issue.. i would appreciate help here...
main.html
<div class="col-md-7">
<textarea rows="2" class="form-control"
ng-model="processDTO.processStatementText"
id="processStatement" placeholder="Process statement" maxlength="4000"
chars-ct-tooltip-focus="{{processDTO.processStatementText}}"
required>
</textarea>
</div>
directive.js
angular.module('App').directive('charsCtTooltipFocus', function () {
'use strict';
return{
restrict: 'A',
template: '<input tooltip tooltip-placement="top" tooltip-trigger="focus">',
replace: true,
link: function (scope, element, attrs) {
attrs.$observe('charsCtTooltipFocus', function (value) {
var modelValue = value;
var maxlength = attrs.maxlength;
modelValue = maxlength-modelValue.length+' characters left';
attrs.$set('tooltip', modelValue);
});
}
};
});
A: I have to change directive because it was set to replace true and supporting only input fields, So below directive i implemented to made it work on textarea and input both.
directive.js
angular.module('riskAssessmentApp').directive('characterCounter', function () {
'use strict';
return{
restrict: 'A',
require: '^ngModel',
link: function (scope, element, attrs, ngmodel) {
var characterCount;
element.after('<p class="character-count" style="display: none;"><span class="characters-left"></span> characters left</p>');
element.focus(function(){
element.next().show();
});
element.focusout(function(){
element.next().hide();
});
scope.$watch(function(){
return ngmodel.$viewValue;
}, function(newVal, oldVal){
if(newVal){
characterCount = parseInt(attrs.maxlength - newVal.length, 10);
} else{
characterCount = parseInt(attrs.maxlength, 10);
}
element.next().find('.characters-left').text(characterCount);
})
}
};
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,516 |
"use strict";
const shell = require('shelljs');
const chalk = require('chalk');
const PACKAGE = `nowzoo-angular-bootstrap-lite`;
const NPM_DIR = `dist`;
const MODULES_DIR = `${NPM_DIR}/modules`;
const BUNDLES_DIR = `${NPM_DIR}/bundles`;
shell.echo(`Start building...`);
shell.rm(`-Rf`, `${NPM_DIR}/*`);
shell.mkdir(`-p`, `./${MODULES_DIR}`);
shell.mkdir(`-p`, `./${BUNDLES_DIR}`);
/* TSLint with Codelyzer */
// https://github.com/palantir/tslint/blob/master/src/configs/recommended.ts
// https://github.com/mgechev/codelyzer
shell.echo(`Start TSLint`);
shell.exec(`tslint -c tslint.json -t stylish src/**/*.ts`);
shell.echo(chalk.green(`TSLint completed`));
/* AoT compilation: ES2015 sources */
shell.echo(`Start AoT compilation`);
if (shell.exec(`ngc -p tsconfig-build.json`).code !== 0) {
shell.echo(chalk.red(`Error: AoT compilation failed`));
shell.exit(1);
}
shell.echo(chalk.green(`AoT compilation completed`));
/* Creates bundles: ESM/ES5 and UMD bundles */
shell.echo(`Start bundling`);
shell.echo(`Rollup package`);
shell.exec(`rollup -i ${NPM_DIR}/index.js -o ${MODULES_DIR}/${PACKAGE}.js --sourcemap`, { silent: true });
shell.exec(`node scripts/map-sources -f ${MODULES_DIR}/${PACKAGE}.js`);
shell.echo(`Downleveling ES2015 to ESM/ES5`);
shell.cp(`${MODULES_DIR}/${PACKAGE}.js`, `${MODULES_DIR}/${PACKAGE}.es5.ts`);
shell.exec(`tsc ${MODULES_DIR}/${PACKAGE}.es5.ts --target es5 --module es2015 --noLib --sourceMap`, { silent: true });
shell.exec(`node scripts/map-sources -f ${MODULES_DIR}/${PACKAGE}.es5.js`);
shell.rm(`-f`, `${MODULES_DIR}/${PACKAGE}.es5.ts`);
shell.echo(`Run Rollup conversion on package`);
if (shell.exec(`rollup -c rollup.config.js --sourcemap`).code !== 0) {
shell.echo(chalk.red(`Error: Rollup conversion failed`));
shell.exit(1);
}
shell.exec(`node scripts/map-sources -f ${BUNDLES_DIR}/${PACKAGE}.umd.js`);
shell.echo(`Minifying`);
shell.cd(`${BUNDLES_DIR}`);
shell.exec(`uglifyjs -c warnings=false --screw-ie8 --comments -o ${PACKAGE}.umd.min.js --source-map ${PACKAGE}.umd.min.js.map --source-map-include-sources ${PACKAGE}.umd.js`);
shell.exec(`node ../../scripts/map-sources -f ${PACKAGE}.umd.min.js`);
shell.cd(`..`);
shell.cd(`..`);
shell.echo(chalk.green(`Bundling completed`));
shell.rm(`-Rf`, `${NPM_DIR}/*.js`);
shell.rm(`-Rf`, `${NPM_DIR}/*.js.map`);
shell.rm(`-Rf`, `${NPM_DIR}/src/**/*.js`);
shell.rm(`-Rf`, `${NPM_DIR}/src/**/*.js.map`);
shell.cp(`-Rf`, [`package.json`, `LICENSE`, `README.md`], `${NPM_DIR}`);
shell.echo(chalk.green(`End building`));
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,974 |
{"url":"https:\/\/physics.stackexchange.com\/questions\/201917\/time-evolution-of-expectation-value-of-an-operator","text":"# Time evolution of expectation value of an operator\n\nI'm studying QM from Sakurai, and I have a doubt regarding the proof given that in the case of time independent Hamiltonian the expectation value of an observable doesn't change with time. The argument goes as follows:\n\nAt any time the expectation value of an observable $B$ is given by:\n\n$$<\\alpha_0|U^{\\dagger}BU|\\alpha_0>$$\n\nWhere $U$ is the time evolution operator and $|\\alpha_0>$ is the ket in our initial state. Suppose that $|\\alpha_0>$ is also an eigenket of some other operator $A$ (which commutes with the Hamiltonian operator), that the Hilbert space is finite-dimensional and that the eigenkets of $A$ form a basis of the Hilbert space. We can use the explicit form of the time evolution operator:\n\n$$U=\\exp{\\frac{-iHt}{\\hbar}}$$\n\nAnd we know that it acts on the eigenkets of $A$ by multiplication by a phase. Sakurai then writes this:\n\n$$<\\alpha_0|\\exp{\\frac{iE_{\\alpha_0}t}{\\hbar}}B\\exp{\\frac{-iE_{\\alpha_0}t}{\\hbar}}|\\alpha_0>$$\n\nMy doubt is: the observable $B$ could send the ket $|\\alpha_0>$ in a superposition of eigenkets, and then the simple multiplication by $\\exp{\\frac{iE_{\\alpha_0}t}{\\hbar}}$ wouldn't hold anymore. What am I missing?\n\nThe left exponential evolves the $\\langle \\alpha_0 \\lvert$ on the left.\nThis is one of the pitfalls of Dirac notation, it would be unambiguous to write $$(\\mathrm{e}^{-\\mathrm{i}E_{\\alpha_0} t} \\lvert \\alpha_0 \\rangle,B \\mathrm{e}^{-\\mathrm{i}E_{\\alpha_0} t} \\lvert \\alpha_0 \\rangle)$$ where $(\\dot{},\\dot{})$ denotes the inner product on the Hilbert space, i.e. $(\\lvert\\psi\\rangle,\\lvert\\phi\\rangle) = \\langle \\psi\\vert\\phi \\rangle$.","date":"2020-05-31 14:53:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.9457998275756836, \"perplexity\": 122.8258910967718}, \"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-2020-24\/segments\/1590347413406.70\/warc\/CC-MAIN-20200531120339-20200531150339-00200.warc.gz\"}"} | null | null |
using System;
using System.Linq;
using System.Linq.Expressions;
using System.Threading.Tasks;
using FluentAssertions;
using GitLabApiClient;
using GitLabApiClient.Models.Issues.Responses;
using GitLabApiClient.Models.MergeRequests.Responses;
namespace GitLabCLI.GitLab.Test
{
public static class GitLabApiHelper
{
private static readonly GitLabClient _client =
new GitLabClient("https://gitlab.com/api/v4", "KZKSRcxxHi82r4D4p_aJ");
public static GitLabClientFactory ClientFactory { get; } = new GitLabClientFactory(new GitLabSettings
{
GitLabAccessToken = "KZKSRcxxHi82r4D4p_aJ",
GitLabHostUrl = "https://gitlab.com/api/v4"
});
public static string ProjectName => "txxxestprojecxxxt";
public static int ProjectId => 4011625;
public static string CurrentUser => "nmklotas";
public static string NonExistingProjectName => Guid.NewGuid().ToString();
public static async Task ShouldHaveIssue(
int issueId,
Expression<Func<Issue, bool>> predicate)
{
var issue = await _client.Issues.GetAsync(ProjectId, issueId);
issue.Should().NotBeNull($"Issue {issueId} does not exists");
issue.Should().Match(predicate);
}
public static async Task ShouldHaveMergeRequest(
int mergeRequestId,
Expression<Func<MergeRequest, bool>> predicate)
{
var mergeRequests = await _client.MergeRequests.GetAsync(ProjectId);
var mergeRequest = mergeRequests.FirstOrDefault(s => s.Iid == mergeRequestId);
mergeRequest.Should().NotBeNull($"Merge request {mergeRequestId} does not exists");
mergeRequest.Should().Match(predicate);
}
public static async Task DeleteAllMergeRequests()
{
var mergeRequests = await _client.MergeRequests.GetAsync(ProjectId);
await Task.WhenAll(mergeRequests.Select(
m => _client.MergeRequests.DeleteAsync(ProjectId, m.Iid)));
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 31 |
Together Through Adversity
With COVID-19 disrupting my normal everyday life, the emergence of racism towards Asians along with the Black Lives uprising over the past couple of months, it has been a battle to find any positivity in the middle of all of this. Despite this, these times have also helped me foster much growth and learning about the power of collectivism and solidarity. It has taught me about the importance of organizing against the racism that threatens our communities, how we can show eachother human decency by taking precautions to protect each other, and how it is crucial to evaluate and challenge one's own role in upholding racist institutions and ideologies. To embody the changes that have occurred in our social discourse and to inspire social change, I created this piece portraying the collective story of my people, the Hmong people, with the purpose of reminding us of the importance of sticking together in the face of adversity. I wanted to spark meaningful reflections in others about how we as a society can combat the pandemic and rampant racism by standing in solidarity with each other's communities who are suffering from the effects of the virus and are also facing injustice at the hands of colonialism and White supremacy.
I chose to focus my piece on this theme because I wanted to convey how human beings no matter our differences in race, ethnicity, religion, sexual orientation, or gender can persevere through any hardship together. Just like how the Hmong people have overcome much adversity throughout the course of history, together we will overcome the pandemic and racism.
In my drawing, I bring to life the stories of the Hmong people and their journey of leaving their agrarian lifestyle to escape persecution and seek asylum in America, which they now call home.
From soldiers of the Secret War to refugees in America; politicians, Olympic gymnasts, and artists representing us; health care workers on the frontlines and activists supporting Black Lives, the Hmong people in America have come a long way. Only with perseverance, change, and creativity can we continue to reimagine and build a better world.
By Alex Yang
18, Lino Lakes, MN
Alex Yang is an aspiring artist and creative storyteller. His work consists of drawings and paintings that depict themes of cultural identity. Typically, he draws and paints in a semi-realistic style and incorporates aspects of Hmong culture to convey the experiences of being a Hmong-American and minority. The purpose of his art is to reflect on the significance of his culture in his life as well as to empower others who empathize with the themes portrayed in his art.
@manchamoon
facebook.com/manchamoonart
@Alex-Yang-87
Amplify Youth Voices: Youth Power Rising
This project is commissioned content for a digital campaign organized by Youthprise in partnership with Juxtaposition Arts, Brooklyn Bridge Alliance, and The SEAD Project. The campaign exists to value and amplify young folks: perspectives, creativity, and thought leadership. We are promoting youth voice as we believe there is wisdom and hope needed to address the dual pandemics of Racism and COVID-19 and their unique impacts on BIPOC communities in Minnesota. Learn more at: youthprise.org/youthpowerrising. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,574 |
Just one example is your ability to combine your busy agenda of travel and work commitments in a way that maintains productivity. We're committed to offering an inclusive workplace that's engaging and which helps you personally develop, setting you up for success as you mentor and grow teams and capabilities. In fact, we believe that by optimizing the environment you and your team work in, as well as the experiences you have, you can lead the organization's transformation and help your team reach operational excellence as well.
What you also encounter as a leader at Philips is a significant element of the unexpected. You may be given the opportunity to drive a new business. To transform an area of the organization you hadn't considered before. To provide coaching and mentoring. Or it may be that you discover the impact of the work that you do is far greater than you could ever have imagined. Regardless of how your legacy develops, working at Philips will give you the platform you need to showcase your true value.
"At Philips, our starting point is always to understand the specific challenges local people face – whether they be a hospital director, a doctor, a nurse, a patient, a consumer, etc."
"Successful executives share one thing you need to know about being a woman in Australian business"
"Philips' Chief Market Leader, Ronald de Jong, talks about healthcare transformation." | {
"redpajama_set_name": "RedPajamaC4"
} | 2,968 |
Praise for SCALE
"David and Jeff, as highly successful business owners of multiple businesses, not only reveal novel ideas that generate huge multiples but also give you tools of _how to think_ so you can invent and design actionable processes for _your_ business. By focusing on best practices from these top business leaders, you too can generate _maximum_ results for your business and your life."
—STEPHANIE HARKNESS, entrepreneur and former chairman of the National Association of Manufacturers
" _Scale_ holds the answers to achieving your business goals. If you are tired of fearing your business will fall apart the minute you step away from the wheel, Hoffman and Finkel know how to take it to the next level."
—SCOTT DUFFY, author, _Launch!_
"I've been building businesses for over thirty years and this is the book I wish I'd read years ago. Jeff and David have laid out a concrete framework to scale a company, along with the practical details to create and sustain rapid growth. Every business owner needs to read this book."
—PATTY DEDOMINIC, entrepreneur and past chairman of the Los Angeles Area Chamber of Commerce
"Rapid growth without a good plan can kill a business faster than no growth at all. This book will show you how to grow smart!"
—JOHN JANTSCH, author, _Duct Tape Marketing_ and _Duct Tape Selling_
"Jeff and David are big-game hunters in a modern business sense. Instead of guiding clients in the art of bagging animals, they stalk and slay fears and indecision. _Scale_ is a competitive weapon! This book gives you exactly what you need to succeed in the big game."
—PHILIP and GAYLE TAUBER, founders, Kashi Company
"Wow! _Scale_ gives you a step-by-step process to grow your business the right way. I've mentored students from some of the best business schools in the nation, including Harvard, USC, and UCLA, and I would give this book to everyone. In fact, I would tell every business owner that they need to read this book. Understanding _Scale_ will save you from reviewing twenty-five other business books."
—SANDY GOOCH, cofounder, Mrs. Gooch's Natural Food Market (later acquired by Whole Foods)
"With so much global attention on founders and start-ups, this is a refreshing read on innovative approaches to scaling companies and reducing their owner dependency. Scale is a must-read for anyone interested in founders and scaling firms."
—JONATHAN ORTMANS, president, Global Entrepreneurship Week, and senior fellow, Kauffman Foundation
"If you're tired of struggling and spending every waking moment consumed by your business, then this is the book for you. Learn how to build a successful and sustainable business that has value without your being involved every minute. Consider the authors your personal mentors and transform the way you think about your business."
—SUSAN SOLOVIC, _New York Times_ best-selling author, award-winning entrepreneur, and media personality
" _Scale_ is a masterwork that encapsulates enough practical wisdom to fill several books. The authors had the experience and talent to craft an exquisite cross-section of the why and how of sales, strategy, and operations in a fresh format for founder-builders. They've distilled the essence of complex management situations into practical summaries, with plenty of anecdotes to bring them to life. This book will ignite years of successful growth for any reader who applies what they share."
—ROB KAUTZ, private investor and strategist, former CEO of Wolfgang Puck Worldwide
"Most owners think that in order to grow and scale a business, they need to work harder and sacrifice their personal lives. Here's a simple road map that will put you on a growth trajectory by working less—a paradox that you'll come to embrace by following this insightful book's path to business success and personal freedom."
—BILLIE DRAGOO, president, National Association of Women Business Owners
PORTFOLIO / PENGUIN
Published by the Penguin Group
Penguin Group (USA) LLC
375 Hudson Street
New York, New York 10014
USA | Canada | UK | Ireland | Australia | New Zealand | India | South Africa | China
penguin.com
A Penguin Random House Company
First published by Portfolio / Penguin, a member of Penguin Group (USA) LLC, 2014
Copyright © 2014 by Jeffrey Hoffman and David Finkel
Penguin supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin to continue to publish books for every reader.
The following trademarks are the exclusive property of Maui Millionaires, LLC and are used with permission: Self-Employment TrapTM and Sweet Spot Analysis. The following trademarks are the exclusive property of Maui Mastermind (NV), LLC and are used with permission: Maui Mastermind® and Level Three Road MapTM.
Charts and graphs courtesy of Maui Mastermind®
Hoffman, Jeff.
Scale : seven proven principles to grow your business and get your life back / Jeff Hoffman and David Finkel.
pages cm
Includes index.
eBook ISBN 978-0-698-15558-9
1. Small business—Growth. 2. Corporations—Growth. 3. Strategic planning. 4. Business planning. 5. Entrepreneurship. I. Finkel, David. II. Title.
HD62.7.H6274 2014 658.4'06—dc23 2014020638
Version_1
JEFF:
To my amazing family, who have endured and supported me throughout the high-speed roller-coaster journey I dragged them on by choosing to be an entrepreneur
DAVID:
To my wife, Heather, and my sons Adam, Matthew, and Joshua—thank you for making my life so much richer
## CONTENTS
Praie for SCALE
Title Page
Copyright
Dedication
Introduction
PART I
Building on a Solid Foundation
PRINCIPLE ONE
Build a Business, Not a Job
PRINCIPLE TWO
Build on the Scalable Base of Systems, Team, and Internal Controls
PRINCIPLE THREE
Understand Why Your Customers _Really_ Do Business with You
PART II
Focusing on Fewer, Better Things
PRINCIPLE FOUR
Create the _Right_ Strategic Plan
PRINCIPLE FIVE
Learn to Read the World So You Build for Tomorrow's Marketplace
PART III
Obstacles to Scaling (and How to Overcome Them)
PRINCIPLE SIX
Remove the Predictable Obstacles to Growth— Pillar by Pillar
YOUR SALES/MARKETING PILLAR
Build Scalable Lead-Generation and Conversion Systems
YOUR OPERATIONS PILLAR
Three Breakthrough Ideas to Scale Your Capacity
YOUR FINANCE PILLAR
CFO Secrets to Manage Cash Flow, Improve Margins, and Fund Growth
YOUR TEAM PILLAR
Attracting, Retaining, and Unleashing Talent
YOUR EXECUTIVE LEADERSHIP PILLAR
Alignment, Accountability, and Leading Your Leadership Team
PART IV
You DO Have the Time
PRINCIPLE SEVEN
You _Do_ Have the Time to Scale Your Company 229
PUTTING IT ALL INTO ACTION
Acknowledgments
Appendix: Seven Online Resources to Help You Scale
About the Authors
Index
## INTRODUCTION
In 1992, a 22-year-old undergrad dropped out of college to launch his own business. He wanted to be his own boss and decided to build a company selling health products. His sisters told him he was a "*$%*&^@" idiot, and his befuddled parents asked him softly if he wouldn't rather finish his degree first. But he was committed to make this, his first real attempt at formally building a business, a massive success. He bought the wholesale rights to a line of health products and invested his life savings ($3,200) in stocking inventory and setting up a small office, then spent long hours marketing the line, personally posting thousands of flyers and making hundreds of sales calls.
Eight months later, the business had failed; his family graciously never said " _I told you so_." But the story doesn't end there. The good ones never do.
After taking some time to lick his wounds and salve his bruised ego, our now 25-year-old recent college grad made another go in a totally different business. His earlier failure had taught him some crucial lessons (including a sorely needed dose of humility). Cataloging his talents and interests, he decided on a new industry—info-marketing. He made a list of the top ten business leaders in this industry and approached each one, asking for the opportunity to interview them about how they had built such successful companies, and asking for any advice they could give to someone just starting out. Initially, these big players ignored him, but he was persistent, hounding them until he finally got eight of the ten to meet with him. In their interviews, these leaders spoke candidly about the lessons they had gleaned after decades in the business world, climbing the same peaks our young businessman wanted to scale. He took their advice to heart, following up on skills and topics they said he would have to understand to be successful. He read dozens of business books and made a study of how to successfully build a business.
Eight years later, he was an "overnight" success, the owner of one of the top real estate training companies in the United States, having trained more than 50,000 investors who went on to buy and sell over $1 billion of properties. With no debt or outside investors, his company generated over $3 million of annual operating profit and was valued at $10 million to $16 million. After he sold this company, he went on to found and build half a dozen other successful companies, each time leveraging the lessons and experiences from his earlier businesses.
In 2009 a mutual friend introduced him to a business superstar—a guy who had helped launch and grow one of the fastest-scaling companies of all time, Priceline.com, which had grown from $0 in sales to over $1 billion _in less than four years_. When the two of them met in Atlanta, they talked for hours, realizing they had a lot in common—especially their shared passion for mentoring and coaching other business owners. One had built several multimillion-dollar companies and personally trained more than 100,000 business owners; the other helped build a multi _billion_ -dollar company that revolutionized an industry. But they both loved coaching and mentoring business owners to scale their companies.
You may have already guessed that the college dropout turned successful entrepreneur was David, and the Priceline.com superstar was Jeff. Like David, Jeff launched his first start-up, a software company, while still in college. Unlike David, Jeff and his team nailed it with their first company, which they later sold to American Express for millions. Priceline.com was actually Jeff's fifth start-up, and after the company went public and Jeff transitioned out of the business, he went on to be the CEO of uBid.com and RedTag.com. He even launched a very successful entertainment company, Black Sky Entertainment, which went on to generate over $100 million in sales.
We've both lived the life of serial entrepreneurs, not just launching companies, but leading them through those all-important middle years when a business must either stay stuck as an extension of the founder, or scale and grow independent of the owner. Over the years, we've grown our friendship and shared the stage teaching at various business conferences. During that time, we came to realize that we shared a complementary skill set and approach to growing companies. And we also learned that we had shared experiences of those scary, absorbing, and exciting years when these businesses began to grow at triple-digit rates or faster. We joked that at times it felt like we were on the back of a rodeo bull, holding on with all our might, praying for that eight-second buzzer to sound. The thing we discovered is that whereas in the rodeo those eight seconds are literal, in your business the buzzer doesn't go off for _years._
We also realized that we shared a frustration with the existing business books that addressed scaling a company. Too many told readers what to do, but didn't have the details on _how_ to do the what to do. That's why we decided to pool our years of experiences to write this book. We wanted to put in one place a structured, systematic approach not just to create growth, but, more important, to show you how to survive and enjoy that growth by leveraging the systems, team, and internal controls necessary to sustain that development over the long term. Following this methodology, not only do you get growth in sales and profits, but you'll also enjoy increased freedom as the business owner. You can build a company that is stable, vibrant, valuable, _and_ a joy to own. This book contains the guiding principles and the concrete formulas you can apply to consistently create the dual results of business growth and personal freedom.
We are primarily serial entrepreneurs, not authors. We love building businesses, and have been fortunate to enjoy some big successes, and to have survived the stupid, messy mistakes that, looking back, were both predictable and avoidable. While we wouldn't trade our paths, we wanted to catalog and share our real-world lessons about how to do it better, smarter, and faster. This is the book we _wished_ we had read when we were struggling to grow our earlier businesses.
Everything you read here has been tested and validated. It is proven to work; we've used it to build more than a dozen successful companies with combined sales in the billions. More important, over the past ten years, we've taught these ideas to more than 100,000 business owners around the world, and the results they've gotten have proved the concepts, strategies, and tools are transferable and get results. We wrote it so that you can read any chapter, put down the book, and immediately apply several of the key ideas the very next day in your business—generating immediate results.
The ultimate goal for a business owner is to build a company that he or she can one day sell, continue to scale, or even own passively. Most business owners reach for growth by working harder and personally trying to produce more. This is a flawed model that at best will lead to only moderate growth. At worst, this strategy can literally put your entire company at risk.
Instead, we will introduce you to the Level Three Road Map, a comprehensive model for building and scaling your business. Essentially the Level Three Road Map will help you go from a Level One business (a start-up), through Level Two (an owner-reliant company), to Level Three (a rapid growth or "exit stage" company).
The bottom line is that scaling your company and reaching the ambitious goals that you really want requires that you move beyond an "owner-reliant" business to one that is systems driven, with a solid management team and intelligent business controls in place—a Level Three business.
Overview of the Book
First and foremost, this is a book about rapid growth, and how you can build a scalable business that not only consistently grows, but also sustains that growth over time. What you'll come to learn is that a direct by-product of building your business the right way is that you'll get your life back. It is a broken model that says you must work 80-hour weeks and sacrifice everything to grow your company.
You don't have to choose between your business and your life. Rather, if you do it right, you get your business _and_ your life. In fact, we strongly believe that the only way to build a truly scalable business is by radically reducing its reliance on you, the owner, and supporting that rapid growth through systems, team, and internal business controls.
Part one of the book focuses on building a solid foundation. You'll learn the essence of the Level Three Road Map. This structured formula to scale your company not only will help you create and sustain rapid growth, but at the same time will help you dramatically reduce the company's reliance on you. It closes with a simple tool to help you clarify your business context—the market you serve, the competition you face, and the position you want to claim in the market. We'll approach all three of these key dimensions of your business context in a radically different way than it's likely you have ever done before.
In part two, you'll identify the fastest leverage points to grow your company and how to turn those leverage points into a winning business strategy. What's more, we'll give you our step-by-step process to create a clear, actionable, one-page plan for executing your strategy on a rolling, 90-day basis. You'll also learn how to read the rapidly changing world around you in order to continually keep your business fresh and relevant to the marketplace.
In part three, we'll walk you through how to overcome the predictable obstacles to scaling, pillar by pillar. From lead generation to lead conversion, operations, finance, HR, and leadership, we'll give you proven solutions to dozens of the toughest challenges you'll face as you scale your company. You'll learn how to build "expert systems" to pull out the key know-how that currently is locked in the heads of a few, expensive employees and replicate that expertise in scalable systems; systematically grow your sales; manage your cash flow; fund your growth; and create a culture of accountability inside your company.
In part four, you'll find out how to leverage your time to scale your company. We'll give you a brand-new model to think about time, and six simple time mastery strategies that will help you upgrade eight hours a week of time you are _already_ working for higher and better uses to grow your company. We'll also give you a clear plan to implement the ideas from the book in a manageable way.
Whether you're a small, 3-person retailer doing $350,000 a year in sales, or a 22-person service business with annual sales of $2.5 million a year, or even a 255-person manufacturing company with $26 million a year in sales, this book will give you clear, actionable insights to grow and scale your company. Not only will the systems we share help you sustain long-term success, but you'll also see fast results.
The road map we've laid out for you to follow will enable you to scale your company the way you've always dreamed you could. We hope you'll reach out to us after you read this book, and share your own stories of success. (Our contact information is at the end of the book.)
7 Free Online Tools to Help You Grow Your Business
* * *
Because we know how important executing on these ideas is to help you enjoy the growth you want, we created a special Web site with seven tools to help you get results. This free value-add for readers like you includes downloadable PDF versions of all the strategy tools shared in this book, along with dozens of valuable video training sessions to help you scale your company and get your life back.
To get immediate access to these tools just visit www.Scale YourBusinessToolkit.com. (See the appendix for full details of all seven valuable tools.)
## PART I
## Building on a Solid Foundation
Sometimes it feels more like my business owns me than me owning my business. I'm reasonably intelligent and self-motivated. I'm willing to put in the work to grow it, I just need the right road map to show me the best path to grow my company and get my life back.
—Comment by anonymous business owner at recent conference
## PRINCIPLE ONE
BUILD A BUSINESS, NOT A JOB
The year was 2007 and Tom was burning out. He was running a successful wholesaling business in Florida, but the 80-hour workweeks were getting to be too much. He had two young kids at home whom he wasn't seeing enough, and he felt torn between the demands of the business and being present with the important people in his life.
And if Tom was tired, so was his wife, Lee. She was tired of Tom missing out on family dinners, not being more involved in family activities, and of watching Tom's long hours impact his health.
Tom and Lee knew they needed to find a way to get out from under the crushing pressures of running the business day-to-day, but they didn't know how. Nor did they want to lose the momentum in the business. They felt the same way so many of us do—trapped by our business, like Atlas, holding the world on our shoulders, afraid to take a step back to reevaluate our position for fear that the whole thing could come crashing down around us. We dream of a way to grow our businesses without sacrificing our lives. But for Tom and Lee, as for a lot of us, it just wasn't obvious _how_ to do this.
So Lee went online to research potential solutions. That's when she learned about a business workshop we were hosting in Arizona to raise money for several nonprofit charities. The goal of this workshop was to teach business owners how to design their businesses both for growth and for reducing their reliance on the owner. Lee wanted to encourage Tom to continue to grow the business but to do so in a way that allowed him to spend more time with the family and didn't take such a harsh toll on his health. She bought him a ticket to the event and arranged his travel.
That workshop taught Tom many of the same tools, strategies, and principles that you'll be learning in this book. When Tom returned home, he immediately began implementing what he had learned and watched as his business blossomed. He also saw his family and his personal life bloom as well, since he finally learned how to manage them both together. Our goal in this book is to help you achieve that same balance of business growth and personal freedom.
Prior to the workshop, Tom thought his business needed him to oversee each and every detail. He believed he had to be there every moment of the day to drive his team forward to produce results. We introduced him to a new approach to scaling his company, one that he could implement in his business in bite-sized chunks that fit in around his responsibilities running the day-to-day. Quarter by quarter, he watched his business grow and its dependencies on him shrink. By 2013 not only had he scaled his $5 million a year wholesaling business into a $23 million per year venture, but he had reduced his hours in the business to under 20 per week.
We share Tom's story with you not because he is special (he is a very nice guy with a lovely family, but he doesn't walk on water), but because he is likely a lot like you. You're driven to succeed but feel stuck as to the best way forward. You carry the weight of your business on your shoulders without complaint and are the linchpin that holds everything together. While you work hard to grow your business, what you may not have realized is that one of the most powerful chains holding you back from succeeding on the scale you want is the way you've designed your business in the first place. Your business's heavy reliance on you, which may have been necessary when you first launched your business, has become a major weakness.
Why is it that some owners can move from one successful company to another, seemingly with a golden touch? Because they have learned to master the critical methodology of building a successful business. It's like the football coach who keeps winning everywhere he goes. He wins not because of the team he has at each new organization, but because he has a _system_ for building winning football programs. While every business is different, just like every team, it's our belief that the commonalities of building a successful business far outweigh their specialized differences. What's more, you likely already know enough of the specialized idiosyncrasies for your industry. If you just mapped them onto a better model of building a business, you'd soon enjoy a radically enhanced success with your business.
Still, there are going to be many owners who say, "But my business is different. It's special." What they never realize is that by asserting their business's specialness, they've locked themselves into being involved in every detail of that business. Their belief that their business can't be weaned off its reliance on them since it is so specialized, complicated, or unique is one of the most expensive limiting beliefs they could ever own. It literally costs them millions of dollars of lost growth. What's more, it also costs them their freedom as they become trapped in the very business they once launched to help them become free. If you think your business is unlike any other company out there, you miss the chance to learn from what other successful companies and business owners have spent years learning and proving. Imagine a football team that had a chance to be coached by the winningest football coach of all time, but instead says, "We don't want your help because you coached other teams, not ours."
So why would a business owner turn his back on a proven methodology to sustainably grow his company, and instead lock himself into an expensive cycle of solo trial and error? Generally, it's because he equates reducing his business's reliance on him with a loss of control. He fears that by his letting go of the "control," major disasters will strike and the business will be irreparably harmed. The irony is that the more the owner feels like he is in "control," the more he isn't. Although the superficial reins of power may be his, at the core he doesn't own the business—the business owns him.
Take Sandy's example. She owned a very successful company that worked for large corporations planning and executing major promotional events and campaigns. When two of her key employees had family emergencies, she was the one who had to step in and fill in all the gaps. A third key employee went away on a previously planned vacation right in the middle of that business storm, leaving Sandy the sole manager left behind to deal with the stress and strain of the massively understaffed company.
It probably won't surprise you that when we asked Sandy what she would do if faced with the same family emergencies as these two key employees, Sandy said she would find a way to work _and_ handle the emergency. As for what she would have done if she had been the one with the vacation planned for this time, well, she didn't even hesitate—she'd cancel the vacation to deal with the business crisis. When you centralize all the control on you, your business doesn't have the depth and strength to operate without you there. You've built a business—and backed yourself into a corner.
The "Hit by a Bus" Test
We want to ask you an extremely important question, one that very few business owners ever allow themselves to consider because often the answer is too painful to contemplate:
If you were hit by a bus tomorrow (or otherwise incapacitated), what would happen to your business?
We surveyed more than 1,000 business owners over the past five years, and our findings revealed that if the average business owner became incapacitated and couldn't work, their business would fail in less than 30 days. Thirty days! Think about what this would mean for their families, employees, and customers. You put in all those years of blood, sweat, and tears to build something that could literally end in an instant.
Even if you are blessed with the health and good fortune never to be hurt or unable to work, that doesn't mean you're home free. The unfortunate reality is that most business owners build a job for themselves, not a business. The businesses they are creating are dependent on their showing up to run them, day in, day out. They have built what we call owner-reliant businesses, with long hours, no real freedom, and no defined exit strategy
Even "successful" businesses have to deal with this challenge. In fact, many owners of successful companies are so busy doing the day-to-day _job_ of their business that they don't have the time or energy left to grow and develop it as a business. Here you are, having done all the hard work of growing sales and production, yet you're likely still so consumed by managing the day-to-day operations that you don't get to truly enjoy all the fruits of your labors. If you're not present each day, your business suffers, in many cases grinding to a halt.
Are you really free if—even as the "business owner"—you don't control your time and what you do with it? Isn't that the entire reason you became a business owner, to control your own schedule and enjoy the time and freedom to do what you wanted, when you wanted? Of course it is.
Furthermore, you simply cannot safely scale your company if you are the critical lodestone for your business. In order to scale your company you will need to build a business, not a job.
Escaping the Self-Employment Trap
All businesses start at Level One (start-ups), feeling their way forward to launch their new venture. Those that survive reach Level Two (owner reliant). It's here that most companies get stuck. A Level Two business is a business that works, but only because you, the business owner, are there every day to make it work. _You_ make most of the decisions. _You_ generate most of the business. _You_ meet with all the key clients and perform most of the important work of the business. _You_ stay in full control. Sure, you have people to help, but they're there to do just that—help—not to lead or take ownership of central parts of your business. The core knowledge of how to manage and direct the business is locked up in your head, and if something should happen to you, your business would crumble. Even if you manage to somehow escape for a short vacation, you probably sneak your iPad or smart phone with you on the trip and check e-mail when your family isn't looking.
The painful reality is that most Level Two business owners get caught in the Self-Employment Trap. They're so busy doing the "job" of their business that they can't step back and focus on growing their business. What's more, because of the way they are building their business, the more success they have, the more trapped they become inside their company.
So what's the way out of the Self-Employment Trap? You've got to work less and get your _business_ to produce more. Remember, the more you do, the more you have to keep doing. The more you get your business to do, the more time you have to grow and build your business. This means building your business with the end in mind, the end being the day when it is no longer reliant on you the owner. We call this type of business a Level Three business.
In the early years of your business, you're naturally the main engine driving your business forward. You'll wear all the hats at various times, and you'll have few formal structures and systems within your organization upon which you can truly rely. But as the business matures, you become more confident you'll generate consistent sales and ensure that your business stays profitable. At this point, you've got a Level Two business.
As you enter Level Two, you'll face a crucial decision point at which you can settle for owning a Level Two company or instead choose to raise your business to be a strong and independent entity that benefits from your involvement but is ultimately independent of it.
This isn't something that "hard work" alone is going to solve. Blindly working hard is part of the problem. The more your growth is based on your personal production, the more dependent your business becomes on you for that production. You've got to make sure that even in the midst of meeting the daily demands of your business, you take some of your energy and invest in the systems, team, and internal controls that will allow you to scale your business beyond just you.
The Paradox of Freedom
We first spoke with Mark Huha in June 2011. He and his wife, Diana, owned a successful $750,000-per-year service business in Oceanside, California. For close to 15 years Mark had worked hard growing the company, commonly putting in 70-hour weeks. He was fast burning out.
We worked with Mark and Diana to apply the same ideas you're learning about in this book to their company. They got clear on their business strategy. They focused on fewer, better activities and cut out the lower-value distractions. They worked with their team to introduce systems and controls into their daily operations, which freed Mark up to focus on generating more sales. This in turn allowed them to bring on more team members, who now were immediately enfolded in a systems-driven culture of accountability and contribution. This sparked another dramatic jump in their business. And each quarter, Mark, Diana, and their key team members stepped away from their business for an off-site planning session to evaluate their progress, brainstorm ideas, and map out the coming quarter.
How did all this pan out for Mark and Diana's business? In a conversation we had with him in June 2013 (24 months after our first phone meeting), Mark shared, "We've grown by over 44 percent in the past two years to sales of over $1.1 million. This is literally ten times faster growth than in the prior 15 years we've had the business. But the thing that amazes me is how well my team is using and building our systems to create results. I've cut my hours back to under 35 hours per week and the business is doing better than ever."
We share this example with you to illustrate a point: Working harder isn't the answer; it's often a big part of the problem. To most owners, working harder equates with personally producing more for their companies. But the more they immerse themselves in the day-to-day operations, trying to produce more, the more they become the limiting factor for the business. The world doesn't pay you for the hours you put in; it pays you for the value you create.
Putting in lots of extra hours may indeed spark a short spurt of growth, but eventually this caps out because you simply can't work any harder, nor can you personally produce any more. This is why the first principle of scaling your company is to understand that the less you do, and the more you get your business to do, the faster the business will grow. Equally important, the easier it will be to sustain that growth over the long term.
There is a time and a place when the hard work and the dedicated, passionate hours of the owner can drive the business forward, but it is just that—a time and a place. The goal of owning a business is not to be needed, but in fact the opposite: to build a business that doesn't need you. When you reach the day when you can be away from the business for a period of time and no one calls or e-mails asking for your help, you've achieved your goal and have built the well-oiled, efficient machine that can continue to scale organically.
The Level Three Road Map
Successful companies go through predictable stages of development. We've refined our observations of hundreds of successful companies—our own and others—into a concrete model that we call the Level Three Road Map. The Level Three Road Map (Figure 1.1) is the defined pathway through the complete life cycle of your business from launch to exit. It takes you through three distinct levels, from the chaos of the start-up, through the owner-reliant years, to the exit-stage phase. Remember, the finish line is not to just be profitable; the real finish line is to build a truly scalable business that creates massive value in the marketplace _without_ needing you to be there every day to run it. Our goal in this book is to get you and your business to Level Three.
FIGURE 1.1: THE LEVEL THREE ROAD MAP
Level One: Pre-Launch Start-up
Focus: Planning your business, raising capital, and getting initial market feedback to see if it's viable.
At Level One, you're designing and planning your start-up. You're gathering your initial team, raising any required start-up capital, and executing your launch plan. Your focus at Level One is to plan your new business and get immediate market feedback to learn if your concept and model are economically viable.
Typical Level One business owners are filled with a mixture of doubts and dreams, fears and ambitions. They work long hours, scrambling to turn their business idea into a tangible, practical, cash-flowing enterprise. And they do all of this with no certainty that the business will be profitable, let alone sustainable.
Level Two Early Stage: Post-Launch Start-up
Focus: Making those early sales and learning to produce and fulfill on its core product or service so that the business reaches profitability.
Fresh in the marketplace, an Early Stage Level Two business has just started actively marketing and selling its products or services. This is the time to learn your business and market, and if needed, discover and fix any fatal flaws in your business model or how your targeted customers perceive the value you're creating.
Your early focus while launching a business isn't on building the perfect product or service, but rather on figuring out how you can get people to buy. Too many entrepreneurs get caught in the trap of making the perfect widget, but never actually sell enough of that widget to ensure a profit. Of course you need to deliver on your promises, but you can iterate and improve your product or service as you go. Without sales, you have no business.
At this stage, you'll be wearing just about every hat in the business. That's OK for now, but as you move toward Middle Stage Level Two, you'll need to find ways to leverage your personal production for the business by hiring staff and building the basic business systems and controls you need.
An Early Stage Level Two business is working to generate sales, establish a market position, and become a sustainable business.
Level Two Middle Stage: The Owner-Reliant Business
Focus: Stabilizing your core and beginning to remove "you" from the center of the business.
Once your business is profitable and you are confident that it is going to survive, it is time to build your business's four core systems:
1. Lead Generation: your system for finding prospects.
2. Lead Conversion: your system for closing sales.
3. Production and Delivery: your system for producing and fulfilling your core product or service.
4. Collections: your system for collecting on what you are owed.
These four systems—for finding leads, closing sales, producing your product or service, and collecting on your receivables—are the four core systems of every business.
Most businesses get stuck in Middle Stage Level Two. Why? Because the owners build their businesses for control based primarily on their personal production. They are the ones who make the key decisions, close the big sales, and manage much of the day-to-day operations. Sadly, these typical Middle Stage Level Two business owners stay stuck at the tactical level of doing the _job_ of the business instead of creating the time and space to step back and _build the business as a business._
Middle Stage Level Two requires building your core systems, team, and controls, while balancing your business's need for you to continue to lead its daily operation. This is a delicate balance between operating the business in the here and now, and building the structure you'll need to grow beyond your current situation.
Once you have established your solid base, you'll progress to Advanced Stage Level Two.
Level Two Advanced Stage: The Rapid-Growth Company
Focus: Scaling your business in earnest.
At Advanced Stage Level Two, you'll begin the important work of refining your systems, building your management team, and scaling your business in earnest. This is the "Rapid Growth" stage of your business.
In this stage you'll also begin to build out the auxiliary systems you need in other parts of your company, from hiring to training to marketing to managing and more. You'll see that your revenue is exploding and often you'll feel like your business is bursting at the seams as you struggle to keep up with the growth.
A key shift at this stage is to enroll your team in building the systems, controls, and scalable solutions your business needs _alongside_ you. Where a Middle Stage Level Two business owner commonly sees her team as a way to leverage her own personal production, an Advanced Stage Level Two business owner sees her team as partners in taking the business to the next level. This essential shift is what allows businesses to successfully make the leap to Level Three.
Level Three: The Exit-Stage Company
Focus: Choosing and executing on your "exit strategy"—to sell, scale, or own passively.
A Level Three business is systems reliant, with a winning management team in place, and strong internal controls that allow the business to run smoothly and effectively independent of any one person. At this point, you truly have built a business, not a self-employed job. The business runs smoothly and your clients look to the business, not the owner, to fulfill on its promises.
Now it is time to determine and execute your exit strategy. This may or may not mean actually selling your business. In fact, many Level Three business owners choose to stay actively engaged. What's the critical distinction? Continued involvement is a personal _choice_ , not a business _requirement_. You get to make that decision and choose any of these three main exit strategies:
1. You can sell the business and move on to your next great adventure.
2. You can scale the business to the big time.
3. You can passively own the business, with a greatly diminished role for yourself in its daily operation, generally fewer than five hours a week.
A Short Quiz to Determine Where You Are on the Level Three Road Map
Now that we've gone through the complete life cycle of a business from Level One to Level Three, at what level and stage is your business right now? To identify where you fall on this path, circle the statement below that presently best describes your business:
**Level One:** You're still working on your business plan, raising your start-up funding, and preparing to launch your new business.
**Early Stage Level Two:** Your business is in its infancy, scrambling to make those early sales and fulfill customer promises.
**Middle Stage Level Two:** Your business is sustainable, but only if you're present to work for it. You're the main producer and director around whom your company revolves.
**Advanced Stage Level Two:** Your business is growing fast, with employee leaders in two or more of your five core pillars (Sales/Marketing, Operations, Finance, Team, and Executive Leadership). You're in the process of refining your business systems and building your management team.
**Level Three:** Your business is firing on all cylinders with employee leaders in at least four of your five core pillars. Your business is systems-reliant and ready for you to choose your exit strategy—to sell, scale, or own passively.
SCALING PRINCIPLE ONE: BUILD A BUSINESS, NOT A JOB.
## PRINCIPLE TWO
BUILD ON THE SCALABLE BASE OF SYSTEMS, TEAM, AND INTERNAL CONTROLS
Imagine you're about to sit down on a stool. You'd want that stool to be a sturdy, stable seat. If it had only two legs or, even worse, one leg, you'd be worried about that stool toppling over.
Most business owners build their companies on a one-legged stool, with that one leg being "team." Now if everything stays in balance and alignment you can sit on that one-legged stool, but it is a wobbly place to sit. When you scale based solely on hiring "key" people, you just create a new problem for yourself. You move the critical dependency from your shoulders to those of your key hires. What happens if you have an employee issue? What happens if a critical team member's spouse is relocated, or someone gets hurt? If the only leg of your stool is "My people will handle it," your stool will topple over the moment your people are not, for whatever reason, there to handle it.
Scaling Principle Two is about building on the stable, scalable, three-legged base of systems, team, and internal controls. Sure, you'll still want great talent on your team, but you'll also create stability and scalability by empowering your team with the structure (i.e., systems and controls) they need to produce more, better, and faster for your company. Scaling on the stable base of systems, team, and controls means that your business can handle the loss of a key team member. Plus, by having the other two legs of systems and controls, it's much easier to find, hire, and onboard new team members because you have a structure into which to integrate them.
Let's briefly define exactly what we mean when we use the terms "systems" and "controls." They are the key ingredients in designing a business that is both capable of scaling and no longer dependent on you alone to achieve that scale.
Systems: The Backbone of Your Level Three Business
Systems are the reliable processes and procedures that enable your business to consistently produce excellent results for your clients or customers. They are the documented best practices and tools that increase your company's efficiency, reduce costly mistakes, and make your business more scalable.
Systems include the checklists your operations manager follows when working with a new client, the orientation process you use to onboard all new hires, the sequential process for producing your core product or service, and the automatic e-mail sequence that goes out to each new prospect. Basically, business systems include any essential company know-how that you have captured in a tangible format as opposed to information locked in the brain of an individual team member.
Done right, systems make life easier for your team and success more predictable for your business. Sometimes when we introduce systems to a company, employees worry that they are "replacing themselves." Well-built systems don't replace people. Rather, they empower your team to consistently create more value.
The Two Layers to Every Successful Business System
Every successful business system has two layers: the process layer and the format layer.
**The process layer** consists of the step-by-step process or procedure you've created to complete any given task or process. Your system should accurately capture the steps of the process so that when you follow it, you consistently get the desired result. It does you no good to formalize poor processes; you want your systems to document your winning moves, making it easier for your company to replicate and scale those successes.
But having a solid process isn't enough. You have to package that process in ways that your team will actually use. This is where the second layer of your system comes into play.
**The format layer** deals with how you package and present your system to your team. Systems should be easy to use, transparent so that team members intuitively understand how to use them, and automated to an extent that much of the work happens via technology instead of manual work.
How do you know if your system has a good, usable format? Ask yourself one simple, unambiguous question: _Is your team using it?_ The real test of a system's utility is whether your team embraces it, ignores it, or, even more tellingly, creates an ad hoc shortcut for the task.
Your team members want to do a good job. If your business systems are simple, intuitive, and effective, they will use them. If they're confusing, complicated, or cumbersome, they'll ignore your systems, even creating their own "cheat sheet" hybrid versions instead. These homespun, individual hybrids normally aren't scalable; in fact, they usually only work for that one team member and only as long as the volume of your business stays relatively level. Plus, even if these private shortcuts work, rarely are they ever captured in a way that the rest of your business can use.
To get the format layer right, watch the way your team members use, or don't use, your systems. Don't argue, preach, or cajole—simply observe their interaction with your systems. Take their behavior as critical feedback to refine and improve your systems, whether it reveals that you need to simply adjust the design of your format, or that you need to revamp the process in order to streamline the steps or more reliably produce the desired result. Remember, systems are meant to leverage, empower, and simplify the lives of your employees, so don't fall in love with any specific system. Rather, fall in love with the result it's intended to generate.
FIGURE 2.1: 32 EFFECTIVE FORMATS TO PACKAGE YOUR SYSTEMS
Different formats to package your systems work better for different situations and roles. Here is a quick list of 32 potential formats for packaging your systems to make them easier and more effective for team members to use:
* Checklists
* Scripts
* Worksheets
* Custom forms
* Written guidelines
* Step-by-step instructions
* Software that automates a process
* Merge documents with precompleted data-entry fields
* Databases of key information
* Pricing lists
* Templates and samples
* Written policies
* Common Q&A sheets
* Written "warnings" for an area, providing how to deal with predictable problems
* Spreadsheets with built-in formulas
* Camera-ready artwork
* Filing system (paper or electronic)
* Preapproved vendors lists
* Standardized equipment and parts
* Online communication tools for effectively sharing information (discussion forums, wikis, whiteboards, social networks, etc.)
* Delivery timetables
* Job or role descriptions
* Instructional videos
* Illustrative picture or diagram
* Budget templates
* Automated data backups
* Project management software with reusable project pathways
* Reporting templates
* Organizational charts
* Preapproved forms and contracts
* A timeline or master calendar
* Complete enterprise management software
The Biggest Myth of Building Business Systems
Does all of this talk about building systems inspire you to step away from your business for a few months to write out all your systems into a complete policies and procedures manual?
**CAUTION!** Don't do it. This is a recipe for disaster. Even if you were to write up all your systems into one long manual, you'd end up with a static and dated policies and procedures manual that no one would actually use. (Between 2009 and 2013 we surveyed more than 1,000 businesspeople, and fewer than 3 percent reported ever using a policies and procedures manual to do their job after the first 30 days on the job.) What's worse is that the moment you write a policies and procedures manual, it is already out of date. For your business systems to work, they must be firmly rooted in reality. The best way to create them is bit by bit as you grow your company.
What's needed is a living, breathing _discipline_ inside your company of creating, refining, following, and, when need be, replacing systems.
In chapter 8, we'll share with you three powerful concepts to help you systematize your business, but what matters now is understanding that building your business systems is an ongoing process, not a onetime project. Early on, the systems you start with are going to be rudimentary and incomplete. That's OK; over time, you'll upgrade and flesh out your business systems so that they are a powerful leg in the stable base of your company. This leads us to the next leg of your stable base—team.
Team: Leveraging the Talent, Time, and Attention of Other People
With all this talk of the importance of systems, and the vulnerability of a "one-legged stool," we don't want you to think that your team doesn't matter. Of course your team matters—they matter a lot. In fact, we're going to spend all of chapter 10 sharing with you our bottom-line best ideas on attracting, retaining, and unleashing the talent you need to win the game of business. But know that having a solid team isn't by itself enough to allow you to sustainably scale your company to Level Three.
You'll also need to define a palpable culture within your organization that bonds your team to one another and to the company. It will help ensure that you hire the kind of people who will work well together and truly execute your vision. What's more, building and maintaining a solid culture helps your team solve problems and deal with novel situations that your systems don't cover. Your company culture is a subtle and powerful force that helps shape your team's behavior and support scaling.
Controls: Specialized Systems That Protect Your Company
Controls are the intelligent processes, procedures, and safeguards that protect your company from uninformed, inappropriate, or just plain careless decisions or actions by any team member. To scale your company, you want your team to have the authority to get tasks done without running everything past you. To do so, you want to make sure there are safeguards in place to protect the business and give your team the feedback it needs to make adjustments and stay on course.
Take, for example, a financial control. To reduce the potential for embezzlement, you could have one person make deposits and a second, unrelated person reconcile your bank statements. Or you might formalize how you give team members levels of decision-making authority that align with their experience and the degree of consequence if they decide poorly. Perhaps you will institute daily and weekly reporting of key indicators that help your team monitor their own performance and proactively respond to changes in your business. These are all examples of business controls.
Remember, every business control is also a "system"—it's just a special kind of system that is designed to safeguard your company. In the next section we want to highlight a crucial distinction about how emphasizing the implementation of sound business controls is not the same thing as you personally trying to maintain control over all aspects of your business.
A Common Question That Could Lead You Astray
During a small-group coaching session for our clients, Cheryl asked an interesting question about growing her business. Cheryl owned a successful owner-reliant Middle Stage Level Two glass installation company with two locations and a steady stream of clients. Like most business owners, Cheryl was dealing with the challenges of delegation. She was jaded by her past experiences of delegating to her staff only to later learn that they ignored the company's systems and missed important details. The more this happened, the more Cheryl micromanaged her team when handing them responsibilities. She mitigated this problem when her business grew from one to two locations by working harder and closely managing her team. But she knew that it would be impossible to grow to three or four locations without getting her team to produce more. She simply didn't have the capacity to manage the staff at these other locations that closely. So in our session, Cheryl shared her primary concern:
"How do I delegate more and more of my day-to-day responsibilities to other people on my team and still maintain control of my business?"
In this simple question lies one of the greatest business dilemmas you'll ever face. On one hand, you need to leverage your team to get more done by delegating. On the other hand, you want to maintain control of your business to avoid negative consequences if team members drop the ball or do things the wrong way.
The solution to this problem isn't choosing either of these: delegating or controlling. Instead, we have to reformulate the question. As it is stated, the question presupposes that you, the business owner, _should_ maintain personal control of your business. That creates a false dilemma, and if you fall prey to it, you'll lose both your freedom and your control over your business. The key is to let the business control itself. Talented people following efficient processes with controls and systems behind them will achieve far more than your business will ever achieve if you are the main point of control.
It's not a choice between "delegate and lose control" or "hang on to things and keep the control." Deciding between those two choices is counterproductive. If you're committed to building a Level Three business, you must learn the distinction between _control_ and _controls._ The former will keep your business stuck in Level Two; the latter is key to scaling your business to Level Three. Here's the difference.
CONTROL IS A LEVEL TWO REACTION:
* It's fear based.
* It's autocratic.
* It means you're gathering more and more of it for _yourself_ versus the business.
* It says, "Check with me" and "I'll make that decision each time it comes up" and "Don't take the next step until I look it over."
CONTROLS (WITH AN S) ARE A LEVEL THREE RESPONSE:
* They're proactive.
* They're enterprise driven (building controls for your _business_ , not for you, the owner).
* They're tools that allow everyone in your business to see the status of a given process, check on results, and self-correct as needed.
* They say, "Check the dashboard" and "Follow our internal process" and "How do we implement a systems-based control here?"
Cheryl's initial question—"How do I delegate more and more of my day-to-day responsibilities to other people on my team and still maintain control of my business?"—was understandable. Prior to reading this book, you may have asked the same question. But now that we have a clearer understanding of control, let's reformulate her question to a more empowering alternative:
"How can I design my company with intelligent controls that safeguard the business as we scale?"
The Three Kinds of Business Controls
There are essentially three kinds of business controls.
**1. Visual controls.** These include checklists, dashboards, scorecards, budgets, etc. They let you _see_ that the right things are happening; if not, they raise a flag that alerts you to fix the situation. The best visual controls empower the person doing the task with real-time feedback to do his job better.
Visual controls aren't just about giving management better control, but ideally give your team autonomous control to do a better job without needing as much oversight. And at the same time, they create a clear accountability structure so that if things aren't going well, management knows about it early enough to work with a team member to improve the situation and get things back on track. We are constantly surprised at how few businesses use the low-cost and highly effective tool of visual controls. A simple real-time dashboard can enable your employees to evaluate and correct their own performance without waiting in line to talk to you and get your feedback.
**2. Procedural controls.** These include things like your standard review process for all new hires, a rule ensuring that two unrelated parties internally check or be involved in the flow of money, and a clear process for escalating customer service issues to get the problem resolved in a smart way. Procedural controls establish a known pathway to a consistently secure result. They are especially important in institutionalizing key learnings so that your entire team follows your best practices. The consistency you get from procedural controls also helps build your company brand by ensuring that customers know what to expect from your business in various situations, and comforts you with the knowledge that in these situations, your team will know how the business wants them to respond.
**3. Embedded controls.** Embedded controls work automatically in the background without someone's having to remember to do something to use them. They make the default behavior the right behavior. These include things like standardized contracts, automated data backups, and automatic tracking and reporting of your sales team's call metrics directly out of a database.
With all this talk of controls, we need to emphasize a critical point. Building strong internal controls is not about you, the business owner, being in control, but rather about enhancing and giving control to your _business_. The best controls make the default behavior the right behavior. And they empower your team to get better results with less effort by giving them immediate feedback and a more defined playing field. You don't want your controls to be a police officer hiding in a speed trap to catch and ticket an unwary team member. Rather, you want your controls to be more like a speedometer or cruise control system that helps individual team members autonomously do better work.
The best controls also empower your managers and leaders with immediately clear and actionable information on how to coach and redirect their team, by letting them know what's going on in an area at any given moment.
Collectively, your systems, team, and controls are what allow you to scale your company.
FIGURE 2.2: 21 EXAMPLES OF BUSINESS CONTROLS
Sales/Marketing ****
* Written list of preapproved concessions or negotiating parameters
* Approvals process for all sales exceptions
* Standardized sales paperwork and contracts
* Sales scripting
* Limited/segmented access to company database of prospects and clients
* Master marketing calendar
* Lead-generation scoreboard
* Automated marketing e-mail sequences
* Standardized sales collateral
Operations ****
* Operating budgets
* Production schedule/timeline
* Customer surveys
* Preapproved vendor list
* Competitive bidding process for all contracts over a certain dollar level
* Quality review checklist
Financial ****
* Cash registers
* Formal employee expensing system
* Formal refund policy
* Written customer credit policy
* A/R write-off approval process
* Prenumbered invoices (with controlled access)
Refining Your Systems and Controls as You Grow
Looking back at the three-legged stool analogy we shared at the beginning of this chapter, it's tempting to think that you can just build that stool once and use it as is forever. But your business and the markets you serve aren't static; they are in a constant state of flux. This means that over time you'll need to refine and even redesign your systems and controls too. In fact, the more you grow, the more that growth will require you to evolve your systems.
Have you ever watched the television show _This Old House_? Imagine you were a guest on the show, working on a 75-year-old house with its original electrical wiring and plumbing. What would happen if you plugged a full complement of modern electrical appliances into the outdated electrical system? You'd blow the fuses, not to mention create the potential for an electrical fire to break out. Similarly, what would happen to your plumbing if you went from a well-water system to tapping into the higher pressure of city water? Can you say rain gear?
Likewise, rapid growth makes increasing demands on the outdated systems in your business. The systems that worked for a $500,000-a-year business are no longer sufficient to cope with a $5 million business, and not even close to being adequate for a $50 million business. At first, the additional sales will cause a few "leaks," but before long, your business will have burst pipes and water everywhere. This is why you need to approach your systems and controls as a work in progress, never a finished, static thing. As you grow your business, your systems and controls must grow with you.
SCALING PRINCIPLE TWO: BUILD ON THE STABLE BASE OF SYSTEMS, TEAM, AND INTERNAL CONTROLS.
## PRINCIPLE THREE
UNDERSTAND WHY YOUR CUSTOMERS REALLY DO BUSINESS WITH YOU
Ray ran a business that sold sand. That's it—sand. He had a big industrial lot and trucks came in all day to load up on sand and pay him. He had started out small, and while growth was slow, it was steady. He was happy but busy, sourcing sand, managing delivery trucks and logistics, negotiating payment terms, dealing with weather issues and raw material shortages, and battling new competitors when they came into the market and undersold him. Even though he could never really just sit back and rest, he made good money and built a loyal customer base.
One day, though, Ray finally maxed out his market. His growth had plateaued; most of the trucks within reasonable driving distance were already his customers. "Now what?" thought Ray. He felt stuck. Why keep working this hard without the hope of further growth?
Ray called us for help. He said he was out of ideas and couldn't see any way to continue to grow. He was so busy running his business that he didn't really even see his customers anymore; they were just trucks and transactions. Ray's real problem was that he had stopped seeing beyond his daily operations, blinding him to the potential growth opportunities.
We came out to visit Ray and decided to run a little experiment. When trucks came to get their sand loaded, we sat with the drivers and asked them many questions about _their_ business, not Ray's. It turns out that over 90 percent of Ray's customers weren't just buying sand to buy sand; they were using the sand to build bridges. Ray knew nothing about the bridge-building business, except that obviously it required sand. From there, we dug deeper and started asking Ray's customers to tell us all about the bridge-building business. When Ray realized the true goal of his business was to help his customers build bridges, not sell them sand, he quickly created and launched other products and services he could offer his customers in addition to selling them sand. For example, Ray's customers mixed his sand with concrete and poured the mix into wooden forms to shape and set it. Ray started to sell those wooden forms to his sand-buying customers. Today, Ray's business is five times the size it used to be, and growing strong.
Knowing about your customers' businesses helps you put your business in context. This is why Scaling Principle Three is to understand the _real_ reason your customers do business with you. Why do they need and want your product or service in the first place? How does it help them solve a painful problem or in some way add real value to their lives? Why do they buy from you rather than one of your competitors? Why do they continue doing business with you over the long haul? The combined answers to these questions frame out your business context.
Let's say you and a friend just arrived in a new city and were making plans to see the sights. Your friend picked up a pad of paper and started listing all the places he or she wanted to go over the three days of your visit. So far so good—brainstorming ideas is a good start. But at this point your friend simply tears off the page with the list, grabs a coat, and heads for the door. "Wait!" you say. "We need to take a look at a map to see the layout of the city and figure out which sites are near each other and which are far apart. And if we can't get to all of them we should prioritize the list so we hit the top sites first in case we want to change plans later."
Most business owners are like this friend—charging off to "do" their business without ever pausing to step back and get a sense of the "map" of the world in which their business operates. They are so consumed by the day-to-day that they don't stop and think about the broader business context in which their business operates.
Understanding your business context—the map of the market in which you sell, produce, fulfill, and grow your business—is crucial in building a scalable business. In this chapter, we'll guide you through a simple process to clearly lay out the three dimensions of your business context—your customers, your competition, and your positioning. When we're done, you'll have a clear, concrete picture of your target market, including the critical "Marketing Markers" that will help you qualify and screen for your best buyers. You'll also get a simple chart, the Competitive Matrix, to determine how you stack up against your competition in eight key criteria, so that you can narrow your focus to the one or two elements that you need to leverage to effectively compete. And finally, we'll give you two powerful tools to help you intentionally craft your market position and brand that work in the stressed-out and resource-strapped world of small business.
Let's get started with the first dimension of your business context—your customers.
Your Customers—Whom You Serve and What They Really Want
As crazy as it seems, most business owners never concretely clarify whom they serve and what this target market really wants. The average owner is so busy marketing, selling, producing, and fulfilling that they never step back and think through precisely whom they serve, much less the deep aspirations, fears, hopes, dreams, and frustrations of this target market. This makes about as much sense as the salesperson who spends all day going door- to-door selling pool care products—in an apartment complex! When you ask him why he'd waste his time selling a product that clearly this group of people is unlikely to buy, he says, "The doors are so close together that I can call on five times as many people than if I go to the neighborhood you suggest."
Can you tell us who the ideal target market for your product or service is? Within that target market, who are your best prospects? Too many businesses take a broad approach to marketing and claim that "everyone" in their target market can use their product or service. This is ineffective. Instead, think of your marketing the way the world's most successful companies do: as a series of concentric circles. In the middle of the target, the center of the bull's-eye, are the people who are most likely to use your product now and have the highest closing ratio. If you approach these people, you'll have the shortest sales cycle. Identifying those "fastest to close" customers in your target market is critical to achieving momentum in growing your sales.
Here is a list of prompts to help you clarify who your target market and ideal customer are by determining what key elements your best customers have in common. While not every question will apply to your business, most will. We encourage you to pause with each item and jot quick notes as you go.
The list of prompts will differ if you sell business to business or business to consumer, so choose the applicable list. Follow the cascading series of questions to determine the key elements that ultimately identify who your best buyers likely are, and the target market you're going to go after (and which tempting markets are distractions you'll avoid for now).
Clarify Your Target Market
List the key elements that your best buyers have in common:
IF BUSINESS TO BUSINESS:
* Industry(-ies)
* Size of customer
* Geographic location of business
* Title of decision maker
* Titles of key influencers in decision process
* Where the decision maker and key influencers spend time (physical locations, online Web sites, periodicals, events, etc.)
* With whom the decision maker and key influencers already have trusted relationships
IF BUSINESS TO CONSUMER:
* Age
* Gender
* Income level
* Marital status
* Educational level
* Where they live, work, and spend time (passions, affiliations, social networks, media favorites, etc.)
* Who commonly recommends or advises them on purchase decisions for your type of product or service
Your Top Five Marketing Markers: Given all you've reviewed about your target market, which five qualifications are your best clues that you've found a great prospect? Which five elements most reliably predict you've found a likely buyer? These clues are your "Marketing Markers." Your goal is to get better at quickly spotting those prospects who are most likely to buy your product or service. You'll use these markers to focus your sales and marketing efforts on your best leads. For example, perhaps you should be focusing on companies with 50 to 100 employees, or steel fabrication companies with sales over $10 million. Or perhaps you should focus on homeowners who recently moved into one of three ZIP codes, or families with children under the age of 12.
Who Your Target Market Isn't: What are the three to five markets that, while tempting, for the time being you are clearly identifying as NOT your current target market? It's important to know these markets in order to help your sales/marketing team focus on your target market and not be distracted and pulled off track by pursuing these other markets. You'll want to revisit your assumptions and decisions on your target market annually, but in your day-to-day marketing, making this decision will help you invest your limited resources where they'll do the greatest good.
* * *
Here is an example of the power of defining your Marketing Markers, not just to help you target your markets, but also to help you qualify your sales prospects.
Several years after his helping build Priceline.com, Jeff was running an online liquidation service for manufacturers and retailers called uBid.com. At the beginning, Jeff helped his sales team put together their sales presentation and went on the road with them to meet with prospects. At the first meeting, they went through all 20 slides with their prospect. At the end of their presentation, she said she would think about their offering and get back to them. At the next sales pitch, with another company, they were only on slide two of the presentation when the prospect said, "I'll take it!"
You don't want to do a bunch of long sales presentations, hoping that your prospect will call you back. What you want is more prospects who say yes on the second slide. To figure out what differentiated their prospects, Jeff and his team created a detailed profile that described all they knew about their customers, and assigned points to the answers to each main question. They asked: Was the customer already looking for a solution to this problem? Did the customer already have money approved to pay for a product or solution? Was the person in the room an actual decision maker? And so on. Sure enough, when Jeff and his team put their early prospects into the profile, the second prospect they had visited (who bought after the second slide) scored an 89 on the 100-point profile, and the first customer only a 37.
The lesson is this: You can identify the prospects you should be pursuing by building a profile of your true target customer. Who is most likely to need your product, understand your sales pitch, and pull out their checkbook _now_? Before you go on any sales calls, run potential customers and market segments through this filter and go after only the ones who score high enough to meet your minimum criteria. We call this a "best-first search." If you had a crystal ball, you would go to the best customers first and not waste your time on customers who take forever to convince. Our tool can work like this crystal ball, by narrowing your focus to those customers most likely to buy your product or service.
The Marketing Markers you identified a moment ago are the center circle of your prospect target. You'll use these elements to both target your marketing efforts _and_ be more strategic about where you invest your finite sales energy.
Now that you know who you serve, we need to dive deep into what these prospects _really_ want. In order to succeed, you need to understand the hopes and aspirations, fears and frustrations, needs and desires that drive and move your target market.
In the early days at Priceline.com, Jeff asked some of his colleagues what business they thought the company was in. Most answered, "We sell airline tickets." But the truth was that none of their customers woke up in the morning wanting to buy an airline ticket and sit on a plane. No, what they really wanted was to be able to afford a way to go to their sister's wedding. They _had_ to buy an airline ticket, but they _wanted_ to go to a wedding.
Do you think you'd run your business differently if you thought your job was getting people to their sister's wedding as opposed to just selling people airline tickets? This insight sparked one of Priceline.com's most effective early marketing campaigns, where they bought blank pages in major U.S. magazines and asked customers to send their own pictures and tell their own stories about using Priceline.com. Knowing that Priceline.com helped others like them afford to go on a trip or get to an event they couldn't have attended otherwise really resonated with Priceline's target market.
Ask yourself: Why do my customers _really_ come to me? What do they worry about when they wake up in the morning? Figure that out and you will be able to market to them much more effectively.
In Figure 3.1, we share a simple tool to help you map out the deep psychological and emotional needs of your target market. To see how this all plays out in real life, we've filled out the answers for these questions for David's company, Maui Mastermind.
FIGURE 3.1: CASE STUDY: MAUI MASTERMIND
You can download a blank PDF version of this tool for free at **www.ScaleYourBusinessToolkit.com/pdf**. Take 15 minutes to fill out your answers for your business. Once you've got your answers down, share this tool with your team. Ask them to answer these same questions. Come up with a one-page "company" version that you share and reinforce throughout your organization. This is one of your key marketing controls.
Clarifying who your target market is and what they really want will help you identify and search out your best prospects, strengthen and focus your marketing message, and qualify and prioritize the leads you generate so you invest your best sales energy on your most qualified prospects. With this concrete picture of your best buyers in hand, it's now time to move to the second dimension of your business context—your competitors.
Your Competitors
While the most important place to start is with your customers, it is essential that you take stock of your competitors too. Whom do you compete against? What are their strengths and weaknesses? How can you best position yourself and your products and services relative to them? Answering these questions will highlight strategic opportunities for you to seize.
Take the example of how Zappos.com used its competition's weaknesses to differentiate its business and grow tremendously as a result. Zappos realized that the traditional model of customer service was to handle as many customer service calls in the call center as quickly as possible. Call center agents were measured by how many calls they processed per hour, which meant getting the customer off the phone as fast as possible, leaving many customers feeling rushed and unfulfilled. Unlike its competitors, Zappos instructed its customer service agents to focus on the happiness of the customer, not minutes on the clock. Its customer service agents were to spend as much time with any one caller as necessary to ensure the caller felt great at the end of their conversation. As a result, word spread quickly that Zappos cared about its customers. Its competitors' weakness became Zappos's biggest strength, and powerfully differentiated it from all its competitors.
You have three types of competitors, two of whom you likely haven't given much thought to. We'll explore all three types of competitors, then look at a simple tool that will help you compare your company—side by side—with your top competitors.
Your Direct Competitors
Generally, when you think of competition, your direct competitors come to mind first. Direct competitors are those companies that are actively in the same market you are in, selling a comparable product or service. For example, Sasha Ablitt owns a dry- cleaning business in Santa Barbara, California, that she has built into the most successful dry cleaner in the area. Her direct competitors include more than a dozen other dry cleaners in her area.
Identify your top three direct competitors. What are their top strengths, weaknesses, pricing structure, and market share? This is the minimum information you need to get a good handle on who your direct competitors are and how you can effectively compete against them.
Your Indirect Competitors
How else does your target market satisfy their needs with respect to the problem your product or service solves other than with your or your direct competitors' solution? We call these competing solutions your "indirect" competitors. Going back to Sasha's dry-cleaning business, her _indirect_ competitors include people doing their own hand wash and new washing machines with specialized cycles for delicate clothing.
Or take Jennifer Lyle, owner of Software Testing Solutions, Inc. Jennifer's company sells a software tool that automates how a hospital blood bank tests and validates its testing process. Her _indirect_ competitors are hospitals manually conducting the testing validation themselves or outside consultants that hospitals hire to come in and manually validate their testing process.
To figure out who your indirect competitors are, ask yourself who or what your target market turns to—other than you or your direct competitors—in order to solve the core challenge your product or service solves.
The mental exercise of determining who your indirect competitors are is a critical one. All too often we see businesses direct their limited marketing resources (i.e., time and money) to luring customers away from their direct competitors, when sometimes it's much cheaper to acquire a customer that no one currently has. Sasha, for example, can focus her energy on getting the woman who does her own hand wash at home as her next customer, as opposed to targeting the customer who is currently loyal to the dry cleaner down the street.
Who or what are your two main indirect competitors? What are the biggest advantages for someone using these indirect competitors? What are their most glaring deficiencies or weaknesses? What is the best way you can position your company, product, or service relative to them to outshine them?
Disruptive Competitors
Step back from your business for a moment. In your worst nightmares, who or what could totally change the way the game is played in your industry? Think of what Apple did to the music industry with iTunes, what Amazon did to brick-and-mortar booksellers, or what FedEx did to the U.S. Postal Service.
Many businesses fail to spot a disruptive competitor in their industry because they put blinders on and choose not to see it. They have so much invested in the status quo that it warps their perceptions and thinking process. In chapter 5, we'll share with you a process to protect yourself from being blindsided by disruptive competitors by learning to systematically read the world around you. A big part of identifying disruptive competitors is watching for trends and new technologies that aren't even being used in your industry—yet.
For example, with Sasha's dry-cleaning business, potential disruptive competitors could be fabric manufacturers who create a memory fabric that allows people to machine-wash dress clothing, which then pops back into its pressed shape. Or an ionization machine from a different industry that could be applied to dry cleaning that totally eliminates the need for solvents or chemicals.
Now it's your turn. Who do you think your two biggest potential disruptive competitors could be? What might it look like if they made a move into your niche? What steps you could take now to preempt or undercut these moves? It doesn't matter that you have the perfect answers to these questions; for now, it is enough that you train yourself to consistently ask the questions.
Building Your Competitive Matrix
We've got one last step—to map out, in one simple chart, how you stack up against your competition. We call it your Competitive Matrix (Figure 3.2). This chart asks you to fill in your top three direct competitors, your top two indirect competitors, and your two scariest potential disruptive competitors. Then, rate your business and that of your competitors on the eight variables (e.g., price, value, service, etc.) listed on a scale of 1 to 5, with 5 being the highest score possible. For example, if your company is very attractive on price to your target market, then rate yourself a 4 or 5. If your brand is very weak (not well known in the market, has a poor reputation, etc.), then you might rate it a 1 or 2, and so on. When completed, this matrix will offer a clear side-by-side comparison between you and your competition.
FIGURE 3.2
At this point, what matters is understanding where you stand relative to your competitors, not that you outshine each of them on every variable (which usually isn't possible anyway). If you know that competitor A is significantly less expensive, then competing on price would likely be a mistake. Instead, you may decide to compete on service, quality, or brand. This is why we are asking you to lay out a visual view of the competitive playing field. You can use this information to pick the one variable that not only matters to a large enough subset of your target market, but is also the one in which you can trounce your competitors. Having laid out your Competitive Matrix, it's time to turn to the final dimension of your business context—your positioning.
Your Positioning
Your market position is made up of two closely related elements: your chosen "parking space" and your brand. Your parking space is the intentional way you have positioned your company (or its products or services). It is the frame of reference through which you want your market to see you. Your brand, on the other hand, is the emotional shortcut you've helped your market develop that links your company, products, or services to a specific emotional response. Your brand becomes a filter that colors how your market interprets each of its interactions with you.
Both your parking space and your brand are two pieces of the same puzzle, and are critical for your business to successfully scale. Let's look at them in detail.
The Parking Space Theory of Positioning
The easiest way to think about strategically positioning your company is using a simple analogy we call the "Parking Space Theory of Positioning." Imagine driving into a parking lot, looking for a space to park your car. What would you look for? An open space, of course! Theoretically, you could move a car out of an existing space so you could park your car there, but that would require quite a bit of muscle power to do.
The same concept applies when you are looking for the right space in which to position your business in the marketplace. You should look for an open parking space. The right parking space for your company lies at the intersection of three factors: your company's biggest strengths (your parking space must rely on what you do really well); your market's deepest desires around your type of product or service (it must be something that your market values); and the open spaces your competitors don't already own in the mind of your market (it is very expensive to move another company out of a space if they truly already own it).
Your "parking space" is the heart of how you position your company, and its products and services, to be seen by your target market. It represents the _single_ thing you want to be known for—more than anything else. Here are several examples with well-known companies and products:
* Walmart ("Lowest prices" parking space)
* Bayer Aspirin ("Gentlest aspirin on your stomach" parking space)
* Toyota Prius ("Most eco-friendly" parking space)
* Prego Spaghetti Sauce ("Thickest spaghetti sauce" parking space)
* Zappos ("Best customer service" parking space)
* Amazon ("Easiest place to buy online" parking space)
* Priceline.com ("Lowest-priced discount travel" parking space)
Why is it so important that you be known for a single thing? After all, you may say you are both the low-cost provider and have the best service in your market—oh, and your product is the best quality too. Why can't you be known as the "lowest price, best service, highest quality" provider? As soon as we spell it out like that, it becomes obvious how watered down and confusing this will likely prove to your market. It's hard enough to own any _one_ _thing_ in the minds of your market, let alone two or three things. Yet if we look at marketing materials for a group of small businesses (e.g., Web sites, sales brochures, advertising, sales scripting, etc.), many are pushing multiple positions at the same time. And their materials probably don't look all that different from their competitors'. Sound familiar?
One of the things we tell the businesses we work with is that in order to achieve exponential success, you need to win a gold medal in something. You need to find a category in which you can truly win—so that you can stand out in your marketplace. Picking your parking space carefully is a critical long-term decision.
Here is a short series of questions to help you triangulate and pick the best parking space for your business:
1. What are your company's top three to five strengths that create value in the market? These should be things that set you apart, or potentially _could_ set you apart from your competition. Does your cost structure allow you to price more attractively than your competitors? Is the technical expertise of your client support team or credibility of your company unique in your market? Just because a parking space is empty doesn't mean it's necessarily a good choice for you and your company. You have to be able to truly deliver on the promise of that space.
2. What are the top three to five things that a sizable chunk of your target market values highly with respect to your type of product or service? What are the key factors that your market wants in your product or service that would trigger them to buy? Is it the speed of delivery? Flexible payment options or financing to make it more affordable? Ease of use?
3. Which parking spaces (best price, best quality, best service, easiest to use, most environmentally friendly, etc.) are empty and not already "owned" by one of your competitors? If you aren't sure if someone owns a specific parking space in the minds of your market, likely no one does. A simple test to determine this is to ask ten prospects in your target market which company (or product or service) comes immediately to mind when they think about **_______** (fill in the blank with the parking space you're curious to see if someone owns). If all ten prospects say the same company delivers the best prices on dry cleaning, for example, then that parking space has a competitor parked in it. If they each have a different answer, or better yet, no single company comes to mind for them, then likely no one is parked in that space and it is potentially open for you to claim.
Now that you've listed the three sides to the parking space triangle, determine which parking space you can claim that leverages one of your business's key strengths, delivers a benefit that your market values highly, and is not already occupied by a competitor.
The goal behind this parking space exercise is to carve out the one singular space where you can beat your competition by doing something you are very good at for a set of customers who will love you for doing it and pay you well.
Now it's time to turn our attention to a closely related subject—branding.
Branding in the Real World
Your brand is simply the emotional associations and gut-level "sense" of how your company, product, or service is intuitively perceived by your market. If your "parking space" is about choosing the one single element you want to relentlessly burn into the minds of your market so that they'll automatically associate your company with that space, then your brand is the broader emotional response that instantly springs to mind when someone in your market imagines what it will be like to interact or work with your company.
The easiest way to make branding real is to determine what we call your top three "brand emotions." We'll walk through exactly what this means in just a moment, but first let's quickly turn to the biggest branding mistakes we've watched small and midsize businesses make.
THE THREE BIGGEST BRANDING MISTAKES
1. They've never given any real thought to their brand or market positioning. They have no real strategic understanding of how they are perceived or how they ideally _want_ to be perceived.
2. They regularly do things that contradict or water down their brand because they are not congruent with their desired brand.
3. If they have given thought to the brand they want to embody, they lack a clear, practical understanding of how to take this idea and translate it into how they do business. Their branding all feels theoretical and academic.
Your brand is the sum total of the emotional, mental, and experiential associations your marketplace has to your company and offerings. The reason brands work is that they help people quickly make meaning out of the world. The value of your brand is in direct relationship to how valuable the market judges your promise, and how much people trust your company to keep its "brand promise," the informal contract you've made with the market to offer a reliable and consistent experience when interacting with or buying from your company.
To make this concept actionable and intuitive in your business, think about your brand promise as a set of three core emotions that you want your market to experience after every interaction they have with your business. We call these three core emotions your "brand emotions." Here are a few examples of various well-known companies and their brand emotions.
APPLE:
1. Sheer pleasure. ("It is a joy to handle and touch these devices.")
2. Cool. ("When I own and pull out my iPhone, I feel cool and hip.")
3. Smooth and easy. ("They just work and are so intuitive to use.")
VIRGIN:
1. Well served. ("They really do take care of my needs.")
2. Great value. ("I know I got a great deal.")
3. Fun. ("Interacting with Virgin is fun.")
LEXUS:
1. Luxurious. ("It just feels so rich to be riding in this car.")
2. Status. ("I feel more important when people see me in this car.")
3. Trust. ("I know I can rely on the car and the dealer to serve my needs.")
What about your business? What are the top three brand emotions that you want your market to experience after every interaction with you?
Now comes the breakthrough part—making your brand emotions real in your market.
How People Make Meaning of the World
The world is a whirling storm of complex forces and experiences that can quickly overwhelm a person. It seems like everyone, everywhere, is trying to get our attention and sell us something. To deal with this complexity and make sense of all the noise, we find shortcuts to simplify and structure our experiences that help us to navigate and thrive. People evaluate a business's performance on its brand promise by applying a powerful quirk of human nature called _inductive perception_.
Inductive perception is a thinking shortcut where a person takes a relatively small sample size of experience and makes that small experience stand for the broader whole. This meaning then becomes a filter that progressively grows more and more pronounced—and through which that person views all his or her interactions with you and your company. For example, let's say one of your company's brand emotions was "trusted." Imagine your salesperson set a 2 P.M. appointment to call on a prospect. If your salesperson is 20 minutes late, what will your prospect likely make that mean? If your salesperson is 5 minutes early, how will your prospect interpret that?
Let's expand this exercise to say that, during the first sales call, your salesperson carefully takes notes, and at the end clarifies the two items she has promised to follow up on and send over to the prospect. How would the prospect perceive it if, later that day, he got an e-mail that said, "Hi, John. I wanted to follow up and get you both items I promised to send by close of business. They are attached to this message. . . ." Can you see how the prospect will start to generalize that your company is one that he can trust?
Understanding that people take symbolic slivers of their experience and inductively expand what they mean by generalizing outward, you can intentionally design your market's interactions with your business to increase the odds that they'll feel your brand emotions. The more consistently you help your client or market feel your brand emotions after they interact with you, the stronger your brand is with those people. Now, how to do that in the real world?
Go back to your brand emotions. Think of simple things you can do to design your client experience so that your client is more likely to feel one or more of your brand emotions. For example, if you owned a Lexus dealership, you could look for simple ways to enhance a customer's sense of status the moment they walked on the lot. You could have them call all staff on the lot by their first name, while your staff would still address the customer as Mr. or Ms. (putting the customer up on a pedestal emotionally). If they needed to get their car fixed, you could help them arrange a "service appointment." You could sponsor high-end arts events and invite your customers. You could even give customers a visible luxury gift like a watch or fancy key chain, to extend the feeling of high status into their lives outside of their car.
During his time with Priceline.com, Jeff worked with an experimental service called Priceline Yard Sale, an attempt to bring its unique "name your own price" model to yard sale items. Early on in this project, Jeff and his development team realized that they wanted their customers to get as close as possible to the excitement of winning the lottery or a big roll of the dice in Las Vegas. Knowing that one of their key brand emotions was "excitement" (as in, "I won!"), they intentionally made subtle shifts in how their site worked. For example, they didn't say, "Thank you for your purchase" (yawn); instead they said, "Congratulations! You won the item you bid on!" And rather than just have a mundane image pop up while a visitor's bid was processed, they intentionally showed a slot machine with numbers spinning on the three dials, landing on 7-7-7 when the response came back that his bid was accepted. These subtle design elements made a big impact on customers' tell-a-friend viral marketing ratio, and helped Priceline firmly establish this key element of its brand.
David's company, Maui Mastermind, recently redesigned its core business coaching platform to help reinforce one of its brand emotions—"confidence." That redesign included building a simple accountability structure into the company's coaching app. The owner visually _knows_ when her team is meeting deliverables or not and the app delivers weekly progress reports from everyone on her team. Clients can now also actually _see_ the progress they are making on a quarter-by-quarter basis relative to their strategic action plan and the other key metrics inside their company. The end result was an upgraded ability to give clients the confidence (brand emotion) of knowing they are on the right road to reaching their business goals.
Select one of your brand emotions and brainstorm possible ideas—large and small—that you could concretely design into your customer experience to _prompt_ them to feel that brand emotion. Now go on to the second and then third of your brand emotions and ask yourself the same question. When you're done, choose from the ideas you've come up with to implement in your business. (In the next chapter, we'll share with you a powerful strategic tool called the "Sweet Spot Analysis" that you can use to pick the best of your ideas to implement.)
The Two Most Important "Reality Checks" for Your Brand
Now we come to the final two reality checks to your company's brand. First, just because you say it's so, doesn't make it so. Your _clients_ ultimately determine your company brand by an accumulative process. Once you've been labeled by the market, it's very hard to stray too far from it. You can tweak it, twist it, and bend it, but rarely break it. You're almost better starting fresh and creating a new brand.
If you are being brutally honest right now, how does your market currently see your company brand?
Second, your brand emotions must be meaningful and real inside your company or they won't last. Smoke and mirrors are great in a stage production but do not last long in the market. You've got to get the substance of your brand emotions into the way your company sees itself and its mission. Your team needs to understand how it is _everyone's_ job to reinforce your brand emotions any way they can. For your brand emotions to be real for the long term, your entire team has to live your brand every day, in every way. That means being consistent in every touch point and every interaction with a client, whether it's a phone call, in-person conversation, e-mail, or advertisement. A brand can really only achieve its goal when it's consistently applied by everyone at your company, in everything they do.
Unfortunately for too many companies, "brand" is just an external marketing message and, internally, merely a poster on the wall. The world's best companies actually do live their brand every day, in everything they do. Remember Zappos? They made the huge decision to build their brand around one powerful brand emotion: delivering happiness. And they're extremely committed to following through on making sure you leave happy when you interact with them.
Jeff was delivering a keynote address at a large business conference and wanted to give the audience a vivid example of what it meant to live your brand promise. He asked the conference staff to bring a speaker phone up onstage. There, live and unscripted, he dialed the Zappos customer service line. When the customer service agent answered, Jeff told her he was shopping for a barbecue grill. She politely informed him that Zappos sells men's and women's shoes and accessories. Jeff replied that he didn't need any of those things, but a new barbecue grill would make him quite happy. To the amazement of the audience, the customer service agent asked him a bunch of questions about the grill he was looking for, went online and researched his specs, and then e-mailed Jeff links to grills that might serve his needs! She then reminded him that if he ever needed shoes and accessories, he knew whom to call. This is an incredible example of being true to your brand.
It All Starts with Your Customer
The place to start is with your customer. Let your customer's needs, desires, fears, and aspirations deeply impact how you run your business. Once you have a clear understanding of whom you are serving and why they choose to do business with you, review your competition. Take the strategic insights found by looking at your Competitive Matrix and use them to help you sharpen your positioning. While you may never nail your parking space and branding as effectively as Zappos does, you can make real improvements that will help you sustain your growth. Your brand isn't just about helping your customers feel a certain way about your company. When your brand is vibrant and real inside your company, it can powerfully shape how your employees see themselves and your business.
SCALING PRINCIPLE THREE: UNDERSTAND WHY YOUR CUSTOMERS REALLY DO BUSINESS WITH YOU.
## PART II
## Focusing on Fewer, Better Things
Courage rather than analysis dictates the truly important rules for identifying priorities: Pick the future against the past; focus on opportunities rather than on problems; choose your own direction, rather than climb on the bandwagon; and aim high, aim for something that will make a difference, rather than for something that is "safe" and easy to do.
—Peter Drucker
## PRINCIPLE FOUR
CREATE THE RIGHT STRATEGIC PLAN
When Jeff was CEO of uBid.com, the team developed a technology engine that could track incoming inventory, manage the dynamic pricing of that inventory, set up and run online auctions for that inventory, and handle all the shipping and logistics to deliver and report on sold inventory. It was an impressive set of assets. But the company wasn't growing. It was positioned as a more trustworthy alternative to eBay, but it was difficult for David to go out and pick a fight with Goliath. eBay had a massive marketing budget and huge brand recognition with consumers; people weren't going to easily leave it.
What was uBid.com to do? First, they asked themselves if their strategic positioning was correct. Turns out, it wasn't. They were pointed at the wrong market, living in eBay's shadow. Acknowledging that their current strategy simply wasn't working, they stepped back and reassessed their strategy.
They concluded that with their assets (product, people, relationships, etc.), they were better suited to pursue the excess inventory market, not the consumer-direct market. Under their new strategy, they offered themselves as a solution to major manufacturers and retailers, filling a need that was underserved. In this new strategic position, they could beat their competitors, unlike when they were challenging eBay. The company signed a series of new contracts with major customers who now knew exactly what value the company could provide to them. They carved out a well-defined niche in which they were positioned to win, and as a result, uBid's stock quintupled in value.
Now that you understand your business context (your market, your competitors, and your positioning, as we discussed in Principle Three), Scaling Principle Four says it's time to determine your strategy to grow and reduce that strategy into a series of rolling, one-page, quarterly action plans that help you execute and get results. In this chapter we'll walk you through a concrete, structured process to do exactly this.
Your strategic plan is the road map that directs your company's focus to the fewer, better things that will allow you to dominate your market niche and create explosive growth. It prompts you to look at the big picture. What really matters? Where are you committed to going? How can you meaningfully and accurately track your progress? Your strategic plan also helps you prioritize and intelligently allocate your company's resources to their best advantage. Finally, your strategic plan helps align your team on the big picture so they can better manage their responsibilities and contribute more to the real needs of the business.
Conventional wisdom says your strategic plan is where you write down your answers; that it is a fixed-in-stone plan to accomplish your key business goals. In theory this sounds great, even alluring—a simple document that contains the secret plan to marshal your resources and smoothly attain your goals. Alas, that's just not how things work in the real world of business.
If a combined 50 years of running companies has taught us anything, it is this: Your strategic plan is not a place for fixed answers, but rather it is a trusted process comprising provocative questions you systematically ask yourself as you iterate your way to success. It is not a onetime exercise but a recurring activity you engage in with your business's best minds to continually learn, evolve, refine, and, at times, radically re-create your business.
Many companies write a strategic plan at the end of each year to serve as their map for the next year. While this is indeed a valuable intellectual exercise, the next time they see that plan is one year later. Imagine a football coach putting together a game plan for Sunday's big game. At halftime they are down 28–0. Would the coach say, "Well, since we write out the plan for each game in advance, we'll stick with the old plan for the next half. We can talk about changes next week"? Of course not. The best coaches make adjustments to their plans at halftime. They change their defense or adjust their offense in response to real-world results. Your business should do the same thing. Schedule your own halftime, go back to the locker room, revisit your plan, and make ongoing adjustments based on the market feedback.
Our model of strategic planning is iterative. We encourage you to frame powerful questions, explore potential answers, test those answers in the market, and willingly (dare we say eagerly) challenge those trial answers every quarter.
As you iterate your way to success, you must find a working balance between change and momentum. Change says, "Hey, let's question our focus and radically redesign our approach." Change is essential to keep your business fresh and relevant in the marketplace. It allows you to nimbly seize opportunities. But change also stresses out your team and puts increased demands on your limited resources. It often causes you to scrap existing systems, products, and infrastructure as they become dated, even obsolete. Going from a static idea to a moving, producing idea in the market demands extra energy (read: time, effort, and money). Meanwhile, momentum says, "I'm working. Feed me! Support my success and build on my breakthroughs." It leverages your known winners and doubles down on your strongest successes by scaling and optimizing what has been proven to work in the market. But momentum can be lazy, complacent, and naïvely oblivious to market shifts that can pull the rug out from under it.
Clearly, you need both to successfully grow your company. You need change to secure your future and best position your resources to go after opportunities; you need momentum to give you time to build on successes and develop the infrastructure that allows you to scale. The two often come into conflict and cause tensions in a company.
How do you maintain this dynamic balance? While there is no perfect, permanent answer to this question, we do have a working solution that has produced amazing results for our businesses, and for tens of thousands of business owners we've shared it with. Once a year (we suggest December or early January), go off-site with your executive team for a one-to-two-day strategic planning retreat. Your assignment is to take a critical look at your long-term plan and examine the state of your business with the following questions: What are the current market conditions? What trends are coming our way? What is our business context? Are our key objectives still the right ones? Is our strategy to get us there still working? Once you know the state of your business and market, make any big-picture corrections needed.
Every quarter, we recommend taking your executive team off-site for a half- to full-day session to revisit this plan and concretely map out the next 90 days. You'll lay out your top three strategic priorities for the coming quarter, and write up a simple one-page plan of action specifying exactly what you must do that quarter to grow and develop your business. At the end of the quarter, you'll repeat the process. At the end of the year, you'll step back and do a fresh pass on your big-picture, long-term plan for your company.
The reason why this process works so well is because it prompts you to look freshly at your business every quarter while also allowing you and your team to dive deep into the execution and actually accomplish meaningful progress on clearly prioritized focus areas of maximal importance to your business. Without this clear framework, too many owners change their focus so often that their teams are left dizzy, feeling the vertigo of too much change, and frustrated because, just when they seem to be on the verge of really getting something big done, the owner shifted the playing field yet again, forcing them to abandon projects prior to completion and wasting hours of their effort. The business owner who finds himself changing things on his team monthly (or even weekly!) may just be addicted to the adrenaline of change or the illusion that this enhances his control. Generally, stepping back once a quarter to determine your priorities and resource allocation is your best bet.
An important part of finding the balance between change and momentum is the concept of "opportunity cost." In addition to constantly reexamining the work you are currently doing in your company, you also need to look at what work you could pursue if you changed your focus. Some businesses get so much momentum going that it would be like stepping in front of a speeding train to change direction in any way now—but those same businesses could potentially be speeding past new opportunities that could make the difference between short-term survival and long-term success. The concept of opportunity cost prompts you to ask, _If we weren't doing the things we are doing right now, are there other goals or activities that we would be free to pursue that are more valuable?_ If not, then keep on truckin'! If yes, you'll need to examine how to pursue these more lucrative opportunities.
Done well, these quarterly 90-day sprints reward your business with the biggest benefits of regular opportunities to change and adapt, _and_ also the results you get from the momentum of disciplined execution on known priorities quarter by quarter.
We've developed a six-part process to craft your big-picture strategic plan and turn that plan into a one-page quarterly action plan. Let's dive in.
PART ONE:
Three Questions to Start Your Big-Picture Planning Process
Take out your notes from chapter 3 clarifying your business context (your market, your competition, and your position). Working from those, here are the next three questions for you to answer about your current position that will help you define the highest-level view of what you are in business to do and why it matters. Take 10 to 30 minutes to thoughtfully answer these three fundamental questions.
Question One: Why is your company in business?
This first question cuts to the heart of why you are in business in the first place. Often it helps to approach this from two perspectives: externally (your customers) and internally (your team).
**Externally:** What is it that your customers _really_ buy from you?
For example, with Jeff's former company Priceline.com, its reason for being was to help Jenny get to her sister's wedding at a price she could afford. With David's company Maui Mastermind, his clients may be superficially buying business coaching services, but what they are really purchasing is a business they love owning. When someone orders from Amazon.com, they might be buying a widget, but what they are really getting is the simplicity and ease of buying from a trusted marketplace that offers just about every type of product available online.
**Internally:** Besides just the money, what inspires you to show up to work each day?
At Jeff's current company, ColorJar, their internal "why" is to help companies maximize the success of a new product launch. They see themselves as mission control of the moon launch; when that Go button is pressed to launch, they know they have done all that is possible to make sure everything has been prepared, tested, and will work flawlessly. Their internal "why" shapes how they see themselves. Sure, they are in business to make a profit, but what really inspires them is to design, orchestrate, and run a perfect product launch for their clients.
What are your company's external and internal reasons for being in business?
Question Two: What is your Singular Goal?
What is the one goal that you are mobilizing all your company's resources to accomplish over the next three to five years? This is your Singular Goal. Another way of coming up with your company's Singular Goal is to ask, "What business are we committed to build over the next three to five years?" Approach your answer two ways: quantitatively and qualitatively.
Quantitatively **:** What does your future business look like in terms of things you can measure and count? What are your annual sales? Your annual number of transactions? Your market share? Your average sales per customer? Your gross profit margin? Your operating profit margin? Your retention rate of clients? Your sales per employee? Pick three to five of the most important _quantitative_ ways to describe your future business.
**Qualitatively:** What does your business look like in other ways that you can't easily measure or count, but that still matter greatly? Who are your key customers? What is your brand? What is your market reputation? Who do you have on your team helping lead the business? Which markets do you serve? What key products or services do you offer? What impact do you have on the lives of your customers? What is your future role as the business owner? Pick three to five of the most important _qualitative_ ways to describe your future business.
After you've combined both the top quantitative and qualitative descriptions of your future business into one single target, whittle away at this picture until you are left with a tight, clear statement that can guide your company's focus over the next several years. Here's an example of what a Singular Goal might look like:
_By December 31, 2020, we've built Growth, Inc., into a thriving Level Three business with $24 million in annual sales and a 20 percent operating profit margin. We have deeply penetrated four distinct verticals, with no single vertical or customer representing over 30 percent of our business. We continue to grow at 25 percent a year or more._
Of course, your Singular Goal depends on your industry, business model, and goals. The key is that you want to end up with a concentrated statement that describes the one target all of your team is focusing on helping the business reach.
By the way, because you took 5 to 15 minutes to clarify the business you are working to build both quantitatively and qualitatively, you have just completed one more key component to help you reach that goal—you have drafted your enterprise-level scoreboard. (And you thought you were just dreaming on paper!) You now have a clear list of the key variables that describe your key business objective. These are the variables that you'll measure at least quarterly. For example, looking at the sample Singular Goal we shared, you can see that for Growth, Inc., their enterprise-level scoreboard would track: gross sales, operating profit margin, number of verticals they have penetrated and how deeply, percent of total sales each customer and vertical represent, and the company's annual growth rate.
Question Three: What's in it for you and your team?
If you accomplish your business's Singular Goal and build the business you've described, what's in it for you and your team? In other words, what would accomplishing this goal do for each of you? Obviously the financial rewards come easily to mind, but don't stop there. What opportunities would accomplishing your Singular Goal create for your team? What kind of impact would you have in the lives of your customers and how would that directly impact you and your team? What kinds of cool new projects would it open up for you? What you're trying to do with this line of questioning is concretely link the rewards and meaning of reaching your business's Singular Goal with your and your team's deeper drives and motivations.
This is a great topic to bring up at a team meeting. Again, make sure you go beyond just the financial rewards to the other nonfinancial benefits that energize and inspire everyone in the company. If one of the keys to successfully growing your business is having a great team of highly motivated people, then the key to motivating those people is to make sure that you as a business owner know what your employees want out of their careers at your company as well.
PART TWO:
Uncover Your Business's Top Leverage Points
If part one of the planning process had to do with clarifying the big-picture view of your business, then part two focuses on spotting the biggest leverage points inside your business to help you accomplish your top goals. A leverage point is a place in your business where a small amount of effort yields a magnified return, and identifying them is one of the keys to scaling your business.
Conduct a "S-O-O-T Review"
A S-O-O-T (Strengths-Obstacles-Opportunities-Threats) Review is a structured way to see where your business stands today. It helps you lay out the key landmarks from which to craft your company's strategy to scale. Here are the four components to review about your company:
**Strengths:** Any strategy that you eventually choose must rely on your strengths. Keeping your company's Singular Goal clearly in mind, what are its top five strengths to accomplish this goal?
**Obstacles:** Each key obstacle is a clue as to what next steps you need to take in your business. What are the five biggest obstacles that you see currently blocking your company from achieving its Singular Goal? When you look at them from this frame, obstacles become stepping stones to help you cross the gap from where you are to where you want to go.
**Opportunities:** Opportunities are where you win the game of business. What are the three biggest opportunities your company could pursue that could potentially help you achieve this goal? One guiding strategic principle is to put your best people and resources on seizing your biggest opportunities rather than fixing your squeakiest wheel.
**Threats:** What are the three biggest threats that could massively harm your business? Look at those things that, if they fell wrong, could literally put you out of business: a harmful market trend, a disruptive competitor, government regulation, or even the loss of a key customer or supplier. Your goal is to take simple, proactive steps now to mitigate these dangers later. There will likely come a day when you'll either say, "Why didn't I do something about this earlier when I had the luxury of time?" or "I'm so glad I prepared for this contingency."
FIGURE 4.1: CONDUCT YOUR S-O-O-T REVIEW
Find Your "Sweet Spot"
Your business's biggest Limiting Factor is the single biggest constraint currently limiting your growth. It's the _one_ ingredient that, if only you had more of it, would allow your business to grow instantly. The more precisely you can identify your Limiting Factor, the easier it is to effectively push it back. For example, if you say your Limiting Factor is "lack of sales," you might come up with a dozen ideas to increase sales.
But before you solve this, dig deeper and see if you can pin your Limiting Factor down more precisely. Is it the need for more leads on the front end? Perhaps your business has enough leads but instead lacks the sales capacity to effectively follow up on all the leads you are already generating? Or is it that you have plenty of sales staff, but lack a proven sales process so your sales team's conversion is too low? As you can imagine, depending on your answer here, you'll need to take an entirely different approach to solve that Limiting Factor. That's why it is so critical that you narrow down your Limiting Factor to the most accurate kernel you can.
While every business has multiple limiting factors, each has _one_ Limiting Factor (capital L; capital F) that does the most to limit its growth in the here and now. Pushing back your biggest Limiting Factor is a major leverage point to grow your company. One key way to grow your business is to identify and push back your current Limiting Factor quarter by quarter. In the context of creating a true Level Three business, you need to build the systems, team, and controls to help you push back your Limiting Factor _long-term_. Doing this usually exposes a new Limiting Factor you'll need to work with. It's like lying in bed on a cold night with a blanket that just isn't big enough. You pull it up to your chin because your chest is cold, and in doing so, leave your feet exposed to the cold. You curl up your feet under the cover and the blanket untucks from your back. Every time you push back your Limiting Factor so it is no longer your biggest constraint, you expose a new Limiting Factor to work on. Good! This is how you grow your business in a leveraged way—by focusing each quarter on pushing back your current Limiting Factor.
Once you've pinpointed your company's current Limiting Factor, it's time to pick the highest-leverage tactics to push this Limiting Factor back. We developed the three-part **Sweet Spot Analysis Tool** to help you do just that.
First, brainstorm a list of all the potential ideas you have to push back your Limiting Factor. Don't settle for five or six ideas, push yourself to come up with at least 10, ideally 15 to 20, ideas. For example, if your Limiting Factor is a lack of sales capacity to follow up effectively on the leads you currently generate, your list of ideas could include: a better lead-qualification system to prioritize your sales efforts, hiring more salespeople, or creating a sales video to do some of the selling for you. The key is to push yourself to come up with as many ideas as you can that could potentially help you push back your Limiting Factor. The best way to come up with a few great ideas is to first come up with a potential list of _a lot_ of ideas.
Next, run your brainstormed list of potential tactics through two filters: the "Low-Hanging Fruit" filter and the "Home Run" filter. A **Low-Hanging Fruit** is a no-brainer opportunity that you're almost certain will work. While it may or may not have a big impact, it is fairly straightforward to implement and you have a very high level of confidence that it will work. A **Home Run** , on the other hand, is an opportunity that if you hit it well and all goes just right, has a _huge_ payoff for your business. Go through each brainstormed idea on the list and ask, _Is this tactic a Low-Hanging Fruit?_ If it is, mark it with "LH" for Low-Hanging Fruit. Then in a second, separate pass, go through your list of brainstormed ideas and ask of each item in turn, _Is this tactic a Home Run?_ If it is, mark it with "HR" for Home Run.
What you're looking for are those tactics that are _both_ Low-Hanging Fruit and Home Runs; these are your Sweet Spot ideas, the highest-leverage choices to push back your Limiting Factor. Low-Hanging Fruit are easy to implement with high odds of success, and Home Runs offer big impact if they work. These Sweet Spots are the best tactics to focus your company's resources on first.
Finally, now that you've identified your Sweet Spot tactics, turn them into a mini–action plan of who does what by when. (See Figure 4.2 for a sample Sweet Spot Analysis.)
Put Your Key Strategic Decisions on Trial
It's been said that the enemy of the best is not the worst, it's the good that wastes resources and distracts you from investing your resources in the best. This directly applies in your business as you likely are frittering away your resources (such as staff time, focus, money, and customer attention) on too many choices.
This is again the concept of opportunity cost. The things you are working on are "good," but there may be "great" things that would yield a better return if you focused your resources there instead. This is why we are coaching you to put your key strategic decisions on trial every six months. While not every choice is critical for every business, generally three to five of them strongly influence your company's growth. Pick the three to five strategic decisions that matter most to your company, and put them on trial twice a year. Here are the seven strategic decisions you can examine:
FIGURE 4.2: THE SWEET SPOT ANALYSIS TOOL
**1. Choice of target** **market**. Who will you focus on selling to and who will you ignore? Which markets are the easiest for you to dominate? Cost the most to reach? Are the most profitable for you? Taking all this into consideration, which clients should you fire? Phase out? Refuse to take on? Aggressively court? Who should your customers _really_ be if you want to reach your business's Singular Goal?
**2. Choice of product or service**. Which products or services will you develop and promote? Which of your products have the best margins? The strongest competitive advantages? Are the easiest for you to scale? Which products or services should you eliminate? Phase out? Aggressively promote?
**3. Choice of business model**. How will you profitably charge for the value you bring to the market, and who will be paying for that value? Will customers buy a product or rent a hosted solution? Will you sell direct to your market or through a wholesale model? Will you use physical locations, mobile locations, conferences, or sell online? If you didn't have your current investment in how you do things and were starting over today totally fresh, what business model for your company would most excite you?
**4. Choice of pricing.** How will you price your product or service? In relationship to your costs? Your competitors? The cost of your customer's pain? Should you price with one large payment due up front or with an ongoing subscription fee paid monthly? Or offer long-term financing to your customers? If you have a limited supply of your product or service and strong demand, how do you use pricing to control demand? It's all too common for business owners to have legacy pricing or pricing models left over from when they first opened their business. Back then, you often had lower costs since you were operating on a shoestring, and priced relative to your competition with the goal of being lower cost than they were. Well, that might have served you once upon a time, but does it still? If you were starting fresh with how you price and the payment options you offer clients, what would you choose?
**5. Choice of marketing channel or sales model.** What is the main means by which you market and sell your product or service? Do you use an inside tele-sales force or a field sales team? Do you use strategic partners who already have existing relationships with your target market as your main means to sell, or do you build your own marketing list? What marketing or sales channels are working well for you that you should scale? Which are ineffective and should be cut or phased out?
**6. Choice of positioning and branding.** What "parking spot" have you chosen for your company? What is your essential brand promise and your top three brand emotions? Is this choice of position and branding consistent with how you are perceived by your market? Is it consistent with your real strengths? If you were a third-party consultant looking at your business, what choice of positioning and branding would you recommend the company go after and why?
**7. Choice of product pathway.** How will your customer likely move through your suite of products or services? Which will be your "gateway" purchase? What is the desired flow of purchases they will make to optimally benefit from your product or service? How does this optimal purchase pathway relate to how you currently do business? What changes or adjustments do you need to make?
All too often business owners get so caught up in the day-to-day adrenaline of running their business that they forget to step back and ask the hard, strategic questions that would have such a dramatic impact on their business. Take Thomas Jordan, the owner of a successful mobile bottling company.
Jordan's company, Peregrine Mobile Bottling, mainly served wineries in Northern California and Oregon. The business was doing well, but he wanted a clearer path forward to continue his growth and knew he needed a better strategy to scale. With a little prompting from us, he stepped back and put his key strategic decisions on trial, especially his choices of market and service offering. Although the fifty-five-foot production trailer that he and his team brought on-site to bottle for his customers was technologically head and shoulders better than his competitors', on many jobs, he wasn't able to price in this greater value because of the customers he was serving.
Thomas recognized that he needed to focus on the niches that would pay a premium for the higher quality and other advantages his equipment offered that his direct competitors simply couldn't match. He chose to focus on the higher-end wineries that valued his company's greater precision and quality. He also went after wineries that produced semisparkling wines because he was the only mobile bottler in the market who could provide this value-add in his services. By leveraging his technological advantage to focus on semisparkling wine producers, Thomas was able to grow his business by over 35 percent in less than 12 months. By putting two of his strategic decisions on trial, he found that his best growth strategy became obvious.
PART THREE:
Choosing Your Business Strategy
Now that you have clarified the bigger picture, explored the biggest leverage points inside your company, and put your key strategic decisions on trial, it is time to pick the strategy that will help you best harness one or more of these leverage points to accomplish your big-picture goals.
Here is a six-question sequence you can use to craft the best strategy for your business:
**1. What criteria would a successful strategy have to satisfy?** In other words, what would a winning strategy need to do for you in order for it to work? For Thomas, a winning strategy would have to help him command a premium price by working with customers who valued his higher quality and technological capabilities, dominate a niche market that was large enough for him to grow 200 percent over a three-year period, and differentiate his value offering relative to any alternatives the market had.
**2. What possible strategies could you choose that would likely meet all or most of the above criteria?** Brainstorm a list of all the potential choices you have. Thomas's potential strategies included aggressively courting semisparkling wine producers and hard cider producers.
Which of the potential strategies seems best suited to give you what you want while relying primarily on your current strengths? Remember that you are trying to win a gold medal in something, so always focus on the things that you and your company do exceptionally well. For Thomas, his strength was his technical advantage to not only bottle liquids, but also control the carbonation during the process.
**3. In order for this strategy to be successful, what assumptions would have to prove true?** For Thomas, this included the assumption that consumers wanted semisparkling wines, and that he could manufacture these wines at attractive prices for the wineries that were his customers.
**4. How could you measure or observe this strategy in action to see if it is in fact working?** What should you pay attention to that would provide objective data about whether your assumptions are true and the strategy is in fact working? Create a dashboard internally for your assumptions so that you can closely monitor whether or not they are proving true, or if adjustments need to be made. Thomas researched wine sales data to verify the upward trend in consumption and sales of semisparkling wines. He also test-marketed his carbonation bottling services to his customers, measuring the price premium he was able to command.
**5. How would you know that your strategy wasn't working, and at what point should you stop and choose another strategy?** What is the stop-loss point that signals you've reached your threshold for investing in this strategy? It's important to determine in advance what success would look like, and also what failure would look like, so you don't keep investing time and money once you've hit your stop-loss threshold.
**6. If you should hit that stop-loss point, what is your rollback plan?** What is your plan B and even plan C to handle the contingency of your chosen strategy not working?
Here is another case study of how this question cascade can help you find your winning business strategy. Genevieve was the founder of the online babysitting and pet-sitting service Sittercity .com. She originally started the company to help solve a problem that she herself was facing—finding reliable sitters that were prescreened and responsible hires. She originally marketed direct to consumers, and the business worked, growing well over the first seven years. Then growth slowed to a halt. Her company had reached a plateau that she didn't know how to break through.
Rather than just keep pushing harder with her existing strategy, or even picking a new one, Genevieve followed the process we just shared with you. She laid out the criteria that a winning strategy would have to produce for her. The strategy needed to give her a way to market one-to-many versus one-to-one, so she could sell one larger group or association on her service and they would promote the service to all of their members; the groups or associations she worked with would have to have a large membership base to promote her service to; those members would have to be people with kids and/or pets; the members would need to be people who traveled a lot.
When she brainstormed all the potential strategies that could match her criteria, she kept coming back to the idea of marketing to the U.S. Department of Defense. Its employees were relatively young, many with kids and pets at home. Plus they regularly had to travel, both on short and long trips. Approaching the Department of Defense was a scary move for Genevieve as she had never sold to the military before, but she knew it was worth trying. She contacted the Department of Defense offering to provide a solution to one of its employees' problems—how to care for their children or pets when they traveled. In essence, Genevieve said, "Hey, check me and my company out thoroughly. Then and only then let your employees know about our Web site. Your employees will get a discount on our services and you'll have happier employees." The strategy worked and worked big. Once she was reviewed and approved, the Department of Defense promoted her site to close to a million of its employees!
In her past, Genevieve would have dismissed this strategy because it intimidated her. But when she went through the structured questions you just learned, she realized that pursuing the Department of Defense really did potentially meet all her key criteria. It paid off handsomely too. She doubled her business over a 24-month period and Sittercity.com has continued to grow.
Genevieve and Thomas are great examples of this seven-step process in action. You can see by these structured questions that we approach crafting your strategy with the real bias of iterating and testing to find the winning strategy. Once you've found your proven winner, we believe strongly in the power of doubling down and pressing that winning strategy forward. The key is that rarely will you know which strategy is the winning one right from the start. Smart entrepreneurs accept that they don't have all the answers and take smart, small steps that yield great results, if they work. And if they don't work, these missteps are survivable and yield great insights to help owners more effectively take their next steps.
Yes, this approach takes a little bit more time—emphasis on "little bit"—but it dramatically increases your chances of success. It protects you from blindly choosing a strategy and jumping in with all you have before you have proven it will work. It prods you to challenge your status quo and think through your strategy freshly. And it keeps you from stubbornly sticking with a failing strategy far past the point where you should have known it was a bust. All of this from a simple six-step process that takes less than 30 minutes of time to think through? We think that's a great return on investment.
PART FOUR:
Applying Reverse Leverage by Saying No
There are two layers to any strategy. The first layer is your choice of big-picture path to accomplish your key objectives and goals. This could be the choice of a new product to develop and launch, the selection of two markets to clearly focus on, the creation of a new pricing structure to sell your service under, or even a combination of several of these elements. This is the "theoretical" part of creating your strategy.
There is a second, deep layer to strategy—the practical, "on the ground" reality of how your company's daily decisions about where it focuses resources cumulatively define what your strategy actually is in the real world. You might say your strategy is X, but if day-to-day you don't back that up by actually investing your key resources (e.g., employee time, customer attention, money, equipment, capacity, etc.) in this area, then this reality isn't your strategy. Your _real_ strategy is the one that gets the resources, regardless of what you _say_ your strategy is.
If we were there with you in your business as informal observers and advisers, what would we actually _see_ your business investing resources in? Whatever your business consistently invests its best resources in defines what your real strategy is. This can be a painful realization when you finally see how much of your resources are being wasted and that your beautiful strategy has no legs. Further, despite what your strategy is, or whatever the posters on your walls say, your team will always respond to the things that you yourself seem focused on. They will spend their best efforts on the things you actually judge them on, not the theoretical plan you created at the beginning of the year. Deploy your best resources to do the things that can actually help you win.
We once worked with a company that had developed a product that could reliably deliver important messages _securely_ to key personnel, and then report back on their engagement around that message, telling the sender if they read it and acted on it. Their official company strategy said they were marketing this tool as an HR solution for delivering highly sensitive communications that needed to be responded to. When we visited the company, however, we saw that their marketing and sales teams weren't actually doing this. Instead, they were spending the majority of their time and money creating collateral material to show why people should use their product instead of e-mail. Their sales team was attending conferences in the e-mail industry. In essence, the marketing and sales teams were acting as if the strategy was to sell to the IT departments of large companies, not through the HR door.
We pointed out to them that if their goal was to get engagement from HR departments, then their people should be spending all their time and effort visiting HR departments. They took our feedback and acted on it, reassigning their best resources to focus on the direct tasks of working with HR managers instead of the indirect tasks of trying to get in through the IT departments. The result of this realignment with their stated strategy was immediate. Within a six-month period they increased sales by over 80 percent.
The idea of "reverse leverage" is to say no to something ordinary so you can free up resources for better, more potent things. Considering your chosen business strategy, what activities or initiatives should your business reduce, discontinue, phase out, or avoid altogether? Actually list these activities or behaviors on a "Stop Doing" list. Every quarter, revisit this list, adding new items that might otherwise tempt you to squander your limited company resources.
PART FIVE:
Your Capabilities—Thinking Ahead About What Ingredients You'll Need to Achieve Your Business's Singular Goal
You've likely heard the expression that the best time to plant a shade tree is 50 years ago. Looking back, what capabilities do you wish you would have cultivated as a business five years ago? Although we can't give you a way to go back in time to have done just that, we can prompt you to make sure that five years from today, you answer that same question very differently.
Ask yourself: Over the next three to five years, what capabilities will my company need to cultivate, acquire, or tap into to achieve its Singular Goal? These capabilities could include the competencies and strengths, the staff, the products or services, the resources, the systems and controls, and the strategic elements necessary for your business to achieve its top long-term goals.
Let's pretend you were an injection-molding plastics manufacturer in the United States in early 2000. You would have seen more and more of your customers ordering from competitors in China and India, driving down prices, shrinking margins, and commoditizing the products you sell. What would you do?
This was exactly the situation in which Stephanie Harkness, owner of Pacific Plastics and Engineering, found herself. She and her husband Jack had spent more than ten years in the business, building their company into a real success, but all of that was threatened by the increased presence of overseas competitors. She realized that her company needed to develop the capacity to compete on price for large orders, which meant that it needed an overseas production partner. In her case, Stephanie decided to co-venture on a facility in India. Now, when her company approached customers, it had the ability to offer both faster turnaround from its United States–based facility and also lower-cost production offshore, if that is what the customer wanted. It was a clear growth move.
Stephanie also knew that her exit strategy—selling the business—called for her company to establish itself as a leader in a lucrative niche that was of much higher value than the traditional injection-molding plastics work they had done in the past. She determined that this niche was the world of medical-device manufacturing. This was not a market they were able to step into overnight; it took several years to develop the skills, experience, and accreditation that allowed them to add this value offering to their mix. Once they did, however, the impact was massive. When Stephanie and Jack went to sell their company, they commanded a huge premium (three times more than their competitors') because of how valuable these enhanced capabilities were to their market.
It's important to think about trends and new developments in this process. For example, digital media was a new technology that seemed to have no home when it first came out, so many companies ignored it, including those in the music industry. If they had wondered more about possible uses of this new technology, they might have predicted the MP3 and digital music formats and started retooling their businesses in advance.
Now it's your turn. Take five minutes to look at your capabilities relative to your Singular Goal. What are the positives, the needed capabilities you have currently that are strengths of your company? What are the negatives, the capabilities you don't currently have but will need if you are somehow going to reach your Singular Goal? What steps can you take this quarter to enhance your capabilities so that you'll be better positioned as a company to achieve your Singular Goal?
The Power of a Rolling, One-Page Quarterly Action Plan
At this point, you've looked at your company from the highest level. You know why you're in business, whom you serve, what they really want from you, and what your company's top objectives are. You've also identified the key leverage points and big-picture strategy you'll use to reach your business goals. So how do you make all of this actually work in the real world?
The key to executing on your strategy to accomplish your business goals lies in the quarter. The quarter is the perfect unit of time to bridge your big-picture goals, which likely have a two- to five-year timeline or longer, and your weekly planning and daily action. It's long enough that you can get meaningful units of work done that collectively bring you closer to your long-term goals, but short enough so that you can frequently course-correct and hold your focus.
We want to walk you through a simple system to create your one-page quarterly action plan. Yes, you read right: one page! You're busy, overwhelmed with competing demands for your time and attention—and so is your team. Your one-page quarterly action plan accepts this reality and lets you have a clear, visual anchor to hold your company's focus true for 90-day sprints. A one-page plan forces you to distill your key action items into a short list of prioritized demands that you can see in one whole place with a single glance.
Here is our three-step process to craft your one-page quarterly action plan.
FIGURE 4.3: SAMPLE 1-PAGE QUARTERLY ACTION PLAN
Step One: Pick Your Top Three "Focus Areas" for the Quarter
Every quarter you'll sit down and decide on what the top three Focus Areas for your business are for that quarter. You may decide that your Focus Areas are increasing your lead flow, improving your sales conversion system, speeding up your collections cycle, or making a specific key hire. Your Focus Areas are the three most important areas for your business to spotlight during the coming quarter. Sure, you'll still have to "take care of business," dealing with your normal operational needs to push the business forward, but your Focus Areas will pinpoint where you will invest a portion of your _best_ resources that quarter because you know that these areas are what will really help you scale and develop your business.
Why do we limit your company to three Focus Areas? Why not four or seven? Because too many top priorities means you have no top priorities. Ninety days comes fast, and if you spread your company too thin, you'll find that you partially do more things instead of fully doing a few key chunks that actually produce value for your company. Not only is this a waste of resources, but it is incredibly frustrating for your team, who crave clear priorities and strategic direction. If you want, choose two or only one Focus Area for the quarter. Just make sure you cap your choices at three.
What should you choose for your Focus Areas? Below is a simple cheat sheet to help you determine what Focus Areas to pick for your business for the coming quarter.
Focus Area One: Pushing back your current Limiting Factor. **** Your first Focus Area should be something that directly helps you push back your number-one Limiting Factor for your business. By definition, your Limiting Factor is the single biggest current constriction to growth, and hence it is a great leverage point. If you take concrete steps each quarter to push back your current number-one Limiting Factor, you'll be putting your resources where they will do great good in growing your business.
Focus Area Two: Seizing one of your biggest opportunities. **** Your second Focus Area should be about seizing one of your company's biggest opportunities. You win the game of business by effectively leveraging big opportunities, not by inching your way along accepting the status quo. That is why each quarter you should choose one of your biggest opportunities to invest some of your best resources in. Often some of the best opportunities will take more than one quarter to seize, and as such you may find yourself working on this Focus Area for several quarters.
Focus Area Three: Mitigate one of your gravest threats. **** As you uncovered in your S-O-O-T Review, every business faces threats that could deeply harm or even kill it. That's why for your third Focus Area, we recommend looking at your short list of the top threats facing your business, and picking one of those threats to mitigate this quarter. Maybe you can't eliminate it completely in one quarter, but you can take definite steps to reduce your business's exposure to this danger.
Step Two: Clarify the Criteria of Success for Each of Your Three Focus Areas
Now that you've picked your three Focus Areas for the quarter, it is time to clarify your criteria of success for each. What would you need to accomplish this quarter in order to feel successful in this Focus Area? Be ruthlessly realistic about what is possible for you to accomplish in 90 days. Generally we suggest you try to pick criteria of success that you have control over (or at the very least over which you have a great deal of influence). It's important to look for criteria that are as objectively and quantitatively measurable as possible. When criteria are too subjective, you may reach the end of the quarter without agreeing on whether or not you succeeded.
Also, we suggest that for every Focus Area you pick one "Key Performance Indicator" (KPI) to track. If you look to this KPI to determine your performance, you'll know if your company is on track to succeed in this Focus Area (more on KPIs in chapter 11). In laying out the criteria of success for each Focus Area, not only are you defining what success will look like in this 90-day sprint, but you now have a clear yardstick against which to measure progress as you go. Plus, laying out your criteria of success for each Focus Area _before_ you map out your action steps provides clear clues to what you'll actually need to do over the quarter. Most of your action steps are evident in your criteria of success.
Step Three: Lay Out Your Key Action Steps and Milestones for This Quarter
Now is the time to lay out the key action steps and milestones you need to take or reach to accomplish your criteria of success for each Focus Area over the coming quarter. In order to keep your plan to one page, you'll likely break each Focus Area down into five to seven action steps and milestones. While your plan must be detailed enough to guide your actions, it must not be so detailed that you feel overwhelmed or lose yourself in the minutiae.
If your quarterly plan creeps into 2, or 3, or 12 pages, you just won't use it. We know this. When you have a one-page plan, you can easily keep a printout of it on your desk. You'll refer to it each week to see what key steps you need to take that week to progress your business. And your team will digest and use the plan too. Of course, you can go into greater detail on one of your Focus Areas to help you plan better, but do that in a separate document and not your one-page plan.
For each action step, pick a specific team member to be ultimately responsible for executing the step by a definite date. While you can have multiple people contribute to a specific step or steps, you need to pick one person who is tasked with the responsibility and authority to get that step done and done well. We say that this person "owns" the task. This sense of ownership is critical to your success. It's hard to hold anyone accountable for missed milestones when it wasn't clear who was really responsible in the first place. With this structure, the owner doesn't have to do all of the work herself—she just needs to be responsible for making sure that it gets done in the best way possible within the company.
Congratulations! You now have your first quarterly action plan. Here's a quick recap of our strategic planning framework.
Annually: Full Big-Picture Strategic Review. Is the destination still the same? Is your overall strategy working? What major adjustments need to be made? How is the world changing around you?
Quarterly: Strategic Execution. How effectively are you executing on your strategy? Is it still the best strategy to follow? What has changed in the world around you? What fewer, better things do you need to focus on this quarter to grow and develop your business? What tactical changes need to be made to best execute your strategy?
Weekly: Accountability on Deliverables. What three items are most important for you and your team to do this week? Did you get your items done for last week? What did you learn? Are you still on plan? What tactical adjustments need to be made? (We'll have quite a bit more to say starting on page 209 about creating the best accountability structure to get results.)
Together, your big-picture strategy, quarterly one-page action plan, and weekly execution will give you and your team a powerful and comprehensive framework to scale your company. In the next chapter, we'll share with you one more critical component (learning to read the world) to make sure your strategic planning doesn't become complacent or anemic.
SCALING PRINCIPLE FOUR: CREATE THE RIGHT STRATEGIC PLAN AND REDUCE THAT PLAN INTO A SERIES OF ROLLING, ONE-PAGE QUARTERLY ACTION PLANS THAT HELP YOU EXECUTE AND GET RESULTS.
## PRINCIPLE FIVE
LEARN TO READ THE WORLD SO YOU BUILD FOR TOMORROW'S MARKETPLACE
In August 2000, Jeff was working as the CEO of an entertainment start-up. He was trying to make a remake of the hit movie _Grease_ with pop sensation 'N Sync in the starring role. He knew the band and was spending the day with them while they did a promotional event at the flagship store of one of the largest music retailers in the world. While his friendship with the band didn't make him one ounce cooler, it did give him a unique view into the inner workings of the music industry.
Because of the immense popularity of the band at that time, the CEO and several other top executives of the music company were in the store for the event. Watching people come in and out of the massive store to buy music, Jeff turned to the company's CEO and key leaders and asked why they thought people shopped in their stores.
"To buy CDs," they told him. When Jeff replied, "I don't think so. Nobody here or anywhere wants to buy a CD," they looked at him like he was nuts. They responded indignantly, "Do you have any idea how many millions of CDs we sell a year?"
Jeff pushed further. "Nobody in the world wakes up in the morning thinking to themselves, 'Wow, I wish I was holding a round piece of plastic with a hole in it right now.' They wake up in the morning thinking, 'I want to hear that new song right now!' They _have_ to buy a CD, but what they _want_ is to listen to a song right now!"
Walking away in disgust at Jeff's apparent stupidity, the CEO said to him, "What's the difference?"
The difference is this: If the music industry had understood that it existed not to sell records or CDs but to offer the fastest, easiest way to let you hear the song you wanted to hear, they would have invented the iPod and iTunes. Instead, they got steamrolled by Apple, a company _not even in the music industry_!
Now Apple makes billions of dollars selling music online, while record companies and music stores have suffered years of massively declining sales. And that major music retailer? They are no longer in business, having filed for bankruptcy the first time in 2004, and finally closed their doors for good in December 2006, six years after Jeff's visit to the store. All because the music industry let its vested interests blind it from seeing what its customers really wanted and the ways in which the world was changing. What they didn't do was constantly scan the world around them to see what was coming next and make the hard choices to reinvent themselves before someone else came along and pulled the rug out from under their feet.
They kept saying, "We're doing fine," right up until the day they went over the fiscal cliff. Those words, "We're doing fine," may just be the three most dangerous words that any business owner can say. In a rapidly changing world, the status quo never stays static for long.
If you think the upheaval of the music industry was a fluke, think again. Remember Arthur Andersen? How about Swissair? Or Circuit City? Or Hollywood Video? For years each of these companies was "doing fine," feeling like they could do no wrong. That is, until they day they went out of business or into bankruptcy.
We are not trying to scare you, but we do want to strongly remind you that your business cannot rest on its laurels. Scaling Principle Five says that you must learn to read the world so that you stay relevant and build your business for tomorrow's marketplace, not yesterday's reality. If you don't, your business will become marginalized and dated. That's why it is so important that you learn to read the world, to connect the dots between trends and ideas from various parts of the world to your business. This ensures that you are building your business with an eye on the future.
In the 1930s in St. Louis, Missouri, Grand National Bank added a new innovation to banking—the drive-up teller. For the first time ever, bank customers could make deposits or withdraw money from the comfort of their car. It took over a decade for In-N-Out Burger founder Harry Snyder to connect the dots and apply this innovation to restaurants when he opened his first In-N-Out restaurant with a drive-through lane in 1948. Can you imagine a fast food restaurant today without drive-through windows? This was an idea that was invented in one industry (banking) and took over a decade to be applied to another industry (restaurants).
When was the last time you left your industry for a day to go explore another, seemingly unrelated one? Have you _ever_ done that? Probably not. Most people don't. It may never have seemed necessary or important for you as a restaurateur to look for ideas at your local bank, or your local medical doctor, or your local car dealer. Likely you felt the problems you faced in your industry were unique. But the world is a smaller place today, and trends tend to move freely across industries and geographic boundaries. If you don't broaden your world to let strange ideas connect, you just might be missing out on your next big innovation—and as a result, you could get left behind by your competition.
In this chapter we are going to share with you a proven two-step process you can use to read the world around you, and to use those insights in growing your company.
Step One: Stimulate Yourself with New Inputs
As humans, our default setting is to fall into habitual patterns. We drive the same way to work each day. We shop at the same stores each week. We visit the same Web sites and read the same magazines each month. It's almost as if we go on autopilot as we live much of our lives.
But creative breakthroughs don't come from following routine. They come from seeing the world in a new way. This is why the first step in this process is to regularly explore the world, letting your natural curiosity pull you along. One of the best ways to be creative is to let outside stimuli help you make a strange connection between two seemingly unrelated ideas. This is what comedians do that startles and delights us as they point out the outrageous craziness in our daily lives. This is what business innovators do too; they let an idea from one part of their world collide with a problem or opportunity they are facing in another area, and then they connect the dots.
So the first step in this innovation process is to consciously make time to stream new ideas and inputs through your life to spark magic.
Set aside 30 minutes a week to:
* **Flip through a magazine that you'd normally never pick up.** Read with a fat Sharpie marker in hand and circle, underline, doodle, and annotate as you go.
* **Watch a Webinar for a totally unrelated industry.**
* **Visit a business that you've never seen before** and ask a ton of questions about why they do what they do.
* **Google a topic that you've never Googled before** and follow the links down the rabbit hole to see where it takes you.
* **Play the "Mash Up" game.** Ask yourself what it would look like if you took two or more unrelated ideas that you saw and applied them to your business.
* **Spend a few hours with a group of people who are from a radically different age, culture, or industry from yours.** Ask them questions about their world (we'll cover several great questions to ask them later in this chapter) and take notes on their answers. You can do this by paying your teenage daughter and four of her friends $20 each to give you an hour of their time. Or you can attend a trade show for another industry and interview three of the people there for their answers. (If you're over 40, we strongly recommend you start by interviewing a group of 16- to 22-year-olds. You'll be blown away by how differently they see the world.)
The goal of this step is to let two things happen. First, to let the magic of randomness mix things up so that when you follow your habitual thinking patterns, you're starting from a different place and working with a slightly different set of initial conditions. You'll be amazed at how radically this can lead you to new conclusions. The second goal of your regular 30-minute explorations is to let new ideas collide, knowing that out of some of those collisions new breakthrough ideas are born.
Keep a journal of the ideas you come across that you find intriguing or moving or frustrating or exciting. David keeps his "reading the world" ideas in the popular Evernote app that collects notes and reminders. He clips articles or information from Web sites and magazines and pastes them there, so these ideas are always with him. Jeff does the same thing. Every time he reads or hears something interesting, he makes a note using Notepad on his phone or computer. A few times a week, he looks at all those interesting data points to see if he can connect all the dots to create something new, a new product or way of thinking that might not even have been possible yesterday.
Consider each new thing you learn as another puzzle piece in the game of business, and constantly shuffle the puzzle pieces around on the table to see what you can make of them. This isn't a big or tedious research project. Just jot down simple one-line bullet points, or quickly cut and paste interesting photos or headlines. The idea is to start a blender of new thoughts and ideas whirling in your head, and to let those ideas mix and recombine.
Step Two: Engage in Structured Innovation Sessions Every Quarter
We suggest that as part of your quarterly strategic planning process, you and your executive team (if you have one; solo if you don't) take 45 minutes to creatively reimagine the world and brainstorm ideas by answering the following six provocative questions.
Question One: "What can I do today that I couldn't do yesterday?"
(E.g., _share a video with the world from my cell phone; have 2,000 people join me for a Webinar; video conference with five of my team members for effectively no cost; access hosted solution software for nominal fees where once I would have had to pay thousands or millions to develop the tool on my own;_ etc.) The goal of this question is to make sure you constantly reassess the landscape of possibilities. The world changes so fast now that every morning when you wake up, something has changed that may create a brand-new opportunity for your business.
Question Two: "What things do I observe in the world that . . ."
**Startle and Surprise Me:** (E.g., _how ubiquitous smart phones have become—I feel lost when mine isn't at hand or when it's out of power; how when a question in conversation comes up that we don't know the answer to, someone just Googles the answer on the spot; how little direct mail I get in the mail anymore;_ etc.) Identifying the things in the world that surprise you prompts you to integrate relevant changes into your business.
**Delight and Intrigue Me:** (E.g., _how Wikipedia makes finding an answer so easy; how convenient it is to order off Amazon.com; how I can be a voyeur in the lives of old acquaintances on Facebook without even talking with them; how much I live my life to a soundtrack of my choosing;_ etc.) Think of ways to apply these intriguing new tools to the work you do for your customers.
**Terrify Me:** (E.g., _how short people's attention spans are now; how overwhelmed with choices I feel; how powerfully the media plays up fear, making the world appear so unsafe and dangerous; how intermixed the economy is and how things that I have no control over can wreck my business;_ etc.) This exercise can help you spot threats from outside your industry or your normal range of competitors that you might otherwise have never seen coming.
Question Three: How has the way people are buying things in the world changed (especially outside of my industry)?
(E.g., _how_ w _e consume bite by bite with micropayments; how we comparison shop online, using aggregated Web sites and comparison engines to find products that we may not have known existed; how we listen to what other people say about the product via customer reviews, and how we no longer rely on the seller or advertising to educate ourselves; how we use new forms of payment that didn't exist a few years ago;_ etc.)
Question Four: What is changing in the lives of our customers (whether it has anything to do with our company or not)?
(E.g., _they are getting older as a whole [on average five to ten years older]; they are spending two hours a day on social media; regulatory restrictions are making their lives much tougher and increasing the time they must spend per site on compliance; they are struggling to find the right workforce domestically;_ etc.)
Question Five: "Wouldn't it be cool if . . ." (Come up with at least ten endings to this statement.)
(E.g., _wouldn't it be cool if we had a waiting list for our service that was booked six months out; wouldn't it be cool if I could eliminate all spam from my inbox; wouldn't it be cool if customers paid us to give them a sales quote; wouldn't it be cool if we had an app that thousands of our customers used that taught us exactly what they loved and used with our product; wouldn't it be cool if I could be onstage playing with a rock band;_ etc.)
All five of these questions are designed to help you spark new ideas, to effectively see the world differently than you normally would. Why? Because we filter our own ideas with our preconceptions of what will work and what won't—and we base that on what worked in the past, not what might work in the future. We need to open our minds and toss out some crazy ideas in order to find true innovations. By themselves, these ideas may not be concrete enough for you to take action on. But when you mix a little creativity with an idea you brainstormed, you can come up with some pretty cool stuff. For example, take that last idea, "Wouldn't it be cool if I could be onstage playing with a rock band?" Think of the ways you could apply this to a business. If you were a concert promoter, could you have a journalist or blogger embedded onstage writing about the experience and sharing that experience in real time? Or maybe you could get a celebrity friend of the band with a big following to do it? Could you hold a contest to give three fans the chance to join the band and let the band's fan base choose the winners online?
What if you didn't have a concert promotion business and instead own a dry-cleaning business (about as far from the cool of a rock band as we can think of at the moment)? Could you have a local reporter spend a day behind the scenes and write about what it really takes to deliver your beautifully cleaned and pressed clothes? Or could you follow the day in the life of a rock band's stage clothing in your video blog with a final press release afterward? (Oh, and don't forget the huge signed poster of the band thanking your cleaners to hang in your store.) Or maybe you do it from the frame of a local business tycoon. (Can you see the suit sitting at the closing of some big deal?) Even if none of these ideas are what you choose, they still move your thinking to new territory, and all the rest of your strategy session will be infused with a new energy.
Question Six: Imagine you were starting to build your business from scratch all over again today. What business would you ideally build? What would it look like? Why?
Pretending that your business didn't exist at all and you were starting from scratch today allows you to get creative about your business without worrying about all the existing worries, challenges, givens, and investments that you normally take for granted.
An amazing example of this is when Apple decided to "cannibalize" its own product line. Apple was making lots of money selling iPods. But they saw that one day someone would be able to add music players to phones. Instead of waiting for someone else to kill its iPod business and then struggling to find a revenue stream to replace it, Apple launched its next growth engine right away by developing iPhones that had music players on them, so customers no longer needed to purchase its previous product. It was a bold and painful move in the present that created a huge future for Apple.
Now that you've stretched the way you normally look at the world and your business, it's time to look for ways to apply your insights to your business. What insights came up through this process that you can no longer afford to ignore? The question is no longer whether or not you can afford to do R&D and risk some potentially unsuccessful experiments. The question is, can you afford _not_ to?
Instead of starting from your existing products and company, or even what other companies in your industry do, pretend none of it exists. When most companies try to innovate, they look at their existing business, products, and industry, and try to find a way to make it better. If you have been in the health care business for 20 years, you won't achieve a breakthrough by staring at 20 years of health care products and services. Instead, you'll make small linear improvements and create health care 21.0.
In order to really stretch your thinking when it comes to your company, imagine what you would do if you were starting your company from scratch _right now_ , using everything you've learned, and all the new technologies and techniques that were not available when you first founded your company. Go a step further—pretend there are absolutely no rules, no regulations, no requirements—even the laws of physics could be broken! Don't edit your ideas by what's possible or practical; imagine anything is possible and there are no constraints. Go wild and think of ideas with the abandon of a seven-year-old child! When you start with the way you would build your company if there were no rules, and then edit it back to reality, you wind up way farther ahead than if you start with your existing business and try to improve it.
Here's a real example from one of the regular innovation sessions that Jeff's company, Colorjar, runs. His team was focused on a client working to innovate in the fast-food industry. The goal was to improve the speed at which you get your lunch in the drive-through lane. Instead of going inside a McDonald's to look at ways to improve the existing systems, his team pretended they were designing a new drive-through lane that would exist in a world with no gravity—literally! With these constraints eliminated, how would an ideal ordering system work? One of the Colorjar team members suggested that customers could text their order to McDonald's, then, without ever stopping the car, unroll their windows as they pulled through the drive-in lane. Restaurant employees would use the same cannons used to fire T-shirts into the crowd at concerts to fire lunches into the moving cars. Crazy, yes, and very unlikely to happen tomorrow. But that wide-open thinking was the starting point, and designing back from this idea led Jeff's team to come up with a new idea for speeding up drive-throughs: installing an electric eye in the drive-through parking lot that could see your car coming, check your order history, prepare your order, and bill you electronically. Not as much fun as shooting lunch from a cannon, but a really cool idea that went straight to prototype. Regularly reimagine your business using this type of blue-sky methodology so you keep your company fresh and relevant. In a rapidly changing world, it's the only way to keep your business safe.
Continue to read the world and gather cool, interesting, terrifying, intriguing data points for your next innovation session. The journey never ends.
What does thinking big inspire you to build or do? What is the first step and when will you complete it by? Do something. Even a small experiment with a new way of doing something or a new product is better than doing nothing at all. Take that first bold step, and see where it leads you. A journey into imagination sometimes yields powerful and unexpected results. Reading the world is a new discipline that will pay dividends over the long term.
SCALING PRINCIPLE FIVE: LEARN TO READ THE WORLD SO THAT YOU STAY RELEVANT AND BUILD YOUR BUSINESS FOR TOMORROW'S MARKETPLACE, NOT YESTERDAY'S REALITY.
## PART III
## Obstacles to Scaling
(and How to Overcome Them)
Obstacles are a blessing. They give you sturdy stepping-stones to cross the gap from where you are to where you want to go.
## PRINCIPLE SIX
REMOVE THE PREDICTABLE OBSTACLES TO GROWTH—PILLAR BY PILLAR
At its core, every business has five functional areas that carry the load and provide a stable base upon which to scale and grow: Sales/Marketing, Operations, Finance, Team, and Executive Leadership. In the next five chapters, we'll share with you dozens of concrete strategies and tools to help you systematically remove the predictable obstacles to growth, pillar by pillar.
Before we go into each pillar in detail, however, we want to give you a high-level view of these five pillars of your business, and then ask you to evaluate your business's performance in each of them using a tool we call the Five-Pillar Audit, which we've included at the end of this chapter (see Figure 6.1).
Pillar 1: Sales/Marketing
No leads, no sales. No sales, no business. It's that simple. The Sales/Marketing pillar finds clients, makes sales, and generates revenue. It's the part of your business that makes it rain cash.
Sales is everything you do to make your offers as effective as possible and to close selling opportunities. Your offers can be delivered in a variety of ways—from trained sales reps in a call center, to a sales force in the field, to direct response sales letters or print ads, to interactive Web sites, to retail locations.
Marketing is everything you do to get one of your offers in front of the right prospective buyers under the best possible conditions. Marketing crafts your company's identity and positions it in the hearts and minds of your marketplace so you consistently generate the volume of sales leads you need. You want leads that are primed and ready for your selling systems to convert into thrilled clients or repurchasers.
Too many entrepreneurs focus on this pillar only because they have to, not because they want to. They feel intimidated by the idea of selling. But understand this: When your business is in Early and Middle Stage Level Two, it's crucial for you as the company founder to focus a majority of your energy on generating profitable sales. If you don't, your new business won't survive, let alone thrive. Only as you grow your business can you relieve yourself of the responsibilities of this pillar. In the early days of your business, you focus on making sure sales happen, which means meeting with clients and closing deals yourself. Later, however, you must shift your focus to creating repeatable and scalable sales and marketing _systems_ that don't depend on your involvement.
Every pillar has clear component responsibilities it must take care of for the business to succeed. While we'll go into depth on every pillar's core functions in the next five chapters, we want to give you a quick summary of what these responsibilities are as we highlight each pillar.
THE SIX KEY RESPONSIBILITIES OF YOUR SALES/MARKETING PILLAR
1. Defining your target market and knowing what it really wants
2. Clarifying your offer(s)
3. Lead generation
4. Lead conversion
5. Planning and strategy
6. Branding and positioning
Pillar 2: Operations
Your Operations pillar is the part of your business that creates the products or delivers the services your business offers, fulfilling on the promises made by Sales/Marketing. Your Operations pillar also performs the general and administrative back-end functions of your business.
No company will thrive without having a well-organized, strong Operations pillar. Sure, you can generate sales, but unless you're able to fulfill on the promises you've made, your business won't last.
As crazy as it may sound, there are many cases of businesses that were killed by success. Their front-end sales/marketing pillar was so successful that their back-end operations pillar couldn't keep pace. This caused an extreme disconnect between the promises these companies made their customers and the execution in fulfilling those promises. The results were predictable—angry customers and enough bad press to put these companies six feet underground.
A very visible example of this was the online retailer eToys.com. They had a great marketing program and sold thousands of toys for the 2000 holiday season. But their operations department was less than a well-oiled machine, and many families got their holiday presents delivered in January, after Christmas. The company's lack of operational efficiency led to bad press, many lawsuits, and ultimately the death of the company.
THE SIX KEY RESPONSIBILITIES OF YOUR OPERATIONS PILLAR
1. General administration
2. Key company infrastructure (e.g., facilities, equipment, Web sites, etc.)
3. Production and quality control
4. Fulfillment and delivery
5. Customer service
6. Purchasing/cost controls
Pillar 3: Finance
The Finance pillar of your business encompasses all the essential functions of collecting, tracking, distributing, and reporting the flow of money in and out of your business. It includes your billing procedures, collection practices, and accounts payable processes. It also includes all financial reporting from balance sheets to profit and loss statements, and the managing of cash flow. As your business matures, it will also include the management of risk, the sourcing of capital, and the forecasting to help your management team make better business decisions.
THE SIX KEY RESPONSIBILITIES OF YOUR FINANCE PILLAR
1. Bookkeeping and financial reporting
2. Collections (AR)
3. Bill paying (AP)
4. Budgeting, planning, risk management
5. Cash flow management
6. Tax planning/filing
Pillar 4: Team
Your Team pillar establishes how you hire, orient, train, assess, compensate, and if necessary, let go of, your staff. It deals with your policies and procedures for team members and the legal requirements of working with employees.
To grow most businesses, you'll need talented team members to both spark and support that growth. Whether it's adding sales team members to increase sales, engineers to design products, or accounting staff to keep track of the money, your company's greatest source of leverage is its ability to attract, hire, integrate, and empower talented, committed people to play on your team.
Scaling your company and growing your freedom depends on your ability to find, hire, and retain talented people to help you build the systems and controls your business needs to scale.
THE SIX KEY RESPONSIBILITIES OF YOUR TEAM PILLAR
1. Hiring and new team-member orientation
2. Regular staff evaluation, coaching, and training
3. Compensation and benefits administration
4. Compliance procedures for all applicable labor laws
5. Troubleshooting and personality challenges
6. Exit process for both friendly partings and expedited exits (firings)
Pillar 5: Executive Leadership
The final pillar of your business is Executive Leadership—the area that leads your leaders and sets the big-picture direction for your company. It's also likely the final area for you to personally let go of. Remember, many Level Three business owners _choose_ to stay in their businesses as CEOs and continue working. The key distinction is that this is a choice, not a requirement for their companies' survival. Successful business owners groom their successors and ready their organizations to successfully transition leadership when the time is right. They intentionally nurture the culture that helps their team successfully operate the business even when the founder is not there in the room.
On a macro level, your job as leader is establishing what your business stands for and how it sees itself; that is, where it focuses its resources and what its key goals and priorities are, and the values through which it makes its daily decisions. Leaders create the narrative through which all stakeholders interpret the business and their relationship to it. As you grow your business, it becomes increasingly important that you shape the stories and traditions that will become part of your company's heritage. These hold your business on course even when you're no longer present each day to drive it.
On a micro level, you want to create a business in which all team members understand their roles, know what they're responsible for, how success in their role will be measured, and how leaders and team members will provide feedback as they go.
Your most important role as a leader is building a workplace where the most talented people want to come to work and stay. When people who could work anywhere want to work for you, that's when you know you got it right.
THE FOUR KEY RESPONSIBILITIES OF YOUR EXECUTIVE LEADERSHIP PILLAR
1. Strategic planning
2. Leadership development/continuity planning
3. Company culture and traditions
4. Communication company-wide
It's time to map your baseline of precisely where your company currently stands in each of the five pillars. The Five-Pillar Audit tool below will give you a quantitative breakdown of how your business is doing. Take five minutes and complete it now.
FIGURE 6.1: THE FIVE-PILLAR AUDIT
Rate your five business pillars in each of the six subareas listed on a scale from 1 to 10, with 1 being lowest and 10 being highest. For example, under "lead generation," if you think your business does a fantastic job at consistently generating new leads, give yourself a 9 or 10. If your business struggles to find new leads, barely generating enough to keep it afloat, give yourself a 2 or 3. Total your score for each pillar (possible high score of 60 for each pillar) and then total your score for the entire audit. When you're done with this audit, we'll walk you through your score and what it means for your business.
SALES AND MARKETING PILLAR:
Lead generation
Lead conversion
Client repeat business
Client "upgrade" business
Revenue growth (Short term)
Revenue growth (Long term)
Subtotal
OPERATIONS PILLAR:
General administrative function
Performance of client work or fulfillment of client orders
Client's rating of your company's performance
Cost controls for operation of your business
Business infrastructure (Web site, physical location, equipment, etc.)
Scalability of your core product or service
Subtotal
TEAM PILLAR:
Communication systems for team to work together
Having the right team in the right positions
Systems for bringing on new team members
Systems for training and reviewing team members
Compensation and benefits administration
Compliance procedures with all applicable laws
Subtotal
FINANCE PILLAR:
Accurate and timely financial reporting
Budgeting
Financial controls
Collection systems for accounts receivable
Effective management and use of financing
Cash flow management in general
Subtotal
EXECUTIVE LEADERSHIP PILLAR:
Each team member has a clear understanding of the mission, values, and goals of the company, and how her individual role contributes to the bigger picture
Current business strategy
Review process for company performance, direction, strategy, and development
Troubleshooting major challenges when they come up
Leadership training
Company culture and tradition
Subtotal
Complete score for your business (scale of 30–300):
It isn't important that you score high now because what matters is not where you start but where you finish. To end up where you want to be, it's critical to have a reliable way of evaluating your positions along the way. This abridged Level Three Audit is a simple tool to help you do just that.
If you have an **Early or Middle Stage Level Two** business, **** chances are your scores are currently quite low in three or more pillars. This is normal. You'll improve these scores rapidly when you follow the Level Three Road Map to progress your business. The two most important pillars at these stages are your Sales/Marketing pillar and your Operations pillar. If either scored below a 20 (out of a possible 60 for each area), then you have concentrated work ahead to improve these two critical pillars. At this time in your business, if you score low in reliably generating leads, closing on sales, fulfilling on your client promises, and collecting on what you are owed, you need to immediately remedy that situation. _These functions are the minimum requirements to have a sustainable business._ Remember, basic survival is the first hurdle to get to Level Three.
If you own an **Advanced Stage Level Two or Level Three** business, chances are your scores in most pillars are high (above 40 for each pillar). Look for any specific pillar in which your score was below 20. This area of your business needs immediate attention.
We encourage you to repeat this Five-Pillar Audit each quarter. Make sure to date and save your scores so you can note your progress over time. This tool is a simple, structured way to know where and how your business is progressing, and determine where it needs more of your attention.
Now it's time to go back to your business—pillar by pillar—and look for ways to refine each core pillar to create and sustain rapid growth.
SCALING PRINCIPLE SIX: REMOVE THE PREDICTABLE OBSTACLES TO GROWTH, PILLAR BY PILLAR.
## YOUR SALES/MARKETING PILLAR
BUILD SCALABLE LEAD-GENERATION AND CONVERSION SYSTEMS
Every company has to generate leads; every business must convert leads into sales. The fact is, if you want to sustainably scale your business, you are going to have to evolve from growth based on your personal production to growth based on the stable base of systems, team, and controls we've been emphasizing throughout this book. With respect to your Sales/Marketing pillar, this means building with the end in mind—the day when you personally are not needed to generate leads or close sales. In this chapter, we'll keep linking our suggestions for tactics and strategies to grow sales with systematic ways to implement each.
Take the example of Dominique Molina, owner of Certified TaxCoach.com, a specialized training company that trains CPAs in tax planning. She's grown her $1.2-million-a-year business primarily based on two things: giving public talks to industry groups (lead generation) and selling her training program memberships via live Webinars (lead conversion). The downside was that her lead generation was heavily reliant on her getting on a plane to go speak at business conferences and industry meetings. As the mother of a young son, she wanted to scale her company without its being so reliant on her being away from her family.
When we met with Dominique during a coaching session, we learned she had never used any kind of tele-sales team to follow up with all the prospects she had gathered from her public talks who either didn't attend her Webinars or didn't buy on them. Our advice to her on her sales challenge was twofold. First, based on our past experiences with her type of business, we recommended that she develop a tele-salesperson (or team) who could follow up with all her leads that weren't buying on the Webinars. At conservative estimates, we projected she could double her sales using her existing lead flow. The second prong of advice we had for Dominique was that developing her tele-sales capacity would also allow her to radically enhance her "back end" sales opportunities to add higher-value offerings to her existing clients. After running the numbers, we calculated she could grow her sales by another 50 percent. These two simple "sweet spot" strategies could create 150 percent growth for her from her existing lead pool. Further, both would dramatically reduce her business's reliance on her to personally close business since they would be done by a trained salesperson following a clearly written sales script. This is the power of enhancing and systematizing your sales/marketing pillar.
The first step in removing your sales/marketing pillar's obstacles to scaling is to pinpoint your company's most costly lead-generation weaknesses. Use the ten-point checklist below to evaluate your company's lead-generation challenges. Check the box for any of the challenges that currently hurt your business. If you check more than three boxes, which is very common if you're an owner doing this assessment for the first time, go back to the list and circle the _one_ pain point that hurts your business the most. This is where you'll put your initial focus.
LEAD-GENERATION TROUBLESHOOTING CHECKLIST
* **You don't have enough leads to sell to.** Or your lead generation is erratic and you don't have a consistent lead stream you can count on.
* **You don't have a system to organize and manage your leads.** As a result you don't consistently follow up with leads in a timely way and many of your sales opportunities slip between the cracks.
* **You don't have a structured lead-scoring system.** You waste your sales efforts by squandering your best sales resources on variable leads rather than being able to quickly and easily sort your lead pool to separate your highest-quality leads for follow-up first.
* **You don't systematically track your lead-generation efforts.** Without hard data you have no way of determining which of your lead-generation tactics are producing and which aren't. Raw intuition or anecdotal observations are not enough.
* **You don't have a** system **to generate leads.** You or a key staff member has all the "know-how" for implementing a marketing tactic in your heads versus a concrete business system. Your company is vulnerable should the key person not be available to do the marketing.
* **Your current lead-generation processes aren't scalable.** Either the prospect pool you work with is too small or the system itself is limited, and as such you can't scale the basic lead- generation system(s) that you have.
* **Your current cost per lead is too high.** The leads you generate are just too expensive to make sense considering your current conversion rates and average unit of sale. You need less expensive ways to generate leads or better conversion systems to get more from each lead you do generate.
* **Your lead quality is just too poor.** It's like looking for a needle in a haystack of junk leads who either aren't qualified or aren't motivated to buy your product or service.
* **You have a ton of lead-generation ideas but you just don't seem to be able to effectively** implement **them.** You have halfdone many different marketing ideas, but never seem to finish implementing the majority of them so that they actually produce leads for your business.
* **Your marketing is too reliant on you, and as such you "start/stop" it as other demands in the business pull you away.** As a result, you regularly go through cycles of feast or famine where you alternate between focusing totally on producing your product or service offering with no time to market, and likely no real capacity to take on more work, followed by periods of panic where you see your funnel running low and you scramble to start up your marketing again to quickly bring in more business. Not only does this constant cycling back and forth between the two exhaust you, but you never seem to build enough momentum to break out of this trap.
In our experience working with small and midsize businesses, many mistakenly believe their lead-generation woes are about volume: "If only we had more raw, fresh leads to work with . . ." However, lack of leads is generally a result, not a cause. It's almost always a by-product of one of the other lead-generation pain points. For example, maybe you don't have an organized sales database so the leads you _do_ generate get a poorly crafted sales effort and almost no organized sales follow-up. Or perhaps because you don't systematically track your marketing tactics, you can't rationally compare lead-generation tactics so that you end up wasting thousands and thousands of dollars on poorly performing tactics when those same dollars could be reinvested in your best lead-generation producers to scale those winning efforts. Still others may find they regularly go through periods where they have to turn attention away from marketing to other areas of the business, and as a result never seem to build any marketing momentum. To build a Level Three business, you need a lead-generation _system_ that you can count on to produce for you regardless of whether or not you personally were there to be the driving and directive force.
Jeff once worked with a company that sold products and services to hotels and hotel chains. They weren't hitting their sales numbers, so they paid to buy more sales leads. The next quarter they still didn't hit their numbers, but by then they had spent so much money on lead generation that their business was in a vulnerable place. Their real problem wasn't that they lacked leads, though; rather, it was that they had no system for scoring or ranking their sales leads. They had no way of knowing which leads should get their best sales efforts and which weren't worth the costly investment of sales energy. Their lead-generation pain point could have been solved with a lead-scoring system that allowed them to filter and rank all incoming sales leads by factors like: Is their budget to buy already approved? Can they afford our product or service? Have they bought this type of product or service before? Can we get to the actual decision maker?
Don't stop at the surface pain point—you've got to go deeper and ask what causes this lead-generation challenge so that your company invests its limited resources in solving the _real_ problem.
With this understanding, it is time to look at your core system for generating leads.
The Five Steps to Building Your Baseline Lead-Generation System
It's not enough that you learn new tactics to generate leads or close sales. You also need to learn how to build the _systems_ that help you generate leads and close sales. That is why we keep emphasizing the importance of systems to help you scale. The problem with informal systems you keep in your head is that they leave your business vulnerable. If you get hurt or busy, no one else knows how to do that key process. Plus, it's hard to refine a system when you can't see it written out. Perhaps most damaging of all, informal systems that are locked in the minds of you and your key employees just aren't scalable.
Here are five steps to building out your baseline lead-generation system. You've likely already created parts of this system but haven't done so formally.
**1. Determine which marketing tactic(s) to focus on first.**
Pick your single most important lead-generation tactic. This is the one you'll want to systematize first. If you're not sure which of your various lead-generation tactics are most important, ask yourself the following: _If you could only do one thing to generate new leads for your business, what one tactic would you pick?_
**2. Draft the "process layer" of how to implement that marketing tactic.**
Grab a pad of yellow sticky notes and lay out the steps to implement this marketing tactic. Put one step on each sticky note. The reason this is such an effective way to document your process is that it frees your mind to lay out all the steps, and even edit the steps as you go, into a simple flow of yellow stickies. Don't like the order or missed a step? No problem; just move around the stickies to suit your needs. Once you have the process of executing this marketing strategy or tactic clearly laid out in your yellow sticky notes, write it up into a simple, step-by-step recipe.
**3. Create a rough scorecard to track this marketing tactic.**
The key here is to make sure you're gathering relevant, objective data that will tell you how well your marketing efforts are going and allow you to compare tactics in order to make smart strategic decisions about where to invest your company's time and money. These numbers will root any forecasts or projections you make in reality versus the "go by the gut" fantasy too many entrepreneurs accept. (We'll have more to say about your sales/marketing metrics in a moment.)
**4. Package your "process" in a way that ensures your team and business consistently follow the steps to get the results you want.**
As we discussed in chapter 2, every system has two layers to it—the process layer and the format layer. In step two, you wrote out your process layer to this system. Now it's time for you to give some thought to the best way to package each of these steps so that the system is easy to use and works. For example, if the system you're focused on is search engine optimization, tools to package your system could include a one-page "cheat sheet" list of the top keywords; a weekly checklist of the seven most important tasks you want your marketing team to do; an instruction document, including annotated screen shots, that shows exactly how to post text or video blog posts to effectively emphasize keywords; etc. If the system you're focused on is direct mail to targeted lists, tools to package your process could include a tracking spreadsheet to measure the results; your direct mail control letter; a list of your key direct mail vendors with past pricing, terms, and notes; etc.
The essence of this step is to ask yourself, "What is the best format to package the key steps of this system so that our team can more easily follow it to consistently produce quality leads for our company?"
**5. Implement, track, and refine.**
By paying close attention to the results you're getting from your lead-generation efforts, you'll spot opportunities to improve your system. Whether it's the addition of a new tool like a sales video for your Web site, or a five-e-mail follow-up sequence that automates part of your process, you and your team will constantly be finding ways to tweak and improve your baseline lead-generation system so that it consistently produces more and better-quality leads. You need to look for ways to improve both the process and format layers. We suggest that each quarter you and your marketing team (if you have one) sit down and conduct a Sweet Spot Analysis to choose the highest-leverage improvements to your lead-generation system(s) to work on that quarter. Then break down and timeline these upgrades over the quarter, staffing out exactly who is responsible for what results and by when. It's this kind of consistent refinement that, over time, will result in a finely tuned lead-generation machine.
The Four Most Important Marketing Controls
As we previously defined, a control is simply a specialized system that your business uses to make sure that the right steps are being taken, at the right time, to get the right results. Here are the four most important marketing controls to make sure that your marketing system is operating smoothly to consistently produce quality leads for your company.
Marketing Control One: Marketing Calendar
Lay out your key lead-generation campaigns for the next 90 days on a standard calendar. Then add the deadlines for any key steps required to successfully run those campaigns. For example, one retail service business we coached laid out its quarterly marketing calendar, which included a "front counter upgrade" campaign to enroll clients in its Concierge Level service, and a campaign for add-on service. This latter campaign showed up on the calendar with deadlines for completing the draft marketing flyers, printing the information sheet for its counter team about the special promotion, and posting the new point-of-purchase signage in the store.
Your marketing calendar is a great visual control to make sure you are on track with all the precampaign steps necessary to make your lead-generation efforts successful. It also helps make sure you keep the most important steps front and center to your business so that other urgencies don't pull you away from the consistent marketing necessary to build momentum. Perhaps the most important thing your marketing calendar does is allow you to delegate ownership of many (dare we say all) of the steps to your team, since it gives everyone a powerful way to make sure that all your marketing efforts are on track.
It's often useful to compare your marketing calendar with external calendars as well. David once spent $15,000 promoting a business workshop in San Diego only to learn that he chose a date when the local football team was competing for the division championship. You can likely anticipate the outcome—dramatically lower attendance that he could have avoided if he had only checked the dates against local and national events. Make sure you run your marketing calendar through the filter of outside events and holidays to avoid this same costly mistake.
Marketing Control Two: Standardized Marketing Collateral
Whether it's a template e-mail sequence you send to everyone who registers on your Web site, the glossy sales brochure your field sales team uses when prospecting at trade shows, or a white paper prospects can download from your Web site, standardized marketing collateral is an effective "embedded" control to make sure your prospects get your best sales messaging. Not only will your marketing collateral help your prospects learn about your products or services, but they are also great tools to help train new team members in the company and product knowledge they need to be effective on your team. Further, future vendors will many times use your collateral to help them learn about and produce a better result for your business. For example, a new online marketing company you hire to help you improve lead flow from your Web site will likely start by looking at your existing marketing collateral to best understand who your target market is, what their most sensitive hot buttons are, and the branding and positioning of your company.
Don't think of this building out of your marketing collateral as an all-or-nothing proposition. Instead, start small and build out your collateral in bite-size bits. What one or two marketing collateral pieces does your business need most? Create your best version of those items first and get your team to consistently use them. Over time you'll flesh out your marketing collateral with better and better brochures, product slicks, Web pages, catalogs, direct mail pieces, point-of-purchase signage, and much more. You'll get more talented people involved in helping you create the refined versions of these marketing tools. The key is to start where you are and take the first step to pick the highest-value marketing piece to create first. The rest will come later.
Marketing Control Three: A Simple Marketing Scoreboard
Building on the theme of simplicity, we want to suggest that you create a simple scoreboard that gives you the high-level results of your marketing efforts. In fact, we'll go one further; we recommend that unless you already have a marketing scoreboard, you start by measuring just three numbers:
**Marketing Metric #1: Your "Cost per Lead."** Measure the total cost of a particular marketing tactic (e.g., pay-per-click advertising, direct mail, trade show booth, etc.) and divide that by the total number of leads that tactic generated in a specific period of time. If you spent $10,000 on your trade show booth, travel, and staff cost and generated 100 new leads, your cost per lead is $100 ($10,000/100 leads). Knowing your cost per lead helps you compare which lead tactics are most cost-efficient, and gives you feedback so you can refine a specific lead-generation tactic to perform better. Plus, any big change in your cost per lead can be a critical leading indicator, good or bad, that you must pay attention to in your business.
**Marketing Metric #2: Your "Cost per Sale."** This measure is simply the total cost of a particular marketing tactic divided by the total number of sales you made from that tactic. Continuing with our trade show example, imagine that of those 100 leads you got from the trade show, you closed 10 sales. That means your cost per sale is $1,000 ($10,000/10 sales). Knowing your cost per sale helps you get a sense of which marketing tactics yield leads that are higher-propensity buyers relative to your cost to acquire these leads.
**Marketing Metric #3: The "Return on Investment (ROI) $1 Spent on Lead Cost."** This final number is a powerful way to equalize various marketing tactics so you can see which one has the greatest return on investment. You calculate it by dividing the total sales you made with this tactic by the total amount you spent on that marketing tactic. Using the trade show example, if your 10 sales each purchased $10,000 from your company, that means that your $10,000 investment in the trade show yielded $100,000 in gross sales (10 sales × $10,000 per sale). In this case, your ROI per $1 spent on lead cost would be $10 ($100,000 in total sales/$10,000 in lead cost for those sales). In essence this means you had a _gross_ return of 10x (1,000 percent) on the money you spent on the trade show. Depending on your margins, and how much future business these 10 customers will yield, and a few other factors, you'll know if that return on your marketing dollars for the trade show was worthwhile or not.
**Marketing Control Four: Your Sales Database or "CRM" Solution.** Your CRM (customer relationship management) solution is the system you use to organize your customer and prospect data and manage those relationships over time. Don't let the term intimidate you if you're unfamiliar with it; you've likely been using one for years now even if you didn't realize it. Whether you use Salesforce.com, Infusionsoft, Zoho, Microsoft Dynamics, a custom Oracle database, or another system, your CRM solution is simply the contact management tool that holds your customer and prospect list and also helps you use that information to create and service sales over the long term.
The most rudimentary sales database is a spreadsheet of prospect information that includes a name, address, e-mail, and phone. As your business grows, the spreadsheet solution isn't likely to be your ultimate solution because it is limited. When multiple people use it, the data often become messy and inconsistent as people fill in certain fields in their own styles. Plus, a true CRM solution allows you to automate and systematize much of your sales process, including building in prompts and automated steps or reminders for follow-up, standardized reporting to tell you how your sales efforts are going, and filters that allow you to easily pull marketing lists together and make sure leads are properly assigned to various salespeople.
This isn't a technical book about sales databases. We simply want you to realize that your sales database (spreadsheet or CRM solution) is a form of a marketing control that, when used correctly, helps your business ensure that leads are properly captured and followed up with on a timely and effective basis, and that all the insights gleaned along the way are recorded for future use in a standardized way.
The Four Fastest and Least-Expensive Marketing Tactics to Generate Additional Sales
With all this talk on lead generation, we want to help you avoid one of the most expensive mistakes that too many business owners make—thinking that their marketing efforts should be focused on finding new, fresh sales prospects.
The "new" and "fresh" labels are seductive. After all, don't you want fresh and new leads versus stale and old? The truth is, though, for most existing businesses, the fastest and least expensive marketing tactics to create rapid growth don't focus on finding new and fresh prospects, but rather on more creatively and systematically getting more from their existing customer base.
When Jeff was building his first company, CTI (which he eventually sold to American Express), it developed and sold desktop travel booking systems to corporations. Every day Jeff's team would search for fresh new leads, hoping to find new corporations to go visit. One day Jeff and one of his salespeople were at a customer site when they heard the customer say, "I wish our expense management vendor was as good as you guys." Jeff and his salesperson both had the same thought. Here they were so busy looking for new customers once they closed a sale that they had never asked their existing customers if there was anything else CTI could do for them. It turned out there was. Adding additional service to their existing contracts was far faster and much cheaper than finding new clients and starting the sales cycle over from the beginning.
Here are four tactics to tap into your _existing_ client base to grow your sales. As you read through each tactic, ask yourself how you could profitably apply it to scale your sales.
Tactic One: Formalized Referral Systems
We all know that referred prospects are generally faster and easier to enroll, buy more frequently, and are more loyal. Most business owners we work with tell us that a good portion of their business comes from customer referrals. Yet when we dig a little deeper, we discover that 95 percent of those referrals came to the business by word of mouth. Word of mouth is _passive_. Your business didn't do anything to spark it other than to deliver on your service or product promises (as if that were an easy thing!). A formalized referral system is an _active_ referral strategy that you intentionally craft to spark and prompt your existing clients to help you find more prospects.
Bill owns a driver's education company that primarily teaches teens safe driving skills through on-the-road, one-to-one driver's instruction. The primary reason parents sign their teens up for and pay for these lessons is because they want to prevent their kids from getting in an accident. We suggested that Bill tap into this desire to protect their teens from auto accidents by creating a "Safety Circle Program." After the second lesson with a teen, Bill meets with the parents and explains that their son or daughter is more likely to be driven by one of their three or four closest friends than they are to be the one driving. To really protect their teen, it's not enough for their teen to get trained; his or her closest friends need to get the same training so that when their teen is driving with these friends, parents can feel confident that they've done all they can to keep their teen safe.
To this end, Bill goes on to explain, his company created the Safety Circle Program. Bill helps the parents identify the four closest friends of their teen and gives them a special coupon certificate that their friends' parents can use when they bring in _their_ teens to get trained. Bill shares the certificate and a simple one-page flyer, then coaches the teen's parents on how to have a direct conversation with the parents of the other teens, suggesting that they work with Bill's company to get their teens trained too. The net result is that Bill turns his customers into allies in generating more business for his driving school.
Tactic Two: Formal Reactivation System
If you've been in business for more than two years and provided great value to your customers during that time, you should strongly consider adding a formal reactivation system to your marketing mix. A formal reactivation system is a process whereby you systematically go back to those past customers or clients who used to buy from you, but for one reason or another haven't purchased from you in a while, to powerfully invite them to buy again.
At uBid.com, Jeff saw lots of customers register an account, then disappear soon after. When Jeff and his team looked at the data, they saw that there were many customers who could not find the product they wanted at the price they wanted, so they left the site without buying anything. But uBid.com knew what these customers had been looking for, so they created a reactivation campaign that sent follow-up e-mails to these people, asking them if they would come back if the site offered a slightly different product but in the price range they were seeking. This generated significant incremental sales from a group of customers who otherwise would not have made these additional purchases.
Creating your formal reactivation system is simple. First, gather your pool of past client leads into one list. Whether you compile a spreadsheet of contacts or create a memorized report in your CRM database, pull your list together and document how you gathered this list so that the next time you repeat this process next quarter or next year, your team can follow this process to do it better and faster.
Next, create your outreach strategy. This is going to depend both on the number of past customers you are going to reach out to, and also on the makeup of that list. If the list is small enough, and the dollars make sense, you might arrange to meet with each of them in person to follow up or perhaps divide up the list for your sales team to call through. Maybe you'll send out a series of letters or e-mails. The key here is to consistently reach out to these former clients and give them a crafted "come back to the fold" offer with a logically compelling reason why you are reaching out to them at this time.
Once you've determined your outreach strategy, the third step is to craft your reactivation message. You need to reconnect first, then make sure to end the conversation with the clear next step you want the customer to take. Do you want them to come back in for an office visit? To have you bid on more work? To place an order right there? Formally script out what you want your team to say when they make this call or visit, or write out the letter or e-mail you'll send.
Once you've gotten your reactivation message ready, it's time to pull the trigger on the campaign. Start making your dials, sending your e-mails, or arranging your on-site visits. Make sure you track your efforts so that you can measure the impact of the campaign.
The final step is to evaluate the campaign to refine it for the future. If you are smart, you'll not only have your team looking to increase sales, but you'll use your interactions with your former customers to gather valuable insights into the hearts and minds of your market. Is there a consistent trigger for why they stopped buying that you could fix? Were there two features they were dying to have that could be the spark for you to innovate around? This way you will profit not only from the sales you make, but also from the insights you gain in digging into why your customers use your services, and why they might leave.
Let's look at a formal reactivation campaign we helped one of our clients create and implement. Dr. Kim owned two successful dental clinics. Over her 15-year history, the practices had accumulated a large number of patients who kept putting off their regular dental examinations and cleanings. Dr. Kim's office team pulled a list of all their past patients who hadn't been in to see them in more than nine months. Then, in quiet times in the day, her office staff would spend a few hours calling these patients, using a simple script with the express goal of scheduling them to come back into the office for a dental exam and cleaning. Of the people her team reached, 25 percent immediately reactivated and scheduled an appointment on the spot. Another 15 percent of them asked the office to call them in 30 days to schedule an appointment at that time. Not only did this generate thousands of dollars of service volume for Dr. Kim's practice every month, but it also helped her business fulfill its mission of caring for the oral health of its patients. Best of all, this formal reactivation system is now a staple in her office that allows her team to fill in any lulls in treatment hours so that the practice always stays full, utilizing its capacity to best effect.
Tactic Three: Retention, Retention, Retention!
Over time, every business sees some of its clients leave. This attrition is expensive because in most cases the cost to attract, close, and support a new customer is many times greater than the cost to service and support an existing one. This is the primary reason why a formal retention system can have a great impact on your business. For example, Brad, who owns a commercial insurance business, has to pay five times greater sales commissions during the first year one of his team sells a policy than in year two, when they get the renewal. Patty, who runs a professional service firm, estimates that her support costs in the first 90 days of a new client are 30 percent higher than for her established clients, as she must train her new clients in how to use her services and communicate with her team.
Unless your business is built on a "one sale" model, you'd do well to revisit how your company retains its existing customers and look for ways to systematically enhance how you keep your best customers doing business with you.
First, measure what your baseline retention rate is. You might measure the average number of months a client stays with your service, or the percentage of customers who continue buying your products a second, fifth, or tenth time. The key is to pick one number to measure that gives you an accurate sense of how well your business is doing retaining its customers. Once you have this number, formalize a process to keep it front and center in the minds of your team, ideally on a weekly basis, but at least on a monthly basis.
Next, pull several of your key team members into a conference room and analyze what you think are the two or three biggest "drop points" where you commonly lose customers. What causes them to drop? Why do other clients instead choose to stay? Conduct a Sweet Spot Analysis to pick the best ways to formally plug these drop points so that you retain more of your clients. Could you strategically schedule a visit or phone call on day X? Pre-complete a key step for them so that you help them skip right over a drop point altogether? Add a visual tool that helps them see the progress they've made?
Finally, implement the Sweet Spot enhancements you come up with and track their effectiveness. Then, in several months' time, repeat the process. The end goal is that over time you'll be able to plug the gaps and keep more of your customers buying from you for longer.
Probably one of the most effective examples of a retention system is the airlines' frequent flyer mileage programs. They are so effective that customers will often choose a more expensive fare just to get the miles on the airline where they have already accumulated the most miles. Airlines have continued to enhance these retention programs over time by allowing customers to use their miles to book hotels or spa services, donate them to friends as gifts or to charity, trade them for gift cards, and more. A loyalty program is a powerful retention tool that rewards your best customers for sticking with you, and makes the switching cost of leaving you even higher.
Tactic Four: Up-sell, Cross-sell, Re-sell!
A company that we worked with in the early days of the Internet specialized in antivirus systems. Their product was one of the best on the market and they sold it well. However, they didn't tell customers that antivirus alone did not provide complete protection for your computer. You also needed a firewall product (which they actually had, but which was sold by a different team), and a malware scanner (which they could have easily offered by partnering with another company). Their customers got annoyed when they discovered that they really needed all three products. As soon as a competitor began bundling and cross-selling these three products, their business quickly died.
Stay vigilant in what you can up-sell, cross-sell, and re-sell to keep your customers committed to the services and products you offer. Strategically map out how you can enhance your customer results by getting customers to buy the optimal selection, choice, and frequency of your product or service.
**Up-sell:** An up-sell is when you offer your customer a richer total package or premium service level or other upgrades that directly relate to what your customer originally purchased from you.
For example, David's business coaching company created a special service level ("Diamond") for companies doing over $10 million a year in revenue, for which his company coaches not only the business owner, but also the entire executive team, and facilitates private on-site company strategic retreats. Most clients come to his business because they want help building their company and reducing its reliance on key team members, and for a percentage of these businesses this upgraded service level delivers an exponentially more valuable result.
Generally an up-sell happens right at the point of sale or soon thereafter. In fact, many times your prospect may come to you looking for your base-level product or service, with no knowledge that you even have an upgraded option. As part of the sales discovery process, you'll need to pull out your prospect's needs, desires, and budget, and with that information make the decision of whether that customer would be best served by your upgraded offer or not. Here's the best part: Done right, even if your prospect turns down your up-sell offer, it can often create the contrast you need to make your base-level product or service feel much more affordable and, as such, enhance your sales closing rate.
Think about your own business now. What upgraded product or service offering could you present to your customers, or perhaps just to a subsection of your customers? Keep in mind that packaging your own products together is not the only option. Are there opportunities to bundle in or offer complementary products from other companies?
Here are a few suggestions to spark your thinking:
* What could you bundle together to make an upgraded package to get your client the optimal result?
* What service(s) could you add to this package to get your client an amazing result?
* What product(s) could you add to radically enhance the client experience?
* If you knew that for 5–15 percent of your customers, price was no barrier, and that they just wanted the highest value and level of service and quality consistent with your brand, what could you profitably offer them that would delight and thrill them?
**Cross-sell:** A cross-sell is when you offer your client complementary and related products or services that may or may not be direct enhancements to what your client originally purchased from you. This often happens at the point of sale or soon thereafter, although the timing of a cross-sell can be much later.
Jeff's former company, Priceline.com, was able to significantly accelerate its revenue growth by adding cross-sell products to its original product line. Its initial focus was on selling airline tickets, but many of its customers also needed a hotel room or rental car for their trip. By offering these complementary products, and later bundling them and offering package deals to make the cross-sell even more attractive, Priceline.com rapidly grew sales.
When someone buys from you, what other products or services will they often go on to buy? What other products or services complement or enhance their purchase from you? Could you offer to sell them these items yourself? Or refer out to a third-party company that you have vetted and with which you have arranged a revenue split or referral fee? Or arrange a "lead swap" whereby you reciprocally share leads with one or more of these third-party providers?
**Re-sell:** A re-sell is when you help your customer repurchase (once or many more times) essentially what they purchased from you the first time. This action requires strategically mapping out the optimal repurchase timeline for your customer, and prompting them to follow that timeline.
For example, Kimberly Ackworth, the owner of two successful beauty salons, knows that to keep her clients looking their best, they should come in for treatments every four to six weeks. That's why her salons have a scripted-out process (re-sell system) to schedule the client's next beauty appointment at the conclusion of the current appointment. She starts this right from the first visit with a new client, measuring her system's efficacy by tracking "New Client Retention Score" and "Rebooking Rate." She has between 30 and 50 percent of her clients book their next appointment right there at the time of service. For those clients who don't book their next session as they leave, her business has a powerful direct mail system of postcards and e-mail that it uses to prompt clients to come back in more frequently.
Too many business owners passively wait for their customers to ask to buy again. Not only does this underserve the customer, but it also potentially costs the business many thousands of dollars of easy, high-margin revenue. If your business is one in which your customers buy the same product or service from you multiple times, take two minutes to ask the following three questions:
1. What is the best purchase timeline to help your customers get the optimal result with your product or service?
2. As your business currently does things, what is the average actual repurchase timeline?
3. What simple, leveraged steps could you take that would effectively prompt more of your customers to follow the optimal purchase timeline? (We suggest you use the Sweet Spot Analysis tool we shared in chapter 4 to pick two to three Sweet Spot ideas you could implement this quarter to grow your repeat business.)
Lead Conversion: Converting Leads into Sales
If the marketing half of your Sales/Marketing pillar is primarily tasked with generating a steady stream of qualified leads to your business, then the sales half is focused on converting those leads into paying customers.
Whether you employ an inside sales team working with customers who come to your location to generate business, an outside sales team that works in the field, a Web-based model that sells online, or even a wholesale network whereby other companies generate most of your sales, every business needs a trusted process whereby it effectively converts leads into sales. We call this trusted process your "lead-conversion system." In a moment we'll walk you through exactly how to build your baseline lead-conversion system, but first we want to step back and look at your current system. You already have one; your system may be ad hoc and wholly housed in your head, but you do have a process of some sort, otherwise you wouldn't have been able to generate the business you have.
Before we share with you our process to formally build out your baseline lead-generation process, use the ten-point lead-conversion checklist below to help you diagnose what your most painful and costly lead-conversion challenges currently are. The more clearly and concretely you can pin this down, the better your lead-generation system will be, since you'll better understand the current constraints and needs of that system. Review the list and check the box beside any of these ten challenges your business is currently struggling with.
LEAD-CONVERSION TROUBLESHOOTING CHECKLIST
* **You don't have a focused understanding of your product or service that you can powerfully communicate to your prospective customers.** Or if you do have a clear understanding, your prospects just don't find the way you communicate it to them compelling or enticing.
* **You don't have a defined sales process to lead prospects through clear stages to close the sale.** You've never strategically laid out all the steps in your sales process in a visual way so that you could formalize your sales process and work to optimize it.
* **You, the owner, are the only salesperson for your business.** All the essential sales know-how is locked in your head (or in the head of a single key employee). If you (or your single key employee) aren't there, no one else would be able to close business.
* **You lack the sales capacity to actually process and sell to all the leads you already have.** As a result, many leads spoil for lack of someone to follow up with them.
* **You have no reliable control in place to accurately track, measure, and refine your sales system over time.** You don't know what your key conversion rates are at each critical step in your sales process, and as a result you're flying blind, not really knowing if a change you make is working as intended or not.
* **Your sales system simply isn't scalable as it exists at present.** It works at your current business volume but it won't scale as you grow.
* **You don't have effective standardized sales tools** (e.g., testimonials, sales collateral, scripting, sales contracts, etc.). As a result, you waste valuable time creating improvised sales tools as you go that never seem to turn into the refined sales tools you know you eventually need.
* **You lack an organized follow-up system to coherently nurture leads that need more time to close.** You attempt to sell once and if they don't buy, you move on to your next prospect.
* **You don't have a back-end offering to up-sell/cross-sell your customers.** Once your customers buy from you, you have no real next-step offering for them to keep buying.
* **Your sales scripting is weak and/or your sales team is poorly trained.** You sell based on price with no real scripted process to develop your prospect's need, pain, and aspirations. You don't systematically help your prospect quantify the cost (emotional and financial) of their status quo, and the real value of a solution.
The Five Steps to Building Your Baseline Sales System
Our guess is that you found the preceding lead-conversion pain-point checklist a bit uncomfortable. It is very common to have checked many of the boxes. Have no fear; this is just your starting point. Now that you are aware of the weaknesses, we can help you deal with them. The place to start is by building out your baseline sales system to convert prospects into customers. Here is a simple five-step process to do just this:
**Lay out your current sales process in a flowchart.** Make each step in your sales process a box and link that box to the next box(es). Every time you draw an arrow from one key step in your sales process to another key step, that arrow represents a key conversion point when your prospect can say yes or no. Label that conversion point with a letter; you'll use these in step two when you build your scoreboard. (See Figure 7.1.)
FIGURE 7.1: SAMPLE TWO-STEP SALES PROCESS VISUALLY MAPPED OUT
**2. Create a rough scorecard of how you track each key conversion point.** Count how many prospects made it into and out of each key conversion point and do a simple division of _leads out_ divided by _leads in_ to calculate the "conversion rate" of that step in your sales process.
**3. Pick** one **conversion point to focus on first and brainstorm leveraged improvements you could make to it over the next 90 days.** For example, if you choose the conversion point B in Figure 7.1 (your closing ratio at your "Evaluation Appointment"), potential improvements could include:
* Refined sales collateral (e.g., brochure, flyer, sales presentation deck, etc.)
* Improved sales scripting (including how to preempt, reframe, or overcome common objections)
* Product samples or photos
* Recorded sales presentation to help your sales team model your best practices
The key here is to start small by focusing on one conversion point at a time. Remember, fewer better ideas that you actually implement are better than a whole host of ideas that you do only partially.
**4. Create a mini–action plan to implement the top leveraged improvements you chose in step three.** Clarify who is responsible for doing what, by when, to what standard, and how they are going to be held accountable. Make sure someone in your company is responsible for this area of your business as a whole, and as such, is tasked with following up to ensure that your team effectively executes your mini-plan.
**5. Implement, track, and refine quarterly**. Each quarter, repeat this process. We know it isn't the sexiest thing in the world, but the sales it will help your company consistently generate sure are. Plus, each quarter you'll be taking powerful, leveraged action to consistently reduce your company's reliance on you or your key salesperson, protecting your business and making it more scalable.
Patricia, a chiropractor building a multi-clinic practice, realized that she was her business's best salesperson. To deal with this bottleneck, she selected two of her doctors to do the initial examination with new patients (where the "sale" occurred for her business) and trained them in how to replicate her sales success. She scripted out the sales process, videoed herself in action in mock exams, role-played with her exam doctors, and designed a simple scorecard that told the doctors how they were doing on their conversion rate at any given moment. Her two exam doctors saw their conversion rates go up by over 40 percent in less than 90 days! Now Patricia has a scalable training program that teaches a new doctor how to sell her way in less than 90 days.
We know you'll use these tools and techniques to help you blow through the obstacles standing in your business's way of scaling its Sales/Marketing pillar. In the next chapter we'll share with you three breakthrough tools to help you scale your Operations pillar without sacrificing your life to do so.
## YOUR OPERATIONS PILLAR
THREE BREAKTHROUGH IDEAS TO SCALE YOUR CAPACITY
The Operations pillar of your company creates your product or delivers on your service. It is responsible for fulfilling on the promises you made to your customers in the sales process, along with performing the general and administrative back-end functions of your business.
In this chapter, you'll learn three breakthrough concepts to efficiently produce more and better value for your market as your business scales. First, we'll give you a powerful blueprint to build out your most important business systems, which we call your "Expert Systems." Then, we'll give you a framework—we call it your "UBS"—to organize, house, access, and refine your business systems. Finally, we'll share with you one more critical component for sustaining growth over the long term, called "business linkages." This final tool will help you reduce the number of things that fall between the cracks as one pillar of your company hands off to another pillar of your company, and speed up the pace at which your business produces results internally as you script out and reinforce critical business linkages.
The place to start is with building your Expert Systems.
Expert Systems
Your ability to scale is directly proportional to your company's ability to create cookie-cutter processes for mission-critical areas. Your Expert Systems capture the hard-earned wisdom of _how_ to perform your company's core business functions. They replicate that expertise in a formalized process and connected set of tools, training, and controls to make it possible for the _business_ to own that expertise versus the know-how being held in the brain of a key employee. Not only does this protect your business from the loss of a key employee, but it also allows you to replicate this formally expert-level-only process in a coherent system. This means you can scale an Expert System because one person is no longer the bottleneck. It also means you can lower your costs as you push down the level of expertise needed to reliably produce the desired result, perhaps even automating a large chunk of it. Because you are freezing the formerly impromptu process into a _formal_ process, you can also optimize it, increasing the speed and value of the output. And finally, because you now have a readily reproducible recipe to focus your efforts on, you can control for consistency and quality, even creating simple business controls to ensure the process runs right.
In a perfect world, you'd love to have a business that runs by itself; while you're at the pool, your company takes orders and delivers products. While this fantasy may not be possible, using this filter to look at your business prompts you to constantly find ways to automate, streamline, and improve operations instead of just adding more people when more work comes in.
Here's an example of how Expert Systems work in the real world. In January 1997, David launched his first business-coaching company. In the beginning, he and his business partner did all the coaching, were responsible for most of the sales, and oversaw all the other core functions of the business. In essence they were the typical start-up entrepreneurs, working long hours and wearing all of the hats. As the business started to grow they faced a major hurdle—finding a way to deliver a world-class business-coaching service that didn't rely too heavily on them to personally do the coaching.
David eventually sold this company in 2005, but not before he scaled the coaching division to work with more than 1,500 clients a year. As you can imagine, he and his business partner maxed out individually coaching at the 50-client mark. To go beyond this number, they had to create a formal process where they froze their method for working with a coaching client to deliver great results in a collection of tools. These tools included the process to find, hire, and train new coaches; the diagnostic tools to work with a new client; the accountability tools to keep clients on track; the ongoing training process to continually grow their coaches over time; and the technology to automate many of these processes. Collectively, all these tools and processes evolved into David's Expert System for producing coaching services. The most amazing part was that once they had built this Expert System that essentially replicated and replaced David and his partner in direct client coaching, by every measure of success (e.g., growth in client sales, growth in client profits, client satisfaction ratings, client referral rate, etc.) the Expert System produced a _better_ result!
Expert Systems don't have to be just about production; they can be critical components of _other_ parts of your business. Take the example of L. H. Thomson Inc., a contract manufacturing company in Macon, Georgia, that makes components for airplanes. One of their Expert Systems is their system for filtering, pricing, and responding to bidding opportunities for new manufacturing contracts. Since most of their aerospace manufacturing contracts run for five years and are for half a million dollars or more a year, choosing which contracts to bid on and at what price is a critical part of their business success. Once upon a time, all this know-how was locked in the heads of their CFO and CEO. Today, they have formal, written criteria for which contracts they will and won't go after, a clear methodology on how to accurately price the bid that ensures the job will be profitable, strong boilerplate contract language that protects them from fluctuations on raw materials, a software tool that speeds up the number crunching and allows them to respond quickly and accurately, and a training process that captures their best practices and brings team members up to speed on effectively using this system. Collectively these tools are their Expert System for bidding on new work.
Your company may have an Expert System for producing its core product or service offering, or for effectively converting leads into sales, or managing your collections process. In fact, most businesses will ultimately want to build three to five Expert Systems that help them clearly lay out how to handle the critical parts of their business in a stable, scalable way.
Here is a six-step formula for building your core Expert Systems.
Step One: Define All Deliverables
A deliverable is any result your Expert System needs to produce to meet the expected or promised outputs of your system. This is just a fancy way of saying that you need to clarify what exactly your system is supposed to produce for its customer. This may include getting a certain number of physical products to your customer by a certain date, making a report or recommendation on a course of action to your client on how to best handle a specific challenge, or some other output your Expert System has promised to fulfill.
There are four types of deliverables you need to concretely define: external deliverables, internal deliverables, phantom deliverables, and reverse phantom deliverables. **External deliverables** are what most of us think about when we think of a deliverable—products or services we've promised to our customer, like fulfilling a hundred-widget order by Friday or completing a full-day training program. **Internal deliverables** are those things that you have to produce, create, or deliver behind the scenes in order to meet your external deliverables for your client. They may include collecting certain customer information and entering that into your system, holding an internal project meeting, handing a project step from person A to person B, or even subcontracting certain portions of the finished "product" and managing your subcontractors to deliver their pieces on time and up to standards.
**Phantom deliverables** are those deliverables that you haven't actually promised your customer but they _think_ you have. This could include their belief that your product will come ready to use out of the box, or that you'll provide follow-up support after the contract formally ends. You need to clarify all your phantom deliverables so that you can either clarify that they aren't included, or make sure that they are delivered, because if your customer thinks you've promised them something and you don't clarify or deliver that deliverable, then they will leave feeling dissatisfied. While this may not be fair, it is accurate. Finally, there are **reverse phantom deliverables** —those things you _think_ you promised your customers that not only you didn't, but that your customer simply doesn't care about. Maybe you deliver weekly reports that your customer just deletes, or product features that your customer never uses. Take the time to reassess all your current deliverables to make sure they are still relevant to your business and add value to your customers. We have seen companies build Expert Systems that incorporate all of their current deliverables, when in fact some of those deliverables were old legacy requirements that nobody had questioned in years.
As you can imagine, the more complicated the Expert System you are working on, the more deliverables you'll capture on your list. While this may seem overwhelming at first, this should actually comfort you. You've already been producing all of these deliverables, but you were doing so in an informal way that one or two key people in your company just "did." By concretely defining your deliverables, you're taking that key first step in building out a system that will effectively produce them. After all, how can you produce what you haven't consciously identified you've promised to create? And without documenting all these deliverables, how can you get customer sign-off that you are actually doing your job well and delivering on your promises?
Step Two: Lay Out the Process
Now the fun begins. Grab a pad of yellow sticky notes and your list of deliverables. Placing one "step" on each sticky note, lay out the process your company will use to create and deliver on all those deliverables. Using sticky notes keeps you fluid and loose as you design and document your process, allowing you to move them around, add steps, combine steps, or delete steps.
Once you've got a rough layout of your steps, it's time to ask yourself a series of questions to refine your draft process:
* Which deliverables _really_ matter? Which deliverables are nice but not essential? Which deliverables do your employees _think_ your customers want or asked for but didn't? How can you eliminate these deliverables that actually just get in the way and are not wanted?
* How can you reduce the steps and still generate the desired results? And generate an _improved_ result?
* How could you decrease the resources needed and still generate the desired results? And generate an _improved_ result?
* How can you speed up this process?
* How can you automate or template this process (or part of this process)?
* How can you lower the costs of doing this process without impacting the value of the output?
* What simple changes or improvements can you make to increase the value of the output? How could you marginally increase your cost to produce but in a way that so enhances the value of the output that you can get a price increase for the value you're offering now?
* Who else in the world has a related process or tool you can learn from to help you better design this process?
* Could you outsource any parts of this system? Does it really make sense long-term to do this?
* How could you make the system more robust? More stable? Less prone to error?
Once you've thought about and answered these questions, return to your sticky notes. Based on your answers, move, add, delete, and play with the steps of your system until you lay out a process that promises to be faster, cheaper, of better quality, of greater impact, and more scalable.
When you've drafted these improvements into your sticky notes, you'll type out the finished process into a complete and neat recipe to produce the desired outputs. This recipe will be a simple longhand list of each step in the process (e.g., Step One . . . Step Two . . . Step Three . . . etc.).
Once you have this clear process laid out it is time to move on to step three.
Step Three: Determine the Optimal Level of Expertise for Each Step
When Jeff consulted to the Cleveland Clinic, one of the world's top-rated medical institutions, their goal was to get every doctor to practice "at the top of their degree." To do so, they had to review everything a doctor in their institution did to identify those nonspecialized things that should have been done by support staff. The closer they got to this goal, the more efficiently their institution ran.
As a business, you want to find ways to relieve your most expensive and experienced employees from doing lower-value work. The hospital example is very clear. A surgeon is most useful to the hospital, and generates the most value and revenue, when she is performing surgery. The same thing applies to your business, even if your "experts" aren't surgeons, but rather just your most experienced and expensive team members. To sustainably scale your business, you must commit to push as much of the work of each Expert System down the value chain so that your experts do less of your Expert System. Not only does this immediately increase your capacity because your expert can now be spread over a larger volume of total work, it also drives down your costs as the work performed at a lower level is much less expensive.
There is a hierarchy of expertise for each step in your Expert System. At the bottom, you have those steps that can be automated, semiautomated, or made into a template. Next, you have those steps that require a person to complete them, but not necessarily a skilled person (e.g., clerical, administrative, unskilled laborer, etc.). The next level up is for those steps that need a semiskilled team member to perform them (e.g., paralegal, nurse, journeyman, etc.). Above this is the level of skilled, which requires a basic expert to produce these steps. Finally, the top level of the pyramid is for those steps that require a top expert to produce for the business, which in most small businesses is the owner or one or two key employees.
Let's look at an example of how this hierarchy of expertise plays out in a law firm to make these levels clear. Automated, semiautomated, or templated refer to things like the standardized engagement letter that gets sent to any new client or the library of boilerplate contract templates on the company's server. Nonskilled tasks include those tasks that a clerical worker without legal training could handle, such as scheduling meetings, collecting client data, and gathering historical documents to give to the attorney. Semiskilled tasks in this context would likely refer to those items that a paralegal could produce as opposed to an actual attorney. Basic expert tasks are those that only a licensed attorney could do, although they could be done by a less expensive, less experienced associate attorney. Top expert tasks are processes and functions that require the best legal talent at that firm in that area—things that likely need years of experience to understand and do properly.
The goal of your Expert System is to identify the best level of expertise to match up with each step of your Expert System. What you'll likely find is that there is a vast gap between what you intellectually know to be the optimal level of expertise for a particular step and the actual level at which you currently perform that step.
For most existing businesses, the biggest immediate reward of drafting their Expert System in a given area is how this exercise reveals where they need to push steps down from the top two levels (Basic Expert and Top Expert) to lower levels. In many cases, a business can quickly increase its capacity by 30 to 50 percent or more simply by staffing down many of the steps in its Expert System to a lower level in the pyramid.
In the example of our hypothetical law firm, this would include things like getting clerical staff involved in scheduling meetings and reminding clients of information they need to get to the firm prior to that meeting, using better software that automates or semiautomates the invoicing of clients for work performed, or standardizing the core legal services to allow a less experienced attorney to do tasks previously performed by the senior partners of the firm (but that don't require the latter's deep expertise).
From all our years coaching business owners, we consistently see that most businesses have their best, most expensive "experts" doing too many of the steps of their informal Expert System. As a result they struggle with capacity issues and poor margins, and are vulnerable to that "expert" getting hurt or otherwise leaving the business, taking with her all the know-how and institutional knowledge she gained from years of being the wizard inside the black box producing in this area of the business. If you want to scale your business exponentially, then you must reduce your company's reliance on any one expert with formalized Expert Systems upon which you have trained and cross-trained your team.
Step Four: Control for Consistency
Now that you've got your written process, and have identified which level of expertise optimally goes best with each step, it's time to refine your process to control for consistency. This is just another way of saying that you now need to look for ways to improve quality and reduce variability. Here are several key thoughts to help you do just this:
* **The more you can automate, semiautomate, and template, the easier it is for you to control for consistency.** All it then takes is a sharp review of your template or automated steps to make sure they are accurate. These processes become great "embedded controls" to protect your business.
* **Streamline the process**. The fewer the steps in any complete process, performed by fewer people, the fewer the potential problems.
* **Pay particular attention to the critical linkages**. Script out the critical linkages between tasks and reinforce them. (More on this at the end of this chapter.)
* **Standardize wherever you can.** This will help you accelerate the process, increase efficiencies, lower costs, increase impact, and improve quality.
* **Create your three "master" documents:** your master timeline, your master checklist, and your master budget.
* **Capture institutional knowledge in a structured, searchable place.** This includes detailed client notes not in the heads of your staff, but in searchable text in your CRM. It can also include an organized file of the associated documents for a specific project or client. If you don't take steps now to capture this essential past history, there will come a day when a key team member leaves your company and you'll have to scramble to re-create the institutional knowledge they took with them. Not only will this be financially expensive, but it will also be incredibly stressful and emotionally painful.
* **Train and cross-train your team.** Make sure that every key role in your Expert System has at least one fully trained understudy. Redundancies aren't glamorous, but they give you incredible peace of mind and business depth.
Step Five: Map Out the Key Components of Your Expert System to Refine First
Likely you don't have any formal Expert Systems established right now, but you do have an informal collection of best practices that are in the heads of your key team members. Now that you've followed steps one through four to formally lay out your Expert System, it's time to flesh out the system with the tools, training, and controls you need to enhance your Expert System.
Don't worry—we won't ask you to do this all at once, as likely your Expert System is a complicated thing, and the tools you want will best be created and phased in over time. For example, when Pete Fowler, owner of a construction-defect litigation company, first applied the Expert System concept to his business, he started by looking at the inspection process his highly trained team used to review a building. He took his top three team members off-site for two days to create both a detailed inspection template and a workflow map of all the steps in this portion of their client engagements. Then, over the next two quarters, they built out the other tools they needed to complete their Inspection Service Expert System.
Pick the piece of your Expert System—the "block"—that you think either would be easiest to refine, or would have the biggest impact for your business. Picking a block that you know you can successfully model in an Expert System gives your staff a visible example of the value and operation of Expert Systems, and increases their confidence in attacking the next, more complex block. Give this block a name (e.g., The "New Client Launch," the "Quality Review Process," the "Bid Selection Step," etc.) to make it easier for you and your team to talk about.
Now we'll approach this block from four specific directions:
**Critical Knowledge to Institutionalize:** What is the critical know-how about this block of your Expert System that is locked in the heads of one or more key team members? Identify this institutional knowledge and brainstorm the best way to capture, store, and share it. Perhaps you could have a checklist of items to note in the client record each week during a longer project. Or you could make sure that you have well-thought-through custom fields in your CRM to input key parameters for a project. Or maybe you build a searchable company wiki or discussion forum to take essential conversations out of e-mail and capture them in a more permanent and accessible way. The idea here is to ask yourself what information or knowledge would be the most costly and painful to replicate if you lost your key expert, and to look for ways to reduce that expert reliance.
**Tools to Enhance and Leverage:** What tools, templates, and automation would make this block of your Expert System faster, cheaper, better? Would an automated sequence of welcome e-mails help bring on a new customer more smoothly? Would an online Web survey allow you to gather key client information? How about a CRM solution that lets you record all client or project e-mail, and select from a library of e-mail templates, to reduce the time and expertise needed for specific steps?
**Training to Design and Implement:** What training and cross-training will team members need in order to be successful in using this Expert System (or at least this block of your Expert System)? How could you formalize or "freeze" that training in easily accessible, updatable, and scalable systems? Could you create a series of short video training modules, a process document with annotated screen-shot images, or maybe a formal mentoring system inside your company? Remember, you are not just looking to one-off train one team member to be able to back up one "expert." This may help reduce your reliance a bit, but it doesn't go far enough. You are looking for longer-term ways of building your training system so that you can train the next, and the next, and the next team member you need to perform in this area of your business. A good test of your eventual training tools is asking a member of your team who hasn't been involved in this area of the business to review your Expert System. If he or she is able to follow the documentation, training, and tools to perform useful work quickly, then you're on the right track.
**Controls to Monitor and Ensure Quality:** Consider what internal controls (visual, procedural, or embedded) would best help your business ensure that this block of your Expert System consistently works exceptionally well. Do you need a master timeline to lay out all the key steps and milestones in a time-coded sequence? Automate key steps into your software so that they happen regardless of which team member is handling that stage of the project? Implement a simple metric to let you see that things are running smoothly at that step? Considering how mission critical your three to five Expert Systems will be to your company's success, you'll need to incrementally develop, test, and refine the business controls you implement for each of them.
Step Six: Each Quarter Reevaluate Your Expert System to Prioritize the Next Block to Enhance and Refine
Each quarter, revisit your Expert System and pick the next "block" to focus on and refine. It's normal to take three to four quarters to really nail down a complete Expert System for your company. Because these are the processes that produce the most value in your business, you'll find your company greatly benefits from the time and attention you invest on each block as you and your team iterate and refine over the long term.
Now that you have a clear process to build out your Expert Systems, it's time to focus on the bigger picture of how you store, organize, and access _all_ your systems.
Building Your Master System— Your "UBS"
One of the most powerful operational secrets we've discovered after decades of building successful companies is a concept called your "UBS." UBS stands for your "Ultimate Business System," and it is the master system for how you structure, organize, store, access, refine, and, if need be, delete, your company's systems.
Most owners will hear this and say, "Oh, you mean our policies and procedures manual?" We most definitely do _not_. After surveying thousands of businesspeople about the usefulness of a policies and procedures manual, we've come to one firm and irrefutable conclusion—no one _uses_ or refers to a policies and procedures manual, especially after the first 30 days on the job.
We want you to think of your UBS as an organized collection of tools that captures the actual, everyday know-how of your business in a searchable, accessible, and editable way. Generally this means a collection of cloud-based file folders into which you organize and put your checklists, your worksheets, your templates, and your archived files. As technology progresses, more small and midsize businesses are able to put part of their business processes into enterprise software that runs their companies. As this trend accelerates, expect that your UBS will increasingly live inside the enterprise software you develop for your company. Even still, there will likely always be a place for the other systems that aren't easily captured in your software: systems for how you hire, how you generate leads, how you process sales, and how you pay your vendors. The UBS concept is timeless.
The goal is to make your UBS a living, breathing way of doing business in your company—an ongoing practice. You'll know you've won when you regularly hear your team say things like, "Did you add that to the UBS?" and "Great solution to that problem, can you UBS it?" This is not something that you sit down and "do" in one sitting. Your UBS is a discipline that your team will cultivate and refine over the long term.
Here is our four-step process for building your UBS.
Step One: Create Your UBS's File Folder Organizational Hierarchy
If you had to cluster all the functions of your business into five to nine main headings, what would those headings be? Further, how would you break each of these folders into five to nine subheadings? This is the first step in creating your company's UBS.
In Figure 8.1, we show you what this might look like. In the example, we've used the five functional pillars of every business. This is a great default place to start from. Too many file folders creates too much complexity, and your team won't know where to put things, where to find things, and soon your UBS will devolve into a mess that people ignore. Instead, limit your UBS to between five and nine headings each, with only five to nine subheadings, so that your team becomes intuitively able to know which main folder or subfolder to use to access and store your systems.
In the sample UBS below, notice how we used the numeric system to label each main folder and subfolder with a number—a useful shortcut to organize your UBS. You may be tempted to extend this numbering system beyond two levels. DON'T! We can tell you from direct experience that if you push the numbering system too far (e.g., "2.3.2.9"), you'll end up with a mess of a UBS. With the ability to search folders easily, and with the way our brains have been trained to think about keyword search, you're better off with a simpler folder hierarchy that allows you to find the tool you want, when you want it. As you name the files and tools in your UBS, standardize the keywords you use in your naming so that people save things in a way that lets other people know how to search for them. The UBS hierarchy should be intuitive. We suggest that you also use complete names that spell out exactly what a document or tool is so that when searching for it brings up a few possibilities, the correct document is obvious. For example, when you need to pull up the lead-conversion scorecard to enter your daily numbers, you'll pull up the spreadsheet named "1.5 Daily Sales Team Tracking Spreadsheet March 20XX," enter your numbers, save the document, and then close it. Now anyone else in your company who wants to review the numbers can pull up the spreadsheet and see where things stand. (We'll get into the software choices to house your UBS in a moment.)
What matters most is that you strictly map out the architecture of how your UBS will be organized. Try to develop a structure that looks more like your actual business than a computer database. Only your IT team thinks in database terms; everyone else just understands the business by category, assignment, or workflow. If you design the UBS with this in mind, no matter which software or hardware you use to store it, you'll have a good start on an organized, useful way to create, store, access, and refine all your business systems.
FIGURE 8.1: A SAMPLE OUTLINE FOR A COMPANY UBS
1.0 SALES/MARKETING
1.1 Lead Generation
1.2 Lead Conversion
1.3 Sales Team Tools
1.4 Sales Planning and Strategy
1.5 Sales Metrics and Reporting
2.0 OPERATIONS
2.1 Production
2.2 Fulfillment
2.3 Purchasing
2.4 Facilities/Infrastructure
2.5 General Administration
3.0 FINANCE
3.1 Accounting/Financial Reporting
3.2 Collections
3.3 Accounts Payable
3.4 Budgeting
3.5 Financial Controls
3.6 Cash Flow Management
4.0 TEAM
4.1 Hiring and Orientation
4.2 Training, Review, and Retention
4.3 Benefits Administration
4.4 Labor Compliance
4.5 Exit Processes
5.0 LEADERSHIP
5.1 Strategic Planning
5.2 Leadership Development/ Succession Planning
5.3 Company Culture and Traditions
5.4 Communications Company-wide
Step Two: Rename and Store Your Existing Systems and Tools
If you've been in business for a while, you likely have a hodgepodge collection of systems and tools in place already. You might have a scheduling spreadsheet you use to organize which staff are on which shifts month by month, a nine-step checklist to open up your store for the day, and even a database of potential suppliers to get bids from on new product orders.
Step two in the UBS process is to go through your computer and files to gather your existing systems and tools and put them into your new UBS. This is a perfect time to identify which tools are outdated or inadequate, which work well, and which tools your system desperately needs. Only save to your UBS those documents and tools that you want your business actually using going forward. (You can always store anything no longer current in an "archive" folder in each section of your UBS on the off chance you need to access it later.) Rename files and tools in the hierarchy and naming pattern you determined previously, so that finding them is easy and fast. Append the word "temp" to the name of each document or tool that you know you need to upgrade to cue your team to refine or replace that system at some point in the future.
Step Three: Create a Prioritized List of the Three Systems You'll Build This Quarter (and get to work building them)
Brainstorm which systems your company desperately needs. Then from that list pick three (yes—just three) to build this quarter. We recommend choosing only three because we've both seen far too many business owners get excited by this concept of building systems and launch 22 different business improvement projects all at the same time, only to never finish any of them. Building systems sucks up resources, so make sure not only that the systems you build are worthwhile, but that you actually finish them and get them to a place where they produce results for your business.
Here is a short list of questions to ask to help you pick which three systems to begin building first:
* Which system would yield the biggest return?
* Which system would best help us push back our current number-one biggest Limiting Factor?
* Which system costs us the most for not having in place?
You've got the rest of the quarter to create your first (or next) version of these three systems.
Step Four: Repeat Step Three Every Quarter
Each quarter, you'll go back to step three and pick the next three systems to build. As you grow your team, you will have help in building out your systems and will likely be able to build many more systems each quarter than just three. Still, we recommend that each quarter you clearly identify the top three systems that your company is committed to building. If the systems are too big to complete in one quarter, which might be true if you're building out one of your core Expert Systems, then frame out which _part_ of the system you'll have done by the end of the quarter. Over time, this process is magic, and you'll find that you consistently make your business more and more scalable and less and less reliant on a few key staff members (including you).
Picking Your Storage Technology to House Your UBS
Every company needs to come up with its own way of storing its systems for its team to use. The problem is that technology changes so fast that the solutions we recommend today will quite likely be out of date in a very short time. That said, here are the four most critical attributes that any technology solution to house your UBS must meet:
1. _Accessible:_ Your UBS needs to be quickly accessible. Usually, this means it's accessible online, but it could be a paper-based system too. In the case of your team's doing work remotely while traveling, ideally your choice would still be accessible when a team member is off-line (with the system automatically syncing up when they go back online). Clearly in today's world, having systems and data accessible even on handheld and mobile devices is a big productivity enhancer.
2. _Searchable:_ People have to be able to quickly find what they want in the UBS; otherwise they'll start to keep their own "cheat sheet" systems at their desks or on their computers. This would eventually mean your UBS won't house the best practices but merely return to being a procedural manual that no one uses. Remember, people are used to keyword search as a tool to find answers on demand. Your UBS tool needs a robust search function.
3. _Version Control:_ Your UBS needs to be collaborative, which means all users need to be able to edit and improve the data. This requires having version control and also constantly pruning your UBS's outdated information and systems. Systems will change and grow, and if you leave the old one to live with the new, it will cloud the waters and make it harder for people to know quickly which systems to use and when. In many ways, the eraser is mightier than the pen for your UBS.
4. _Security:_ **** Your UBS needs to have security features that allow you to protect your intellectual property both from outside parties and from internal misuse. Your UBS is one of your business's most valuable assets. Ideally, you'd have simple, granular control over who has access to what parts of your UBS. Also, this is a good place to suggest that all your employees and contractors who have access to part or all of your UBS sign strong confidentiality agreements that make clear that this system is proprietary to your company and as such protected under that agreement.
Systems and controls cannot be a fad, but must be an ingrained way of doing things inside your business. If you start the process and don't see it through, your efforts will wither and you'll lose credibility with your team. Your team must see why systems and controls matter to the business, to you, and to them. And they must see you staying the course by making them a fundamental part of your business.
Here are four more quick suggestions to make systems a part of your company culture:
**1. Train your team.** Don't just expect that your team will know how to build systems, or even use them. You need to make training on how to create the UBS, and how to access and use it, a core part of your business. Explain to them why the UBS is so important, and how critical it is to your company's ability to grow.
**2. Involve your team.** You don't have to do all the work of creating the UBS alone. In fact, if your business already has a much better "process thinker" than you are, get this person to lead the project of creating your UBS. As for building your systems, the best leverage point is to enroll your team to build your systems with you. Your most important job is to cultivate the _discipline_ of systems throughout your organization.
**3. Systems must make your team's lives better.** Be on the lookout for silly systems or steps that just add red tape or offer no compelling benefit to the company. A system must make the company and/or your team's lives better.
**4. Start right from the point of hire.** Emphasize your culture of creating and using systems. Make editing and using the UBS part of any new hire's job. Train them on it from day one. Your team will never be as open to doing things the "company" way as they will be at the point of hire.
The bottom line is that you want your team to internalize the discipline and understanding of the value of creating, using, and refining your systems and controls over time.
Reinforcing Critical Linkages
The final predictable obstacle you'll need to blow through as you scale your Operations pillar is a direct outgrowth of your increased size—making sure details don't slip through the cracks as more of what your company does crosses functional areas (i.e., pillars) of the business. That's why we want to share with you one more tool to help you successfully scale—"linkages."
Linkages are the connections between two or more pillars of your company. Whenever a process crosses over from one pillar to another, there's heightened potential for something to slip between the cracks. The best companies script out and reinforce their critical linkages so that they preempt things from falling between the cracks.
Think of it like a relay race. You have one sprinter racing to hand the baton to the next racer, then the next. The goal is to beat their competition to the finish line. While it seems so simple—just run as fast as you can and pass the baton—it is far from easy. In fact, if you were to spend time watching Olympic- level relay teams train, you'd notice they spend a disproportionate amount of their training time practicing the handoff, because they know that it's the biggest point of failure for a relay team. It is the one place in the race where everything they have worked for can be lost in a single fumble.
In your business, critical linkages include the handoff between marketing and sales when you generate a new lead and ask your sales team to follow up on that lead, the handoff between your operations team and finance when it's time to make sure regular client billing is completed in a timely and accurate way, and the handoff of a new customer order from sales to operations. At these points, it's key to have a foolproof system to make sure the transition is smooth and effective as work flows between pillars in your company.
The keys to a strong linkage include:
* **Get a moving start.** The pillar who is handing off the process to the next recipient pillar of the business has to make sure that certain information is clearly captured and passed along in a structured way. This could mean that your salesperson formally introduces your operations team to the client, or that your account rep sets manageable and realistic expectations with your customer when they turn over the client to your production team.
* **Make the handoff explicit.** In racing, you'll hear sprinters shouting "Baton!" or see them slap the baton into the hand of the recipient so they know the transition has happened. In business, it is just as essential to make sure your team always knows who owns which stages in the process and, most important, which stage the process is currently at. Define each stage and make it part of your process to formally declare which stage the process is at and get both halves of the linkage to acknowledge this.
* **Define the handoff boundaries.** In racing, there is a specific section of the track in which the handoff must occur or the team will be disqualified. In business, you also need to have a clear, formal process that lays out the parameters for the handoff and ensures that all parties have signed off on these rules.
* **Explicitly ask both halves of the handoff what their needs are (and why).** What does the person who is receiving the handoff need to help them succeed? What information, materials, introductions, setup, or support? Why do these items matter to them? What happens if they don't get it? And what about the party doing the handoff—what do they need to succeed? What response, feedback, timeliness, or assurances? And what is the impact if they don't get these items? It may seem obvious that this conversation needs to take place, but in our experience it rarely does. When your team understands the other side of the handoff's needs and the "why" behind those needs, you'll notice your team going above and beyond to be better team members and meet those needs. This in turn will lead to fewer fires and better teamwork.
* **Formally process out the handoff.** Pull out your sticky notes and process out the exact steps in the ideal handoff. Try out this draft process and observe where the wasteful steps and rough edges are. Then, in version two, simplify the process and sand down the rough edges. Consider adding a simple control (e.g., checklist, visual timeline with marker of where you stand in the process, etc.) to make sure the process is operating smoothly.
So there you have three powerful operational tools to help you scale your operations pillar: Expert Systems, your UBS, and critical linkages. In the next chapter we'll look closely at the obstacles to scaling in your Financial pillar.
## YOUR FINANCE PILLAR
CFO SECRETS TO MANAGE CASH FLOW, IMPROVE MARGINS, AND FUND GROWTH
The Finance pillar of your business encompasses all the essential functions of collecting, tracking, distributing, and reporting the flow of money in and out of your business. It includes your billing procedures, collection practices, accounts payable processes, all financial reporting from balance sheets to profit and loss statements, and your management of cash flow. As your business matures, this pillar will also include the sourcing of capital, management of risk, and the forecasting necessary to help your management team make better business decisions.
At Level One and Early Stage Level Two, chances are you have few financial systems in place. This is normal for a small business scrambling to survive _._ At the very least, we suggest that you outsource your bookkeeping to a part-time service so your financial transactions are accurately entered into an accounting software program.
Once your business enters Middle Stage Level Two, you need to put your financial house in order. Whether you outsource your accounting to a part-time controller or employ someone on staff full-time, it's essential that you enlist help to organize your Finance pillar and begin establishing the intelligent financial controls that your business will need as it grows.
At Advanced Stage Level Two, you'll most likely have a full-time controller overseeing the finance part of your business; as you enter Level Three, you'll probably transition to a full-blown CFO (Chief Financial Officer), not just a controller. What's the difference? A controller is expert in following the financial systems you have in place; a CFO is expert in helping you build them. A controller can help you make sure you have accurate reporting and can maintain your existing financial infrastructure; a CFO can help you with higher-order financial thinking, such as running pro forma analyses and managing your capital sources. A controller can help you execute existing plans in the here and now; a CFO will help you plan for the future.
Regardless of which stage your business is at or whom you have helping the financial area of your company, as the business owner you still must understand how to manage cash flow, how to fund growth, and the basics of financial controls. While you can leverage your financial team, you cannot abdicate responsibility. In this chapter we'll give you the essential finance short course you need to safely and intelligently scale your business. Let's begin with cash flow.
Managing Cash Flow: The Seven Cash Flow Commandments
One of the biggest obstacles to scaling your business is poor management of cash flow. That makes sense when you consider how a typical product-based business works. The business pays to manufacture or purchase its products and then inventories the products until customers purchase them. Buying or producing that inventory ties up the business's cash and increases its risk (e.g., damage to inventory, obsolescence when upgraded, etc.). And when they _do_ ask for payment, most business owners never create a clear, measurable collections system that sends invoices in a timely manner and relies on a formalized follow-up system to make sure they are paid.
Here are seven Cash Flow Commandments, along with our suggestions to effectively apply them, to give you a rock-solid financial base upon which to scale your company.
Cash Flow Commandment #1: Thou shalt collect more of what you're owed.
It's our observation that most business owners are simply uncomfortable or even afraid to look clearly at their collection practices. They bury their heads in the sand and passively wait to get paid. Many are even afraid of upsetting clients by asking for payment and push it off onto their bookkeeper or another poorly equipped member of their team.
When you are delivering value to your customers you should never be afraid or apologetic about asking to get paid. Your customers wouldn't pull up to the gas station and expect to get their tank filled for free. Likewise, they knew they would be paying for your product or service when they ordered it.
The costs to most small businesses of not collecting money due to them are huge, and only compound as the business grows. If a business is struggling with cash flow, one of the most common causes isn't lack of sales, it's poor collections. Generally, by focusing on this area of a business, an owner can get a 5 to 15 percent boost in his _collected_ gross income within 90 days.
Think about this for a moment. If your business operates at a 33 percent profit margin, then that 5 to 15 percent increase in gross income actually translates into a 15 to 45 percent increase in operating profit. That's the power of Cash Flow Commandment #1, and why you _must_ pay attention to it.
Take the example of Brett, who owns an IT services company in Colorado. He came up to us at a break during a workshop we were hosting and asked our advice on a collections issue he was facing. He was embarrassed to admit that he was guilty of ignoring this part of his business. It turned out that Brett had become so busy in the day-to-day of his business that he hadn't billed several large clients for more than six months! He went on to say that he didn't want to upset them with a large, late bill so he was planning on writing off the past-due invoices. We strongly pushed him to send the past-due invoices right away, along with a letter and phone call to explain the situation and ask for full payment. He collected more than $30,000 in immediate revenue by doing so.
Despite what you might think, most people just want to do the right thing. Your customers are not trying to avoid paying you—you just didn't bill them. Nor are they trying to cheat you by not paying. So while a late invoice and an apology might be a little annoying, they're more likely to get you paid than you might think. After all, it's money that they truly owe you.
The first foundation of your collections process is to collect all that you are owed. The only thing worse than not making the sale is making the sale, but not getting paid. After all, the moment you make the sale you now have your cost of goods sold, any sales commissions you're paying, and other expenses. If you don't collect on what you're owed, the sale is a negative drain on your company's cash flow. Here are some specific suggestions to help you collect more of what you are owed:
* **Accurately track who owes what.** Many collections issues aren't about poor practices in asking for the money, they are about poor operational systems to track what money is actually owed, and where you stand in collecting that money. If this is true of your business, make sure that your collections Expert System is something you focus on building.
* **Invoice more frequently.** Asking for more frequent payments means that the amount of payment is smaller and therefore easier for your customer to pay. It establishes a professional pattern of asking for and getting the money you're owed. Furthermore, by invoicing more frequently you get the added benefit of a steady stream of income for your own cash management, instead of struggling along between each large payment over time.
* **If you have to escalate your collections efforts, compress the timeline.** Don't let slow-paying customers stretch out payment for another 60 to 90 days. Front-load your collections efforts.
* **Be smart about the people you task with asking for payment.** By default, most companies assign their bookkeepers or controllers to ask for payment. But these people tend to have weak interpersonal skills. Instead, consider involving a mix of people in your business to collect what's owed. For example, when the receivable is fresh, ask your sales team to make collection calls. Because many of them are compensated on a commission basis, they have a strong financial incentive to help you collect on the sales they make. Also, paying salespeople when money is actually collected, as opposed to paying them the day a customer just says yes, is a smart way to keep salespeople vested in the collections process.
* **Pay attention!** As simple as it seems, just paying close attention to your A/R reporting on a weekly basis helps your business collect thousands of dollars more of the money it's owed. The result? Increases to your cash flow and better effective operating margins for your business.
Cash Flow Commandment #2: Thou shalt collect faster.
One of the most important lessons with respect to managing cash flow is this: **The faster you collect from your customers, the easier it will be to manage your cash flow.** Most businesses carelessly fall into cash crunches because they let their receivables slip longer and longer before being paid. Not only does this put the revenue in jeopardy because the longer it takes you to collect, the lower your odds of getting paid, but also because your collections costs rise, cutting into your margins. That is why you need to do all you can to accelerate the pace of your "collections cycle," the average time from the moment you have cash going out the door for "cost of goods sold" to the time you collect on the sale of that product or service.
Consider a manufacturer like Brian Thomson. His company designs and manufactures a high-end line of bicycle parts. Their collections cycle starts when they buy raw materials to produce their parts and continues until three to four months later, when they collect from their distributors and other wholesale customers. As you can imagine, Thomson and his team do everything they can to reduce the collections cycle to free up a tremendous amount of cash with which to operate and grow the business.
The same concept of a collections cycle applies to service businesses too. Far too many service providers make a sale, perform the work, and only then invoice for payment, often waiting 30 to 90 days to be paid after the work has been completed. In many cases, this means that the service business has had the cost of goods sold from labor, materials, and other out-of-pocket expenses to fulfill on its service offering for 60 to 90 days or longer. Remember, the longer your collections cycle, the more operating capital you need to run your business. Our goal with Cash Flow Commandment #2 is to shorten the collections cycle and free up operating capital to grow your business. Here are some concrete suggestions to do just that:
* **Get paid in advance!** The best way to reduce your collections cycle is to get paid up front. If you can't collect in full, can you at least collect a deposit, retainer, or up-front fee? Getting paid before you fulfill your product or service helps you eliminate the hassle and additional cost of chasing down payment later.
* **Consider incentivizing customers to pay in advance if you struggle to collect in advance otherwise.** You could offer a discount on advance payment, moving the customer to the front of the delivery line, or adding a bonus to their purchase they wouldn't otherwise get. More customers are willing to pay in advance than you might think if you offer them an appealing incentive. Many business owners just assume customers will say no, so they don't even ask. Don't answer for your customers—give it a shot and see what happens.
* **Don't wait to bill; collect right at time of service or delivery.** If you can't bill before you fulfill, why not ask for payment upon completion of the work or at the time you deliver the product to your customer? Smile and say you'll wait while they grab their checkbook or credit card. At the very least, give your customers a bill at the time the services are rendered as opposed to waiting several weeks.
* **Build a "cost" for your clients into your standard contracts.** If you are financing your client's purchases, then you should get paid for your trouble. Make sure your contract includes a monthly financing charge for all accruing bills. This clause should also state that the client is responsible for all reasonable costs of collection. Finally, where possible, get the business owner to sign individually and not just in the name of the business.
* **Accelerate your production and delivery cycle.** Assuming your business gets paid in part or entirely after you have produced and delivered your product or service, then the faster you finish that production and delivery cycle, the sooner you'll get paid.
* **Make it easy for your clients to pay.** Accept credit cards, PayPal, and other online payment options. Set clients up on auto-pay through an ACH bank draft or credit card. Take payments by phone or through your Web site. Make sure your invoice clearly says to whom they should make the check out and for how much. Include a self-addressed envelope with your invoice. The easier and more convenient you make it to pay you, the faster you'll get your money.
Cash Flow Commandment #3: Thou shalt maintain timely and accurate financials.
Your cash flow is like blood circulating through your body—it is the very essence of life flowing through your business. Yet too many businesses have poor bookkeeping practices that either distort their real financial picture or, worse, keep them operating totally in the dark.
We spoke with the owner of an $11-million-a-year promotions company who shared with us how she discovered that her financial bookkeeping was off by more than $3 million because her controller had mis-entered critical information into their accounting software! Another business owner with a $5-million-a-year convenience store business found out, after he fired his bookkeeper, that she hadn't reconciled the company bank statements for over a full year. Sadly, these examples are all too common.
Successfully scaling your company requires that you have good information with which to make smart decisions. Your financial information is one very important view of your business. So let us ask you a blunt question: Is your financial record keeping up-to-date and accurate? Is it timely? In other words, can you look up your profit and loss statement from last month and know that it accurately reflects your income and expenses for that period? What about your balance sheet and statement of cash flow?
If you don't have accurate and timely financials, managing cash flow is almost impossible. You need this information to base your decisions on and to guide the moves you and your team make.
Cash Flow Commandment #4: Thou shalt work to stabilize income and expenses so that they are more predictable and consistent.
The more accurately you can predict your income and expenses, the easier it is to effectively manage your cash flow. As you make financial projections of income and expenses, work hard to learn from each round of predictions so that with time you become more and more accurate. Think of your financial projections like your weather forecast. Back in the 1970s, weather forecasts were notorious for being more like fiction than fact. But with the advent of satellite weather data and more powerful computers with which to model the weather, today the forecast is incredibly accurate—in the short term. With some effort, your business cash flow forecasts can become very accurate in the short term (over 90 to 120 days), but the farther out you go, the more that can go wrong with them. This is why you must update your projections regularly—monthly in normal times; weekly in crunch times.
Once you have an accurate picture of your cash flow—past and projected—you can begin to clearly identify and deal with patterns you see. For example, do you have seasonality issues that cause your business to slow down in certain seasons? If so, you can make sure you don't build up assets, inventory, or staffing at a time when your business is about to slow down. Also, when you know about a downtime, you can proactively come up with ideas to stabilize your income.
Klayton Tapley, who owns The Fireplace Place, an indoor/outdoor fireplace and patio grill company, did just that. His company's busy season starts in spring and runs through fall, which is tough on his business as he has a full-time sales and technical staff with all the overhead that entails. This past year, Tapley implemented a new strategy to go after the multifamily building market, which included both service and repair work as well as major capital improvements for these buildings. In his first four months of implementing his seasonal strategy, his dedicated multiunit salesperson had signed up more than $250,000 of business, much of it during his historically slow season.
Another important exercise to stabilize income and expenses is to make a list of external factors outside of your control that could affect your cash flow and your forecast. We worked with a company that sold tires for cars, and it had developed a solid sales and cash flow forecast. But then something unexpected happened—there was a huge rubber shortage in the primary countries that produced rubber. As a result, prices for raw materials were projected to dramatically increase in the next quarter. Not only did this create a problem for the company's projected margins for the next quarter, it also created an important strategic opportunity. Should they use some of their cash reserves to purchase a larger stock of inventory before the prices changed? While this would impact short-term cash flow, it would also possibly give them an edge over their competitors in the next quarter when everyone else was buying tires at higher prices. Looking at external events and their potential impact is an important step in forecasting cash flow.
Here are some tips on stabilizing your income and expenses:
* **Work to get longer-term contracts in place with your customers.** Can you go from month-to-month or order-by-order arrangements to annual contracts? How about multiyear contracts? If you must have a cancellation clause to let your customer cancel, see if you can include a provision that requires 60- to 90-day (or longer) notice.
* **The longer your sales cycle or production cycle, the more important it is for you to have "clues" that tell you where you stand on closing the business or producing your product or service.** We call these critical clues "leading indicators." For example, one of our clients does a monthly calculation of the potential sales in his prospect pipeline to give him his expected sales volume over the next 180 days. The idea is that if you can spot a problem early enough, you can do something about it. It's those "surprises" that can derail your business.
* **Look for products or services you could cross-sell or up-sell.** If you sell a product or service that is a "onetime" purchase, what can you do to expand on this sales opportunity? For example, is there any service plan that would enhance customers' ownership of your product? Could you create and offer a next-phase service to follow your product offering?
Cash Flow Commandment #5: Thou shalt sharply manage your expenses.
Costs matter. Early on, business owners know this and agonize over every penny spent. But as the business grows, you will no longer be able to be the hawk watching closely over every dollar spent. Here are several suggestions to help your company control costs as you scale:
* **Consolidate your purchases and negotiate better pricing.** This is especially important for companies that have gone through a recent burst of growth. Too often we see companies paying prices based on purchase volumes that they far exceed. Renegotiate frequently. Also, check around your community for local buying organizations that gather all the local businesses in the area and use their collective buying power on behalf of all the members.
* **Get vendors to compete for your business.** Make sure they know about each other, without rubbing their faces in it. It's amazing how much better your pricing can be when your vendors feel the hot breath of their competition on their necks. Even if you plan on staying with your current vendor, the very fact that you know and they know that you're getting outside bids will keep their pencils sharp and help ensure you get better pricing.
* **Review your vendors regularly.** Make this a standard practice in your company. Also make sure you flag all automatically renewing contracts to pop up for review and rebidding 60 to 90 days before their renewal. Better yet, cross out the boilerplate language from your vendor's contract that calls for an automatic renewal and instead write in that you have the option to renew, but not the obligation.
* **Train your staff to ask for and get discounts.** A short negotiation course on how your team can get discounts from your vendors, plus consistent recognition for team members who do this, pays off handsomely in increased cash flow. This practice alone could reduce your variable expenses by 5 to 10 percent.
* **Wherever possible, make expenses variable versus fixed.** You can change variable expenses when you need to, dialing them up or down to suit your cash flow situation and business needs. This flexibility is highly valuable. For example, could you use performance-based compensation versus guaranteed payments? Can you rent, not purchase? Can you lock in an option to renew, instead of a contractual obligation?
* **Cultivate fiscal discipline as a core company value.** Symbolic choices you make or allow as the business owner will find their way into the culture of your company. Sure, you can buy that fancy car or travel first-class on your company's dime, but just know that your team is _always_ watching your example.
Cash Flow Commandment #6: Thou shalt relentlessly manage and improve your margins.
The purpose of your business is to profitably create value for the market in a scalable way. In order for you to serve your customers, pay your employees, and reward your investors (yourself or outside investors), your business must be profitable.
Your margins are a measurement of your profitability. There are two "margins" that you the owner must focus on. The first and most easily understood is your **operating profit margin**. This number is simply a calculation of how much of every dollar in sales ends up as operating profit (pretax) for your business. For example, if you had $1 million in sales and ended up with a pretax profit of $250,000, your operating profit margin would be 25 percent. Your operating profit margin is a great measure of how profitable your business is overall. Building on our fictitious $1-million-per-year company, if you were able to go from a 25 percent to a 30 percent operating margin by better managing your expenses, you'd earn $50,000 more profit from that same $1 million of gross revenue. That 5 percent increase in operating profit margin equals a 20 percent increase in profit. Don't worry about the math too closely; what matters is to get a _feel_ for the concept of your operating profit margin and why it matters to your business.
The second margin you must understand is your **gross profit margin**. This is perhaps the most misunderstood and least leveraged number in your business. Your gross profit margin is a measure of how much money you have left over from every sale after you take out what it cost you to produce or acquire the product or service you just sold. It's calculated as follows:
Gross Sales (i.e., total sales before any expenses)
−COGS (the "cost of goods sold" for the sales you made)
* * *
= Your Gross Profit
Continuing with our imaginary $1-million-a-year in sales company, if you had a COGS of $250,000, that would mean that your gross profit was $750,000. When you express that gross profit as a percentage, you get your gross profit margin of 75 percent.
Your gross profit margin tells you exactly how much money you have left after you pay the cost to produce and fulfill on a sale. This lets you know how much you can spend on marketing, sales, fixed overhead, and so on—and still have enough left to make a reasonable profit for your time, effort, and risk. This number is also a good indicator of the overall efficiency of your business. Knowing this number helps you look strategically at your pricing. It lets you know which customers, products, or projects are the best margin business to go after, and which you should consider phasing out (or even immediately cutting).
Here are some tips to improve your margins over the long term:
* **Velocity matters.** The faster your turnaround time (from order to delivery), the lower your overhead cost per unit produced. This in turn means improved profit margins.
* **Up-sell and cross-sell to increase your average unit of sale.** In general, when you increase the amount you sell to your customer at one time, you'll improve your margins because you'll be increasing the purchase velocity and therefore lowering your cost per sale in terms of overhead burden. Plus you're amortizing your marketing cost over a larger unit of sale, which dilutes your marketing cost for each sale.
* **Encourage your customers to upgrade to higher-value products or services.** Explore whether you can get your customer to upgrade to higher-value offerings that serve them better and that generally have a better margin for your business.
* **Cut low-margin clients, products, or services, and invest the saved time and money in higher-producing parts of your business.** This presupposes that you have accurate and timely reporting that shows you which clients, products, or services produce what margins.
* **Retention, retention, retention.** Attrition costs. Do all you can to keep your clients actively purchasing from you. Study the most common "drop points" in your client's purchase history. Can you strategically reinforce your business system to reduce that attrition? Perhaps you need to better communicate with them how to use your product or service? Or give them a well-timed "gift" or make a well-timed visit or phone call? Courting your current customers eliminates or greatly reduces the acquisition or marketing cost on that second and all later transactions.
* **Watch out for scrap, spoilage, and wastage.** Is it a quality issue on production? Are you poor at forecasting, and keep too much supply on hand for an order? Does it take you too long to sell your inventory and you lose part of it to obsolescence? This can also be an issue in areas of your business outside of operations. David once owned a real estate training company that purchased a thousand leads a month from Foreclosures.com at $5 per lead. He thought his sales team was following up on each of these leads with an outbound phone call, but later learned that 90 percent of these leads just weren't followed up on. His sales team had _said_ they were calling them, but the reality was that they were earning so much money off the easy inbound leads that they ignored the harder-to-follow-up-with outbound leads. The lack of sound sales controls meant his company wasted $5,000 per month to buy leads that spoiled because they were not effectively followed up on.
Cash Flow Commandment #7: Thou shalt make smart strategic decisions on purchasing and pricing.
The final part of managing your cash flow is making smart big-picture financial decisions. This includes things like your pricing model, capital investments, staffing, and other strategic investment decisions. Now that you have good financials, tight controls on spending, and a solid understanding of how to manage your margins, you're ready to make the big financial decisions that so heavily impact both cash flow and the long-term success of your company. This is often where the skills and experience of a CFO prove so valuable.
Here are several concrete suggestions to make better strategic financial decisions:
* **Get accurate and timely financial data** before **making long-term financial decisions.** Smart business owners let accurate data inform their mission-critical moves.
* **Review your strategic pricing decisions.** Most businesses set their prices when the business was new and desperately needed business and, as a result, set pricing levels low. Over time, the business may have made nominal increases to pricing every few years, but rarely did the owner ever sit down and fundamentally rethink his pricing model. Do you price in relationship to your costs and your competitors? The most successful companies take both of these factors into consideration, but they also price in relationship to the cost of the status quo for their customers. How much is the problem that your product or service solves already costing them? What is the real value of your product or service? What is the "frame of reference" you could give your customers that would help them immediately see your product or service as both the logically sound and emotionally satisfying solution? The more you are able to provide your market with solutions that other companies can't, the greater your ability to price in relationship to the true value of your solution instead of the race-to-the-bottom commoditization that so many businesses suffer from. While some business owners fear increasing prices for loyal long-term customers, the truth is that customers fear "switching costs"—the cost to leave you, train a new vendor, and go through the whole learning curve all over again with someone new. Often the switching cost is higher than your increase in prices, so they won't leave you just for making changes.
* **Look for the three clues that your pricing may be too low.** (1) You have limited production capacity and a large and hungry demand that exceeds your capacity to produce. (2) Your price is well below market without a compelling reason for it to be so low. (3) You have never _systematically_ looked at your competitor's value proposition or your customer's true costs for the problem you solve.
* **Consider changing how you charge.** Is there a way you can move from a one-time charge to an ongoing revenue stream? Perhaps you do have a onetime charge for the initial purchase, but is there a way you can provide ongoing value to service your client on an ongoing basis? The smartest business models allow companies to annuitize their business relationships.
* **Find the optimal staffing level and manage your hiring intelligently.** Look for a simple heuristic that helps you know when you need to hire more production and operational staff (e.g., sales per employee, projects per operations staff, etc.) and when you are too heavy. What are the indicators that alert you to the need to staff up or staff down? What investments could you make in technology, systems, and training that would allow you to produce more with fewer people? Note that generally A players produce _multiples_ more value than B or C players, yet cost only a percentage more. Look for ways you can upgrade your team over time so that you can produce more with less.
* **Get perspective before you make a major capital investment.** All too often, business owners find that a succession of small commitment steps lead them over the edge of the cliff when making the big infrastructure and capital decisions. They let sunk costs and vested interests that they are afraid of losing push them to chase bad money with good. After you have gathered all the relevant facts, step back with your leadership team and ask the question fresh: "Knowing all we know today and imagining that we had no sunk costs at this point at all, what is the best decision for the business over the short, medium, and long term?"
* **Know the difference between strategic expenses and nonstrategic expenses.** Strategic expenses are those things that directly help you sell more or produce better. They include marketing campaigns that work, salespeople who sell, technology upgrades that reap real returns and ongoing advantages of significant value, and intellectual property barriers that give you a sustainable advantage for which the market will pay. Nonstrategic expenses essentially include everything else. Outspend your competition for strategic expenses—in good times and bad. Relentlessly cut nonstrategic expenses. And repeat this over and over!
Funding Rapid Growth
Growth almost always sucks up cash. The faster your growth, the greater your need for sources of capital to fund that growth. Plan in advance of that growth to secure the capital reserves or funding sources to afford that growth.
Here are five sources of capital to fund your growth with important lessons for using each that we paid dearly to learn over the past 20 years.
Sources to Fund Your Growth
Customers (the people who buy your products or services):
* Redesign your collections system so you get paid up front, or at least partially paid up front with progress payments paid along the way instead of at the end.
* Let your customers pay you to be first in line for new features or services that they dearly want.
* Offer a premium deal for cash up front (with a clear reason why you want it). This could include an upgraded value offering or a discounted price for prepayment.
* Revisit your pricing. How could you increase your value and hence increase your pricing?
Vendors (includes equipment leasing or purchases, supplier terms, etc.):
* Build your relationship, strategically share information, and ask for input and help from your vendors. They are partners in your growth.
* Get your vendors to increase your available credit and lengthen the period before you must pay them.
* Barter. Find ways to trade products or services to conserve cash for growth.
* Consider engaging your vendor as a strategic partner when you have a big opportunity. After all, your growth will benefit them too.
* Build healthy relationships with your vendors and, if needed, work out win-win ways they can help you get through a cash flow crunch. They can be excellent champions to have on your side.
Lenders (the traditional institutions from which to borrow money):
* Make your business bankable before you need the money. Normalize expenses, improve your margins, lock in better customer contracts—then go to your bank for more money.
* Learn how to package your application so banks are competing to lend you money. Smart businesses flip the process on its head by creating competition and soliciting loan proposals from multiple lenders. The better you make your loan package, the easier it is to create a feeding frenzy.
* Understand how a banker looks at your business. (For more on this, see the free training video titled _Business by the Numbers: Margins, Ratios, Key Indicators, and Cost of Goods Sold_. You can access this and two dozen other training videos at **www.ScaleYourBusinessToolkit.com/videos**.)
* Look for ways to enhance your personal credit and financial statement to grow your other bank sources of funding (including credit cards, signature loans, home equity lines of credit, and so on).
* Lines of credit are great sources of liquidity to finance operations, but they can be fickle. **** If your business relies heavily on short-term credit to finance operations, deepen your banking relationships by consistently and accurately communicating with your bankers. Bankers hate surprises more than anything. Also, plan your contingencies in case your credit line gets reduced or, worse, closed.
Internal (reinvesting the profits and cash flow of your business back into the business to fund future growth):
* Build up a capital reserve to help you seize opportunities to grow. Liquid capital at the right time is worth its weight in gold.
* The faster you can accelerate your production and collections cycles the less working capital you need, and the faster your business can grow without external sources of funding. So do all you can to speed up production and reduce your collections cycle.
* Reduce your sales cycle. This will help you generate a faster return from your marketing dollars and hence also lower your working capital requirements.
Investors (venture capital, private equity, crowdfunding, etc.):
* Make sure that the extra capital and the growth it can potentially enable you to gain is worth sharing ownership. Does investment capital from venture capital or private equity really serve the long-term goals for your company?
* Money follows people first, and the ideas second. If you are going after investor capital, understand that they will be placing their bet on your team, not just on your business ideas.
* Every capital infusion requires that you set a value for your company. Learn how your business (in your industry and with your model) is valued and take steps early to enhance its value so that when you go for outside capital you get more money while giving up less ownership.
* Do your homework. Research your potential investors. Make sure you are approaching people who have invested in businesses in your industry. Don't try to raise capital to expand your financial-services business from an investor who focuses on retailing.
* Learn to tell a compelling story. Raising capital is a sales process, and you need to capture investors' attention with a story that they understand and covet. Speak from the perspective of your prospective investor's needs, wants, and desires, not your own. Also, if raising money is partially about telling a compelling story, remember the best characters have flaws. Don't hide your business's warts; expose them to your potential investors along with your concrete plans to fix them. This will build your credibility (they'll likely find them anyway before they stroke that check).
* Because you're looking for money to fund growth of your existing business, make sure you use your existing track record as the springboard to investors' return. Show them how their capital will increase sales and pay them a healthy return.
Safeguard Your Company from Fraud and Theft
In this final section on removing obstacles to scaling your Financial pillar, we want to look at proven ways to systematically protect your company from fraud and theft. Private companies are prime targets for fraud. According to the Association of Fraud Examiners, in 2010, 42 percent of fraud cases happened in privately held companies. The majority of these cases occurred in companies with fewer than a hundred employees, with the average fraud continuing for 18 months before it was even detected! The median loss in all these fraud cases was devastating: a whopping $231,000 per case. Since your company may be vulnerable to fraud and theft, let's dive in and look at how you can _systematically_ protect yourself and your company from being blindsided.
The "Big Three" Financial Control Concepts
Financial Control Concept #1: Always have two unrelated parties involved in any money flow in or out of your business.
This could include one person who opens the mail, lists all checks received on a spreadsheet, and preps these checks for depositing, and a second, unrelated person who double-checks the math and reviews the deposit amounts before it goes to the bank. One person who sets up the ACH payments for the payables period, and another who goes in and approves the payments to go out.
The idea behind this financial control concept is pairing two unrelated team members on a given financial job so they can be a check and balance for each other. This does two things. First, it reduces temptation for fraud as both parties know that the other will likely quickly see any bad behavior and report it immediately. Second, it makes it so two people would both have to collude _and keep the secret_ before any serious theft could happen in those areas.
Financial Control Concept #2: Create permanent footprints in your financial system that cannot be erased so that you always have a clear audit trail.
This includes numbering your invoices and keeping a separate invoice log of who has which invoice number series. Footprints can also mean keeping checks under lock and key, with a check log clearly showing who is in control of and responsible for which check series, or making sure that your financial software is set up with individual log-ins for any team member who needs to access that information, and that the software permanently tracks any changes to the accounting and who made those changes. (This is an option you can turn on in QuickBooks, one of the most popular small business accounting programs. Other accounting software programs like Peachtree have similar functionality.) We also suggest that you require your financial staff to have private, robust passwords that they change at least twice a year.
What this financial control concept does is make it much more difficult for a person to cover up theft, whether by making it clear who had which checks or invoices, or by keeping a permanent record of who altered what accounting records. By making these things obvious and traceable back to the offending party, you lower the temptation of bad behavior and increase the odds of its being spotted faster.
Financial Control Concept #3: Show vigilance and actively question things that strike you as strange or out of the ordinary.
As the business owner, it is your responsibility to actively and regularly review the financial records of your business. Sure, you will have financial staff who do the heavy lifting, but you still need to pay close attention too. Spot-check reports and highlight any line item or general question you might have. Follow up with your staff to get answers. If the answers don't satisfy you, _dig_. Listen to that still, small voice inside that sounds the alarm. At the very least you'll keep your financial team on its toes and at least _thinking_ that you are watching closely. Most business owners know roughly what the numbers should look like, and what expenses and sales figures make sense. If you see something that is out of the ordinary, question it.
Here are several more powerful financial controls under four categories for you to consider to make your Financial pillar stronger and more resistant to fraud or theft.
CASH CONTROLS:
* Bank statements should be mailed to the business owner's house. The business owner should open and review the bank statement. (And she should mark up the statement so that her controller or bookkeeper sees that she's looked closely at the statement!)
* Perform monthly bank reconciliations, preferably with someone other than the person doing the deposits.
* Use a lockbox or separate post-office box for accounts receivable payments instead of having them sent to a general company address that many more people have access to.
* Consider using a sweep account to take money over a certain amount out of your business's main operating business account. This second sweep account should have much tighter financial controls around who can access it (usually with the owner being the only person authorized to move money out of that account).
* Do not keep signed blank checks for future use. This practice leaves you vulnerable to misuse.
COLLECTIONS CONTROLS:
* Any collections write-offs should be approved in writing by the owner or senior-level manager.
* When possible, use prenumbered invoices and maintain an invoice log.
* Collections A/R reports should be reviewed _weekly_ by management.
* The owner and senior management should periodically review the master client list for potential fake customers.
* Look for outside ways to verify your sales receipts. For example, compare the attendance numbers at a workshop against the gross income for that event, or the sales for the week against the traffic flow in your store. If something is out of line, investigate.
ACCOUNTS PAYABLE CONTROLS:
* All vendor invoices over a certain amount should be reviewed by the owner or senior manager.
* Restrict access to corporate credit cards and require receipts and detailed invoices for all credit card charges. Have a formal expensing system that includes a clear, written policy of what expenses are and are not reimbursable, and who needs to sign off on what types of expenses. Also, require that all expenses submitted include a copy of the original receipt. Require your employees to sign their expense reports warranting that they are true and accurate.
* The owner should periodically review the master vendor list for fake vendors (a common way employees steal from a business). This is a simple report you can pull from your accounting software.
* Hand out paychecks at a job site to see if you have any fictitious employees (especially for businesses with large off-site projects that have employees or contractors there whom they are paying).
* All overtime should be approved by management prior to its being worked.
OTHER FINANCIAL CONTROLS:
* Make sure that any software or other purchase that the company pays for be owned and titled in the company name. This includes checking that the company is the one named with the software vendor or other supplier.
* Keep a close eye on your margins and key numbers. For example, if your cost of goods sold or gross margin is off, find out why.
* Thoroughly check employees and independent contractors before you hire them. Consider criminal background checks, drug testing, and credit checks.
* Consider bonding financial team members or getting appropriate insurance coverages.
The reason we laid out this final section of this chapter is not to turn you into a paranoid business owner who is suspicious of all your employees. Rather, the reason is to empower you to be a _responsible_ business owner who radically reduces the temptation of some team members' doing something inappropriate because they know that they will be caught. This gives you peace of mind knowing that one bad apple isn't hurting things for your business as a whole.
Now that you have a solid understanding of how to improve your Finance pillar, it's time to turn our attention to removing the obstacles to scaling from your Team pillar.
## YOUR TEAM PILLAR
ATTRACTING, RETAINING, AND UNLEASHING TALENT
Tim was a good salesperson. He understood the retail flooring industry and could help a homeowner find and purchase the perfect flooring solution for his or her home. But as he grew his flooring company past the $1 million mark, he increasingly found himself behind the scenes in his store, hiring and managing his staff. He had no training or experience to draw on to help him design compensation plans and get his team to buy into the company vision and values, and he certainly suffered his share of expensive staff turnover issues. He was a strong technician who now found that his business needed him to learn and master a completely new skill set.
The Team pillar of your company handles the hiring, orienting, training, assessment, compensation, development, and, if necessary, the exiting of your team. It is also responsible for keeping your company in compliance with all the legal requirements of working with employees.
Most small and midsize companies' Team pillars are fundamentally weak. The causes for this vary from the founder's fear of hiring better, smarter people than himself, to a faulty belief that employees are just a hassle and expense, to a lack of training and experience with hiring and managing employees. As a consequence, most business owners spend too little time strategically deciding whom they need to hire and when, and how to best retain and tap into these talented team members to accomplish the goals and objectives of the company. Yet when this is done well, your team is the ultimate source of leverage for you to create and sustain rapid growth. Their passion, skills, and experiences are often the missing link between your scaling successfully and reaching Level Three, or getting mired in the Self-Employment Trap. Remember the three-legged stool we shared with you in chapter 2? Your team is one of the three essential components of creating a stable base upon which to scale (the other two being systems and controls). In this chapter, we'll share with you our best lessons on recruiting, retaining, and unleashing great talent.
Common Sense—Where Have You Gone?
When Jeff was in his twenties and the CEO of his first company, he hadn't had much experience managing people. Here he was, this newbie, when one of his older, more seasoned team members asked him about the company's employee manual. Jeff's company didn't have one, so he called everyone from the small company together into a conference room. He asked them all to take out a blank sheet of paper and draw a vertical line down the middle of the page. He asked them to list everything they disliked at other jobs on the left side of the page, from the way people treated them, to the way the company expected them to behave, to all the things that made them want to leave the company. Then, on the right- hand side of the page, he asked them to list the exact opposite: the things they had loved about the companies they had worked for, the policies they wished the companies had had, and the way they wanted to be treated by management and their peers.
Everyone in the room now had a single sheet of paper with two detailed columns, one a list of what they hated about past employers and the other listing how they wanted to be treated. Jeff then instructed the team to all tack their lists outside of their office or cubicle. They had just outlined the company's employee manual. "If I catch anyone doing anything on the left side of the page, including me, you're in trouble," he said, smiling. "And we'll all work hard to do the things on the right side of the page. In fact, I'll make it a point to have myself and any future managers read through the right side of the page every Monday morning as a refresher of how you want to be treated."
While not formally an "employee manual," the two lists gave Jeff a clear outline as to what specifically his team loved and hated. They gave him the simple outline of how he could empower, encourage, and embrace them over the long term.
Here is one of the most interesting parts: Most of what his team had listed on the right-hand side of the page had nothing to do with financial compensation. They wanted recognition, both as valuable people and as contributors to the team. They wanted simple freedoms that kept bureaucratic red tape from wasting their time and hobbling their choices when those choices only helped (or at the very least, in no way hurt) the company reach its goals. They wanted the company to trust their judgment and to actually allow them to make decisions that mattered. And they wanted to grow professionally.
This story illustrates how uncommon common sense has become. When was the last time you asked each of your team members to think about what they wanted from their job? Or the last time you looked at your company's internal systems and controls to see if they helped your employees do their best work? Asking employees what things management can do to motivate them and make them feel valued, and then comparing that list with what your company actually does day to day, is often a shocking wake-up call to leaders.
Making Your Company a Place Where Great Talent Wants to Work
Companies compete for top talent. One of the most important insights we can give you to help your company create and sustain rapid growth is for you to systematically make your business a place where great talent wants to work. Here are some simple suggestions you can apply today to do just this.
* **Deliberately define and profile your ideal team member.** Establish a profile of your business's perfect team member. Obviously the specific qualifications will vary from position to position, but think about the common traits, beliefs, and drives of your ideal team member. Who would they need to be to fit well into your company culture? How could you build simple filters into your hiring process to weed out any candidates who clearly are not a strong cultural fit for your business?
* **Be selective about whom you invite on your team.** Great talent thrives when working with other talented team members. Keep your standards high as you hire and consider upgrading weak team members as you have the cash flow and opportunity to do so. A great side benefit of this is that birds of a feather flock together. In many of our companies, our best new employees were often friends and referrals of our existing talented employees.
* **See the whole person and manage individually.** Sure, your company needs to have standard HR policies, but use common sense and humanity in applying those standards. Clarify what matters most to each individual team member and manage them in light of their preferences. The goal isn't to treat all team members the same—the goal is to produce amazing results by getting your team to perform at its best. There is a reason why NBA super coach Phil Jackson coached Michael Jordan differently from Scottie Pippen and Steve Kerr—they all responded best to different coaching and different motivations. And while we're at it, why not have a one-page "cheat sheet" that defines how to coach each individual player on your team? All it takes is a willingness to ask your team what works best for them, what they like least, and to observe them over time.
* **Affirm your team by sharing information, holding high standards, and believing in their ability to deliver results.** Top talent wants to be respected and challenged. Sharing information sends the message you trust them to be competent and integral enough to understand and use the information. Holding high standards and believing in your team's ability to perform is contagious, and helps you create a results-oriented culture inside your business.
* **Remove poor performers quickly.** Give a poorly performing employee proper guidance, coaching, and training, but if after a time it becomes clear that they just can't perform at the level of those around them, remove them swiftly and decisively. It is very demotivating to high performers when they see complacent, low-performing coworkers get a free pass from management.
* **Reward achievements when they happen—immediacy, enthusiasm, and authenticity matter.** The old-school model of evaluating and recognizing performance at annual review time is outdated. When a team member does something exceptional, find a way to applaud that performance right away. It doesn't have to be a raise or promotion, or even cash. Instead, stop everyone in the halls and have them give a team member a standing ovation for work well done, announce the exceptional work in a quick company-wide e-mail, or walk over to a team member's office and tell him or her in person how impressed you are. The key is to show how much you appreciate what the team member did _right now_ , when it just happened.
The Inverted Pyramid of Leadership
Once you've hired great talent, your job as a leader in your business is to serve your talent so that they get great results for your business. Sometimes this means connecting resources; at other times this means removing obstacles. There are times when it will even mean taking on mundane details that support your team in accomplishing its big goals.
Early in the life of his first company, CTI, a travel software business, Jeff was checking in with a development team who had moved into a conference room to finish the final sprint to complete a new product release on time. They were working diligently and Jeff, as CEO, popped in the room and asked, "Is there anything I can do for you to help you meet the deadline?" At this point CTI was a successful business doing $5.8 million per year in sales and had just been named to the _Inc_. magazine list of the 500 fastest-growing small businesses; Jeff was the CEO and owner of the business. One of the junior team members said, "Yeah, Jeff, can you pick up my dry cleaning for me?" There was a palpable pause in the room as the team watched how Jeff would react. Jeff didn't miss a beat. He simply asked for the dry-cleaning ticket and went and got the clothing from the cleaners.
Here's the point: The best thing Jeff could do for his business at that moment was to keep his talent in that room, working on the product, so that they could meet their looming deadline and wow their waiting customers. If you have to go run a few errands so your company wins, then that's what you need to do—ego and personal glory be damned! CTI tripled sales to over $16 million the very next year, and Jeff later sold the company to American Express for an eight-figure payday. That's why one of the most important lessons Jeff shares at large CEO conferences is: "Build a culture and environment where really talented people want to work, and then go get their dry cleaning!"
Compensation—A Game You Can Only Tie or Lose (Rarely Win)
One of the most stressful areas for most business owners in the Team pillar of their company has to do with compensation: how much they will pay their team. The two prevailing beliefs on compensation are either pay as little as you can to secure the talent that you want or pay handsomely for the best, never letting money stop you from bringing the most talented team on board. We think both approaches are flawed.
First, as to the belief that you should pay the least you can while still being able to sign on the team members you want—we feel that this view doesn't get the way people really work. Over time, people will become resentful if they don't feel like they are being compensated fairly. This may mean they'll leave when a suitor company woos them or that they'll do their work begrudgingly day to day if they stay. Your team is not an expense; they are one of the most essential strategic investments your business will ever make, so use compensation to establish the right relationship with them from the start.
Years ago, Jeff wanted to hire a new programmer for one of his companies. He brought the man into his office. On his desk was a blank sheet of paper. Jeff said, "I understand that you're making $70,000 a year where you're working now and we would like to give you a reasonable increase to take this job and switch companies. There's a sheet of paper on the corner of my desk. I want you to take that paper and write down the salary you would like to join our team." The programmer wrote down $80,000. Jeff then asked him to flip the paper over and read aloud what was written on the other side—$85,000. Jeff explained that prior to the meeting, he had written down what he thought this programmer was really worth to the company. At first, the programmer didn't understand. He said, "But I only asked for eighty thousand." Once Jeff asked him if he really wanted to continue that conversation, the programmer smiled and said no, he didn't.
Business owners who view employees as expenses will say that Jeff wasted $5,000. After all, the programmer would have been happy to take $80,000. But we look at employees as strategic expenses, which made that $5,000 an investment that paid off handsomely. That programmer helped Jeff recruit four of his programmer buddies at zero cost to the company. He also often stayed late to do great work and prove that Jeff was right to pay him the extra $5,000. All things considered, Jeff's company got a huge return on this $5,000 investment.
The point is not that you should pay your team extra, but that you should do your best to nail down the rough range of what each team member is _really_ worth to your company and, over time, make sure that you are paying them roughly that amount. By fighting over the last dollar, sometimes you win the battle but lose the war.
Now let's turn our attention to the second most prevailing (and in our opinion, flawed) business owner belief on compensation: that you need to overpay to get a team of stars. To the person who follows this line of reasoning, we would just say that if you overpay, you can often turn a great talent into an entitled, spoiled prima donna who doesn't live up to his or her own hype. Since many small businesses need to be smart with their capital and stretch their dollars, they really shouldn't let money be the leading lure to work for their company. Employees motivated more by money than any other factor are typically not the best employees to have when your company experiences the common ups and downs that all growing businesses go through.
You rarely win with money. The best you can hope for is not to blow it completely. You've got to be in the fair neighborhood. If you come in much too low or too high you'll lose (or pollute) that player. So be fair, and with top talent be generous, but also be strategic. Money is just one of a string of rewards you can "pay" your people with; in many cases it is midway down their list of wants or lower. As a smaller player in the market you can run circles around your competition with the nonfinancial rewards. These include things like:
* Giving your team cool projects and interesting work.
* Letting them _earn_ autonomy and real control of many of the details of their work and environment. (Does it really matter how they dress? Can they bring their dog into work with them? Can they work from home several days a week? Can they adjust their hours to make their daughter's soccer game next Tuesday?)
* Sharing information and respecting their insights.
* Helping them find meaning in the work they do.
* Treating them fairly and letting them see you treat others fairly (positively and negatively).
A Simple Technique to Help Your Team Make the Connection
People thrive on meaning. Companies thrive by creating value. Linking how each team member's role adds value to the company and serves the customer helps your team focus on the right things even when no one is around to "manage" them.
We want to share with you a little-known technique to help team members self-manage and eradicate wasted efforts. We call this technique "making the connection."
Several years ago, Jeff was being shown around the headquarters of 1-800-Flowers by its CEO and founder, Jim McCann. Jim opened up his first flower shop in 1976, and had grown it into one of the world's largest flower and gift companies with annual sales of $700 million. As they walked around the production facility, Jim stopped an employee and asked him what he was doing, and how what he was doing helped the company sell more flowers. After the employee responded, Jim thanked him and told him to get back to work. Jim turned to Jeff and told him that if an employee can't tell him immediately how his actions at any moment helped achieve the goal of selling more flowers, he tells that employee to stop doing what he's doing, and find something to do that actually sells more flowers.
They continued walking, making their way to the garage. Jim stopped a mechanic there who was installing fuel filters on their delivery vans. When Jim asked him how installing fuel filters helped sell more flowers, the mechanic responded that he had done research and found a fuel filter that improved the efficiency of every tank of gas by 11 percent. If he could help the company save 11 percent of the cost of each delivery van's fuel, the company could take some of the savings, and give the rest to the customer in the form of lower prices, which would yield increased flower sales.
As you can imagine, Jeff was floored by how clearly each 1-800-Flowers team member could connect what he or she was doing to the one goal of selling more flowers. He took the idea back to his company and started stopping everyone in the halls to ask what they were doing and how it directly helped the company achieve its main goal.
The core of this technique is crafting your "connection question." What is the single question that you want your team to use internally as a filter to focus them on work that matters and to eliminate work that doesn't?
For 1-800-Flowers, their connection question is, "How does what I'm doing help us sell more flowers?"
For David's business coaching company, Maui Mastermind, their connection question is, "How does what I'm doing help us create and keep a business coaching client?"
For Jeff's innovation company, Colorjar, their connection question is, "How does what I'm doing help us startle and delight our client?"
What is your company's connection question? Of course, coming up with the question is just the first step. You've got to explain the question's importance to your team and give them permission to use it as a benchmark against which to measure the work you're asking them to do at any given moment. This means you've got to also give them the power to call the company on tasks that just don't make sense, and to use their initiative to better direct themselves to create more value for the company. Building a culture where employees feel free to ask piercing questions and challenge the status quo is critical to your ability to have continuous improvement.
Tapping into Your Team for Ideas to Scale
There is one last practice we want to share with you. It's one that we've used for a while now inside our companies, and we've seen it make a real difference for the companies we coach. We call this practice the "Three Best Ideas Tool," and it helps you enlist your team in developing ideas to scale and improve your company.
Every six months, give your entire team the tool shown in Figure 10.1 (you can also get a full, fancy-formatted two-page PDF version of this tool by going to **www.ScaleYourBusiness Toolkit.com/pdf** ). Ask each individual to go through the company area by area and write their three best ideas in each section of the tool. Once they've completed this, take their answers and spend several hours going over them to compile the strongest answers. Hold a team meeting to go over the results and involve your team in picking several of the best ideas to immediately implement in the company. Make sure you regularly report back to your team over time how the company is doing on actually _using_ their ideas to improve the business. Repeat this process over time and watch not only how your company benefits and improves from these ideas, but how your team responds and grows when they see the company takes their input and ideas seriously.
FIGURE 10.1: THE THREE BEST IDEAS TOOL
In the next chapter we'll focus on the final pillar of your company—the Executive Leadership pillar.
## YOUR EXECUTIVE LEADERSHIP PILLAR
ALIGNMENT, ACCOUNTABILITY, AND LEADING YOUR LEADERSHIP TEAM
The final pillar of your company is the Executive Leadership pillar. This is the pillar that makes the big-picture, strategic decisions and enrolls your entire team in that vision. It is the pillar that sets the tone for your company culture, including establishing a pervading sense of accountability within your business. And finally, this is the pillar that grooms and leads your leadership team, developing your leaders so that the company is radically less reliant on you to make decisions and manage execution.
In this chapter, we'll share our best ideas to help you align your entire team, upgrade the level of accountability inside your company, and lead your leadership team.
Leaders Clarify the Big Picture
One of your core functions as the leader of your company is to set its big-picture vision. In chapter 5, you began this process by laying out the big-picture view of why you're in business, what your company's Singular Goal is, and how you and your team will pursue that goal. You also uncovered your company's top leverage points (in doing the S-O-O-T and Sweet Spot analyses). You put your seven key strategic decisions on trial, and crafted your big-picture business strategy. This is a great start, but in order to make this vision real inside your business, you need to both communicate it throughout your company and engage your team—quarter by quarter—in taking the small action steps that will, over time, result in you all reaching your goals. Remember, the most effective form of leadership is leading by example. Everyone is watching what you, the business owner, are doing, not just what you are saying. Your responsibility is to effectively communicate your vision, and then live it.
Here are seven concrete suggestions for you to communicate your big-picture vision with your team so that you enlist them in achieving your top business goals:
* **Put your big-picture vision for the company in writing and share it with your team.** We suggest that you edit it down to one page. It will have more impact.
* **Repetition works (so say it again and again).** You need to regularly reinforce your vision with your team. Take every opportunity to bring it up in meetings, in conversation, and in the normal course of your business. When you see an opportunity to highlight how a recent event or action at the company is in direct alignment with that vision, don't miss a chance to point it out.
* **Ask your team to tell you what** they **think the big-picture vision for the company is.** You'll be surprised by what they come up with. This is a great way to check to see if you've been successful in communicating your vision for the company. When David did this a few years ago at a company retreat, he got many different answers, some of which nailed it, others of which were totally different from the company vision. This became a great coaching moment to clarify the big picture with his team.
* **Use the one-page quarterly planning tool company-wide.** Bring your key leaders together each quarter to create your company's 90-day plan. Share the one-page plan with everyone on your team. Make sure that you give them regular updates on how you as a company are doing on progressing on the steps it outlines. Sharing information is a sign that your company is serious about enlisting your team's best efforts in making the company's vision a reality. And the reverse of that—not sharing company information or progress—is demotivating to your team at any level.
* **Make sure that all team members understand their roles, responsibilities, what success for them in fulfilling those responsibilities looks like, and how they will be measured and given feedback as they go.** As obvious as this seems, in our experience it is a rare company whose entire team understands each of these things, let alone the business that has a formal system to make sure that as they grow, each new team member does too. Publicly reward behavior that directly supports the company's vision.
* **Actions speak louder than words, so make sure your company lives its vision.** Do you regularly use your company values as a filter through which to make your big decisions? Do you put your best resources on projects that are geared to move you closer to your Singular Goal, or do you fritter them away on putting out your brightest momentary fires? Do you regularly assess how your company is progressing on its most important goals, using the feedback to improve the business, or do you just react to things as they hit the business? Your team is watching everything. Make sure what they see reinforces how seriously you take your own big-picture vision for the company.
* **It is your responsibility to make sure you look to the future and ensure that your company invests some of its resources in dreaming up what comes next.** The world is changing so fast; you must keep looking toward tomorrow and thinking about what comes next. You don't have to be the source of the ideas, but as the leader in charge of the big picture, it is ultimately your responsibility to make sure your company asks the right questions so that it stays vibrant and relevant to the world. Complacency in a changing world is fatal.
Leaders Shape Meaning and Establish Culture
Leaders create the narrative through which their team interprets the business and their relationship to it. As you grow your business, it becomes increasingly more important that you shape the stories and traditions that will become part of your company's culture. Stories and traditions empower employees to make the right decision for the company when no one is around to help and the customer wants an answer immediately. They hold your business on course even when you're no longer present each day to drive it, which is what you want in a Level Three business.
Here are our best tips to shape meaning and establish the company culture you want so that your business thrives long-term:
* **Look for small, symbolic stories that transmit the values of your company and share these stories at every opportunity.** If you see Mark, a client support rep, creatively solve an issue in a way that delights your client, share this story internally with your team again and again. The best of these stories will be absorbed into the mythology of your company and shape the way it sees itself for years to come.
* **Intentionally seize moments to take an action or make a decision that illustrates the best of how you want your team to behave.** Maybe it's always calling your customer by name as they walk back into your store, letting your team see you write out a thank-you card after a sales call, or recapping who owns which action steps after a meeting and by when. Your company's culture is built by the accumulation of thousands of small decisions and examples that happen over and over and over. In your business, there are no throwaway moments; your team watches and absorbs _everything_ you do.
* **What you do at stressful moments in the business has a magnified impact on your team and on your company culture.** Think of stress and the heightened emotions that go with it as a magnifying glass that powerfully enhances the impact—good or bad—of your behavior. Use these moments as the golden opportunities they are. Leaders face the hard realities, but do so in a spirit of positive faith that they can find a way through them to the results they want. Whatever you do in these stressful moments will cumulatively impact the tone of how your team will respond to these same circumstances in the future.
* **Help your team see the impact of what they do and what the company does on the lives of one another and your customers.** When a client shares a story of how your product or service impacted their life or business, share that story company-wide. Enroll your team in gathering and sharing these same stories. Make this part of your regular meetings. Archive these stories so that the best of them get told and retold inside your company. Also, make sure that you encourage your team to gather stories of how other team members really helped them out and added to their lives through something they did. Encourage your team to share these stories—publicly and privately—too.
* **Clarify your company's values and make them the filter through which you make all your business's tough decisions.** Encourage your team to use them the same way, and celebrate when they do. From time to time, ask your employees how they used the company's values to make big decisions in the heat of the moment. It's a great barometer of just how deeply your values have been absorbed into the company.
Leaders Demand Accountability
Great businesses are built on teams that take full ownership of individual responsibilities. Too many businesses let people slide by partially doing things, and accepting excuses for delays and mistakes that could have been avoided. A culture of accountability is one of the most valuable institutional habits that your business can form, and will empower your company to grow and develop.
Accountability starts with you and your leadership team. If you don't move heaven and earth to honor your word and meet all your commitments, your team will learn that they don't have to either. Here are seven time-tested tips on your role in creating accountability in your company:
* **Clarify your commitments in writing at the end of every meeting.** Not only does this make sure that you've captured all your action items, but it is also a powerful way to role- model how you want your team to behave. Wherever possible, try to enumerate all your commitments to make it even clearer what you have agreed you'll do: "So, summing up, I've got three action items here. Item one . . . item two . . . and item three . . ." __ (while visibly writing each of them down in your notes). Teach your team to employ this same tactic with customers as well. Customers love doing business with companies that follow through.
* **Circle back with the team members who are involved with your action steps and give them feedback on your progress and completion at regular intervals.** This is a great way to model the behavior you want from them, and highlight how important it is inside your company to meet deliverables. "Just wanted to give you a quick update. As I committed to, I did [item one] and [item two] today, and will get [item three] done by close of business Friday." Even when the task isn't complete, don't leave people guessing. Circle back and say, "I haven't solved the problem yet, but I haven't forgotten you and I'm actively working on a solution."
* **Clearly state what you can't commit to so that you don't lower the accountability bar in your company by missing a "phantom deliverable."** Remember back in chapter 8 when we talked about "phantom deliverables"? These are things that the other person _thinks_ you committed to but you didn't. As a leader, you need to exhibit great communication by making any phantom deliverables you see come out of a meeting explicit. That way if you can commit to that deliverable, you do so, and if you can't, you clarify that you are not committing to it.
* **Be on time, all the time.** Being on time—all the time—is a simple behavior that your team will generalize to mean that you take your commitments seriously and live with integrity. It is one behavior with a huge return on investment in terms of modeling accountability inside your company. Too many companies implement respect in a hierarchical manner. Your time is _not_ more important than an employee's time or a customer's time in their eyes. Being on time shows respect, and it makes a big difference to the receiver.
* **Credibility is a marathon, not a sprint.** It doesn't help if you take off out of the gate gung ho in your desire to model accountability, only to let it slip a few weeks later. If you want accountability to be a real and lasting part of your company culture, you've got to maintain your behavior over time.
* **How you own your failures is as important as how you model your successes.** You're human and you _will_ mess up. To think otherwise is just not realistic. How you own your missed deliverables is incredibly important to the culture you are building. Do you make excuses? Sweep them under the rug? Melodramatically beat yourself up? Don't! Instead, show your team how mistakes are a part of being in business, and often can lead to profitable insights. When you make a mistake, publicly take responsibility, share what you learned and how you'll apply it, and implement a better solution going forward.
* **Most breakdowns in accountability come from incomplete or poor handoffs.** At the moment any deliverable is created, it needs to be assigned to someone who will be responsible for seeing that it happens. We call this assignment of a deliverable a "handoff." As a leader, you need to show that every handoff clearly details who is responsible for what, by when, what success in meeting that deliverable looks like, and how he or she will be held accountable for that deliverable. "Sarah, you own this deliverable and it includes doing _x_ and _y_ by Friday close of business. Can you also please make sure to send a quick recap to Tom and me on Monday that shares how client _z_ responded?"
If these are the behaviors that help you and your leadership team model accountability inside your business, here are two additional tools to help your business systematically weave accountability into the fabric of your company culture.
Accountability Tool #1: Key Performance Indicators
Key performance indicators (KPIs) are the critical metrics that measure how your business is performing in a specific area at any given moment. The right KPIs direct your team's attention to what matters most, and give you and your team the timely feedback you need to make adjustments on the fly.
The best KPIs are "leading indicators." This means that they are numbers that help you predict whether or not you will hit your target and achieve your goals in a specific area. KPIs are critical to track and regularly review because they give you feedback early enough in the game for you to use the information to do something about what they reveal to influence the outcome. For example, if your goal is to increase sales this quarter, potential leading indicators include the number of dials your sales team makes each day, the average dollar per sale, the average number of hours per week your sales team spends on direct sales activities, or your average contact rate per dial. Given that you could measure any of these leading indicators, or in fact dozens more, your choice of the right KPI for a specific area requires you to narrow your selection to the one or two numbers that, more than any other, measure the most important leverage point in helping your company hit its goal in this area.
Don't think of KPIs as absolute numbers, though. Think of them over time as levers you can pull to fine-tune your business. _If I double this number and lower that one, does my business perform better or worse?_ It's important to use KPIs as a way not just to measure your current performance, but also to shape and enhance your future performance.
The best KPIs:
* **Measure the most meaningful leading indicator in that area of your business, or in a specific team member's area of responsibility.** KPIs direct attention to what matters most. The very choice of your KPIs helps your company narrow its focus to those fewer, better things where its efforts will have the biggest payoff.
* **Are either automatically tabulated or self-scored so that they give you accurate feedback on a timely basis.** Think about the game of golf—a golfer knows how she is doing at any moment in the round because she is keeping score in real time on a small scorecard. Smart businesses help each team member pick one or two KPIs to individually focus on that give them regular feedback, at least weekly, and ideally daily.
* **Make the score easy to understand**. The best KPIs are concrete and simple such as "percent complete," "visitors to our store per day," or "conversion rate of our main Web landing page." In order for the feedback a KPI generates to help direct behavior, the KPI must be obvious and intuitive to understand. Make sure that your team understands what the KPI measures and how increasing or decreasing the number impacts the business.
* **Are leading indicators, as opposed to trailing indicators.** KPIs should measure key leverage points that _predict_ your current status and trend in hitting an important goal. In most cases, it is easiest to think of the key goal you are trying to accomplish as the _trailing_ indicator. Trailing indicators (also known as results indicators) clarify outcomes that have already happened and as such tell you the score after the game is over. By focusing your KPIs on leading indicators, you give your team feedback in time for them to use that information to influence the eventual outcome in that area.
So how do you best use KPIs to scale your company? In the real world, the two biggest mistakes we watch businesses make with respect to KPIs are either to not have any or to have too many. The businesses that don't use this tool at all lose out on the power that a few carefully chosen metrics can have to direct attention and give timely feedback. Using a KPI doesn't have to be complicated or difficult. It can be as simple as asking the question, "What one variable could we measure that, if we focused on improving that one variable, would have the biggest positive impact on helping us hit our most important goals in this area of our business?"
For example, Sasha, whom we met earlier, owned the dominant dry cleaning business in her area. One of two KPIs she chose for her business was the number of enrollments per week her counter team secured for their "concierge" level of cleaning (a free service that included pickup and delivery and auto billing of all charges to a credit card on account). She knew that her concierge-level clients on average did a much higher volume of business and were more loyal to her company, which meant that increasing the number of enrollments her team signed from walk-in customers sustainably increased sales.
Having too many KPIs is also counterproductive. While it might make sense to measure 15 different variables in your sales and marketing, too much data dilutes the impact that one or two carefully chosen KPIs could have on your sales efforts. By focusing on one or two KPIs in an area, you'll concentrate your efforts on making a few carefully chosen, high-leverage changes to improve performance.
Remember Thomas, who owned the mobile bottling company from chapter 4? One of his company's most important goals was to improve its gross profit margin (the trailing or results indicator). He measured many variables, including his cost of raw materials, his shipping costs, his storage costs, and more. But when he looked closely at all the factors that impacted his gross profit margin, he realized that his top leverage point was controlling overtime worked by his staff, since that overtime added an additional 50 percent per hour to his labor cost. Knowing this, Thomas chose the total number of overtime hours worked per week as one of his KPIs. Tracking this KPI helped his team improve their systems, tighten up on scheduling, and better manage their employees. The bottom-line result was that they reduced overtime hours by _30 percent_ within the first six months of implementing this KPI. This wouldn't have been nearly as easy to do if he tried reporting reams of data back to his team. It was the clarity of having that _one_ simple KPI for this area that focused attention and helped them get such dramatic results so quickly.
Here are some tips to implement KPIs in your business tomorrow morning:
* **Pick one KPI for each of your three Focus Areas in your quarterly one-page plan of action.** What is your most important result for this Focus Area this quarter? Choose the KPI that is the best leading indicator to help you accomplish this result. For example, if the most important result is to lower your production costs, then a possible KPI could be the percentage of your jobs that require overtime, or the average number of staff hours required to complete a service call. Your goal is to pick a KPI that will help give you feedback on your progress toward your most important result for that Focus Area that quarter.
* **Help each of your team members choose one or two KPIs to focus on each quarter.** For example, you might have your two salespeople focus on the KPI of the number of qualified sales appointments they set each week, or you could have your customer service reps focus on the average number of hours it takes to close out a service ticket. Wherever possible, coach your team members to pick the best KPIs for them to focus on rather than just dictating to them from on high. You'll not only get much deeper buy-in this way, you'll also help your team grow as businesspeople. Your team's KPIs both direct their focus and give them a powerful tool to self-manage their behavior.
* **Create simple** visual **ways that your company and individual team members can see the top KPIs.** Whether you use a graph, a dial, or a thermometer, people understand numbers best when they are put into a simple visual frame.
Accountability Tool #2: The "Big Rock" Huddle
It's a Monday morning, 9 A.M. Kevin Bassett and the nine members of his tax team gather for a quick team huddle. He briefly shares a recent company victory, then they go around the group with each person succinctly sharing what their top two or three action commitments were for the prior week, and whether they completed them or not. The team then quickly outlines their top two or three action commitments for the coming week, end the huddle, and get back to work.
All totaled, this huddle took less than 15 minutes and ensured that each team member started the week with a clear focus on what mattered most. At the same time Kevin led this meeting for his tax team, his partner, Dane, held the same meeting with the "financial and accounting" group within their CPA firm, Bassett and Byers. We call this structured start to your week the "Big Rock Huddle."
"Big Rocks" are those important and valuable chunks of work that actually help the business make real progress on its most important goals. Every week, we strongly encourage you to ask each team member to identify the two to three Big Rocks they can do that week that will have the greatest positive impact on progressing the business toward accomplishing its most important objectives. Sure, your team is still going to have all the to-dos on their plates to operate the business day to day, but the discipline of identifying and committing to your weekly Big Rocks makes sure that, in the rush of the week, some of your best resources go to doing those key things that actually progress the business.
The best Big Rocks are small enough tasks that they can be completed in 30 minutes to three hours, yet important enough that when accomplished they help your company take a real step forward toward reaching your big business goals. Remember the example of 1-800-Flowers? Make sure the Big Rocks you choose really do help you sell more flowers.
Here are a few other tips to help you use the "Big Rock Huddle" in your company:
* **Keep your Big Rock Huddle short—ideally under 15 minutes.** This may mean you have to break into smaller groups as Kevin and Dane do in their midsize CPA firm.
* **Consider making it a** standing **huddle.** This will force you to keep it short.
* **Teach your team what Big Rocks are, and how to pick the best ones each week.** Don't assume your team will immediately be able to choose the right Big Rocks. For the first 30 to 60 days you implement this idea, don't worry about the Big Rocks they choose. It is more about building the muscle than choosing the perfect Big Rocks. As you progress, make sure you and your management team coach team members to choose better and better Big Rocks. This is a great way to make sure your team understands the real goals of the company and what each team member can do to support those goals.
* **Use your KPIs to help you choose your Big Rocks.** Any Big Rock that has a real impact on a KPI is likely a great choice.
* **Have each team member send a quick summary e-mail at the end of the week**. This Friday e-mail should go out to the entire team or department (depending on how big your company is). In it, your team members should each list what their Big Rocks were and give a bulleted recap of how they did—rock by rock. Your team should also share what their two or three Big Rocks are for the coming week. If you need to up the accountability ante, consider holding a second huddle on Thursday morning to get a quick status check-in on everyone's Big Rocks.
* **Make sure you still have each team member share their Big Rocks and quick update at the huddle.** There is power in people standing before their peers and verbally sharing what they are doing that is most important, and articulating how they are doing on these key commitments.
Leaders Grow Other Leaders
The final area we want to address in your Executive Leadership pillar is the need to grow and groom your leadership team. For most businesses, directing the leadership team is one of the last responsibilities you the owner will let go of when transitioning to a Level Three business. Prior to that, you're going to need strong leaders heading up each pillar of your business.
Remember, you'll flesh out your leadership team one by one as you progress from Middle Stage Level Two, through Advanced Stage Level Two, and into Level Three. Here are eight important insights to help you grow, integrate, and leverage your leadership team:
* **Generally, your first key management-level hire will be either in the Operations or Sales/Marketing pillar of your company.** If you are more of an operations person yourself, then likely your first management hire will be a sales and/or marketing leader. If you are more sales focused, your first key hire will likely be in operations. You want this first key hire to balance your best abilities and stabilize your company.
* **The right management hire is an investment, not an expense.** For example, David remembers one of his companies that had plateaued at the $1-million-a-year mark. After he brought on a new COO ($100,000 per year) it grew by 600 percent over the next 24 months. That $100,000 annual investment helped the business increase its sales by $5 million per year. You can't play every position on the field. You need talented leaders in your business who are as good as or better than you are in a specific pillar of your company. This frees you up to continue to grow and develop the business.
* **The management team you build today may not be the team you need tomorrow.** As your company grows larger, it is common for it to outgrow the capabilities of its leadership team. Some of your team will continue to grow over time and you'll enjoy their contribution for their entire career. Others on your leadership team stop growing and at some point need to be moved to a different role or replaced. This can be scary and uncomfortable to do, yet ultimately you need to let the needs of the business and the requirements of the position direct these decisions, not your own discomfort with making a tough call.
* **Beware hiring to make yourself feel better about yourself.** Secure business leaders hire better, smarter, more capable people than themselves. They don't fear looking poor in comparison. They realize that the better their team, the more successful their business will be.
* **When you hire smart, talented, experienced people, give them the authority to do what you hired them to do.** We always say hire hard, manage easy. Invest your time up front to get the right people, then let them do their job. Of course, you need to have controls in place that give you regular feedback on your team's results, but when they are performing, let them. Too many business owners undercut their own talent by micromanaging them.
* **Just because you've hired smart, talented, experienced people, don't just abdicate responsibility to them or allow yourself to lose touch with what is happening in the business.** We know this may seem like the opposite of the tip above, but it isn't. Both micromanaging and abdication are recipes for a brutal wake-up call at some point down the line. Your business needs you to make sure that you are using the internal controls you have in place like KPIs, regular progress updates, and spot checks to make sure that problems are discovered early and dealt with while they are small. Since you're the one steering the ship at the highest level, it helps for you to have a sense of what's going on everywhere else on the ship to ensure that you are all moving in the same direction.
* **Make sure each leader in your business has his or her own one-page action plan each quarter.** You can use this plan to help you manage your executive team and hold them accountable. Use the same format you already learned about in chapter 4. Get regular updates (we suggest weekly) to make sure you both stay on track and spot opportunities to coach and mentor your executive team. In order to really set your executive team up to win, you need to invest the time to support them with the direction and feedback to help them win.
* **Invest in your executive team.** Get them together each quarter off-site (even if it is just at a nice location in your city) to take a full day to do your strategic planning. Bring them to industry events or business training programs. Not only will they benefit from the content of the training, but you'll also be able to use the time together outside the business to enhance your relationship. Consider hiring outside coaching for your executive team, especially if mentoring and coaching your leaders is not something that is a personal strength of yours.
Ultimately, your Executive Leadership pillar is the key strategic director that integrates and empowers all the other pillars of your business to work smarter and accomplish more.
## PART IV
## You DO Have the Time
Start by doing what is necessary; then doing what is possible; and suddenly you are doing the impossible.
—St. Francis of Assisi
## PRINCIPLE SEVEN
YOU DO HAVE THE TIME TO SCALE YOUR COMPANY
Wearing so many hats in your business and being responsible for so many demands to keep it going, it's hard to imagine scaling your company without working longer hours. As one business owner we talked with put it, "I just don't have any more time. I feel like the only way I can grow my business is by taking away time with my family or time for myself."
This is a false dilemma. You don't have to choose between long hours and growth _or_ a lesser business with time. Throughout this book, we've encouraged you to focus on those fewer, better things that make a magnified contribution to your company's growth, knowing that by focusing on less you can accomplish more. Time is one of the most powerful variables you can control in the business success equation. In this chapter, we'll share a revolutionary new time mastery system we developed for business owners like you. This system will help you free up eight hours every week to reinvest in your business in upgraded ways.
Remember the example of Mark from chapter 1? Mark owns a commercial maintenance company that contracts with home-owner associations. When we first started working with him, he was working 70-hour weeks, including many nights and weekends. After over a decade of these long hours—week after week—Mark was getting burned out. Over a 24-month period, Mark adopted these ideas in his company and grew his business from $750,000 a year in sales to over $1.2 million a year. The best part was that Mark reduced his working hours to 35 a week! Mark's story isn't an anomaly. Remember Tom from that same chapter? Tom reduced his working hours from 80 hours per week to under 20 at the same time he scaled his company from $5 million per year in sales to $23 million.
These stories are powerful reminders that it's not about more, it's about _better._ Scaling your company won't require you to work any more hours—you're already working plenty of hours. Instead, allow us to help you be _smarter_ with the hours you are _already_ working The more you do for your business, the more you have to keep doing. The more you get the business to do for the business, the more you can step back and grow your company the _right_ way.
Many years ago, famed college basketball coach John Wooden said, "It's what you learn after you think you know it all that really matters." You'll get the most out of these time mastery strategies if you keep Coach Wooden's advice in mind and approach this section of the book fresh. You don't have to work nights and weekends to build your business. Working longer hours is rarely the answer. Making better choices with your time and fundamentally changing how you structure your use of time are the best ways to sustainably create the time you need to grow your company. Here are six powerful time mastery strategies to help you do just that.
Time Mastery Strategy #1
To upgrade your use of time, first identify what you do that truly creates value for your business.
As a business owner, you don't get paid for time and effort; you get paid for creating value. So as you build your business, look for ways to create value independent of your personal time. This essentially is what it means to build a business, not a job.
If you've read anything on time management, you've come across Pareto's Principle, inspired by the work of nineteenth-century economist Vilfredo Pareto. Commonly called the 80-20 Rule, Pareto's Principle states that 20 percent of your actions generate 80 percent of your results (high value) and 80 percent of your actions generate the other 20 percent of your results (low value). This useful distinction becomes the basis of a refined model for using your time to create massive value, independent of the hours you put in.
If you take the 20 percent of your actions that generate 80 percent of your results and apply the same distinction a second time, then 20 percent of that 20 percent produces 80 percent of 80 percent of your results. That means 4 percent of your effort (the 20 percent of 20 percent) generates 64 percent of your results (80 percent of 80 percent).
And if you can bear with us for one more math moment, apply this distinction one final time.
Only 1 percent of your effort (20 percent of 20 percent of 20 percent) generates 50 percent of your results! That's right—a tiny fraction of your highest-leverage work produces _half_ of all your results.
No, this is not an exact science. Nor does this just work automatically. But Pareto's Principle illustrates a valuable point: All time is not spent equally. Six hours of work on Monday may not produce the same value as six hours of work you did on Tuesday, depending on what you actually did in a given day. A surgeon who answers e-mail on Monday, does a heart surgery on Tuesday, and is a guest on Dr. Oz's television show on Wednesday may have worked the same hours each day, but she did not add the same value to her business each day.
We want to help you focus your time on the most valuable activities. That's why we used this distinction to create the Time Value Matrix—a visual hierarchy of the four types of time: A Time, B Time, C Time, and D Time. (See Figure 12.1.)
D **time** is the 80 percent of unleveraged, wasteful time that produces only 20 percent of your total return. We call this the "80 Percent Mass." We've given it a relative value of 1.
C **time** is the leveraged 20 percent of your time that produces 80 percent of your results. We call this "Leveraged Time." It has a relative value of 16 (1/4 less input generating 4 × more output).
B **time** is the highly focused 4 percent that generates 64 percent of your results. We call this time the "4 Percent Sweet Spot." (It has a relative value of 64. That means one hour of B time produces 64 times the value of the same time spent on a D activity.)
A **time** is the top of the pyramid—the "Magic 1 Percent." Fully 50 percent of your results come from these activities. ( _A_ time has a relative value of 200 times that of _D_ time.)
Most business owners have no clue which of their activities fall into these four categories. How in the world can you create more value in the same or less time if you don't know what activities constitute _A_ and _B_ time for you?
FIGURE 12.1: THE TIME VALUE MATRIX
Before we share with you some examples of our _ABCD_ -level activities, understand that one person's _D_ activity may be another person's _A_ \- or _B_ -level activity. It's all relative. The examples from our business lives are illustrations that are not to be taken as absolute benchmarks of value. For example, one of David's _D_ -level activities is scanning documents. However, people in his company have this activity as one of their _C_ \- or _B_ -level activities. Your _A/B/C/D_ -level activities are only comparable to your own value and expertise, not to other people's.
Take action and examine what you do that creates the highest value for _your_ business at this moment in time.
Identify your A/B/C/D activities and learn what you do that truly creates value for your business. List five of the tasks or activities you currently do at each level.
D **Time:** The 80 percent mass of unleveraged, wasteful time that produces only 20 percent of your total return.
Examples of Jeff's _D_ activities include: reading e-mails to route them internally, reviewing bills to spot discrepancies, scheduling meetings and calls, reviewing meeting notes to determine required follow-ups, and writing replies to low-level e-mails.
C **Time:** The leveraged 20 percent that produces 80 percent of your results.
Examples of David's _C_ activities include: delegating to his assistant, dictating a letter, holding a group meeting versus talking with several people one at a time, updating his master to-do list, and sending out an e-mail update to his executive team.
B **Time:** The highly focused 4 percent "sweet spot" that generates 64 percent of your results.
Examples of Jeff's _B_ activities include: meeting with key clients to solidify the relationship, coaching his management team to be better leaders, sharing company stories/successes/challenges in their biweekly staff meetings, reviewing his company's quarterly progress, instituting a systemic solution to a recurring problem.
A **Time:** The magic 1 percent that generates more than 50 percent of your total results.
Examples of David's _A_ activities include: making executive-level hiring decisions; decision meetings with key joint venture partners to secure high-value, win-win strategic partnerships; making strategic decisions that set the direction of the business; holding the executive team accountable for their deliverables.
Now that you've identified your current _ABCD_ -level activities, it's important to understand that what you currently list as an _A_ \- or _B_ -level activity will change. For example, if meeting one-on-one with a prospective client is currently an _A_ -level activity for you, make sure that in the next six to twelve months, you've increased the value you create for your business so this activity is pushed down to a _B_ \- or _C_ -level activity. Ideally, working with a joint venture partner who can generate dozens of leads for you every month will become an _A_ -level activity, or training your sales team to meet with prospective clients one-on-one, or creating a sales DVD that generates passive sales. By that point, meeting one-on-one with a prospective client will no longer be important for you to do personally. It will likely still be someone _else's_ _A_ \- or _B_ -level activity, just not yours. This is good; this is growth. And it's how you both grow your business and reduce its reliance on you.
When you get this distinction in your bones and shift your focus from "putting in hours" to _upgrading_ the type of work you do (more _A_ and _B_ time and less _D_ time), the results will be amazing.
Take the example of Dr. Gurpreet Padda, a surgeon in St. Louis who also owns more than a dozen other successful businesses. Following the advice in this chapter, Dr. Padda reviewed his _A/B/C/D_ -level activities and developed a clear map that delineated those activities he needed to get off his plate, and those he needed to find more time to focus on. By realigning how he spent his time, Dr. Padda added over $1 million of operating profit to his businesses within the first six months! Imagine the potential impact on your business when you do the same.
In order to move from limited Level Two thinking to a smarter Level Three mind-set, you need to stop believing that the solution to growth is working harder and longer, and instead upgrade how you use your time and get rid of the time-sucking _D_ -time activities. For example, if you were an attorney who charges $300 an hour, you shouldn't spend your time doing things like fixing a computer glitch, making copies, sorting mail, or little things you can't bill a client for. Instead, you might focus on more useful _C_ -time activities, such as any time that's billable, like working on a legal brief, reviewing a contract, or updating a client on legal considerations.
But be careful. While _C_ time can provide you with a great income, you'll always have to work exceptionally hard to earn it. This is the trap that catches most high-income professionals. They seek to increase their earnings by cranking out more hours. Mistake! Working more hours will only take you so far. It's just not scalable past a certain point, not to mention that when you get there, you'll be exhausted from so much work, and a stranger to your family too.
The real, scalable solution lies in _A_ and _B_ time. For an attorney, _B_ time might include building relationships with other professionals who can refer valuable business, or putting systems in place so staff can get better results without tapping too much into the attorney's time.
_A_ time could be speaking at a large conference where this attorney can generate new client relationships worth hundreds of thousands of dollars in billable services. _A_ time might be spent creating an accounts receivable system that increases the collection on all the firm's billings by 10 percent or introducing a new billing model whereby the firm charges a fixed price for a standardized service offering.
See the difference? You want to get _D_ -level activities off your plate; _C_ time is needed to do your work more efficiently. _A_ and _B_ time, however, are when you step out of the "job" and do something that improves your capacity to create results, or significantly pushes back your biggest limiting factor.
So here's the big question: _How can you have more_ A _and_ B _time?_ You won't get it by "trying harder" or by sitting down and saying, "OK, let's have an _A_ moment right now." It just doesn't work that way. That would be like a parent saying to his three-year-old, "Let's have an hour of quality time right now, Junior." __ How well do you think that would work?
To get more _A_ and _B_ time, you have to fundamentally alter the way you structure your day and your week, which is exactly what the next four time mastery strategies will help you do.
Time Mastery Strategy #2
To "find" the time, focus first on your D-level activities.
This one may seem counterintuitive, but the best place to look _first_ to upgrade your use of time isn't to your _A_ -, _B_ -, or _C_ -level activities. It's looking at your _D_ -level activities. Not only by definition do you spend the most time at this level, but it's the easiest place to make changes because the consequences of dropping _D_ -time activities are small.
List all the _D_ -level activities you do on a weekly basis. Keep a time log for a week or two to figure out the time spent on low-value _D_ activities. Once you've identified them, you can apply the "Four _D_ s" to get them off your plate.
THE FOUR DS
**1. Delete it**. Some _D_ activities shouldn't be done by anyone. Look at the action item and ask yourself what the consequences would be if no one did it. If it's small, then consider just crossing it off your list altogether. Remember our 1-800-Flowers story? If it doesn't help you sell more flowers, then it's an activity you can possibly delete.
**2. Delegate it.** Maybe it's a task that needs to get done but not necessarily by _you_. Hand it off to your assistant, a staff member, or a vendor. Anytime you can hand off a _D_ -level activity to someone, you free up both your time and your focus to do more valuable work for your business. This is the surgeon letting her administrative assistant do e-mail responses, freeing up her time to perform another surgery.
**3. Defer it.** Maybe this task needs to be done and done by you, but that doesn't mean it should happen right now. Sometimes delaying the action is the smartest choice.
**4. Design it out.** If you find yourself handling a recurring _D_ activity over and over, improve the process or system to keep the task from coming up in the first place. **** For example, if you get the same seven customer questions repeatedly, post an FAQ page with the answers on your Web site. Perhaps you can preempt these questions by giving new clients a "quick start" booklet or instructional video that proactively answers these questions. Designing out a recurring activity is the very essence of building a systems-reliant Level Three business. It simplifies processes and empowers your team to get consistently great results with less and less reliance on you, the business owner.
Look closely at your _D_ activities as the place to mine more time. By applying the Four _D_ s of Deleting, Delegating, Deferring, and Designing Out, you'll free up eight or more hours each week to reinvest in _A_ and _B_ activities.
Time Mastery Strategy #3
Structure your week to reinvest your "saved" D time in A and B activities.
It's not enough to free up eight to ten hours each week by clearing the clutter of your _D_ activities; you have to fill your freed-up time with _A_ and _B_ activities. Nature abhors a vacuum, and if you don't fundamentally change how you structure your week, you'll find yourself squandering the time you supposedly saved on more _D_ -level "junk."
This brings us to a powerful concept to get more _A_ and _B_ time and minimize the _D_ time that gets in the way: "Focus Days" and "Push Days." A "Focus Day" is a specific day of the week you set aside to primarily work on a few key _A_ \- or _B_ -level projects. "Push Days" are all the other days of the week that you use to just "push" your normal projects another step forward. Focus Days help you create long-term impact on your business; Push Days help you keep your day-to-day operations rolling forward. The key to succeeding with this technique is discipline. When it's your Focus Day, you need to _focus_. Don't get distracted by your Push tasks. We suggest that you set aside a full day each week for your Focus Day and make sure your team supports you in keeping this time clear so you can invest in these high-leverage activities.
On your Focus Day, get outside of your normal environment and work on the highest-leverage, highest-value, highest-return part of your business. This could mean building out a baseline operational process to use with new clients, spending the day creating a hiring system to consistently supply you with quality team members for your sales team, or calling on your two most important prospective customers or joint venture partners to deepen the relationship or close the sale.
Here's an example of how this works in the real world. Mondays and Wednesdays are David's Push Days. These are the days he plows through his project list and takes care of the day-to-day fires that come up in any business and need to be put out. On those days, he is very accessible to his staff and the outside world by phone and e-mail, and gets the bulk of his "job" in his business done.
David sets aside his Tuesdays and Thursdays as his Focus Days. That's when he turns off the phones and e-mail for the majority of the day (usually until 1 or 2 P.M.) and focuses on doing those few, highest-value activities that create the most value for his company. These _A_ \- and _B_ -level activities might include holding a key meeting on a new joint venture, working on a new book, or being in the studio recording a new business-owner video training program.
Three to four hours on his Focus Days can result in more value to his business than _an entire week_ living in _C_ \- or _D_ -level activities. Focus Days give David a way to create the space in which to get high-value work done. If a true emergency comes up his staff can reach him, but they've learned to support his Focus Days because they understand how protecting that time produces big results for the company. Plus, David encourages many team members in his company to set aside at least one half day each week for their focus time. This practice has contributed to the triple-digit growth of his company.
Here are a few practical tips to help you implement this strategy:
* Even if you can't set aside two full days a week as Focus Days like David, you _can_ find one day or at least _half_ a day every week and use it as your Focus Day. If you choose to start with a half day, we strongly suggest you use the first half of the day. You're much less likely to have urgent fires pull you away from your _A_ and _B_ activities.
* Generally we don't suggest you choose the first or last day of your workweek for your Focus Day. Too many urgencies need your attention on those days.
* Communicate your need for help to your staff so they understand both what you are doing and how it will help the business succeed. Enlist their help. When your team understands how important Focus Days are to the company's ability to achieve its goals, they will rally to support that effort.
* Get out of your office and away from distractions. Work from a quiet conference room, a local café, or the lobby of a nice hotel. Bring only the key project(s) you want to work on for your Focus Day. One component of willpower is controlling your environment to best support you. Leave the other work and distractions behind. That way, you couldn't work on them even if you wanted to.
* Consider prompting your key team members to take their own Focus Days too. You'll quickly see the positive increase in production this will bring.
* Finally, remember that Focus Days aren't about being off by yourself—they are about focusing on your top _A_ \- or _B_ -level activities that truly create value for your company. For many people this can mean meetings, phone calls, or critical e-mail. It's not time off or time away from the hard work; the key distinction is that on your Focus Day you are primarily doing your _A_ \- or _B_ -level activities.
Time Mastery Strategy #4
Work first "above the line" and live by the Results Rule.
Most business owners start each day with high hopes. They take a moment to write down the list of tasks for that day—a list that often grows to 15 or 20 items. Then the day hits and they find themselves pulled off track to deal with customer challenges, operational fires, and sales emergencies.
Here is a better way to organize each day: When you sit down in the morning (or the night before, if you prefer), choose three bottom lines for that day and write them at the top of your to-do list. Draw a line under them to visually mark them as different and special. Make two of them business related and one of them personal. Your bottom lines are the action steps you'll take that day that will create the most value for your business. Generally these are _A_ \- and _B_ -level activities. Here is an example of a time mastery to-do list.
SAMPLE TIME MASTERY TO-DO LIST
* Draft the marketing calendar for Q1.
* Call Tom Smith about expanding joint venture.
* Write my wife a love letter.
* E-mail Shirley.
* Check on Collin's project status with Angela.
* Review Web PPC proposal.
* Call Larry (Webinar glitches).
* Call Jenna (her event questions).
* Etc.
Live by the Results Rule
The Results Rule says that by 10:30 A.M., you have either completed each of your three bottom lines for the day or have scheduled a definite appointment on your calendar during which you'll complete them. This technique is powerful because it pushes you to do what matters most _first_ , and to treat your daily bottom lines with the respect they deserve.
Most business owners put off these bottom lines to deal with the urgent requests that come at them during the day. In doing so, they sacrifice far more than they'll ever know. Avoid this trap by following through on your bottom lines first thing in your day whenever possible. For added benefit, combine Time Mastery Strategy #4 with Time Mastery Strategy #5.
Time Mastery Strategy #5
Every "Push Day," schedule a "Prime Time" block to work on A or B activities.
Everyone has a certain time in the day when they're at their best. A Prime Time block is a 60- to 90-minute appointment that you set for yourself at your peak effectiveness time. That's when you work only on your highest-value items (usually your "bottom lines" for the day). By blocking out this time as an actual appointment on your calendar, you guarantee yourself at least one hour each Push Day to have a focused block of time to create real value for your business.
For example, David schedules his Prime Time in the morning from approximately 9 A.M. until 10:30 A.M. every Monday and Wednesday, his Push Days. He doesn't take inbound phone calls or answer e-mail during that time unless he deems that specific e-mail or phone call to be the highest use of his time. Jeff sets aside every Monday and Thursday morning from 7:30 A.M. to 8:30 A.M. as his Prime Time to do the same.
Don't demand perfection in honoring your Prime Times; you don't need to reach that standard. Even 80 percent consistency here will be enough to create a dramatic increase to your personal productivity. You'll find that setting aside this regular appointment to do your highest-value work for the day allows you to create more value for your business.
Time Mastery Strategy #6
Create a "Stop Doing" List and add to it weekly.
Too many people live their lives based on a to-do list to which they keep adding more and more tasks. But they rarely make the hard choices of what to let go of, what to delay, what to delegate, what to delete altogether.
For example, could you delegate vendor issues, or scheduling, or customer-service issues to your staff? Outsource your bookkeeping or computer issues? Hire help to run your personal errands, do yard work, or organize home repairs?
Look at your to-do lists from the past 60 days and identify the activities you can add to your "stop doing" list. How much time will eliminating these tasks save you? Each week, pick a few more activities you deliberately choose to add to your "stop doing" list. You'll find that the items you put on it tend to be tasks you find draining—maybe ones you put on your to-do list out of obligation or inertia. When you get rid of them, you'll not only enjoy a sense of elation and energy but will truly free up your time for more valuable activities.
The Real Secret to Unshakable Time Discipline
Most people shudder when thinking about the need for greater discipline because they've always associated discipline with pain and effort. It doesn't have to be this way. Instead, we suggest that you link discipline to two very important concepts: Accountability and Environment.
Who holds you accountable for your actions and decisions within your business? Who coaches you to help you develop as a business owner? Who helps keep you on track? It's usually difficult to have this person be someone who works for you (after all, there is a power imbalance to the relationship). If you want to build a Level Three business, it's vital to have an accountability structure and at least one third-party mentor who helps keep you on track.
Discipline also results from working in a structured environment. Take the example of Andrea Butter, owner of Maui By Design, a gift product design and wholesale company. Andrea came to her own business late in her professional life, after a long and successful career in Silicon Valley as a high-level marketing executive. She struggled letting go of things, and found e-mail the irresistible lure that tempted her into personally handling too many of the low-level details of operating her business. This was even harder for her as she worked out of her home and, because she worked with many suppliers in Asia, she found herself responding to work e-mails in real time during their workday (her nighttime).
Andrea took control of her environment by pulling the cable out of her Internet modem each day by 6 P.M. to reduce the temptation of checking e-mail one last time. During the first year Andrea did this, not only did she grow her business by over 40 percent, but she got her nights back again.
By controlling your environment, you make healthy, profitable behaviors for your business much easier. Willpower can win a sprint, but rarely a marathon. To win the marathon of building a Level Three business, take control of your environment to ensure it supports your goals each day.
The Top Ten E-mail Best Practices Every Company Should Adopt
Speaking of distractions, we need to address what has grown to be one of the most difficult and addictive time distractions in the business world— _e-mail_! Here are our top ten e-mail best practices we think every company should adopt:
**1. Use powerful subject lines to streamline the time it takes for your team to process and find e-mail.** No more blank subject lines, or "Hello . . ." Instead, you and your team should make your subject line a clear, concise description of the e-mail. This helps you screen messages and it helps you later search for e-mails you need to find after the fact. If you are forwarding the e-mail, don't be lazy; redo the subject line to make sense to your recipient and ask that your team do that for you too. Also, consider using the "1-2-3" system in your subject line:
1 = Time sensitive and important. Read and take action ASAP.
2 = Action required. Read and take noted action in a reasonable time frame.
3 = FYI. No action required. Scan for content when convenient.
Here's how this might look:
_2: Notes from Franklin call 2/5/15_ (This tells recipient they need to take action on the e-mail.)
_2 Mark; 3 Sarah: Two follow-up items still needed to complete Sullivan Project_ (This tells Mark he needs to take action and Sarah that this is just FYI for her.)
**2. Don't mass "CC" or "BCC."** Only CC or BCC if the person really needs the information. Remember, it's not just that one e-mail, but all the subsequent e-mails in that chain that you'll likely include that person on.
**3. Turn off your auto send-and-receive function (or at least reduce the frequency it downloads new e-mail).** Contrary to the way it feels, you don't need to see every e-mail the instant it comes in. Also, turn off e-mail alerts (audio and visual). Instead, intentionally check e-mail when you choose versus when someone hits Send to you. E-mail alerts only promote compulsive behaviors that kill productivity.
**4. To get less e-mail, send less.** The more you send, the more you get.
**5. Consider aging your e-mail before you respond to get less of it.** If something isn't time sensitive or a critical relationship, consider waiting a few days or (gulp!) a few weeks before you reply. We've all had the experience of immediately replying to an e-mail only to get a reply to our reply ninety-four seconds later. (If you like answering right away, consider using the "delayed delivery" option in most e-mail programs to answer now but send the e-mail later.)
**6. If you're involved in a frustrating back-and-forth conversation by e-mail due to hazy understanding on either side, just pick up the phone or speak in person.** E-mails are nogood as a nuanced conversation tool and shouldn't replace all conversations.
**7. In replying to a long conversation thread, pull up the key information to the top of the e-mail.** Make it easier for your recipient to quickly get what you are communicating. **** Also, if you are creating a longer e-mail with multiple items, consider numbering your items to make them easier for your reader to follow and respond to your e-mail.
**8. If you think the topic may be a sensitive one, or that the reader may be upset or offended by your e-mail,** don't __**send it.** Talk with them instead (even if you then send a summary or confirming e-mail after). Also, never say something in an e-mail that you wouldn't be willing to say directly to the person you are speaking to in the e-mail. This goes double for your team. Quickly deal with any inappropriate e-mails.
**9. Don't use e-mail to manage your "tasks" or to manage your team's tasks.** Use a project list on a spreadsheet, or a shared task management or project management tool instead. E-mail is a poor place to keep a running list. What comes today is washed away by what comes later today (let alone tomorrow). There are simple, inexpensive project management tools available online and on mobile devices that allow you to list, categorize, prioritize, and share your open action items. It's a worthwhile investment to prevent tasks and follow-ups from falling through the cracks.
**10. Learn your top five e-mail recipients' preferences.** Just sort your "Sent" folder by recipient and pick out the five people you send the most e-mail to. These will likely be internal team members. Ask them if they prefer wide or shallow e-mails (i.e., one e-mail per subject as it comes up, or a grouped e-mail that has more items in one single e-mail). When are their e-mail-free times? What do they want to and not want to be CC'd on? What are the three things they like best about how you communicate with them by e-mail? What three things would they like you to do differently about how you communicate by e-mail to make their life better? Then reverse the conversation and share your e-mail preferences with them.
Accelerate Your Upgrade of Time—Get a Personal Assistant
So far in this chapter, we've asked you to clearly identify the things you do that create the most value for your company and to structure your day and week so that you upgrade eight hours of " _D_ time" to _A_ and _B_ uses.
There is one final element that will help you massively upgrade the value you create for your company—finding, hiring, and leveraging a personal assistant. Whether you bring on a full-time assistant, outsource to a virtual assistant who works remotely, or do something in between, it is our belief that just about any business owner who runs a successful company can leverage a personal assistant to help her radically grow her business.
Figure 12.2 shows a list of 25 things you can ask your assistant to do. While we have a lot more we wanted to share with you about this subject, for the sake of brevity we recorded a full-hour training video on how to find, hire, and leverage your personal assistant and included it free for you to access at **www.Scale YourBusinessToolkit.com/videos**. (See the appendix for full details.)
FIGURE 12.2: 25 THINGS YOU CAN ASK YOUR ASSISTANT TO DO TO LEVERAGE YOUR TIME
1. Organize your travel (including learning your travel preferences)
2. Handle billing disputes
3. Help set up bills on auto payment on your credit card
4. Address and mail cards, letters, and packages
5. Update your contact manager (or CRM database)
6. Screen your e-mail and handle low-level responses
7. Update your blog and social media accounts
8. Organize and manage your filing system, both paper-based and scanned e-files
9. Take dictation (either live or via recordings)
10. Set up phone appointments
11. Daily cleanup of your office, including refilling items
12. Screen phone and e-mail
13. Take notes at key meetings and follow up with attendees on key deliverables
14. Keep a master chart/list/calendar of your projects and deadlines and set reminders
15. Tickler all birthdays and anniversaries, holidays, or other important dates, and even arrange for gifts, cards, or phone calls that make you look good
16. Update his or her own "Project List" so that all the tasks and deliverables they are responsible for are in one place for you to review
17. Get, open, sort, forward, handle, and if need be shred your mail
18. Coordinate with outsourced vendors when you have an IT issue
19. Order things online for you and handle any product returns or service issues
20. Handle any personal errands or schedule any household repairs
21. Notarize your documents
22. Help you to streamline your office—filing, sorting, and systematizing work flow
23. Perform basic updates to your Web sites
24. Create and continue to refine the "expert system" for how to be your assistant (this one should be part of their job function right from the start)
25. Take on any parts of your projects that he or she is capable of doing for you
In the end, we all have the same number of hours in a day. The most successful business owners have learned to apply these time mastery strategies to create more value in the same or less time.
SCALING PRINCIPLE SEVEN: YOU DO HAVE THE TIME TO SCALE YOUR BUSINESS AND GET YOUR LIFE BACK.
## PUTTING IT ALL INTO ACTION
Derek was a mid-twenties college grad who had played linebacker for his college football team. After graduating, he struck out on his own, launching a real estate business. All his life he had achieved his goals by outworking and outhustling everyone else. When he took this same recipe into the world of business ("Put your head down and just outwork everyone else"), however, he soon hit a plateau in his business. Derek had a few team members helping to leverage his time, but he was the critical linchpin that held his company together. Without him, nothing happened. And he did a lot! He listed and sold homes, wholesaled investor deals, and even did a handful of rehab projects each year. In 2006, Derek had hit his limit, completing the most transactions he could imagine himself personally doing in one year—65. He was financially successful, earning six figures, but he was burned out and knew that he just couldn't run any faster.
We began working with Derek in 2007. He locked on to the ideas you've been learning about in this book and applied them to his business. By 2009, his real estate investment company was buying, rehabbing, and profitably re-selling more than 200 houses a year and Derek was the largest player in his city. By 2012, his real estate brokerage company was doing 500 transactions a year. He had literally scaled his businesses by over 1,000 percent. This was a guy who used to think that growing required him to work harder and longer than everybody else. It would have been impossible for him to grow to this level if he hadn't let go of the belief that he had to work harder and personally produce more.
Now Derek is the proud father of twin boys and enjoys a strong marriage to an amazing spouse. Each summer, they go out to Southern California for six to eight weeks to enjoy a beachfront house and plenty of family time together. He still contributes great value to his company but is no longer the linchpin who holds everything together. His systems, team, and internal controls have helped him scale well beyond what his old limitations were back in the day when he thought he had to do it all himself.
Jennifer, successful owner of a medical software company in Tucson, Arizona, was also at a different place in her business life when we first met her in June 2008. A few years prior, she had her "all in" moment when she bought out her business partners and took sole ownership of the company. At that point, not only did she wear most of the hats in the business, but she was the financial backstop for the company with her life savings literally on the line. Though the company was profitable, it was vulnerable because it was almost completely owner-reliant.
Fast forward to today: Jennifer's company has a core leadership team, strong systems, and simple controls that have greatly lessened its reliance on her. She's grown sales by more than 500 percent and the company is worth several million dollars more than it was back when she bought out her partners. Best of all, Jennifer loves her life and her role inside the company, and enjoys watching her company continue to grow by over 35 percent a year.
It _is_ possible for you to scale your business and get your life back. You don't have to work harder or longer. You don't have to sacrifice your life to do it. You _can_ enjoy rapid growth and greater freedom. And you now have the concrete road map to do it.
Let's walk through a quick recap of the seven principles you'll use to scale your company, and then conclude with five final action steps to apply what you've learned in this book starting tomorrow morning.
Quick Recap of the Seven Proven Principles to Grow Your Business and Get Your Life Back
Scaling Principle One: Build a business, not a job.
The best way to build your business is not by working harder, but rather by reducing your business's reliance on you so that your _business_ can produce more. For all the artificial drama implied by the "Hit by a Bus" test, the core lesson is very real: If you ever want to escape the Self-Employment Trap you ****_must build a business, not a job._ ****
Scaling Principle Two: Build on the stable, scalable, three-legged base of systems, team, and internal controls.
It's not enough to just hire good people. Sustainable growth demands that you support your talented team with the smart business systems and intelligent internal controls that reliably help them produce more and better.
Scaling Principle Three: Understand why your customers really do business with you.
This principle boils down to truly knowing your customers—their desires, hopes, fears, aspirations, needs, pain, and perspective—so that you can find a profitable way to create value for them in a niche that you can dominate. This requires clarifying the three dimensions of your business context: your customers, your competitors, and your position and brand.
Scaling Principle Four: Create the right strategic plan and reduce that plan into a series of rolling, one-page quarterly action plans that help you execute and get results.
It is only by focusing your company's limited resources on those fewer, better things that you are able to grow your business. Follow our structured five-step process to determine your strategy, and how to best apply it quarter by quarter.
Scaling Principle Five: Learn to read the world so that you stay relevant and build your business for tomorrow's marketplace, not yesterday's reality.
In a rapidly changing world, the status quo is never safe. This is why it's crucial that you learn to read the world so that you can innovate and seize opportunities for growth (not to mention avoid being blindsided by a disruptive change and put out of business!).
Scaling Principle Six: Remove the predictable obstacles to growth—pillar by pillar.
Every pillar of your business—Sales/Marketing, Operations, Finance, Team, and Executive Leadership—has predictable obstacles that you must move past as you scale your company. Because these obstacles are known, you can proactively chip away at them until they are no longer in the way of your growth.
Scaling Principle Seven: You do have the time to scale your company and get your life back.
You don't have to work nights and weekends to grow your company. You are _already_ working enough hours to grow your business. You just need to upgrade your use of time to focus on the things that you do that truly create value for your company.
Scaling your company is a process, not a light switch you flip in one moment. While it won't happen overnight, when you stay the course, it will happen faster than you could have imagined.
Tomorrow Morning
We get asked all the time for the "magic bullet" to immediately make an impact in growing a company. Here, at the end of the book, we want to give you the five highest-leverage action steps you can take starting tomorrow morning to apply what you've learned in this book to make an immediate impact on your company. Each takes less than 30 minutes.
Action Step #1: Know Your Customers (pp. 39–45). Write out the one-page summary of your target customers. Who are they? What are their hopes, fears, aspirations, and challenges? The most important place to start is to dive deep into the lives of your customers.
Action Step #2: Conduct a Sweet Spot Analysis (pp. 73–79). Determine your business's current Limiting Factor. Then conduct a Sweet Spot Analysis to pick the highest-leverage tactics to push your Limiting Factor back over the next 90 days. Turn your ideas into a mini-plan and get to work.
Action Step #3: Build Your Own Time Value Matrix **** (p. 233). The only way to consistently upgrade your time is to identify what you do that truly creates value for your company. List your _A_ -, _B_ -, _C_ -, and _D_ -level tasks so you have a written reminder. Then pick one day each week to be your Focus Day (or at least a half day) and schedule that day to be about _A_ \- and _B_ -level items.
Action Step #4: Create Your First One-Page Quarterly Action Plan **** (pp. 87–92). Don't worry if the process is messy, or if you stumble your way through it. You'll still get great value from doing it and, each quarter going forward, you'll get better and better at the process. Of course you'll still have to take care of the daily operational needs of your business, but your one-page action plan ensures that you consistently invest a portion of your best resources on those things that will actually help you reach your business goals instead of just reacting to the pressing urgencies that will otherwise dominate your day.
Action Step #5: Register at www.ScaleYourBusinessToolkit.com and Invest Thirty Minutes a Week on the Site. **** We've designed this powerful Web site to help you grow as a business owner and successfully scale your business. Not only will you find all the tools we've shared in this book available for you to download as PDFs, but you'll also be able to access more than two dozen proprietary training videos. Plus you'll be able to connect with Jeff and David via our private LinkedIn business owner mastermind group.
Worth the Journey
When Jeff was ten years old, he went over to his friend Mike's house, where a few other friends were gathered, fawning over his newest poster of a red Ferrari. Later that day, Jeff asked his mom what the big deal was about the car. He didn't understand why Mike and his friends seemed so impressed by it. Jeff's mom replied that people often find themselves fascinated by things they'll never have. "What do you mean that they'll never have it?" Jeff asked. His mom responded, "Jeff, that car probably costs the same as our house. Very few people will ever be financially successful enough to be able to afford that car."
Jeff clearly remembers saying to her, "But someone must be able to afford that car, why not me?" His mom tried to explain that it just wasn't realistic for him to think he could ever own a car like that. As Jeff walked away, he said one last time, almost as if to himself, "Yeah, but somebody gets that car. . . ."
In retrospect, it was clear that Jeff's mom wanted to protect him from having unrealistic expectations that would lead to disappointment. But Jeff left that conversation thinking that if somebody gets to drive that car, why not him? What if he worked hard enough and smart enough to earn the money to buy that Ferrari? Jeff never forgot that experience, and over time the phrase "Why not me?" became his personal mantra.
Over the years, Jeff never stopped thinking about the car, or the lesson he took from it about the relationship of goals, hard work, focus, and success. Whenever work was hard and he felt like quitting, he thought about that car. The dream of the car gave him the energy to keep going. It wasn't about the money or the material thing. That car become a symbol for him of what he could achieve if he kept working toward his goals, shutting out all the naysayers in his life.
It was a great day for Jeff when he finally bought that car and put on the license plate that read "PCLN," the stock ticker symbol for Priceline.com, the company he'd helped scale whose success had allowed him to own that car. Jeff would often go outside, sit in the car, close his eyes, and just let the sound of the engine drown out the sound of all the nos he had heard in his life. "Somebody drives that car, why not me?"
Many business owners have built thriving Level Three businesses— _why not you?_ They successfully scaled their companies and enjoy an incredible lifestyle— _why not you?_ They have impacted the world through their businesses, employed people, served their customers, supported their vendors— _why not you?_ You now have the road map, your business is the vehicle, and it's just up to you to hit the gas and go.
We believe in you and know you have what it takes to build the company you want to build and create the life you want to enjoy. You are not alone on your journey. Hundreds of thousands of other business owners are committed to the same goals, leveraging the same tools. If you ever doubt yourself, then borrow our faith—we know you can do it. Savor the journey. It truly is worthwhile.
Your friends,
Jeff & David
## ACKNOWLEDGMENTS
_Scale_ is about building a business that is so much more than just a reflection of the owner. In much the same way, this book is much more than just the direct experiences of the two authors. This book benefited from the enormous skill and experience of a number of people. While the list is too long to thank everyone, we do want to make special mention of several people.
First, thank you to the team at Portfolio. Your passion for the project and editorial insights made the book more than we could have hoped for. A special thank-you to Natalie Horbachevsky for all the hours you pored over the manuscript, challenging our assumptions and making key suggestions to radically improve the book, and to the promotional team at Penguin Random House, who helped us get the word out about the book so successfully. Thank you also to our book agent, Jim Levine, who took the mystery out of the publishing equation and helped us pare our message down to its essential core (and to Andrea who introduced us to you). Thank you to Kevin Bassett, CPA, for his keen insights and great suggestions that helped make chapter 9 ("Your Finance Pillar") more concrete and easier to apply.
To make the book flow more smoothly we decided that, rather than laboriously label every story as Jeff's or David's, wherever we felt that the story was one that either of us had had similarly experienced with multiple business owners we've advised, we would use the collective "we." We hope this contrivance enhanced your enjoyment of the book. We would like to thank all the brilliant business owners featured in this book who shared their experiences with us, which resulted in strong examples that any reader could relate to and learn from. Also, we would like to thank the tens of thousands of business owners who've attended our live events over the past ten years and given us such great feedback as to how to best share the _Scale_ concepts for fastest implementation.
In addition, Jeff would like to thank: "My parents for always saying 'Why not?' instead of 'You can't.' And to many of you who have worked for me, with me, and next to me over the years, proving that a great team made of people with passion and values can do anything they set their minds to. And finally, to my amazing family, who jumped off so many cliffs with me, knowing I had no parachute, and clung tightly to me anyway because you knew it was the only way I could truly feel alive. Your love and support have always been the wind beneath my wings."
David would like to thank: "My entire team at Maui Mastermind. Your commitment to helping our business coaching clients grow their businesses and reduce their companies' reliance on them, the owners, has directly touched thousands of lives. I am exceptionally proud that as of this printing, your support has helped our clients enjoy an average annual growth rate of 32.4 percent. Also, thank you to all our business coaching clients for being proof positive that any business owner can build a business they love to own again if they are committed and coachable. To my friends and family, thank you for being such a source of joy and support in my life. And finally to Matthew, Adam, and Joshua—you make me so grateful and proud to be your dad; and to Heather—thank you for your love, your friendship, and for sharing your life with me."
## APPENDIX:
SEVEN ONLINE RESOURCES TO HELP YOU SCALE
Congratulations on finishing the book. We've created these seven free online tools to help you apply the _Scale_ concepts in the book to grow your business and get your life back. All of them can be accessed at: **www.ScaleYourBusinessToolkit.com** **.** ****
Here Are the Seven Tools
Tool #1: www.ScaleYourBusinessToolkit.com/videos
More than **two dozen video training modules** to help make building your business _easier_ and _faster_. These include the following five "short courses":
* The _Grow Your Sales_ Short Course (6 training videos and PDF action guide)
* The _Scale Your Operations_ Short Course (3 training videos and PDF action guide)
* The _Financial Pillar_ Short Course (6 training videos and PDF action guide)
* The _Strategic Planning_ Short Course (4 training videos and PDF action guide)
* The _Time Mastery_ Short Course (3 training videos and PDF action guide)
Tool #2: www.ScaleYourBusinessToolkit.com/pdf
**Free** **PDF downloads** of the business growth tools we highlighted in the book (including the four-page _Strategic Planning Document_ , the _Sweet Spot Analysis Tool_ , and several others).
Tool #3: www.ScaleYourBusinessToolkit.com/studykit
**Complete Small Group Study Kit** to guide you and your business mastermind team through applying the key concepts from the book to grow your businesses. Designed as a self-contained 90-day launch kit, it will help you support each other to quickly implement the key concepts from this book to scale your respective companies.
Tool #4: www.ScaleYourBusinessToolkit.com/coaching
**Free private 60- to 90-minute business coaching session** (if your business qualifies) to customize the strategies from the book to your specific situation so you can grow your company the right way. This is your chance to ask questions, learn best practices, and get the one-to-one coaching you need to quickly apply the _Scale_ concepts to your company.
Tool #5: www.ScaleYourBusinessToolkit.com/interviews
Exclusive access to private interviews we did with some of our peers who have each personally built and scaled extremely successful Level Three companies in various industries. Listen as they candidly share the things that helped them succeed and the most painful lessons they had to learn the hard way. You'll be both inspired and empowered as you hear their stories and apply their lessons to grow your business.
You'll meet:
* Gayle and Phil Tauber, cofounders of Kashi®. Hear the inside story of how they scaled this company into one of the premier food brands and then sold the company to Kellogg.
* Stephanie Harkness, former chairman of the National Association of Manufacturers, and a serial entrepreneur who took a small, struggling manufacturing company and grew it into one of the most successful medical device contracting manufacturers in the world.
* Morgan Smith, founder of Morgan Financial, a mortgage brokerage company that he built to 212 offices before he sold it to a private equity company.
* Patrice and Rob Mabary, cofounders of Consensus Medical Communications. Patrice and Rob have built and still own seven Level Three businesses with sales in the tens of millions.
* Bill Shopoff, founder of The Shopoff Group. You'll hear how he recovered from a total financial "do over" and built one of the premier land development investment firms in the United States, and in the process literally completed over $1 billion of deals.
* Patty DeDominic, former president of the National Association of Women Business Owners and founder of PDQ Careers, an HR consulting company that she grew to more than 600 staff in 30 states before she sold it.
Tool #6: www.ScaleYourBusinessToolkit.com/subscribe
Opt in and **receive the "Business Owner Success E-letter."** Join the more than 100,000 business owners worldwide who regularly benefit each week. This short digest is filled with tips, strategies, and concrete examples to help you apply the _Scale_ concepts to grow your business and reduce its reliance on you the owner.
Tool #7: www.ScaleYourBusinessToolkit.com/innercircle
**Insider access to Jeff and David's private LinkedIn business-owner mastermind group.** Come hang out with us and ask us your toughest business questions, share your experiences as you apply the strategies in the book, and connect with an upgraded peer group of other business leaders who, like you, are committed to take their companies to the next level.
To access these Web tools to scale your business and get your life back just go to
www.ScaleYourBusinessToolkit.com
## ABOUT THE AUTHORS
JEFF HOFFMAN:
Jeff is a successful entrepreneur and motivational speaker. In his career, he has been the founder of multiple start-ups and the CEO of both public and private companies, and has served as a senior executive in many capacities. Jeff has been part of a number of well-known companies, including Priceline.com, uBid.com, CTI, and ColorJar.
Today Jeff serves on the boards of companies in the United States, Europe, South America, and Asia. He supports entrepreneurs and small businesses on a worldwide basis, serving on the global advisory boards of Global Entrepreneurship Week (supporting entrepreneurship in more than 130 countries), the U.S. State Department's GIST program (Global Innovation through Science and Technology, working in 49 emerging nations), the APEC Start-up Accelerator Initiative (the Asia Pacific Economic Council's 21-member-nation association), and many more. He supports the White House, the State Department, USAID, and similar organizations internationally on economic growth initiatives and entrepreneurship programs.
Jeff is a frequent keynote speaker, having been invited to speak in more than 50 countries. He speaks on the topics of innovation, entrepreneurship, and business leadership. Jeff also teaches innovation workshops to major corporations on a regular basis.
Jeff is a business expert featured on Fox News, Fox Business, CNN, CNN International, Bloomberg News, CNBC, ABC, and NPR, and in publications including _Forbes_ , _Inc._ , _Time_ , _Fast Company_ , and the _Wall Street Journal_.
Jeff received a Lifetime Achievement Award from the national Collegiate Entrepreneurship Organization (CEO) council for his contributions to the field of entrepreneurship, as well as the 2012 Champion of Entrepreneurship Award from JPMorgan Chase, Citibank, and Rising Tide Capital.
Outside the world of technology, Jeff has produced movies in Hollywood; has produced musical events, including concerts, tours, and charity events with such artists as Elton John, Britney Spears, and 'N Sync; and serves on numerous charity and nonprofit boards.
DAVID FINKEL:
David is an ex–Olympic level athlete turned serial entrepreneur. He is the _Wall Street Journal_ and _BusinessWeek_ best-selling author of ten business and financial books, including the wildly successful _Build a Business, Not a Job_ , and a coauthor of _The Maui Millionaires for Business_.
David is the founder and CEO of Maui Mastermind®, one of the world's premier business coaching companies, which helps businesses in the $1 million to $20 million range build their companies to sell, scale, or own passively.
Over the past 20 years, David and the other Maui coaches have personally started, scaled, and sold over $2 billion in businesses.
Maui Mastermind helps business owners build companies they love owning again—for the value they create, the lives they touch, the profits they earn, the teams they employ, and the freedom they enjoy. Its clients have enjoyed an average annual growth rate of 32.4 percent while at the same time radically reducing their companies' reliance on them as the owners.
To learn more about working with his company to get the structured business coaching you need to take your company to the next level, call them toll-free at 866-214-6610, or visit them on the Web at www.MauiMastermind.com/coaching.
David's weekly business owner e-letter is read by 100,000 business owners around the world, and his business articles have been featured in more than 6,000 newspapers and other publications.
He and his wife, Heather, and their three young sons live a very simple life in Jackson Hole, Wyoming.
## INDEX
The page numbers in this index refer to the printed version of this book. To find the corresponding locations in the text of this digital version, please use the "search" function on your e-reader. Note that not all terms may be searchable.
Ablitt, Sasha, 46, 47, 48, 219
accountability, 214–23
Big Rock Huddle and, 221–23
commitments, enumerating and clarifying, 214–15
failures, owning, 216
feedback on progress, providing, 215
handoffs and, 216
key performance indicators (KPI) and, 217–21
phantom deliverables, avoiding, 215
timeliness, fostering, 215
time mastery system and, 244–45
accounts payable controls, 195–96
Ackworth, Kimberly, 139
action steps for immediate impact on growing company, 255–56
additional sales, tactics to generate, 131–40
cross-selling as, 138–39
formalized referral systems as, 132–33
reactivation system as, 133–35
re-selling as, 139–40
retention system as, 135–36
up-selling as, 137–38
Amazon, 47, 51, 68
American Express, 202
Apple, 47, 54, 96, 103
Arthur Anderson, 96
Association of Fraud Examiners, 192
base of business, building on, 23–35, 253
internal controls, building, 28–35
refining and redesigning systems and controls, as company grows, 34–35
systems, building, 24–28, 34–35
team, building, 28
Bassett, Kevin, 221
Bassett and Byers, 221
Bayer Aspirin, 51
best-first search, 43
big-picture planning process, 68–71
rewards and meaning of reaching Singular Goal, identifying, 70–71
Singular Goal, identifying, 69–70
why is company in business, determining, 68–69
big-picture vision, 209–12
company lives its vision, making sure, 211
example, leading by, 210
future, investing in, 212
planning process for ( _See_ big-picture planning process)
quarterly planning tool and, 211
repetition and, 210
sharing vision with team, 210
team's roles and responsibilities, clarifying, 211
team's thoughts as to, 210–11
written, 210
Big Rock Huddle, 221–23
Black Sky Entertainment, 2–3
bottom line tasks, 241–42
branding, 50, 53–59
choice of, trial for, 78
emotions and, 54–59
inductive perception and, 55–58
mistakes in, 53–54
reality checks for, 58–59
reinforcing, 58–59
business model, choice of, 77
business strategy, 79–83
questions to help in crafting, 79–81
Sittercity.com case study, 81–82
business systems. _See_ systems
business to business, common elements of best buyers for, 41
business to consumer, common elements of best buyers for, 41
business versus job, building, 9–22, 253
freedom, paradox of, 15–17
"hit by a bus" test and, 13–14
Level Three Road Map and ( _See_ Level Three Road Map)
loss of control, fear of, 12
self-employment trap, escaping, 14–15
specialness of business belief, as trap for owners, 11–12
working harder as part of problem, 16–17
Butter, Andrea, 244–45
Byers, Dane, 221
capabilities company will need in future, 85–87
capital investments, 188
cash controls, 194–95
cash flow, 172–88
collecting more of what is owed, 173–75
expenses, management of, 181–82
faster collections, 175–78
income/expenses, predictability and consistency of, 179–81
margins, managing, 183–85
pricing and purchasing decisions, 186–88
timely and accurate financials, maintaining, 178–79
CertifiedTaxCoach.com, 119–20
change, and strategic plan, 65–67
chief financial officer (CFO), 172
Circuit City, 96
Cleveland Clinic, 153
collecting more of what is owed, 173–75
frontloading of collection efforts, 175
invoicing, frequency of, 174–75
parties responsible for collections, assigning, 175
paying attention to A/R reporting, 175
tracking what is owed, 174
collection practices
collecting more of what is owed ( _See_ collecting more of what is owed)
collections cycle, accelerating ( _See_ collections cycle, accelerating)
controls for, 195
collections cycle, accelerating, 175–78
advance payments, 176–77
financing charge for accruing bills, use of, 177
incentivizing advance payments, 177
making payments, ease and convenience of, 178
production and delivery cycles, accelerating, 177
service or delivery, collecting at time of, 177
ColorJar
big-picture planning process at, 68–69
connection question for, 206
innovation sessions at, 104–5
company culture, 212–14
action/decision moments that illustrate way team should behave, seizing, 213
impact on client's life or business, sharing stories illustrating, 213
leader's establishing and shaping of, 212–14
stressful moments, handling of, 213
symbolic stories that transmit company values, sharing, 212
team and, 28
values of company, clarifying, 214
compensation, 203–5
Competitive Matrix, 48–49
competitors, 45–49
Competitive Matrix, building, 48–49
direct, 46
disruptive, 47–48
indirect, 46–47
strengths and weaknesses, identifying, 45–46
connection question, 205–7
consistency, controlling for, 156–57
control for consistency, 156–57
controller, 171–72
controls
financial controls ( _See_ financial controls)
internal ( _See_ internal controls)
marketing ( _See_ marketing controls)
converting leads into sales, 140–45
baseline sales system, building, 142–45
checklist for evaluating lead-conversion challenges, 141–42
conversion rates, calculation of, 143–44
leveraged improvements, identifying, 144
mapping out sales process, 143
mini-action plan to implement leveraged improvements, creating, 144
quarterly implementation, tracking and refining, 144–45
scorecard for conversion points, development of, 143–44
cost per lead, 129
cost per sale, 129
cross-selling, 138–39
margins and, 184
stabilizing income and expense and, 181
CTI, 131, 202
culture. _See_ company culture
customer relationship management (CRM) solution, 130
customers
as funding source, 189
real reason for doing business with you ( _See_ customer's real reason for doing business with you)
retention of, and margins, 185
customer's real reason for doing business with you, 37–60, 253–54, 255
competitors, analysis of, 45–49
context of business, understanding, 39
knowing customer's businesses, 38
positioning and, 50–60
target market, identifying, 39–45
delegating versus controlling dilemma, 29–31
deliverables, 150–52
external, 150
internal, 150–51
phantom, 151, 215
reverse phantom, 151
direct competitors, 46
discipline, and time mastery system, 244–45
disruptive competitors, 47–48
dollar sold per dollar spent on lead cost, 129–30
drive-up teller, 97
Drucker, Peter, 61
eBay, 63, 64
80-20 Rule, 231
e-mail best practices, 245–48
aging e-mail before responding, 246
auto-send-and-receive function, turning off, 246
conversation in lieu of e-mail, 246–47
inappropriate e-mails, dealing with, 247
mass CC or BCC, avoiding, 246
preferences of most commonly e-mailed recipients, learning, 247–48
send less e-mail, 246
subject lines for e-mails and, 245–46
task management via e-mail, avoiding, 247
embedded controls, 32–33
emotions, brand, 54–59
eToys.com, 111
Evernote app, 99
Executive Leadership, 113–14, 209–25
accountability and, 214–23
big-picture vision of company set by, 209–12
company culture, establishing and shaping, 212–14
Five-Pillar Audit tool to map baseline of, 114–17
key responsibilities of, 114
leadership team, growing and grooming, 223–25
expenses
better pricing, negotiating for, 181
discounts, asking for, 182
fiscal discipline as core company value, cultivating, 182
management of, 181–82
stabilizing ( _See_ stabilizing income and expenses)
strategic versus nonstrategic, 188
variable versus fixed, 182
vendors, review of and competition by, 181–82
Expert Systems, 148–60
control for consistency, 156–57
deliverables, defining, 150–52
key component to refine first, mapping out, 157–59
optimal level of expertise for each step in, determining, 153–55
process used to create deliverables, laying out, 152–53
quarterly reevaluation to prioritize next block of system to refine, 159–60
external deliverables, 150
FedEx, 47
Finance, 112, 171–96
cash flow, managing, 172–88
Chief Financial Officer (CFO), role of, 172
components of, 171
controller, duties of, 171–72
financial controls to safeguard against fraud and theft, 192–96
Five-Pillar Audit tool to map baseline of, 114–17
funding rapid growth, sources of capital for, 188–92
key responsibilities of, 112
financial controls, 192–96
accounts payable controls, 195–96
cash controls, 194–95
collections controls, 195
financial records, reviews of, 194
footprints in financial system, creating, 193–94
money flows, having two unrelated parties involved in, 192–93
Finkel, David, 1–2, 57, 99, 127, 223–24, 234, 239–40
The Fireplace Place, 179–80
Five-Pillar Audit tool, 114–17
fixed expenses, 182
focus areas for quarterly action plans, 89–91
criteria defining success for, 90–91
key performance indicator (KPI), tracking, 91, 220
Limiting Factor, pushing back, 89–90
mitigating gravest threat as, 90
opportunities, seizing, 90
selecting, 89–90
focus days, 238–41
focusing on fewer, better things, 61–105
learning to read world to build for tomorrow's marketplace, 95–105, 254
strategic plan, creating, 63–93, 254
formalized referral systems, 132–33
formal reactivation system, 133–35
formal retention system, 135–36
format layer, of business systems, 25–27
foundation for business, building, 7–60
base of business, building on, 23–35, 253
business versus job, building, 9–22, 253
customer's real reason for doing business with you, understanding, 37–60, 253–54
Fowler, Pete, 157
fraud, financial controls to safeguard against. _See_ financial controls
freedom, paradox of, 15–17
frequent flyer mileage programs, 136
funding rapid growth, sources of capital for, 188–92
customers, 189
internal, 190–91
investors, 191–92
lenders, 190
vendors, 189
Grand National Bank, 97
_Grease_ (movie), 95
gross profit margins, 183–84
handoffs
accountability and, 216
linkages and, 168–70
Harkness, Stephanie, 86
"hit by a bus" test, 13–14
Hoffman, Jeff, 2–3, 42–44, 57, 59, 63, 95–96, 131, 133, 153, 198–99, 202–6, 234, 256–57
Hollywood Video, 96
Home Run filter, for identifying sweet spot tactics, 74–75
income, stabilizing. _See_ stabilizing income and expenses
indirect competitors, 46–47
inductive perception, 55–58
In-N-Out Burger, 97
innovation sessions, 100–105
internal controls, 28–35
control and controls distinguished, 30–31
delegating versus controlling dilemma, 29–31
embedded, 32–33
examples of, 29, 33–34
procedural, 32
refining and redesigning of, as company grows, 34–35
visual, 31–32
internal deliverables, 150–51
internal funding, 190–91
investors, 191–92
iPhone, 103
iPod, 96, 103
iTunes, 47, 96
Jackson, Phil, 201
Jordan, Michael, 201
Jordan, Thomas, 78–81, 219–20
Kerr, Steve, 201
key performance indicators (KPI), 217–21
accurate feedback on regular basis, providing, 218
easy to understand, selecting indicators that are, 218
focus areas, tracking KPI for, 91, 220
leading indicators, 180–81, 217–18
mistakes companies make with respect to, 218–19
selection of, 217–18
visuals for, 220–21
Kim, Dr., 134–35
KPI. _See_ key performance indicators (KPI)
L. H. Thompson Inc., 149–50
leadership
executive ( _See_ Executive Leadership)
supporting team as role of, 202–3
team for, growing ( _See_ leadership team)
leadership team, 221–23
authority, delegation of, 224
company growth, and changes to, 224
first key management hires for, 223
as investment, 223–24, 225
oversight of, 224–25
quarterly action plans for, 225
lead-generation and conversion. _See_ Sales/Marketing
leading indicators, 180–81, 217–18
learning to read world to build for tomorrow's marketplace, 95–105, 254
innovation sessions, 100–105
music industry's failure to, 95–96
new inputs, exposure to, 98–100
trends, cross-industry/geographic boundaries movement of, 96
lenders, 190
Level One business (start-ups), 4, 14, 18
Level Three business (rapid growth/exit stage company), 4, 14–15
Level Three Road Map, 4, 17–22
determining where your are in business lifecycle, quiz for, 21–22
Five-Pillar Audit tool and, 114–17
Level One, 18
Level Three, 21
Level Two, 18–20
Level Two business (owner-reliant company), 4, 14, 15, 18–20
owner-reliant business phase of, 19–20
post-lauch start-up phase, 18–19
rapid-growth phase of, 20
leverage points of business, 71–79
Limiting Factor, identifying and pushing back, 73–75, 89–90, 255
S-O-O-T Review of, 71–73
strategic decisions, examination of, 75–79
Sweet Spot Analysis Tool and, 74–75, 76, 255
Lexus, 55, 56
Limiting Factor, 73–75, 89–90, 255
linkages, 168–70
boundaries for handoffs, establishing, 169
explicit handoffs, 169
moving start, getting, 169
needs of both halves of handoff, determining, 169–70
processing out steps in handoff, 170
loss of control, fear of, 12
Low-Hanging Fruit filter, for identifying sweet spot tactics, 74–75
loyalty programs, 136
Lyle, Jennifer, 47
margins, 183–85
gross profit, 183–84
low-margin clients, products or services, cutting, 185
operating profit, 183
retention of customers and, 185
scrap, spoilage and waste, avoiding, 185
upgrading of customers to higher-value products or services, 184
up-selling and cross-selling to increase average unit of sale, 184
velocity (turnaround time) and, 184
marketing calendar, 126–27
marketing channel, choice of, 78
marketing collateral, standardization of, 127–28
marketing controls, 126–30
customer relationship management (CRM) solution, 130
marketing calendar, 126–27
marketing collateral, standardization of, 127–28
scoreboard for marketing efforts, 128–30
Marketing Markers, 41–42, 43
market position. _See_ positioning
Maui By Design, 244
Maui Mastermind
branding and, 57
connection question for, 206
target market needs, mapping out, 44
why company is in business, determining, 68
McAnn, Jim, 205–6
Microsoft Dynamics, 130
Molina, Dominique, 119–20
momentum, and strategic plan, 65–67
music industry, 95–96
new inputs, exposure to, 98–100
'N Sync, 95
obstacles
to scaling ( _See_ obstacles to scaling)
S-O-O-T Review and, 72
obstacles to scaling, 107–225, 254
Executive Leadership pillar and, 113–14, 209–25
Finance pillar and, 112, 171–96
Operations pillar and, 111–12, 147–70
Sales/Marketing pillar and, 109–11, 119–45
Team pillar and, 113, 197–208
1-800-Flowers, 205–6, 222
operating profit margins, 183
Operations, 111–12, 147–70
Expert Systems, building, 148–60
Five-Pillar Audit tool to map baseline of, 114–17
key responsibilities of, 112
linkages, reinforcing, 168–70
Ultimate Business System (UBS), building, 160–67
opportunities, 72
opportunity cost, 67, 75
Oracle, 130
Pacific Plastics and Engineering, 86
Padda, Gurpreet, 235
Pareto, Vilfredo, 231
Pareto's Principle, 231
parking space theory of positioning, 50–53
examples of, 51
factors relevant to determining, 50
questions to help pick best parking space for business, 52
as single thing you want to be known for, 51
Peachtree, 193
Peregrine Mobile Bottling, 78–79
personal assistant, 248–49
phantom deliverables, 151, 215
Pippen, Scottie, 201
policies and procedures manuals, 27–28, 160
positioning, 50–60
branding and, 50, 53–59
choice of, trial for, 78
parking space theory of, 50–53
Prego Spaghetti Sauce, 51
Priceline.com, 2, 257
branding and, 57
cross-selling at, 138–39
parking space of, 51
profiling of target customers at, 43–44
why company is in business, determining, 68
Priceline Yard Sale, 57
pricing, choice of, 77
pricing decisions, 186–88
changing how you charge, 187
clues pricing may be too low, 187
review of, 186–87
Prime Time block, scheduling, 242–43
procedural controls, 32
process layer, of business systems, 25
product, choice of, 77
product pathway, choice of, 78
profile of target customer, building, 42–43
purchasing decisions, 186–88
push days, 238–41
qualitative goals of future business, 69–70
quantitative goals of future business, 69–70
quarterly action plans, 66–67, 87–93, 255–56
criteria defining success for focus areas, 90–91
example of, 88
focus areas, selecting and tracking, 89–91
innovation sessions, 100–105
key action steps and milestones, laying out, 91–93
key performance indicator (KPI) for focus areas, tracking, 91, 220
for leadership team, 225
responsibility for action steps, assigning, 92
QuickBooks, 193
reactivation system, 133–35
RedTag.com, 2
referral systems, 132–33
reinforcing brands, 58–59
re-selling, 139–40
Results Rule, 242
retention system, 135–36
reverse leverage, 83–85
reverse phantom deliverables, 151
St. Francis of Assisi, 227
sales database, 130
salesforce.com, 130
Sales/Marketing, 109–11, 119–45
additional sales, tactics to generate, 131–40
baseline lead-generation system, building, 124–26
checklist for evaluating lead-generation challenges, 121–22
converting leads into sales, 140–45
Five-Pillar Audit tool to map baseline of, 114–17
key responsibilities of, 110–11
marketing controls and, 126–30
weaknesses in lead generation, evaluating, 120–23
sales model, choice of, 78
scaling principles
base of business, building on, 23–35, 253
business versus job, building, 9–22, 253
customer's real reason for doing business with you, understanding, 37–60, 253–54
learning to read world to build for tomorrow's marketplace, 95–105, 254
obstacles to growth, removing, 109–18, 254
strategic plan, creating and executing on, 63–93, 254
time to scale business, 229–50, 254
scoreboard for marketing efforts, 128–30
cost per lead, 129
cost per sale, 129
dollar sold per dollar spent on lead cost, 129–30
scrap, 185
self-employment trap, escaping, 14–15
service, choice of, 77
Singular Goal, 69–70
Sittercity.com, 81–82
Snyder, Harry, 97
S-O-O-T (Strengths-Obstacles-Opportunities-Threats) Review, 71–73
specialness of business belief, as trap for owners, 11–12
spoliage, 185
stabilizing income and expenses, 179–81
cross-selling or up-selling opportunities, look for, 181
external events, monitoring of, 180
leading indicators, monitoring, 180–81
long-term contracts, use of, 180
Stop Doing list, 85, 243–44
storage technology for UBS, selection of, 164–67
strategic decisions, examination of, 75–79
branding, choice of, 78
business model, choice of, 77
marketing channel, choice of, 78
positioning, choice of, 78
pricing, choice of, 77
product, choice of, 77
product pathway, choice of, 78
sales model, choice of, 78
service, choice of, 77
target market, choice of, 76–77
strategic plan, 63–93, 254
big-picture planning process, 68–71
business strategy, choosing, 79–83
capabilities company will need in future and, 85–87
change and momentum, balancing, 65–67
innovation sessions, 100–105
investment of key resources defines actual strategy, 83–84
iterative model of, 65
leverage points of business, determining, 71–79
long-term plan, annual review of, 66
opportunity cost and, 67, 75
purpose of, 64
quarterly action plans, 66–67, 87–93, 255–56
as recurring activity, 64–65
reverse leverage and, 83–85
Stop Doing list, creating, 85
strengths, 72
SugarCRM, 130
Sweet Spot Analysis Tool, 74–75, 76, 255
Swissair, 96
switching costs, 187
systems, 24–28
Expert Systems ( _See_ Expert Systems)
format layer of, 25–27
internal controls ( _See_ internal controls)
list of potential formats for packaging of, 26–27
policies and procedures manual, creation of, 27–28
process layer of, 25
purpose and components of, 24
refining and redesigning of, as company grows, 34–35
team members interaction with, as feedback on, 25–26
Ultimate Business System (UBS). _See_ Ultimate Business System (UBS)
Tapley, Klayton, 179–80
target market, 39–45
business to business, common elements of best buyers for, 41
business to consumer, common elements of best buyers for, 41
fastest to close customers, identifying, 40
Marketing Markers, 41–42, 43
Maui Mastermind case study, 44
non-target market, identifying, 42
profile of target customer, building, 42–43
prompts for identifying, 40–42
tool for mapping out psychological and emotional needs of, 44–45
target market, choice of, 76–77
Team, 28, 113, 197–208
achievements, immediacy and authenticity in rewarding, 201–2
common sense, and employee manual, 198–99
company culture and, 28
company where talent wants to work, creating, 200–202
compensation of, 203–5
connecting team with company goals, 205–7
Five-Pillar Audit tool to map baseline of, 114–17
ideal team member, defining and profiling of, 200
key responsibilities of, 113
leadership and, 202–3
managing team members individually, 200–201
poor performers, removing, 201
selectivity in hiring talent, 200
sharing information with, 201
standards for, 201
Three Best Ideas Tool and, 207–8
theft, financial controls to safeguard against. _See_ financial controls
_This Old House_ (TV show), 34
Thomson, Brian, 176
threats, 73
Three Best Ideas Tool, 207–8
time mastery system, 229–50, 254
_A_ / _B_ / _C_ / _D_ activities, identifying, 234–36
accountability and, 244–45
bottom line task to-do list, 241–42
discipline and, 244–45
_D_ -level activities, focusing on and eliminating, 237–38
e-mail best practices and, 245–48
focus days, 238–41
Pareto's Principle and, 231
personal assistant, hiring, 248–49
Prime Time block, scheduling, 242–43
push days, 238–41
reinvesting saved _D_ time in _A_ and _B_ activities, 238–41
Results Rule and, 242
Stop Doing list, creating, 243–44
Time Value Matrix and, 232–33, 255
value you create for business, identifying, 231–36
Time Value Matrix, 232–33, 255
Toyota Prius, 51
trailing indicators, 218
uBid.com, 2
formal reactivation system at, 133
profiling of target customers at, 42–43
strategic plan of, 63–64
Ultimate Business System (UBS), 160–67
company culture of creating and using, 167
existing systems and tools, renaming and storing, 163–64
organizational hierarchy for UBS file folders, creating, 161–63
policies and procedures manuals distinguished, 160
prioritized list of systems to build in current quarter, creating, 163–64
sample outline for, 162–63
storage technology for, 164–67
team training and involvement in, 167
up-selling, 137–38
margins and, 184
stabilizing income and expense and, 181
U.S. Postal Service, 47
value you create for business, identifying, 231–36
variable expenses, 182
vendors
expenses, review of and competition by vendors to reduce, 181–82
as funding source, 189
Virgin, 55
visual controls, 31–32
Walmart, 51
waste, 185
Wooden, John, 230
www.ScaleYourBusinessToolkit.com, 6, 256, 261–64
Zappos.com, 45–46, 51, 59
Zoho, 130
That event Tom attended raised over $250,000 for several charities, including the Justa Center in Phoenix. One of the best parts of building a successful business is the causes and groups you can support.
You can download a free four-page PDF version of our Strategic Planning Tool discussed in this chapter at **www.ScaleYourBusinessToolkit.com/pdf**. See appendix for full details.
There are two fantastic books that helped us put to words what our combined decades of practical experience as entrepreneurs taught us on this subject. They are _Just Start_ (Schlesinger et al.) and _How Will You Measure Your Life?_ (Christensen et al.). In fact, on **www.ScaleYourBusinessToolkit .com** we've posted what we consider to be the ten must-read books for all business owners.
How you use your time is so critical to your success as a business owner that we've recorded a one-hour online video training to help you get more done in less time. You can access this free video at **www.ScaleYourBusinessToolkit.com/videos**.
| {
"redpajama_set_name": "RedPajamaBook"
} | 2,492 |
Verizon Unlimited customers will get 6 months of Apple Music for free.
It seems Verizon is taking a page out of T-Mobile's handbook in terms of adding value to its unlimited plans.
Verizon is entering into an exclusive, multi-step partnership with Apple. Step one is to offer a musical perk. Both new and current Verizon Unlimited customers will get 6 months of Apple Music at no extra charge.
Even better, if you've already taken advantage of the free three months of the streaming service that Apple offers, you're still eligible for this deal. This means that Verizon Unlimited customers could have up to 9 months of Apple Music for free.
This offer begins Thursday, Aug. 16, and currently has no expiration date. Verizon will also unveil the redemption process on that day. I have a feeling it might involve you linking your Apple ID.
In a prepared statement, Angie Klein, vice president of marketing at Verizon, said, the offer "gives our customers exactly what they want: Apple's best-in-class music streaming experience, paired with an unlimited plan tailored to them, on the network they deserve."
Your guess is as good as ours, but I wouldn't expect this to be a hardware move in the future. Apple generally likes to keep the playing field level in terms of iPhone, iPad, and Apple Watch availability.
However, customers in the future could get exclusive discounts when using Apple Pay. Alternatively, Verizon and Apple could solve a pain point for unlimited customers by increasing iCloud Storage, like the 50GB upgrade free for a year.
Apple might expand this deal with other carriers, but this would likely happen after the exclusive period runs dry.
In the meantime, this is a clearly a move to drive more customers to Verizon's Unlimited offering, which expanded to three plans that can be mixed and matched on the same account earlier this summer.
All eyes are on AT&T, T-Mobile, and Sprint for a checkmate move (hmmm, maybe give away Spotify for free?). | {
"redpajama_set_name": "RedPajamaC4"
} | 733 |
Nelojalna konkurencija je epizoda Dilan Doga objavljena u Srbiji premijerno u svesci #11. u izdanju Veselog četvrtka. Koštala je 150 din (2,3 €; 2,7 $). Imala je 94 strane. Na kioscima se pojavila 20. novembra 2008.
Originalna epizoda
Originalna epizoda pod nazivom Concorreanza sleale objavljena je premijerno u #220. regularne edicije Dilana Doga u Italiji u izdanju Bonelija. Izašla je 23. decembra 2004. Naslovnu stranicu je nacrtao Anđelo Stano, scenario napisao Pasquale Ruju, a nacrtao Giampiero Casertano. Koštala je 3,9 €.
Kratak sadržaj
Dilan dobija konkurenciju u vidu Debi Dojl. Debi je odlučila je Dilanu otme mušterije. Otvorila je kancelariju u Craven Roadu i svoje usluge počela da naplaćuje 10 funti dnevno manje. Dilan se oseća ugroženo i odmah ulazi u sukob sa Debi. Međutim, uskoro kod Dilana dolazi nova mušterija, gospodin Campbell, koji želi da mu Dilan pomogne da pronađe zamak koji ne nasledio. Problem je u tome što je zamak nestao. Dilan ponovno doživljava šok kada saznaje da je g. Campbell istovremeno angažovao i Debi za isti posao.
Pojava crtača
U poslednjem kadru crtač Đampiero Kasertano prolazi pored Dilanovog stana.
Reference
Бонели
Италијански стрип јунаци
Дилан Дог | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,912 |
\section{Introduction}
The galaxy, NGC\,1275, harbors the radio source 3C\,84 with both a
parsec scale jet and a kpc scale jet as well as a large low frequency
radio halo \citep{ped90,wal00}. It has been the brightest
extragalactic radio source at high frequency with flux densities of
45\,Jy $-$ 65\,Jy in 1980 at 270\,GHz and $\sim 40-50$\,Jy at 90\,GHz
from 1965$-$1985 \citep{nes95}. The flare subsided in the 1990s, but a
new strong flare began in 2005 \citep{nag10,tri11}. Due to its
proximity ($z = 0.0176$), and its brightness, 3C\,84 along with M\,87
are the best candidates for resolving the jet near the launching
region with high frequency Very Long Baseline Interferometry
(VLBI). High resolution observations of M\,87 and 3C\,84 have
displayed extremely wide jet opening angles within 0.1\,mas of the
core, $\sim 127^{\circ}$ and $138^{\circ}$, respectively and both jets
are extremely edge brightened near the base
\citep{kim18,gio18,pun19}. The large degree of edge brightening and
the enormous (the maximum possible geometrically is $180^{\circ}$)
opening angles were not anticipated in seminal works on simple jet
theory \citep{bla79}. Numerical models of jet formation have been
designed to explain these extreme properties in M\,87. It was hoped
that choosing lines of sight (LOS) that are nearly aligned with the
jet axis and plasma emissivity/plasma enthalpy profiles (injected by
numerical researchers) that were specifically designed to produce the
high resolution radio images would resolve the conflict
\citep{mos16,cha19}. However, current numerical simulations still
produce synthetic jet images that are far too narrow and not nearly
edge brightened enough near their bases when compared directly to the
highest resolution interferometric images of the jet base
\citep{pun19}. This wide base seems to abruptly transition to a highly
collimated inner jet. The powerlaw fit to the jet width, $W(z)$, as a
function of axial displacement along the jet, $z$, is $W(z) \propto
z^{k}$, $k=0.230\pm 0.049$ for $0.06 \rm{mas} < z < 0.3 \rm{mas}$ in M87, even though the jet might be parabolic (k=0.5) at
larger z \citep{pun19}. A similar abrupt transition also seems to
occur in the jet of 3C 84 juxtaposed to its wide base, see
\citet{gio18} or Figures 9 and 10 of this paper. Furthermore, 3C\,84
was shown to have a triple nucleus in the highest resolution radio
observation, 30\,$\mu$as, performed with RadioAstron at
22\,GHz\footnote{We define the multiplicity of the nucleus,
i.e. double or triple, as the apparent number of components that
can be discerned from visual inspection of the image. This
designation is a function of the resolution of the telescope. If
the resolution were much higher then there could be many
small components that could not be seen at lower resolution.}
\citep{gio18}. The triple is distributed primarily along the east-west
direction roughly orthogonal to the jet that is directed towards the
south and is defined by two bright ridges, beginning 150\,$\mu$as to
the south (see the schematic diagram in Figure~1). These circumstances
suggest that there is jet launching physics to be discovered with high
resolution of VLBI, and not just the verification of existing theory.
Thus motivated, this study explores the nature of the multi-component
nucleus that has been seen with the highest resolution VLBI, $\sim$
30\,$\mu$as in the east-west direction with RadioAstron on September
21-22, 2013 \citep{gio18}. Our method is to use the lower resolution
7\,mm Very Long Baseline Array (VLBA) data that is created for the
purpose of approximately monthly monitoring by a Boston University
based research effort, the VLBA-BU Blazar Monitoring
Program\footnote{http://www.bu.edu/blazars/VLBAproject.html}
\citep{jor17}. Thus, we can, in principle, detect time evolution of
the nuclear region. A partially resolved double nucleus has appeared
in the CLEAN images from August 26, 2018 to April 7, 2020. We rely on
the publicly available files from the BU website to extract the
highest resolution data, corresponding to the longest baselines
associated with the 43\,GHz VLBA.
\begin{figure}
\begin{center}
\includegraphics[width=120 mm, angle= 0]{f1.eps}
\caption{The taxonomy of prominent features in the nuclear region of
3C\,84. The position angle of the southerly directed jet swings back
and forth from east to west by $\sim 35^{\circ}$ over periods of
about 5-10 years.}
\end{center}
\end{figure}
We begin with a description of the very complex nuclear inner light
year of the jet that has been revealed in recent high resolution VLBI
campaigns. Figure~1 is a schematic diagram of the numerous nuclear
features that have been detected in this nearby very bright radio
source. A schematic is much clearer than overlaying numerous images
made at various frequencies. An approximate scale is indicated at the
bottom, both in angular size and physical dimension (projected on the
sky plane). The primary feature is the triple nucleus (enclosed by the
ellipse) from the 22\,GHz RadioAstron observations \citep{gio18}. This
region is the focus of this study. This nuclear feature is apparently
not time stationary. A more recent 86\,GHz global VLBI observation on
May 16, 2015 revealed this region to be a double nucleus along the
east-west direction \citep{kim19}. With 43\,GHz VLBA, a double nucleus
has never been identified previously. Based on the aforementioned
higher resolution VLBI, the east-west separation is $\sim
0.1-0.15$\,mas, similar to the nominal east-west restoring beam of
the uniformly weighted 10 station VLBA of $0.15$\,mas. Thus, we
expect to see partially resolved structure in epochs of wide
separation.
Figure~1 shows other complicating features as well and helps to
illustrate the ultimate goal of finding a unifying picture that
incorporates all of these important elements. First, there is the
faint counter-jet that has been seen at numerous frequencies
\citep{wal00,lis01,fuj17}. Then there are the very prominent east and
west ridges that frame the jet headed to the south. They are of varied
prominence from epoch to epoch and observation to observation. In the
high resolution 22\,GHz RadioAstron observation they are significantly
brighter than the almost hollow interior of the jet. This ``double
rail" configuration has also been seen with 43\,GHz VLBA
\citep{nag14}. A similar double rail configuration has been detected
in high resolution VLBI images of M\,87 adjacent to the nucleus as
well \citep{had14,kim18}. The most curious aspect of these ridges in
the 43\,GHz BU VLBA monitoring data is that most of the time one ridge
is much brighter than the other for periods of time that last
years. The fundamental question that we wish to gain some insight into
is: what is the relationship between the multiple east-west nucleus
and these vacillating bright ridges of emission that emerge almost
orthogonal to the axis of nuclear emission, in the southern direction?
Some of these features are evident in the magnified views of the
nuclear region from the total intensity images taken from the BU
website in Figure~2. We note one other aspect of the nuclear region. A
modest feature has emerged to the west of the nucleus in 2018, about
0.25\,mas away. It appears to be real because it is persistent and is
seen a few contours above the noise level. It also seems to be
stationary. With all the components emerging in almost every
direction, the situation is not well explained by a simple pencil-beam
jet. We are actually getting close to the jet launching region, so we
are starting to uncover some of the messy details of the physical
mechanism. We do not attempt to resolve these complexities in the
present paper. We take a more modest approach and merely try to establish and
quantify the rate of nuclear expansion within the central ellipse of
Figure~1.
Section~2 describes our methods of extracting the nuclear separation
from the data. In Section~3, we describe how we use the CLEAN
component (CC) models from the BU program to extract the east-west
resolution associated with the longest baselines. Section~4 presents
our estimate of the apparent speed of separation of the double
nucleus. In Section~5, we explore the non-trivial methods of
fitting the nuclear components in the $(u,v)$ plane using Gaussian
models. In Section~6, we compare the nuclear configuration observed
by the 43\,GHz VLBA in 2018-2020 with the nuclear triple observed by
RadioAstron in 2013. Throughout this paper, we adopt the following
cosmological parameters: $H_{0}$=69.6 km s$^{-1}$ Mpc$^{-1}$,
$\Omega_{\Lambda}=0.714$ and $\Omega_{m}=0.286$ and use Ned Wright's
Javascript Cosmology Calculator website \citep{wri06}. In our adopted
cosmology we use a conversion of 0.360 pc to 1 mas.
\begin{figure*}
\begin{center}
\includegraphics[width=55 mm, angle=-90]{f2.eps}
\includegraphics[width=55 mm, angle= -90]{f3.eps}
\includegraphics[width=55 mm, angle= -90]{f4.eps}\\
\includegraphics[width=55 mm, angle= -90]{f5.eps}
\includegraphics[width=55 mm, angle= -90]{f6.eps}
\includegraphics[width=55 mm, angle= -90]{f7.eps}\\
\includegraphics[width=55 mm, angle= -90]{f8.eps}
\includegraphics[width=55 mm, angle= -90]{f9.eps}
\includegraphics[width=55 mm, angle= -90]{f10.eps}\\
\includegraphics[width=55 mm, angle= -90]{f11.eps}
\includegraphics[width=55 mm, angle= -90]{f12.eps}
\includegraphics[width=55 mm, angle= -90]{f13.eps}
\caption{\footnotesize{The nucleus in the 43\,GHz image starts
elongating in the east-west direction in August 2018. The nucleus
appears almost fully resolved by November 2019. The eastern
component seems to veer towards the large scale southerly directed
jet in the 2020 images. The details of the degree of degradation
of many compromised observations in 2019 can be found in Table
1. The June and July 2019 images reflect the significant
degradation indicated in Table 1. The contours start from 10
mJy/beam and increase in steps of 2x in all the plots. The
restoring beam is indicated by the cross in the lower left.} }
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=40 mm, angle= 0]{f14.eps}
\includegraphics[width=40 mm, angle= 0]{f15.eps}
\includegraphics[width=40 mm, angle= 0]{f16.eps}
\includegraphics[width=40 mm, angle= 0]{f17.eps}
\includegraphics[width=40 mm, angle= 0]{f18.eps}
\includegraphics[width=40 mm, angle= 0]{f19.eps}
\includegraphics[width=40 mm, angle= 0]{f20.eps}
\includegraphics[width=40 mm, angle= 0]{f21.eps}
\includegraphics[width=40 mm, angle= 0]{f22.eps}
\includegraphics[width=40 mm, angle= 0]{f23.eps}
\includegraphics[width=40 mm, angle= 0]{f24.eps}
\includegraphics[width=40 mm, angle= 0]{f25.eps}
\caption{\footnotesize{East-west intensity cross sections created from
the CLEAN images. They are fixed in the north-south direction by
locating them to pass through the peak intensity. Notice the
emergence of a local maximum to the east of the peak on August 26,
2018. A double peak is clearly seen on October 15, 2018. A third
local maximum emerges in October 2019. Notice that January 10,
2019 has a different cross-section with just a single peak. This
epoch requires special attention (see Appendix B).}}
\end{center}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[width=85 mm, angle= 0]{f26.eps}
\includegraphics[width=85 mm, angle= 0]{f27.eps}
\includegraphics[width=85 mm, angle= 0]{f28.eps}
\includegraphics[width=85 mm, angle= 0]{f29.eps}
\caption{The location of the nuclear CCs in four epochs. The two
examples in the top row are used to illustrate our methods of
defining which CCs determine the WN and EN. The WN and EN are
defined by the CCs enclosed within red and blue ellipses,
respectively. The red and blue triangles are the
corresponding centroids of the flux density for each physical
component. The details of these constructions are provided in the
text. Note that the CCs outside the solid ellipses are weak
(20$-$35\,mJy) and do not affect the centroid estimates, but we cut
the CC distribution off once it approaches 0.1\,mas in east-west
width as motivated in the text. The two examples in the bottom row
show the emergence of a weaker (green) component in the fall of
2019. We consider EN as the first ejection, ejection A, and the
green cluster of CCs as a second ejection, ejection B.}
\end{center}
\end{figure}
\section{Method of Estimating Nuclear Expansion}
It is fortuitous that the nucleus expands primarily in the east-west
direction, this is also the direction of maximum resolution of the
VLBA at 43\,GHz (the major axis of the restoring beam has a position
angle of $-10^{\circ}$). The full width at half maximum (FWHM) of the
beam is approximately 0.28\,mas\,$\times$\,0.15\,mas for uniformly
weighted $(u,v)$ data. However, when the data have sufficiently high
signal-to-noise ratio, the interferometer can resolve structures
significantly smaller than the nominal beam size. In our case the
longest baselines provide a typical resolution of approximately
0.08\,mas in the east-west direction. The resolution of the
interferometer is defined here as the FWHM of a circular Gaussian
brightness distribution that gives a visibility amplitude of 50\%
times the zero-baseline value. Mathematically, this is expressed as
$\rm{resolution} = (91 M\lambda/$baseline-length)\,mas
\citep{mar85}. It is known that higher resolution can be
achieved by analyzing the visibility domain and these resolution
limits depend on the noise in the $(u,v)$ plane \citep{mar12}. However,
working in the image plane has the advantage of less ambiguity as to
whether the model chosen for the source is appropriate
\citep{fom99}. Since we are measuring small changes to a small
separation on a complex background, the ambiguities in the models
can cause a significant degradation of accuracy. In Appendix A, we provide
simulations in the visibility domain to motivate the \citet{mar85}
resolution limit as appropriate to this study. These simulations incorporate all the details of the observations that degrade the very high resolution limits achievable with model-fitting the visibility data with near perfect circumstances. We incorporate the $(u,v)$ coverage, SNR, and the complexity of the source structure.
Even though this analysis is in the $(u,v)$ plane,``analysis of a properly made CLEAN image via good analysis techniques should produce values and errors which are equal to those of model fitting visibility data"\citep{fom99}. The CLEAN component based method utilized in this paper will be shown to be an example of a ``good analysis technique" in the image plane. We are able to find Gaussian fitting methods in Section 5 that yield similar resolution limits to our CLEAN component based methods. This forms a bridge between the simulations in Appendix A and the CLEAN component based method presented here.
So, in theory, there is information within the observations that can
resolve east-west features larger than 0.1\,mas, provided their
signal-to-noise ratio (SNR) is high. The 0.28\,mas\,$\times$\,0.15\,mas uniformly weighted restoring beam is
too large to do this cleanly, the resultant blurring prevents a
clear east-west determination. The images in Figure~2 show this to be
the case as there appears to be two heavily blurred components that
are separating over time. We need to extract these partially resolved
structures from the observations in order to quantify the east-west
separation over time. There are three possible methods, super-resolved
images based on a restoring beam smaller than the central peak of the conventional interferometer beam, direct analysis of the
clean component (CC) model, and fitting the visibilities with simple
models of the core structure. The super-resolved images highlight the
multiple nucleus, but create large artifacts at angles more than
$45^{\circ}$ from the east-west direction. The net result is a
suggestive image, but Gaussian fits to the multiple components fail to
converge to a unique decomposition. The same circumstance exists with
efforts to fit the visibilities with simple Gaussian models (see
Section 5). The other alternative is to use the CC models. Even though
the models cannot be considered robust in a single epoch, they should
be suitable for identifying a trend that persists over time,
especially with regards to very bright features as is the case
here. This is the primary guiding principle of the following analysis.
\begin{deluxetable*}{cccccc}
\tablecaption{Compromised Observations in 2019}
\tabletypesize{\small}
\tablehead{\colhead{Date} & \colhead{Missing Stations} & \colhead{Severely Compromised Stations} & \colhead{Comments} & \colhead{Quality}}
\startdata
02/03/2019 & MK and BR & ... & east-west resolution severely degraded & un-useable \\
02/08/2019 & NL & BR and HN much data deleted & only $\sim 7$ stations disperses the & un-useable \\
& & due to bad weather & distribution of CCs in image plane & \\
03/31/2019 & HN & LA missing half of the data & good east-west resolution & useable \\
& & & from SC and MK, low SNR & \\
05/12/2019 & SC & ... & MK and HN still provided & useable \\
& & ... & good east-west resolution & \\
06/18/2019 & SC & ... & inaccurate pointing of & useable \\
& & ... & the antennae, low SNR & \\
07/01/2019 & SC & ... & inaccurate pointing of & useable \\
& & ... & the antennae, low SNR & \\
10/19/2019 & BR & ... & low SNR & useable \\
\enddata
\tablecomments{Details of the observations generously provide by
A. Marscher, 2019. Station acronyms: MK (Mauna Kea), BR (Brewster,
Washington), NL (North Liberty, Iowa), HN (Hancock, New Hampshire),
LA (Los Alamos, New Mexico), SC (St. Croix)}
\end{deluxetable*}
Before describing our models we first note that most observations in
2019 are degraded to varying degrees by bad weather, down observing
stations and inaccurate antennae pointing. The issues are summarized
in Table 1. Modest degradation of observations in 2019 adversely
affect the CC models and one of the main tasks of this analysis is to
quantify this degradation in terms of an uncertainty in our results.
In order to transition from the images in Figure~2 to the CC models,
we investigate intensity profiles of the CLEAN image along a
particular direction (east-west). We define a pixel size as
0.005\,mas\,$\times$\,0.005\,mas. We center an array of pixels on the
peak of the intensity. We place a circular Gaussian beam with a
0.1\,mas FWHM at the location of peak intensity and measure the
intensity. This is about 25\% larger angular resolution than the
angular resolution of the longest baselines of the array. We then
slide the circular Gaussian beam to the east and the west in discrete
0.005\,mas increments in order to generate the intensity
cross-sections in Figure~3. The intensity profiles were generously
provided by Alan Marscher. There is a clear local maximum to the east
of the peak on August 26, 2018. This corresponds to the slight
elongation towards the east in the first panel of Figure~2. A double
peak is clearly seen on October 15, 2018 and December 9, 2018 which
appears as increasing elongation of the nucleus towards the east in
Figure~2. The spacing between the peaks keeps increasing and on March
31, 2019 and May 12, 2019 there is clearly two distinct components in
the image plane. Notice that January 10, 2019 has a different
cross-section with just a single peak. This is out of family with the
other epochs, including the epoch just before and the epoch just after
in December and in March, respectively. Consequently, we analyze this
observation separately in Appendix B in order to assess the reason
that the uniform cross-sectional procedure failed to pick up the
second peak and nature of the root cause.
\section{The CLEAN Component Models}
CC models are provided for each observation epoch on the BU
website. During the first year of monitoring in Figure 2, the western
component of the double nucleus (WN, hereafter) is brighter and
fortunately is predominantly a tight east-west distribution of the
bright CCs that always includes the brightest CC. We face two primary
technical issues that we illustrate in Figure~4 with four
examples. The issues are:
\begin{enumerate}
\item How should the cluster of components be defined for both the
eastern component of the nucleus (EN, hereafter) and the WN?
\item How do we define the coordinates of the cluster of components
and the uncertainty in this location?
\end{enumerate}
We explore these issue below.
\begin{deluxetable*}{cccccccccccc}
\tablecaption{Details of the CLEAN Component Models of the Double Nucleus}
\tabletypesize{\tiny}
\tablehead{
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)}
& \colhead{(7)} & \colhead{(8)} & \colhead{(9)} & \colhead{(10)} & \colhead{(11)} \\
\colhead{Date} & \colhead{Day} & \colhead{$F_{\nu, \mathrm{WN}}$} & \colhead{$F_{\nu,\mathrm{EN}}$} &
\colhead{Uniformly Weighted Beam} & \colhead{$I_\mathrm{peak}$} & \colhead{Min.\,Neg.}
& \colhead{Smallest} & \colhead{$\Delta \rm{RA}$} & \colhead{$\Delta \rm{dec}$}
& \colhead{Separation} \\
& & & & \colhead{(Naturally Weighted Beam)} & & & & & & \\
\colhead{} & \colhead{} & \colhead{} & \colhead{} &
\colhead{Size/PA} & \colhead{} & \colhead{Cont.}
& \colhead{CC} & \colhead{} & \colhead{} & \colhead{} \\
\colhead{} & \colhead{} & \colhead{(Jy)} & \colhead{(Jy)} & \colhead{(mas/degrees)} & \colhead{(Jy/beam)} &
\colhead{(mJy/beam)} & \colhead{(mJy)} & \colhead{(mas)} & \colhead{(mas)} & \colhead{(mas)}
}
\startdata
08/26/2018 & 0 & 1.448 & 0.479 & 0.303 x 0.179/+11.0 & 1.572 &-15 &20 & $0.116 \pm 0.013$& $0.010\pm 0.021$ & $0.116 \pm 0.013$ \\
& & & & (0.370 x 0.231/+13.3) & & & & & & \\
10/15/2018 & 50 & 1.856 & 0.838 & 0.314 x 0.154/+3.6 & 2.462 & -14 &25 & $0.133 \pm 0.011$ & $0.001\pm 0.022$ & $0.133 \pm 0.011$ \\
& & & & (0.352 x 0.175/+1.1) & & & & & & \\
12/09/2018 & 105 & 2.707 & 1.441 & 0.345 x 0.166/+14.9 & 2.875 & -10 & 20 & $0.144 \pm 0.013$ & $-0.030\pm 0.024$ & $0.147 \pm 0.014$ \\
& & & & (0.381 x 0.190/+10.3) & & & & & & \\
01/10/2019 & 137 & 3.349 & 1.079 & 0.361 x 0.160/+21.4 & 2.974 & -15 & 20 & $0.147 \pm 0.014$ & $-0.075\pm 0.024$ & $0.165 \pm 0.017$ \\
& & & & (0.405 x 0.187/+21.0) & & & & & & \\
03/31/2019 & 217 & 1.325 & 0.901 & 0.306 x 0.153/+8.9 & 2.204 & -24 & 25 & $0.175 \pm 0.011$ & $-0.031\pm 0.021$ & $0.178 \pm 0.012$ \\
& & & & (0.344 x 0.185/+7.4) & & & & & & \\
05/12/2019 & 259 & 1.356 & 0.894 & 0.407 x 0.176/-13.2 & 2.064 & -17 & 20 & $0.178 \pm 0.014$ & $-0.045\pm 0.028$ & $0.183 \pm 0.015$\\
& & & & (0.435 x 0.197/-10.8) & & & & & & \\
06/18/2019 & 296 & 2.337 & 0.799 & 0.397 x 0.183/-16.4 & 2.462 & -29 & 35 & $0.173 \pm 0.015$ & $0.003 \pm 0.027$ & $0.173 \pm 0.015$\\
& & & & (0.432 x 0.211/-14.5) & & & & & & \\
07/01/2019 & 309 & 3.448 & 2.116 & 0.425 x 0.173/-14.3 & 4.030 & -36 & 35 & $0.149 \pm 0.014$ & $0.019 \pm 0.029$ & $0.150 \pm 0.014$\\
& & & & (0.434 x 0.210/-12.9) & & & & & & \\
10/19/2019 & 419 & 0.793 & 0.823 & 0.329 x 0.142/-5.2 & 1.840 & -29 & 35 & $0.193 \pm 0.010$ & $-0.043\pm 0.023$& $0.198 \pm 0.011$\\
& & & & (0.386 x 0.159/-8.3) & & & & & & \\
11/03/2019 & 434 & 0.973 & 0.988 & 0.318 x 0.153/+3.6 & 1.490 & -13 & 20 & $0.206 \pm 0.011$ & $-0.030\pm 0.022$ & $0.208 \pm 0.011$\\
& & & & (0.387 x 0.183/+0.6) & & & & & & \\
01/04/2020 & 496 & 1.311 & 1.268 & 0.330 x 0.165/+2.1 & 1.633 & -13 & 20 & $0.220 \pm 0.012$ &$-0.067\pm 0.023$ & $0.230 \pm 0.013$\\
& & & & (0.399 x 0.205/+0.1) & & & & & & \\
04/07/2020 & 590 & 1.634 & 1.613 & 0.316 x 0.144/+4.5 & 2.127 & -10 & 20 & $0.238 \pm 0.010$ & $-0.068\pm 0.022$ & $0.248 \pm 0.012$\\
& & & & (0.342 x 0.160/+6.4) & & & & & &
\enddata
\end{deluxetable*}
\subsection{Defining the Clustering of the CLEAN Components}
This is a slightly subjective exercise at times in which we use the
visual evidence from Figures~2 and 3 and the structure of the
components in the previous epoch and the subsequent epoch. The peak
surface brightness is about $1.5 - 3.0$\,Jy/beam in each image. We
consider any CC with a flux density larger than $\sim 20-35$\,mJy as
significant. This varies depending on the quality of the
observation. This choice typically scales with the magnitude of the
strongest negative contour in the FITS image file (see Table~2). In
practice, components less than 40\,mJy never contribute significantly
to our flux density related results (component centroids and
uncertainties). Figure~4 plots all the components larger than 20\,mJy
in the vicinity of the nucleus in four epochs. The red ellipse defines
all the components that we associate with the WN and the blue ellipse
encapsulates all the components associated with the EN. We
crudely estimate that the east-west spread of the CCs in the WN and
EN will be distributed within a length $< 0.1$\,mas. This follows
from de-convolving the 0.1\,mas FWHM restoring beam from the FWHM of
the intensity cross-sections peaks in Figure~3. We get 20 estimates
of the FWHM of peaks that range from 0.023\,mas to 0.112\,mas with a
mean of 0.076\,mas. Truncating the east-west width of the CC
distribution $< $0.1\,mas is the fundamental constraint for
estimating which components should be included in the WN and
EN. This approximate limit guarantees that the size of the
distributions of CCs in the EN and WN are small enough that the
nuclear components can be clearly differentiated in our efforts to
estimate separations as small as 0.1\,mas. If this condition is not
obeyed by the data then the CC clustering method cannot be used to
measure the smallest separations, $\sim 0.1$\,mas. A priori we do
not know if this assumption is consistent with the data. Ultimately,
this assumption is verified empirically. For example, we present 7
CC scatter plots of the nuclear region in Figures~4, 7, 8 and
9. There is a strong tendency for the adjacent background to be
almost devoid of any CCs above 20\,mJy in the surrounding regions as
well as regions that exist between clusters of strong CCs $\leq
0.1$\,mas across in the east west direction. The north-south
resolution is worse making the constraint on the north-south
distribution of CCs comprising the WN and EN more subjective in
nature. The north-south distributions of CCs in Figures~4, 7, 8 and
9 also seem to naturally cutoff at a size $ < 0.15$\,mas, similar to
the resolution of the longest north-south baselines ($\sim 0.13 -
0.14$\,mas).
The details of our CC models of the WN and EN can be found in
Table~2. The first two columns are the date of the observation and the
number of days since the nuclear separation was first detected. This
is not the same as the time from the epoch of physical separation due
to the finite resolution of the array. The next two columns are the
flux densities of the WN, EN and the total of these numbers,
respectively. The absolute flux calibration accuracy of the BU data is
about 5\% \citep{jor17}\footnote{Normally, the VLBA-BU-BLAZAR
fluxes are accurate to within $\sim5\%$, but from early 2019 to
Septamber 2019 it is quite a bit more uncertain, at least 30\%,
because of pointing errors. Given the diffuse nature of 3C 84, we
can only roughly estimate corrections through use the Mets\"ahovi monitoring program fluxes at 37\,GHz \citep[for a
description of the Mets\"ahovi monitoring, see][]{ter04}. The
VLBA resolves out too much of the structure to do that accurately,
as we can for more compact sources.}. Column 5 provides
the properties of the uniformly weighted beam on top and the
naturally weighted beam below in parenthesis. Column (6) is the
peak intensity from the CLEAN images (Figure 2) for comparison with
the numbers in the previous columns. Columns (7) and (8) are the
largest magnitude negative contour values from the CLEAN image and the
CC cutoff used in or models, respectively. Columns (9) - (11) are the
uncertainties in the nuclear separation that are computed per the
methods of Section~3.2.
First, let us consider the epoch of August 26 2018. Since the absolute
astrometry is lost in the phase self-calibration step of the VLBI data
reduction, we are only interested in the relative positions of the EN
and WN and we place the brighter WN at the origin. We include
all the CCs within a region $\leq 0.1$\,mas in the east-west
direction. The blue (red) ellipse encircles all the CCs that are
associated with the EN (WN). The EN involves considerably weaker CCs
than the WN and the distribution tends to be more elongated in the
north-south direction. This is typical during the first year of our
monitoring of the time evolution of the nuclear region. The May 12
2019 observation in the top righthand side of Figure~4 shows a larger
separation between the EN and WN.
The bottom two panels of Figure~4 show the emergence of a third
component in the fall of 2019. It was clearly seen in the October 2019
observation, but the SNR was poor so we show the high SNR epochs of
November 2019 and January 2020. The bottom right hand panel shows that
the centroids of the three clusters of CCs (see the next subsection)
are nearly colinear. Thus, it makes sense to consider them all part of
the same flow. As such, we tentatively relabel the three components as
follows. Based on the early epochs, the WN is associated with the
point of origin. It is the brightest feature in 2018 and the much
weaker EN seems to flow out of the partially resolved cluster in the
time frame of August to December. We therefore consider the EN as the
first ejection, or ejection A. In the fall of 2019, the new feature
shown inside the green ellipse is about half the brightness of the EN
and WN, we call it ejection B. We actually never see it separate from
the WN, so this scenario is not verified, but suggested by the pattern
evolution. We cannot rule out that the red ellipse represents
a new ejection in the western direction and the green component is the
point of origin. This is disfavored because it would require the EN to
be contracting towards the point of origin and the point of origin
would change from being a much brighter component than the EN to much
fainter than the EN.
\begin{figure}
\begin{center}
\includegraphics[width=87 mm, angle= 0]{f30.eps}
\includegraphics[width=87 mm, angle= 0]{f31.eps}
\includegraphics[width=87 mm, angle= 0]{f32.eps}
\caption{\footnotesize{The separation between WN and EN versus time is
demonstrated in two ways. The top left panel is a plot of
total displacement versus time. The east-west displacement is
plotted in the top right panel. The two methods yield a
similar separation velocity. The solid black line is a least
squares fit with uncertainty in the vertical variable
\citep{ree89}. The dashed lines define the standard error of the
fit. The equation for the best fit is printed on the plot. We also
provide a crude fit for ejection B in green. The top
right panel also plots the east-west separation from the
intensity profiles (in red) with no information from January 10,
2019 available from Figure~3. We consider these data less rigorous
and are shown as a consistency check. The peak separation of the
intensity cross-sections lies below the uncertainty of the CC
model fit for April 2020 (day 590). This is a consequence of a
curved trajectory of the EN. As can be seen in Figure 2, the EN
starts veering south relative to the declination of the WN in 2020
and is most pronounced in April. Thus, there is no east-west cross
section that captures intensity peaks of both the EN and WN. The
April data indicates the breakdown of the simple east-west cross
section method for a curved trajectory for the
ejection. For completeness, we include the southern
displacement of the EN with its fit in the bottom panel. The
magnitude of the trend is small compared to the uncertainties
and this is reflected in the standard error of the best
fit. Thus, the result should be viewed with caution. The three
outliers are January 10, 2019 (that is discussed in Appendix B)
and the two low SNR observations in June and July 2019.}}
\end{center}
\end{figure}
\subsection{The Position of the Features}
We calculate the position of the features by taking the flux-density
weighted mean (centroid) of the CC positions in the feature. In Figure
4, the red triangle inside the red ellipse and the blue triangle inside the blue ellipse are the locations of the
centroids of the flux density of the CCs encapsulated within each
ellipse. We assign an uncertainty to the centroid location. The origin of the coordinates will be the centroid of the WN and all displacements are measured relative to this. The uncertainty in the positional locations is $\sim 10\%$ of the
synthesized beam FWHM for these bright components
\citep{lis09} \footnote{see also
https://science.nrao.edu/facilities/vla/docs/manuals/oss/performance/positional-accuracy}. In
Table 2, the uncertainties in coordinate separation, $\sigma_{\Delta
X}$ and $\sigma_{\Delta Y}$ ($X$ is right ascension and $Y$ is
declination), are the uncertainties on the individual centroid
coordinates added in quadrature. Similarly, the uncertainty of the
total displacement between the EN and WN, $\sigma_{D}$, is calculated
as the error propagation of the centroid uncertainties. The random noise contribution to the separation uncertainty can be
estimated by $\sigma_{D} (\rm{noise})\approx$ 0.5 d
$\sqrt{\sigma_{\rm{rms}}/S_{\rm{peak}}}$, where $\sigma_{\rm{rms}}$
is the post-fit rms noise of the image and $S_{\rm{peak}}$ is the
flux density of a putative unresolved component of size $d$
\citep{fom99,lee08}. $\sigma_{\rm{rms}}<$ 1 mJy for all epochs
except June and July 2019 where it is closer to 2 mJy. We determine
in Table 2 that $S_{\rm{peak}}> 479$ mJy so all the SNR are $> 500$
except June 2019 which is $\gtrsim 400$. It is therefore clear
that positional uncertainty is not dominated by the random noise
component. The error estimate adopted in Table~2 is much more conservative than
stochastic noise errors and incorporates systematic uncertainty.
\section{The Separation Rate of the Double Nucleus}
The motivation for defining the locations of the EN and WN and the
corresponding uncertainty is to track the time evolution of the
separation of the components. This is performed in Figure~5. The top left panel is the plot of the displacement versus
time. However, we note that the displacement is primarily east-west
and the highest resolution is in this direction. We plot east-west (x
direction) displacement in the top right panel for direct
comparison to the intensity cross-sections in Figure~3.
The data were fit by a least squares fit with uncertainty in the
vertical variable \citep{ree89}. The solid line is the fit and the
dashed lines represent the standard error to the fit. Based on the fit
and standard error, the time averaged nuclear separation speed is
$0.086 \pm 0.008$\,c. The fit to the total displacement yields a
separation rate of $73 \pm 7$\,$\mu\rm{as}$ per year compared to $70
\pm 6$\,$\mu\rm{as}$ per year for pure east-west displacement. Note
that we also plot (in green) the separation of the second ejection,
ejection B, in the lower right hand corner of the two panels. We also
provide very crude four point fits to the ejection velocity.
It is apparent that the EN-WN separations in the spring and summer of
2019 (days 217 $-$ 309) deviate from the least squares fits in
Figure~5. This is not unexpected from the Tables~1 and 2. These are
the most compromised observations. However, the July 2019 estimated
separation lies particularly far from the fit. It is possible to
explain this behaviour to be due to the ejection``B" that is shown in
the bottom frames of Figure~4. This central component is resolved from
EN and WN in our CC cluster models first in October 2019, but it is
most likely present already in earlier epochs. If this is indeed the
case than we may just not resolve it in July 2019, but it is there and
it blends with WN, moving its measured position slightly towards
EN. This can cause the small inward motion seen at that epoch. The
results of this paper do not rely on this interpretation.
By considering the east-west displacement, we are able to directly
compare the results of the intensity profile description in Figure~3
with the CC models of the EN and WN in Section~3. The
distribution of intensity cross-section separations is generally
consistent with both the standard error of the least squares fit and
the uncertainty of the CC model fits to the EN-WN separation (the
measurement uncertainty given in Table~2) except for two regions, in
June and July 2019 for which the SNR was poor and the last two
points, January and April 2020. Figure~2 and the bottom frame of
Figure 5 seem to indicate that beginning in January 2020, the EN no
longer appears to follow a predominantly east-west trajectory, but
begins to veer southward towards the southerly directed large scale
jet. The EN appears to propagate in a southeasterly direction
rather than an easterly direction. Thus, a pure east-west cross
section is unable to properly capture the intensity peaks of both
components. The situation is more extreme in April and this could
cause the measurement using the intensity cross section to be less
than the true east-west displacement. This circumstance illustrates
the limitation of the simple east-west cross section method of
measuring separation if the trajectory curves.
For the sake of completeness, we plot the southern
displacement of the EN as it separates from the WN in the bottom
panel of Figure~5. The putative trend is far less pronounced than
the east-west separation and the uncertainty is much larger (the
synthesized beam is elongated in this direction). Consequently,
there is no robust quantitative assessment that can be made due to
the limitations of the CC based method of this paper.
One can also perform a long extrapolation back in time
(assuming that the separation velocity was approximately constant in
time) of the linear fits in order to find the epoch of zero
separation from the nucleus. From Figure~5, we have that ejection A,
the WN, emerged on January 1, 2017 with a large uncertainty that
places it anywhere between September 21, 2016 and March 22, 2017. We
can crudely estimate the zero separation epoch for ejection B from
Figure~5 as $\sim$\,November 6, 2017. The two ejections originated at
least 7 months apart and represent two distinct nuclear events. Our
estimate of the January 1, 2017 ejection is not tightly constrained,
but we would be remiss not to mention that there was a major gamma
ray flare reaching TeV energies that was detected by the Large Area Telescope on board the Fermi satellite
and MAGIC telescopes that peaked around December 31, 2016 to January
1, 2017 \citep{bag17,ans18}.
\section{The Double Nucleus in the uv-Plane}
It is customary for astronomers to fit the visibility data
with a small number of Gaussians in the order to track component
separations. Thus, we explore the possibility of evidence of the
double nucleus in the $(u,v)$ plane by making Gaussian fits to the
nuclear region. However, there are two concerns for this method in the
context of the separating nuclear features in
Figure~2. Firstly, the radio source 3C 84 is extremely complex
compared to a typical blazar nucleus and it is extended. So,
one needs to use a CC model to represent it, but that creates a
problem regarding how large of an area around the core should be
cleared from CCs before model-fitting. This is always a bit subjective and can influence the results
significantly. The second issue is that the initial stages of
the expansion in Figure~2 are at the limits of the resolution of the
VLBA. There is no unique solution to the fitting
method. Each ambiguity and assumption adds a possible error to
the fitting process. After some initial attempts to make Gaussian
fits to the nuclear region we found two methods that give reasonable
results.
\subsection{Gaussian Fitting Method 1}
The first method addresses the issue of the complex background
emission by a simple excision method that can be uniformly applied
to each epoch. We took the CLEAN model and automatically removed CCs
from the area three times the uniformly weighted beam size (from
Table~2) around the core. The excised region is then modelled with
circular Gaussians. We added new components in the model if a
clear peak was seen in the residual image after fitting. The
background subtraction is not perfect so it introduces additional
Gaussian components in addition to those associated with the EN, WN
and ejection B. This method typically needs many components to
represent the core, which can make the fit unstable.
\begin{deluxetable*}{cccccccccc}
\tablecaption{Gaussian Fits to the Double Nucleus}
\tabletypesize{\tiny} \tablehead{ \colhead{(1)} & \colhead{(2)} &
\colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} &
\colhead{(7)} & \colhead{(8)} & \colhead{(9)} \\ \colhead{Date} &
\colhead{Model} & \colhead{$F_{\nu, \mathrm{WN}}$} &
\colhead{$F_{\nu,\mathrm{EN}}$} & \colhead{$\Delta \rm{RA}$} &
\colhead{$\Delta \rm{dec}$} & \colhead{Separation} &
\colhead{Gaussian FWHM WN} & \colhead{Gaussian FWHM EN}
\\ \colhead{} & \colhead{} & \colhead{(Jy)} & \colhead{(Jy)} &
\colhead{(mas)} & \colhead{(mas)} & \colhead{(mas)} &
\colhead{(mas)} & \colhead{(mas)} }
\startdata
08/26/2018 & 1 & 1.137 & 1.437 & 0.093 & 0.046 & 0.104 & 0.073 & 0.156 \\
08/26/2018 & 2 & 1.146 & 0.840 & 0.135 & 0.001 & 0.135 & 0.068 & 0.102 \\
10/15/2018 & 1 & 2.525 & 1.138 & 0.157 & -0.053 & 0.166 & 0.104 & 0.127 \\
10/15/2018 & 2 & 2.219 & 1.459 & 0.146 & -0.008 & 0.146 & 0.088 & 0.134 \\
12/09/2018 & 1 & 3.211 & 1.435 & 0.166 & -0.027 & 0.168 & 0.113 & 0.052 \\
12/09/2018 & 2 & 2.819 & 1.856 & 0.151 & -0.034 & 0.155 & 0.085 & 0.082 \\
01/10/2019 & 1 & 1.179 & 4.634 & 0.139 & 0.019 & 0.140 & 0.085 & 0.190 \\
01/10/2019 & 2 & 2.030 & 2.321 & 0.127 & 0.018 & 0.128 & 0.076 & 0.072 \\
03/31/2019 & 1 & 1.193 & 1.576 & 0.169 & -0.009 & 0.169 & 0.068 & 0.132 \\
03/31/2019 & 2 & 1.819 & 0.932 & 0.166 & 0.022 & 0.167 & 0.066 & 0.071 \\
05/12/2019 & 1 & 2.735 & 0.593 & 0.200 & -0.007 & 0.200 & 0.143 & 0.018 \\
05/12/2019 & 2 & 2.601 & 0.786 & 0.199 & -0.043 & 0.204 & 0.119 & point source \\
06/18/2019 & 1 & 2.794 & 0.970 & 0.170 & -0.010 & 0.170 & 0.091 & 0.011 \\
06/18/2019 & 2 & 2.470 & 0.795 & 0.151 & 0.018 & 0.152 & 0.074 & 0.107 \\
07/01/2019 & 1 & 2.716 & 1.961 & 0.162 & 0.007 & 0.162 & 0.019 & 0.055 \\
07/01/2019 & 2 & 3.876 & 2.032 & 0.151 & 0.028 & 0.153 & 0.071 & 0.029 \\
10/19/2019 & 1 & 2.505 & 1.629 & 0.183 & -0.066 & 0.195 & 0.155 & 0.100 \\
10/19/2019 & 2 & 1.284 & 1.454 & 0.221 & -0.026 & 0.223 & 0.091 & 0.092 \\
11/03/2019 & 1 & 0.688 & 1.041 & 0.234 & -0.040 & 0.237 & 0.063 & 0.053 \\
11/03/2019 & 2 & 1.193 & 1.120 & 0.222 & -0.031 & 0.224 & 0.061 & 0.042 \\
01/04/2020 & 1 & 1.370 & 1.109 & 0.232 & -0.066 & 0.242 & 0.104 & 0.059 \\
01/04/2020 & 2 & 1.222 & 1.523 & 0.240 & -0.064 & 0.249 & 0.075 & 0.081 \\
04/07/2020 & 1 & 2.517 & 0.706 & 0.264 & -0.111 & 0.286 & 0.144 & point source \\
04/07/2020 & 2 & 1.878 & 1.286 & 0.262 & -0.080 & 0.274 & 0.109 & 0.046
\enddata
\end{deluxetable*}
\subsection{Gaussian Fitting Method 2}
We took the CLEAN model, but removed only those CCs that
roughly correspond to the double (triple) nucleus in a
super-resolved image (created with a 0.1\,mas circular beam). This
CC excision was done ``by eye", so there is some subjectivity
here. We modelled this region with 2 or 3 circular Gaussian
components. The method is less subjective than the first as a result
of incorporating additional information from the super-resolved
image. However, the method relies significantly on the CLEAN model,
and it is therefore not independent.
\begin{figure}[htp!]
\begin{center}
\includegraphics[width=120 mm, angle= 0]{f33.eps}
\includegraphics[width=120 mm, angle= 0]{f34.eps}
\footnotesize{\caption{The Gaussian components produced by the Difmap
fit in the visibility domain are indicated by blue and orange
symbols, for Method 1 (described in Section~5.1) and Method 2
(described in Section~5.2). These are overlayed on the plots from
Figure~5. Method 1A and 2A points are the distances from the
Gaussian representing the EN from the Gaussian representing the
WN. Similarly, method 1B and 2B points are the distances from the
Gaussian representing the ejection B from the Gaussian
representing the WN. We do not present error bars on the Gaussian
fit data since it would clutter the graph making it
unintelligible.}}
\end{center}
\end{figure}
\subsection{Comparison of Methods}
In Figure~6 we reproduce Figure~5 with the addition of the
separation derived from the Gaussian models of the nuclear region
described above. There are two things to note. First of all, the
east-west separation in 2018 and early 2019 displayed in the bottom
panel deviates much more from the cross-sectional method of Figure 3
than the CC approach used in this paper. Secondly, because Model 2
is derived using super-resolved images, it agrees better with the CC
model data. The Gaussian fits to the EN and WN are listed in Table
3. The two fits can be compared to each other as well as the CC
based method in Table 2. There are some large differences in the
flux density of the EN and WN between the two models in columns (3)
and (4). This might be a consequence of the different methods of
removing the background and the relatively large Gaussian FWHM in
columns (8) and (9) that are comparable to the separations in column
(7). If we perform a linear fit to the separation from the Gaussian
fit we find a similar separation rate for both Gaussian fitting
schemes, about 15\% larger than what was found in Section 4,
$0.100$\,c. This is surprising considering that on a point by point
level the agreement is not that tight.
In Figure~7, the Gaussian components for both models given in
Table~3 are overlayed on CC distribution for two adjacent epochs. We
define EN and WN as in Figure~4. The image is qualitative, but it
gives the reader a feel for the difference between the two model
results and the difference between the models and the CC based
methods. Method 2, which utilizes information from the
super-resolved images, seems very consistent with the EN and WN
identifications in May 2019. Figure~6 and Tables~2 and 3 indicate
some significant discrepancies between the results of the CC method
of Sections 3 and 4 and the Gaussian models during the epoch April
7, 2020. This is most pronounced for ejection B. In Figure~8, we
overlay the Gaussian fits on the CC scatter plot from
Figure~4. There is clearly disagreement between the two
fits. Furthermore, Model 1 seems to blend the EN and ejection B and
Model 2 seems to ignore all the CCs at the north end of the EN. This
suggests that the Gaussian fit methods have difficulty resolving 3
components within a total extent of 0.24\,mas. The figure clearly
shows the origin of the larger displacements for ejection B in the
April 2020 epoch of Figure~6 given by the Gaussian models compared
to the CC based analysis.
The main objective of this study is to track the time
evolution of the double separation as close to its inception as
possible in order to estimate the trajectory and separation speed as
accurately as possible. There is arbitrariness in the Gaussian
fitting procedure that is accentuated in the early epochs (in 2018)
when the component separation was smaller. The intensity
cross-sections were introduced in Figure~3 to provide a clear
diagnostic of the model fitting schemes. Based on Figure~6, both
Gaussian decomposition schemes give worse fits to the east-west
separation derived from the intensity cross-sections in the early
epochs than the CC based methods of Section 3 and 4. We do not
advocate extrapolating the Gaussian fitting method to the first
three epochs of small component separation. Similarly, Figure~8
seems to indicate that the second ejection crowds the narrow
0.24\,mas field making it difficult for the Gaussian fits to
segregate components uniquely or with high accuracy. For this
reason, we consider the CC method of this paper preferable to
conventional Gaussian fitting for exploring the partially resolved
compact nuclear region of 3C\,84. The Gaussian fitting does provide
qualitative agreement with CC model fitting. As such, along with the
intensity cross-sections, this corroborates the results of the CC
based analysis. We have also demonstrated that with proper care in
the definition of the Gaussian models, one can achieve a similar
estimate (within 15\%) to the nuclear separation.
\begin{figure}
\begin{center}
\includegraphics[width=85 mm, angle= 0]{f35.eps}
\includegraphics[width=85 mm, angle= 0]{f36.eps}
\includegraphics[width=85 mm, angle= 0]{f37.eps}
\includegraphics[width=85 mm, angle= 0]{f38.eps}
\footnotesize{\caption{The foreground Gaussian components from
Table~3 are indicated by the dashed black circles with the
inscribed black crosses. Both Model 1 (described in Section 5.1)
and Model 2 (described in Section 5.2) are plotted, but
separately for clarity. The diameter of the circle represent the
FWHM of the Gaussian component. These are superimposed on the CC
component models as defined in Figure~4. The flux densities are
in black near or inside each Gaussian component. The eastern
component of Gaussian Model 2, in May, is a point source and it
is represented by a black cross.}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=85 mm, angle= 0]{f39.eps}
\includegraphics[width=85 mm, angle= 0]{f40.eps}
\footnotesize{\caption{The figure compares the Gaussian models
with the CC derived components during the epoch April 7,
2020. The figure is formatted the same as Figures~4 and 7. The
three bright components in close proximity to each other seems
to provide a challenge for the Gaussian models as evidenced by
the significant differences between the models from methods 1
and 2. The eastern component of Gaussian Model 1 is a point
source and it is represented by a black cross. We have not
included the 0.1\,mas rulers in this figure in order to avoid
distracting clutter.}}
\end{center}
\end{figure}
\section{The Nuclear Separation in the Context of RadioAstron}
In order to create more context for these results in 2018-2020, we
compare the nuclear structure to the triple nucleus observed on
September 21-22, 2013 by RadioAstron \citep{kar13,kar17}. First of
all, the RadioAstron observations were accompanied by a 43\,GHz
observation with an array of the VLBA combined with the phased VLA
\citep[][Savolainen et al., in prep.]{gio18}. We use the clean
component model from imaging of these data together with the methods
of this paper to detect evidence of a triple nucleus using the
east-west resolution of the VLBA. There were no September or October
observations by the BU monitoring program. The top panel of Figure~9
is a nuclear clustering model similar to Figure~4. It shows all the
CCs $>20$ mJy in a nuclear region that is a square, 0.4\,mas on a
side. Without the RadioAstron results, this compact configuration
would have been more ambiguous to decompose into three
components. Based on the RadioAstron images, we cluster the CC
components into three regions of concentration, RA East (blue), RA
Central (red) and RA West (green). The nuclear region is very clean,
there are relatively few CCs compared to the other epochs considered
and the background CC field is sparse near the triple nucleus in the
innermost 0.2 mas. This fortuitous circumstance facilitates the
identification of the components of the triple nucleus. The distance
from RA East to RA Central is $\approx 0.08$ mas and RA Central to RA
West is $\approx 0.09$ mas.
\begin{figure}
\begin{center}
\includegraphics[width=125mm, angle=0]{f41.eps}
\includegraphics[trim={5.3cm 0 0 0}, clip, width=120mm, angle=0]{f42.eps}
\caption{The top panel shows the location of the nuclear CCs
$>20$\,mJy on September 22, 2013 based on a 43\,GHz VLBA plus phased
VLA observation that was quasi-simultaneous with the 22\,GHz RadioAstron observation in the bottom panel. We cluster
the CCs as three physical components RA East (blue), RA Central
(red) and RA West (green). Their positions are overlaid on the
RadioAstron image from \citet{gio18}. We cross-identify the brightest
component, RA Central, with the bright component in 2018/2019, the
WN. Based on this identification, we also show the positions of the
EN in August 26, 2018 and May 12, 2019 relative to the RadioAstron
image in the bottom panel. The contour levels are (-10, -7.50, -3, 7.500, 10, 30, 50, 100, 150, 200, 300, 500)mJy.}
\end{center}
\end{figure}
We overlay the locations of the RA East, RA Central and RA West as
blue, red and green triangles, respectively from the top
panel on the RadioAstron image. The locations appear to be close to
what is expected from the 22\,GHz image. The flux densities of the RA
East, RA Central and RA West are 456\,mJy, 2106\,mJy and 845\,mJy,
respectively in the 43\,GHz CC decomposition of the top panel of
Figure~9. Gaussian fits to the nuclear region of the
RadioAstron data yield flux densities of the RA East, RA Central and
RA West of 564\,mJy, 775\,mJy and 389\,mJy, respectively at 22
GHz. RA Central and RA West have a highly inverted (rising) spectrum
between 22\,GHz and 43\,GHz either from synchrotron self-absorption
or from free-free absorption \citep{wal00}.
We try to assess the degree to which the nuclear region has changed in
terms of morphology and size in 2018-2020 compared to 2013. The
cross-identification of components is rather uncertain due to 5 years
between the observations. We cannot make any robust
cross-identifications of the components from epoch to epoch due to
the known variability of the morphology and the effects of blending
with low resolution. We simply give two plausible
cross-identifications in the bottom panel of Figure~9 and Figure~10
to help elucidate the large qualitative changes that might be
occurring over time. First, we identify the brightest component of
VLBA with the brightest component of RadioAstron. With this chosen
scenario, we tentatively identify RA Central as the same physical
feature as the WN in Figure~4. We also overlay the locations of the EN
in August 26, 2018 and May 12, 2019 on the RadioAstron image in the
bottom panel of Figure~9. In this cross-identification scenario, the
nuclear structure has widened by $\sim 40\mu\rm{as}$ and the whole
pattern is shifted to the east.
\begin{figure}
\begin{center}
\includegraphics[trim={5cm 0 0 0}, clip, width=110mm, angle=0]{f43.eps}
\caption{We consider an alternative (to Figure~9) cross-identification
of components between our 43 GHz VLBA monitoring of the multiple
nucleus and the RadioAstron image. This interpretation is based on
the CC model for January 4, 2020 in the bottom right hand panel of
Figure~4. There is a triple nucleus in that epoch as well. The color
coding of the centroid locations are the same as in Figure~4. One
very significant difference is that in January 2020 the cental
component is the weakest and in September 2013 it is the
brightest. The multiple nucleus is wider in 2020. The contour levels
are (-10, -7.50, -3, 7.500, 10, 30, 50, 100, 150, 200, 300,
500)mJy.}
\end{center}
\end{figure}
The first cross-identification scenario is straightforward,
but just one of many possible choices. Thus, in Figure~10 we show a
different plausible cross-identification based on the high SNR January
4, 2020 observation of the nuclear triple. The color coding of the
triangles corresponds to the color coding in the CC model in
the bottom right hand panel of Figure~4. Note that the
triangles have a different identification than in
Figure~7. In this scenario the nuclear structure has widened by $\sim
60\mu\rm{as}$ and the point of origin of the ejections is the red
triangle at the far west edge of the jet. The triple axis is
tilted relative to September 22, 2013. In summary, both the
interpretation in Figure~9 and Figure~10, indicate a highly dynamic
nuclear region that evolved to a wide state in 2019. The physical
identification of the components is not obvious, but our time
evolution study clearly identified luminous plasma ejected towards the
east.
\section{Conclusion}
In this paper, we consider three methods for resolving the nuclear
structure in 43\,GHz VLBA observations of 3C\,84, intensity
cross-section, Gaussian fitting in the $(u,v)$ plane and CC
models. The time averaged separation speed from August 2018 to April
2020 is estimated as $0.086\pm 0.008$\,c. A second ejected component
was identified in October 2019 to April 2020 with a similar speed.
The multiple nucleus has been detected before with RadioAstron and
86\,GHz global VLBI, but the dynamical evolution has never been seen
previously. This separation speed estimate accomplishes the primary
goal of this paper.
The most relevant comparison are the component measurements at the
base (inner $\sim 0.2$\,mas) of the 43\,GHz VLBA southerly directed
jet \citep{dha98}. They found apparent velocities in the range
0.035\,c $-$ 0.1\,c. However, there was uncertainty in the
identification of the location of nucleus. Independent of any of the
nuclear identification scenarios, the apparent velocity was slow by
relativistic standards $\lesssim 0.1$\,c, very similar to what was
found in this study of the multiple east-west nucleus. The apparent
jet speed accelerates as the jet propagates southward with $\sim
0.25$\,c at $\sim 1$\,mas and the highest estimates approach 0.47\,c
$\sim 2$\,mas away \citep{suz12}.
It is tempting to try to understand the observed motion in the context
of the complicated circum-nuclear structure depicted in
Figure~1. However, we choose not to speculate and use our result as
one piece of the puzzle that must be incorporated into future efforts
to explain the base of the jet in 3C\,84. It is certainly odd to see
an inner flow or expansion that appears to be orthogonal to the large
scale jet that is clearly delineated only $150$\,$\mu$as downstream
\citep{gio18,kim19}. One thing that we can say is that the double
separation is $\sim 0.3$\,lt-yrs on the sky plane in April 2020. This
corresponds to an intrinsic distance of $\approx
0.3/\sin{\theta}$\,lt-yrs, where $\theta$ is the line of sight (LOS)
angle to the component motion. This equates to $\sim
(1900/\sin{\theta}) M$ where $M\sim 10^{9}M_{\odot}\sim 1.5\times
10^{14}\rm{cm}$ is the black hole mass in geometrized units
\citep{nag19,pun18,sch13}. Thus, this slow apparent separation speed
persists very far from the source. The LOS angle has been estimated at
$11^{\circ} - 65^{\circ}$ based on jet/counter-jet asymmetry
\citep{wal94,asa06,lis09,fuj17}. However, these viewing angle
estimates are for the north-south jet. The direction of the east-west
core motion may differ significantly from this. It is perhaps more
relevant that historical data indicates that 3C\,84 sometimes appears
moderately blazar-like based on broad band variability and optical
polarization as high as 6\% \citep{ver78,ang80,chu85,nes95}. In
current nomenclature, it is a slightly ``off angle blazar" at times
\citep{pun18}. For the sake of comparison, estimates for blazars are
typically $\theta \approx 1^\circ - 3^{\circ}$
\citep{hov09,jor17}. However, the blazar-like properties have been far
more benign for the last 35 years \citep{pun18}. As an example, in the
last 12 years the optical polarization has been consistently between
1\%-3\% with the very rare jumps to
4\% \footnote{http://www.bu.edu/blazars/VLBAproject.html}. So we
cannot rule out $\theta \sim 10^{\circ}$, but this behavior does not
favor an extreme blazar-like line of sight. Based on the published
ranges of $\theta$, the physical distance between the EN and WN is
$\sim 0.3- 1.5$ lt-yrs, or $\sim 1900M - 9800$ M. What is extremely
odd is that at this large distance, the VLBA images seem to indicate a
dramatic change in the jet direction. The most straightforward
interpretation is that we are detecting the internal dynamics of a jet
that is $\sim 2000$\,M wide. The lateral expansion near the central
engine proceeds sub-relativistically at $\sim 0.086$c. The jet is
collimated on scales $> 2000$\,M. The RadioAstron image and the April
2020 image in Figure 2 seem to indicate a continuous flow between the
central, wide core component and bright components moving along the
edges of the jet going south. \par Based on RadioAstron and this
analysis, the bifurcating nuclei are bright ejections of plasma that
can go primarily to the east or west. In the epochs considered here
there is an ejection to the east. In 2020, the images in Figure 2
appear to show that the EN trajectory starts to bend toward the
south. It might be joining the highly collimated portion of the jet
that is directed almost due south. However, this is not so simple
based on Figure~2 because the western ridge is much brighter at this
time. The ejection might traverse (twist around) the front face of the
southerly directed jet and join the western ridge.
The discussion above shows how confusing the dynamics of the inner jet
are likely to be. Considering the variations in the ridge brightness
distribution over time, suggests that the jet dynamics are not even
consistent from epoch to epoch. A year long series of target of
opportunity, 43\,GHz relative astrometry VLBA observations of
$>8$\,hours with two calibrators triggered by a $>0.15$\,mas
separation of the nuclear double could be a feasible method of
tracking the time evolution of the ejections accurately. The
observations could be triggered by a separation determined by the
methods of this paper from the BU data. Perhaps 86\,GHz observations
with global VLBI (including the Atacama Large Millimeter Array) spread
out over 6 months that are triggered simultaneous with the VLBA
observations might shed light on what kind of dynamics are occurring.
These are challenging observations, but this might be one of our best
laboratories for studying the base of a powerful extragalactic jet.
\begin{acknowledgments}
This study makes use of 43\,GHz VLBA data from the VLBA-BU Blazar
Monitoring Program VLBA-BU-BLAZAR funded by NASA through the Fermi
Guest Investigator
Program\footnote{http://www.bu.edu/blazars/VLBAproject.html}. The Very Long Baseline Array (VLBA) is an instrument of the National Radio Astronomy Observatory. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated by Associated Universities, Inc. TS was supported by the Academy of
Finland projects 274477 and 315721. HN is supported by JSPS KAKENHI grant No. JP18K03709.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,876 |
Q: openshift & optaplanner rostering I have been following instructions at https://github.com/kiegroup/optashift-employee-rostering (readme). I was able to deploy roster app in openshift and dont see any errors in log. I have also created external route (localhost).
but when i access roster application as http://localhost/roster I get error "Not Found". But i see pod running and service active.
Any help on this?
A: Changed the "route" path from "/roster" to "/"
Now I can access roster application using http://localhost/gwtui/gwtui.html
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,856 |
\section{Introduction}
Every morning, we agonize in front of the closet over what to wear, how to dress up, and imagine ourselves with different clothes on. To generate mental images~\cite{eysenck06} of ourselves wearing clothes on a hanger is an effortless work for our brain. In our daily lives, we ceaselessly perceive visual scene or objects, and often transfer them to different forms by the mental imagery.
Our focus of this paper lies on the problem; to enable a machine to transfer a visual input into different forms and to visualize the various forms by generating a pixel-level image.
Image generation has been attempted by a long line of works~\cite{hinton06,salakhutdinov09,vincent08} but generating realistic images has been challenging since an image itself is high dimensional and has complex relations between pixels. However, several recent works have succeeded in generating realistic images~\cite{goodfellow14,gregor15,sohl15,theis15}, with the drastic advances of deep learning. Although these works are similar to ours in terms of image generation, ours is distinct in terms of \textit{image-conditioned image generation}. We take an image as a conditioned input lying in a domain, and re-draw a target image lying on another.
In this work, we define two domains; a source domain and a target domain. The two domains are connected by a semantic meaning. For instance, if we define an image of a dressed person as a source domain, a piece of the person's clothing is defined as the target domain. Transferring an image domain into a different image domain has been proposed in computer vision~\cite{saenko10,kulis11,gopalan11,oquab14,chen15,huang15}, but all these adaptations take place in the feature space, i.e. the model parameters are adapted. However, our method directly produces target images.
We transfer a knowledge in a source domain to a pixel-level target image while overcoming the semantic gap between the two domains. Transferred image should look realistic yet preserving the semantic meaning. To do so, we present a pixel-level domain converter composed of an encoder for semantic embedding of a source and a decoder to produce a target image. However, training the converter is not straightforward because the target is not deterministic~\cite{yan15}. Given a source image, the number of possible targets is unlimited as the examples in \Fref{fig:nonunique} show. To challenge this problem, we introduce two strategies as follows.
\begin{figure}[t]
\centering
\begin{tabular}{ccccc}
\includegraphics[width=0.17\linewidth]{./figs/nonunique_1.png}&
\includegraphics[width=0.18\linewidth]{./figs/nonunique_2.png}&
\includegraphics[width=0.19\linewidth]{./figs/nonunique_3.png}&
\includegraphics[width=0.20\linewidth]{./figs/nonunique_4.png}&
\includegraphics[width=0.18\linewidth]{./figs/nonunique_5.png}\\
A source image.&\multicolumn{4}{c}{Possible target images.}
\end{tabular}
\caption{A real example showing non-deterministic property of target image in the pixel-level domain transfer problem.}
\label{fig:nonunique}
\end{figure}
To train our converter, we first place a separate network named \textit{domain discriminator} on top of the converter. The domain discriminator takes a pair of a source image and a target image, and is trained to make a binary decision whether the input pair is associated or not. The domain discriminator then supervises the converter to produce associated images. Both of the networks are jointly optimized by the adversarial training method, which Goodfellow~\etal~\cite{goodfellow14} propose for generating realistic images. Such binary supervision solves the problem of non-deterministic property of the target domain and enables us to train the semantic relation between the domains. Secondly, in addition to the domain discriminator, we also employ the discriminator of~\cite{goodfellow14}, which is supervised by the labels of ``real'' or ``fake'', to produce realistic images.
Our framework deals with the three networks that play distinct roles. Labels are given to the two discriminators, and they supervise the converter to produce images that are realistic yet keeping the semantic meaning. Those two discriminators become unnecessary after the training stage and the converter is our ultimate goal. We verify our method by quite challenging settings; the source domain is a natural human image and the target domain is a product image of the person's top. To do so, we have made a large dataset named LookBook, which contains in total of 84k images, where 75k human images are associated with 10k top product images. With this dataset, our model succeeds in generating decent target images, and the evaluation result verifies the effectiveness of our \textit{domain discriminator} to train the converter.
\clearpage
\subsubsection{Contributions}
In summary, our contributions are,
\begin{enumerate}
\item Proposing the first framework for semantically transferring a source domain to a target domain in pixel-level.
\item Proposing a novel discriminator that enables us to train the semantic relation between the domains.
\item Building a large clothing dataset containing two domains, which is expected to contribute to a wide range of domain adaptation researches.
\end{enumerate}
\section{Related Work}
Our work is highly related with the image-generative models since our final result from an input image is also an image. The image-generative models can be grouped into two families; generative parametric approaches~\cite{hinton06,salakhutdinov09,vincent08} and adversarial approaches~\cite{goodfellow14,mirza14,Denton15,Radford15}. The generative parametric approaches often have troubles in training complexities, which results in a low rate of success in generating realistic natural images. The adversarial approaches originate from Generative Adversarial Nets (GAN) proposed by Goodfellow~\etal~\cite{goodfellow14}. GAN framework introduces a generator (i.e. a decoder), which generates images, and a discriminator, which distinguishes between generated samples and real images. The two networks are optimized to go against each other; the discriminator is trained to distinguish between real and fake samples while the generator is trained to confuse the discriminator. Mirza and Osindero~\cite{mirza14} extend GAN to a class conditional version, and Denton~\etal~\cite{Denton15} improve the image resolution in a coarse-to-fine fashion. However, GAN is known to be unstable due to the adversarial training, often resulting in incomprehensible or noisy images. Quite recently, Radford~\etal~\cite{Radford15} have proposed architectures named Deep Convolutional GANs, which is relatively more stable to be trained, and have succeeded in generating high quality images. As approaches focusing on different network architectures, a recurrent network based model~\cite{gregor15} and a deconvolutional network based model~\cite{dosovitskiy15} have also been proposed.
The recent improvements of GAN framework and its successful results motivate us to adopt the networks. We replace the generator with our converter which is an image-conditioned model, while \cite{mirza14} is class-conditional and \cite{yan15} is attribute-conditional. The generator of Mathieu~\etal~\cite{mathieu15} is similar to ours in that it is conditioned with video frames to produce next frames. They add a mean square loss to the generator to strongly relate the input frames to the next frames. However, we cannot use such loss due to the non-deterministic property of the target domain. We therefore introduce a novel discriminator named domain discriminator.
Our work is also related with the transfer learning, also called as the domain adaptation. This aims to transfer the model parameter trained on a source domain to a different domain. For visual recognition, many methods to adapt domains \cite{saenko10,kulis11,gopalan11} have been proposed. Especially for the recent use of the deep convolutional neural network~\cite{lecun89}, it has been common to pre-train a large network~\cite{krizhevsky12} over ImageNet~\cite{russakovsky15} and transfer the parameters to a target domain~\cite{oquab14,razavian14,yoo15}. Similar to our clothing domains, Chen~\etal~\cite{chen15} and Huang~\etal~\cite{huang15} address a gap between fashion shopping mall images and unconstrained human images for the clothing attribute recognition~\cite{chen15} and the product retrieval~\cite{huang15}. Ganin and Lempitsky~\cite{ganin15} also learns domain-invariant features by the adversarial training method. However, all these methods are different from ours in respect of cross-domain \textit{image generation}. The adaptation of these works takes place in the feature space, while we directly produce target images from the source images.
\section{Review of Generative Adversarial Nets}
\label{sec:gan}
Generative Adversarial Nets (GAN) \cite{goodfellow14} is a generalized framework for generative models which \cite{Denton15,Radford15,mathieu15} and we utilize for visual data. In this section, we briefly review GAN in the context of image data. GAN is formed by an adversarial setting of two networks; a generator and a discriminator. The eventual goal of the generator is to map a small dimensional space $Z$ to a pixel-level image space, i.e., to enable the generator to produce a realistic image from an input random vector $z\in Z$.
To train such a generator, a discriminator is introduced. The discriminator takes either a real image or a fake image drawn by the generator, and distinguishes whether its input is real or fake. The training procedure can be intuitively described as follows. Given an initialized generator $G^0$, an initial discriminator $D_R^0$ is firstly trained with real training images $\{I^i\}$ and fake images $\{\hat{I}^j=G^0(z^j)\}$ drawn by the generator. After that, we freeze the updated discriminator $D_R^1$ and train the generator $G^0$ to produce better images, which would lead the discriminator $D_R^1$ to misjudge as real images. These two procedures are repeated until they converge. The objective function can be represented as a minimax objective as,
\begin{equation}
\min_{\Theta^G}\max_{\Theta_R^D}\mathbb{E}_{I\sim p_{\text{data}}(\mathbf{I})}[\log(D_R(I))]+
\mathbb{E}_{z\sim p_{\text{noise}}(\mathbf{z})}[\log(1-D_R(\hat{I}))],
\end{equation}
where $\Theta^G$ and $\Theta_R^D$ indicate the model parameters of the generator and the discriminator respectively. Here, the discriminator produces a scalar probability that is high when the input $I$ is real but otherwise low. The discriminator loss function $\mathcal{L}_R^D$ is defined as the binary cross entropy,
\label{eq:gan}
\begin{multline}
\mathcal{L}_R^D\left(I\right)=-t\cdot \log[D_R(I)] + (t-1)\cdot\log[1-D_R(I)],\\
\text{s.t.}\;\;t=\left\{
\begin{matrix}
1&\;\;\text{if}\;\;I\in\{I^i\}\\
0&\;\;\;\text{if}\;\;I\in\{\hat{I}^j\}.
\end{matrix}\right.
\label{eq:d}
\end{multline}
One interesting fact in the GAN framework is that the model is trained under the lowest level of supervision; real or fake. Without strong and fine supervisions (e.g. mean square error between images), this framework succeeds in generating realistic images. This motivates us to raise the following question.
Under such a low-level supervision, would it be possible to train a connection between distinct image domains? If so, could we transform an image lying in a domain to a realistic image lying on another? Through this study, we have succeeded in doing so, and the method is to be presented in \Sref{sec:method}.
\section{Pixel-Level Domain Transfer}
\label{sec:method}
In this section, we introduce the pixel-level domain transfer problem. Let us define a source image domain $S\subset \mathbb{R}^{W\times H\times3}$ and a target image domain $T\subset \mathbb{R}^{W\times H\times3}$. Given a transfer function named a converter $C$, our task is to transfer a source image $I_S\in S$ to a target image $\hat{I}_T\in T$ such as
\begin{equation}
\hat{I}_T=C(I_S|\Theta^C),
\end{equation}
where $\Theta^C$ is the model parameter of the converter. Note that the inference $\hat{I}_T$ is not a feature vector but itself a target image of $W\times H\times3$ size. To do so, we employ a convolutional network model for the converter $C$, and adopt a supervised learning to optimize the model parameter $\Theta^C$. In the training data, each source image $I_S$ should be associated with a ground-truth target image $I_T$.
\subsection{Converter Network}
\label{ssec:converter}
Our target output is a \textit{pixel-level} image. Furthermore, the two domains are connected by a \textit{semantic} meaning. Pixel-level generation itself is challenging but the semantic transfer makes the problem even more difficult. A converter should selectively summarize the semantic attributes from a source image and then produce a transformed pixel-level image.
\begin{figure}[t!]
\centering
\includegraphics[width=1\linewidth]{./figs/architecture.pdf}
\caption{Whole architecture for pixel-level domain transfer.}
\label{fig:arch}
\end{figure}
\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
Layer&\bigcell{c}{Number\\of filters}&\bigcell{c}{Filter size\\(w$\times$h$\times$ch)}&Stride&Pad&\bigcell{c}{Batch\\norm.}&\bigcell{c}{Activation\\function}\\\hline\hline
Conv. 1&128&5$\times$5$\times$\{3, 3, 6\}&2&2&$\times$&L-ReLU\\
Conv. 2&256&5$\times$5$\times$128&2&2&$\ocircle$&L-ReLU\\
Conv. 3&512&5$\times$5$\times$256&2&2&$\ocircle$&L-ReLU\\
Conv. 4&1,024&5$\times$5$\times$512&2&2&$\ocircle$&L-ReLU\\
Conv. 5&\{64, 1, 1\}&1$\times$1$\times$1,024&1&0&$\{\ocircle,\times,\times\}$&\{L-ReLU, sigmoid, sigmoid\}\\\hline
\end{tabular}\\
(a) Details of the \{encoder, real/fake discriminator, domain discriminator\}.\\$\;$\\
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
Layer&\bigcell{c}{Number\\of filters}&\bigcell{c}{Filter size\\(w$\times$h$\times$ch)}&Stride&Pad&\bigcell{c}{Batch\\norm.}&\bigcell{c}{Activation\\function}\\\hline\hline
Conv. 1&4$\times$4$\times$1,024&1$\times$1$\times$64&1&0&$\ocircle$&ReLU\\
F-Conv. 2&1,024&5$\times$5$\times$512&1/2&-&$\ocircle$&ReLU\\
F-Conv. 3&512&5$\times$5$\times$256&1/2&-&$\ocircle$&ReLU\\
F-Conv. 4&256&5$\times$5$\times$128&1/2&-&$\ocircle$&ReLU\\
F-Conv. 5&128&5$\times$5$\times$3&1/2&-&$\times$&tanh\\\hline
\end{tabular}\\
(b) Details of the decoder.\\$\;$\\
\caption{Details of each network. In (a), each entry in \{$\cdot$\} corresponds to each network. L-ReLU is leaky-ReLU. In (b), F denotes fractional-stride. The activation from the first layer is reshaped into 4$\times$4$\times$1,024 size before being fed to the second layer.}
\label{tab:arch}
\end{table}
The top network in \Fref{fig:arch} shows the architecture of the converter we propose. The converter is a unified network that is end-to-end trainable but we can divide it into the two parts; an encoder and a decoder. The encoder part is composed of five convolutional layers to abstract the source into a semantic 64-dimensional code. This abstraction procedure is significant since our source domain (e.g. natural fashion image) and target domain (e.g. product image) are paired in a semantic content (e.g. the product). The 64-dimensional code should capture the semantic attributes (e.g. category, color, etc.) of a source to be well decoded into a target. The code is then fed by the decoder, which constructs a relevant target through the five decoding layers. Each decoding layer conducts the fractional-strided convolutions, where the convolution operates in the opposite direction. The reader is referred to \Tref{tab:arch} for more details about the architectures of the encoder and the decoder.
\subsection{Discriminator Networks}
\label{ssec:discrim}
Given the converter, a simple choice of a loss function to train it is the mean-square error (MSE) such as $||\hat{I}_T-I_T||_2^2$. However, MSE may not be a proper choice due to critical mismatches between MSE and our problem. Firstly, MSE is not suitable for pixel-level supervision for natural images. It has been well known that MSE is prone to produce blurry images because it inherently assumes that the pixels are drawn from Gaussian distribution \cite{mathieu15}. Pixels in natural images are actually drawn from complex multi-modal distributions. Besides its intrinsic limitation, it causes another critical problem especially for the pixel-level domain transfer as follows.
Given a source image, the target is actually not unique in our problem. Our target domain is the lowest pixel-level image space, not the high-level semantic feature space. Thus, the number of possible targets from a source is infinite. \Fref{fig:nonunique} is a typical example showing that the target is not unique. The clothing in the target domain is captured in various shapes, and all of the targets are true. Besides the shapes, the target image can be captured from various viewpoints, which results in geometric transformations. However, minimizing MSE always forces the converter to fit into one of them. Image-to-image training with MSE never allows a small geometric miss-alignment as well as various shapes. Thus, training the converter with MSE is not a proper use for this problem. It would be better to introduce a new loss function which is tolerant to the diversity of the pixel-level target domain.
In this paper, on top of the converter, we place a discriminator network which plays a role as a loss function. As in \cite{goodfellow14,Denton15,Radford15}, the discriminator network guides the converter to produce realistic target under the supervision of real/fake. However, this is not the only role that our discriminator plays. If we simply use the original discriminator replacing MSE, a produced target could look realistic but its contents may not be relevant to the source. This is because there is no pairwise supervision such as MSE. Only real/fake supervision exists.
Given arbitrary image triplets ($I_S^+, I_S^{\oplus}, I_S^-$) in the source domain $S$, where $I_S^+$ and $I_S^{\oplus}$ are about the same object while $I_S^-$ is not, a converter transfers them into the images ($\hat{I}_T^+, \hat{I}_T^{\oplus}, \hat{I}_T^-$) in the target domain $T$. Let us assume that these transferred images look realistic due to the real/fake discriminator. Beyond the realistic results, the best converter $C$ should satisfy the following condition,
\begin{equation}
s\left(\hat{I}_T^+, \hat{I}_T^{\oplus}\right) > s\left(\hat{I}_T^+, \hat{I}_T^-\right) \;\;\text{and}\;\; s\left(\hat{I}_T^+, \hat{I}_T^{\oplus}\right) > s\left(\hat{I}_T^{\oplus}, \hat{I}_T^-\right),
\label{eq:inequality}
\end{equation}
where $s$($\cdot$) is a semantic similarity function. This condition means that an estimated target should be semantically associated with the source. One supervision candidate to let the converter $C$ meet the condition is the combined use of MSE with the real/fake loss. However, again, it is not the best option for our problem because the ground-truth $I_T$ is not unique. Thus, we propose a novel discriminator, named domain discriminator, to take the pairwise supervision into consideration.
The domain discriminator $D_A$ is the lowest network illustrated in \Fref{fig:arch}. To enable pairwise supervision while being tolerant to the target diversity, we significantly loosen the level of supervision compared to MSE. The network $D_A$ takes a pair of source and target as input, and produces a scalar probability of whether the input pair is associated or not. Let us assume that we have a source $I_S$, its ground truth target $I_T$ and an irrelevant target $I_T^-$. We also have an inference $\hat{I}_T$ from the converter $C$.
We then define the loss $\mathcal{L}_A^D$ of the domain discriminator $D_A$ as,
\begin{multline}
\mathcal{L}_A^D(I_S,I)=-t\cdot \log[D_A(I_S, I)] + (t-1)\cdot\log[1-D_A(I_S, I)],\\
\text{s.t.}\;\;t=\left\{
\begin{matrix}
1&\;\text{if}\;\;I=I_T\\
0&\;\text{if}\;\;I=\hat{I}_T\\
0&\;\;\text{if}\;\;I=I_T^-.
\end{matrix}\right.
\label{eq:dd}
\end{multline}
The source $I_S$ is always fed by the network as one of the input pair while the other $I$ is chosen among ($I_T^-$, $\hat{I}_T$, $I_T$) with equal probability. Only when the source $I_S$ and its ground-truth $I_T$ is paired as input, the domain discriminator is trained to produce high probability whereas it minimizes the probability in other cases. Here, let us pay more attention to the input case of ($I_S$, $\hat{I}_T$).
The produced target $\hat{I}_T$ comes from the source but we regard it as an unassociated pair ($t$=0) when we train the domain discriminator. Our intention of doing so is for \textit{adversarial training} of the converter and the domain discriminator. The domain discriminator loss is minimized for training the domain discriminator while it is maximized for training the converter. The better the domain discriminator distinguishes a ground-truth $I_T$ and an inference $\hat{I}_T$, the better the converter transfers the source into a relevant target.
In summary, we employ both of the real/fake discriminator and the domain discriminator for adversarial training. These two networks play a role as a loss to optimize the converter, but have different objectives. The real/fake discriminator penalizes an unrealistic target while the domain discriminator penalizes a target being irrelevant to a source. The architecture of the real/fake discriminator is identical to that of \cite{Radford15} as illustrated in \Fref{fig:arch}. The domain discriminator also has the same architecture except for the input filter size since our input pair is stacked across the channel axis. Several architecture families have been proposed to feed a pair of images to compare them but a simple stack across the channel axis has shown the best performance as studied in \cite{zagoruyko15}. The reader is referred to \Tref{tab:arch} for more details about the discriminator architectures.
\subsection{Adversarial Training}
\label{ssec:training}
In this section, we present the method for training the converter $C$, the real/fake discriminator $D_R$ and the domain discriminator $D_A$. Because we have the two discriminators, the two loss functions have been defined. The real/fake discriminator loss $\mathcal{L}_R^D$ is \Eref{eq:d}, and the domain discriminator loss $\mathcal{L}_A^D$ is \Eref{eq:dd}. With the two loss functions, we follow the adversarial training procedure of \cite{goodfellow14}.
Given a paired image set for training, let us assume that we get a source batch $\{I_S^i\}$ and a target batch $\{I^i\}$ where a target sample $I^i$ is stochastically chosen from $(I_T^i, I_T^{i-}, \hat{I}_T^i)$ with an equal probability. At first, we train the discriminators.
We train the real/fake discriminator $D_R$ with the target batch to reduce the loss of \Eref{eq:d}. The domain discriminator $D_A$ is trained with both of source and target batches to reduce the loss of \Eref{eq:dd}.
After that, we freeze the updated discriminator parameters $\{\hat{\Theta}_R^D,\hat{\Theta}_A^D\}$, and optimize the converter parameters $\Theta^C$ to \textit{increase} the losses of both discriminators.
The loss function of the converter can be represented as,
\begin{equation}
\mathcal{L}^C(I_S,I)=-\frac{1}{2}\mathcal{L}_R^D\left(I\right)-\frac{1}{2}\mathcal{L}_A^D(I_S,I),\;\;\text{s.t.}\;\;I=\text{sel}\left(\{I_T,\hat{I}_T,I_T^-\}\right),
\end{equation}
where sel($\cdot$) is a random selection function with equal probability. The reader is referred to \Aref{alg:train} for more details of the training procedures.
\begin{algorithm}[t]
\SetAlgoLined
\textbf{Set} the learning rate $\eta$ and the batch size $B$.\\
\textbf{Initialize} each network parameters $\Theta^C, \Theta_R^D, \Theta_A^D$,\\
\KwData{Paired image set $\{I_S^n, I_T^n\}_{n=1}^N$.}
\While{not converged}{
Get a source batch $\left\{I_S^i\right\}_{i=1}^B$ and a target batch $\left\{I^i\right\}_{i=1}^B$,\\
$\;\;\;\;\;\;$where $I^{i}$ is a target sample randomly chosen from $(I_T^i, I_T^{i-}, \hat{I}_T^i)$.\\
\textbf{Update the real/fake discriminator $D_R$:}\\
$\;\;\;\;\;\;\Theta_R^D\leftarrow\Theta_R^D-\eta\cdot\frac{1}{B}\sum_{i=1}^B\frac{\partial\mathcal{L}_R^D\left(I^i\right)}{\partial\Theta_R^D}$\\
\textbf{Update the domain discriminator $D_A$:}\\
$\;\;\;\;\;\;\Theta_A^D\leftarrow\Theta_A^D-\eta\cdot\frac{1}{B}\sum_{i=1}^B\frac{\partial\mathcal{L}_A^D\left(I_S^i,I^i\right)}{\partial\Theta_A^D}$\\
\textbf{Update the converter $C$:}\\
$\;\;\;\;\;\;\Theta^C\leftarrow\Theta^C-\eta\cdot\frac{1}{B}\sum_{i=1}^B\frac{\partial\mathcal{L}^C\left(I_S^i,I^i\right)}{\partial\Theta^C}$\\
}
\caption{Adversarial training for the pixel-level domain transfer.}
\label{alg:train}
\end{algorithm}
\section{Evaluation}
In this section, we verify our pixel-level domain transfer by a challenging task; a natural human image belongs to the source domain, and a product image of that person's top belongs to the target domain. We first give a description on the dataset in~\Sref{sec:db}. We then provide details on the experimental setting in \Sref{sec:setting}, and we demonstrate and discuss the results in \Sref{sec:quali}$\sim$\ref{sec:quanti2}.
\subsection{LookBook Dataset}
\label{sec:db}
We make a dataset named LookBook that covers two fashion domains. Images of one domain contain fashion models, and those of the other domain contain top products with a clean background. Real examples are shown in \Fref{fig:lookbook}. We manually associate each product image with corresponding images of a fashion model fitting the product, so each pair is accurately connected with the same product. LookBook contains 84,748 images where 9,732 top product images are associated with 75,016 fashion model images. It means that a product has around 8 fashion model images in average. We collect the images from five on-line fashion shopping malls\footnote{\{bongjashop, jogunshop, stylenanda\}.com, \{smallman, wonderplace\}.co.kr} where a product image and its fashion model images are provided. Although we utilize LookBook for the pixel-level domain transfer, we believe that it can contribute to a wide range of domain adaptation researches.
Chen~\etal~\cite{chen15} also has presented a similar fashion dataset dealing with two domains. However, it is not suitable for our task since the domains are differently defined in details. They separate the domain into user taken images and on-line shopping mall images so that both domains include humans.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{./figs/lookbook.png}
\caption{Example images of LookBook. A product image is associated with multiple fashion model images.}
\label{fig:lookbook}
\end{figure}
\subsection{Experiment Details}
\label{sec:setting}
Before training, we rescale all images in LookBook to have 64 pixels at a longer side while keeping the aspect ratio, and fill the margins of both ends with 255s. Pixels are normalized to a range of $[-1, 1]$ according to the tanh activation layer of the converter. We then randomly select 5\% images to define a validation set, and also 5\% images for a test set. Since LookBook has 9,732 products, each of the validation set and the test set is composed of 487 product images and their fashion model images. The remaining images compose a training set.
The filters of the three networks are randomly initialized from a zero mean Gaussian distribution with a standard deviation of 0.02. The leak slope of the LeakyReLU in~\Tref{tab:arch}-(a) is 0.2. All models were trained with Stochastic Gradient Descent with mini-batch of 128 size. We also follow the learning rate of 0.0002 and the momentum of 0.5 suggested by \cite{Radford15}. After 25 epochs, we lessen the learning rate to 0.00002 for 5 more epochs.
\Tref{tab:notation} shows the notations and the descriptions of the 4 baselines and our method. The training details of all the baselines are identical to those of ours.
\subsection{Qualitative evaluation}
\label{sec:quali}
First, we show qualitative results in~\Fref{fig:comp}, where the examples are chosen from the test set. Our results look more relevant to the source image and more realistic compared to those of baselines. Boundaries of products are sharp, and small details such as stripes, patterns are well described in general. The results of ``C+RF'' look realistic but irrelevant to the source image, and those of ``C+MSE'' are quite blurry.
\Fref{fig:invariance} verifies how well the encoder of the converter encodes clothing attributes under the various conditions of source images. The source images significantly vary in terms of backgrounds, viewpoints, human poses and self-occlusions. Despite these variations, our converter generates less varying targets while reflecting the clothing attributes and categories of the source images. These results imply that the encoder robustly summarizes the source information in a semantic level.
\begin{table}[t]
\begin{center}
\begin{tabular}{|l|l|}\hline
Notations&Descriptions\\\hline\hline
C+RF&A converter trained only with the real/fake discriminator.\\\hline
C+MSE&A converter trained only with the mean square loss.\\\hline
C+RF+DD$-$Neg&\bigcell{l}{A converter trained with both of the discriminators.\\Negative pairs are not used. Only positive pairs are used.}\\\hline
Retrieval by DD-score&\bigcell{l}{Retrieving the nearest product image in the training set.\\The queries are the human images in the test set.\\The retrieval scores come from the domain discriminator.}\\\hline
C+RF+DD (Ours)&A converter trained with both of the discriminators.\\\hline
\end{tabular}
\end{center}
\caption{Notations and descriptions of baselines and our method.}
\label{tab:notation}
\end{table}
\begin{table}[t]
\begin{center}
\begin{tabular}{|l|c|c|c||l|c|c|}\hline
\multicolumn{4}{|c||}{User study score}&\multicolumn{3}{c|}{Pixel-level (dis)similarity}\\\hline
Methods&Real&Att&Cat&Methods&RMSE&C-SSIM\\\hline\hline
C+RF&0.40&0.21&0.06&C+RF&0.39&0.18\\
C+MSE&0.28&0.60&0.60&C+MSE&\textbf{0.26}&0.20\\
C+RF+DD (Ours)&\textbf{0.82}&\textbf{0.67}&\textbf{0.77}&C+RF+DD$-$Neg&0.32&0.18\\
&&&&Retrieval by DD-score&0.44&0.19\\
&&&&C+RF+DD (Ours)&0.32&\textbf{0.21}\\\hline
\end{tabular}
\end{center}
\caption{Quantitative evaluations. All the values are normalized to a range of $[0, 1]$.}
\label{tab:eval}
\end{table}
\subsection{Quantitative evaluation by user study}
\label{sec:quanti1}
Since the target domain is not deterministic, it is difficult to quantitatively analyze the performance. Thus, we conduct a user study on our generation results as a primary evaluation. We compare our method with the top two baselines in \Tref{tab:notation}. For this study, we created a sub-test set composed of 100 source images randomly chosen from the test set. For each source image, we showed users three target images generated by the two baselines and our method. Users were asked to rate them three times in accordance with three different evaluation criteria as follows. A total of 25 users participated in this study.
\begin{enumerate}
\item How realistic is each result? Give a score from 0 to 2.
\item How well does each result capture the attributes (color, texture, logos, etc.) of the source image? Give a score from 0 to 2.
\item Is the category of each result identical to that of the source image? Give a binary score of 0 or 1.
\end{enumerate}
The left part of \Tref{tab:eval} shows the user study results. In the ``Realistic'' criteria, it is not surprising that ``C+MSE'' shows the worst performance due to the intrinsic limitation of the mean square loss for image generation. Its assumption of Gaussian distribution results in blurry images as shown in~\Fref{fig:comp}. However, the strong pairwise supervision of the mean square loss relatively succeeds in representing the category and attributes of a product.
When the converter is supervised with the real/fake discriminator only, the generated images are more realistic than those of ``C+MSE''. However, it fails to produce targets relevant to inputs and yields low attribute and category scores.
The user study results demonstrate the effectiveness of the proposed method. For all valuation criteria, our method outperforms the baselines. Especially, the ability to capture attributes and categories is better than that of ``C+MSE''. This result verifies the effectiveness of our domain discriminator.
Another interesting observation is that our score of ``Realistic'' criteria is higher than that of ``C+RF''. Both of the methods include the real/fake discriminator but demonstrate distinct results. The difference may be caused by the domain discriminator which is added to the adversarial training in our method. When we train the domain discriminator, we regard all produced targets as ``unassociated''. This setting makes the the converter better transfer a source image into a more \textit{realistic} and relevant target image.
\subsection{Quantitative evaluation by pixel-level (dis)similarity}
\label{sec:quanti2}
For each method, we measure a pixel-level dissimilarity by Root Mean Square Error (RMSE) between a generated image and a target image over the test set. We also measure a pixel-level similarity by Structural Similarity (SSIM), since SSIM is known to be more consistent with human perception than RMSE. We use a color version of SSIM by averaging SSIMs for each channel.
The right part of \Tref{tab:eval} shows the results. As we can expect, ``C+MSE'' shows the lowest RMSE value because the converter is trained by minimizing the mean square loss. However, in case of SSIM, our method outperforms all the baselines.
To verify the effectiveness of the ``associated/unassociated'' supervision when we train the domain discriminator, we compare ours with ``C+RF+DD$-$Neg''. In \Tref{tab:eval}, our method outperforms this method. Without the irrelevant input pairs, the generation results could look realistic, but relatively fail to describe the attributes of items. This is why we added the irrelevant input pairs into supervision to encourage our model to capture discriminative attributes.
To verify the generalization capability of our model, we also compare ours with ``Retrieval by DD-score''. If our model fails in generalization (i.e. just memorizes and copies training items which are similar to query), our generation results could not be better than the retrieved items which are real. However, our method outperforms the retrieval method. It verifies the capability of our model to draw unseen items.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{./figs/invariance.pdf}
\caption{Generation results under varying input conditions. The odd rows are inputs, and the even rows are generation results. Each image is in 64$\times$64$\times$3 dimensions.}
\label{fig:invariance}
\end{figure}
\begin{figure}[h!]
\centering
{\small Source$\;\;\;$RF$\;\;\;\;$MSE$\;\;\;$Ours$\;\;\;\;\;\;$Source$\;\;\;$RF$\;\;\;\;$MSE$\;\;\;$Ours$\;\;\;\;\;\;$Source$\;\;\;$RF$\;\;\;\;$MSE$\;\;\;$Ours$\;$}
\includegraphics[width=1\linewidth]{./figs/comparison.png}
\caption{Qualitative comparisons. Each image from the left to the right respectively corresponds to a source image, a ``C+RF'' result, a ``C+MSE'' result and our result. Each image is in 64$\times$64$\times$3 dimensions.}
\label{fig:comp}
\end{figure}
\begin{figure}[h!]
\footnotesize
\centering
\begin{tabular}{cc|cc|cc|cc|cc|cc}
Source&Ours&Source&Ours&Source&Ours&Source&Ours&Source&Ours&Source&Ours\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000005S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000005T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000008S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000008T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000010S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000010T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000012S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000012T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000014S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000014T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000015S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000015T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000017S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000017T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000020S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000020T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000021S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000021T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000025S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000025T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000027S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000027T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000028S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000028T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000032S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000032T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000033S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000033T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000035S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000035T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000036S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000036T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000038S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000038T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000039S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000039T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000040S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000040T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000041S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000041T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000045S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000045T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000047S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000047T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000051S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000051T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000053S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000053T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000055S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000055T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000056S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000056T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000060S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000060T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000061S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000061T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000064S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000064T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000065S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000065T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000067S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000067T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000069S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000069T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000072S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000072T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000074S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000074T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000075S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000075T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000076S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000076T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000078S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000078T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000081S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000081T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000083S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000083T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000085S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000085T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000087S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000087T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000094S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000094T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000096S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000096T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000098S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000098T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000100S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000100T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000101S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000101T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000102S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000102T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000103S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000103T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000104S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000104T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000106S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000106T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000108S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000108T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000110S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000110T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000111S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000111T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000112S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000112T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000113S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000113T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000114S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000114T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000115S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000115T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000116S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000116T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000117S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000117T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000118S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000118T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000119S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000119T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000121S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000121T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000123S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000123T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000126S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000126T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000127S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000127T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000128S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000128T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000131S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000131T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000132S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000132T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000133S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000133T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000134S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000134T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000136S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000136T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000137S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000137T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000138S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000138T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000139S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000139T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000140S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000140T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000143S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000143T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000144S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000144T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000145S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000145T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000147S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000147T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000148S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000148T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000149S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000149T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000152S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000152T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000153S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000153T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000154S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000154T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000155S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000155T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000158S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000158T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000159S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000159T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000160S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000160T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000161S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000161T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000162S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000162T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000163S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000163T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000164S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000164T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000165S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000165T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000166S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000166T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000167S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000167T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000168S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000168T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000169S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000169T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000171S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000171T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000172S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000172T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000173S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000173T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000174S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000174T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000175S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000175T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000176S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000176T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000177S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000177T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000178S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000178T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000180S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000180T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000181S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000181T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000184S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000184T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000185S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000185T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000186S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000186T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000188S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000188T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000191S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000191T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000192S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000192T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000193S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000193T.png}\\
\includegraphics[width=0.066\linewidth]{./figs/inverse/000194S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000194T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000196S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000196T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000197S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000197T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000198S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000198T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000199S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000199T.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000200S.png}&\includegraphics[width=0.066\linewidth]{./figs/inverse/000200T.png}\\
\end{tabular}\\
\caption{100 chosen results of ``product to human''. Each image is shown in 64$\times$64$\times$3 dimensions.}
\label{fig:inverse}
\end{figure}
\Fref{fig:inverse} shows the results of \textit{``product to human''} setting. Since generating human is a more complex task, 65 epochs for initial training and 5 more epochs for fine-tuning are required for these results. All the other details are identical to those of the original setting.
\section{Conclusion}
We have presented pixel-level domain transfer based on Generative Adversarial Nets framework. The proposed domain discriminator enables us to train the semantic relation between the domains, and the converter has succeeded in generating decent target images. Also, we have presented a large dataset that could contribute to domain adaptation researches. Since our framework is not constrained to specific problems, we expect to extend it to other types of pixel-level domain transfer problems from low-level image processing to high-level synthesis.
\clearpage
\bibliographystyle{splncs03}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,948 |
package com.itellity.sys.conf;
import java.util.ArrayList;
import java.util.Arrays;
import javax.xml.bind.JAXBContext;
import javax.xml.bind.Marshaller;
import junit.framework.TestCase;
public class ConfigurationTest extends TestCase {
public void testConfiguration() {
try {
assertTrue(Configuration.getInstance().getEntries().size() > 0);
} catch (Exception e) {
fail(e.getMessage());
}
}
public void testOuputConfiguration() {
try {
ConfigurationEntry entry = new ConfigurationEntry();
entry.setKey("rootPath");
entry.setValue("/u01/tomcat");
ConfigurationEntry entry2 = new ConfigurationEntry();
entry2.setKey("thumbsHost");
entry2.setValue("http://localhost:8080/");
Configuration conf = new Configuration();
conf.setEntries(new ArrayList<ConfigurationEntry>(Arrays.asList(entry2, entry)));
JAXBContext jc = JAXBContext.newInstance(Configuration.class );
Marshaller m = jc.createMarshaller();
m.setProperty(Marshaller.JAXB_FORMATTED_OUTPUT, Boolean.TRUE);
m.marshal( conf, System.out );
System.out.println(conf.getValue("rootPath"));
} catch (Exception e) {
fail(e.getMessage());
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,321 |
Laura Järnefelt eller Laura Kolehmainen Järnefelt, född 1904 i Tusby i Finland, död 17 september 1985, var en finländsk bildkonstnär. Hon var dotter till konstnären Eero Järnefelt. Hon debuterade som konstnär 1930 och använde namnet Järnefelt som sitt konstnärsnamn.
Hon studerade vid Konstuniversitetets Bildkonstakademi 1923–1925, Helsingfors universitets ritsal 1924–1925, Académie Othon Friesz och Académie Colarossi i Paris 1926 samt vid Accademia Inglese och Lipinsky i Rom 1928 och 1932.
Hon är känd för sina målningar av idylliska landskap från en period, då modernismen började vinna terräng inom den finländska konsten, men följde inte själv de modernistiska strömningarna. Hon hämtade ofta sina motiv från naturen kring sitt föräldrahem Suviranta, där hon senare bodde med sin egen familj, belägen i Träskända vid Tusby träsk.
Laura Järnefelt har gjort monumentalmålningen Landskap från Kakkovuori i Pielavesi sparbank och hon har målat ett stort antal porträtt, som finns bland annat i Helsingfors universitet, Aalto-universitetet och hos Träskända stad.
Källor
Noter
Externa länkar
Pauliina Laitinen-Littorin: Laura Järnefelt maalaa idyllistä luontoa ("Laura Järnefelt målar idyllisk natur") i webbtidskriften Taloustaito den 3 februari 2017. Läst den 17 mars 2019.
Finländska porträttmålare
Finländska landskapsmålare
Finländska målare under 1900-talet
Födda 1904
Avlidna 1985
Kvinnor
Personer från Tusby | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 101 |
System.register([], function (exports, module) {
'use strict';
return {
execute: function () {
exports('a', fn);
function fn () {
console.log('lib1 fn');
}
}
};
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,020 |
\section{Introduction}
In many modern systems, the system learns the behavior of the users by adaptively interacting with the users and adjusting to their responses. For example, in online advertisement, a website may show a certain type of ads to a user, observe whether the user clicks on the ads or not, and based on this feedback adjust the type of ads it serves the user. As part of this learning process, the system faces a dilemma between ``exploring'' new types of ads, which may or may not seem interesting to the user, and ``exploiting'' its knowledge of what types of ads historically seemed interesting to the user. This concept is very well studied in the context of online learning.
In principle, users benefit from a system that automatically adapts to their preferences. However, users may naturally worry about a system that observes all of their actions, and worry that the system may use this personal information against them or mistakenly reveal it to hackers or untrustworthy third parties. There therefore arises an additional dilemma between providing a better personalized user experience and acquiring the users' trust.
We study the problem of online learning in multi-user systems under a version of anonymity inspired by \emph{$k$-anonymity} \citep{sweeney2002k}. In its most general form, $k$-anonymity is a property of anonymous data guaranteeing that for every data point, there exist at least $k-1$ other indistinguishable data points in the dataset. Although there exist more recent notions of privacy and anonymity with stronger guarantees (e.g. various forms of differential privacy), $k$-anonymity is a simple and practical notion of anonymity that remains commonly employed in practice and enforced in various legal settings \citep{goldsteen2021data, SafariN2019, SLIJEPCEVIC2021102488}.
To explain our notion of anonymity, consider the online advertisement application mentioned earlier. In online advertisement, when a user visits a website the website selects a type of ad and shows that user an ad of that type. The user may then choose to click on that ad or not, and a reward is paid out based on whether the user clicks on the ad or not (this reward may represent either the utility of the user or the revenue of the online advertisement system). Note that the ad here is chosen by the website, and the fact that the website assigns a user a specific type of ad is not something we intend to hide from the website. However, the decision to click on the ad is made by the user. We intend to protect these individual decisions, while allowing the website to learn what general types of ads each user is interested in. In particular, we enforce a form of group level $C$-anonymity on these decisions, by forcing the system to group users into groups of at least $C$ users and to treat each group equally by assigning all users in the same group the same ad and only observing the total aggregate reward (e.g. the total number of clicks) of these users.
More formally, we study this version of anonymity in a simple model of online learning based on the popular multi-armed bandit setting. In the classic (stochastic) multi-armed bandits problem, there is a learner with some number $K$ of potential actions (``arms''), where each arm is associated with an unknown distribution of rewards. In each round (for $T$ rounds) the learner selects an arm and collects a reward drawn from a distribution corresponding to that arm. The goal of the learner is to maximize their total expected reward (or equivalently, minimize some notion of regret).
In our multi-user model, a centralized learner assigns $N$ users to $K$ arms, and each rewards for each user/arm pair are drawn from some fixed, unknown distribution. Each round the
learner proposes an assignment of users to arms, upon which each user receives a reward from the appropriate user/arm distribution. However, the learner is only allowed to record feedback about these rewards (and use this feedback for learning) if they perform this assignment in a manner compatible with \emph{$C$-anonymity}. This entails partitioning the users into groups of size at least $C$, assigning all users in each group to the same arm, and only observing this group's aggregate rewards for this arm. For example, if $C=3$ in one round we may combine users $1$, $2$ and $4$ into a group and assign them all to arm $3$; we would then observe as feedback the aggregate reward $r_{13} + r_{23} + r_{43}$, where $r_{ij}$ represents the reward that user $i$ experienced from arm $j$ this round. The goal of the learner is to maximize the total reward by efficiently learning the optimal action for each user, while at the same time preserving the anonymity of individual rewards users experienced in specific rounds. See Section~\ref{sec:prelim} for a more detailed formalization of the model.
\subsection{Our results}
In this paper we provide low-regret algorithms for the anonymous bandit setting described above. We present two algorithms which operate in different regimes (based on how users cluster into their favorite arms):
\begin{itemize}
\item If for each arm $j$ there are at least $U \geq C+1$ users for which arm $j$ is that user's optimal arm, then Algorithm \ref{alg:main} incurs regret at most $\tilde{O}(NC\sqrt{\alpha KT})$, where $\alpha = \max(1, \lceil K(C+1)/U \rceil)$.
\item If there is no such guarantee (but $N > C$), then Algorithm \ref{alg:nou} incurs regret at most $\tilde{O}(C^{2/3}K^{1/3}T^{2/3})$.
\end{itemize}
We additionally prove the following corresponding lower bounds:
\begin{itemize}
\item We show (Theorem \ref{thm:main_lb}) that the regret bound of Algorithm \ref{alg:alg1} is tight for any algorithm to within a factor of $K\sqrt{C}$ (and to within a factor of $\sqrt{C}$ for $U \geq K(C+1)$).
\item We show (Theorem \ref{thm:lb_nou}) that the dependence on $T^{2/3}$ in Algorithm \ref{alg:nou} is necessary in the absence of a lower bound on $U$.
\end{itemize}
The main technical contribution of this work is the development/analysis of Algorithm \ref{alg:main}. The core idea behind Algorithm \ref{alg:main} is to use recent algorithms developed for the problem of \textit{batched bandits} (where instead of $T$ rounds of adaptivity, users are only allowed $B \ll T$ rounds of adaptivity) to reduce this learning problem to a problem in combinatorial optimization related to decomposing a bipartite weighted graph into a collection of degree-constrained bipartite graphs. We then use techniques from combinatorial optimization and convex geometry to come up with efficient approximation algorithms for this combinatorial problem.
\subsection{Related work}
The bandits problem has been studied for almost a century~\citep{thompson1933likelihood}, and it has been extensively studied in the standard single user version~\citep{audibert2009minimax,audibert2009exploration,audibert2010regret,auer2002finite,auer2010ucb,bubeck2013bounded,garivier2011kl,lai1985asymptotically}. There exists recent work on bandits problem for systems with multiple users~\citep{bande2019adversarial,bande2019multi,bande2021dynamic,vial2021robust,buccapatnam2015information,chakraborty2017coordinated,kolla2018collaborative,landgren2016distributed,sankararaman2019social}. These papers study this problem from game theoretic and optimization perspectives (e.g. studying coordination / competition between users) and do not consider anonymity. To the best of our knowledge this paper is the first attempt to formalize and study multi-armed bandits with multiple users from an anonymity perspective.
Learning how to assign many users to a (relatively) small set of arms can also be thought of as a clustering problem. Clustering of users in multi-user multi-arm bandits has been previously studied. \cite{maillard2014latent} first studied sequentially clustering users, which was later followed up by other researchers~\citep{nguyen2014dynamic,gentile2017context,korda2016distributed}. Although these works, similar to us, attempt to cluster the users, they are allowed to observe each individual's reward and optimize based on that, which contradicts our anonymity requirement.
One technically related line of work that we heavily rely on is recent work on batched bandits. In the problem of batched bandits, the learner is prevented from iteratively and adaptively making decisions each round; instead the learning algorithm runs in a small number of ``batches'', and in each batch the learner chooses a set of arms to pull, and observes the outcome at the end of the batch. Batched multi-armed bandits were initially studied by \cite{perchet2016batched} for the particular case of two arms. Later \cite{gao2019batched} studied the problem for multiple arms. \cite{esfandiari2019regret} improved the result of Gao et al. and extended it to linear bandits and adversarial multi-armed bandits. Later this problem was studied for batched Thompson sampling~\citep{kalkanli2021batched,karbasi2021parallelizing}, Gaussian process bandit optimization~\citep{li2021gaussian} and contextual bandits~\citep{zhang2021almost,zanette2021design}.
In another related line of work, there have been several successful attempts to apply different notions of privacy such as differential privacy to multi-armed bandit settings ~\citep{tossou2016algorithms,shariff2018differentially,dubey2020differentially,basu2019differential}. While these papers provide very promising guarantees of privacy measures, they focus on single-user settings. In this work we take advantage of the fact that there are several similar users in the system, and use this to provide guarantees of anonymity. Anonymity and privacy go hand in hand, and in a practical scenario, both lines of works can be combined to provide a higher level of privacy.
Finally, our setting has some similarities to the settings of stochastic linear bandits ~\citep{dani2008stochastic, rusmevichientong2010linearly, abbasi2011improved} and stochastic combinatorial bandits ~\citep{chen2013combinatorial, kveton2015tight}. For example, the superficially similar problem of assigning $N$ users to $K$ arms each round so that each arm has at least $C$ users assigned to it (and where you get to observe the total reward per round) can be solved directly by algorithms for these frameworks. However, although such assignments are $C$-anonymous, there are important subtleties that prevent us from directly applying these techniques in our model. First of all, we do not actually constrain the assignment of users to arms -- rather, our notion of anonymity constrains what feedback we can obtain from such an assignment (e.g., it is completely fine for us to assign zero users to an arm, whereas in the above model no actions are possible when $N < CK$). Secondly, we obtain more nuanced feedback than is assumed in these frameworks (specifically, we get to learn the reward of each group of $\geq C$ users, instead of just the total aggregate reward). Nonetheless it is an interesting open question if any of these techniques can be applied to improve our existing regret bounds (perhaps some form of linear bandits over the anonymity polytopes defined in Section \ref{sec:lp_decomp}).
\section{Model and preliminaries}\label{sec:prelim}
\paragraph{Notation.} We write $[N]$ as shorthand for the set $\{1, 2, \dots, N\}$. We write $\tilde{O}(\cdot)$ to suppress any poly-logarithmic factors (in $N$, $K$, $C$, or $T$) that arise. We say a random variable $X$ is $\sigma^2$-subgaussian if the mean-normalized variable $Y = X - \mathbb{E}[x]$ satisfies $\mathbb{E}[\exp(sY)] \leq \exp(\sigma^2s^2/2)$ for all $s \in \mathbb{R}$.
Proofs of most theorems have been postponed to Appendix \ref{app:proofs} of the Supplemental Material in interest of brevity.
\subsection{Anonymous bandits}
In the problem of \textit{anonymous bandits}, there are $N$ users. Each round (for $T$ rounds), our algorithm must assign each user to one of $K$ arms (multiple users can be assigned to the same arm). If user $i$ plays arm $j$, they receive a reward drawn independently from a 1-subgaussian distribution $\mathcal{D}_{i, j}$ with (unknown) mean $\mu_{i, j} \in [0, 1]$. We would like to minimize their overall expected regret of our algorithm. That is, if user $i$ receives reward $r_{i, t}$ in round $t$, we would like to minimize
$$\mathbf{Reg} = T\sum_{i=1}^{N}\max_{j}\mu_{i, j} - \mathbb{E}\left[\sum_{t=1}^{T}\sum_{i=1}^{N} r_{i, t}\right].$$
Thus far, this simply describes $n$ independent parallel instances of the classic multi-armed bandit problem. We depart from this by imposing an \textit{anonymity constraint} on how the learner is allowed to observe the users' feedback. This constraint is parameterized by a positive integer $C$ (the minimum group size). In a given round, the learner may partition a subset of the users into groups of size at least $C$, under the constraint that the users within a single group must all be playing the same arm during this round. For each group $G$, the learner then receives as feedback the total reward $\sum_{i \in G} r_{i, t}$ received by users in this group. Note that not all users must belong to a group (the learner simply receives no feedback on such users), and the partition into groups is allowed to change from round to round.
Without any constraint on the problem instance, it may be impossible to achieve sublinear regret (see Section \ref{sec:lbs}). We therefore additionally impose the following \textit{user-cluster assumption} on the users: each arm $j$ is the optimal arm for at least $U$ users. Such an assumption generally holds in practice (e.g., in the regime where there are many users but only a few classes of arms). This also prevents situations where, e.g., only a single user likes a given arm but it is hard to learn this without allocating at least $C$ users to this arm and sustaining significant regret. Typically we will take $U > C$; when $U \leq C$ the asymptotic regret bounds for our algorithms may be worse (see Section \ref{sec:nou}).
\subsection{Batched stochastic bandits}\label{sec:batched}
Our main tool will be algorithms for \textit{batched stochastic bandits}, as described in \cite{gao2019batched}. For our purposes, a batched bandit algorithm is an algorithm for the classical multi-armed bandit problem that proceeds in $B$ stages (``batches'') where the $b$th stage has a predefined length of $D_{b}$ rounds (with $\sum_{b} D_{b} = T$). At the beginning of each stage $b$, the algorithm outputs a non-empty subset of arms $A_b$ (representing the set of arms the algorithm believes might still be optimal). At the end of each stage, the algorithm expects at least $ D_b/|A_b|$ independent instances of feedback from arm $j$ for each $j \in S$; upon receiving such feedback, the algorithm outputs the subset of arms $A_{b+1}$ to explore in the next batch.
In \cite{gao2019batched}, the authors design a batched bandit algorithm they call $\textsf{BaSE}$ (batched successive-elimination policy); for completeness, we reproduce a description of their algorithm in Appendix \ref{app:base}. When $B = \log\log T$, their algorithm incurs a worst-case expected regret of at most $\tilde{O}(\sqrt{KT})$. In our analysis, we will need the following slightly stronger bound on the behavior of $\textsf{BaSE}$:
\begin{lemma}\label{lem:base}
Set $B = \log\log T$. Let $\mu^{*} = \max_{j \in [K]} \mu_{j}$, and for each $j \in [K]$ let $\Delta_{j} = \mu^* - \mu_{j}$. Then for each $1 \leq b \leq B$, we have that:
$$D_{b} \cdot \mathbb{E}\left[ \max_{j \in A_{b}} \Delta_{j} \right] = \tilde{O}(\sqrt{KT}).$$
\end{lemma}
In other words, Lemma \ref{lem:base} bounds the expected regret in each batch, even under the assumption that we receive the reward of the worst active arm each round (even if we ask for feedback on a different arm).
It will also be essential in the analysis that follows that the total number of rounds $D_{b}$ in the $b$th batch only depends on $b$ and is independent of the feedback received thus far (in the language of \cite{gao2019batched}, the grid used by the batched bandit algorithm is \textit{static}). This fact will let us run several instances of this batched bandit algorithm in parallel, and guarantees that batches for different instances will always have the same size.
\section{Anonymous Bandits}
\subsection{Feedback-eliciting sub-algorithm}\label{sec:fesa}
Our algorithms for anonymous bandits will depend crucially on the following sub-algorithm, which allows us to take a matching from users to arms and recover (in $O(C)$ rounds) an unbiased estimate of each user's reward (as long as that user is matched to a popular enough arm). More formally, let $\pi:[N] \rightarrow [K]$ be a matching from users to arms. We will show how to (in $2C+2$ rounds) recover an unbiased estimate of $\mu_{i, \pi(i)}$ for each $i$ such that $|\pi^{-1}(\pi(i))| \geq C+1$ (i.e. $i$ is matched to an arm that at least $C$ other users are matched to).
The main idea behind this sub-algorithm is simple; for each user, we will get a sample of the total reward of a group containing the user, and a sample of the total reward of the same group but minus this user. The difference between these two samples is an unbiased estimate of the user's reward. More concretely, we follow these steps:
\begin{enumerate}
\item Each round (for $2C+2$ rounds) the learner will assign user $i$ to arm $\pi(i)$. However, the partition of users into groups will change over the course of these $2C+2$ rounds.
\item For each arm $j$ such that $|\pi^{-1}(j)| \geq C+1$, partition the users in $\pi^{-1}(j)$ into groups of size at least $C+1$ and of size at most $2C+1$. Let $G_1, G_2, \dots, G_S$ be the set of groups formed in this way (over all arms $j$). In each group, order the users arbitrarily.
\item In the first round, the learner reports the partition into groups $\{G_1, G_2, \dots, G_S\}$. For each group $G_s$, let $r_{s, 0}$ be the total aggregate reward for group $G_s$ this round.
\item \sloppy{In the $k$th of the next $2C+1$ rounds, the learner reports the partition into groups $\{G_{1, k}, G_{2, k}, \dots, G_{S, k}\}$, where $G_{s, k}$ is formed from $G_s$ by removing the $k$th element (if $k > |G_{s}|$, then we set $G_{s, k} = G_{s}$). Let $r_{s, k}$ be the total aggregate reward from $G_{s, k}$ reported this round.}
\item
If user $i$ is the $k$th user in $G_{s}$, we return the estimate $\hat{\mu}_{i, \pi(i)} = r_{s, 0} - r_{s, k}$ of the average reward for user $i$ and arm $\pi(i)$.
\end{enumerate}
\begin{lemma}\label{lem:fesa}
In the above procedure, $\mathbb{E}[\hat{\mu}_{i, \pi(i)}] = \mu_{i, \pi(i)}$ and $\hat{\mu}_{i, \pi(i)}$ is an $O(C)$-subgaussian random variable.
\end{lemma}
\subsection{Anonymous decompositions of bipartite graphs}
The second ingredient we will need in our algorithm is the notion of an anonymous decomposition of a weighted bipartite graph. Intuitively, by running our batched stochastic bandits algorithm, at the beginning of each batch we will obtain a demand vector for each user (representing the number of times that user would like feedback on each of the $m$ arms). Based on this, we want to generate a collection of assignments (from users to arms) which guarantee that we obtain (while maintaining our anonymity guarantees) the requested amount of information for each user/arm pair.
Formally, we represent a weighted bipartite graph as a matrix of $nm$ non-negative entries $w_{i, j}$ (representing the number of instances of feedback user $i$ desires from arm $j$). We will assume that for each $i$, $\sum_{j} w_{i, j} > 0$ (each user is interested in at least one arm). A \textit{$C$-anonymous decomposition} of this graph is a collection of $R$ assignments $M_1, M_2, \dots, M_{R}$ from users to arms (i.e., functions from $[N]$ to $[K]$) that satisfies the following properties:
\begin{enumerate}
\item A user is never assigned to an arm for which they have zero demand. That is, if $M_r(i) = j$, then $w_{i, j} \neq 0$.
\item If $M_{r}(i) = j$, and $|M_{r}^{-1}(j)| \geq C + 1$, we say that matching $M_{r}$ is \textit{informative} for the user/arm pair $(i, j)$. (Note that this is exactly the condition required for the feedback-eliciting sub-algorithm to output the unbiased estimate $\hat{\mu_{i, j}}$ when run on assignment $M_{r}$.) For each user/arm pair $(i, j)$, there must be at least $w_{i, j}$ informative assignments.
\end{enumerate}
The weighted bipartite graphs that concern us come from the parallel output of $N$ batched bandit algorithms (described in Section \ref{sec:batched}) and have additional structure. These graphs can be described by a positive total demand $D$ and a non-empty demand set $A_i \subseteq [K]$ for each user $i$ (describing the arms of interest to user $i$). If $j \in A_{i}$, then $w_{i,j} = D / |A_i|$; otherwise, $w_{i,j} = 0$. To distinguish graphs with the above structure from generic weighted bipartite graphs, we call such graphs \textit{batched graphs}.
Moreover, for each user $i$, let $j^*(i)$ be the optimal arm for user $i$ (i.e., $j^*(i) = \arg\max_{i}\mu_{i}$). In the algorithm we describe in the next section, with high probability, $j^*(i)$ will always belong to $A_{i}$. Moreover, a user-cluster assumption of $U$ implies that, for any arm $j$, $|j^{*-1}(j)| \geq U$. This means that in batched graphs that arise in our algorithm, there will exist an assignment where each user $i$ is assigned to an arm in $A_{i}$, and each arm $j$ has at least $U$ assigned users. We therefore call graphs that satisfy this additional assumption \textit{$U$-batched graphs}. Note that this assumption also allows us to lower bound the degrees of arms in this bipartite graph. Specifically, for each arm $j$, let $B_{j} = \{i \in [N] \mid j \in A_{i}\}$. Then a user-cluster assumption of $U$ directly implies that $|B_{j}| \geq U$.
In general, our goal is to minimize the number of assignments $R$ required in such a decomposition (since each assignment corresponds to some number of rounds required). We call an algorithm that takes in a $U$-batched graph and outputs a $C$-anonymous decomposition of that graph an \textit{anonymous decomposition algorithm}, and say that it has approximation ratio $\alpha(C, U) \geq 1$ if it generates an assignment with at most $\alpha(C, U)\cdot D + O(NK)$ total assignments. This additive $O(NK)$ is necessary for technical reasons, but in our algorithm, $D$ will always be much larger than $K$ (we will have $D \geq \sqrt{T}$), so this can be thought of as an additive $o(D)$ term.
Later, in Section \ref{sec:decomp_algs}, we explicitly describe several anonymous decomposition algorithms and their approximation guarantees. In the interest of presenting the algorithm, we will assume for now we have access to a generic anonymous decomposition algorithm $\textsf{Decompose}$ with approximation ratio $\alpha(C, U)$.
\subsection{An algorithm for anonymous bandits}
We are now ready to present our algorithm for anonymous bandits. The main idea behind this algorithm (detailed in Algorithm \ref{alg:alg1}) is as follows. Each of the $N$ users will run their own independent instance of $\textsf{BaSE}$ with $B = \log\log T$ synchronized batches. During batch $b$, $\textsf{BaSE}$ requires each user $i$ to get a total of $D_{b}$ instances of feedback on a set $A_{i, b}$ of arms which are alive for them. These sets $A_{i, b}$ (with high probability) define a $U$-batched graph, so we can use an anonymous decomposition algorithm to construct a $C$-anonymous decomposition of this graph into at most $\alpha(C, U)D_{b}$ assignments. We then run the feedback-eliciting sub-algorithm on each assignment, getting one unbiased estimate of the reward for each user/arm pair for which the assignment is informative.
The guarantees of the $C$-anonymous decomposition mean that this process gives us $D_{b}$ total pieces of feedback for each user, evenly split amongst the arms in $A_{i, b}$. We can therefore pass this feedback along to $\textsf{BaSE}$, which will eliminate some arms and return the set of alive arms for user $i$ in the next batch.
\begin{algorithm2e}[h]
\caption{Low-regret algorithm for anonymous bandits.}\label{alg:main}
\textbf{Input:} Anonymity parameter $C$, a lower bound on the user-cluster assumption $U$, time-horizon $T$, number of users $N$, and number of arms $K$. We additionally assume we have access to an anonymous decomposition algorithm $\textsf{Decompose}$ with approximation factor $\alpha = \alpha(C, U)$. \\
For each user $i \in [N]$, initialize an instance of $\textsf{BaSE}$ with a time horizon of $T' = \frac{T}{\alpha(2C+2)}$ and $B = \log\log T$ batches. Let $D_{1}, D_{2}, \dots, D_{b}$ be the corresponding batch sizes (with $\sum_{b} D_{b} = T'$). Let $A_{i, b}$ be the set of active arms for user $i$ during batch $b$.\\
\For{$b \gets 1$ \KwTo $B$}{
If $D_{b}$ and $(A_{1, b}, A_{2, b}, \dots, A_{n, b})$ do not define a $U$-batched graph, abort the algorithm (this means we have eliminated the optimal arm for a user, which can only happen with negligible probability). \\
Run $\textsf{Decompose}$ on $D_{b}$ and $(A_{1, b}, A_{2, b}, \dots, A_{n, b})$ to get a $C$-anonymous decomposition into at most $R_b = \alpha D_b$ assignments. Let $M_{r}$ be the $r$th such assignment.\\
\For{$r \gets 1$ \KwTo $R_{b}$}{
Over $(2C+2)$ rounds, run the feedback-soliciting sub-algorithm (Section \ref{sec:fesa}) on $M_{r}$ to get estimates $\hat{\mu}_{i, M_{r}(i)}$ for all users $i$ for which $M_{r}$ is informative.
}
For each user $i$, we are guaranteed to receive (by the guarantees of $\textsf{Decompose}$) at least $\frac{D_b}{|A_{i, b}|}$ independent samples of feedback $\hat{\mu}_{i, j}$ for each arm $j \in A_{i,b}$. Pass these samples to user $i$'s instance of $\textsf{BaSE}$ and receive $A_{i, b+1}$ in response (unless this is the last batch).
}
\label{alg:alg1}
\end{algorithm2e}
\begin{theorem}\label{thm:main_alg}
Algorithm \ref{alg:main} incurs an expected regret of at most $\tilde{O}(NC\sqrt{\alpha KT})$ for the anonymous bandits problem.
\end{theorem}
One quick note on computational complexity: note that we only run $\textsf{Decompose}$ once every \textit{batch}; in particular, at most $\log\log T$ times. This allows us to efficiently implement Algorithm \ref{alg:alg1} even for complex choices of $\textsf{Decompose}$ that may require solving several linear programs.
\subsection{Algorithms for constructing anonymous decompositions} \label{sec:decomp_algs}
\subsubsection{A greedy method}
\label{sec:greedy_decomp}
We begin with perhaps the simplest method for constructing an anonymous decomposition, which achieves an approximation ratio $\alpha(C, U) = K$ as long as $U \geq C+1$. To do this, for each arm $j \in [K]$, consider the assignment $M_{j}$ where all users $i$ with $j \in A_{i}$ (i.e., users with any interest in arm $j$) are matched to arm $j$, and other users are arbitrarily assigned to arms in their active arm set. Our final decomposition contains $D$ copies of $M_{j}$ for each $j \in [K]$ (for a total of $KD$ assignments).
Note that since $U \geq C+1$, there will be at least $C+1$ users matched to arm $j$ in $M_{j}$, and therefore $M_{j}$ will be informative for all users $i$ with $j \in A_{i}$. Since we repeat each assignment $D$ times, we will have at least $D$ informative assignments for every valid user/arm pair, and therefore this is a valid $C$-anonymous decomposition for the original batched graph.
Substituting this guarantee into Theorem \ref{thm:main_alg} gives us an anonymous bandit algorithm with expected regret $\tilde{O}(NCK\sqrt{T})$.
\subsubsection{The anonymity polytope}
\label{sec:lp_decomp}
As $U$ grows larger than $C$, it is possible to attain even better approximation guarantees. In this section we will give an anonymous decomposition algorithm that applies techniques from combinatorial optimization to attain the following guarantees:
\begin{itemize}
\item If $U \geq K(C+1)$, then $\alpha(C, U) = 1$.
\item If $(C+1) \leq U \leq K(C+1)$, then $\alpha(C, U) = \left\lceil \frac{K(C+1)}{U} \right\rceil$.
\end{itemize}
To gain some intuition for how this is possible, assume $U = K(C+1)$, and consider the randomized algorithm which matches each user $i$ to a random arm in $A_{i}$ each turn. In expectation, after $D$ rounds of this, user $i$ will be matched to each arm $j \in A_{i}$ exactly $D/|A_{i}|$ times (as user $i$ desires). Moreover, since $U \geq K(C+1)$, each arm $j$ has at least $K(C+1)$ candidate users that can match to it. Each of these users matches to arm $j$ with probability at least $1/K$, so in expectation at least $C+1$ users match to arm $j$, and therefore the feedback from arm $j$ is informative ``in expectation''.
The catch with this method is that it is possible (and even reasonably likely) for fewer than $C+1$ users to match to a given arm $j$, and in this case we receive no feedback for user $i$. While it is possible to adapt this method to work with high probability, this requires additional logarithmic factors in either $\alpha$ or the user-cluster bound $U$, and even then has some probability of failure. Instead, we present a deterministic algorithm which can exactly achieve the guarantees above by geometrically ``rounding'' the above randomized matching into a small weighted collection of deterministic matchings.
We define the \textit{$C$-anonymity polytope} $\mathcal{P}_{C} \subseteq \mathbb{R}^{N \times K}$ to be the convex hull of all binary vectors $v \in \{0, 1\}^{N \times K}$ that satisfy the following conditions:
\begin{itemize}
\item For each $i$, $\sum_{j} v_{ij} \in \{0, 1\}$.
\item For each $j$, either $\sum_{i} v_{ij} \geq C+1$ or $\sum_{i} v_{ij} = 0$.
\end{itemize}
We can interpret each such vertex $v$ as a single assignment in a $C$-anonymous decomposition, where $v_{ij} = 1$ iff we get feedback on the user/arm pair $(i, j)$ (so we must match user $i$ to $j$, and at least $C+1$ users must be matched to arm $j$).
Now, for a fixed $U$-batched graph $G$, let $w \in [0, 1]^{N \times K}$ be the weights of this $G$ normalized by the demand $D$: so $w_{ij} = 1/|A_{i}|$ if $j \in A_{i}$, and $w_{ij} = 0$ otherwise. It turns out that we can reduce (via Caratheodory's theorem) the problem of finding a $C$-anonymous decomposition of $G$ into finding the maximal $\beta$ for which $\beta w \in \mathcal{P}_{C}$.
\begin{lemma}\label{lem:cara}
If for some $\beta > 0$, $\beta w \in \mathcal{P}_{C}$, then there exists a $C$-anonymous decomposition of $G$ into at most $\frac{1}{\beta}D + NK + 1$ assignments. Similarly, if there exists a $C$-anonymous decomposition of $G$ into $\alpha D$ assignments, then $\frac{1}{\alpha}w \in \mathcal{P}_{C}$.
\end{lemma}
In a sense, Lemma \ref{lem:cara} provides an ``optimal'' algorithm for the problem of finding $C$-anonymous decompositions. There are two issues with using this algorithm in practice. The first -- an interesting open question -- is that we do not understand the approximation guarantees of this decomposition algorithm (although they are guaranteed to be at least as good as every algorithm we present here).
\begin{problem}
For a $U$-batched graph $G$, let $\beta(G)$ be the maximum value of $\beta$ such that $\beta w \in \mathcal{P}_{C}$ (where $w$ is the weight vector associated with $C$). What is $\max \beta(G)$ over all $U$-batched graphs? Is it $\Omega(1)$?
\end{problem}
The second is that, computationally, it is not clear if there is an efficient way to check whether a point belongs to $\mathcal{P}_{C}$ (let alone write it as a convex combination of the vertices of $\mathcal{P}_{C}$). We will now decompose $\mathcal{P}_{C}$ into the convex hull of a collection of more tractable polytopes; while it will still be hard to e.g. check membership in $\mathcal{P}_{C}$, this will help us efficiently compute decompositions that provide the guarantees at the beginning of this section.
For a subset $S \subseteq [K]$ of arms, let $\mathcal{P}_{C}(S) \subseteq \mathbb{R}^{N \times K}$ be the convex hull of the binary vectors that satisfy the following conditions:
\begin{itemize}
\item For each $i$, $\sum_{j} v_{ij} \in \{0, 1\}$.
\item If $j \in S$, then $\sum_{i} v_{ij} \geq C+1$.
\item If $j \not\in S$, then $\sum_{i} v_{ij} = 0$.
\end{itemize}
By construction, each vertex of $\mathcal{P}_{C}$ appears as a vertex of some $\mathcal{P}_{C}(S)$ and each vertex of $\mathcal{P}_{C}(S)$ belongs to $\mathcal{P}_{C}$, so $\mathcal{P}_{C} = \conv(\{\mathcal{P}_{C}(S) \mid S \subseteq [K]\})$. We now claim that we can write each polytope $\mathcal{P}_{C}(S)$ as the intersection of a small number of halfspaces (and therefore check membership efficiently). In particular, we claim that $x \in \mathbb{R}^{N \times K}$ belongs to $\mathcal{P}_{C}(S)$ iff it satisfies the following linear constraints:
\begin{equation} \label{eq:lp}
\begin{array}{ll@{}ll}
& 0 \leq & x_{ij} \leq 1, &\forall\; i \in [N], j \in [K]\\
& \displaystyle\sum_{j\in[K]} &x_{ij} \leq 1, &\forall\; i \in [N]\\
& \displaystyle\sum_{i\in[N]} &x_{ij} \geq C+1, &\forall\; j \in S\\
& \displaystyle\sum_{i\in[N]} &x_{ij} = 0, &\forall\; j \in [K] \setminus S.
\end{array}
\end{equation}
\begin{lemma}
\label{lem:member_pcs}
A point $x \in \mathbb{R}^{N \times K}$ belongs to $\mathcal{P}_{C}(S)$ iff it satisfies the constraints in \eqref{eq:lp}.
\end{lemma}
We can now prove the guarantees at the beginning of the section. We start with the case where $U \geq K(C+1)$. Here we show (via similar logic to the initial randomized argument) that $w \in \mathcal{P}_{C}([K])$.
\begin{lemma}\label{lem:kc}
If $U \geq K(C+1)$, then $w \in \mathcal{P}_{C}([K])$.
\end{lemma}
Applying Caratheodory's theorem, we immediately obtain a $C$-anonymous decomposition from Lemma \ref{lem:kc}
\begin{corollary}\label{cor:anon_decomp}
If $U \geq K(C+1)$, there exists a $C$-anonymous decomposition into at most $D+KC+1$ assignments. Moreover, it is possible to find this decomposition efficiently.
\end{corollary}
When $C+1 \leq U \leq K(C+1)$, we first arbitrarily partition our arms into $\alpha = \lceil K(C+1)/U \rceil$ blocks $S_1, S_2, \dots, S_{\alpha}$ of at most $U/(C+1)$ vertices each. We then show how to write $w$ as a linear combination of $\alpha$ points, one in each of the polytopes $\mathcal{P}_C(S_a)$.
\begin{lemma}\label{lem:midc}
There exist points $w^{(a)} \in \mathcal{P}_{C}(S_{a})$ for $1 \leq a \leq \alpha$ such that $w \leq \sum_{i=1}^{\alpha} w^{(a)}$.
\end{lemma}
Likewise, we can again apply Caratheodory's theorem to obtain a $C$-anonymous decomposition from Lemma \ref{lem:midc}.
\begin{corollary}\label{cor:anon_alpha_decomp}
If $(C+1) \leq U \leq K(C+1)$, there exists a $C$-anonymous decomposition into at most $\alpha D+NK+1$ assignments, where $\alpha = \lceil K(C+1)/U \rceil$. Moreover, it is possible to find this decomposition efficiently.
\end{corollary}
\subsection{Settings without user-clustering} \label{sec:nou}
The previous algorithms we have presented for anonymous bandits rely heavily on the existence of a user-cluster assumption $U$. In this section we present an algorithm (Algorithm \ref{alg:nou}) for anonymous bandits which works in the absence of any user-cluster assumption as long as $N > C$. This comes at the cost of a slightly higher regret bound which scales as $T^{2/3}$ -- as we shall see in Section \ref{sec:lbs}, this dependence is in some sense necessary.
\begin{algorithm2e}[h]
\caption{Explore-then-commit algorithm for anonymous bandits without a user-clustering assumption.}\label{alg:nou}
\textbf{Input:} Anonymity parameter $C$, time-horizon $T$, number of users $N$, and number of arms $K$. \\
Set $T_{exp} = 10C^{2/3}K^{1/3}T^{2/3}(\log NKT)^{1/3}$.\\
For each user $i \in [N]$ and arm $j \in [K]$ initialize two variables $n_{ij} = 0$ and $\sigma_{ij} = 0$.\\
\For{$r \gets 1$ \KwTo $\lceil T_{exp}/(2C+2) \rceil$}{
Divide $[N]$ arbitrarily into $S$ groups $G_{1}, \dots, G_{S}$ of size $C+1$ (adding all remaining users to the last group $G_{S}$). \\
For each group $G_{s}$, set $j_{s} = (r \bmod K)$. \\
Run the feedback-eliciting sub-algorithm on the assignment $\pi$ induced by the groups $G_{s}$ (where if $i \in G_{s}$ then $\pi(i) = j_{s}$). \\
\For{$i \gets 1$ \KwTo $N$}{
$n_{i,\pi(i)} \gets n_{i,\pi(i)} + 1$ \\
$\sigma_{i, \pi(i)} \gets \sigma_{i, \pi(i)} + \hat{\mu}_{i, \pi(i)}$. Here $\hat{\mu}_{i, \pi(i)}$ is the unbiased estimate for $\mu_{i, \pi(i)}$ produced by the feedback-eliciting sub-algorithm.
}
}
\For{remaining rounds $t$}{
Match user $i$ to $\arg\max_{j} \sigma_{ij}/n_{ij}$.
}
\end{algorithm2e}
Algorithm \ref{alg:nou} follows the standard pattern of Explore-Then-Commit algorithms (see e.g. Chapter 6 of \cite{lattimore2020bandit}). For approximately $O(T^{2/3})$ rounds, we run the feedback-eliciting sub-algorithm on random assignments from users to arms (albeit ones which are chosen to guarantee each arm with any users matched to it has at least $C+1$ users matched to it), getting unbiased estimates of the means $\mu_{i, j}$. For the remaining arms, we match each user to their historically best-performing arm.
\begin{theorem}\label{thm:nou_alg}
Algorithm \ref{alg:nou} incurs an expected regret of at most $\tilde{O}(NC^{2/3}K^{1/3}T^{2/3})$ for the anonymous bandits problem.
\end{theorem}
\subsection{Lower bounds}\label{sec:lbs}
We finally turn our attention to lower bounds. In all of our lower bounds, we exhibit a family of hard distributions over anonymous bandits problem instances, where any algorithm facing a problem instance randomly sampled from this distribution incurs at least the regret lower bound in question.
We begin by showing an $\Omega(N\sqrt{CKT})$ lower bound that holds even in the presence of a user-cluster assumption (and in fact, even when $U \geq K(C+1)$). Since Algorithm \ref{alg:main} incurs at most $\tilde{O}(NC\sqrt{KT})$ regret for $U \geq K(C+1)$, this shows the regret bound of our algorithm is tight in this regime up to a factor of $\sqrt{C}$ (and additional polylogarithmic factors).
\begin{theorem}\label{thm:main_lb}
Every learning algorithm for the anonymous bandits problem must incur expected regret at least $\Omega(N\sqrt{CKT})$, even when restricted to instances satisfying $U \geq K(C+1)$.
\end{theorem}
We now shift our attention to settings where there is no guaranteed user-cluster assumption. We first show that the $O(T^{2/3})$ dependency of the algorithm in Section \ref{sec:nou} is necessary.
\begin{theorem}\label{thm:lb_nou}
There exists a family of problem instances satisfying $N=C+1$ where any learning algorithm must incur regret at least $\Omega(T^{2/3})$.
\end{theorem}
Finally, we show the assumption in Section \ref{sec:nou} that $N > C$ is in fact necessary; if $N=C$ then it is possible that no algorithm obtains sublinear regret.
\begin{theorem}\label{thm:lb_lin}
There exists a family of problem instances satisfying $N=C$ where any learning algorithm must incur regret at least $\Omega(T)$.
\end{theorem}
\section{Simulations}
Finally, we perform simulations of our anonymous bandits algorithms -- the explore-then-commit algorithm (Algorithm \ref{alg:nou}) and several variants of Algorithm \ref{alg:alg1} with different decomposition algorithms -- on synthetic data. We observe that both the randomized decomposition and LP decomposition based variants of Algorithm \ref{alg:alg1} significantly outperform the explore-then-commit algorithm and the greedy decomposition variant, as predicted by our theoretical bounds. We discuss these in more detail in Section \ref{app:experiments} of the Supplemental Material.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,264 |
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"redpajama_set_name": "RedPajamaC4"
} | 106 |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-solve-and-graph-3-7c-4-18","text":"# How do you solve and graph -3<=7c+4<18?\n\n##### 1 Answer\nNov 2, 2017\n\nSee a solution process below:\n\n#### Explanation:\n\nFirst, subtract $\\textcolor{red}{4}$ from each segment of the system of inequalities to isolate the $c$ term while keeping the system balanced:\n\n$- 3 - \\textcolor{red}{4} \\le 7 c + 4 - \\textcolor{red}{4} < 18 - \\textcolor{red}{4}$\n\n$- 7 \\le 7 c + 0 < 14$\n\n$- 7 \\le 7 c < 14$\n\nNow, divide each segment by $\\textcolor{red}{7}$ to solve for $c$ while keeping the system balanced:\n\n$- \\frac{7}{\\textcolor{red}{7}} \\le \\frac{7 c}{\\textcolor{red}{7}} < \\frac{14}{\\textcolor{red}{7}}$\n\n$- 1 \\le \\frac{\\textcolor{red}{\\cancel{\\textcolor{b l a c k}{7}}} c}{\\cancel{\\textcolor{red}{7}}} < 2$\n\n$- 1 \\le c < 2$\n\nOr\n\n$c \\ge - 1$ and $c < 2$\n\nOr, in interval notation:\n\n$\\left[- 1 , 2\\right)$\n\nTo graph this we will draw vertical lines at $- 1$ and $2$ on the horizontal axis.\n\nThe line at $- 1$ will be a solid line because the inequality operator contains an \"or equal to\" clause. The line at $2$ will be a dashed line because the inequality operator does not contain an \"or equal to\" clause.\n\nWe will shade between the lines to show the interval:","date":"2020-09-18 14:47:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 17, \"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\": 0.6003344058990479, \"perplexity\": 581.0668023918314}, \"config\": {\"markdown_headings\": true, \"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-2020-40\/segments\/1600400187899.11\/warc\/CC-MAIN-20200918124116-20200918154116-00580.warc.gz\"}"} | null | null |
Wehay, It's The Bank Holiday Weekend - What Wine Are You Drinking Tonight?
Blossom Hill red for me (I'm not a classy bloke).
Cheeky little fruity number though with lots of berry fruits. It's doing the job for me.
7 out of 10, I reckon.
I only drink cheap plonk on the meagre pension I get so you would not be interested in the name.
Veuve Cliquot for us. 9/10. Punches well above its weight.
Oh alright then it`s Sainsburys Basics Spanish White Wine Vino de Mesa wine for the table not the cellar a medium dry table wine with fresh fruit flavours @ £2-68 a bottle.
Not a bad french merlot from a 5 litre box cost 8 euros 95 cents.....buy two get one free.
Don`t worry at my pub meet you can choose what you like to drink as a special Forum friend.
No plonk tonight. Nice cup of tea.
btw, jack daniels - OK out of 10.
I know 'Bart' was being ironic but I am genuine in saying I never touch the stuff. Tastes like a classy alcopop to me; far too strong and far too sweet.
Doom Bar, Tribute and HSD tonight. With a view across the bay and a low-to-the-horizon yellow moon and good company. It gets no better than this.
Where have the posters' names in the quotes gone? Sort it out, I remain yours faithfully etc. etc.
... and far too sweet.
Well, some of it is but there's much that's as astringent as.
I have a great sensitivity to sweet. When Starr Carr was excavated they found a lot of discarded sloe seeds which it was said evidenced that Neolithic man had a much more robust palate than us.
Well I happily much them as I go walking at this time of year, so I don't find them at all challenging. Slightly dry / alkaline if not ripe.
Bart's use of inverted commas confuses the software. A problem for HPC's often troublesome technical side. Other IP based fora I use do not suffer. Go figure.
The home brew ran out weerks ago and I've not got round to putting any more on so have to buy my bloody alcohol at present. Boulstering the "recovery" though, I suppose.
Chateau le Vinata 2007 (Bordeaux Red) €3.5 at Leclerc.
As an alternative Thatchers Katy Cider. This stuff actually tastes of apples unlike the chemical cocktail of loopy juice that masquerades as Cider in most boozers. It is a class act that converted me from being a 30 year hater of Cider into quite a fan.
Will I be able to see the sea from Callington?
Ah, good you made it down.
Well you will if you watch Home and Away. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,767 |
package com.opengamma.analytics.financial.interestrate.future.provider;
import com.opengamma.analytics.financial.interestrate.future.calculator.FuturesPriceCurveSensitivityIssuerCalculator;
import com.opengamma.analytics.financial.interestrate.future.calculator.FuturesPriceIssuerCalculator;
import com.opengamma.analytics.financial.interestrate.future.derivative.FuturesSecurity;
import com.opengamma.analytics.financial.provider.description.interestrate.ParameterIssuerProviderInterface;
import com.opengamma.analytics.financial.provider.sensitivity.multicurve.MulticurveSensitivity;
/**
* Interface to generic futures security pricing method for multi-curve and issuer provider.
*/
public class FuturesSecurityIssuerMethod extends FuturesSecurityMethod {
/** The futures price calculator **/
private static final FuturesPriceIssuerCalculator FPIC = FuturesPriceIssuerCalculator.getInstance();
/** The futures price calculator **/
private static final FuturesPriceCurveSensitivityIssuerCalculator FPCSIC = FuturesPriceCurveSensitivityIssuerCalculator.getInstance();
/**
* Computes the quoted price of a futures from a multicurve provider.
* @param futures The futures security.
* @param multicurve The multicurve provider.
* @return The price.
*/
public double price(final FuturesSecurity futures, final ParameterIssuerProviderInterface multicurve) {
return futures.accept(FPIC, multicurve);
}
/**
* Computes the quoted price curve sensitivity of a futures from a multicurve provider.
* @param futures The futures security.
* @param multicurve The multicurve provider.
* @return The price curve sensitivity.
*/
public MulticurveSensitivity priceCurveSensitivity(final FuturesSecurity futures, final ParameterIssuerProviderInterface multicurve) {
return futures.accept(FPCSIC, multicurve);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 0 |
package com.andbase.im.activity;
import java.util.ArrayList;
import java.util.List;
import android.os.Bundle;
import android.view.View;
import android.widget.AdapterView;
import android.widget.AdapterView.OnItemClickListener;
import android.widget.ListView;
import com.ab.activity.AbActivity;
import com.ab.db.storage.AbSqliteStorage;
import com.ab.db.storage.AbSqliteStorageListener.AbDataSelectListener;
import com.ab.db.storage.AbStorageQuery;
import com.ab.util.AbToastUtil;
import com.ab.view.pullview.AbPullToRefreshView;
import com.ab.view.pullview.AbPullToRefreshView.OnFooterLoadListener;
import com.ab.view.pullview.AbPullToRefreshView.OnHeaderRefreshListener;
import com.ab.view.titlebar.AbTitleBar;
import com.andbase.R;
import com.andbase.global.MyApplication;
import com.andbase.im.adapter.MessageListAdapter;
import com.andbase.im.dao.IMMsgDao;
import com.andbase.im.model.IMMessage;
/**
* 消息列表界面
*
*/
public class MessageActivity extends AbActivity implements OnHeaderRefreshListener,OnFooterLoadListener{
private MyApplication application;
private List<IMMessage> list = null;
private AbPullToRefreshView mAbPullToRefreshView = null;
private ListView mListView = null;
private MessageListAdapter myListViewAdapter = null;
//每一页显示的行数
public int pageSize = 10;
//当前页数
public int pageNum = 1;
//数据库操作类
private AbSqliteStorage mAbSqliteStorage = null;
private IMMsgDao mIMMsgDao = null;
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setAbContentView(R.layout.im_message_list);
AbTitleBar mAbTitleBar = this.getTitleBar();
mAbTitleBar.setTitleText(R.string.message_name);
mAbTitleBar.setLogo(R.drawable.button_selector_back);
mAbTitleBar.setTitleBarBackground(R.drawable.top_bg);
mAbTitleBar.setTitleTextMargin(10, 0, 0, 0);
mAbTitleBar.setLogoLine(R.drawable.line);
application = (MyApplication) getApplication();
//初始化AbSqliteStorage
mAbSqliteStorage = AbSqliteStorage.getInstance(this);
//数据业务类
mIMMsgDao = new IMMsgDao(this);
//获取ListView对象
mAbPullToRefreshView = (AbPullToRefreshView)this.findViewById(R.id.mPullRefreshView);
mListView = (ListView)this.findViewById(R.id.mListView);
//打开关闭下拉刷新加载更多功能
mAbPullToRefreshView.setOnHeaderRefreshListener(this);
mAbPullToRefreshView.setOnFooterLoadListener(this);
//设置进度条的样式
mAbPullToRefreshView.getHeaderView().setHeaderProgressBarDrawable(this.getResources().getDrawable(R.drawable.progress_circular));
mAbPullToRefreshView.getFooterView().setFooterProgressBarDrawable(this.getResources().getDrawable(R.drawable.progress_circular));
//mAbPullListView.getHeaderView().setHeaderProgressBarDrawable(this.getResources().getDrawable(R.drawable.progress_circular2));
//mAbPullListView.getFooterView().setFooterProgressBarDrawable(this.getResources().getDrawable(R.drawable.progress_circular2));
//ListView数据
list = new ArrayList<IMMessage>();
//使用自定义的Adapter
myListViewAdapter = new MessageListAdapter(this,list);
mListView.setAdapter(myListViewAdapter);
//item被点击事件
mListView.setOnItemClickListener(new OnItemClickListener(){
@Override
public void onItemClick(AdapterView<?> parent, View view,
int position, long id) {
}
});
}
@Override
protected void onResume(){
//第一次下载数据
list.clear();
queryData(0);
super.onResume();
}
@Override
public void onHeaderRefresh(AbPullToRefreshView view) {
pageNum = 1;
list.clear();
queryData(0);
}
@Override
public void onFooterLoad(AbPullToRefreshView view) {
pageNum ++;
queryData(1);
}
public void queryData(final int query){
//查询数据
AbStorageQuery mAbStorageQuery = new AbStorageQuery();
mAbStorageQuery.equals("type", IMMessage.SYS_MSG);
AbStorageQuery mAbStorageQuery2 = new AbStorageQuery();
mAbStorageQuery2.equals("type", IMMessage.ADD_FRIEND_MSG);
mAbStorageQuery.or(mAbStorageQuery2);
mAbStorageQuery.setLimit(pageSize);
mAbStorageQuery.setOffset((pageNum-1)*pageSize);
//无sql存储的查询
mAbSqliteStorage.findData(mAbStorageQuery, mIMMsgDao, new AbDataSelectListener(){
@Override
public void onFailure(int errorCode, String errorMessage) {
AbToastUtil.showToast(MessageActivity.this,errorMessage);
}
@Override
public void onSuccess(List<?> paramList) {
if(query==0){
if(paramList!=null && paramList.size()>0){
list.addAll((List<IMMessage> )paramList);
myListViewAdapter.notifyDataSetChanged();
}
mAbPullToRefreshView.onHeaderRefreshFinish();
}else{
if(paramList!=null){
list.addAll((List<IMMessage>)paramList);
myListViewAdapter.notifyDataSetChanged();
}
mAbPullToRefreshView.onFooterLoadFinish();
}
}
});
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,245 |
Q: Role-based REST API? I'm building a REST API for which several users with different roles will have access to the resources it contains.
To keep the scope simple let's take the "student/teacher/class" domain:
GET /students is the resource to access.
Users might have roles like Student and/or Teacher
Students will only have access to students of their classes. Teachers will have access to students of the classes they teach. Some uses may be a student AND teach other classes too. They must have access to the students of their classes AND the students of the classes they teach.
Ideally I want to implement this as two functions - one per role and then "union" if a user have multiple roles.
My question is: Which pattern should I use for implementing this?
Externally
*
*Should I split up my API per role? GET /teacher/students and GET /student/students It doesn't seems right to me.
*Keep it all I'm one resource (preferred)
Internally
How should it be implemented internally?
*
*Should every method start with a BIG switch/if per role?
*Should I implement a repository per role?
*Is there a design pattern that will help me in achieving this?
As a side comment: I'm using ASP.NET Web API and Entity Framework 6, but it really doesn't matter for the conceptual implementation.
A: You should architect the API around resources, not around roles, e.g.:
/rest/students
should be accessible to anyone with a role that allows them to see students.
Internally, you are implementing role-based security. How you go about that depends on the details of your application, but let's say you have a role table, each person has one or more roles, and those roles determine what each person can access. You have already stated the rules for accessing students:
*
*students can access students in the classes they take
*teachers can access students in the the classes they teach
So when a person calls:
/rest/students
you call a method that accesses students, passing in the role of the person. Here is some pseudo code:
roles = person.roles; //array
students = getStudents( roles );
return students;
and in that method, you could get the students for each role with separate calls, e.g.:
factory = getFactory();
classes= [];
students = [];
for( role in roles ){
service = factory.getService( role );
// implementation details of how you get classes for student/teacher are hidden in the service
classes = classes.merge( service.getClasses( person ) );
// classes[] has class.students[]
// loop on classes and add each student to students, or send back classes with nested students? depends on use case
}
}
That's a very rough idea for what you could do and isn't necessarily going to fit your specific needs, but it should give you a sense of the pieces involved. If you want to return the classes with each student listed, this is a good approach. If you just want the students, you could extract them from each class and merge them into a collection of students.
No, you should not have a separate repository per role. All the role does is determine how you get the data, and maybe what you can do with the data (e.g. Teachers can enter Student grades). The data itself is the same.
As for patterns, this approach is using Factory Pattern to abstract away the service that gets data based on role. It may or may not be appropriate to have separate services by role. I like this approach because it minimizes the amount of code at each stage of the program and makes it more readable than a switch or if block.
A: Find a pen and a paper and start modelling your system.
You will find that you probably need a domain entity called PERSON. Since both STUDENTS and TEACHER "is-a" PERSON, you could create an abstract entity called PERSON with generic attributes like firstname, lastname, etc. A TEACHER -> is-a -> Person. Now you can try to find characteristics for a TEACHER that doesn't apply to STUDENTS; e.g. A TEACHER teaches CLASS(es) regarding one or more SUBJECT(s).
Enforcing security is considered a non-functional aspect of your application. It is a cross-cutting concern that should be handled outside of your "business logic". As @Robert Munn points out, the ROLE(s) should all be maintained in one place. Using roles to limit access to certain functions is rather coarse-grained, and the concept is called role-based access control (RBAC).
To verify whether or not a teacher should be allowed to see a students grades, should to be expressed in your domain model. Say a teacher has a class on the subject programming. You would probably express in your model that students attends classes for different subjects. This is where the application/business logic kicks in. This is logic that you can verify using test-driven development.
You should split your resources to make your application testable and modular.
Anyway, the best way to really show what I mean is to show it with code :) Here is a GitHub page: https://github.com/thomasandersen77/role-based-rest-api
Good luck :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,045 |
Улун Дану () — крупный индуистский храм в районе Бедугул, округ Табанан, Бали, Индонезия. Храм расположен на берегах . Этот и более мелкие водные храмы участвуют в орошении земель.
Сооружение было построено в 1633 году. Храм используется для жертвоприношений богине воды, рек и озёр .
11-ярусная башня посвящена божеству Шиве и его супруге Парвати. В храме также находится статуя Будды и ступа.
Храм также называется Храмом на воде, поскольку во время прилива создаётся ощущение того, что храм находится на воде.
Посмотреть 3D-модель ступы можно здесь.
Примечания
См. также
Индуизм в Индонезии
Литература
Индуистские храмы Индонезии
Храмы Шивы
Здания и сооружения, построенные в 1663 году | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,284 |
{"url":"https:\/\/vsoch.github.io\/2013\/sparse-k-means-clustering-sparcl\/","text":"Tibshirani and Witten introduced a variation of K-Means clustering called \u201cSparcl.\u201d \u00a0What gives it this name? \u00a0It is a method of sparse clustering that clusters with an adaptively chosen set of features, by way of the lasso penalty. \u00a0This method works\u00a0best when we have more features than data points, however it can be used in the case when data points > features as well. \u00a0The paper talks about the application of Sparcl to both K-Means and Hierarchical Clustering, however we will review it based on just K-Means. \u00a0We will start with doing what the paper does, which is reviewing previous work in sparse clustering. Previous Work in Sparse Clustering\n\n\u2022 Dimensionality Reduction Algorithms:\n\u2022 Use PCA to reduce the dimensionality of your data, and cluster the reduced data\n\u2022 Use Non-Negative Matrix factorization, and cluster the resulting component matrix\n\nWhat are the problems with the dimensionality reduction proposals above? \u00a0They don\u2019t allow for sparse features, and we cannot be sure that our signal of interest is in the resulting decomposition. \u00a0Why? \u00a0The components with the largest eigenvalues do not necessarily do the best to distinguish classes. \u00a0Let\u2019s talk about other ideas.\n\n\u2022 Model-based clustering framework:\n\u2022 Model the rows of the data, X, as independent multivariate observations from a mixture model with K components, and fit with the EM model. \u00a0The picture below \u00a0shows this approach:\n\nThe function, fk is a is Gaussian density, and it is parameterized by its mean and covariance matrix. \u00a0We iterate from 1 to K in the inside loop, one for each Gaussian, and the outside loop from 1 to n is going through each of our data objects. \u00a0Fitting with EM means fitting by expectation maximization. \u00a0The problem with this algorithm for this application is in the case when our number of features (p) is much bigger than our number of observations (n). \u00a0In this case, we can\u2019t estimate the p x p covariance matrix. \u00a0There are ways around this (see paper), however they involve using dimensionality reduction, and the resulting components, in being combinations of all features, do not allow for sparsity. \u00a0Poozle. \u00a0Is there another way we can use model-based clustering? Yes.\n\n\u2022 Use model-based clustering (above), except instead of maximizing the log likelihood, maximize the log likelihood\u00a0subject to a penalty that will enforce sparsity. \u00a0What does that look like?\n\nAbove, you will see the same equation as before except\u2026 oh hello, L1 penalty! \u00a0The last double summation above is basically summing up the means for each feature (j) across all Gaussians (k). \u00a0The parameter out front, lambda, is a fudge factor that gives us some control over the penalty. \u00a0When lambda is really big, this means that we are penalizing more, and so some of the Gaussian means will be exactly equal to zero. \u00a0When this is the case for some feature, j, over all Gaussians, K, this means that we don\u2019t include the feature in the clustering. \u00a0This is what makes it sparse. \u00a0 \u00a0It isn\u2019t clear in the paper why this approach isn\u2019t sufficient, but let\u2019s move into their proposed \u201cSparcl.\u201d\n\n### Sparse Clustering, Sparcl\n\nLet\u2019s start simply, and generally. \u00a0Unsupervised clustering problems are trying to solve the following problem:\n\nXj denotes some feature, j, and so the function is \u201csome function\u201d that involves only the j-th feature. \u00a0We are trying to maximize our parameters for this function. \u00a0For example, for K-Means, this function is the between cluster sum of squares for feature j. \u00a0The paper proposes sparse clustering as the solution to the following problem:\n\nThe function, f, is the between cluster sum of squares, j is still an index for a particular feature, and wj is the weight for that feature. \u00a0We can say a few things about the equation above:\n\n\u2022 If all of our weights are equal, the function above reduces to the first (without the subject to addition) and we are using all features weighted equally.\n\u2022 s is again a tuning parameter. \u00a0When it is small, this means that we are more stringent, and more of our weights will be set to zero.\n\u2022 The L2 norm, which means that we square each weight, sum the squares, and then take the square route of that, is important as well, because without it, at most one weight would be set to zero.\n\u2022 The value of the weights, as usual, is reflective of how meaningful a particular feature, j, is to the sparse clustering. \u00a0A large weight means that the feature is important, while a small one means the opposite. \u00a0A weight of zero means that we don\u2019t use it at all. \u00a0Dear statistics, thank you for making at least\u00a0one thing like this intuitive!\n\u2022 Lastly, the last term in the \u201csubject to\u201d says that we can\u2019t have all of our weights equal to zero, in which case we have no selected features at all!\n\nHow do we solve the equation above? \u00a0We first hold our weights, w fixed, and optimize with respect to the parameter. \u00a0We then hold our parameters fixed, and optimize w. \u00a0The paper notes that when we hold the parameters fixed and optimize w, the problem can be rewritten as:\n\nFor the above, the \u201csubject to\u201d parameters are equivalent, and since our parameters are fixed, we are just multiplying the weights by a, where a is a function of X and the fixed parameter. \u00a0The paper notes that this can be solved with soft-threholding, something that I need to read up on. \u00a0It is detailed in the paper if you are interested. \u00a0Now let\u2019s talk about this approach specifically applied to K-Means.\n\n### Sparse K-Means Clustering (Sparckl?) :)\n\nThis is going to be a good method for three reason:\n\n1. the criteria takes on a simple form\n2. easily optimized\n3. the tuning parameter controls the number of features in the clustering.\n\nLet\u2019s first talk about standard K-Means. \u00a0For standard K-Means, our goal is to minimize the within cluster sum of squares (WCSS). This means that, for each cluster centroid, we look at the points currently assigned to that cluster, square each one, and add them up. \u00a0If this value is small, our cluster is tight and awesome. \u00a0The equation below is saying exactly this: \u00a0on the inside we calculate the distance between each cluster centroid and each member across all features j, and then we make sure to divide by the number of members in the cluster (nk) to normalize for differences in cluster sizes. \u00a0We then add up the summed distances for each cluster, K, to get a \u201cfinal\u201d total distance.\n\nWe want to minimize this total distance to yield the \u201ctightest\u201d clusters, i.e., the clusters that have their points closest to their centroids. \u00a0The equation above uses all features, j, and so there is not sparsity. \u00a0Before we add a penalty and weights, let\u2019s talk about the between cluster sum of squares, BCSS.\n\nThe term on the right is exactly what we had before, the within cluster sum of squares for all clusters K. \u00a0However, what we are trying to get at is the between\u00a0cluster sum of squares, meaning that we want to minimize the distance between all the points in a cluster and all the other points\u00a0not\u00a0in the cluster. \u00a0So the equation above, in the first term that does not take a cluster k into account, is basically saying \u201cok, let\u2019s imagine that we don\u2019t have any clusters, and just calculate the pairwise distance for all of our observation, n, and then get an average. \u00a0 We can then subtract the within cluster sum of squares, leaving only the distances for the pairs of points from different clusters. \u00a0The between cluster sum of squared distances. \u00a0And we do this for all of our features, p. \u00a0Maximizing the between cluster sum of squares (above, meaning that points in different clusters are far apart) is equivalent to minimizing the within cluster sum of squares.\n\nNow, let\u2019s add weights and penalty terms! \u00a0The paper notes that if we were to add these parameters to the equation that just specifies the WCSS, since each element of the weighted sum is negative (see that big negative sign in front of the WCSS equation?) this would mean that all of our weights would be zero. Fail! \u00a0Instead, let\u2019s add the weights and penalty terms to the BCSS equation:\n\n### The Sparse K-Means Clustering Criterion\n\nHoly crap, that looks ugly. \u00a0But it\u2019s really not as scary as we think - all that we\u2019ve done is taken the BCSS equation from above, and appended a weight (w) to the front for each feature, j. \u00a0As before, we can say some things about this criterion:\n\n\u2022 When the weights are all equal, we are back to basic BCSS\n\u2022 The smaller s, the more sparse the solution\n\nIntuitively, we are assigning a weight to each feature, j, based on the increase in BCSS that the feature can contribute. \u00a0Why does this make sense? If BCSS is greater, we would want our weight to be larger to result in a larger value. \u00a0If BCSS is smaller, we don\u2019t get so much out of the feature, and since we are constrained in our values for w, we choose a smaller w and (hopefully) other features will be more meaningful.\n\n### How to use the criterion to solve for the optimal weights\n\n1. Initialize each weight equal to 1\/sqrt(p)\n2. Iterate until convergence {\n\nForget about the first term in the BCSS equation, and just think of standard K-Means. \u00a0Our first step is to minimize the WCSS for our current (fixed) weights:\n\nThis is how we are going to come up with our cluster assignments and centroids - we want points getting matched to their closest centroid, and on each iteration we re-define the cluster centroids based on the assigned points. \u00a0Yes, this is standard K-Means, out of the box!\n\nWhen we finish with the above, we have defined our centroids, cluster member indices, and this selection has resulted in a minimum WCSS distance. \u00a0We now hold these cluster assignments fixed (C1 to Ck), and can look at the left side of the BCSS equation. \u00a0As was stated previously, we want to maximize this, and we are going to use something called \u201csoft thresholding.\u201d \u00a0Again, see the paper, I will try to discuss soft thresholding in another post. \u00a0This results in a new set of weights. \u00a0We then define a stopping criterion (below), and keep iterating through these two steps until this stopping criterion is reached:\n\nwr refers to the set of weights obtained at some iteration r, so we are stopping when the change in these weights from one iteration to the next is tiny. \u00a0Yes, this is convergence.\n\n}","date":"2019-12-07 19:58:47","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.87700355052948, \"perplexity\": 826.5496129333299}, \"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-2019-51\/segments\/1575540501887.27\/warc\/CC-MAIN-20191207183439-20191207211439-00007.warc.gz\"}"} | null | null |
Blepharoneurinae zijn een onderfamilie van insecten uit de orde vliegen en muggen of tweevleugeligen (Diptera).
Taxonomie
De volgende taxa zijn bij de onderfamilie ingedeeld:
Geslacht Baryglossa Bezzi, 1918
Baryglossa bequaerti Bezzi, 1924
Baryglossa emorsa Munro, 1957
Baryglossa histrio Bezzi, 1918
Baryglossa mimella Munro, 1957
Baryglossa oldroydi Munro, 1957
Baryglossa tersa Munro, 1939
Baryglossa trulla Munro, 1957
Geslacht Blepharoneura Loew, 1873
Blepharoneura amazonensis Lima & Leite, 1952
Blepharoneura amplihyalina Norrbom & Condon, 2010
Blepharoneura apaapa Norrbom & Condon, 2010
Blepharoneura aspiculosa Norrbom & Condon, 2010
Blepharoneura atomaria (Fabricius, 1805)
Blepharoneura bidigitata Norrbom & Condon, 2010
Blepharoneura bipunctata Norrbom & Condon, 2010
Blepharoneura biseriata Wulp, 1899
Blepharoneura bivittata Norrbom & Condon, 2010
Blepharoneura brevivittata Norrbom & Condon, 2010
Blepharoneura chaconi Norrbom & Condon, 2010
Blepharoneura cornelli Norrbom & Condon, 2010
Blepharoneura cyclantherae Norrbom & Condon, 2010
Blepharoneura diva Giglio-Tos, 1893
Blepharoneura femoralis Wulp, 1899
Blepharoneura fernandezi Norrbom & Condon, 2010
Blepharoneura furcifer Hendel, 1914
Blepharoneura hirsuta Bates, 1933
Blepharoneura hyalinella Norrbom & Condon, 2010
Blepharoneura impunctata Hendel, 1914
Blepharoneura io Giglio-Tos, 1893
Blepharoneura isolata Norrbom & Condon, 2010
Blepharoneura longicauda Hendel, 1914
Blepharoneura lutea Norrbom & Condon, 2010
Blepharoneura macwilliamsae Norrbom & Condon, 2010
Blepharoneura manchesteri Condon & Norrbom, 1994
Blepharoneura marshalli Norrbom & Condon, 2010
Blepharoneura mexicana Norrbom & Condon, 2010
Blepharoneura mikenoltei Norrbom & Condon, 2010
Blepharoneura multipunctata Norrbom & Condon, 2010
Blepharoneura nigriapex Norrbom & Condon, 2010
Blepharoneura nigrifemur Norrbom & Condon, 2010
Blepharoneura nigripilosa Hering, 1935
Blepharoneura osmundsonae Norrbom & Condon, 2010
Blepharoneura parva Hendel, 1914
Blepharoneura perkinsi Condon & Norrbom, 1994
Blepharoneura poecilogastra (Loew, 1873)
Blepharoneura poecilosoma (Schiner, 1868)
Blepharoneura pulchella (Wulp, 1899)
Blepharoneura punctistigma Norrbom & Condon, 2010
Blepharoneura quadristriata Wulp, 1899
Blepharoneura quetzali Norrbom & Condon, 2010
Blepharoneura regina Giglio-Tos, 1893
Blepharoneura ruptafascia Norrbom & Condon, 2010
Blepharoneura rupta (Wulp, 1899)
Blepharoneura septemdigitata Norrbom & Condon, 2010
Blepharoneura sinepuncta Norrbom & Condon, 2010
Blepharoneura splendida Giglio-Tos, 1893
Blepharoneura tau Norrbom & Condon, 2010
Blepharoneura thetis Hendel, 1914
Blepharoneura unifasciata Norrbom & Condon, 2010
Blepharoneura variabilis Norrbom & Condon, 2010
Blepharoneura wasbaueri Norrbom & Condon, 2010
Blepharoneura zumbadoi Norrbom & Condon, 2010
Geslacht Ceratodacus Hendel, 1914
Ceratodacus longicornis Hendel, 1914
Ceratodacus priscus Norrbom & Condon, 2000
Geslacht Hexaptilona Hering, 1941
Hexaptilona hexacinioides (Hering, 1938)
Hexaptilona palpata (Hendel, 1915)
Geslacht Problepharoneura Norrbom & Condon, 2000
Problepharoneura antiqua Norrbom & Condon, 1999
Boorvliegen | {
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\section{Introduction}
Toroidal compactification of the heterotic string~\cite{narain}
preserves the full spacetime supersymmetry, giving the algebras
$N$$=$$4$ in $D$$=$$4$, $N$$=$$2$ (nonchiral) in $D$$=$$6$, and
$N$$=$$1$ in $D$$=$$8$. These maximally supersymmetric
compactifications have played a major role in recent discussions of
strongly coupled string theory.\footnote {For reviews and recent
discussions see refs.~\cite{sdual},
\cite{ssdual},~\cite{eddual}.}
Banks and Dixon\cite{bd} showed that maximal supersymmetry in the
heterotic string requires that the right-moving (supersymmetric)
target space be a
$(10$$-$$D)$-dimensional torus. For some time the only known
example was that of ref.~\cite{narain} in which the left-moving
degrees of freedom were also toroidal. Recently, Chaudhuri,
Hockney, and Lykken (CHL) have used fermionization to construct
many new exact conformal field theory solutions\footnote{We will
use the term \lq\lq solutions'' since it not clear at present
whether all solutions to the heterotic string consistency
conditions can be interpreted as backgrounds or compactifications
of the ten-dimensional string, though this is widely assumed.}
having maximal supersymmetry in any spacetime dimension
$D$$<$$10$~\cite{chl}. These solutions are characterized by a
reduction of the rank of the left-moving gauge group relative to
the toroidally compactified ten dimensional heterotic string. CHL
have found maximally supersymmetric $N$$=$$4$ theories in $D$$=$$4$
spacetime dimensions with gauge groups of rank
\begin{equation}
(r_L, r_R)=(26-D-k,\ 10-D), \quad k=8,12,14,16,18,20,22 ~ .
\end{equation}
Here $k$ is the reduction of the rank. The first two members of
this series can be followed up to (even) spacetime dimension
$D$$=$$8$ and $D$$=$$6$, respectively. Since the moduli in these
maximally supersymmetric theories appear in vector multiplets, the
number of inequivalent marginal deformations at any point in the
moduli space is fixed to be $r_L r_R$. It was shown in
\cite{chl},\cite{sc} that such moduli spaces contain points of both
non-simply-laced and simply-laced enhanced symmetry, as well as
higher level realizations of the gauge symmetry.\footnote
{We should also note that analogous solutions for the Type II
string, four dimensional theories with extended spacetime
supersymmetry including
$N$$=$$4_R$$+$$ 0_L$, $4_R$$+$$ 1_L$,
$4_R $$+$$ 2_L$, and
$2_R $$+$$ 2_L$, and with reduced dimension moduli spaces, were
discovered some time ago by Ferrara and Kounnas using the free
fermionic construction~\cite{costas}. The Type II solutions differ
from those of
\cite{chl} in using left-right symmetric Majorana fermions, which
are excluded in the heterotic case by maximal supersymmetry
\cite{bd}. The heterotic solutions require chiral Majorana
fermions. A target space interpretation of these Type II conformal
field theory backgrounds, and the identification of their duals,
remains to be explored. We would like to thank Costas Kounnas for
informing us of this work.}
The purpose of this paper is to clarify the nature of these
theories. While fermionization has proven to be a powerful
technique for discovering qualitatively new classes of string
compactifications it can give only isolated pieces of their moduli
space, generally at points of higher symmetry. We will describe an
orbifold construction which yields the moduli space of the first
theory in this series, recovering, in particular, all of the $k=8$
solutions given by the fermionic construction. We also discuss the
implications for strong-weak coupling duality in string theory. For
the conjectured self-duality of the heterotic string in $D=4$ we
find new evidence: gauge groups always appear with their
electric-magnetic duals, for example $Sp(20)
\times SO(9)$ and $Sp(8) \times SO(21)$ in the same moduli space.
However, the CHL theories present a puzzle for the conjectured
string-string duality in $D=6$.
\section{Construction}
A clue to the nature of these theories comes from considering the
decompactification limit. Moduli $j \tilde j$ can always be
constructed from the Cartan subalgebra of the gauge group. At
large values of these moduli the spectrum becomes continuous, and
as long as the left-moving rank is at least
$10-D$ one can reach in this way a theory with 10 large
translationally invariant dimensions, which must be a toroidal
compactification of either the $E_8$$\times$$E_8$ or $SO(32)$
heterotic string. This seems a puzzle at first, since the
construction of ref.~\cite{narain} appears to be general, but there
is at least one additional possibility. Consider the effect of
transporting a string around a toroidal dimension. The string must
return to its original state up to a symmetry $g$. For
$g$ a general gauge transformation (inner automorphism of the gauge
group), this is equivalent to a Wilson line background and is
therefore included in the classification of ref.~\cite{narain}.
However, the $E_8 \times E_8$ theory also has an outer isomorphism
which interchanges the two $E_8$'s.\footnote {The outer isomorphism
has been considered previously in refs.~\cite{outer}, and very
recently in \cite{andy}, but always in conjunction with right-moving
twists which reduce the spacetime supersymmetry.} Modding out by
the outer isomorphism alone does not lead to a new theory, because
the gauge bosons of the other $E_8$ are recovered in the twisted
sector. The only additional possibility in $D$$=10$ is to combine
this action with a non-trivial twist on the world-sheet fermions,
but this breaks the spacetime supersymmetry to
$N$$=$$0$ \cite{outer}. Since our interest is in $Z_2$ actions that
leave the supersymmetry unbroken, twists on the world-sheet
fermions are excluded. Compactifying on a torus to any spacetime
dimension $D$$<$$10$ opens up a new option: an accompanying
translation in the torus. Begin with an ordinary toroidal
compactification to $D$ spacetime dimensions and twist by $RT$,
where $R$ is the outer isomorphism that interchanges the two $E_8$
lattices and $T$ is a translation in the spacetime torus. This has
just the necessary action: it eliminates one linear combination of
the two $E_8$'s, leaving the diagonal $E_8$ at level~2.
By a similarity transformation
$T$ and $R$ can be taken to commute. Without loss of generality we
assume that $(RT)^2 = T^2$ is a symmetry of the original lattice,
else we could twist by $T^2$ to abtain a different lattice in the
same moduli space. Also we can assume that $T$ is not a symmetry of
the lattice; if it were then $RT$ would be equivalent to
$R$ and so would actually act trivially.
Denote a general momentum state by $|p_1,p_2,p_3\rangle$.
Here
\begin{equation}
p_1^I = {\textstyle{{{1}\over{\sqrt 2}}}} (p^I - p'^I), \quad
p_2^I = {\textstyle{{{1}\over{\sqrt 2}}}} (p^I + p'^I), \qquad I = 1, \ldots, 8
\end{equation}
are the linear combinations of momenta in the two $E_8$
lattices which are respectively reflected and left invariant
by $R$, while
\begin{equation}
(p_{3}^m ; \tilde p_{3}^m), \qquad m = D, \ldots, 9
\end{equation}
are the momenta of the torus. Taking $T$ to be a translation by
$2\pi(0, v_2^I, v_{3}^m ; \tilde v_{3}^m)$, $RT$ acts as
\begin{equation}
RT |\, p_1,\ p_2,\ p_3\rangle = e^{i2\pi v \cdot p}
|-p_1,\ p_2,\ p_3\rangle
\end{equation}
where the inner product has signature~$(26-D, 10-D)$.
We start with the $E_8$$\times$$E_8$ theory,
with $10 - D$ dimensions compactified on a given torus without
background gauge fields.
The momentum lattice of such a theory is of the form
\begin{equation}
\Gamma = \Gamma^8 \oplus \Gamma^8 \oplus \Gamma',
\end{equation}
where $\Gamma'$, is even, self-dual, and of Lorentzian
signature $(10$$-$$D,10$$-$$D)$.
In the basis above $\Gamma$
takes the form
\begin{equation}
\left| {\textstyle{{{1}\over{\sqrt 2}}}} (p^I - p'^I),\ {\textstyle{{{1}\over{\sqrt 2}}}} (p^I + p'^I),\ p_3 \right\rangle,
\end{equation}
where $p^I, p'^I \in \Gamma^8$ and $p_3 \in \Gamma'$.
Specifically,
\begin{equation}
\Gamma^8: \quad p^I = {\textstyle{1\over2}} m^I, \qquad m^I \in {\bf Z},\ \ m^I -
m^J
\in 2{\bf Z},\ \ \sum_I m^I \in 4{\bf Z}.\nonumber\\
\end{equation}
General points in the moduli space can then be reached by boosts of
the momentum lattice as in ref.~\cite{narain}. In order to allow a
twist by $R$ the boost $\Lambda$ must commute with
$R$, leaving $p_1$ invariant.\footnote {More generally there is the
possibility that $\Lambda R \Lambda^{-1}$ be a nontrivial discrete
symmetry (duality) of $\Gamma$, leading to a disconnected moduli
space. We will not consider this here.} This subgroup is $SO(18 -
D, 10 - D)$, so the moduli space of inequivalent vacua is locally
of the form
\begin{equation}
{SO(18-D, 10-D)}\over{SO(18-D)\times SO(10-D)}
\end{equation}
as required by considerations of low energy supergravity.
The twist by $RT$ produces an asymmetric orbifold \cite{nsv}. We
review the relevant results from that paper. The lattice $I$ is
defined to consist of those momenta invariant under $R$. Here, this
implies
\begin{equation}
I:\quad p_1 = 0,\ \ p_2 \in {\textstyle{\sqrt 2}} \Gamma^8,\ \ p_3 \in \Gamma'.
\end{equation}
The dual lattice, in the subspace invariant under $R$, is
\begin{equation}
I^*:\quad p_2 \in {\textstyle{{{1}\over{\sqrt 2}}}} \Gamma^8 ,\ \ p_3
\in \Gamma'.
\end{equation}
The number of twisted sectors is $D$ where
\begin{equation}
D^2 = \det{}'(1-R) |I/I^*| = 2^{8-8} = 1.
\end{equation}
The momenta in the twisted sector are
\begin{equation}
p \in I^* + v \label{twistmom}
\end{equation}
and the twisted sector spectrum consists of the level-matched
states. The untwisted sector simply retains the $RT$-invariant
states. The left-moving zero point energies are the usual $-
\frac{24}{24} = -1$ in the untwisted sector and $-\frac{16}{24} +
\frac{8}{48} = -\frac{1}{2}$ in the twisted sector.
\subsection{Compactification to $D=8$}
Consider first the case $D=8$. Let
$\Gamma'$$=$$\Gamma^{SU(2)}$$\oplus$$\Gamma^{SU(2)}$, two
copies of the free boson at the $SU(2)$ radius:
\begin{equation}
\Gamma^{SU(2)} :\quad p_{3} = {\textstyle{{{1}\over{\sqrt 2}}}} \left( n ;\ {\tilde n}\right) ,
\qquad
n, \tilde n \in {\bf Z},\ \ n + \tilde n \in 2{\bf Z}.
\end{equation}
We take $T$ to be a translation by half
the spacetime periodicity in the $m$$=$$9$ direction,
\begin{equation}
v_2 = 0, \quad v_3 = \left( 0,{\textstyle{{{1}\over{2 \sqrt 2}}}} ;\ 0,-{\textstyle{{{1}\over{2 \sqrt 2}}}} \right).
\end{equation}
Let us determine the gauge symmetry. The untwisted sector contains
ten neutral left-moving gauge bosons\footnote {The right moving
gauge symmetry is always $U(1)^{10-D}$ in a maximally
supersymmetric compactification of the heterotic string \cite{bd}.}
\begin{equation}
{\textstyle{{{1}\over{\sqrt 2}}}} (\alpha^I_{-1} + \alpha'^I_{-1})
|0,0,0\rangle ,
\qquad \alpha^m_{-1} |0,0,0\rangle, \quad m=8,~9, \label{neutral}
\end{equation}
the eight antisymmetric neutral combinations being removed by the
$RT$ projection. Let $r$ denote any root of $E_8$. The charged gauge
bosons in the $E_8$$\times$$E_8$ theory which are invariant under
$RT$ are the symmetric combinations
\begin{equation}
{\textstyle{{{1}\over{\sqrt 2}}}} \Bigl(\,\Bigl| {\textstyle{{{1}\over{\sqrt 2}}}} r ,\ {\textstyle{{{1}\over{\sqrt 2}}}} r ,\ 0 \Bigr\rangle
+ \Bigl| - {\textstyle{{{1}\over{\sqrt 2}}}} r ,\ {\textstyle{{{1}\over{\sqrt 2}}}} r ,\ 0 \Bigr\rangle\Bigr) \label{e8l2}
\end{equation}
and the $SU(2)$ gauge bosons
\begin{equation}
p_1 = p_2 = 0,\ p_3 = \left( \pm {\textstyle{\sqrt 2}} , 0 ;\ 0,0 \right).
\label{su28}
\end{equation}
In the twisted sector there are no massless states because the
right-moving component of the momentum~(\ref{twistmom}) will not
vanish. The states~(\ref{e8l2}) form the $E_8$ root lattice, but the
momentum~$p_2$ to which the neutral gauge bosons~(\ref{neutral})
couple is scaled down by a factor of $\sqrt{2}$ so the current
algebra realization of $E_8$ is at level~2. The gauge
bosons~(\ref{su28}) are the remnant of one of the original
$SU(2)$'s, the other being removed by the $RT$ projection, so that
the gauge symmetry originates in the left-moving current algebra
$(E_8)_2 \times SU(2)_1 \times U(1)$.
General points in the moduli space are reached by $SO(10,2)$ boosts.
Generically the left-moving symmetry is broken to $U(1)^{10}$, but
there are also points of enhanced gauge symmetry. Consider for
example the vectors
\begin{eqnarray}
e_{(1)}:&& p_1 = p_2 = 0,\ p_3 = {\textstyle{\sqrt 2}} \left( 1,0;\ 0,0 \right), \nonumber\\
e_{(2)}:&& p_1 = p_2 = 0,\ p_3 = {\textstyle{{{1}\over{\sqrt 2}}}} \left( 0,3;\ 2,1 \right).
\end{eqnarray}
These survive the $RT$ projection in the untwisted sector
and satisfy
\begin{equation}
e_{(i)} \cdot e_{(j)} = 2 \delta_{ij}. \label{orth}
\end{equation}
Making an $SO(2,2)$ boost to a frame in which ${\textstyle{{{1}\over{\sqrt 2}}}} e_{(i)}$ form a
left-moving orthonormal basis, these become $SU(2)$$\times
SU(2)$$=$$ SO(4)$ weights. The vectors
\begin{equation}
\pm {\textstyle{1\over2}} e_{(1)} \pm' {\textstyle{1\over2}} e_{(2)} ,
\end{equation}
where all four independent choices of sign are included, appear in
the twisted sector~(\ref{twistmom}). These momenta have
length-squared~1 and so with the twisted zero-point energy give
rise to massless states, combining with $\pm e_{(i)}$ to form the
root lattice of $SO(5)$. The left-moving current algebra is
$(E_8)_2 \times SO(5)_1$.
Notice in this example that short roots of a level~1
non-simply-laced algebra and long roots of a level~2 algebra both
have $p_2 \cdot p_2 + p_3 \cdot p_3 = 1$, and that they can arise
in the Hilbert space in two ways: as untwisted states with momentum
$p_1 \cdot p_1 = 1$ or as twisted states. Note also that in this
moduli space non-simply-laced algebras can only appear with root
lengths in the ratio $1:{\textstyle{\sqrt 2}}$, and only as level~1 realizations,
and that simply-laced algebras can appear only at levels~1 or~2.
To find further enhanced symmetry points it is useful to focus first
on the long roots, which have $p_1 = 0,\ p_2 \in {\textstyle{\sqrt 2}} \Gamma^8$. We
will now construct an orthonormal set of long roots, with inner
product $e_{(i)}$$\cdot$$e_{(j)}$$=$$2 \delta_{ij}$, that extends
the basis $( e_{(1)}, e_{(2)})$ described above to the root-lattice
of $(SU(2))^{10}$. To begin with, we identify a useful basis of
vectors contained in the euclidean lattice ${\textstyle{\sqrt 2}} \Gamma^8$. These
are
\begin{eqnarray}
u_{(i)}:&& p^1_2 = {\textstyle{\sqrt 2}} , ~~~ p^{i+1}_2 = {\textstyle{\sqrt 2}} , \quad i= 1, \ldots,
7 \nonumber\\
u_{(8)}:&& p^I_2 = {\textstyle{{{1}\over{\sqrt 2}}}} , \quad I=1, \ldots 8
\end{eqnarray}
with the property $u_{(i)} \cdot u_{(j)} = 2 + 2 \delta_{ij}$.
The vectors $e_{(1)}$, $e_{(2)}$, and
\begin{equation}
e_{(i)}:\qquad p_2 = u_{(i-2)},\ p_3= (0,{\textstyle{\sqrt 2}} ;\ {\textstyle{\sqrt 2}} ,{\textstyle{\sqrt 2}} ),
\quad i= 3,
\ldots, 10 ,
\end{equation}
satisfy the orthonormality condition~(\ref{orth}) and so by a
Lorentz transformation can be taken to a left-moving basis, at
which point they form the $(SU(2))^{10}$ root lattice normalized to
level~1. The lattice
$(SU(2))^n$ is the long root lattice of the non-simply laced group
$Sp(2n)$. The linear combinations
\begin{equation}
\pm {\textstyle{1\over2}} e_{(i)} \pm' {\textstyle{1\over2}} e_{(j)}
\end{equation}
are vectors of length $1$ filling out the short root lattice of the
group $Sp(20)_1$. These states can all be found in the untwisted
sector, except for $i=2$ or $j = 2$ which are contributed by the
twisted sector.
Since we lack a more elegant characterization of the allowed
momentum lattices, we will follow the above procedure in looking for
points of enhanced symmetry. Fortunately, it suffices for the
examples at hand. The possible enhanced symmetry points that can
appear in the D=8 moduli space are limited. In the appendix we show
that at any point in this moduli space the long roots are always
orthogonal, so they can only form products of $SU(2)$'s. The only
non-simply-laced groups that can appear are therefore $Sp(2n)$,
including $Sp(4) = SO(5)$, while the only simply laced group that
can appear at level~1 is a product of $SU(2)$'s.
Including an inner automorphism (Wilson line) in the translation
leads to nothing new. That is, twist by $RT'$ where now
$v_2$ is nonzero. The requirement that $T'^2$ be a symmetry of
$\Gamma$ implies that $v_2 \in I^*$, but then the asymmetric
orbifold generated by $RT'$ has the same spectrum as that generated
by $RT$.
It is interesting to consider the decompactification limit of the
$Sp(20)$ theory in which the radius of the 9-direction is taken to
infinity. Initially the gauge symmetry is broken to $Sp(18)$, but
in the limit the twist becomes irrelevant and the antisymmetric
combinations
\begin{equation}
{\textstyle{{{1}\over{\sqrt 2}}}} (\alpha^I_{-1} - \alpha'^I_{-1}) |0,0,0\rangle ,
\qquad
{\textstyle{{{1}\over{\sqrt 2}}}} \Bigl(\,\Bigl| {\textstyle{{{1}\over{\sqrt 2}}}} r ,\ {\textstyle{{{1}\over{\sqrt 2}}}} r ,\ 0 \Bigr\rangle
- \Bigl| - {\textstyle{{{1}\over{\sqrt 2}}}} r ,\ {\textstyle{{{1}\over{\sqrt 2}}}} r ,\ 0 \Bigr\rangle\Bigr)
\end{equation}
become massless. These lie in the antisymmetric tensor representation
of $Sp(18)$, combining with the $Sp(18)$ adjoint to form the adjoint of
$SU(18)$, a Narain compactification. Decompactifying the remaining
direction along a line of unbroken $SU(16)$ leads in the limit to $SO(32)$,
so these theories can also be regarded as compactifications of the $SO(32)$
string.
\subsection{Compactification to $D < 8$}
As $D$ decreases, more gauge groups become possible. It is
convenient to take $\Gamma' = \Gamma^{SU(2)}$$\oplus$$ \Gamma^{9 -
D,9 - D}$. Here $\Gamma^{SU(2)}$ is the self-dual $SU(2)$ lattice
described above, $(p^9_3 ;\ \tilde p^9_3)$. Let $w$, $\tilde w$
denote vectors in the weight lattice of $SO(18 - 2D)$. Then
$\Gamma^{9 - D,9 - D}$ is defined as the lattice $(w ;\ \tilde w)$
such that
$w - \tilde w$ is in the root lattice of $SO(18- 2D)$. Momenta
$p_3$ are thus labeled $(w, p^9_3 ;\ \tilde w , \tilde p^9_3 )$.
Define an orthonormal set $w_{(i)}$ of $SO(18- 2D)$ vector
weights,
\begin{equation}
w_{(i)} \cdot w_{(j)} = \delta_{ij}.
\end{equation}
For $T$ take the shift\footnote
{For $D=8$ this is equivalent to the earlier construction under a
boost and Weyl reflection.}
\begin{equation}
v_2 = 0, \qquad v_3 = ( w_{(1)}, 0 ;\ 0, 0).
\end{equation}
Define the following:
\begin{eqnarray}
f_{(i)}:&& p_2 = u_{(i)}, \quad p_3 = (0,0 ;\ 0, {\textstyle{\sqrt 2}} ),
\qquad i = 1, \ldots, 8, \nonumber\\
f_{(9)}:&& p_2 = 0, \quad p_3 = (0, {\textstyle{\sqrt 2}} ;\ 0,0),
\nonumber\\
g_{(i)}:&& p_2 = u_{(i)},
\quad p_3 = (0,0 ;\ w_{(i)}, {\textstyle{\sqrt 2}} ),\qquad i = 1, \ldots, 9 - D,
\nonumber\\
h_{(i)}:&& p_2 = 0, \quad p_3 = ( w_{(i)} , 0 ;\ 0, 0),
\qquad i = 1, \ldots, 9 - D.
\end{eqnarray}
All of these are allowed momenta, $g$ and $h$ being in the twisted
sector. For $1 \leq n \leq 10 - D$ the set
\begin{equation}
f_{(i)},\ i = n, \ldots, 9, \quad
g_{(i)}, \ i = 1, \ldots n- 1, \quad
h_{(i)}, \ i = 1, \ldots 9 - D,
\end{equation}
is a basis of $18$$-$$D$ orthogonal vectors, with the $f$'s having
length-squared~2 and the $g$'s and $h$'s having length-squared~1.
By an $SO(18 - D,10 - D)$ transformation, bring the given set to
lie fully on the left. The $f$'s form the long roots of
$Sp(20-2n)_1$ and the $g$'s and $h$'s form the short roots of
$SO(17 - 2D + 2n)_1$. The remaining roots of each group (the sums
and differences of the short roots, and
${\textstyle{1\over2}}$ the sums and differences of the long roots) are all in the
untwisted sector. The product $Sp(20)_1$$\times$$SO(17 - 2D)_1$,
obtained first in the fermionic construction, is also in the same
moduli space. Replacing $h_{(9 - D)}$ with
\begin{equation}
f_{(10)}:\quad p_2 = {\textstyle{{{1}\over{\sqrt 2}}}} (5,1^7), \quad
p_3 = (2 w_{(9-D)} , 0 ; 0, 3 {\textstyle{\sqrt 2}} )
\end{equation}
gives the needed long root of $Sp(20)$, the additional short roots
being in the twisted sector. Thus
we have found
\begin{equation}
Sp(20 - 2n)_1 \times SO(17 - 2D + 2n)_1
\qquad n = 0, \ldots, 10 - D
\label{sosp}
\end{equation}
at special points in a single moduli space.
The maximal symplectic group is $Sp(20)$. The central charge of
$Sp(2n)_1$ is $(2n^2 + n)/(n+2)$. The minimum central charge,
including an additional $18$$-$$n$$-D$ from $U(1)$'s needed to
saturate the left-moving rank, exceeds the available $26 - D$ if $n
> 10$. We also strongly believe that $SO(37 - 4D)$ is maximal. It
might appear that one could get larger $SO(2n + 1)$ groups by
decompactification, say $SO(25)$ in
$D=4$ from $SO(25)$ in $D=3$. However, this idea would lead to a
contradiction in going from $D=7$ to $D=8$, where we have already
shown that
$SO(5)$ is maximal. In fact, study of these decompactifications
shows that decompactifying while preserving the larger orthogonal
symmetry gives a limit in which additional vectors become massless,
leading to a rank~16 Narain theory. We expect that the $D=8$
result can be extended to show that $SO(37 - 4D)$ is maximal.
While larger $Sp$ and $SO$ groups cannot arise, there are other
possibilities. For example, twisting the $D$$=$$6$
$Sp(16)_1$$\times$$SO(9)_1$ theory by the additional
$T'$,
\begin{equation}
v' = {\textstyle{1\over2}}(f_{(2)} + f_{(3)} + f_{(4)} + f_{(5)})
\end{equation}
leads to the gauge group $(F_4)_1 \times (F_4)_1 \times SO(9)_1$.
This is in the same moduli space: we could twist by
$T'$ before
$RT$, producing a different toroidal starting point which must
therefore be related by a boost~\cite{narain}. Similarly the group
$(F_4)_1 \times (F_4)_1 \times Sp(8)_1$ can be obtained~\cite{chl}.
\section{Discussion}
Twisting by the outer isomorphism reduces the rank by eight
relative to toroidal compactification, giving the moduli space of
the rank $-$$8$ theory. This orbifold construction appears to
reproduce all of the rank $-$$8$ solutions obtained in the
fermionic construction \cite{chl},\cite{sc} and shows that they
are indeed special points within a single moduli space. A more
elegant characterization of the allowed lattices analogous to that
of ref.~\cite{narain} would be helpful.\footnote {The global {\it
reduction} of the rank (dimension) of a maximally supersymmetric
string moduli space via orbifolding should be contrasted with
familiar field theoretic mechanisms for symmetry enhancement at
specific points in a moduli space, such as Higgsing, whether by
fundamental scalars or more exotic composites.}
As mentioned in the introduction, this is only the first member in
a series of maximally supersymmetric theories characterized by
further reduction of the dimension of the moduli space, and it
remains to find a construction of the moduli spaces of the
remaining theories. Since Majorana fermion field theories only
contain $Z_2$ spin fields, it is possible that an asymmetric
orbifold construction of these moduli spaces will turn up
additional reduced rank moduli spaces that are absent in the
fermionic construction. Conversely, it is also possible that the
solutions with rank $r_L$$<$$10$$-$$D$ in four dimensions have no
orbifold realization. In these cases, the marginal deformations of
the six dimensional abelian torus are either absent or partly
constrained. These backgrounds do not appear to have a large radius
limit.
It is quite interesting to consider the implications of the CHL
string for weak/strong coupling duality. Dual theories describing
the strongly coupled limit of the heterotic string in various
dimensions have been proposed; see refs.~\cite{sdual},
\cite{ssdual},~\cite{eddual} and references therein. In $D=4$ the
conjectured dual is the heterotic string itself~\cite{fontdual},
with the evidence being strongest in the case of toroidal
compactification~\cite{sdual},~\cite{eddual}. What are the duals
of the CHL theories? This presents a new challenge, because the
S-duality transformation includes electric-magnetic duality of the
low energy theory~\cite{montol}, and the groups $Sp(2n)$ and
$SO(2n+1)$ are not invariant under electric-magnetic duality but
rather are interchanged~\cite{gno}. Thus, S-duality cannot leave
the individual points in the CHL moduli space invariant, but
requires that for each solution there be a heterotic string
solution with the dual group. The construction in the previous
section seems in no way to single out
$D=4$, and it is evident from the list~(\ref{sosp}) that it does not
automatically give dual groups. But remarkably, just in $D=4$ where
required by S-duality, the series~(\ref{sosp}) is dual: the
strongly coupled behavior at one of these special points in moduli
space can be described by the weakly coupled theory with $n \to
6-n$, the point $n$$=$$3$ being self-dual. This is further evidence
for S-duality, apparently independent of previous results.
Moreover it is evidence for duality of the full string theories,
not the low energy effective theories, because stringy phenomena
(enhanced gauge symmetries) play an essential role. It would be of
course interesting to compare the spectrum of short multiplets of
these two theories.
In $D=6$ there is a puzzle, however. There is mounting evidence
for string-string duality, the heterotic string on $T^4$ being
equivalent to the IIA string on $K_3$
\cite{ssdual},~\cite{eddual}. It is natural to look for a related
dual in the present case. Indeed, Ferrara, Harvey, Strominger, and
Vafa~\cite{andy} have recently considered a similar situation, in
which
the IIA string is compactified on $K_3$$\times$$T^2$ modded by a
$Z_2$ isomorphism. They argue that the dual theory is the
heterotic string on an asymmetric orbifold of $T^6$, where the
reflection includes an outer isomorphism of $E_8$$\times$$E_8$. In
that case the reflection also acts on the right-movers, breaking
half the supersymmetry, but it suggests that the dual in our case
as well should be some twisting of the $K_3$ compactification of
the IIA string.\footnote{We would like to thank Paul Aspinwall and
Andy Strominger for independently suggesting this and pointing out
the parallel with ref.~\cite{andy}.} The following argument of
Seiberg~\cite{seiberg} appears to exclude this attractive
possibility. In order to obtain $N$$=$$2$, $D$$=$$6$ supersymmetry
it is necessary to preserve the $(4,4)$ world-sheet supersymmetry
of the $K_3$ theory~\cite{bd}. This is also a good background for
the IIB string~\cite{seiberg}, giving chiral
$N$$=$$2$ $D$$=$$6$ supergravity. Spacetime anomaly cancellation
then determines completely the massless spectrum, in particular the
number of massless vectors and moduli in the IIB theory. For the
IIA on the same background this implies a rank~24 gauge group,
exactly as found in toroidal compacitication of the heterotic
string but inconsistent with theories of reduced rank. But there
are other possibilities; for example, the dual could be a type~II
theory with both spacetime supersymmetries right-moving,
corresponding to a right-moving torus times a generic $N$$=$$1$
left-moving theory. The full picture of duality of the CHL
theories remains to be discovered.
\subsection*{Acknowledgments}
We would like to thank Paul Aspinwall, Jeff Harvey, Costas Kounnas,
Joe Lykken, and Andy Strominger for illuminating discussions. S.C.
acknowledges the warm hospitality of the CERN Theory division during
the completion of this work. This work is supported by NSF grants
PHY91-16964 and PHY94-07194.
\subsection*{Appendix}
\setcounter{equation}{0}
\defA.\arabic{equation}{A.\arabic{equation}}
For the lattice used in $D=8$ we show that if
\begin{eqnarray}
&& p_1 = p_1' = 0 \label{p1van}\\
&& p \cdot p = 2 \bmod 4,\quad
p' \cdot p' = 2 \bmod 4, \label{mod4}
\end{eqnarray}
then
\begin{equation}
p \cdot p' = 0 \bmod 2. \label{dot}
\end{equation}
Eq.~(\ref{p1van}) implies that $p_2$ and $p_2'$ lie on $\sqrt{2}
\Gamma^8$ and so contribute to the lengths~(\ref{mod4}) only modulo
4 and to the dot product~(\ref{dot}) only modulo 2. Similarly the
$RT$ projection implies that $n_9 + \tilde n_9$ is a multiple of~4,
hence $p^9$ contributes in the same way. Finally, $n_8 = 2r-\tilde
n_8$ for integer $r$. Then $p \cdot p \bmod 4$ is ${\textstyle{1\over2}}
(n_8^2 - \tilde n_8^2) = 2r(r - \tilde n_8)$. It follows
from~(\ref{mod4}) that $r$ is odd and
$n_8$ and $\tilde n_8$ even. Similarly $n'_8$ and $\tilde n'_8$
are even, and so eq.~(\ref{dot}) holds.
For long roots $p_1 = 0$ and $p \cdot p =2$ with $p$ purely
left-moving and so having a positive inner product. It follows from
this and eq.~(\ref{dot}) that distinct long roots are orthogonal.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,271 |
Crowell House may refer to several houses in the United States:
Alphabetical by state, then town
Crowell House (Denver, Colorado), on the list of the oldest buildings in Colorado
C. C. Crowell House, Somerville, Massachusetts, listed on the US National Register of Historic Places (NRHP)
Crowell–Bourne Farm, West Falmouth, Massachusetts, NRHP-listed
C.C. Crowell Jr. House, Blair, Nebraska, NRHP-listed
Crowell House (Sea Cliff, New York), NRHP-listed
J. B. Crowell and Son Brick Mould Mill Complex, Wallkill, New York, NRHP-listed
Crowell House (Durham, North Carolina), NRHP-listed
Warren-Crowell House, Terrell, Texas, NRHP-listed | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,707 |
Q: Is there a way to change stlyes position:'relative' to position:'absolute' depending on Y scroll value in a Animated.ScrollView React-Native I have an animated header that gets smaller and opacity increments when the scroll view is being scrolled down.. Under that there is a filter bar that needs to stick just below the animated header after it has fully clamped. I got it to do just that with position:"absolute" and making that view follow the y value. The problem is that it seems LAGGY when the animated header has not fully clamped yet, and when it is scrolled up it looks much worse. On top of that the scroll indicator gets hidden behind the position:'absolute' filter bar view... I think the solution would be to have that view with position relative then change it to absolute.. but I need to know if there is a way to change the style position:"relative" to position:"absolute" after it gets done clamping so it can look smooth until it clamps, or atleast a way to do this without it looking laggy, any help is very apreciated.
some code here (some tags may not be closed but you get the gist):
constructor(props) {
super(props);
this.scrollY = new Reanimated.Value(0);
}
render() {
const translateY = multiply(min(this.scrollY, 0),-1);
const headerHeight = this.scrollY.interpolate({
inputRange: [0, HEADER_MAX_HEIGHT - HEADER_MIN_HEIGHT],
outputRange: [HEADER_MAX_HEIGHT, HEADER_MIN_HEIGHT],
extrapolate: Extrapolate.CLAMP
});
this.NewHeaderHeight= this.scrollY.interpolate({
inputRange: [0, HEADER_MIN_HEIGHT],
outputRange: [0, HEADER_MIN_HEIGHT + 1],
extrapolate: Extrapolate.CLAMP
});
const headerOpacity = this.scrollY.interpolate({
inputRange: [0, HEADER_SCROLL_DISTANCE / 3, HEADER_SCROLL_DISTANCE],
outputRange: [0, 0, 1],
extrapolate: Extrapolate.CLAMP
});
return(<View style={{flex:1}}>
{/* contenido */}
<Reanimated.ScrollView
style={[{flex:1}]}
bounces={false}
scrollEventThrottle={16}
onScroll={Reanimated.event([
{ nativeEvent: { contentOffset: { y: this.scrollY } } }
])}
>
{/* Start of animated HEADER */}
<View style={{height: HEADER_MAX_HEIGHT, width:'100%'}}>
{/* IMAGEN DEL HEADER Y ICONS*/}
<ImageBackground style={{flex:1, justifyContent:'center' }} source={{uri: 'https://images.unsplash.com/photo-etc}}>
//THERE WOULD BE SOME ICON AND OTHER CONTENT HERE DOESNT MATTER THOUGH
</ImageBackground>
</View>
*//there would be more children here doesnt matter either since filter toolbar is position:absolute for the moment*
</Reanimated.ScrollView>
{/*THIS IS FILTERBAR (here is where want to change the position style depending on scroll value)*/}
<Reanimated.View
style={[styles.categoryContainer, styles.shadow, { transform: [{translateY}],position: 'absolute', top: headerHeight, elevation:1, backgroundColor:'white'}]}
>
<FlatList
horizontal
data={infoDataFilter}
showsHorizontalScrollIndicator={false}
renderItem={({item}) => (
<TouchableOpacity
key={item.id}
style={[styles.categoryItem]}
activeOpacity={.7}
onPress={() => console.log(item.screen)}
>
<Text style={{paddingVertical: 5, fontSize:12, fontWeight:'bold', color: '#444444'}} >
{item.name.toUpperCase()}
</Text>
</TouchableOpacity>
)}
/>
</Reanimated.View>
that does this:
https://www.youtube.com/watch?v=4GdnkUEdmko
It scrolls perfectly when that view has position:'relative' and i would like to have it like that but change to position:'absolute' when the animated header has clamped..
EDIT: i am using react native version .61.5
reanimated ^1.7.0, gesture-handler ^1.6.0, react navigation x4 libraries.
this was tested on android samsung galaxy s6 devmode (its the same on production)
A: I don't think changing the position prop is a good idea. To fix the "laggy" behavior, you can use useNativeDriver on your Animated.event and move the animation over to native side. It usually does wonders. Here's how you do that:
Reanimated.event(
[{ nativeEvent: { contentOffset: { y: this.scrollY } } }],
{ useNativeDriver: true })
Although, you have to make sure you don't drive any layout animations (width, height, etc) with scrollY or you'll get an error, since those are not supported.
EDIT
I noticed you ARE using scrollY to drive a layout animation (headerHeight). To make layout animations smooth, you have to use LayoutAnimation. It's a very good library. Here's a link.
| {
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} | 2,492 |
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For the Serbian footballer, see Jelena Dimitrijević (footballer).
Bust of Jelena J. Dimitrijević
Jelena Dimitrijević (27 March 1862 – 10 April 1945) was a Serbian short story writer, novelist, poet, traveller, social worker, feminist, and a polyglot. She is considered to be the first Serbian woman to publish a work of prose in 1894.[1]
2 Last years and death
This section relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. (December 2016)
Dimitrijević was born in Kruševac on 27 March 1862, and featured as a prominent Serbian writer of the late 19th- and early 20th-century. She taught herself to speak French, English, Russian, Italian, Greek and Turkish. She was raised in respected and wealthy family, in the spirit of Serbian cultural heritage and Orthodox religion.
From an early age she dedicated herself to writing - notwithstanding a childhood eye injury that forced her to leave school, and against medical advice forbidding her to read.[2] She had a great support in her husband Jovan Dimitrijević. Besides supporting her writing and social activities, he was often her fellow-traveler and the person she could completely rely on. Dimitrijević travelled widely, describing her experiences of Greece, India, Egypt, and America in a series of books. When he died, she was in mourning for the rest of her life.[3]
In 1881 they moved to Niš, which had been part of the Ottoman Empire until 1878. She devoted her energies to the study of Muslim women, and in 1897 published Pisma iz Niša o haremima (Letters from Niš Regarding Harems) - "...the first Serbian prose book written and published by a female author."[2] Among her achievements were gaining an understanding of the lives of Turkish women, including access to the private world of the harem, and undertaking a journey round the world in her sixties. Such portraits are a valuable counter to the narrow conceptions of nineteenth- and early twentieth-century feminism which sees it firmly rooted in north-west Europe and North America. For example, "Jelena was proud to have met Mrs Hoda Sha'arawi, the founder of the Egyptian Feminist Union.[2] Her most important novel Nove (New Women); deals with the dilemmas facing educated Muslim women in the twentieth century in relation to their traditional way of life. For Nove Dimitrijevic won the prestigious Matica Srpska prize for literature in 1912.
She also wrote lyric poetry as well as novels, but is possibly most famous for her Pisma iz Nisa o Haremima, a semi-fictionalised, semi-historical, anthropological narrative containing portraits of life in the Turkish harems 50 years before her birth when the south-Serbian city of Niš was still a part of the Ottoman Empire, and Pisma iz Soluna/Letters from Salonica, a genuine travelogue from the Ottoman Empire during the Young Turk Revolution in 1908, of which Salonica was the centre. The Letters were published first in Srpski književni glasnik (Serbian Literary Review) in 1908-09, and then as a separate book in 1918 in Sarajevo.
By the beginning of the 20th Century she and her husband were living in Belgrade and she was a member of the Serbian Writers' Society.[2]
Last years and death[edit]
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She also wrote Pisma iz Indije/Letters from India in 1928, Pisma iz Misira/Letters from Egypt in 1929, and Novi svet ili u Americi godinu dana/The New World, alias: In America for a Year in 1934. Along with Isidora Sekulić, Dimitrijević is one of the first feminist authors in Serbia. She died in Belgrade on 10 April 1945, aged 83.
Works[edit]
Jelenine pesme (Pesme Jelene Jov. Dimitrijevića), 1894.
Pisma iz Niša o haremima, 1897.
Đul-Marikina prikažnja, short stories, 1901.
Fati-Sultan, Safi-Hanum,Mejrem-Hanum, short stories, 1907.
Nove, 1912.
Amerikanka, 1918.
Pisma iz Soluna, 1918.
Pisma iz Indije, 1928.
Pisma iz Misira, 1929.
Novi svet ili u Americi godinu dana, 1934.
Une vision, 1936.
Sedam mora i tri okeana. Putem oko sveta, 1940.
Mir-Jam
Jelena Dimitrijević (1986). "Pisma iz Niša o haremima". National Library of Serbia. Retrieved 2 December 2016.
Slobodanka Peković (2008). "Putopisi Jelene Dimitrijević kao mogućnost viđenja Drugog (117-35)" [Jelena Dimitrijević's travellogues] (in Serbian). Belgrade, Serbia: Institute for Literature and Arts. Retrieved 2 December 2016.
Hawkesworth, Celia, Voices in the Shadows: Women and Verbal Art in Serbia and Bosnia, published by Central European University Press (Budapest, New York, 2000).
^ Stjelja, Ana (2012). "Tragom prve priznate srpske književnice" (PDF). Književna istorija: 344. Retrieved 14 August 2017.
^ a b c d Stevanović, Ana (27 March 2019). "Jelena Dimitrijević, Serbia's first feminist author". Europeana (CC By-SA). Retrieved 1 May 2019.
^ Stjelja, Ana, "The case of Serbian writer Jelena J. Dimitrijević (1862-1945)", Lamed-E, Autumn 2013, Number 20 (ed. Ivan Ninić), p. 21.
Skerlić, Jovan, Istorija Nove Srpske Književnosti/A History of New Serbian Literature (Second Edition, 1921), p. 476.
NKC: js2018981461
Retrieved from "https://en.wikipedia.org/w/index.php?title=Jelena_Dimitrijević&oldid=895024855"
Serbian feminists
Women travel writers
Serbian travel writers
Serbian women short story writers
Serbian novelists
Serbian poets
Disease-related deaths in Serbia
People from Kruševac
Women novelists
20th-century Serbian people
20th-century Serbian women
20th-century Serbian women writers
Use dmy dates from July 2013
CS1 Serbian-language sources (sr) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,599 |
\section{Introduction}
\label{intro}
Electromagnetic calorimeters are fundamental elements of numerous experiments ranging from nuclear to hadronic and high-energy
physics. Their main goal is to measure precisely the energy of the detected electrons and photons. The depth of the calorimeters
is generally taken to be large enough to avoid energy leaks and to allow a full development of the electromagnetic shower created
by the incident particles. In addition to the energy measurement, the impact position of particles, corresponding to the shower
central position in case of normal incidence, is usually required in order to provide refined information as particle identification.
Indeed, the knowledge of the photon or the electron energy and both the impact and interaction vertex positions allow the determination
of the particle four-vector. This is of direct relevance for instance in the experimental determination of the origin of two photon
events from the decay of $\pi^0$- or $\eta$-mesons where the 2$\gamma$-invariant mass allows identifying the meson nature. It has been
shown is some experiments, that the spatial resolution is of equal importance than the energy resolution for particle identification~\cite{ref1}.
The knowledge of the particle impact position requires a laterally segmented calorimeter where the energy is released in a cluster
of adjacent blocks. The impact position can then be determined using the energy deposited in each block. Laterally segmented
calorimeters such as lead-fluoride (PbF$_2$) are frequently employed to determine the energy and the position of the showering particle~\cite{{ref2},{ref3},{ref4}}. This high-density material has a short radiation length ($X_{0}=0.93$ cm) and a small Moli\`ere
radius ($r_{M}=2.12$ cm) leading to compact detector geometries~\cite{{ref5},{ref14}}.
Many efforts are still developed for optimizing the performances of such lead-fluoride calorimeters, in particular their energy and
spatial resolution~\cite{ref16}. We investigate in this work, on the basis of GEANT4 Monte Carlo simulations~\cite{ref6}, the spatial resolution
of a PbF$_2$ calorimeter for several block sizes and different particle energies. Two approaches are successively considered, discussed, and optimized to obtain a parametrization of the spatial resolution in terms of the block size and the particle energy.
Finally, these approaches are compared and the effect of the energy resolution on the spatial resolution is addressed.
\section{Simulation}
\label{sec2}
\begin{figure}[h!]
\begin{center}
\epsfig{file=fig1.eps, width=5.5cm, angle =-90}
\end{center}
\caption{Schematic of the geometry of the simulated calorimeter.}
\label{fig1}
\end{figure}
The electromagnetic calorimeter is simulated within the GEANT4 framework following a $15 \times 15$ PbF$_2$ block matrix.
Each block has a transverse square shape with dimensions $d \times d$~cm$^2$ and large enough depth (50$X_{0}$) along
the $z$-axis to avoid energy leaks. Different configurations corresponding to 13 different transverse size varying from
$0.3 r_{M}$ to $5 r_{M}$ are studied. The minimum block size ($0.3 r_{M}$) is consistent with a large enough minimal calorimeter
($4.5 r_M$$\times$$4.5 r_M$) to contain the electromagnetic shower. For each configuration, photons of energy $E$ are sent parallel to
the $z$-axis and impinge normally on the central region of the calorimeter (Fig.~\ref{fig1}). The known impact position ($x$,$y$)
of incident particles is distributed uniformly over the area of the central block $(-d/2,-d/2) < (x,y) < (d/2,d/2)$. Thirteen
photon energies varying from 100~MeV to 20~GeV are generated, leading to a total of 169
configurations. In each case, the response on the calorimeter is studied for $10^{5}$ generated initial photons: the event-by-event
energy deposit in each individual block is recorded and used to reconstruct the shower central position ($x_{c}$,$y_{c}$) as
described in the following sections. Focus is put here only on the $x_{c}$ coordinate since the $y_{c}$ coordinate can be deduced
following the same methods.
Two approaches can be used for reconstructing the shower central position $x_{c}$. The first method is based on a
numerical formula depending on the $x_{i}$-coordinate of the $i^{th}$ block center and the energy deposit $E_{i}$ in this block.
The second method developed in the present work relies on fitting the calorimeter block response with a known
profile function where $x_{c}$ becomes the free parameter of the fit.
\section{Formula based method}
\label{sec3}
Various formulas have been proposed to determine the shower central position~\cite{{ref7},{ref8},{ref9},{ref10},{ref11}}.
The center of gravity method is one of the most common formula
\begin{equation}
x_{c} = \frac{\sum_{i}w_{i}x_{i}}{\sum_{i}w_{i}}~~,
\label{eq1}
\end{equation}
where the sum runs over the number of blocks in the shower cluster and $w_{i}$ is a weight factor depending of the energy $E_{i}$.
In the simplest case where $w_{i}=E_{i}$, it has been shown that the obtained $x_{c}$ depends on $x$ in a non-linear way. A term,
depending of the size $d$ of the blocks and the exponential radial falloff of the shower, has to be added to Eq.~\ref{eq1} to
correct from the correlation between $x_{c}-x$ and $x$~\cite{ref7}, the so-called {\it S-curve}. Many other formulas and algorithms
have been discussed and compared in reference~\cite{ref9}. For example, the un-shifted estimate of $x_{c}$ based on here-after
expressions (Eq.~\ref{eq2}-\ref{eq3}) reduces significantly the non-linear correlation between $x$ and $x_{c}$ and gives a satisfying reconstruction of the shower position
\begin{equation}
x_{c} = x_{m}+\frac{\sigma_{1}}{2}\ln{\left(\frac{E_{m+1}}{E_{m-1}}\right)}~~,
\label{eq2}
\end{equation}
\begin{equation}
x_{c} = x_{m}\pm d \mp \sigma_{2}\ln{\left(\frac{1}{2}\left[1+\frac{E_{m}}{E_{m\pm1}}\right]\right)}~~.
\label{eq3}
\end{equation}
In the previous relations $x_{m}$ represents the coordinate of the block with a maximal energy deposition and $\sigma_{1}$ and
$\sigma_{2}$ are related to the transverse exponential behavior of the shower~\cite{ref9}. This un-shifted estimate of $x_{c}$ uses
only the information of two particular blocks: the ones with the largest energy deposition (Eq.~\ref{eq3}) and the ones adjacent to
the block with maximal energy (Eq.~\ref{eq2}). In comparison to the simplest energy weighting case, the logarithmic weighting of
energies in these expressions procures a stronger influence of low energy deposit blocks in the calculation of $x_{c}$. Lately, a
simple method based on Eq.~\ref{eq1} has been proposed and is still largely employed nowadays in many experimental analysis. It has
been shown that it gives results similar or superior in quality to all those discussed previously and does not need position correction~\cite{ref11}. In this method the energy weight $w_{i}$ is given by
\begin{equation}
w_{i}=\max \left\{ 0; W_{0}+\ln\left( \frac{E_{i}}{E} \right) \right\}
\label{eq4}
\end{equation}
where $W_0$ is a free dimensionless parameter. Consequently, only the blocks having an energy deposition higher than $E e^{-W_0}$ are taken into
account in the calculation of $x_{c}$. In addition $W_0$ allows to set the relative weight of the blocks, with a small energy
deposition, used in the sum of Eq.~\ref{eq1}. Indeed, high $W_0$ values attribute almost an equal weight to the blocks entering the
sum, while small $W_0$ values favor the highest energy blocks. It exists therefore an optimal $W_0 \approx 4$ giving the best
position reconstruction~\cite{ref11}.
\begin{figure}[h]
\begin{center}
\epsfig{file=fig2.eps, width=8cm}
\vspace*{-10pt}
\caption{Relative position resolution $\sigma/d$, for $d=r_M$ and $E=10$ GeV, as a function of $W_0$ (statistical uncertainties are smaller than the point size).}
\label{fig2}
\end{center}
\end{figure}
Actually the optimal $W_0$ value, called $W_0^{for}$ hereafter to indicate the formula method origin of this parameter, depends on
the size $d$ of the blocks as well as the energy $E$ of the incident particle. Fig.~\ref{fig2} shows a typical behavior of the position resolution $\sigma$, defined by the root mean square (RMS) of the $(x_c-x)$ distribution, as a function of $W_0$. The minimal value of
$\sigma$ defines $W_0^{for}$ whereas the worst resolution obtained for $W_0=0$ or $W_0\rightarrow+\infty$ equals $d/\sqrt{12}$ and corresponds to the RMS of a uniform distribution of width $d$. The optimal $W_0^{for}$ and corresponding position resolution $\sigma^{for}$ are determined for each geometry and energy configuration previously described (Sec.~\ref{sec2}). Fig.~\ref{fig3} shows the energy dependence of $W_0^{for}$ for different block sizes. At a given size, the energy deposit in the blocks surrounding the central block becomes small and very sensitive to the sampling fluctuations of the shower when the energy $E$ decreases. Including these blocks in the calculation of $x_c$ could then degrades the resolution. These blocks are removed by small $W_0$ values as shown on Fig.~\ref{fig3}. At given energy $E$, $W_0^{for}$ increases as the block size $d$ becomes larger. Indeed, for large blocks the energy of the shower is essentially deposited in the central block. Having a high threshold with small $W_0$ excludes the remaining blocks and the $x_c$ position becomes the central block coordinate with a resolution relative to the block size. An empirical parametrization of $W_0^{for}$ as function of $d$ and $E$ is proposed here following the expression
\begin{equation}
W_0^{for}=\ln \left(\frac{100~E(\mathrm{GeV})}{2.01~e^{-\frac{d}{r_M}} + \left[4.95~e^{-\frac{d}{r_M}}+0.307\right]E(\mathrm{GeV})}\right)
\label{eq5}
\end{equation}
where the three numerical constants are determined from a global fit of the 169 configurations. Fig.~\ref{fig3} shows the result of this parametrization for some particular values of $d$.
\begin{figure}[t]
\begin{center}
\epsfig{file=fig3.eps, width=8cm}
\vspace*{-10pt}
\caption{Energy dependence of $W_0^{for}$. The block size $d$ relative to the Moli\`ere radius is reported in the legend
for each configuration. Dashed lines represent the parametrization of Eq.~\ref{eq5}. The 0.1 uncertainty on $W_0^{for}$ corresponds to the bin size of Fig.~\ref{fig2}.}
\label{fig3}
\end{center}
\end{figure}
The obtained resolution $\sigma^{for}$ is represented on Fig.~\ref{fig4} as function of $E$ for different block sizes.
$\sigma^{for}$ is expected to scale with $1/\sqrt{E}$ at high energies~\cite{ref12}: the relative energy
resolution $\sigma_i/E_i$ for each block $i$ is proportional to $1/\sqrt{E_i}$ and then to $1/\sqrt{E}$, assuming
the same average shower profile; all the terms in Eq.~\ref{eq1} having then a relative precision
proportional to $1/\sqrt{E}$, $\sigma^{for}$ is also scaling as $1/\sqrt{E}$. At low energies ($E\ll$ 1~GeV),
this proportionality becomes not valid since the worst resolution one could obtain cannot exceed $d/\sqrt{12}$.
The $E$-dependent $\sigma^{for}$ graphs for each block size are fitted at high energies with the expression
\begin{equation}
\frac{\sigma^{for}}{d}=\frac{\alpha}{\sqrt{E}}+\beta~~,
\label{eq6}
\end{equation}
\begin{figure}[t]
\begin{center}
\epsfig{file=fig4.eps, width=8.cm}
\vspace*{-10pt}
\caption{Energy dependence of the relative resolution. Dashed lines represent Eq.~\ref{eq8} parametrization.}
\label{fig4}
\end{center}
\end{figure}
where $\alpha$ and $\beta$ are the free parameters of the fit. The resulting coefficient $\alpha$ is represented on
Fig.~\ref{fig5} as function of the block size. The $d$-dependence of $\alpha$ can be parametrized as
\begin{equation}
\alpha=a^{for}\left(\frac{d}{r_M}\right)^{b^{for}}
\label{eq7}
\end{equation}
\begin{figure}[h]
\begin{center}
\epsfig{file=fig5.eps, width=8.cm}
\vspace*{-10pt}
\caption{The fitted coefficient $\alpha$ as a function of $d$ (in $r_M$ units).}
\label{fig5}
\end{center}
\end{figure}
where $a^{for}$ and $b^{for}$ are two constants relative to the formula method. The global fit of the $\sigma^{for}$
resolution for each configurations leads to the semi-empirical parametrization of the relative resolution
\begin{equation}
\frac{\sigma^{for}}{d}=\frac{a^{for}\left(\frac{d}{r_M}\right)^{b^{for}}}{\sqrt{E(\mathrm{GeV})}}+c^{for}\left(\frac{d}{r_M}\right)+d^{for}
\label{eq8}
\end{equation}
where $a^{for}=0.110$, $b^{for}=-0.334$, $c^{for}=3.55 \times 10^{-3}$ and $d^{for}=4.02 \times10^{-3}$. The results
of this expression are shown in Fig.~\ref{fig4} for some particular block sizes. It is obvious from Eq.~\ref{eq8} that
the resolution $\sigma^{for}$ becomes better at high energies and for small block sizes. However, the $d$-dependence of
the relative resolution $\sigma^{for}/{d}$ at fixed energy is more involved. It is worth noting that Eq.~\ref{eq8} suggests
the existence of an optimum block size corresponding to an optimum relative resolution at a given $E$, allowing optimizing
the calorimeter block size for a given energy measurement range.
\section{Fit based method}
\label{sec4}
\begin{figure}[h]
\begin{center}
\epsfig{file=fig6.eps, width=8.cm}
\vspace*{-10pt}
\caption{Average shower profile function for $E$=1~GeV photons.}
\label{fig6}
\end{center}
\end{figure}
A second approach for reconstructing the shower central position relies on the knowledge of the lateral profile of the
electromagnetic shower. Fig.~\ref{fig6} shows the profile function $F(x)$ obtained from the simulated response of an infinite
lead fluoride calorimeter to $N_{sim}$=$10^5$ photons of 1~GeV. This response is represented as function of the distance
from the shower center along $x$-axis. The energy deposit is here determined per generated photon, as the transverse dimension
integral normalized by the initial photon energy
\begin{equation}
F(x) = \frac{1}{N_{sim}} \, \frac{1}{E} \, \int_{-\infty}^{+\infty} \frac{d^{2}E(x,y)}{dxdy} \, dy = \frac{1}{N_{sim}}\, \frac{1}{E} \, \frac{dE(x)}{dx}
\end{equation}
where $dE(x)$ is the energy released by the shower in a vertical column calorimeter centered at $x$, having a width $dx$=1~mm
and an infinitely large height. This profile function turns out to be the same for photons and electrons and no energy dependence
is expected according to the definition of the Moli\`ere radius.
In Ref.~\cite{ref9}, the shower profile was approximated by a single exponent term and a least-squares fit algorithm was developed
for extracting the shower position taking into account all of the responding blocks of the calorimeter. It was shown that the results
are similar to those obtained with Eq.~\ref{eq2} and Eq.~\ref{eq3}~\cite{ref9}, and do not present any significant advantage relatively
to the fast and simple method based on Eq.~\ref{eq1}-\ref{eq4}~\cite{ref11}. Nevertheless, it is clear from Fig.~\ref{fig6}, that the
lateral profile of the shower is a more intricate function different from a simple exponent or a combination of two exponent contributions.
A more realistic form $F(x)$ of the profile function is proposed in the present work, deduced from the bin-to-bin linear interpolation of
the simulated profile function (Fig.~\ref{fig6}).
In the calorimeter described in section~\ref{sec2}, the expected energy deposit for a given simulated event in a column $i$ identified by its central coordinate $x_i$ writes
\begin{equation}
E_i^{exp}= E \, \int_{x_i-d/2}^{x_i+d/2} \, F(x-x_c) \, dx
\label{eq9}
\end{equation}
where $x_c$ is the unknown shower central position, and $E$ is the incident particle energy corresponding to the total energy deposit in the calorimeter if we neglect the energy resolution effect.
The clear energy deposit in that column for the same event expresses
\begin{equation}
E_i=\sum_{j}E_{ij} \,\, \Theta(E_{ij}-E~e^{-W_0})
\label{eq10}
\end{equation}
where $W_0$ is a dimensionless parameter related to the energy threshold applied on the blocks, and $\Theta(x)$=1 when $x>0$ and $0$
otherwise is the Heaviside function. In the previous equation, $E_{ij}$ represents the energy deposit in the block belonging to column $i$ and row $j$.
Fitting for each event, with the MINUIT package~\cite{ref13}, the $E_i$ distribution (Eq.~\ref{eq10}) with the expected distribution $E_i^{exp}$ (Eq.~\ref{eq9}) allows
to extract the single free parameter of the fit $x_c$. Following Eq.~\ref{eq10}, this algorithm rejects the blocks with energy deposit
smaller than the threshold $E~e^{-W_0}$.
\begin{figure}
\begin{center}
\epsfig{file=fig7.eps, width=8cm}
\vspace*{-10pt}
\caption{$W_0$-dependence of the relative position resolution $\sigma/d$, for $d$=$r_M$ and $E$=10~GeV.}
\label{fig7}
\end{center}
\end{figure}
Fig.~\ref{fig7} shows the influence of $W_0$ on the obtained resolution $\sigma$ given by the RMS of the $(x_c-x)$ distribution for a
particular value of $E$ and $d$. As expected, larger the block number contributing to Eq.~\ref{eq10}, better the relative resolution.
However, the resolution remains constant after a certain $W_0$ since very small energy deposit blocks do not contribute significantly
to $x_c$ calculation and consequently not change the fit result. The optimum $W_0$, denoted $W_0^{fit}$ here-after, is defined as the
value above which the resolution does not improve by more than 1\%.
\begin{figure}
\begin{center}
\epsfig{file=fig8.eps, width=8cm}
\vspace*{-10pt}
\caption{Block size dependence of $W_0^{fit}$ for two different photon energies; the dashed line corresponds to Eq.~\ref{eq11} parametrization.}
\label{fig8}
\end{center}
\end{figure}
No real energy dependence of $W_0^{fit}$ is observed but logarithmic scaling with the block size is demonstrated on Fig.~\ref{fig8},
following the parametrization
\begin{equation}
W_0^{fit}=1.61~\ln\left(\frac{d}{r_M}\right)+5.55~~.
\label{eq11}
\end{equation}
For large block size, the shower energy is mainly deposited in one single block so the energy threshold must be reduced to
include the surrounding blocks in order to improve the spatial resolution.
\begin{figure}[h]
\begin{center}
\epsfig{file=fig9.eps, width=8cm}
\vspace*{-10pt}
\caption{The relative resolution $\sigma^{fit}/d$ as a function of the shower energy $E$. The block size $d$ is indicated
in the legend for each configuration. Dashed lines represent the parametrization of Eq.\ref{eq12}.}
\label{fig9}
\end{center}
\end{figure}
The relative resolution of the fit method here developed is represented in Fig.~\ref{fig9} as function of the shower energy
$E$ for different block sizes. As for the formula method, the resolution can be parametrized following the expression
\begin{equation}
\frac{\sigma^{fit}}{d}=\frac{a^{fit}\left(\frac{d}{r_M}\right)^{b^{fit}}}{\sqrt{E(\mathrm{GeV})}}+c^{fit}\left(\frac{d}{r_M}\right)+
d^{fit}~~,
\label{eq12}
\end{equation}
where $a^{fit}=0.121$, $b^{fit}=-0.349$, $c^{fit}=1.78 \times 10^{-3}$ and $d^{fit}=1.89 \times 10^{-3}$. These four constants
are deduced from a global fit of the studied configurations.
The fit algorithm described in this section is obviously more computing time consuming at the data analysis level than the
formula based method. Nevertheless, modern computers can easily and efficiently handle this problem: for instance, more than
$10^4$ events per second can be analyzed with a modest 2~GHz processor.
\section{Discussion}
\label{sec5}
\subsection{Comparison of the formula and fit methods}
\label{sec5.1}
\begin{figure}[h]
\begin{center}
\epsfig{file=fig10.eps, width=8cm}
\vspace*{-10pt}
\caption{Reconstructed shower position $x_c$ (in block size units) using the formula method (solid line) and the fit method (dashed
line) for $d=r_M$ and $E=5$~GeV.}
\label{fig10}
\end{center}
\end{figure}
The formula and fit methods are compared on Fig.~\ref{fig10} from a typical distribution of the shower central position obtained for a
given configuration. In opposition to the fit determination, the formula determination of $x_c$ does not exhibit the constant behavior
expected from the uniform generation of the shower $x$ origin. This appears as a consequence of the number of blocks taken into account
in Eq.~\ref{eq1}. It was found in this study that the average number of blocks entering Eq.~\ref{eq1} is approximately 4 but could
slightly differ depending on the configuration type (for instance, 4.2 in the configuration shown on Fig.~\ref{fig10}). Therefore on
average, two different block coordinates only are contributing to the determination of $x_c$ in Eq.~\ref{eq1}, which slightly favors the region
between the center and the boundaries of the central block. This restriction does not show-up in the fit method where $W_0^{fit}$ is
always larger than $W_0^{for}$ and allows a larger number of blocks to contribute to the determination of the shower central position.
\begin{figure}[h]
\begin{center}
\epsfig{file=fig11.eps, width=8cm}
\vspace*{-10pt}
\caption{Energy dependence of the relative difference between the formula and fit methods. The block size $d$ is indicated in the legend
for each configuration.}
\label{fig11}
\end{center}
\end{figure}
Fig.~\ref{fig11} shows the relative difference between the spatial resolution obtained with the two previously described methods. For small block sizes, the formula based method tends to be slightly preciser than the fit based method. However, when the block size comes closer or higher than the Moli\`ere radius, the fit based method can provide as high as 40\% better resolution at high energies. Within the present work, particles with normal incidence only are considered whereas experimentally one can have different incident angles. It has been shown that the position resolution obtained with the formula based method is rather insensitive to this angle for moderate values~\cite{ref11}. For large incidence angles, a simulation optimizable geometrical correction depending on the shower depth must be taken into account since the impact position of the particle is shifted relatively to the shower central position~\cite{ref15}. Similarly, different lateral shower profile functions depending on the incidence angle can be simulated and exported within the fit method to obtain the shower position and the corrected impact position.
\subsection{Effect of the energy resolution}
\label{sec5.2}
The resolution of the energy measurement per block originates solely from the shower sampling fluctuations in the current simulation approach. Experimentally, lead-fluoride calorimeters are based on the detection of \v{C}erenkov light and the fluctuations in the number of collected photo-electrons dominates the energy resolution~\cite{ref12}. The generation and tracking of \v{C}erenkov photons is not performed here because of unrealistic computing times and strong sensitivity to exact optical properties of crystal and wrapping surfaces which are known to differ from an experimental device to another. This effect can globally be symbolized by an additional smearing of the energy deposition $E_i$ in each block $i$. This is done by adding to $E_i$ a random number following a Gaussian distribution centered at zero and having the following width
\begin{equation}
\sigma^{s} = s \, \sqrt{E_i} \,\, ,
\label{eq13}
\end{equation}
where $s$ is a constant related to the global relative energy resolution of the calorimeter. Several $s$-values in the range 1\%-20\% are used in the following to study the effect of the energy resolution on the position resolution. The degradation of the position resolution $\sigma$ is defined as
\begin{equation}
\sigma^{deg} = \sqrt{\sigma^2(s)-\sigma^2(s=0)}
\label{eq14}
\end{equation}
where $\sigma(s)$ is the position resolution at a given $s$-value, and $\sigma(s$=$0)$ is the position resolution in absence of energy
smearing, both determined for any of the two methods. This study shows that $\sigma^{deg}$ is proportional to $s$.
\begin{figure}
\begin{center}
\epsfig{file=fig12.eps, width=8cm}
\vspace*{-10pt}
\caption{$\sigma^{deg}/s/d$ as a function of $1/\sqrt{E}$ for the $d=r_M$ configuration. The same behavior is observed for other
configurations which are omitted sake of clarity.}
\label{fig12}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfig{file=fig13.eps, width=8cm}
\vspace*{-10pt}
\caption{The fitted parameter $a^{deg}$ (Eq.~\ref{eq15}) as a function of $d/r_M$. The dashed line corresponds to the relation
$a^{deg}$=$ 0.112 \left(\frac{d}{r_M}\right)+1.12$ and represents the first order polynomial fit of the distribution.}
\label{fig13}
\end{center}
\end{figure}
Fig.\ref{fig12} shows the typical energy dependence of the relative average resolution $\sigma^{deg}/s/d$ for a particular block size.
As expected from Eq.~\ref{eq13}, the degradation of the position resolution can be parametrized at high energies
\begin{equation}
\frac{1}{s} \, \frac{\sigma^{deg}}{d} = \frac{a^{deg}}{\sqrt{E}}
\label{eq15}
\end{equation}
where $a^{deg}$ is a parameter depending linearly on the block size $d$ as shown on Fig.~\ref{fig13}. Finally the obtained position
resolution $\sigma$ taking into account energy resolution effects can be expressed
\begin{equation}
\frac{\sigma}{d} = \frac{\sigma^{for,fit}}{d} \oplus ~s~\frac{0.112\left(\frac{d}{r_M}\right)+1.12}{\sqrt{E(\mathrm{GeV})}}
\label{eq16}
\end{equation}
where $\sigma^{for,fit}$ is either given by Eq.~\ref{eq8} or Eq.~\ref{eq11} according to the chosen method of the position determination.
Experimentally, other effects such as the physics and electronics background, energy calibration, radiation damage... can contribute to
the energy resolution and then degrades the position resolution. However, in a well designed experiment these effects are not dominating the position resolution. Finally, the systematic uncertainty coming from the experimental knowledge of the coordinates $x_i$ of the block centers has to be added quadratically to Eq.~\ref{eq16} to obtain the final position resolution but this contribution can generally be neglected, as connected to the accuracy of the mechanical design and mounting. In Ref.~\cite{ref3}, the obtained experimental resolution on the
reconstructed position using the center of gravity method is 2~mm, in good agreement with the 1.9~mm predicted by Eq.~\ref{eq16}.
\section{Conclusion}
\label{sec6}
The present work discussed the determination of the shower central position in a laterally segmented electromagnetic calorimeter
following the widely used center of gravity method and a new fit method here developed. A semi-empirical parametrization of the
relative position resolution as a function of the incident particle energy and the calorimeter block size is proposed and
optimized for each case. The fit method is shown to significantly improve the position resolution as compared to the formula
method, particularly at high energies and for large block sizes. Finally, energy resolution effects on the position resolution
are also discussed and quantified.
Despite the present application to lead-fluoride calorimeters, the results presented here can be exported to several electromagnetic
calorimeter materials through their parametrization in terms of the Moli\`ere radius. In addition, as far as simulations are
concerned, the profile function at the heart of the fit method has energy and material universality features. This global study
not only provides useful parametrization for the design optimization of electromagnetic calorimeters in future experiments looking
for a good spatial resolution, but procures also a new algorithm to improve position resolution at the data analysis level.
\section*{References}
| {
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Review THE HOW & THE WHY by Cynthia Hand @CynthiaHand @HarperTeen
November 12, 2019 By Mary
The How and the Why by Cynthia Hand
Published by HarperTeen on November 5, 2019
Imprint: HarperTeen (November 5, 2019)
On Sale: 11/05/2019
List Price: 17.99 USD Age: 13+
Praise for THE HOW AND THE WHY
"The novel's great strength is the emotional depth of its characters and the complexity of their relationships. A heartfelt and hopeful story about coming of age as an adoptee." ―School Library Journal
"Hand explores adoption's multiple dimensions with great insight and sensitivity. Inclusive and illustrative: an engaging lesson in timeless family values." ―Kirkus Reviews
"Cynthia Hand is the master of pulling at your heartstrings. The How AND
the Why tells both sides of an adoption story with love, compassion, and care. You'll be reaching for your tissue box with this one — if you can stop turning pages long enough, that is." ―Brigid Kemmerer, New York Times Bestselling Author of Letters to the Lost
"Beautifully rendered and superbly shaped. Hand has crafted an absorbing novel that focuses on family, friendship, teen pregnancy, adoption, personal choices, and serious health issues. Give this exquisite novel to readers seeking an emotionally intricate story." ―Booklist (starred review)
A poignant exploration of family and the ties that bind, perfect for fans of Far From the Tree, from New York Times bestselling author Cynthia Hand.
Today Melly had us writing letters to our babies…
Cassandra McMurtrey has the best parents a girl could ask for. They've given Cass a life she wouldn't trade for the world. She has everything she needs—except maybe the one thing she wants. Like, to know who she is. Where she came from. Questions her adoptive parents can't answer, no matter how much they love her.
But eighteen years ago, someone wrote Cass a series of letters. And they may just hold the answers Cass has been searching for.
Alternating between Cass's search for answers and letters from the pregnant teen who gave her up for adoption, this voice-driven narrative is the perfect read for fans of Nina LaCour and Jandy Nelson.
ALL THE FEELS.
What a gorgeous and heartfelt story about adoption, family, and friendship. Cynthia Hand has always been one of my favorite authors and I have waited patiently for this latest one and I wasn't disappointed. This story is told in alternating voices between Cassandra, now 18 and a senior in high school paired with her birth mother, S at age 16. At her side is her best friend Nyla, who was adopted at age 3 from Liberia so the two share a lot in common. The story is based on Hand's own search for her birth mother. It is written with such love and emotion that you won't be able to forget this story quickly. While the story is so true to life, there are some helpful coincidences that take place but are necessary to move the plot forward.
Photo Content from Cynthia Hand
Cynthia Hand is the New York Times bestselling author of several books for teens, including the UNEARTHLY trilogy, THE LAST TIME WE SAY GOODBYE, MY LADY JANE and MY PLAIN JANE (with fellow authors Brodi Ashton and Jodi Meadows), THE AFTERLIFE OF HOLLY CHASE, and the upcoming novel THE HOW AND THE WHY (Fall 2019). Before turning to writing for young adults, she studied literary fiction and earned both an M.F.A. and a Ph.D. in fiction writing. She currently resides in Boise, Idaho, with a husband who's addicted to typewriters, two kids, two cats, one crazy dog, and a entourage of imaginary friends.
What is on your nightstand?
Refuge, by Alan Gratz, which I am reading to my kids, A Sorcery of Thorns by Margaret Rogerson, which I just finished and was so good it gave me goosebumps, The Silent Patient by Alex Michaelides, which I am reading for my book club this month and is totally freaking me out, and the big fat tome, Mary, Queen of Scots by John Guy, which I am reading to research on our latest Jane book.
What author would you totally fan?
Laurie Halse Anderson. I have fangirled all over her twice, as it happens. I was at Calvacade of Authors West with her last spring and watched her go through an exhausting day with hundreds of high school students, and then take about an hour after the event was done taking pictures with the other authors at the event, who of course were all such huge fans of hers, too. She was so generous and super gracious that it made me love her even more.
What makes you cringe?
People being unkind or catty to each other. That's the worst. We live in a time where it is so easy to be unkind to people without having to see their hurt in person, and I hate that.
Do you obsessively plot out each point or just go with the flow?
I do a bit of a combination of those things. I generally have a basic idea of the shape of the story. I know the beginning, a few points in the middle, and how it ends. But then I give my characters the freedom to do what they're going to do, and so often that ends up taking the story in directions that I never could have foreseen. It's a mysterious, magical process.
Is there a word you love to use?
Quintessential. For some reason I love this word, but I don't use it very often.
I also love ephemeral. I used that a bit when I was writing the Unearthly series, which has so much philosophical discussion of life and death, but I haven't used it much since then.
Also, if you have any pictures of your pet you would like to share, please attach them. My readers love animals!
Yes! We have a ton of pets in this house: one crazy but sweet Golden Irish named Poesy, two Italian mobster cats, Frank and Stella, and one beautiful beta named Max, who we only realized months after naming him that he is BETTA-MAX.
Poe thinks she's a lap dog.
Poe the wonder hound
There for the snuggles.
Frank's favorite pastime is sitting on knitting.
Stella likes to cuddle in bed.
The cats expect food NOW. At the table. (Frank on the left with the Phantom of the Opera mask, Stella on the right with the little mustache.)
Betta-Max, who is actually a pretty cool fish. He has personality.
By Cynthia Hand
#Giveaway 1-2-3-4, I DECLARE A THUMB WAR by Lisi Harrison & Daniel Kraus @LisiHarrison @DanielDKraus @UnionSquareKids @RockstarBkTours Ends 9.13
Review The Shelterlings by Sarah Beth Durst @sarahbethdurst @ClarionBooks
#Giveaway Review ANYTHING BUT PINK by J.C. Benthin @jcbenthin @RockstarBkTours Ends 6.20
#Giveaway Review I WANT TO BE A VASE by Julio Torres & Julian Glander #IWantToBeAVase, #JulioTorres, @glanderco, @SimonKIDS, & @RockstarBkTours Ends 6.13
#Giveaway Review THE LAST TIME WE SAID GOODBYE by CYNTHIA HAND @CynthiaHand @HarperTeen
#Giveaway Review THE HAIR BOOK by LaTonya Yvette & Amanda Jane Jones @latonyayvette1 @amandajonesj @UnionSquareKids @RockstarBkTours Ends 5.9
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Sunday Post: Book Pre-order Campaigns & Giveaways Galore – 11/24/19 – Feed Your Fiction Addiction says:
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Home Streaming News TuneIn Radio Infringed Sony & Warner's Copyrights, High Court Rules
TuneIn Radio Infringed Sony & Warner's Copyrights, High Court Rules
Free Tech
In 2017, Sony and Warner sued US-based radio service TuneIn, claiming the company infringed its copyrights in the UK. A judgment handed down today by the High Court states that while TuneIn does not offer content itself, the provision of hyperlinks to content not officially licensed in the UK constitutes a communication to the public and is therefore infringement.
TuneIn is one of the most prominent and recognizable providers of radio content in the world.
Available for free or on a premium basis, the service offers access to well over 100,000 radio stations and millions of podcasts. It doesn't provide this content itself but acts as an indexer ("audio guide service", according to TuneIn) for those looking to access third-party streams.
In 2017 it emerged that Sony Music UK and Warner Music UK had sued the US-based company in the UK, claiming that since many of the TuneIn-indexed stations are unlicensed to play music in the region, linking to them amounts to infringement of the labels' copyrights.
Today, the High Court of England Wales handed down its decision and it doesn't look good for TuneIn. The judgment begins by stating the opposing positions of the labels and TuneIn, which are particularly familiar in these types of disputes concerning hyperlinking.
"The claimants say that a finding for the defendant will fatally undermine copyright. The defendant says that a finding for the claimants will break the internet," Justice Birss writes.
The labels argued that TuneIn needs a license, an assertion "strongly disputed" by TuneIn. The company argued that it does not "store any music, and merely provides users of TuneIn Radio with hyperlinks to works which have already been made freely available on the internet without any geographic or other restriction."
In other words, TuneIn presents itself as not unlike Google search but instead of indexing websites, it indexes and links to radio streams. However, Justice Birss declared the service to be "much more than that", in part due to its curation and search features.
"I find therefore that the activity of TuneIn does amount to an act of communication of the relevant works; and also that that act of communication is to a 'public', in the sense of being to an indeterminate and fairly large number of persons," he writes.
The ruling, which was first published by a blog connected to Bird and Bird, the law firm that represented TuneIn, runs to 47 pages and is both extremely detailed and complex. However, the conclusion to Judge Birss' judgment can be summarized in a straightforward manner.
When TuneIn supplied UK users with links to radio stations that are already licensed in the UK, the company did not infringe Sony or Warner's copyrights.
However, when TuneIn supplied UK users with links to radio stations that are not licensed for the UK or are not licensed at all, the company did infringe the labels' rights.
Noting that TuneIn cannot rely on the safe harbor defenses under the E-Commerce Directive, Judge Birss declared TuneIn, "liable for infringement by authorization and as a joint tortfeasor."
The full judgment can be found here (pdf)
Update: Comment from TuneIn CEO Juliette Morris:
"Today in the U.K., a judgment was announced in a lawsuit involving TuneIn, the leading directory service identifying freely available audio content on the Internet, and Sony Music and Warner Music regarding the availability of music radio stations to TuneIn users in the U.K. The U.K. Court found in favor of TuneIn on the most important claim, confirming that music radio stations licensed in the U.K. can be made available through the TuneIn service to TuneIn's U.K. users.
"While we continue to evaluate the ruling and consider all options, including appeal, we believe the judgment will have very little impact on the company's revenue and ongoing growth strategies. We won on the most important element of the case, which was the right to provide U.K. users with access to U.K.-authorized radio stations. TuneIn is committed to complying with all applicable laws in the countries we serve and will continue to defend the right to operate a directory service providing listeners access to content freely available on the Internet."
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{"url":"https:\/\/electronics.stackexchange.com\/questions\/347223\/voltage-drop-calculation-ac","text":"# Voltage Drop Calculation (AC)\n\nHope someone can help me out and point me the right direction to solving this.\n\nI am trying to do a voltage drop calculation to see the VD % for the cable selected. The following are the values,\n\nCurrent: 140A (I)\nResistance of Cable: 0.0001 Ohm\/m (R)\nReactance of Cable: 0.0000704 Ohm\/m (X)\nPF: 1\nDistance: 300m (L)\nVoltage: 20 kV (20000V)\n\nI am using the following voltage drop formula that I found in a very old school notes (can someone pls verify this formula)?,\n\nVd = (I x (R cos theta + X sin theta) x L) \/ 1000\n\nIf I input the values into the formula, my answer is 0.266V. This does not look right for me as I will get a weird VD %.\n\n\u2022 Shouldn't theta = pf? \u2013\u00a0SunnyBoiz Dec 29 '17 at 8:43\n\u2022 @SunnyBoiz no, \u2018Cos theta\u2019 equals the power factor! \u2013\u00a0rfkortekaas Dec 29 '17 at 9:49\n\u2022 Ic! What does 'sin theta' equals to? (where do I obtain the value?) \u2013\u00a0SunnyBoiz Dec 30 '17 at 3:53\n\nI think that the 1000 in your formula is placed incorrect and probably used when the length is in kilometers. So basically for a one phase load the formula is:\n\n$$V_d = 2I \\bigl(R \\cos(\\theta) + X \\sin(\\theta)\\bigr) L$$ Note: the first 2 is to get the result from source to load and the return as the return cable also influences the voltage drop\n\nFor a three phase system the formula is: $$V_d = \\sqrt{3} I \\bigl(R \\cos(\\theta) + X \\sin(\\theta)\\bigr) L$$\n\nWhere:\n\n\\begin{align} V_d&=\\text{voltage drop in volts}\\\\ I&=\\text{current in amperes}\\\\ R&=\\text{conductive resistance in ohms\/m}\\\\ X&=\\text{conductor inductive reactance in ohms\/m}\\\\ L&=\\text{one way length of circuit in m (or km\/1000 in your formula)}\\\\ \\theta&=\\text{phase angle of the load}\\\\ PF&=\\cos(\\theta)\\\\ \\end{align}\\\\\n\n\\begin{align} PF&=1\\\\ \\theta&=\\arccos(PF)=0\\\\ \\\\ V_d&= 2I \\bigl(R \\cos(\\theta) + X \\sin(\\theta)\\bigr) L\\\\ V_d&= 2\\cdot140\\cdot\\bigl(0.0001 \\cdot\\cos(0) + 0.0000704\\cdot\\sin(0)\\bigr)\\cdot 300\\\\ V_d&= 2\\cdot140\\cdot\\bigl(0.0001 \\cdot1 + 0.0000704\\cdot0\\bigr)\\cdot 300\\\\ V_d&= 2\\cdot140\\cdot\\bigl(0.0001\\bigr)\\cdot 300\\\\ V_d&= 2\\cdot4.2\\\\ V_d&= 8.4\\\\ \\\\ \\end{align}\\\\\n\nSo this gives a voltage drop for one run of 4.2 V and for a double run 8.4 V.\n\nBecause a PF of 1 in an AC circuit is not a real world example I will show the influence of a PF of 0.8 on the voltage drop:\n\n\\begin{align} PF&=0.8\\\\ \\theta&=\\arccos(PF)=36.8699\u00b0\\\\ \\\\ V_d&= 2I \\bigl(R \\cos(\\theta) + X \\sin(\\theta)\\bigr) L\\\\ V_d&= 2\\cdot140\\cdot\\bigl(0.0001 \\cdot\\cos(36.8699\u00b0) + 0.0000704\\cdot\\sin(36.8699\u00b0)\\bigr)\\cdot 300\\\\ V_d&= 2\\cdot140\\cdot\\bigl(0.0001 \\cdot0.8 + 0.0000704\\cdot0.6\\bigr)\\cdot 300\\\\ V_d&= 2\\cdot140\\cdot\\bigl(0.00008+0.00004224\\bigr)\\cdot 300\\\\ V_d&= 2\\cdot140\\cdot\\bigl(0.00012224\\bigr)\\cdot 300\\\\ V_d&= 2\\cdot5.13408\\\\ V_d&= 10.26816\\\\ \\\\ \\end{align}\\\\\n\nAs a result of the reactive power the voltage drop will increase.\n\n\u2022 Hi! Thanks for the response. After checking the school notes again, you are right! I misplaced my 1000 division and it is meant to be used for L. Can you enlighten how you obtain 4.524V in your calculation? I input the values into the formula but my answer is 266.112V. \u2013\u00a0SunnyBoiz Dec 29 '17 at 15:17\n\u2022 Hi rfkorekass, thanks for the steps. Can I ask one more thing? How come it is 2pi? I thought if the pf is given, in this case is 1, I just have to sub in to cos and sin respectively. And when I key it in my calculator it will be cos(inverse)1 and sin(inverse)1. Can you show me how to derive to 2pi? If this is a three phase load, I have to multiply sqrt 3? \u2013\u00a0SunnyBoiz Dec 30 '17 at 3:57\n\u2022 Thank you so much everyone. I am almost there to understanding the equation. One more thing, for the sin(theta) value, do I just simply sub in the arccos(PF) value of 36.86 into the equation? \u2013\u00a0SunnyBoiz Dec 31 '17 at 2:38\n\u2022 Yes, you need to replace $\\theta$ with $\\arccos(PF)$. As in the example $PF=0.8$ then $\\sin(\\theta)=\\sin(\\arccos(0.8)=0.6$. \u2013\u00a0rfkortekaas Dec 31 '17 at 7:20\n\u2022 Thank you! For this particular voltage drop calculation, the 2 is necessary? I thought the multiplication of 2 is only for single phase\/DC. If this is a three phase, is there any addition I must add into equation? \u2013\u00a0SunnyBoiz Jan 1 '18 at 2:57\n\nLet's do this step by step. Shoving everything into one formula usually obscures what's happening.\n\n0.0001ohms\/s is quite big cable. 1mm2 cable is about 17mohms\/m, so 0.1mohm\/m means an area of 170mm2, which is a diameter of about 15mm. Is your cable that big?\n\n140A in 0.1mohm gives 14mV voltage drop per metre.\n\n300m of cable at 14mV\/m is a total drop of 4.2v. Way above your figure, and we've not included the reactance yet, which will increase it.\n\nWhat's that \/1000 doing in your formula? Unless you understand why it's there, where it's come from, you might get answers that are 1000 times too small!\n\n\u2022 Hi Neil. Yes the cable used in the example is a huge cable. I have no idea what is the purpose of the division of a thousand. Hence I will like someone to verify the above formula (afterall its taken from my old school notes). Couldn't find anything online. T_T \u2013\u00a0SunnyBoiz Dec 29 '17 at 8:46\n\nHow the voltage drop formulas derived; Simply voltage drop of cable is the voltage occured due to the resistance (R) and inductance (X) characteristic of the cable itself which depend on load (amp) and total cable length (meter). Taken both R and X as impedance (Z).\n\nV = IZ V cable drop = Icable x Z cable Phytagoras formula, Z\u00b2= R\u00b2+X\u00b2 Z = R\u00b2\/Z + X\u00b2\/Z Apply trigonometric function; cos \u03f4 = R\/Z sin\u03f4 = X\/Z Thus Z = R(cos \u03f4) + X(sin \u03f4)\n\nFor 1 phase 2 wire, we need to multiply Z by 2 since it is for live and neutral cable. Vd= I x Z x Length of 2 cable (or L x 2)\n\nFor 3 phase, we need to multiply by \u221a3\n\nAssuming the voltage, current and power factor are receiving end quantities the magnitude difference in voltage obtained is 4.2V which is no surprise because, the cable impedance is very low.\n\nYour formula is correct, the \"\/1000\" is most probably because the required answer is in kiloVolts rather than in Volts. On practical transmission lines the voltage drop due to reactance is usually in the range of kilovolts.\n\n\u2022 4.2 is your final answer? I assume you forgot about reactance? \u2013\u00a0MCG Dec 29 '17 at 11:03\n\u2022 Please look at what power factor is given \u2013\u00a0spaul Dec 29 '17 at 11:04\n\u2022 Oh right..... which means you didn't answer the question then? \u2013\u00a0MCG Dec 29 '17 at 11:07\n\u2022 I've improved my ans \u2013\u00a0spaul Dec 29 '17 at 11:15\n\u2022 Look at the other 2 answers. They give explanations, all you did was say omit the 1000, and the formula is wrong. You did not explain how\/why you came to this conclusion \u2013\u00a0MCG Dec 29 '17 at 11:25","date":"2019-08-23 16:08:19","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\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 3, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8263582587242126, \"perplexity\": 2395.447332060988}, \"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-2019-35\/segments\/1566027318894.83\/warc\/CC-MAIN-20190823150804-20190823172804-00477.warc.gz\"}"} | null | null |
\section{Introduction}
\section{Introduction}\label{sec-1}
Lossless source codes are classified into
fixed-to-variable-length (FV) codes and variable-to-fixed-length (VF) codes,
which can be represented by code trees and parse trees, respectively.
It is well known that the Huffman coding \cite{Huffman} and Tunstall coding \cite{Tunstall} can attain the best compression rate
in FV codes and VF codes, respectively, for stationary memoryless sources if a single code tree or a single parse tree is used.
But, Yamamoto and Yokoo \cite{Y-Y-01} showed that the AIVF (almost instantaneous variable-to-fixed length) coding can attain better compression rate than the Tunstall coding.
An AIVF code uses $|{\cal X}|-1$ parse trees for a source alphabet ${\cal X}$ and
codewords are assigned to incomplete internal nodes in addition to leaves in each parse tree. Although instantaneous encoding is not possible since incomplete internal nodes are used for encoding, the AIVF code is devised such that the encoding delay is at most one source symbol, and hence the code is called {\em almost instantaneous}.
Furthermore, Yoshida and Kida \cite{Y-K-10}\cite{Y-K-12} showed that any AIVF code can be encoded and decoded by a single virtual multiple parse tree and the total number of nodes can be considerably reduced by the integration.
In the case of FV codes, it is well known by Kraft and McMillan Theorems \cite{Kraft}\cite{McMillan}\cite{C-T-05} that
any uniquely decodable FV code must satisfy Kraft's inequality, and such a code can be realized by an instantaneous FV code, i.e., a prefix FV code. Hence, the Huffman code, which can attain the best compression rate in the class of instantaneous FV codes, is also the best code in the class of uniquely decodable FV codes.
However, it is assumed implicitly in the above argument
that the best code in uniquely decodable FV codes can be constructed by a fixed set of codewords (in other words, a single fixed code tree) for stationary memoryless sources.
But, this assumption is not correct generally. Actually, Yamamoto and Wei \cite{Y-W-2013} showed that we can devise more efficient FV codes than Huffman codes if multiple code trees can be used in the same way as the AIVF codes, and they called such FV codes $K$-ary AIFV (almost instantaneous fixed-to-variable length) codes when the size of code alphabet is $K$.
The $K$-ary AIFV code requires $K-1$ code trees to realize that the decoding delay is at most one code symbol. Hence, in the binary case with $K=2$, multiple code trees cannot be realized. To overcome this defect, they also proposed the binary AIFV code such that the decoding delay is at most two bits.
Although they proposed a greedy algorithm to construct a good AIFV code for a given source in
\cite{Y-W-2013}, it is complicated and the optimal AIFV code cannot always be derived. Furthermore, only a sketch is described for the binary AIFV codes, which are important practically, although $K$-ary AIFV codes for $K\geq 3$ are treated relatively in detail.
In this paper, we refine the definition of the binary and $K$-ary AIFV codes.
The binary (resp.~$K$-ary for $K\geq 3$) AIFV code uses two (resp.~$K-1$) code trees,
in which source symbols are assigned to incomplete internal nodes in addition to leaves.
Although the AIFV codes are not instantaneous codes, they are devised such that the decoding delay is at most two bits (resp.~one code symbol) in the binary (resp.~$K$-ary) case.
Furthermore, for the binary and ternary AIFV codes, we give an algorithm based on integer programing to derive the optimal AIFV code for a given source.
In Section \ref{sec-2}, we show some simple examples of ternary AIFV codes, which can attain better compression rate than the ternary Huffman codes.
Then, after we give the formal definition of $K$-ary AIFV codes for $K\geq 3$, we derive the Kraft-like inequality for the AIFV code trees.
Binary AIFV codes are treated in Section \ref{sec-3}.
Furthermore, we show in Section \ref{sec-4} that the optimal AIFV codes can be derived by solving 0-1 integer programming problems for the binary and ternary AIFV codes. Finally, the compression rates of the AIFV codes are compared numerically with the Huffman codes for several source distributions in Section \ref{sec-5}.
\section{$K$-ary AIFV codes for $K\geq 3$.}\label{sec-2}
\subsection{Examples of ternary AIFV codes} \label{sec-2-1}
We first consider a simple ternary FV code which encodes a source symbol $x\in{\cal X}=\{a, b, c, d, e\}$ to a codeword in ${\cal Y}^*=\{0, 1,2\}^*$.
If the source distribution is uniform, i.e., $P(x)=1/5$ for all $x\in{\cal X}$, then the
entropy of this source is $H_3(X) = \log_3 5\approx1.465$. The code tree of the Huffman code is given by Fig.~\ref{fig2} for this source, and the average code length $L_H$ of the Huffman code is $L_H=1.6$.
\begin{figure}[b]%
\begin{center}
\includegraphics[height=3.5cm]{fig2.eps}
\caption{The Ternary Huffman code for ${\cal X}=\{a, b, c, d, e\}$.} \label{fig2}
\end{center}
\end{figure}%
\begin{figure}[tb]%
\begin{center}
\includegraphics[height=3.5cm]{fig1.eps}
\caption{A ternary AIFV code for ${\cal X}=\{a, b, c, d, e\}$.} \label{fig1}
\end{center}
\end{figure}%
Next we consider a ternary AIFV code given by Fig.~\ref{fig1},
which satisfies the following properties.
\begin{definition}[Ternary AIFV codes]\label{3AIFV}
\begin{itemize}
\item[(A)] A ternary AIFV code consists of two code trees $T_0$ and $T_1$.
\item[(B)] Each complete internal node has three children connected by code symbols `0', `1', and `2', and each incomplete internal node has only one child connected by code symbol `0'
\footnote{
For simplicity, we say ``a node has a child connected by code symbol `$j$' " if the child is connected to the node by a branch with code symbol `$j$'.}.
\item[(C)] The root of $T_1$ must have two children connected by code symbols `1' and `2'.
\item[(D)] Source symbols are assigned to incomplete internal nodes in addition to leaves. But no source symbols are assigned to complete internal nodes.
\end{itemize}
\end{definition}
The AIFV code encodes a source sequence $x_1x_2x_3\cdots$ as follows.
\begin{procedure}[Encoding of ternary AIFV codes] \label{e-3AIFV}
\begin{enumerate}
\item[(a)] Use $T_0$ to encode the initial source symbol $x_1$.
\item[(b)] If $x_i$ is encoded by a leaf (resp.~an incomplete internal node), then
use $T_0$ (resp.~$T_1$) to encode the next source symbol $x_{i+1}$.
\end{enumerate}
\end{procedure}
When $T_0$ given by Fig.~\ref{fig1} is used, the codewords of $a, b, c, d, e$ are 0, 1, 2, 10, 20, respectively. But, they are 1, 10, 20, 21, 22, respectively, when $T_1$ is used.
For instance, source sequence `$abac$' is encoded to `$0.1.1.20$' and
source sequence `$cdebac$' is encoded to `$2.21.20.1.1.20$', where dots `$.$' are
inserted for the sake of human readability, but they are not necessary in the actual codeword sequences.
In the decoding of a codeword sequence $\mbox{\boldmath $y$}=y_1y_2y_3\cdots\in {\cal Y}^*$, code trees $T_0$ and $T_1$ are used in the same way as the encoding.
\begin{procedure}[Decoding of ternary AIFV codes] \label{d-3AIFV}
\begin{enumerate}
\item[(a)] Use $T_0$ to decode the initial source symbol $x_1$ from $\mbox{\boldmath $y$}$.
\item[(b)]
Trace $\mbox{\boldmath $y$}$ as long as possible from the root in the current code tree.
Then, output the source symbol assigned to the reached incomplete internal node or leaf.
\item[(c)]
Remove the traced prefix of $\mbox{\boldmath $y$}$, and
if the reached node is a leaf (resp.~an incomplete internal node), then use $T_0$ (resp.~$T_1$) to decode the next source symbol.
\end{enumerate}
\end{procedure}
For instance, if $\mbox{\boldmath $y$}=10020$, then the decoded sequence is $dae$
because `10', `0', and `20' correspond to leaves $d$, $a$, and $e$, respectively, in $T_0$. But, if $\mbox{\boldmath $y$}=1120$,
$b$ is decoded from `1' in $T_0$ first because there is no path with `$11\cdots$' in $T_0$. Then, the current code tree transfers to $T_1$ because `1' corresponds to the incomplete internal node of $b$ in $T_0$. By removing `1' from $\mbox{\boldmath $y$}$, we have $\mbox{\boldmath $y$}=120$.
Next, $a$ is decoded from `1' in $T_1$ because there is no path with `$12\cdots$' in $T_1$, and the current code tree keeps $T_1$ because `1' corresponds to the incomplete internal node of $a$ in $T_1$. Finally, $c$ is decoded from `20' in $T_1$.
Note that when a source symbol assigned to a leaf is decoded, the decoding is instantaneous. On the other hand, the decoding is not instantaneous when a source symbol assigned to an incomplete internal node is decoded. But the decoding delay is only one code symbol even in such cases.
We now evaluate the average code length of the ternary AIFV code given by Fig.~\ref{fig1}.
Let $L_0$ and $L_1$ be the average code length of $T_0$ and $T_1$, respectively. Then, we can easily show from Fig.~\ref{fig1} that $L_0=1.4$ and $L_1=1.8$.
The transition probability of code trees are given by
\begin{align}
Q(T_1|T_0)&= P_X(b)+P_X(c)=0.4,\\
Q(T_0|T_1)&= P_X(b)+P_X(c)+P_X(d)+P_X(e)=0.8,
\end{align}
and the stationary probabilities of $T_0$ and $T_1$ are given by $Q(T_0)=2/3$ and $Q(T_1)= 1/3$, respectively. Hence, the average code length of the ternary AIFV code is given by
\begin{align}
L_{AIFV}=\frac{2}{3} L_0 + \frac{1}{3}L_1
= \frac{4.6}{3} \approx 1.533, \label{eq-1}
\end{align}
which is shorter than the average code length of the Huffman code $L_H=1.6$.
Now we explain the reason why the AIFV code can beat the Huffman code.
Since incomplete internal nodes are used in addition to leaves for encoding in $T_0$,
$L_0=1.4$ smaller than the source entropy $H_3(X)\approx 1.465$ can be realized.
On the other hand,
$L_1=1.8$ is larger than $L_H=1.6$ because the root of $T_1$ has only two children. But, from $Q(T_0) > Q(T_1)$, $L_{AIFV}$ is smaller than $L_H$ in the above example.
If $|{\cal X}|$ is even, the loss of the ternary Huffman code becomes larger because the Huffman code tree must have an incomplete node. Consider the case that ${\cal X}=\{a, b, c, d\}$ and $P_X(x)=1/4$ for all $x\in {\cal X}$. Then the Huffman and AIFV code trees are given by Fig.~\ref{fig2-2} and Fig.~\ref{fig1-2}, respectively. In this case, the entropy of this source is $H_3(X)=\log_3 4\approx 1.262$, and the average code length is given by $L_H=1.5$ and $L_{AIFV}=4/3\approx1.333$.
\begin{figure}[t]%
\begin{center}
\includegraphics[height=3.5cm]{fig2-2.eps}
\caption{The ternary Huffman code for ${\cal X}=\{a, b, c, d\}$.} \label{fig2-2}
\end{center}
\end{figure}%
\begin{figure}[tb]%
\begin{center}
\includegraphics[height=3.5cm]{fig1-2.eps}
\caption{A ternary AIFV code for ${\cal X}=\{a, b, c, d\}$.} \label{fig1-2}
\end{center}
\end{figure}%
It is well known that if we construct the Huffman code for ${\cal X}^2$ as shown in Fig.~\ref{fig2-3}, the average code length per source symbol $L_H^{{\cal X}^2}$ can be improved compared with $L_H$.
In the case of $P_X(x)=1/4$, we have $L_H^{{\cal X}^2}=43/32\approx 1.344 <L_H=1.5$. But, the Huffman code for ${\cal X}^2$ has demerits such that
the size of the code tree increases to roughly $|{\cal X}|^2$, and the encoding and decoding delay becomes long as $|{\cal X}|$ becomes large.
On the other hand,
by concatenating $T_1$ to incomplete nodes of $T_0$ and $T_1$ in Fig.~\ref{fig1-2}, we obtain a code tree shown by Fig.~\ref{fig3}. Hence, the AIFV code can realize a flexible code tree for ${\cal X}^*$ by using only two code trees $T_0$ and $T_1$. We note that the total size of AIFV code trees is roughly $2|{\cal X}|$, and $L_{AIFV}=4/3\approx1.333$ is better than $L_H^{{\cal X}^2}=43/32\approx 1.344$.
Furthermore, the encoding delay is zero and the decoding delay is at most one code symbol in the case of AIFV codes.
\begin{figure}[t]%
\begin{center}
\includegraphics[height=4cm]{fig2-3.eps}
\caption{A ternary Huffman code tree for ${\cal X}^2$.} \label{fig2-3}
\end{center}
\end{figure}%
\begin{figure}[t]%
\begin{center}
\includegraphics[height=4cm]{fig3.eps}
\caption{The concatenated code tree of AIFV code.} \label{fig3}
\end{center}
\end{figure}%
We note from Definition \ref{3AIFV} that the root of $T_1$ must have two children.
But, the root of $T_0$ can become an incomplete node.\footnote{The idea of assigning a source symbol to the root of $T_0$ was suggested by
Prof. M.~Nishiara at the presentation of \cite{Nishiara}.}
For instance, consider a source such that $P_X(a)=0.8$, $P_X(b)=0.1$, and $P_X(c)=P_X(d)=0.05$. In this case, the entropy is given by $H_3(X)\approx 0.6448$, and
the Huffman code of this source is given by Fig.~\ref{fig2-2} which attains $L_H(X)=1.1$. On the other hand, the ternary AIFV code shown in Fig.~\ref{fig1-3} attains $L_{AIFV}=Q(T_0)L_0+Q(T_1)L_1=(5/9)0.4+(4/9)1.2
\approx 0.7556$ for this source.
Note that $L_H$ cannot become shorter than 1 in any case while $L_{AIFV}$ can become shorter than 1 by assigning the source symbol $x$ with $P_X(x)\gg0.5$ to the root of $T_0$.
For instance, if we use the AIFV code shown in Fig.~\ref{fig1-3},
a source sequence $aabaaacd$ is encoded to `$\lambda.1.00.\lambda.1.\lambda.21.02$', where
$\lambda$ represents the null codeword and dots `.' are not necessary in the actual codeword sequence.
Hence the codeword sequence $\mbox{\boldmath $y$}$ is given by $\mbox{\boldmath $y$}=10012102$.
Although the code length of $a$ in $T_0$ is zero, we can decode $x_1=a$ from the codeword sequence $\mbox{\boldmath $y$}$ because we begin the decoding with $T_0$, and there is no path with `$1\cdots$' in $T_0$, which means that $x_1$ is $a$.
Similarly we can decode `$aabaaacd$' from $\mbox{\boldmath $y$}=10012102$.
\footnote{Refer Remark \ref{remark-2} for how to detect the end of a source sequence. }
\begin{figure}[t]%
\begin{center}
\includegraphics[height=4cm]{fig1-3.eps}
\caption{A ternary AIFV code.} \label{fig1-3}
\end{center}
\end{figure}%
\subsection{$K$-ary AIFV codes}
In this subsection, we generalize ternary AIFV codes to $K$-ary AIFV codes with code alphabet ${\cal Y}=\{0, 1, 2, \cdots, K-1\}$ for $K\geq 3$.
\vspace{0.1cm}
\begin{definition}[$K$-ary AIFV code]\label{def-K-AIFV}\footnote{This definition is slightly different from \cite[Definition 1]{Y-W-2013} because the root of code tree $T_k$, $0\leq k\leq K-3$, can become incomplete in this paper although it must be complete in \cite[Definition 1]{Y-W-2013}.}
\label{KAIFV}
\begin{itemize}
\item[(A)] A $K$-ary AIFV code consists of $K-1$ code trees, $T_0, T_1, \cdots$, $T_{K-2}$.
\item[(B)]
Each complete internal node has $K$ children connected by code symbols
`0', `1', $\cdots$, `$K-1$'. Every incomplete internal node has at least one and at most $K-2$ children connected by code symbols `0', `1', $\cdots$, `$K_c-1$', where $K_c$ is the number of the children.
\item[(C)]
The root of $T_k$ is called complete if it has $K-k$ children.
For $0\leq k\leq K-2$, the root of $T_k$ has $K-k$ children connected
by code symbols `$k$', `$k+1$', $\cdots$ `$K-1$' if the root is complete.
For $0\leq k\leq K-3$, the root of $T_k$ can become incomplete,
and the incomplete root of $T_k$ must have at least one and at most $K-k-2$ children connected by `$k$', `$k+1$', $\cdots$, `$K_c-1$', where $K_c-k$ is the number of the children of the incomplete root. We regard the incomplete root of $T_k$ with $K_c-k$ children as an incomplete internal node with $K_c$ children.
\item[(D)]Source symbols are assigned to incomplete internal nodes in addition to leaves. But no source symbols are assigned to complete internal nodes.
\end{itemize}
\end{definition}
\vspace{0.2cm}
A $K$-ary AIFV code can encode a source sequence $x_1x_2x_3\cdots$
and decode a codeword sequence $\mbox{\boldmath $y$}=y_1y_2y_3\cdots$
in the same way as ternary AIFV codes.
\newpage
\begin{procedure}[Encoding of $K$-ary AIFV codes] \label{e-KAIFV}
\begin{enumerate}
\item[(a)] Use $T_0$ to encode the initial source symbol $x_1$.
\item[(b)] When $x_i$ is encoded by a leaf (resp.~an incomplete internal node with $j$ children), then use $T_0$ (resp.~$T_{j}$) to encode the next source symbol $x_{i+1}$.
\end{enumerate}
\end{procedure}
\begin{procedure}[Decoding of $K$-ary AIFV codes] \label{d-KAIFV}
\begin{enumerate}
\item[(a)] Use $T_0$ to decode the initial source symbol $x_1$ from $\mbox{\boldmath $y$}$.
\item[(b)]
Trace $\mbox{\boldmath $y$}$ as long as possible from the root in the current code tree.
Then, output the source symbol assigned to the reached incomplete internal node or leaf.
\item[(c)]
Remove the traced prefix of $\mbox{\boldmath $y$}$, and
if the reached node is a leaf (resp.~an incomplete internal node with $j$ children), then use $T_0$ (resp.~$T_j$) to decode the next source symbol.
\end{enumerate}
\end{procedure}
As an example, an AIFV code is shown in Fig.~\ref{Exof4aryAIFV} for the case of $K=4$ and
${\cal X}=\{a, b, c, d, e, f, g, h, i, j\}$. When source sequence
`$abacgcebbd$' is encoded by this AIFV code, the codeword sequence and the transition of code trees are given in Table \ref{T-4aryAIFV}.
Note that when source symbol $x_i$ is encoded (or decoded) at a node with $j$ children,
then $x_{i+1}$ is encoded (or decoded) by $T_j$.
Furthermore we can easily check that every $x_i$ can be uniquely decoded.
For instance, $x_2=b$ is encoded to codeword `1' at incomplete internal node $b$ in $T_0$.
In this case, $x_3$ is encoded in $T_1$ because the incomplete internal node $b$ has one child in $T_0$.
This means that the codeword of $x_3$ does not begin with `0'.
In the decoding, we obtain $\mbox{\boldmath $y$}=113130\cdots$ after the decoding of $x_1=a$ in $T_0$ and removing decoded codeword `$0$' from $\mbox{\boldmath $y$}$.
Then we can decode $x_2=b$ because there is no path with $\mbox{\boldmath $y$}=11\cdots$ in $T_0$ but
the path `$1$' corresponds to node $b$ in $T_0$.
Another example of 4-ary AIFV code trees for ${\cal X}=\{a, b, c, d, e, f, g, h\}$ is shown in Fig. ~\ref{Exof4aryAIFV-2}, in which the roots of $T_0$ and $T_1$ are incomplete.
The codeword sequence for `$badbacgaec$' is shown in Table \ref{T-4aryAIFV-2},
where `$\lambda$' represents the null codeword.
Note that the incomplete root of $T_k$ with $K_c-k$ children is regarded as an incomplete internal node with $K_c$ children as explained in Definition \ref{def-K-AIFV}-(C).
Hence, for instance, node $x_2=a$ is the incomplete root with one child in $T_1$, and it is regarded as an incomplete internal node with 2 children. Hence $x_3$ is encoded (or decoded) in $T_2$.
In the decoding, we can decode $x_1=b$ from $\mbox{\boldmath $y$}=03210\cdots$ in $T_0$ because there is no path with $\mbox{\boldmath $y$}=03\cdots$ in $T_0$, but path `0' corresponds to node $b$.
In the decoding of $x_2$, we have $\mbox{\boldmath $y$}=3210\cdots$ in $T_1$. But, there is no path which begins with `3'. Hence we obtain $x_2=a$ because `no path' means the root in $T_1$.
\begin{figure}[t]%
\begin{center}
\includegraphics[height=5cm]{fig4ary.eps}
\caption{An example of 4-ary AIFV code trees.} \label{Exof4aryAIFV}
\end{center}
\end{figure}%
\begin{table}[t]
\begin{center}
\caption{An example of codeword sequence for 4-ary AIFV code.}\label{T-4aryAIFV}.
\begin{tabular}{c|cccccccccc}
$i$ & 1 &2& 3& 4& 5& 6& 7& 8& 9& 10\\
\hline
Current code tree& $T_0$ & $T_0$& $T_1$& $T_2 $& $T_1$&$T_0$&$T_2$&$T_2$&$T_0$&$T_1$\\
Source symbol $x_i$& $a$& $b$& $a$& $c$ & $g$ & $c$ & $e$ & $b$ & $b$ & $d$\\
Codeword &0& 1& 1& 31& 30& 2& 33& 30& 1& 11\\
Number of children of node $x_i$ & 0& 1& 2 &1&0& 2&2&0&1&0
\end{tabular}
\end{center}
\end{table}
Since some source symbols are assigned to incomplete internal nodes, the AIFV code is not
an instantaneous code. But since the following theorem holds, this code is
{\em almost instantaneously decodable}.
\begin{theorem}
The $K$-ary AIFV codes defined in Definition \ref{def-K-AIFV} and Procedures \ref{e-KAIFV} and \ref{d-KAIFV} are uniquely decodable, and the maximum decoding delay is at most one code symbol.
\end{theorem}
{\em Proof:}
From Procedure \ref{e-KAIFV}-(b) and Procesure \ref{d-KAIFV}-(c),
both encoding and decoding have the same transition of code trees.
Hence, each source symbol $x_i$ is decoded in the same code tree used in the encoding.
It is clear from Procedure \ref{d-KAIFV}-(b) that if $x_i$ is encoded at a leaf in $T_k$, then
$x_i$ is uniquely decodable.
If $x_i$ is encoded at an incomplete internal node with $j$-children in $T_k$, then the children are connected
by one of code symbols $\{0, 1, \cdots, j-1\}$ from the incomplete internal node.
On the other hand, $x_{i+1}$ is encoded in $T_j$, in which any path begins with one of code symbols $\{j, j+1 \cdots, K-1\}$. Hence the node reached in Procesure \ref{d-KAIFV}-(b) is the same incomplete internal node used in the encoding.
It is obvious that when $x_i$ is encoded at a leaf, then it can be decoded instantaneously.
But, when $x_i$ is encoded at an incomplete internal node in $T_k$, we must read one more code symbol to judge whether the incomplete internal node corresponds to the longest path in $T_k$. Hence the maximum decoding delay is at most one code symbol.
\begin{flushright} Q.E.D.\end{flushright}
\begin{figure}[t]%
\begin{center}
\includegraphics[height=5cm]{fig4ary-2.eps}
\caption{An example of 4-ary AIFV code trees with incomplete roots.} \label{Exof4aryAIFV-2}
\end{center}
\end{figure}%
\begin{table}
\begin{center}
\caption{An example of codeword sequence for 4-ary AIFV code with incomplete roots.}\label{T-4aryAIFV-2}.
\begin{tabular}{c|cccccccccc}
$i$ & 1 &2& 3& 4& 5& 6& 7& 8& 9& 10\\
\hline
Current code tree& $T_0$ & $T_1$& $T_2$& $T_1 $& $T_0$&$T_2$&$T_0$&$T_1$&$T_2$&$T_1$\\
Source symbol $x_i$& $b$& $a$& $d$& $b$ & $a$ & $c$ & $g$ & $a$ & $e$ & $c$\\
Codeword &0&$\lambda$& 32& 10& $\lambda$& 31& 13& $\lambda$& 33& 31\\
Number of children of node $x_i$ & 1& 1 &1&0& 2&0& 1&1&2&0
\end{tabular}
\end{center}
\end{table}
\vspace{0.2cm}
\begin{remark} \label{remark-1}
If there are no incomplete internal nodes with $j$ children in all code trees, we can delete the code tree $T_{K-j}$. Furthermore, if we use only the incomplete internal nodes with $j$ children for a fixed $j$, $1\leq j \leq K-2$, then the code trees can be reduced to two code trees $T_0$ and $T_{K-j}$
even for the case of $K>3$. Such restriction worsens the compression rate of the $K$-ary AIFV codes. But, the construction of code trees becomes easy as shown in Section IV.
\end{remark}
\vspace{0.2cm}
\begin{remark} \label{remark-2}
In the decoding described in Procedures \ref{d-3AIFV} and \ref{d-KAIFV}, we assumed that the end of codeword sequence can be detected by another mechanism. In the case that the end cannot be detected and/or the null codeword is assigned to an incomplete root, we add a special symbol EOF to ${\cal X}$, and we assign EOF to a leaf in each $T_k$.
By encoding EOF at the end of a source sequence, we can know the end of the decoding.
The end of decoding can also be detected by adding the length of a source sequence
encoded by e.g.~Elias $\delta$ code \cite{Elias} into the prefix of the codeword sequence.
These worsen the compression rate a little. But, the degradation is negligible if $|{\cal X}|$ is not small and the length of a source sequence is sufficiently large.
\end{remark}
\subsection{Kraft-like inequalities for $K$-ary AIFV code trees}\label{Kraft-KAIFV}
In this subsection, we derive lower and upper bounds of average code length $L_k$ for code tree $T_k$, $0\leq k \leq K-2$.
Let ${\cal N}_0^{(k)}$ (resp.~${\cal N}_j^{(k)}$) be the set of leaves (resp.~incomplete internal nodes with $j$ children) in code tree $T_k$, $0\leq k \leq K-2$, and
let $n_x$ be the incomplete internal node or leaf corresponding to a source symbol $x\in {\cal X}$. Furthermore, let $l_k(x)$ be the code length of $x\in {\cal X}$ in $T_k$.
We first consider $T_0$. If $n_x\in{\cal N}^{(0)}_j$, then we can change the node $n_x$ to a complete internal node by adding $K-j$ children at depth $l_0(x)+1$ of $T_0$.
Hence, we have from Kraft's inequality that
\begin{equation}
\sum_{j=0}^{K-2} \sum_{x: n_x\in {\cal N}^{(0)}_j}(K-j) K^{-[l_0(x)+1]} =1. \label{eq-10}
\end{equation}
In the case of $k>0$, since the root of $T_k$ has $K-k$ children, $K^{-[l_0(x)+1]}$ should become
$(K-k)^{-1}K^{-l_k(x)}$. Therefore, we have
\begin{equation}
\sum_{j=0}^{K-2} \sum_{x: n_x\in {\cal N}^{(k)}_j} (K-j) (K-k)^{-1}K^{-l_k(x)} =1. \label{eq-11}
\end{equation}
Let $\hat{P}_X(x) = (K-j) (K-k)^{-1}K^{-l_k(x)}$ for $n_x\in {\cal N}^{(k)}_j$.
Then, from $\sum_{x\in {\cal X}} \hat{P}_X(x)=1$
and $-\log_K \hat{P}_X(x) = l_k(x) + \log_K (K-k) - \log_K (K-j)$,
we have
\begin{align}
0 &\leq D(P_X\|{\hat P}_X)
= \sum_{x\in {\cal X}} P_X(x) \log_K \frac{P_X(x)}{\hat{P}_X(x)} \nonumber\\
& = -H_K(X) -\sum_{x\in {\cal X}} P_X(x) \log_K \hat{P}_X(x) \nonumber\\
&= -H_K(X) + \sum_{j=0}^{K-2} \sum_{n_x\in {\cal N}^{(k)}_j} P_X(x)
\left[ l_k(x) + \log_K (K-k) - \log_K (K-j)\right] \nonumber\\
& =-H_K(X) + \left[\sum_{x\in {\cal X}} P_X(x)l_k(x)\right] +\log_K (K-k)
-\sum_{j=0}^{K-2} \sum_{x: n_x\in {\cal N}^{(k)}_j} P_X(x)\log_K (K-j) \nonumber\\
& =-H_K(X) + L_k + \log_K (K-k)
- \sum_{j=0}^{K-2} P({\cal N}^{(k)}_j) \log_K (K-j),\label{eq-12}
\end{align}
where $P({\cal N}^{(k)}_j)= \sum_{x: n_x\in {\cal N}^{(k)}_j}P_X(x)$. Hence, $L_k$ must satisfy that
\begin{equation}
L_k \geq H_K(X) +\sum_{j=0}^{K-2} P({\cal N}^{(k)}_j) \log_K \frac{K-j}{K-k}
\label{eq-13}
\end{equation}
Next we derive an upper bound of $L_k$.
If we allow that there exist leaves and/or incomplete internal nodes with no source symbol assigned in $T_k$, \eqref{eq-11} becomes
\begin{equation}
\sum_{j=0}^{K-2} \sum_{x: n_x\in {\cal N}^{(k)}_j} (K-j) (K-k)^{-1}K^{-l_k(x)} \leq 1. \label{eq-14}
\end{equation}
Clearly, the original $T_k$ can attain better compression rate than such a relaxed code tree $\hat{T}_k$.
We can easily check that $\hat{T}_k$ can be constructed if it satisfies \eqref{eq-14}
and incomplete internal nodes can be arranged to satisfy the following condition.
\begin{condition}\footnote{Refer Section \ref{IP-tAIFV} to see how this condition can be represented by equations. }
\label{cond-1}
Every node $n\in {\cal N}^{(k)}_j$ has $j$ children.
\end{condition}
We now define $l_k(x)$ as
\begin{align}
l_k(x)& =\left\lceil -\log_K P_X(x) +\log_K \frac{K-j}{K-k}\right\rceil\nonumber\\
& < -\log_K P_X(x) +\log_K \frac{K-j}{K-k} +1. \label{eq-14-2}
\end{align}
Then, this $l_k(x)$ satisfies \eqref{eq-14}, and Condition \ref{cond-1} can
be satisfied by setting $j$ appropriately for each $x$ because it can always be satisfied for any $x$ by $j=0$.
Hence, for appropriately selected $j$,
we can construct $\hat{T}_k$ with average code length $\hat{L}_k$ satisfying that
\begin{align}
L_k \leq \hat{L}_k &=\sum_{x\in{\cal X}} P_X(x) l_k(x) \nonumber\\
&< H_K(X) +\sum_{j=0}^{K-2} P({\cal N}^{(k)}_j) \log_K \frac{K-j}{K-k} +1. \label{eq-15}
\end{align}
Note that the term $\log_K (K-j)/(K-k)$ in \eqref{eq-13} and \eqref{eq-15} is negative
if $j> k$ although it is positive if $j< k$. Especially, in the case of $L_0$,
the second term of \eqref{eq-13} and \eqref{eq-15} is always negative.
The global average code length $L_{AIFV}$ is given by
\begin{equation}
L_{AIFV} =\sum_{k=0}^{K-2} Q(T_k) L_k, \label{eq-16}
\end{equation}
where $Q(T_k)$ is the stationary probability of $T_k$, and $Q(T_k)$ is determined from
$Q(T_j|T_k)=P({\cal N}^{(k)}_j)$, $0\leq k \leq K-2$, $0\leq j \leq K-2$.
Generally, it is difficult to evaluate the term in \eqref{eq-16} given by
\begin{align}
\sum_{k=0}^{K-2} Q(T_k) \sum_{j=0}^{K-2} P({\cal N}^{(k)}_j) \log_K \frac{K-j}{K-k}
&= \sum_{k=0}^{K-2} \sum_{j=0}^{K-2} Q(T_k) Q(T_j|T_k)\log_K \frac{K-j}{K-k}.
\label{eq-16-2}
\end{align}
But, in the case of $K=3$ or the case
such that only two code trees are used for $K>3$ as described in Remark \ref{remark-2},
it holds that $Q(T_k) Q(T_j|T_k)=Q(T_j) Q(T_k|T_j)$. Hence, in these cases,
\eqref{eq-16-2} becomes zero, and the following bound is obtained from \eqref{eq-13} and \eqref{eq-15}--\eqref{eq-16-2}.
\begin{equation}
H_K(X) \leq L_{AIFV} < H_K(X)+1 \label{eq-17}
\end{equation}
Unfortunately, the upper bound in \eqref{eq-17} is the same as the well known bound of the Huffman code. But, this fact does not mean that the performance of AIFV code with two code trees is the same as the performance of the Huffman code.
The AIFV code trees are more flexible than the Huffman code tree. The term `$+1$' in \eqref{eq-14-2} can be decreased by selecting $j$ appropriately for each $x\in{\cal X}$ in the case of AIFV code trees.
Actually, as we will show in Section \ref{sec-6}, the AIFV codes can attain better compression rate than the Huffman codes.
\section{Binary AIFV codes}\label{sec-3}
\subsection{Definition of binary AIFV codes}
The $K$-ary AIFV codes treated in the previous section can be constructed only for $K\geq 3$, and the binary represented codewords of $K$-ary AIFV codes
are not so short as binary Huffman codes. But, we show in this section that if decoding delay is allowed at most two bits,
we can construct a binary AIFV code that attains better compression rate than the binary Huffman code.
We first show a simple example of a binary AIFV code in Fig.~\ref{fig6}, which satisfies the following properties.
\begin{figure}[t]%
\begin{center}
\includegraphics[height=3.5cm]{fig6.eps}
\end{center}
\vspace*{-0.5cm}
\caption{A binary AIFV code.}
\label{fig6}
\vspace*{-0.2cm}
\end{figure}%
\begin{definition}[Binary AIFV codes]\label{2AIFV}
\begin{itemize}
\item[(A)] A binary AIFV code consists of two code trees $T_0$ and $T_1$.
\item[(B)] Each complete internal node has two children connected by code symbols `0', and `1'. Incomplete internal nodes, each of which has one child, are divided into two categories, say master nodes and slave nodes.
The child of a master node must be a slave node, and
the master node is connected to its grandchild by code symbols `00'.
\item[(C)] The root of $T_1$ must have two children
connected by code symbols `0' and `1'.
The child connected by `0' is a slave node and the root cannot have a grandchild connected by code symbols `00'.
\item[(D)] Source symbols are assigned to master nodes in addition to leaves.
But no source symbols are assigned to neither complete internal nodes nor slave nodes.
\end{itemize}
\end{definition}
The binary AIFV code encodes a source sequence $x_1x_2x_3\cdots$ as follows.
\begin{procedure}[Encoding of binary AIFV codes] \label{e-2AIFV}
\begin{enumerate}
\item[(a)] Use $T_0$ to encode the initial source symbol $x_1$.
\item[(b)] When $x_i$ is encoded by a leaf (resp.~a master node), then
use $T_0$ (resp.~$T_1$) to encode the next source symbol $x_{i+1}$.
\end{enumerate}
\end{procedure}
If we use the binary AIFV code shown in Fig.~\ref{fig6}, then for instance, a source sequence
`$cbcaab$' is encoded to `$11.10.11.01.0.10$', and
source sequence `$cadbca$' is encoded to `$11.01.1100.10.11.01$', where dots `$.$' are not necessary in the actual codeword sequences.
A codeword sequence $\mbox{\boldmath $y$}=y_1y_2y_3\cdots\in {\cal Y}^*$ can be decoded by using code trees $T_0$ and $T_1$ as follows.
\begin{procedure}[Decoding of binary AIFV codes] \label{d-2AIFV}
\begin{enumerate}
\item[(a)] Use $T_0$ to decode the initial source symbol $x_1$ from $\mbox{\boldmath $y$}$.
\item[(b)]
Trace $\mbox{\boldmath $y$}$ as long as possible from the root in the current code tree.
Then, output the source symbol assigned to the reached master node or leaf.
\item[(c)]
Let $\hat{\mbox{\boldmath $y$}}$ be the path from the root to the reached master node or leaf.
Then, remove $\hat{\mbox{\boldmath $y$}}$ from the prefix of $\mbox{\boldmath $y$}$.
If the reached node is a leaf (resp.~a master node), then use $T_0$ (resp.~$T_1$) to decode the next source symbol.
\end{enumerate}
\end{procedure}
For instance, from $\mbox{\boldmath $y$}=11101101010$, we can decode $x_1=c$ when `111' is read
because there is no path `111' from the root in $T_0$ but the master node $c$ is reached by `11'. Similarly, in the case of $\mbox{\boldmath $y$}=11011100101101$, we can decode $x_1=c$ when `1101' is read because there is no path `1101' in $T_0$.
We can easily check that `$cadbca$' can be decoded from $\mbox{\boldmath $y$}=11011100101101$.
We note that $x_i$ is decoded instantaneously if $x_i$ is encoded by a leaf,
and it is decoded with two-bit delay if $x_i$ is encoded by a master node.
Hence, the decoding delay of the binary AIFV codes is at most two bits.
Now consider a source such that
${\cal X}=\{a, b, c, d\}$, and $P_X(a)=0.45$, $P_X(b)=0.3$, $P_X(c)=0.2$, $P_X(d)=0.05$.
In this case, the entropy and the average code length of the binary Huffman code
are given by $H_2(X)\approx1.7200$ and $L_H=1.8$, respectively.
If we use the binary AIFV code shown in Fig.~\ref{fig6}, the average code length are given by $L_0=1.65$ and $L_1=2.1$ for $T_0$ and $T_1$, respectively. Since $T_1$ is used only just after $c$ is encoded in Fig.~\ref{fig6}, we have $Q(T_1|T_0)=0.2$ and $Q(T_0|T_1)=0.8$ which mean that $Q(T_0)=0.8$ and $Q(T_1)=0.2$. Therefore, we have $L_{AIFV}=1.65\times 0.8+ 2.1\times 0.2 =1.74$, which is better than $L_H=1.8$.
Note that the root of $T_0$ can become a master node although the root of $T_1$ must have two children. Such an AIFV code is shown in Fig.~\ref{2ary-2} for ${\cal X}=\{a, b, c\}$.
For instance, source sequence $x_1x_2x_3=aaab$ is encoded to codeword sequence `$\lambda.1.\lambda.010$' by this AIFV code, which means $\mbox{\boldmath $y$}=1010$.
We can decode $x_1x_2x_3$ uniquely from $\mbox{\boldmath $y$}=1010$. First, we decode $x_1=a$ because there is no path with `$1\cdots$' in $T_0$. This means that $x_1$ is encoded at the root of $T_0$, and hence $x_1=a$. Next we move to $T_1$, and we obtain $x_2=a$ from $\mbox{\boldmath $y$}=1010$. Then, we move to $T_0 $ with $\mbox{\boldmath $y$}=010$. Since there is no path with `$1\cdots$' in $T_0$, we decode $x_3=a$. Finally we move to $T_1$ with $\mbox{\boldmath $y$}=010$, and we obtain $x_4=b$.
When $P_X(a)=0.9$ and $P_X(b)=P_X(c)=0.05$, this AIFV code have that $Q(T_1|T_0)=0.9$,
$Q(T_0|T_1)=1$, $Q(T_0)=10/19$, $Q(T_1)=9/19$, $L_0=0.3$, $L_1=1.2$, and $L_{AIFV}=Q(T_0)L_0+Q(T_1)L_1\approx 0.7263$. On the other hand, this source has $H(X)\approx 0.5690$
and the average code length of the Huffman code is $L_H=1.1$.
In the binary case, $L_H$ cannot become shorter than one while $L_{AIFV}$ can become shorter than one as shown in this example.
\begin{figure}[t]%
\begin{center}
\includegraphics[height=3.5cm]{fig2ary-2.eps}
\end{center}
\vspace*{-0.5cm}
\caption{A binary AIFV code with the incomplete root in $T_0$.}
\label{2ary-2}
\vspace*{-0.2cm}
\end{figure}%
\subsection{Kraft-like inequalities for binary AIFV codes}\label{Kraft-bAIFV}
In the same way as Section \ref{Kraft-KAIFV},
we can derive Kraft-like inequalities for binary AIFV codes.
Let ${\cal N}_0^{(k)}$ (resp.~${\cal N}_1^{(k)}$) be the set of leaves (resp. master nodes) in code tree $T_k$, $k=0, 1$. Furthermore, let $n_x$ be the master node or leaf assigned a source symbol $x$, and let $l_k(x)$ be the code length of $x\in {\cal X}$.
Note that since a master node has only one grandchild, the master node becomes a complete node if we add three grandchildren to the master node.
Hence we have the following relation for $T_0$.
\begin{align}
\sum_{x: n_x\in {\cal N}_0^{(0)}} 2^{-l_0(x)} + \frac{3}{4}
\sum_{x: n_x\in {\cal N}_1^{(0)}} 2^{-l_0(x)}=1 . \label{eq-20}
\end{align}
Similarly, the following relation holds for $T_1$ because the root of $T_1$ can have only three grandchildren.
\begin{align}
\sum_{x: n_x\in {\cal N}_0^{(1)}} 2^{-l_1(x)} + \frac{3}{4}
\sum_{x: n_x\in {\cal N}_1^{(1)}} 2^{-l_1(x)}=\frac{3}{4} \label{eq-20-1}
\end{align}
or
\begin{align}
\frac{4}{3}\sum_{x: n_x\in {\cal N}_0^{(1)}} 2^{-l_1(x)} + \sum_{x: n_x\in {\cal N}_1^{(1)}}
2^{-l_1(x)}=1 .\label{eq-20-2}
\end{align}
Furthermore, the global average code length $L_{AIFV}$ is given by
\begin{align}
L_{AIFV}&=Q(T_0)L_0+Q(T_1)L_1\nonumber\\
& =\frac{P({\cal N}_0^{(1)})L_0 +P({\cal N}_1^{(0)})L_1}{P({\cal N}_0^{(1)})+P({\cal N}_1^{(0)})}. \label{eq-20-3}
\end{align}
Then, in the same way as \eqref{eq-13}, \eqref{eq-15},
and \eqref{eq-17}, we can derive the following bounds.
\begin{align}
H_2(X) -P({\cal N}_1^{(0)})(2-\log_2 3)&\leq L_0
< H_2(X) -P({\cal N}_1^{(0)})(2-\log_2 3) +1, \label{eq-21}\\
H_2(X) +P({\cal N}_0^{(1)})(2-\log_2 3) &\leq L_1
< H_2(X) +P({\cal N}_0^{(1)})(2-\log_2 3) +1, \label{eq-22}\\
H_2(X) \leq L_{AIFV} <H_2(X)+1, \label{eq-23}
\end{align}
where the upper bounds of the above inequalities must satisfy the following condition.
\begin{condition}\footnote{Refer Section \ref{IP-bAIFV} to see how this condition can be represented by equations. }
\label{cond-2}
Every node $n\in {\cal N}^{(k)}_1$, $k=0,1$, has one grandchild.
\end{condition}
Note that $L_0$ can become smaller than the source entropy $H_2(X)$ but $L_1$ is larger than $H_2(X)$.
Although the upper bound $H_2(X)+$1 in \eqref{eq-23} is the same as the case of Huffman codes, the term `+1' can be decreased than the Huffman codes for individual sources because the binary AIFV code trees are more flexible than the Huffman code tree.
\section{Construction of AIFV code trees based on integer programming}\label{sec-4}
In this section, we propose a construction method of AIFV code trees based on integer programming (IP) for AIFV codes with two code trees. Although the IP problem is generally NP hard, the IP is used to solve more practical problems as the hardware of computers and the software of IP solvers develop.
Before we treat AIFV code trees, we first consider the case of binary Huffman code trees.
Let ${\cal X}=\{a_1, a_2, \cdots, a_{|{\cal X}|}\}$, $p_t=P_X(a_t)$, and $d_t=l(a_t)$. Then,
the problem to obtain the binary Huffman code tree is
equivalent to obtain $\{d_t\}$ that minimizes $\sum^{|{\cal X}|}_{t=1} p_t d_t$ under
the Kraft inequality
\begin{align}
\sum^{|{\cal X}|}_{t=1} 2^{-d_t} \leq 1. \label{Kraft}
\end{align}
In this case, the inequality `$\leq$' in \eqref{Kraft} can be replaced with equality `$=$'
because the optimal $\{d_t\}$ always satisfies the equality in \eqref{Kraft}.
In order to formalize this optimization problem as a 0-1 IP problem,
we introduce binary variables $u_{t,d}$ such that
$u_{t,d}=1$ if source symbol $a_t$ is assigned to a leaf of depth $d$ in a code tree,
and $u_{t,d}=0$ otherwise.
Then, the optimization problem can be formalized as follows.
\begin{ip}\label{ip-1}
\begin{align}
\mbox{minimize} \quad&\sum_{t=1}^{|{\cal X}|}\ \sum_{d = 1}^{D} u_{t,d}\, p_t \,d \label{eq3-1} \\
\mbox{subject to} \quad&\sum_{t=1}^{|{\cal X}|}\ \sum_{d = 1}^{D} 2^{-d} u_{t,d}=1,\label{eq3-2} \\
&\sum_{d=1}^{D} u_{t,d}= 1, \qquad t = 1,2,\dots ,|{\cal X}|, \label{eq3-3}
\end{align}
where $D$ is a positive integer constant, which represents the maximum depth considered in the IP problem.
\end{ip}
\vspace{0.2cm}
Condition \eqref{eq3-3} guarantees that each $a_t$ is assigned to only one $d$,
and $d_t$ is determined as $d_t=d$ for $u_{t,d}=1$.
$D$ must be sufficiently large. But, large $D$ consumes computational time and memory.
In many cases, it is sufficient that $D$ is several times as large as $\log_2 |{\cal X}|$.
\subsection{IP problem for binary AIFV code trees}\label{IP-bAIFV}
In order to obtain the optimal binary AIFV code for a given probability distribution $\{p_t\}$, we need to construct an IP problem that minimizes $L_{AIFV}=Q(T_0)L_0+Q(T_1)L_1$.
However, in such IP problems, we need a lot of variables because we must treat two code trees at once. Furthermore, since $Q(T_0)L_0$ and $Q(T_1)L_1$ include nonlinear terms, many additional variables and conditions are required to linearize nonlinear terms.
Hence, although we can formalize an IP problem to obtain the global optimal solution, it becomes impractical or can treat only a small size of ${\cal X}$.
Therefore, in this subsection, we derive individual IP problems for $T_0$ and $T_1$ that can
attain near-optimal $L_{AIFV}$, and we show in Section \ref{GP} that
the global optimal AIFV code can be obtained by solving the individual IP problems finite times.
Since we can assign source symbols to master nodes in addition to leaves in the case of binary AIFV code, we introduce binary variables $v_{t,d}$, in addition to $u_{t,d}$, such that
$v_{t,d}=1$ if source symbol $a_t$ is assigned to a master node of depth $d$,
and $v_{t,d}=0$ otherwise.
Then, an IP problem to construct $T_0$ can be formalized
as follows.
\begin{ip}\label{ip-2}
\begin{align}
\mbox{minimize} \quad&\sum_{t=1}^{|{\cal X}|}\
\sum_{d = 0}^{D} p_t \left(u_{t,d}\,d+v_{t,d}\left(d+C_2\right)\right) \label{eq3-1-1} \\
\mbox{subject to} \quad&\sum_{t=1}^{|{\cal X}|}\
\sum_{d = 0}^{D} 2^{-d} \left(u_{t,d}+\frac{3}{4} v_{t,d}\right)=1,\label{eq3-2-1} \\
&\sum_{d=0}^{D} (u_{t,d}+v_{t,d})= 1, \hspace{6.8cm} t= 1,2,\dots ,|{\cal X}|, \label{eq3-3-1} \\
&\sum_{t=1}^{|{\cal X}|} \left( v_{t,d}+\frac{1}{2}v_{t,d+1}\right)
-\sum_{\ell =d+2}^{D} \sum_{t=1}^{|{\cal X}|} 2^{d+2-\ell} \left( u_{t,\ell}+\frac{3}{4}v_{t,\ell} \right) \leq 0,
\qquad d = 0,1,\cdots ,D-2,\quad \label{eq3-4-1}
\end{align}
where $C_2=2-\log_23 \approx 0.405$.
\end{ip}
Furthermore, an IP problem to derive $T_1$ is obtained by setting $u_{t,0}=v_{t,0}=0$
for all $t$ (or removing the case of $d=0$ in \eqref{eq3-1-1}--\eqref{eq3-4-1})
and replacing \eqref{eq3-2-1}
with the following condition:
\begin{align}
\sum_{i=1}^{|{\cal X}|}\
\sum_{d = 1}^{D} 2^{-d} \left(u_{i,d}+\frac{3}{4} v_{i,d}\right)=\frac{3}{4}. \label{eq3-2-2}
\end{align}
Condition \eqref{eq3-2-1} comes from \eqref{eq-20}, and condition \eqref{eq3-3-1} guarantees that each $a_t$ is assigned to only one of either leaves or master nodes.
The code trees are obtained by assigning $a_t$ to a leaf (resp.~a master node) of depth $d$ if the solution has $u_{t,d}=1$ (resp.~$v_{t,d}=1$).
Note that $C_2$ in \eqref{eq3-1-1} and Eq.~\eqref{eq3-4-1} are newly introduced in IP problem \ref{ip-2} compared with IP problem \ref{ip-1}. We first consider why $C_2$ is required.
A leaf of depth $d$ has weight $2^{-d}$ in \eqref{eq3-2-1}
while a master node of depth $d$ has weight $(3/4)2^{-d}$.
Hence, average code lengths $L_0$ and $L_1$ can be decreased
by making many master nodes in $T_0$ and $T_1$, respectively.
On the other hand, this increases $P({\cal N}_1^{(0)})$ and $P({\cal N}_1^{(1)})$,
and hence $Q(T_1)$ because of
$Q(T_1|T_0) = P({\cal N}_1^{(0)})$ and $Q(T_1|T_1) = P({\cal N}_1^{(1)})$.
Note that the global average code length is given by $L_{AIFV}=Q(T_0)L_0+Q(T_1)L_1$, and
$L_1$ is much larger than $L_0$ because the root of $T_1$ cannot have a grandchild
with code symbols `00'.
Therefore, $L_{AIFV}$ is not always minimized even if $L_0$ and $L_1$ are minimized individually.
Note that if a master node is used to encode a source symbol, we must use $T_1$, instead of $T_0$, to encode the next source symbol. This means that master nodes have the cost $L'_{AIFV}-L_{AIFV}$ compared with leaves, where $L'_{AIFV}$ is the average code length of the case that we start
the encoding with $T_1$ instead of $T_0$.
Since we derive the code trees $T_0$ and $T_1$ by solving separate IP problems,
it is hard to embed the exact cost into each IP problem.
But, the optimal code trees have a good property such that every child of a node has approximately half probability weight of its parent node. So, as an approximation of exact cost,
we can use the cost of the ideal case such that every node has two children with equal probability weight.
In this case, the cost is given by $C_2=2-\log_23$ because
the root of $T_0$ can have four grandchildren while the root of $T_1$ can have only three grandchildren.
Therefore, cost $C_2$ is added for master nodes in \eqref{eq3-1-1}.
Next we consider \eqref{eq3-4-1}. This comes from Condition \ref{cond-2} shown in Section \ref{Kraft-bAIFV}. Each master node of depth $d$ requires a slave node of depth $d+1$ and a node or leaf of depth $d+2$.
Therefore, we cannot make master nodes of depth $d$ if there are not sufficient number
of nodes or leaves at depth $d+2$.
Let $N^{m}_d$ and $N^{nl}_{d+2}$ be the number of master node of depth $d$
and the number of nodes and leaves of depth $d+2$, respectively.
Then, $N^{m}_d$ is given by
\begin{align}
N^{m}_d=\sum_{t=1}^{|{\cal X}|} v_{t,d}. \label{eq3-4-2}
\end{align}
On the other hand,
we can know the number of nodes and leaves of depth $d+2$
by calculating the Kraft's weight at depth $d+2$. Hence,
$N^{nl}_{d+2}$ is given by
\begin{align}
N^{nl}_{d+2}=\sum_{\ell =d+2}^{D} \sum_{t=1}^{|{\cal X}|} 2^{d+2-\ell} \left( u_{t,\ell}+\frac{3}{4}v_{t,\ell} \right).
\label{eq3-4-3}
\end{align}
Furthermore, there are $N^{m}_{d+1}$ master nodes of depth $d+1$, each of which
requires one node or leaf of depth $d+3$. Since a node or leaf of depth $d+3$ has weight $2^{-1}$ at depth $d+2$, we must use $2^{-1}N^{m}_{d+1}$ out of $N^{nl}_{d+2}$
for master nodes of depth $d+1$. This means that the remaining $N^{nl}_{d+2}-2^{-1}N^{m}_{d+1}$ nodes and leaves of depth $d+2$ can be used for $N^{m}_d$ master nodes of depth $d$.
Hence, the condition \eqref{eq3-4-1} is required.
\subsection{IP problem for ternary AIFV code trees}\label{IP-tAIFV}
In order to obtain near-optimal ternary AIFV code,
we can formalize an IP problem for ternary AIFV code trees in the same way as binary AIFV code trees.
\begin{ip}\label{ip-3}
\begin{align}
\hspace{-0.2cm}\mbox{minimize} \hspace{0.1cm}
&\sum_{i=1}^{|{\cal X}|}\ \sum_{d = 0}^{D} p_i\left(u_{i,d}d+v_{i,d}\left(d+C_3\right)\right) \label{eq:3_objective} \\
\hspace{-0.2cm}\mbox{subject to} \hspace{0.1cm}
&\sum_{d = 0}^{D} 3^{-d}
\left( z_d +\sum_{t=1}^{|{\cal X}|} \left(u_{t,d}+\frac{2}{3}v_{t,d} \right) \right)
=1 \label{eq:3_T0}\\
&\sum_{d=0}^{\ell_{\max}} \left(u_{t,d}+v_{t,d}\right) = 1, \hspace{5.8cm} t = 1,2,\dots ,|{\cal X}|, \label{eq:3_00} \\
&\sum_{t=1}^{|{\cal X}|} v_{t,d} -\sum_{\ell =d+1}^{D}3^{d+1-\ell} \biggl( z_\ell
+ \sum_{t=1}^{|{\cal X}|}\left(u_{t,\ell}+\frac{2}{3}v_{t,\ell}\biggr)
\right) \leq 0,
\qquad d= 0,1,\cdots ,D-1. \label{eq:3_continuous} \end{align}
where $C_3=1-\log_32 \approx 0.369$.
\end{ip}
Furthermore, an IP problem to derive $T_1$ is obtained by
setting $u_{t,0}=v_{t,0}=0$ for all $t$ (or removing the case of $d=0$ in \eqref{eq:3_objective}--\eqref{eq:3_continuous})
and replacing \eqref{eq:3_T0} with
the following condition:
\begin{align}
\sum_{d = 1}^{D} 3^{-d}
\left( z_d +\sum_{t=1}^{|{\cal X}|} \left(u_{t,d}+\frac{2}{3}v_{t,d} \right) \right)
=\frac{2}{3}. \label{eq:3_T0-1}
\end{align}
The cost $C_3$ for incomplete internal nodes is given by $L'_{AIFV}-L_{AIFV}$ in
the ideal case such that every child of each node has equal probability weight.
Since the roots of $T_0$ and $T_1$ can have three and two children, respectively,
in the ternary case, we have $C_3= \log_33-\log_32$.
Condition \eqref{eq:3_continuous} is required from Condition \ref{cond-1} shown in
Section \ref{Kraft-KAIFV}, and it can be
derived in the same way as \eqref{eq3-4-1}.
But, since slave nodes do not exist in the ternary case, we do not need
$\frac{1}{2}v_{t, d+1}$ in the first term of \eqref{eq3-4-1}.
A new binary variable $z_d$ is introduced in IP problem \ref{ip-3} compared with IP problem \ref{ip-2}.
Note that the ternary Huffman code has one incomplete node in the code tree when $|{\cal X}|$ is even. Similarly a ternary AIFV code may have one incomplete node in $T_0$ and/or $T_1$,
which is not assigned any source symbol. Variable $z_d$ represents the pruned leaf of such an incomplete node. $z_d=1$ if there is the pruned leaf at level $d$, and $z_d=0$ otherwise.
We can represent the condition \eqref{eq:3_T0} without using $z_d$ as follows.
\begin{align}
\sum_{d = 0}^{D}
\sum_{t=1}^{|{\cal X}|} 3^{-d}\left(u_{t,d}+\frac{2}{3}v_{t,d} \right)
\leq 1. \label{eq:3_T0-2}
\end{align}
But, since the condition \eqref{eq:3_continuous} cannot be represented without $z_d$,
\eqref{eq:3_T0} is used rather than \eqref{eq:3_T0-2}.
Since the pruned leaf must have the longest depth if it exists,
we have $z_{\hat{d}}=1$ for $\hat{d}\equiv \max\{d : u_{t,d}=1, t=1,2, \cdots |{\cal X}|\}$ and $z_{\hat{d}}=0$ for $d\neq \hat{d}$ in the optimal $T_0$ and $T_1$.
But these conditions are not explicitly included in IP problem \ref{ip-3} because the optimal code trees can be obtained without these conditions.
\begin{remark}
IP problem \ref{ip-3} can be applied to the $K$-ary AIFV codes with two code trees $T_0$ and $T_{K-j}$
explained in Remark \ref{remark-1} by modifying 2, 3, $C_3$ and $z_d$ in
\eqref{eq:3_objective}-\eqref{eq:3_continuous} as follows:
\begin{align*}
& 3\rightarrow K, \qquad 2 \rightarrow K-j, \qquad C_3 \rightarrow C_{K,j}=1-\log_K (K-j),\\
& z_d \in\{0,1\} \rightarrow z_d \in\{0,1, \cdots, K-2\}.
\end{align*}
We can also construct IP problems for general $K$-ary AIFV code trees
by using binary variables $v_{t,d}^{(j)}$ to represent incomplete internal nodes with $j$ children for $1\leq j \leq K-2$ instead of $v_{t,d}$ used in IP problem \ref{ip-3}.
But, the necessary number of variables increases and each condition described in `subjet to'
becomes long as $K$ becomes large. Therefore, it is hard to treat large $K$ practically because of
time and/or space complexity.
\end{remark}
\subsection{Global Optimaization} \label{GP}
In IP problems \ref{ip-2} and \ref{ip-3}, costs $C_2$ and $C_3$ are determined
based on the ideal code trees such that every child of each node has equal probability weight. But, since the code trees $T_0$ and $T_1$ obtained by IP Problem \ref{ip-2} (or \ref{ip-3}) do not attain
the perfect balance of probability weight, they are not the optimal AIFV code trees generally.
So, we calculate new cost $C$ based on the obtained code trees $T_0$ and $T_1$, and we derive new code trees for the new cost by solving again IP Problem \ref{ip-2} (or \ref{ip-3}). In this section, we show that the global optimal code trees can be obtained by repeating this procedure.
Let $C^{(m-1)}$ is the $(m-1)$-th cost and let $T_0^{(m)}$ and $T_1^{(m)}$ be the $m$-th AIFV code trees obtained by solving the IP problem for cost $C^{(m-1)}$. $C^{(0)}$ is the initial cost.
Furthermore,
let $L_0^{(m)}$ and $L_1^{(m)}$ be the average code length of $T_0^{(m)}$ and $T_1^{(m)}$,
respectively, and let $q_0^{(m)}$ and $q_1^{(m)}$ be the transition probabilities of code trees $T_0^{(m)}$ and $T_1^{(m)}$, which are defined by $q_0^{(m)}\equiv Q(T_1^{(m)}|T_0^{(m)})=P(\mbox{${\cal N}_1^{(0)}$ in $T_0^{(m)}$})$
and $q_1^{(m)}\equiv Q(T_0|T_1)=P(\mbox{${\cal N}_0^{(1)}$ in $T_1^{(m)}$})$.
Then, we consider the following algorithm.
\begin{algorithm} \label{alg-1}
\begin{description}
\item{1.} Set $m=1$ and $C^{(0)}=C$ for given initial cost $C$.
\item{2.} Obtain $T_0^{(m)}$ and $T_1^{(m)}$ by solving IP problem \ref{ip-2} (or \ref{ip-3})
for cost $C^{(m-1)}$.
\item{3.} Calculate $(L_0^{(m)}, q_0^{(m)})$ for $T_0^{(m)}$ and $(L_1^{(m)}, q_1^{(m)})$ for $T_1^{(m)}$.
\item{4.} Update cost as follows.
\begin{align}
C^{(m)}=\frac{L_1^{(m)} -L_0^{(m)}}{q_0^{(m)}+q_1^{(m)}} \label{eq4-5}
\end{align}
\item{5.} If $C^{(m)}=C^{(m-1)}$, then exit. Otherwise, increment $m$ and go to step 2.
\end{description}
\end{algorithm}
\vspace{0.2cm}
We can use any $C$ for the initial cost. But, if we use $C_2=2-\log_23$ and $C_3=1-\log_32$
as the initial cost in the binary and ternary cases, respectively, $T_0^{(1)}$ and $T_1^{(1)}$ become near-optimal code trees.
\vspace{0.2cm}
\begin{theorem}
The binary AIFV code and the ternary AIFV code obtained by Algorithm \ref{alg-1} are optimal.
\end{theorem}
\vspace{0.2cm}
{\it Proof} \quad
We first prove that Algorithm \ref{alg-1} stops after finite iterations.
First note that for $T_0^{(m)}$, the objective function \eqref{eq3-1-1} in IP problem \ref{ip-2}
(or \eqref{eq:3_objective} in IP problem \ref{ip-3} ) can be represented as
\begin{align}
L_0^{(m)}+C^{(m-1)} q_0^{(m)}. \label{eq4-1}
\end{align}
Similarly, the object function for $T_1^{(m)}$ can be represented as
\begin{align}
L_1^{(m)}+C^{(m-1)} (1-q_1^{(m)}). \label{eq4-2}
\end{align}
Since $C^{(m-1)}$ is fixed in the IP problem used in step 2 of Algorithm \ref{alg-1},
the minimization of \eqref{eq4-2} is equivalent to the minimization of
\begin{align}
L_1^{(m)} - C^{(m-1)} q_1^{(m)}. \label{eq4-3}
\end{align}
On the other hand, the global average code length $L_{AIFV}^{(m)}$ for
$T_0^{(m)}$ and $T_1^{(m)}$ is given by
\begin{align}
L_{AIFV}^{(m)}=\frac{q_1^{(m)} L_0^{(m)}+ q_0^{(m)}L_1^{(m)}}{q_0^{(m)}+q_1^{(m)}}.
\end{align}
Since $T_0^{(m)}$ and $T_1^{(m)}$ are optimal trees that minimize \eqref{eq4-1}
and \eqref{eq4-3} for $C^{(m-1)}$,
the following inequalities hold for any code trees $T_0$ with $(L_0, q_0)$ and $T_1$
with $(L_1, q_1)$.
\begin{align}
L_0^{(m)}+C^{(m-1)} q_0^{(m)}\leq L_0 +C^{(m-1)}q_0, \label{eq4-4}\\
L_1^{(m)}-C^{(m-1)} q_1^{(m)}\leq L_1 -C^{(m-1)}q_1. \label{eq4-4-1}
\end{align}
Hence if we substitute $T_0=T_0^{(m-1)}$ and $T_1=T_1^{(m-1)}$ into
\eqref{eq4-4} and \eqref{eq4-4-1}, respectively, we have the following inequalities.
\begin{align}
L_0^{(m)}+C^{(m-1)} q_0^{(m)}
&\leq L_0^{(m-1)}+C^{(m-1)}q_0^{(m-1)} \nonumber\\
& =L_{AIFV}^{(m-1)} \label{eq4-6}\\
L_1^{(m)}-C^{(m-1)} q_1^{(m)}
&\leq L_1^{(m-1)} -C^{(m-1)}q_1^{(m-1)} \nonumber\\
& = L_{AIFV}^{(m-1)} \label{eq4-7}
\end{align}
If $C^{(m)}< C^{(m-1)}$, we obtain from \eqref{eq4-6} that
\begin{align}
L_{AIFV}^{(m)}&=L_0^{(m)}+C^{(m)} q_0^{(m)} \nonumber\\
&< L_0^{(m)}+C^{(m-1)} q_0^{(m)}\nonumber\\
&\leq L_{AIFV}^{(m-1)} \label{eq4-8}.
\end{align}
Similarly, if $C^{(m)}> C^{(m-1)}$, we have from \eqref{eq4-7} that
\begin{align}
L_{AIFV}^{(m)}&=L_1^{(m)}-C^{(m)} q_1^{(m)} \nonumber\\
&< L_1^{(m)}-C^{(m-1)} q_1^{(m)}\nonumber\\
&\leq L_{AIFV}^{(m-1)} \label{eq4-9}.
\end{align}
Therefore, if $C^{(m)}\neq C^{(m-1)}$, we have that $L_{AIFV}^{(m)}< L_{AIFV}^{(m-1)}$.
Since $L_{AIFV}^{(m)}>0$ for any $m$, we can conclude that $L_{AIFV}^{(m)}$ converges as $m\rightarrow \infty$. Furthermore, since the number of code trees is finite, the convergence is
achieved with finite $m$, i.e. $C^{(m)}= C^{(m-1)}$ occurs and Algorithm 1 stops after finite iterations.
Next we prove that the obtained AIFV code trees are optimal when Algorithm 1 stops.
Assume that Algorithm 1 stops at $m=\hat{m}$, and $T_0^{(\hat{m})}$ and $T_1^{(\hat{m})}$ are the obtained AIFV code trees that satisfy $C^{(\hat{m})}=C^{(\hat{m}-1)}$.
If this pair $(T_0^{(\hat{m})}, T_1^{(\hat{m})})$ is not globally optimal, there exists the optimal pair of code trees $(T_0^*, T_1^*)$ with $(L_0^*, L_1^*, q_0^*, q_1^*)$ such that
\begin{align}
L_{AIFV}^{(\hat{m})}> L^*_{AIFV}. \label{eq4-10}
\end{align}
Then, we have for $C^*\equiv (L_1^*-L_0^*)/(q_0^*+q_1^*)$ that
\begin{align}
L^*_{AIFV}= L_0^* +C^* q_0^* = L_1^* - C^* q_1^*. \label{eq4-11}
\end{align}
Hence, if $C^*\geq C^{(\hat{m})}$, we have
\begin{align}
L_{AIFV}^{(\hat{m})}&=L_0^{(\hat{m})}+ C^{(\hat{m})} q_0^{(\hat{m})} \nonumber\\
&= L_0^{(\hat{m})}+ C^{(\hat{m}-1)} q_0^{(\hat{m})}\nonumber\\
& \leq L_0^*+ C^{(\hat{m}-1)}q_0^* \nonumber\\
& \leq L_0^*+ C^*q_0^*\nonumber\\
& =L^*_{AIFV}, \label{eq4-13}
\end{align}
where the first inequality and the last equality hold from \eqref{eq4-4} and \eqref{eq4-11},
respectively.
Similarly if $C^*\leq C^{(\hat{m})}$, we have
\begin{align}
L_{AIFV}^{(\hat{m})}&=L_1^{(\hat{m})}- C^{(\hat{m})} q_1^{(\hat{m})} \nonumber\\
&= L_1^{(\hat{m})}- C^{(\hat{m}-1)} q_1^{(\hat{m})}\nonumber\\
& \leq L_1^* - C^{(\hat{m}-1)}q_1^* \nonumber\\
& \leq L_1^*- C^*q_1^*\nonumber\\
& =L^*_{AIFV}. \label{eq4-14}
\end{align}
Since \eqref{eq4-13} and \eqref{eq4-14} contradict with \eqref{eq4-10},
the pair of obtained code trees $(T_0^{(\hat{m})}, T_1^{(\hat{m})})$ must be globally optimal.
\begin{flushright}Q.E.D.\end{flushright}
\section{Performance of binary and ternary AIFV codes}\label{sec-5}
In this section, we compare numerically the performance of AIFV codes with Huffman codes.
For ${\cal X}=\{a_1, a_2 \cdots, a_{|{\cal X}|}\}$,
we consider the following three kinds of source distributions:
\begin{align}
P_X^{(0)}(a_t)&=\frac{1}{|{\cal X}|}, \label{eq5-1}\\
P_X^{(1)}(a_t) &=\frac{t}{A_1}, \label{eq5-2} \\
P_X^{(2)}(a_t)&=\frac{t^2}{A_2}, \label{eq5-3}
\end{align}
where $A_1=\sum_{t=1}^{|{\cal X}|} t$ and $A_2=\sum_{t=1}^{|{\cal X}|} t^2$ are normalizing constants.
The performance of AIFV codes is compared with
Huffman codes and Huffman codes for ${\cal X}^2$
in Figs.~\ref{fig10}--\ref{fig11} (resp.~Figs.~\ref{fig12}--\ref{fig14})
for the binary (resp.~ternary) case\footnote{Figures 3--6 and 8 in \cite{Y-W-2013} are not correct
although the algorithms shown in \cite{Y-W-2013} are correct.}.
The comparison for $P_X^{(0)}$ is omitted in the binary case because the compression rate of AIFV codes is equal to the one of Huffman codes. The AIFV codes are derived by Algorithm
\ref{alg-1}.
In the figures, the vertical line represents the normalized compression rate defined by
$L_{AIFV}/H_2(X)$ and $L_H/H_2(X)$ (resp.~$L_{AIFV}/H_3(X)$ and $L_H/H_3(X)$) for the binary
(resp.~ternary) case. The horizontal line stands for the size of source alphabet.
We note from Figs.~\ref{fig10}--\ref{fig14}
that the AIFV codes can attain better compression rate than the Huffman codes in all cases.
Furthermore, in the cases of $P_X^{(1)}$ and $P_X^{(2)}$,
the binary AIFV codes can beat even the Huffman codes for ${\cal X}^2$ and the ternary AIFV codes can attain almost the same compression rate as the Huffman codes for ${\cal X}^2$.
\begin{figure}[t]
\begin{center}
\includegraphics[clip, scale = 0.35]{b-p1.eps}
\caption{Comparison between binary AIFV coding and Huffman coding for $P_X^{(1)}$.}
\label{fig10}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[clip, scale = 0.35]{b-p2.eps}
\caption{Comparison between binary AIFV coding and Huffman coding for $P_X^{(2)}$.}
\label{fig11}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[clip, scale = 0.35]{t-p0.eps}
\caption{Comparison between ternary AIFV coding and Huffman coding for $P_X^{(0)}$}
\label{fig12}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[clip, scale = 0.35]{t-p1.eps}
\caption{Comparison between ternary AIFV coding and Huffman coding for $P_X^{(1)}$.}
\label{fig13}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[clip, scale = 0.35]{t-p2.eps}
\caption{Comparison between ternary AIFV coding and Huffman coding for $P_X^{(2)}$.}
\label{fig14}
\end{center}
\end{figure}
The Huffman coding for ${\cal X}^2$ has demerits such that
the size of Huffman code tree increases to roughly $|{\cal X}|^2$, and the encoding and decoding delay of the first source symbol of $(x_1, x_2) \in {\cal X}^2$ becomes large as $|{\cal X}|$ becomes large.
On the other hand, in AIFV coding, the size of code trees
is roughly $2|{\cal X}|$ for these binary and ternary cases\footnote{In the $K$-ary case for $K\geq 3$, the size of AIFV code trees is roughly $(K-1)|{\cal X}|$.}, and encoding delay is zero and decoding delay is at most two bits
(resp.~one code symbol) in binary (resp.~$K$-ary for $K\geq 3$) case.
Hence, from the viewpoints of coding delay and memory size, AIFV coding is superior to Huffman coding for ${\cal X}^2$ when $|{\cal X}|$ is large.
Finally we remark that
if we use $C=C_2=2-\log_23$ (resp.~$C=C_3=1-\log_32$) as the initial cost
in Algorithm \ref{alg-1} for the binary (resp.~ternary) case,
$L_{AIFV}^{(1)}$ is often optimal without iteration. Furthermore, even if $L_{AIFV}^{(1)}$ is not optimal, the improvement by the iteration of Algorithm \ref{alg-1} is within only 0.1\% compared with $L_{AIFV}^{(1)}$ in all the cases of $P_X^{(0)}$, $P_X^{(1)}$, and $P_X^{(2)}$.
This means that if we use $C=C_2$ (resp.~$C=C_3$) in IP problem \ref{ip-2} (resp.~IP problem \ref{ip-3}),
we can obtain the optimal or near-optimal AIFV codes by solving the IP problem for $T_0$ and $T_1$ only once without using Algorithm \ref{alg-1}.
\section{Conclusion}\label{sec-6}
In this paper, we proposed binary and $K$-ary (for $K\geq 3$) AIFV coding for stationary memoryless sources,
and we showed that the optimal AIFV codes can be obtained by solving integer programing problems for the binary and ternary cases.
Furthermore, by calculating the compression rate numerically for several source distributions,
we clarified that the AIFV coding can beat Huffman coding.
The following are open problems: obtain a tight upper bound of $L_{AIFV}$ given in \eqref{eq-16},
obtain a simple algorithm to derive the optimal binary AIFV codes and/or the optimal $K$-ary AIFV codes.
The AIFV codes proposed in this paper are devised such that decoding delay is at most one code symbol (resp.~two bits) in $K$-ary (resp.~binary) case. But, if decoding delay is allowed more than one code symbol (resp.~two bits), it may be possible to construct non-instantaneous FV codes that can attain better compression rate than the AIFV codes. It is also an interesting open problem to obtain the best non-instantaneous FV codes for a given maximum decoding delay.
\ifCLASSOPTIONcompsoc
\else
\fi
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,601 |
package org.drools.workbench.screens.guided.dtable.client.widget.table;
import com.google.gwt.resources.client.ImageResource;
import org.drools.workbench.models.guided.dtable.shared.model.ActionCol52;
import org.drools.workbench.models.guided.dtable.shared.model.BaseColumn;
import org.drools.workbench.models.guided.dtable.shared.model.ConditionCol52;
import org.drools.workbench.screens.guided.dtable.client.resources.GuidedDecisionTableResources;
import org.kie.workbench.common.widgets.decoratedgrid.client.widget.ResourcesProvider;
import org.drools.workbench.screens.guided.dtable.client.resources.GuidedDecisionTableResources;
/**
* A class to provide different resources for Decision Tables
*/
public class DecisionTableResourcesProvider
implements
ResourcesProvider<BaseColumn> {
public int rowHeight() {
return GuidedDecisionTableResources.INSTANCE.css().rowHeight();
}
public int rowHeaderHeight() {
return GuidedDecisionTableResources.INSTANCE.css().rowHeaderHeight();
}
public int rowHeaderSplitterHeight() {
return GuidedDecisionTableResources.INSTANCE.css().rowHeaderSplitterHeight();
}
public int rowHeaderSorterHeight() {
return GuidedDecisionTableResources.INSTANCE.css().rowHeaderSorterHeight();
}
public int sidebarWidth() {
return GuidedDecisionTableResources.INSTANCE.css().sidebarWidth();
}
public int borderWidth() {
return GuidedDecisionTableResources.INSTANCE.css().borderWidth();
}
public int borderWidthThick() {
return GuidedDecisionTableResources.INSTANCE.css().borderWidthThick();
}
public String cellTableColumn( BaseColumn column ) {
if ( column instanceof ConditionCol52 ) {
return GuidedDecisionTableResources.INSTANCE.css().conditionColumn();
} else if ( column instanceof ActionCol52 ) {
return GuidedDecisionTableResources.INSTANCE.css().actionColumn();
}
return GuidedDecisionTableResources.INSTANCE.css().metaColumn();
}
public String cellTable() {
return GuidedDecisionTableResources.INSTANCE.css().cellTable();
}
public String cellTableEvenRow() {
return GuidedDecisionTableResources.INSTANCE.css().cellTableEvenRow();
}
public String cellTableOddRow() {
return GuidedDecisionTableResources.INSTANCE.css().cellTableOddRow();
}
public String cellTableCell() {
return GuidedDecisionTableResources.INSTANCE.css().cellTableCell();
}
public String cellTableCellSelected() {
return GuidedDecisionTableResources.INSTANCE.css().cellTableCellSelected();
}
public String cellTableCellMultipleValues() {
return GuidedDecisionTableResources.INSTANCE.css().cellTableCellMultipleValues();
}
public String cellTableCellOtherwise() {
return GuidedDecisionTableResources.INSTANCE.css().cellTableCellOtherwise();
}
public String cellTableCellDiv() {
return GuidedDecisionTableResources.INSTANCE.css().cellTableCellDiv();
}
public String cellTableGroupDiv() {
return GuidedDecisionTableResources.INSTANCE.css().cellTableGroupDiv();
}
public String cellTableTextDiv() {
return GuidedDecisionTableResources.INSTANCE.css().cellTableTextDiv();
}
public String headerRowBottom() {
return GuidedDecisionTableResources.INSTANCE.css().headerRowBottom();
}
public String headerRowIntermediate() {
return GuidedDecisionTableResources.INSTANCE.css().headerRowIntermediate();
}
public String headerText() {
return GuidedDecisionTableResources.INSTANCE.css().headerText();
}
public String headerSplitter() {
return GuidedDecisionTableResources.INSTANCE.css().headerSplitter();
}
public String headerResizer() {
return GuidedDecisionTableResources.INSTANCE.css().headerResizer();
}
public String selectorSpacer() {
return GuidedDecisionTableResources.INSTANCE.css().selectorSpacer();
}
public String selectorSpacerOuterDiv() {
return GuidedDecisionTableResources.INSTANCE.css().selectorSpacerOuterDiv();
}
public String selectorSpacerInnerDiv() {
return GuidedDecisionTableResources.INSTANCE.css().selectorSpacerInnerDiv();
}
public String selectorCell() {
return GuidedDecisionTableResources.INSTANCE.css().selectorCell();
}
public ImageResource arrowSpacerIcon() {
return GuidedDecisionTableResources.INSTANCE.images().arrowSpacerIcon();
}
public ImageResource downArrowIcon() {
return GuidedDecisionTableResources.INSTANCE.tableImageResources().downArrow();
}
public ImageResource smallDownArrowIcon() {
return GuidedDecisionTableResources.INSTANCE.tableImageResources().smallDownArrow();
}
public ImageResource upArrowIcon() {
return GuidedDecisionTableResources.INSTANCE.tableImageResources().upArrow();
}
public ImageResource smallUpArrowIcon() {
return GuidedDecisionTableResources.INSTANCE.tableImageResources().smallUpArrow();
}
public ImageResource toggleUnmergeIcon() {
return GuidedDecisionTableResources.INSTANCE.images().toggleUnmergeIcon();
}
public ImageResource toggleMergeIcon() {
return GuidedDecisionTableResources.INSTANCE.images().toggleMergeIcon();
}
public ImageResource selectorAddIcon() {
return GuidedDecisionTableResources.INSTANCE.itemImages().newItem();
}
public ImageResource selectorDeleteIcon() {
return GuidedDecisionTableResources.INSTANCE.itemImages().deleteItemSmall();
}
public ImageResource collapseCellsIcon() {
return GuidedDecisionTableResources.INSTANCE.collapseExpand().collapse();
}
public ImageResource expandCellsIcon() {
return GuidedDecisionTableResources.INSTANCE.collapseExpand().expand();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,452 |
\section{Introduction}
\label{sec:intro}
Please follow the steps outlined below when submitting your manuscript to the IEEE Computer Society Press.
This style guide now has several important modifications (for example, you are no longer warned against the use of sticky tape to attach your artwork to the paper), so all authors should read this new version.
\subsection{Language}
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\subsection{Mathematics}
Please number all of your sections and displayed equations as in these examples:
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Just because you did not refer to it in the text does not mean some future reader might not need to refer to it.
It is cumbersome to have to use circumlocutions like ``the equation second from the top of page 3 column 1''.
(Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers).
All authors will benefit from reading Mermin's description of how to write mathematics:
\url{http://www.pamitc.org/documents/mermin.pdf}.
\subsection{Blind review}
Many authors misunderstand the concept of anonymizing for blind review.
Blind review does not mean that one must remove citations to one's own work---in fact it is often impossible to review a paper unless the previous citations are known and available.
Blind review means that you do not use the words ``my'' or ``our'' when citing previous work.
That is all.
(But see below for tech reports.)
Saying ``this builds on the work of Lucy Smith [1]'' does not say that you are Lucy Smith;
it says that you are building on her work.
If you are Smith and Jones, do not say ``as we show in [7]'', say ``as Smith and Jones show in [7]'' and at the end of the paper, include reference 7 as you would any other cited work.
An example of a bad paper just asking to be rejected:
\begin{quote}
\begin{center}
An analysis of the frobnicatable foo filter.
\end{center}
In this paper we present a performance analysis of our previous paper [1], and show it to be inferior to all previously known methods.
Why the previous paper was accepted without this analysis is beyond me.
[1] Removed for blind review
\end{quote}
An example of an acceptable paper:
\begin{quote}
\begin{center}
An analysis of the frobnicatable foo filter.
\end{center}
In this paper we present a performance analysis of the paper of Smith \etal [1], and show it to be inferior to all previously known methods.
Why the previous paper was accepted without this analysis is beyond me.
[1] Smith, L and Jones, C. ``The frobnicatable foo filter, a fundamental contribution to human knowledge''. Nature 381(12), 1-213.
\end{quote}
If you are making a submission to another conference at the same time, which covers similar or overlapping material, you may need to refer to that submission in order to explain the differences, just as you would if you had previously published related work.
In such cases, include the anonymized parallel submission~\cite{Authors14} as supplemental material and cite it as
\begin{quote}
[1] Authors. ``The frobnicatable foo filter'', F\&G 2014 Submission ID 324, Supplied as supplemental material {\tt fg324.pdf}.
\end{quote}
Finally, you may feel you need to tell the reader that more details can be found elsewhere, and refer them to a technical report.
For conference submissions, the paper must stand on its own, and not {\em require} the reviewer to go to a tech report for further details.
Thus, you may say in the body of the paper ``further details may be found in~\cite{Authors14b}''.
Then submit the tech report as supplemental material.
Again, you may not assume the reviewers will read this material.
Sometimes your paper is about a problem which you tested using a tool that is widely known to be restricted to a single institution.
For example, let's say it's 1969, you have solved a key problem on the Apollo lander, and you believe that the CVPR70 audience would like to hear about your
solution.
The work is a development of your celebrated 1968 paper entitled ``Zero-g frobnication: How being the only people in the world with access to the Apollo lander source code makes us a wow at parties'', by Zeus \etal.
You can handle this paper like any other.
Do not write ``We show how to improve our previous work [Anonymous, 1968].
This time we tested the algorithm on a lunar lander [name of lander removed for blind review]''.
That would be silly, and would immediately identify the authors.
Instead write the following:
\begin{quotation}
\noindent
We describe a system for zero-g frobnication.
This system is new because it handles the following cases:
A, B. Previous systems [Zeus et al. 1968] did not handle case B properly.
Ours handles it by including a foo term in the bar integral.
...
The proposed system was integrated with the Apollo lunar lander, and went all the way to the moon, don't you know.
It displayed the following behaviours, which show how well we solved cases A and B: ...
\end{quotation}
As you can see, the above text follows standard scientific convention, reads better than the first version, and does not explicitly name you as the authors.
A reviewer might think it likely that the new paper was written by Zeus \etal, but cannot make any decision based on that guess.
He or she would have to be sure that no other authors could have been contracted to solve problem B.
\medskip
\noindent
FAQ\medskip\\
{\bf Q:} Are acknowledgements OK?\\
{\bf A:} No. Leave them for the final copy.\medskip\\
{\bf Q:} How do I cite my results reported in open challenges?
{\bf A:} To conform with the double-blind review policy, you can report results of other challenge participants together with your results in your paper.
For your results, however, you should not identify yourself and should not mention your participation in the challenge.
Instead present your results referring to the method proposed in your paper and draw conclusions based on the experimental comparison to other results.\medskip\\
\begin{figure}[t]
\centering
\fbox{\rule{0pt}{2in} \rule{0.9\linewidth}{0pt}}
\caption{Example of caption.
It is set in Roman so that mathematics (always set in Roman: $B \sin A = A \sin B$) may be included without an ugly clash.}
\label{fig:onecol}
\end{figure}
\subsection{Miscellaneous}
\noindent
Compare the following:\\
\begin{tabular}{ll}
\verb'$conf_a$' & $conf_a$ \\
\verb'$\mathit{conf}_a$' & $\mathit{conf}_a$
\end{tabular}\\
See The \TeX book, p165.
The space after \eg, meaning ``for example'', should not be a sentence-ending space.
So \eg is correct, {\em e.g.} is not.
The provided \verb'\eg' macro takes care of this.
When citing a multi-author paper, you may save space by using ``et alia'', shortened to ``\etal'' (not ``{\em et.\ al.}'' as ``{\em et}'' is a complete word).
If you use the \verb'\etal' macro provided, then you need not worry about double periods when used at the end of a sentence as in Alpher \etal.
However, use it only when there are three or more authors.
Thus, the following is correct:
``Frobnication has been trendy lately.
It was introduced by Alpher~\cite{Alpher02}, and subsequently developed by
Alpher and Fotheringham-Smythe~\cite{Alpher03}, and Alpher \etal~\cite{Alpher04}.''
This is incorrect: ``... subsequently developed by Alpher \etal~\cite{Alpher03} ...'' because reference~\cite{Alpher03} has just two authors.
\begin{figure*}
\centering
\begin{subfigure}{0.68\linewidth}
\fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}}
\caption{An example of a subfigure.}
\label{fig:short-a}
\end{subfigure}
\hfill
\begin{subfigure}{0.28\linewidth}
\fbox{\rule{0pt}{2in} \rule{.9\linewidth}{0pt}}
\caption{Another example of a subfigure.}
\label{fig:short-b}
\end{subfigure}
\caption{Example of a short caption, which should be centered.}
\label{fig:short}
\end{figure*}
\section{Formatting your paper}
\label{sec:formatting}
All text must be in a two-column format.
The total allowable size of the text area is $6\frac78$ inches (17.46 cm) wide by $8\frac78$ inches (22.54 cm) high.
Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them.
The main title (on the first page) should begin 1 inch (2.54 cm) from the top edge of the page.
The second and following pages should begin 1 inch (2.54 cm) from the top edge.
On all pages, the bottom margin should be $1\frac{1}{8}$ inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper;
for A4 paper, approximately $1\frac{5}{8}$ inches (4.13 cm) from the bottom edge of the
page.
\subsection{Margins and page numbering}
All printed material, including text, illustrations, and charts, must be kept
within a print area $6\frac{7}{8}$ inches (17.46 cm) wide by $8\frac{7}{8}$ inches (22.54 cm)
high.
Page numbers should be in the footer, centered and $\frac{3}{4}$ inches from the bottom of the page.
The review version should have page numbers, yet the final version submitted as camera ready should not show any page numbers.
The \LaTeX\ template takes care of this when used properly.
\subsection{Type style and fonts}
Wherever Times is specified, Times Roman may also be used.
If neither is available on your word processor, please use the font closest in
appearance to Times to which you have access.
MAIN TITLE.
Center the title $1\frac{3}{8}$ inches (3.49 cm) from the top edge of the first page.
The title should be in Times 14-point, boldface type.
Capitalize the first letter of nouns, pronouns, verbs, adjectives, and adverbs;
do not capitalize articles, coordinate conjunctions, or prepositions (unless the title begins with such a word).
Leave two blank lines after the title.
AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title
and printed in Times 12-point, non-boldface type.
This information is to be followed by two blank lines.
The ABSTRACT and MAIN TEXT are to be in a two-column format.
MAIN TEXT.
Type main text in 10-point Times, single-spaced.
Do NOT use double-spacing.
All paragraphs should be indented 1 pica (approx.~$\frac{1}{6}$ inch or 0.422 cm).
Make sure your text is fully justified---that is, flush left and flush right.
Please do not place any additional blank lines between paragraphs.
Figure and table captions should be 9-point Roman type as in \cref{fig:onecol,fig:short}.
Short captions should be centred.
\noindent Callouts should be 9-point Helvetica, non-boldface type.
Initially capitalize only the first word of section titles and first-, second-, and third-order headings.
FIRST-ORDER HEADINGS.
(For example, {\large \bf 1. Introduction}) should be Times 12-point boldface, initially capitalized, flush left, with one blank line before, and one blank line after.
SECOND-ORDER HEADINGS.
(For example, { \bf 1.1. Database elements}) should be Times 11-point boldface, initially capitalized, flush left, with one blank line before, and one after.
If you require a third-order heading (we discourage it), use 10-point Times, boldface, initially capitalized, flush left, preceded by one blank line, followed by a period and your text on the same line.
\subsection{Footnotes}
Please use footnotes\footnote{This is what a footnote looks like.
It often distracts the reader from the main flow of the argument.} sparingly.
Indeed, try to avoid footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence).
If you wish to use a footnote, place it at the bottom of the column on the page on which it is referenced.
Use Times 8-point type, single-spaced.
\subsection{Cross-references}
For the benefit of author(s) and readers, please use the
{\small\begin{verbatim}
\cref{...}
\end{verbatim}} command for cross-referencing to figures, tables, equations, or sections.
This will automatically insert the appropriate label alongside the cross-reference as in this example:
\begin{quotation}
To see how our method outperforms previous work, please see \cref{fig:onecol} and \cref{tab:example}.
It is also possible to refer to multiple targets as once, \eg~to \cref{fig:onecol,fig:short-a}.
You may also return to \cref{sec:formatting} or look at \cref{eq:also-important}.
\end{quotation}
If you do not wish to abbreviate the label, for example at the beginning of the sentence, you can use the
{\small\begin{verbatim}
\Cref{...}
\end{verbatim}}
command. Here is an example:
\begin{quotation}
\Cref{fig:onecol} is also quite important.
\end{quotation}
\subsection{References}
List and number all bibliographical references in 9-point Times, single-spaced, at the end of your paper.
When referenced in the text, enclose the citation number in square brackets, for
example~\cite{Authors14}.
Where appropriate, include page numbers and the name(s) of editors of referenced books.
When you cite multiple papers at once, please make sure that you cite them in numerical order like this \cite{Alpher02,Alpher03,Alpher05,Authors14b,Authors14}.
If you use the template as advised, this will be taken care of automatically.
\begin{table}
\centering
\begin{tabular}{@{}lc@{}}
\toprule
Method & Frobnability \\
\midrule
Theirs & Frumpy \\
Yours & Frobbly \\
Ours & Makes one's heart Frob\\
\bottomrule
\end{tabular}
\caption{Results. Ours is better.}
\label{tab:example}
\end{table}
\subsection{Illustrations, graphs, and photographs}
All graphics should be centered.
In \LaTeX, avoid using the \texttt{center} environment for this purpose, as this adds potentially unwanted whitespace.
Instead use
{\small\begin{verbatim}
\centering
\end{verbatim}}
at the beginning of your figure.
Please ensure that any point you wish to make is resolvable in a printed copy of the paper.
Resize fonts in figures to match the font in the body text, and choose line widths that render effectively in print.
Readers (and reviewers), even of an electronic copy, may choose to print your paper in order to read it.
You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic.
When placing figures in \LaTeX, it's almost always best to use \verb+\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below
{\small\begin{verbatim}
\usepackage{graphicx} ...
\includegraphics[width=0.8\linewidth]
{myfile.pdf}
\end{verbatim}
}
\subsection{Color}
Please refer to the author guidelines on the CVPR\ 2022\ web page for a discussion of the use of color in your document.
If you use color in your plots, please keep in mind that a significant subset of reviewers and readers may have a color vision deficiency; red-green blindness is the most frequent kind.
Hence avoid relying only on color as the discriminative feature in plots (such as red \vs green lines), but add a second discriminative feature to ease disambiguation.
\section{Final copy}
You must include your signed IEEE copyright release form when you submit your finished paper.
We MUST have this form before your paper can be published in the proceedings.
Please direct any questions to the production editor in charge of these proceedings at the IEEE Computer Society Press:
\url{https://www.computer.org/about/contact}.
{\small
\bibliographystyle{ieee_fullname}
\section{Introduction}
After receiving paper reviews, authors may optionally submit a rebuttal to address the reviewers' comments, which will be limited to a {\bf one page} PDF file.
Please follow the steps and style guidelines outlined below for submitting your author response.
The author rebuttal is optional and, following similar guidelines to previous CVPR conferences, is meant to provide you with an opportunity to rebut factual errors or to supply additional information requested by the reviewers.
It is NOT intended to add new contributions (theorems, algorithms, experiments) that were absent in the original submission and NOT specifically requested by the reviewers.
You may optionally add a figure, graph, or proof to your rebuttal to better illustrate your answer to the reviewers' comments.
Per a passed 2018 PAMI-TC motion, reviewers should refrain from requesting significant additional experiments for the rebuttal or penalize for lack of additional experiments.
Authors should refrain from including new experimental results in the rebuttal, especially when not specifically requested to do so by the reviewers.
Authors may include figures with illustrations or comparison tables of results reported in the submission/supplemental material or in other papers.
Just like the original submission, the rebuttal must maintain anonymity and cannot include external links that reveal the author identity or circumvent the length restriction.
The rebuttal must comply with this template (the use of sections is not required, though it is recommended to structure the rebuttal for ease of reading).
\subsection{Response length}
Author responses must be no longer than 1 page in length including any references and figures.
Overlength responses will simply not be reviewed.
This includes responses where the margins and formatting are deemed to have been significantly altered from those laid down by this style guide.
Note that this \LaTeX\ guide already sets figure captions and references in a smaller font.
\section{Formatting your Response}
{\bf Make sure to update the paper title and paper ID in the appropriate place in the tex file.}
All text must be in a two-column format.
The total allowable size of the text area is $6\frac78$ inches (17.46 cm) wide by $8\frac78$ inches (22.54 cm) high.
Columns are to be $3\frac14$ inches (8.25 cm) wide, with a $\frac{5}{16}$ inch (0.8 cm) space between them.
The top margin should begin 1 inch (2.54 cm) from the top edge of the page.
The bottom margin should be $1\frac{1}{8}$ inches (2.86 cm) from the bottom edge of the page for $8.5 \times 11$-inch paper;
for A4 paper, approximately $1\frac{5}{8}$ inches (4.13 cm) from the bottom edge of the page.
Please number any displayed equations.
It is important for readers to be able to refer to any particular equation.
Wherever Times is specified, Times Roman may also be used.
Main text should be in 10-point Times, single-spaced.
Section headings should be in 10 or 12 point Times.
All paragraphs should be indented 1 pica (approx.~$\frac{1}{6}$ inch or 0.422 cm).
Figure and table captions should be 9-point Roman type as in \cref{fig:onecol}.
List and number all bibliographical references in 9-point Times, single-spaced,
at the end of your response.
When referenced in the text, enclose the citation number in square brackets, for example~\cite{Alpher05}.
Where appropriate, include the name(s) of editors of referenced books.
\begin{figure}[t]
\centering
\fbox{\rule{0pt}{0.5in} \rule{0.9\linewidth}{0pt}}
\caption{Example of caption. It is set in Roman so that mathematics
(always set in Roman: $B \sin A = A \sin B$) may be included without an
ugly clash.}
\label{fig:onecol}
\end{figure}
To avoid ambiguities, it is best if the numbering for equations, figures, tables, and references in the author response does not overlap with that in the main paper (the reviewer may wonder if you talk about \cref{fig:onecol} in the author response or in the paper).
See \LaTeX\ template for a workaround.
\subsection{Illustrations, graphs, and photographs}
All graphics should be centered.
Please ensure that any point you wish to make is resolvable in a printed copy of the response.
Resize fonts in figures to match the font in the body text, and choose line widths which render effectively in print.
Readers (and reviewers), even of an electronic copy, may choose to print your response in order to read it.
You cannot insist that they do otherwise, and therefore must not assume that they can zoom in to see tiny details on a graphic.
When placing figures in \LaTeX, it is almost always best to use \verb+\includegraphics+, and to specify the figure width as a multiple of the line width as in the example below
{\small\begin{verbatim}
\usepackage{graphicx} ...
\includegraphics[width=0.8\linewidth]
{myfile.pdf}
\end{verbatim}
}
{\small
\bibliographystyle{ieee_fullname}
\section{Introduction}\label{sec:intro}
\vspace{-2mm}
Estimating rotations is a crucial problem in visual perception that has broad applications, \textit{e.g.}, in object pose estimation, robot control, camera relocalization, 3D reconstruction and visual odometry~\cite{kendall2015posenet,bui20206d,wang2019densefusion,gojcic2020learning,dong2020robust}. Recently, with the proliferation of deep neural networks, learning to accurately regress rotations is attracting more and more attention. However, the non-Euclidean characteristics of rotation space make accurately regressing rotation very challenging.
As we know, rotations reside in a non-Euclidean manifold, $\mathrm{SO}(3)$ group, whereas the unconstrained outputs of neural networks usually live in Euclidean spaces.
This gap between the neural network output space and $\mathrm{SO}(3)$ manifold becomes a major challenge for deep rotation regression, thus tackling this gap becomes an important research topic.
One popular research direction is to design learning-friendly rotation representations, \textit{e.g.}, 6D continuous representation from \cite{zhou2019continuity} and 10D symmetric matrix representation from \cite{peretroukhin_so3_2020}.
Recently, Levinson \textit{et al.}\cite{levinson2020analysis} adopted the vanilla 9D matrix representation discovering that simply replacing the Gram-Schmidt process in the 6D representation~\cite{zhou2019continuity} with symmetric SVD-based orthogonalization can make this representation superior to the others.
Despite the progress on discovering better rotation representations, the gap between a Euclidean network output space and the non-Euclidean $\mathrm{SO}(3)$ manifold hasn't been completely filled.
One important yet long-neglected problem lies in optimization on non-Euclidean manifolds\cite{absil2009optimization}:
to optimize on $\mathrm{SO}(3)$ manifold, the optimization variable is a rotation matrix, which contains nine matrix elements; if we naively use \textit{Euclidean gradient}, which simply computes the partial derivatives with respect to each of the nine matrix elements, to update the variable, this optimization step will usually lead to a new matrix off $\mathrm{SO}(3)$ manifold.
Unfortunately, we observe that all the existing works on rotation regression simply rely upon
\textit{vanilla auto-differentiation} for backpropagation, exactly computing Euclidean gradient and performing such off-manifold updates to predicted rotations.
We argue that, for training deep rotation regression networks, the off-manifold components will lead to noise in the gradient of neural network weights, hindering network training and convergence.
To tackle this issue, we draw inspiration from differential geometry, where people leverage \emph{Riemannian optimization} to optimize on the non-Euclidean manifold, which finds the direction of the steepest geodesic path on the manifold and take an on-manifold step.
We thus propose to leverage Riemannian optimization and delve deep into the study of the backward pass.
Note that this is a fundamental yet currently under-explored avenue, given that most of the existing works focus on a holistic design of rotation regression that is agnostic to forward/backward pass.
However, incorporating Riemannian optimization into network training is highly non-trivial and challenging. Although methods of Riemannian optimization allow for optimization on $\mathrm{SO}(3)$~\cite{taylor1994minimization, blanco2010tutorial}, matrix manifolds~\cite{absil2009optimization} or general Riemannian manifolds \cite{zhang2016riemannian, udriste2013convex}, they are not directly applicable to update the weights of the neural networks that are Euclidean. Also, approaches like~\cite{hou2018computing} incorporate a Riemannian distance as well as its gradient into network training, however, they do not deal with the \emph{representation} issue.
In this work, we want to \textit{propose a better manifold-aware gradient in the backward pass of rotation regression that directly updates the neural network weights}.
We begin by taking a Riemannian optimization step and computing the difference between the rotation prediction and the updated rotation, which is closer to the ground truth.
Backpropagating this "error", we encounter the mapping function (or orthogonalization function) that transforms the raw network output to a valid rotation.
This projection, which can be the Gram-Schmidt process or SVD orthogonalization\cite{levinson2020analysis}, is typically a many-to-one mapping.
This non-bijectivity provides us with a new design space for our gradient: if we were to use a gradient to update the raw output rotation, many gradients would result in the same update in the final output rotation despite being completely different for backpropagating into the neural network weights.
Now the problem becomes: \textit{which gradient is the best for backpropagation when many of them correspond to the same update to the output?}
We observe that this problem is somewhat similar to some problems with ambiguities or multi-ground-truth issues. One example would be the symmetry issue in pose estimation: a symmetric object, \textit{e.g.} a textureless cube, appears the same under many different poses, which needs to be considered when supervising the pose predictions.
For supervising the learning in such a problem,
Wang \textit{et. al.}\cite{wang2019normalized} proposed to use min-of-N loss\cite{fan2017point}, which only penalizes the smallest error between the prediction and all the possible ground truths.
We therefore propose to find the gradient with the smallest norm that can update the final output rotation to the goal rotation.
This \emph{back-projection} process involves finding an element closest to the network output in the inverse image of the goal rotation and projecting the network output to this inverse image space. We therefore coin our gradient \textit{projective manifold gradient}.
One thing to note is that this projective gradient tends to shorten the network output, causing the norms of network output to vanish. To fix this problem, we further incorporate a simple regularization into the gradient, leading to our full solution \textit{regularized projective manifold gradient} (RPMG).
Note that our proposed gradient layer operates on the raw network output and can be directly backpropagated into the network weights. Our method is very general and is not tied to a specific rotation representation. It can be coupled with different non-Euclidean rotation representations, including quaternion, 6D representation \cite{zhou2019continuity}, and 9D rotation matrix representation \cite{levinson2020analysis}, and can even be used for regressing other non-manifold variables.
We evaluate our devised projective manifold gradient layers on a diverse set of problems involving rotation regression: 3D object pose estimation from 3D point clouds/images, rotation estimation problems without using ground truth rotation supervisions, and please see Appendix \ref{sec:reloc} for more experiments on camera relocalization. Our method demonstrates significant and consistent improvements on all these tasks and all different rotation representations tested.
Going beyond rotation estimation, we also demonstrate performance improvements on regressing unit vectors (lie on a unit sphere) as an example of an extension to other non-Euclidean manifolds.
We summarize our contribution as below:
\begin{itemize}
\setlength\itemsep{-1mm}
\item We propose a novel manifold-aware gradient layer, namely \textit{RPMG}, for the backward pass of rotation regression, which can be applied to different rotation representations and losses and used as a ``plug-in'' at no actual cost.
\item Our extensive experiments over different tasks and rotation representations demonstrate the significant improvements from using RPMG.
\item Our method can also benefit regression tasks on other manifolds, \textit{e.g.} $\mathcal{S}^2$.
\end{itemize}
\section{Related Work}
\vspace{-2mm}
\label{sec:related}
Both rotation parameterization and optimization on SO(3) are well-studied topics.
Early deep learning models leverage various rotation representations for pose estimation, \textit{e.g.}, direction cosine matrix (DCM)~\cite{huang2021multibodysync,yi2019deep}, axis-angle \cite{DeMoN17, deep-6dpose18, occlusion18}, quaternion \cite{xiang2017posecnn, geometric_loss17, posenet,zhao2020quaternion,deng2020deep} and Euler-angle \cite{viewpoints_and_keypoints15, renderforcnn15, 3d-rcnn18}.
Recently, \cite{zhou2019continuity} points out that Euler-angle, axis-angle, and quaternion are not continuous rotation representations, since their representation spaces are not homeomorphic to SO(3). As better representations for rotation regression, 6D~\cite{zhou2019continuity}, 9D~\cite{levinson2020analysis}, 10D~\cite{peretroukhin_so3_2020} representations are proposed to resolve the discontinuity issue and improve the regression accuracy. A concurrent work \cite{regression_manifold21} examines different manifold mappings theoretically and experimentally, finding out that SVD orthogonalization performs the best when regressing arbitrary rotations.
Originating from general Riemannian optimization, \cite{taylor1994minimization} presents an easy approach for minimization on the $SO(3)$ group by constructing a local axis-angle parameterization, which is also the tangent space of $SO(3)$ manifold. They backpropagate gradient to the tangent space and use the exponential map to update the current rotation matrix.
Most recently, \cite{teed2021tangent} constructs a PyTorch library that supports tangent space gradient backpropagation for 3D transformation groups, (\textit{e.g.}, $SO(3)$, $SE(3)$, $Sim(3)$). This proposed library can be used to implement the Riemannian gradient in our layer.
\section{Preliminaries}
\label{sec:prelim}
\vspace{-1mm}
\subsection{Riemannian Geometry}
\vspace{-1mm}
Following~\cite{birdal2018bayesian,birdal2019probabilistic}, we define an $m$-dimensional \textit{Riemannian manifold} embedded in an ambient Euclidean space $\mathcal{X} = \mathbb{R}^d$ and endowed with a \textit{Riemannian metric} $\mathbf{G}\triangleq (\mathbf{G}_{\mathbf{x}})_{\mathbf{x}\in\mathcal{M}}$ to be a smooth curved space $(\mathcal{M},G)$. A vector $\v\in\mathcal{X}$ is said to be \emph{tangent} to $\mathcal{M}$ at $\mathbf{x}$ iff there exists a smooth curve $\gamma:[0,1]\mapsto\mathcal{M}$ s.t. $\gamma(0)=\mathbf{x}$ and $\dot{\gamma}(0)=\v$. The velocities of all such curves through $\mathbf{x}$ form the \emph{tangent space} $\mathcal{T}_{\x}\Man=\{ \dot{\gamma} (0) \,|\, \gamma:\mathbb{R}\mapsto\mathcal{M} \text{ is smooth around $0$ and } \gamma(0)=\mathbf{x}\}$.
\vspace{-1mm}
\begin{dfn}[Riemannian gradient]
For a smooth function $f:\mathcal{M}\mapsto\mathbb{R}$ and $\forall (\mathbf{x},\v)\in \mathcal{T}\mathcal{M}$, we define the \emph{Riemannian gradient} of $f$ as the unique vector field $\grad{f}$ satisfying~\cite{boumal2020introduction}:
\vspace{-2mm}
\begin{equation}
\mathrm{D}f(\mathbf{x})[\v] = \langle \v, \grad{f(\mathbf{x})} \rangle_\mathbf{x}
\vspace{-1mm}
\end{equation}
where $\mathrm{D}f(\mathbf{x})[\v]$ is the derivation of $f$ by $\v$. It can further be shown (see Appendix \ref{sec:supp2.1}) that an expression for $\grad{f}$ can be obtained through the projection of the \emph{Euclidean} gradient orthogonally onto the tangent space
\vspace{-2mm}
\begin{equation}
\grad{f(\mathbf{x})} = \nabla f(\mathbf{x})_{\|} = \Pi_{\mathbf{x}}\big( \nabla f(\mathbf{x})\big).
\vspace{-1mm}
\end{equation}
where $\Pi_{\mathbf{x}}:\mathcal{X}\mapsto\mathcal{T}_{\x}\Man\subseteq \mathcal{X}$ is an orthogonal projector with respect to $\langle \cdot,\cdot \rangle_{\mathbf{x}}$.
\end{dfn}
\vspace{-2mm}
\begin{dfn}[Riemannian optimization]
We consider gradient descent to solve the problems of $\min_{\mathbf{x}\in\mathcal{M}}f(\mathbf{x})$. For a local minimizer or a \emph{stationary point} $\mathbf{x}^\star$ of $f$, the Riemannian gradient vanishes $\grad{f(\mathbf{x}^\star)}=0$ enabling a simple algorithm, \emph{Riemannian gradient descent} (RGD):
\vspace{-2mm}
\begin{align}
\mathbf{x}_{k+1} = R_{\mathbf{x}_k}(-\tau_k\,\grad{f(\mathbf{x}_k)})
\end{align}
where $\tau_k$ is the step size at iteration $k$ and $R_{\mathbf{x}_k}$ is the \emph{retraction} usually chosen related to the exponential map.
\end{dfn}
\vspace{-2mm}
\subsection{Rotation Representations}
\vspace{-1mm}
\label{sec:rot_rep}
There are many ways of representing a rotation: classic rotation representations, \eg Euler angles, axis-angle, and quaternion; and recently introduced regression-friendly rotation representations such as \eg 5D~\cite{zhou2019continuity}, 6D~\cite{zhou2019continuity}, 9D~\cite{levinson2020analysis} and 10D~\cite{peretroukhin_so3_2020} representations. A majority of deep neural networks can output an \emph{unconstrained}, arbitrary $n$-dimensional vector $\mathbf{x}$ in a Euclidean space $\mathcal{X}=\mathbb{R}^n$. For Euler angle and axis-angle representations which use a vector from $\mathbb{R}^3$ to represent a rotation, a neural network can simply output a 3D vector; however, for quaternions, 6D, 9D or 10D representations that lies on non-Euclidean manifolds, manifold mapping function $\pi:\mathbb{R}^n\mapsto \mathcal{M}$ is generally needed for normalization or orthogonalization purposes to convert network outputs to valid elements belonging to the representation manifold. This network Euclidean output space $\mathcal{X}$ is where the representation manifolds reside and therefore are also called ambient space.
\begin{dfn}[Rotation representation]
One rotation representation, which lies on a representation manifold $\mathcal{M}$, defines a surjective rotation mapping $\phi: \hat{\mathbf{x}} \in \mathcal{M} \rightarrow \phi(\hat{\mathbf{x}}) \in \mathrm{SO}(3)$ and a representation mapping function $\psi : \mathbf{R} \in \mathrm{SO}(3) \rightarrow \psi(\mathbf{R}) \in \mathcal{M}$, such that $\phi(\psi) = \mathbf{R} \in \mathrm{SO}(3)$.
\end{dfn}
\begin{dfn}[Manifold mapping function]
From an ambient space $\mathcal{X}$ to the representation manifold $\mathcal{M}$, we can define a manifold mapping function $\pi: \mathbf{x} \in \mathcal{X} \rightarrow \pi(\mathbf{x}) \in \mathcal{M}$, which projects a point $\mathbf{x}$ in the ambient, Euclidean space to a valid element $\hat{\mathbf{x}} = \pi(\mathbf{x})$ on the manifold $\mathcal{M}$.
\end{dfn}
We summarize the manifold mappings, the rotation mappings and representation mappings for several non-Euclidean rotation representations below.
\noindent\textbf{Unit quaternion.} Unit quaternions represent a rotation using a 4D unit vector $\mathbf{q} \in \mathcal{S}^{3}$ double covering the non-Euclidean $3$-sphere \ie $\mathbf{q}$ and $-\mathbf{q}$ identify the same rotation.
A network with a final linear activation can only predict $\mathbf{x} \in \mathbb{R}^{4}$. The corresponding manifold mapping function is usually chosen to be a normalization step, which reads $\pi_{q}(\mathbf{x}) = \mathbf{x}/\|\mathbf{x}\|$. For rotation and representation mapping, we leverage the standard mappings between rotation and quaternion (see Appendix \ref{sec:q_r_trans}).
\noindent\textbf{6D representation and Gram-Schmidt orthogonalization.} 6D rotation representation\cite{zhou2019continuity}, lying on Stiefel manifold $\mathcal{V}_2(\mathbb{R}^{3})$, uses two orthogonal unit 3D vectors $(\hat{\mathbf{c}}_1, \hat{\mathbf{c}}_2)$ to represent a rotation, which are essentially the first two columns of a rotation matrix.
Its manifold mapping $\pi_{6D}$ is done through Gram-Schmidt orthogonalization. Its rotation mapping $\phi_{6D}$ is done by adding the third column $\hat{\mathbf{c}}_3 = \hat{\mathbf{c}}_1 \times \hat{\mathbf{c}}_2$. Its representation mapping $\psi_{6D}$ is simply getting rid of the third column $\hat{\mathbf{c}}_3$ from a rotation matrix.
\noindent\textbf{9D representation and SVD orthogonalization.}
To map a raw 9D network output $\mathbf{M}$ to a rotation matrix, \cite{levinson2020analysis}
use SVD orthogonalization as the manifold mapping function $\pi_{9D}$, as follows: $\pi_{9D}$ first decomposes $\mathbf{M}$ into its left and right singular vectors $\{\mathbf{U},\mathbf{V}^\top\}$ and singular values $\Sigma$, $\mathbf{M}=\mathbf{U}\Sigma \mathbf{V}^\top$; then it replaces $\Sigma$ with $\Sigma'=\mathrm{diag}(1,1,\mathrm{det}(\mathbf{U}\mathbf{V}^\top))$ and finally, computes $\mathbf{R}=\mathbf{U}\Sigma'\mathbf{V}^\top$ to get the corresponding rotation matrix $\mathbf{R} \in$ $\mathrm{SO}(3)$. As this representation manifold is $\mathrm{SO}(3)$, both the rotation and representation mapping functions are simply identity.
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth]{figure/figure1.pdf}
\caption{\textbf{Projective Manifold Gradient Layer.} In the forward pass, the network predicts a raw output $\mathbf{x}$, which is then transformed into a valid rotation $\mathbf{R} = \phi(\pi(\mathbf{x}))$. We leave this forward pass unchanged and only modify the backward pass. In the backward pass, we first use Riemannian optimization to get a goal rotation $\mathbf{R}_g$ and map it back to $\hat{\mathbf{x}}_g$ on the representation manifold $\mathcal{M}$. After that we find the element $\mathbf{x}_{gp}$ which is closest to the raw output in the inverse image of $\hat{\mathbf{x}}_g$, and finally get the gradient $\mathbf{g_{RPM}}$ we want.\vspace{-4mm}}
\label{fig:pip}
\end{figure*}
\noindent\textbf{10D representation.}
~\cite{peretroukhin_so3_2020} propose a novel 10D representation for rotation matrix. The manifold mapping function $\pi_{10D}$ maps $\boldsymbol{\theta}\in\mathbb{R}^{10}$ to $\mathbf{q}\in\mathcal{S}^3$ by computing the eigenvector corresponding to the smallest eigenvalue of $\mathbf{A}(\boldsymbol{\theta})$, expressed as $\pi_{10D}(\mathbf{x})=\underset{\mathbf{q}\in\mathcal{S}^3}{\min}~\mathbf{q}^\top\mathbf{A}(\mathbf{x})\mathbf{q}$, in which
\begin{equation}
\vspace{1mm}
\mathbf{A}(\boldsymbol{\theta})~=~\left[
\begin{array}{cccc}
\theta_1 & \theta_2 & \theta_3 & \theta_4 \\
\theta_2 & \theta_5 & \theta_6 & \theta_7 \\
\theta_3 & \theta_6 & \theta_8 & \theta_9 \\
\theta_4 & \theta_7 & \theta_9 & \theta_{10} \\
\end{array}
\right].
\vspace{1mm}
\end{equation}
Since the representation manifold is $\mathcal{S}^3$, the rotation and representation mapping are the same as unit quaternion.
\subsection{Deep Rotation Regression}
\vspace{-1mm}
We conclude this section by describing the ordinary forward and backward passes of a neural network based rotation regression, as used in \cite{zhou2019continuity, levinson2020analysis}. \vspace{1mm} \\
\textbf{Forward and backward passes.} Assume, for a rotation representation, the network predicts $\mathbf{x} \in \mathcal{X}$, then the manifold mapping $\pi$ will map $\mathbf{x}$ to $\hat{\mathbf{x}} = \pi(\mathbf{x}) \in \mathcal{M}$, followed by a rotation mapping $\phi$ that finally yields the output rotation $\mathbf{R} = \phi(\hat{\mathbf{x}}) = \phi(\pi(\mathbf{x}))$. Our work only tackles the backward pass and keeps the forward pass unchanged, as shown in the top part of Figure \ref{fig:pip}. The gradient in the backward-pass is simply computed using Pytorch autograd method, that is $\mathbf{g}=f'(\mathbf{R})\phi'(\hat{\mathbf{x}})\pi'(\mathbf{x})$. \vspace{1mm}\\
\textbf{Loss function.} The most common choice for supervising rotation matrix is L2 loss, $\|\mathbf{R}-\mathbf{R}_{gt}\|^{2}_{F}$ , as used by \cite{zhou2019continuity, levinson2020analysis}. This loss is equal to $4-4\cos(<\mathbf{R}, \mathbf{R}_{gt}>)$, where $<\mathbf{R}, \mathbf{R}_{gt}>$ represents the angle between $\mathbf{R}$ and $\mathbf{R}_{gt}$.
\section{Method}
\label{sec:method}
\vspace{-1mm}
\noindent\textbf{Overview.}
In this work, we propose a \emph{projective manifold gradient layer}, without changing the forward pass of a given rotation regressing network, as shown in Figure \ref{fig:pip}. Our focus is to find a better gradient $\mathbf{g}$ of the loss function $\mathcal{L}$ with respect to the network raw output $\mathbf{x}$ for backpropagation into the network weights.
Let's start with examining the gradient of network output $\mathbf{x}$ in a general case -- regression in Euclidean space. Given a ground truth $\mathbf{x}_{gt}$ and the L2 loss $\|\mathbf{x} - \mathbf{x}_{gt}\|^2$ that maximizes the likelihood in the presence of Gaussian noise in $\mathbf{x}$, the gradient would be $\mathbf{g} = 2(\mathbf{x} - \mathbf{x}_{gt})$.
In the case of rotation regression, we therefore propose to find a proper $\mathbf{x}^*\in \mathcal{X}$ for a given ground truth $\mathbf{R}_{gt}$ or a computed goal rotation $\mathbf{R}_{g}$ when the ground truth rotation is not available, and then simply use $\mathbf{x} - \mathbf{x}^*$ as our gradient to backpropagate into the network.
Note that finding such an $\mathbf{x}^*$ can be challenging.
Assuming we know $\mathbf{R}_{gt}$, finding an $\mathbf{x}^*$ involves inverting $\phi$ and $\pi$ since the network output $\mathbf{R} = \phi(\pi(\mathbf{x}))$.
Furthermore, we may not know $\mathbf{R}_{gt}$ under indirect rotation supervision (\textit{e.g.}, flow loss as used in PoseCNN\cite{xiang2017posecnn}) and self-supervised rotation estimation cases (\textit{e.g.}, 2D mask loss as used in \cite{wang2020self6d}).
In this work, we introduce the following techniques to mitigate these problems: (i) we first take a Riemannian gradient to compute a goal rotation $\mathbf{R}_{g} \in \mathrm{SO}(3)$, which does not rely on knowing $\mathbf{R}_{gt}$, as explained in Section \ref{sec:rg}; (ii) we then find the set of all possible $\mathbf{x}_g$s that can be mapped to $\mathbf{R}_{g}$, or in other words, the inverse image of $\mathbf{R}_{g}$ under $\pi$ and $\phi$; (iii) we find $\mathbf{x}_{gp}$ which is the element in this set closest to $\mathbf{x}$ in the Euclidean metric and set it as ``$\mathbf{x}^*$''. We will construct our gradient using this $\mathbf{x}^*$, as explained in \ref{sec:pmg}. (iv) we add a regularization term to this gradient forming $\mathbf{g}_{RPMG}$ as explained in \ref{sec:rpmg}. The whole backward pass leveraging our proposed regularized projective manifold gradient is shown in the lower half of Figure \ref{fig:pip}.
\vspace{-1mm}
\subsection{Riemannian Gradient and Goal Rotation}
\vspace{-1mm}
\label{sec:rg}
To handle rotation estimation with/without direct rotation supervision, we first propose to compute the Riemannian gradient of the loss function $\mathcal{L}$ with respect to the output rotation $\mathbf{R}$ and find a goal rotation $\mathbf{R}_{g}$ that is presumably closer to the ground truth rotation than $\mathbf{R}$.
Assume the loss function is in the following form $\mathcal{L}(f(\mathbf{R}))$, where $\mathbf{R} = \pi(\phi(\mathbf{x}))$ is the output rotation and $f$ constructs a loss function that compares $\mathbf{R}$ to the ground truth rotation $\mathbf{R}_{gt}$ directly or indirectly. Given $\mathbf{R}(\mathbf{x})$ and $\mathcal{L}(f(\mathbf{R}(\mathbf{x})))$, we can perform one step of Riemannian optimization yielding our goal rotation $\mathbf{R}_{g} \gets R_{\mathbf{R}}(-\tau\,
\grad{~\mathcal{L}(f(\mathbf{R}}))),$
where $\tau$ is the step size of Riemannian gradient and can be set to a constant as a hyperparameter or varying during the training. For L2 loss $\|\mathbf{R} - \mathbf{R}_{gt}\|_F^2$, the Riemannian gradient is always along the geodesic path between ${\mathbf{R}}$ and $\mathbf{R}_{gt}$ on SO(3)\cite{huynh2009metrics}. In this case, $\mathbf{R}_{g}$ can generally be seen as an intermediate goal between $\mathbf{R}$ and $\mathbf{R}_{gt}$ dependent on $\tau$.
Gradually increasing $\tau$ from 0 will first make $\mathbf{R}_{g}$ approach $\mathbf{R}_{gt}$ starting with $\mathbf{R}_{g}=\mathbf{R}$, and then reach $\mathbf{R}_{gt}$ where we denote $\tau = \tau_{gt}$, and finally going beyond $\mathbf{R}_{gt}$.
Although, when $\mathbf{R}_{gt}$ is available, one can simply set $\mathbf{R}_{g} = \mathbf{R}_{gt}$, we argue that this is just a special case under $\tau =\tau_{gt}$.
For scenarios where $\mathbf{R}_{gt}$ is unavailable, \textit{e.g.}, in self-supervised learning cases (see in Section \ref{sec:wo_gt}), we don't know $\mathbf{R}_{gt}$ and $\tau_{gt}$, thus we need to compute $\mathbf{R}_{g}$ using Riemannian optimization.
In the sequel, we only use $\mathbf{R}_{g}$ for explaining our methods without loss of generality.
See Section \ref{sec:rpmg} for how to choose $\tau$.
\subsection{Projective Manifold Gradient}
\vspace{-1mm}
\label{sec:pmg}
Given $\mathbf{R}_{g}$, we can use the representation mapping $\psi$ to find the corresponding $\hat{\mathbf{x}}_g = \psi(\mathbf{R}_g)$ on the representation manifold $\mathcal{M}$. However, further inverting $\pi$ and finding the corresponding $\mathbf{x}_g \in \mathcal{X}$ is a non-trivial problem, due to the projective nature of $\pi$.
In fact, there are many $\mathbf{x}_g$s that satisfy $\pi(\mathbf{x}_g) = \hat{\mathbf{x}}_g$.
It seems that we can construct a gradient $\mathbf{g} = (\mathbf{x} - \mathbf{x}_g)$ using any $\mathbf{x}_g$ that satisfies $\pi(\mathbf{x}_g) = \hat{\mathbf{x}}_g$. No matter which $\mathbf{x}_g$ we choose, if this gradient were to update $\mathbf{x}$, it will result in the same $\mathbf{R}_g$.
But, when backpropagating into the network, those gradients will update the network weights differently, potentially resulting in different learning efficiency and network performance.
Formally, we formulate this problem as \emph{a multi-ground-truth problem} for $\mathbf{x}$: we need to find the best $\mathbf{x}^*$ to supervise from the inverse image of $\hat{\mathbf{x}}_g$ under the mapping $\pi$.
We note that similar problems have been seen in pose supervision dealing with symmetry as in \cite{wang2019normalized}, where one needs to find one pose to supervise when there are many poses under which the object appears the same. \cite{wang2019normalized} proposed to use a min-of-N strategy introduced by \cite{fan2017point}: from all possible poses, taking the pose that is closest to the network prediction as ground truth. A similar strategy is also seen in supervising quaternion regression, as $\mathbf{q}$ and $-\mathbf{q}$ stand for the same rotation. One common choice of the loss function is therefore $\min \{\mathcal{L}(\mathbf{q}, \mathbf{q}_{gt}), \mathcal{L}(\mathbf{q},-\mathbf{q}_{gt})\}$\cite{peretroukhin_so3_2020}, which penalizes the distance to the closest ground truth quaternion.
Inspired by these works, we propose to choose our gradient among all the possible gradients with the lowest level of redundancy, \textit{i.e.}, we require
$\mathbf{x}^*$ to be the one closest to $\mathbf{x}$, or in other words, the gradient to have the smallest norm, meaning that we need to find the projection point $\mathbf{x}_{gp}$ of $\mathbf{x}$ to all the valid $\mathbf{x}_g$:
\begin{equation}
\mathbf{x}_{gp} = \underset{\pi(\mathbf{x}_g) = \hat{\mathbf{x}}_g}{\text{argmin}}~\|\mathbf{x} - \mathbf{x}_g\|_2
\end{equation}
We then can construct our \textit{projective manifold gradient} (PMG) as $\mathbf{g}_{PM} = \mathbf{x} - \mathbf{x}_{gp}$. We will denote the naive gradient $\mathbf{g}_{M} = \mathbf{x} - \hat{\mathbf{x}}_{g}$ as \textit{manifold gradient} (MG).
Here we provide another perspective on why a network may prefer PMG. In the case where a deep network is trained using stochastic gradient descent (SGD), the final gradient used to update the network weights is averaged across the gradients of all the batch instances.
If gradients from different batch instances contain different levels of redundancy, then the averaged gradient may be biased or not even appropriate.
This argument is generally applicable to all stochastic optimizers (\textit{e.g.}, Adam~\cite{Adams2020AFD})
\begin{figure}
\hspace{2mm}
\begin{minipage}{0.4\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/illustration.png}
\end{minipage}\hfill
\begin{minipage}{0.45\linewidth}
\centering
\includegraphics[width=1\columnwidth]{figure/quaternion_gp.pdf}
\end{minipage}
\hspace{2mm}
\caption{\textbf{Illustration for regularized projective manifold gradient.} \textbf{Left}: In the forward pass, we simply project $\mathbf{x}$ to $\hat{\mathbf{x}}$ by $\pi$. In the backward pass, first we compute a Riemannian gradient, which is shown as the \textit{green} arrow. After getting a next goal $\hat{\mathbf{x}}_{g}\in\mathcal{M}$ by Riemannian optimization, we find the inverse projection $\mathbf{x}_{gp}$ of $\hat{\mathbf{x}}_g$, which leads to our \textit{projective manifold gradient}, shown as the \textit{blue} arrow. With a regularization term, we can get our final \textit{regularized projective manifold gradient}, as the \textit{purple} arrow. \textbf{Right}: Projection point $\hat{\mathbf{x}}_{gp}$ in the case of quaternion.}
\vspace{-1mm}
\label{fig:reg}
\end{figure}
\noindent\textbf{Inverting $\pi$.} There are many ways to solve this projection problem for different manifold mapping functions $\pi$. For example, we can formulate this as a constrained optimization problem. For the manifold mapping functions we consider, we propose the following approach: we first solve for the inverse image $\pi^{-1}(\hat{\mathbf{x}}_g)$ of $\hat{\mathbf{x}}_g$ in the ambient space $\mathcal{X}$ analytically, which reads
$\pi^{-1}(\hat{\mathbf{x}}_g) = \{\mathbf{x}_g \in \mathcal{X}~|~\pi(\mathbf{x}_g) = \hat{\mathbf{x}}_g\}$; we then project $\mathbf{x}$ onto this inverse image space.
Note that, sometimes only a superset of this inverse image can be found analytically, requiring certain constraints on $\mathbf{x}_{gp}$ to be enforced.
Here we list the inverse image $\pi^{-1}(\hat{\mathbf{x}}_g)$ and the projection point $\mathbf{x}_{gp}$ for different rotation representations and their corresponding manifold mapping $\pi$. Please refer to Appendix \ref{sec:inverse_proj} for detailed derivations.
\noindent\textbf{Quaternion.} With $\pi_q(\mathbf{x}) = \mathbf{x}/{\|\mathbf{x}\|}$, $\mathbf{x} \in \mathbb{R}^4$, and $\hat{\mathbf{x}}_g \in \mathcal{S}^{3}$: $\pi^{-1}_{q}(\hat{\mathbf{x}}_g) = \{\mathbf{x}~|~\mathbf{x} = k\hat{\mathbf{x}}_g, k\in \mathbb{R}~\text{and}~k>0\}$, which is a ray in the direction of $\hat{\mathbf{x}}_g$ starting from the origin. Without considering the constraint of $k>0$ , an analytical solution to this projection point $\mathbf{x}_{gp}$ of $\mathbf{x}$ onto this line can be derived: $\mathbf{x}_{gp} = (\mathbf{x}\cdot\hat{\mathbf{x}}_g)\hat{\mathbf{x}}_g$.
\noindent\textbf{6D representation.}
With $\pi_{6D}$ as Gram-Schmidt process, $\mathbf{x}=[\u, \v] \in \mathbb{R}^6$, and $\hat{\mathbf{x}_g}\in \mathcal{V}_2(\mathbb{R}^{3})$: $\pi^{-1}_{6D}(\hat{\mathbf{x}}_g) = \{[k_1\hat{\u}_g, k_2\hat{\u}_g+k_3\hat{\v}_g]~|~k_1,k_2,k_3 \in \mathbb{R}~\text{and}~k_1, k_3>0\}$ (the former is a ray whereas the latter spans a half plane). Without considering the constraint of $k_1,k_3>0$, the projection point $\mathbf{x}_{gp}$ can be analytically represented as $\mathbf{x}_{gp} = [(\u\cdot\hat{\u}_g)\hat{\u}_g, (\v\cdot\hat{\u}_g)\hat{\u}_g+(\v\cdot\hat{\v}_g)\hat{\v}_g]$
\noindent\textbf{9D representation.}
With $\pi_{9D}(\mathbf{x})$ as SVD orthogonalization, $\mathbf{x} \in \mathbb{R}^{3\times 3}$, and $\hat{\mathbf{x}}_g\in$ SO(3), the analytical expression for $\pi_{9D}^{-1}$ is available when we ignore the positive singular value constraints, which gives $\pi^{-1}_{9D}(\hat{\mathbf{x}}_g) = \{\mathbf{S}\hat{\mathbf{x}}_g~|~\mathbf{S}=\mathbf{S^\top}\}$. We can further solve the projection point $\mathbf{x}_{gp}$ with an elegant representation $\mathbf{x}_{gp} = \frac{\mathbf{x}\hat{\mathbf{x}}_g^{T}+\hat{\mathbf{x}}_g\mathbf{x}^{T}}{2}$.
\noindent\textbf{10D representation.}
Please refer to Appendix \ref{sec:inverse_proj} for the derivation and expression of $\mathbf{x}_{qp}$.
\vspace{-1mm}
\subsection{Regularized Projective Manifold Gradient}
\vspace{-1mm}
\label{sec:rpmg}
\noindent\textbf{Issues in naive projective manifold gradient.}
\label{issues}
In the right plot of Figure \ref{fig:reg}, we illustrate this projection process for several occasions where $\mathbf{x}$ takes different positions relative to $\mathbf{x}_g$.
We demonstrate that there are two issues in this process.
First, no matter where $\mathbf{x}$ is in, the projection operation will shorten the length of our prediction because $\|\mathbf{x}_{gp}\| < \|\mathbf{x}\|$ is always true for all of 4D/6D/9D/10D representation. This will cause the length norm of the prediction of the network to become very small as the training progresses (see Figure \ref{fig:length}). The shrinking network output will keep increasing the effective learning rate, preventing the network from convergence and leading to great harm to the network performance (see Table \ref{tab:ablation} and Figure \ref{fig:length} for ablation study).
Second, when the angle between $\mathbf{x}$ and $\hat{\mathbf{x}}_g$ becomes larger than $\pi/2$ (in the case of $\mathbf{x} = \mathbf{x}_3$), the naive projection $\mathbf{x}_{gp}$ will be in the opposite direction of $\hat{\mathbf{x}}_g$ and can not be mapped back to $\hat{\mathbf{x}}_g$ under $\pi_{q}$, resulting in a wrong gradient. The same set of issues also happens to 6D, 9D and 10D representations. The formal reason is that the analytical solution of the inverse image assumes certain constraints are satisfied, which is usually true only when either $\hat{\mathbf{x}}_g$ is not far from $\mathbf{x}$ or the network is about to converge.
\noindent\textbf{Regularized projective manifold gradient.}
To solve the first issue, we propose to add a regularization term $\mathbf{x}_{gp} - \hat{\mathbf{x}}_g$ to the projective manifold gradient, which can avoid the length vanishing problem. The \textit{regularized projective manifold gradient} then reads:
\begin{equation}
\mathbf{g}_{RPM} = \mathbf{x} - \mathbf{x}_{gp} + \lambda(\mathbf{x}_{gp} - \hat{\mathbf{x}}_g),
\end{equation}
where $\lambda$ is a regularization coefficient. See the left plot of Figure \ref{fig:reg} for an illustration.
\noindent\textbf{Discussion on the hyperparameters $\lambda$ and $\tau$.}
Our method apparently introduces two additional hyperparameters, $\lambda$ and $\tau$, however, we argue that this doesn't increase the searching space of hyperparameters for our method.
For $\lambda$, the only requirement is that $\lambda$ is small (we simply set to 0.01), because: (1) we want the projective manifold gradient ($\mathbf{x} - \mathbf{x}_{gp}$) to be the major component of our gradient; (2) since this regularization is roughly proportional to the difference in prediction length and a constant, a small lambda is enough to prevent the length from vanishing and, in the end, the prediction length will stay roughly constant at the equilibrium under projection and regularization. In the ablation study of Section \ref{sec:pc_rotation}, we show that the performance is robust to the change of $\lambda$. Note that, on the other extreme, when $\lambda = 1$, $\mathbf{g}_{RPM}$ becomes $\mathbf{g}_{M}$.
For $\tau$, we propose a ramping up schedule which is well-motivated.
To tackle the second problem of reversed gradient, we need a small $\tau_{init}$ to keep $\mathbf{R}_g$ close to $\mathbf{R}$ at the beginning of training. But when the network is about to converge, we will prefer a $\tau_{converge}$ which can keep $\mathbf{R}_g$ close to $\mathbf{R}_{gt}$ for better convergence. We cannot directly set $\tau_{converge}$ to $\tau_{\mathrm{gt}}$, which is introduced in \ref{sec:rg}, because $\tau_{\mathrm{gt}}$ is not a constant and cannot be used in Riemannian Optimization. However, if we want to tackle the problem of reversed gradient, we must need Riemannian Optimization and $\tau_{init}$. Thus we need a constant approximation of $\tau_{\mathrm{gt}}$ when the angle between $\mathbf{R}$ and $\mathbf{R}_{gt}$ converges to 0. Note that $\tau_{converge}$ can be derived analytically when the loss function is the most widely used L2 loss or geodesic loss(please refer to Appendix \ref{sec:supp2.1} for details), and therefore doesn't need to be tuned. Therefore we propose to increase $\tau$ from a small value $\tau_{init}$, leading to a slow warm-up and, as the training progresses, we gradually increase it to the final $\tau=\tau_{converge}$ by ten uniform steps. This strategy further improves our performance.
\section{Experiments}
\vspace{-1mm}
\label{sec:exp}
\begin{table*}[htbp]
\caption{\textbf{Pose estimation from ModelNet40 point clouds.} Left: a comparison of methods by mean, median, and 5$^\circ$ accuracy of (geodesic) errors after 30k training steps. Mn, Md and Acc are abbreviations of mean, median and 5$^\circ$ accuracy. Right: median test error of \textit{airplane} in different iterations during training. }
\vspace{-4mm}
\begin{minipage}[h]{0.7\columnwidth}
\resizebox{\columnwidth}{!}{
\begin{tabular}{l|ccc|ccc|ccc|ccc|ccc}
\multirow{2}{*}{Methods} &&Airplane&&&Chair&&&Sofa &&&Toilet&&&Bed&\\
\cmidrule{2-16}
& Mn$\downarrow$ & Md$\downarrow$ & Acc$\uparrow$
& Mn$\downarrow$ & Md$\downarrow$ & Acc$\uparrow$
& Mn$\downarrow$ & Md$\downarrow$ & Acc$\uparrow$
& Mn$\downarrow$ & Md$\downarrow$ & Acc$\uparrow$
& Mn$\downarrow$ & Md$\downarrow$ & Acc$\uparrow$ \\
\midrule
Euler & 125 & 131& 0 &13.6&9.0&17&120&125&0&127&133&0&113&122&0 \\
Axis-Angle & 10.8 & 8.2 & 22 &16.4&10.9&9 &24.1&14.6&6&21.9&13.0&9&25.5&11.0&16 \\
Quaternion & 9.7 & 7.6 & 27 & 16.7& 11.4& 12&20.4& 12.7& 10& 16.0& 9.3& 17 &27.8& 11.3& 14\\
6D & 5.5 & 4.7 &54 &9.8& 6.4& 35&14.6& 9.5& 15&9.3& 6.8& 33&24.7& 9.6& 17\\
9D & 4.7 & 3.9 &67 & 7.9& 5.4& 44&15.7& 10.0& 14&10.3& 6.9& 30&22.3& 8.5& 20\\
9D-Inf~(MG-9D) &3.1 & 2.5 &90 & 5.3& 3.7& 69&7.8& 5.0& 50 &4.2& 3.3& 75 &12.9& 4.6& 55 \\
10D & 5.3 & 4.2 & 61 &8.9& 6.0& 38&15.1& 10.3& 13&10.7& 6.5& 35& 23.1& 8.7& 19\\
\midrule
RPMG-Quat & 3.2 & 2.4 & 88 &6.3 &3.7 &67 &8.1& 4.5& 57&4.9& 3.5& 74&13.3& 3.6& 70\\
RPMG-6D & 2.6 & 2.1&\textbf{94} &\textbf{5.0} &\textbf{3.1}& 74 &6.6& 3.6& 70&\textbf{3.8}& 2.9& \textbf{83}& 13.5& 2.7& 81\\
RPMG-9D & \textbf{2.5} & \textbf{2.0}&\textbf{94} & 5.1&\textbf{3.1} & \textbf{76} &\textbf{6.1}& \textbf{3.1}& \textbf{77}&4.3& \textbf{2.7}& \textbf{83}&
\textbf{10.9}& \textbf{2.5}& \textbf{86}\\
RPMG-10D & 2.8 & 2.2 & 93 &5.1 &3.2 &75& 6.5& 3.2& 72&4.9& 2.8& 82&13.5& 2.7& 82\\
\midrule
\end{tabular}
}
\end{minipage}
\begin{minipage}[h]{0.29\columnwidth}
\includegraphics[width=\columnwidth]{figure/Training_curve_full.pdf}
\end{minipage}
\label{tab:rotation_complete}
\end{table*}
We investigate popular rotation representations and find our methods greatly improve the performance in different kinds of tasks. For our regularized projective manifold gradient (\textbf{RPMG}), we apply it in the backpropagation process of Quaternion, 6D, 9D and 10D, without changing the forward pass, leading to three new methods \textbf{RPMG-Quat}, \textbf{RPMG-6D}, \textbf{RPMG-9D} and \textbf{RPMG-10D}.
We compare the following seven baselines: \textbf{Euler angle}, \textbf{axis-angle}, \textbf{Quaternion}, \textbf{6D} \cite{zhou2019continuity}, \textbf{9D} \cite{levinson2020analysis}, \textbf{9D-Inf} \cite{levinson2020analysis} and \textbf{10D} \cite{peretroukhin_so3_2020}.
We adopt three evaluation metrics: mean, median, and 5$^\circ$ accuracy of (geodesic) errors between predicted rotation and ground truth rotation.
For most of our experiments, we set the regularization term $\lambda=0.01$ and increase $\tau$ from $\tau_{init}=0.05$ to $\tau_{converge}=0.25$ by ten uniform steps. We further show and discuss the influence of different choices of these two hyperparameters in our ablation studies.
\vspace{-1mm}
\subsection{3D Object Pose Estimation from Point Clouds}
\vspace{-1mm}
\label{sec:pc_rotation}
\noindent\textbf{Experimental setting.} As in \cite{chen2021equivariant}, we use the complete point clouds generated from the models in ModelNet-40 \cite{wu2015modelnet}. We use the same train/test split as in \cite{chen2021equivariant} and report the results of \textit{airplane}, \textit{chair}, \textit{sofa}, \textit{toilet} and \textit{bed} those five categories because they exhibit less rotational symmetries.
Given one shape point clouds of a specific category, the network learns to predict the 3D rotation of the input point clouds from the predefined canonical view of this category\cite{wang2019normalized}.
We replace the point clouds alignment task used in \cite{zhou2019continuity, levinson2020analysis} (which has almost been solved) by this experiment since it is more challenging and closer to real-world applications (no canonical point clouds is given to the network).
We use a PointNet++ \cite{qi2017pointnetplusplus} network as our backbone, supervised by L2 loss between the predicted rotation matrix $\mathbf{R}$ and the ground truth rotation matrix $\mathbf{R}_{gt}$. To facilitate a fair comparison between multiple methods, we use the same set of hyperparameters in all the experiments. Please see Appendix \ref{sec:imp_detail} for more details.
\noindent\textbf{Analysis of results.}
The results are shown in Table \ref{tab:rotation_complete}. We see a great improvement of our methods in all three rotation representations. In this experiment, one may find \textbf{9D-Inf} also leads to a good performance, which is actually a special case of our method with $\lambda=1$, or in other words, it is MG with $\tau=\tau_{gt}$. Nonetheless, in Table \ref{tab:modelnet_all}, we can observe a larger gap. Also, this simple loss may lead to bad performance when $\mathbf{R}_{gt}$ is unavailable in Section \ref{sec:self-supervised}.
\begin{figure}[t]
\vspace{-4mm}
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/lambda0.pdf}
\end{minipage}\hfill
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[width=1\linewidth]{figure/lambda0.01.pdf}
\caption{\textbf{Average L2 norm of the network raw output $\mathbf{x}$} during training. Left: PMG-4D/6D/9D/10D (w/o reg. $\lambda=0$). Right: RPMG-4D/6D/9D/10D (w/ reg. $\lambda=0.01$)}
\label{fig:length}
\end{minipage}
\vspace{-3mm}
\end{figure}
\noindent\textbf{Ablation study on $\lambda$.} As mentioned in Section \ref{sec:rpmg}, naively using \textbf{PMG} without any regularization, corresponding to setting $\lambda = 0$, will lead to length vanishing; To maintain the length of prediction roughly constant, we only need to add a small $\lambda$. In Figure \ref{fig:length}, We show the length vanishing problem without regularization and stabilized length with a small regularization. In Table \ref{tab:rotation_complete}, we show that the network performs much better when we have a small $\lambda$ (\textbf{RPMG}) than $\lambda=0$ (\textbf{PMG}) or $\lambda=1$ (\textbf{MG}), which deviates too far away from the desired projective manifold gradient. As for the exact value of $\lambda$, our experiments show that our method is robust to the choice of $\lambda$ as long as it is small. Table \ref{tab:ablation} also shows that $\lambda= 0.01, 0.005, 0.05$ all lead to similar performance, thus freeing us from tuning the parameter $\lambda$.
\noindent\textbf{Ablation study on $\tau$.}
For the choices of $\tau$, Table \ref{tab:ablation} shows that our proposed strategy, which ramps up $\tau$ from a small $\tau_\text{init}$ to $\tau_\text{converge}$, works the best. The reason is that: a big $\tau$, when training begins, may cause the problem of reversed gradient discussed in Section \ref{issues}. On the other side, a small $\tau$ at the end of training will slow down the training process and can do harm to convergence.
Note that, the performance is not very sensitive to the exact value,
which means we don't require a parameter tuning for $\tau$ even in general cases. We are good even with simply setting $\tau = \tau_{gt}$.
\begin{table}[htbp]
\centering
\vspace{-2mm}
\caption{\textbf{Ablation study of pose estimation from \textit{airplane} point clouds.} Here MG stands for manifold gradient $\mathbf{x}-\hat{\mathbf{x}}_g$, corresponding to set $\lambda=1$; PMG stands for projective manifold gradient $\mathbf{x}-\mathbf{x}_{gp}$, corresponding to set $\lambda=0$.}
\begin{minipage}[h]{\columnwidth}
\resizebox{\columnwidth}{!}{
\begin{tabular}{p{40pt}ccccc}
\multicolumn{3}{c}{Methods}&\multicolumn{1}{c}{Mean ($^\circ$)$\downarrow$} & \multicolumn{1}{c}{Med ($^\circ$)$\downarrow$} & \multicolumn{1}{c}{5$^\circ$Acc ($\%$)$\uparrow$} \\
\midrule
L2 6D &-&-
&5.50&4.67&54.4\\
\midrule
\multirow{2}{*}{MG-6D} &\multirow{2}{*}{$\lambda=1$}&$\tau_{converge}$
&3.51&2.95&85.2
\\
&&$\tau_{gt}$
&3.19 &2.72 &87.8
\\
\midrule
\multirow{2}{*}{PMG-6D} &\multirow{2}{*}{$\lambda=0$}&$\tau_{converge}$
&57.65&45.22&0.2
\\
&&$\tau_{gt}$
&133&136&0.0
\\
\midrule
\multirow{6}{*}{RPMG-6D} &\multirow{4}{*}{$\lambda=0.01$}&$\tau_{init}$
&2.67&2.18&93.1
\\
&&$\tau_{converge}$
& 2.71&2.14&93.2
\\
&&$\tau_{\mathrm{gt}}$&
3.02 & 2.14 & 89.5
\\
&&$\tau_{init}\rightarrow{}\tau_{converge}$
&2.59&2.07&93.6
\\
\cmidrule{2-6}
&$\lambda=0.05$&\multirow{2}{*}{$\tau_{init}\rightarrow{}\tau_{converge}$}&
2.73 & 2.23 & 92.9
\\
&$\lambda=0.005$ &
&\textbf{2.52}&\textbf{2.05} & \textbf{94.3}
\\
\midrule
\end{tabular}}
\end{minipage}
\label{tab:ablation}
\end{table}
\begin{table*}[htbp]
\caption{\textbf{Pose estimation from ModelNet10 images.} Left: a comparison of methods by mean($^\circ$), median($^\circ$), and 5$^\circ$ accuracy($\%$) of (geodesic) errors after 600k training steps. Mn, Md and Acc are abbreviations of mean, median and 5$^\circ$ accuracy. Right: median test error of \textit{chair} in different iterations during training.}
\begin{minipage}[h]{0.65\columnwidth}
\resizebox{\columnwidth}{!}{
\begin{tabular}{l|ccc|ccc|ccc|ccc}
\multirow{2}{*}{Methods}&& Chair &&&Sofa&&& Toilet&&&Bed&\\
\cmidrule{2-13}
& Mn$\downarrow$ & Md$\downarrow$ & Acc$\uparrow$
& Mn$\downarrow$ & Md$\downarrow$ & Acc$\uparrow$ & Mn$\downarrow$ & Md$\downarrow$ & Acc$\uparrow$ & Mn$\downarrow$ & Md$\downarrow$ & Acc$\uparrow$
\\
\midrule
Euler & 21.5 & 10.9 & 10 & 27.5 & 12.0 & 9 & 14.9 & 8.5 & 19 &27.6 & 9.6 & 17 \\
Axis-Angle & 25.7 & 14.3 & 7 & 30.3 & 14.6 & 6 & 20.3 & 13.0 & 8 & 36.3&16.7&4 \\
Quaternion & 25.8 & 15.0 & 6 & 30.0 & 15.7 & 6 & 20.6&13.0&8&34.1&15.5&5 \\
6D & 19.6 & 9.1 & 19 & 17.5 & 7.3 & 27 & 10.9&6.2&37&32.3&11.7&11 \\
9D & 17.5 & 8.3 & 23 & 19.8 & 7.6 & 25 & 11.8&6.5&34&30.4&11.1&13 \\
9D-Inf & 12.1 & 5.1 & 49& 12.5 & 3.5 & 70 & 7.6&3.7&67&22.5&4.5&56 \\
10D & 18.4 & 9.0 & 20 & 20.9 & 8.7 & 20 & 11.5& 5.9&39& 29.9&11.5&11 \\
\midrule
RPMG-Quat & 13.0 & 5.9 & 40 & 13.0 & 3.6 & 67 & 8.6&4.2&61& 23.2&4.9&51 \\
RPMG-6D & 12.9 & 4.7 & 53 & 11.5 & 2.8 & 77 & 7.8&3.4&71&20.3&3.6&67 \\
RPMG-9D & \textbf{11.9} & \textbf{4.4} & \textbf{58} & \textbf{10.5} & \textbf{2.4} & \textbf{82} & 7.5&3.2&75&20.0&\textbf{2.9}&\textbf{76} \\
RPMG-10D & 12.8& 4.5& 55 & 11.2&\textbf{2.4}&\textbf{82}& \textbf{7.2}&\textbf{3.0}&\textbf{76}&\textbf{19.2}&\textbf{2.9}&75\\
\midrule
\end{tabular}}
\end{minipage}
\begin{minipage}[h]{0.34\columnwidth}
\includegraphics[width=\columnwidth]{figure/img_training_curve_chair.pdf}
\end{minipage}
\label{tab:modelnet_all}
\end{table*}
\vspace{-3mm}
\subsection{3D Rotation Estimation from ModelNet Images}
\label{sec:image}
In this experiment, we follow the setting in \cite{levinson2020analysis} to estimate poses from 2D images. Images are rendered from ModelNet-10 \cite{wu2015modelnet} objects from arbitrary viewpoints \cite{liao19sphere}. A MobileNet \cite{mobilenet} is used to extract image features and three MLPs to regress rotations.
We use the same categories as in Experiment \ref{sec:pc_rotation} except \textit{airplane}, since ModelNet-10 doesn't have this category.
We didn't quote the numbers from \cite{levinson2020analysis} since we conduct all the experiments using the same set of hyperparameters to ensure a fair comparison. Please see Appendix \ref{sec:img_lr} for more details.
The results are shown in Table \ref{tab:modelnet_all}. Our RPMG layer boosts the performance of all three representations significantly. See the curves with the same color for comparison.
\subsection{Rotation Estimation without Supervision}
\label{sec:wo_gt}
\noindent\textbf{Self-supervised instance-level rotation estimation from point clouds.}
\label{sec:self-supervised}
For one complete chair instance $Z$, given a complete observation $X$, we estimate its pose $\mathbf{R}$. We then use Chamfer distance between $Z$ and $\mathbf{R}^{-1}X$ as a self-supervised loss. The network structure and training settings are all the same as Experiment \ref{sec:pc_rotation}, except here we use $\tau=2$. See Appendix \ref{sec:flow loss} for how to find a suitable $\tau$.
The interesting thing here is that vanilla \textbf{9D-Inf} fails while our methods still perform very well. We think that this is because the Chamfer distance loss will greatly enlarge the effect of the noisy part (which is introduced by $\lambda$) in gradient, leading to a very bad performance.
\begin{table}[htbp]
\caption{\textbf{Self-supervised Instance-Level Rotation Estimation from Point Clouds.} We report mean, median and 3$^\circ$ accuracy of (geodesic) errors after 30K iterations.}
\resizebox{0.6\columnwidth}{!}{
\begin{tabular}{lccc}
Methods& \multicolumn{1}{c}{Mean ($^\circ$)$\downarrow$} & \multicolumn{1}{c}{Med ($^\circ$)$\downarrow$} & \multicolumn{1}{c}{3$^\circ$Acc ($\%$)$\uparrow$} \\
\midrule
Euler & 131.9 & 139.1 & 0.0 \\
Axis-Angle & 4.5 & 3.8 & 34.5 \\
Quaternion & 4.3 & 3.5 & 37.5 \\
6D & 55.1 & 6.7 & 20.0 \\
9D & 1.8 & 1.6 & 88.0 \\
9D-Inf & 118.2 & 119.5 &0.0\\
10D & 1.6 & 1.5 & 91.0 \\
\midrule
RPMG-Quat & 3.5 & 2.4 & 70.0 \\
RPMG-6D & 15.0 & 2.9 & 55.0 \\
RPMG-9D & \textbf{1.3} & \textbf{1.2} & \textbf{97.5} \\
RPMG-10D & 1.5 & 1.4 & 97.0
\end{tabular}}
\end{table}
\vspace{-3mm}
\subsection{Regression on Other Non-Euclidean Manifolds}
\label{sec:other manifolds}
In addition to SO(3), our method can also be applied for regression on other non-Euclidean manifolds as long as the target manifold meets some conditions: 1) the manifold should support Riemannian optimization. 2) the inverse projection $\pi^{-1}$ should be calculable, although it doesn't need to be mathematically complete. Here we show the experiment of \textit{Sphere manifold }$\mathcal{S}^2$.
\noindent\textbf{Unit vector regression.}
For rotational symmetric categories (e.g., \textit{bottle}), the pose of an object is ambiguous. We'd rather regress a unit vector for each object indicating the \textit{up} direction of it.
We use the ModelNet-40\cite{wu2015modelnet} \textit{bottle} point cloud dataset.
The network architecture is the same as in Experiment \ref{sec:pc_rotation} except the dimension of output is 3.
L2-loss-w/-norm computes L2 loss between the normalized predictions and the ground truth. L2-loss-w/o-norm computes L2 loss between the raw predictions and the ground truth, similar to $\lambda=1$ and $\tau=\tau_{gt}$. For MG-3D, PMG-3D and RPMG-3D, We increase $\tau$ from 0.1 to 0.5 since here $\tau_{converge}=0.5$ (please see Appendix \ref{sec:supp3} for the derivation).
The results are shown in Table \ref{tab:unit}.
MG-3D performs on par with L2-loss-w/o-norm, and PMG-3D leads to a large error since the length vanishing problem similar to Figure \ref{fig:length}. RPMG-3D outperforms all the baselines and variants.
\begin{table}[htbp]
\centering
\vspace{-1mm}
\caption{\textbf{Unit vector estimation from ModelNet bottle point clouds.} We report mean, median, and 1$^\circ$ accuracy of (geodesic) errors after 30K iterations.}
\vspace{-1mm}
\begin{minipage}[h]{0.7\columnwidth}
\resizebox{\columnwidth}{!}{
\begin{tabular}{lccc}
Methods & \multicolumn{1}{l}{Mean ($^\circ$)$\downarrow$} & \multicolumn{1}{l}{Med ($^\circ$)$\downarrow$} & \multicolumn{1}{l}{1$^\circ$Acc ($\%$)$\uparrow$} \\
\midrule
L2 loss w/ norm & 8.73 & 2.71 & 0.0 \\
L2 loss w/o norm & 5.71 & 1.10 & 37.4 \\
\midrule
MG-3D ($\lambda$=1) & 5.37 & 1.20 & 22.2 \\
PMG-3D ($\lambda$=0) & 21.96 & 14.79 & 0.0 \\
\midrule
RPMG-3D ($\lambda$=0.01) & \textbf{4.69} & \textbf{0.76} & \textbf{72.7} \\
\end{tabular}}
\end{minipage}
\label{tab:unit}%
\end{table}%
\vspace{-2mm}
\section{Conclusion and Future Work}
\label{sec:conclusion}
Our work tackles the problem of designing a gradient layer to facilitate the learning of rotation regression. Our extensive experiments have demonstrated the effectiveness of our method coupled with different rotation representations in diverse tasks dealing with rotation estimation.
The limitation of our methods mainly lies in two fronts: 1) we introduce two new hyperparameters, \textit{i.e.}, $\tau$ and $\lambda$, though our performance is not sensitive to them, as long as they are in a reasonable range; 2) as discussed in Sec \ref{sec:other manifolds}, our method can only be applied to manifolds with certain constraints. We leave how to relax those to future works.
\section{More on Riemannian Geometry}
\label{sec:supp1}
In this part, we supplement the definitions in Section \ref{sec:prelim} to allow for a slightly more rigorous specification of the exponential map for interested readers.
We denote the union of all tangent spaces as the \emph{tangent bundle}: $\mathcal{T}\mathcal{M} = \cup_{\mathbf{x}\in\mathcal{M}}\mathcal{T}_{\x}\Man$.
Riemannian metric $\mathbf{G}_\mathbf{x}$ induces a norm $\|\u\|_\mathbf{x}\,,\forall \u\in\mathcal{T}_{\x}\Man$ locally defining the geometry of the manifold and allows for computing the \emph{length} of any curve $\gamma : [0,1] \rightarrow \mathcal{M}$, with $\gamma(0) = \mathbf{x}$ and $\gamma(1) = \mathbf{y}$ as the integral of its speed: $\ell(\gamma) = \int_{0}^1 \|\dot{\gamma}(t)\|_{\gamma(t)}dt$. The notion of length leads to a natural notion of distance by taking the infimum over all lengths of such curves, giving the \emph{Riemannian distance} on $\mathcal{M}$, $d(\mathbf{x},\mathbf{y})=\inf_{\gamma}\ell(\gamma)$. The constant speed \emph{length minimizing} curve $\gamma$ is called a \emph{geodesic} on $\mathcal{M}$.
By the celebrated Picard Lindelöf theorem~\cite{coddington1955theory}, given any $(\mathbf{x},\v)\in\mathcal{T}\mathcal{M}$, there exists a unique \emph{maximal}\footnote{\emph{maximal} refers to the fact that the curve is as long as possible.} geodesic $\curve_\v$ such that $\curve_\v(0)=\mathbf{x}$ and $\dcurve_\v(0)=\v$. Hence, we can define a unique diffeomorphism or \emph{exponential map}, sending $\mathbf{x}$ to the endpoint of the geodesic: $\exp_\mathbf{x}(\v)=\curve_\v(1)$. We will refer to the well-defined, smooth inverse of this map as the \emph{logaritmic map}: $\log_{\mathbf{x}}{\mathbf{y}}\triangleq \exp^{-1}_\mathbf{x}(\v)$. Note that the geodesic is not the only way to move away from $\mathbf{x}$ in the direction of $\v$ on $\mathcal{M}$. In fact, any continuously differentiable, smooth map $R_{\x}:\mathcal{T}_{\x}\Man\mapsto \mathcal{M}$ whose directional derivative along $\v$ is identity, \ie $\mathrm{D} R_{\x}(\mathbf{0})[\v]=\v$ and $R_{\x}(\mathbf{0}) = \mathbf{x}$ allows for moving on the manifold in a given direction $\v$. Such $R_{\x}$, called \emph{retraction}, constitutes the basic building block of any on-manifold optimizer as we use in the main paper. In addition to those we also speak of a \emph{manifold projector} $\pi:\mathcal{X}\mapsto\mathcal{M}$
is available for the manifolds we consider in this paper. Note that, most of these definitions directly generalize to matrix manifolds such as Stiefel or Grassmann~\cite{absil2009optimization}.
\section{Projective Manifold Gradient on $SO(3)$}
\subsection{Details of Riemannian Optimization on $SO(3)$}
\label{sec:supp2.1}
\paragraph{Riemannian gradient on $SO(3)$.}
Since we mainly focus on the $\mathrm{SO}(3)$ manifold in this paper, we will further show the specific expression of some related concepts of $\mathrm{SO}(3)$ below.
Firstly, $\mathrm{SO}(3)$ is defined as a matrix subgroup of the general linear group $GL(3)$:
\begin{equation}
\mathrm{SO}(3)=\{\mathbf{R}\in\mathbb{R}^{3\times3}:\mathbf{R}^\top\mathbf{R}=\mathbf{I}, \det(\mathbf{R})=1\}.
\end{equation}
The tangent space of a rotation matrix in $\mathrm{SO}(3)$ is isomorphic to $\mathbb{R}^3$ making $\mathrm{SO}(3)$ an embedded submanifold of the ambient Eucldiean space $\mathcal{X}$. Hence, $\mathrm{SO}(3)$ \emph{inherits} the metric or the inner product of its embedding space, $\mathcal{X}$.
Since $\mathrm{SO}(3)$ is also a Lie group, elements of the tangent space $\boldsymbol{\phi}^\wedge \in \mathcal{T}_{\Id}\Man$ can be uniquely mapped to the manifold $\mathcal{M}$ through the exponential map:
\begin{equation}\label{eq:expso3}
\exp_{\mathbf{I}}(\boldsymbol{\phi}^\wedge) = \mathbf{I} + \boldsymbol{\phi}^\wedge + \frac{1}{2!}(\boldsymbol{\phi}^\wedge)^2+\frac{1}{3!}(\boldsymbol{\phi}^\wedge)^3+... \quad ,
\end{equation}
where $\mathbf{I}\in\mathrm{SO}(3)$ is the identity matrix and $^\wedge$ is a skew-symmetric operator $^\wedge:\mathbb{R}^3\to\mathcal{T}_\mathbf{I}\mathcal{M}$ as
\begin{equation}
\boldsymbol{\phi}^\wedge=\begin{pmatrix}
0 & -\phi_z & \phi_y \\ \phi_z & 0 & -\phi_x \\ -\phi_y & \phi_x & 0
\end{pmatrix}
\end{equation}
Due to the nature of the Lie group, we can expand the formula in~\cref{eq:expso3} from the tangent space of the identity, $\mathcal{T}_{\Id}\Man$, to $\mathcal{T}_{\mathbf{R}}\mathcal{M}$ by simply multiplying by an $\mathbf{R}$:
\begin{equation}
\exp_\mathbf{R}(\boldsymbol{\phi}^\wedge)
=\mathbf{R}\left(\sum_{n=0}^{\infty} (\frac{1}{n!}(\boldsymbol{\phi}^\wedge)^n)\right)
\end{equation}
If the vector $\boldsymbol{\phi}$ is rewritten in terms of a unit vector $\boldsymbol{\omega}$ and a magnitude $\theta$, the exponential map can further be simplified as
\begin{equation}
\mathrm{Exp}_\mathbf{R}(\boldsymbol{\phi}) = \mathbf{R}(\mathbf{I} + \sin\theta~\boldsymbol{\omega}^\wedge+(1-\cos\theta)(\boldsymbol{\omega}^\wedge)^2)
\end{equation}
\noindent which is well known as the Rodrigues formula~\cite{rodrigues1840lois}.
Following \cite{taylor1994minimization}, we have
\small
\begin{equation}
\left.\frac{\partial}{\partial \phi_x}\mathrm{Exp}_\mathbf{R}(\boldsymbol{\phi})\right|_{\boldsymbol{\phi}=\mathbf{0}}=\left.\mathbf{R}\left(\cos\theta~\frac{\partial\theta}{\partial\phi_x}\boldsymbol{\omega}^\wedge\right)\right|_{\boldsymbol{\phi}=\mathbf{0}}=\mathbf{R} \mathbf{x}^\wedge
\end{equation}
\normalsize
where $\mathbf{x}=(1,0,0)\in\mathbb{R}^3$. For $\phi_y$ and $\phi_z$, there are the similar expressions of the gradient. Finally we can have
\begin{equation}
\grad~\mathcal{L} f(\mathbf{R})
=\left(\left.\frac{\partial f(\mathbf{R})}{\partial \mathbf{R}}\frac{\partial}{\partial \boldsymbol{\phi}} \mathrm{Exp}_\mathbf{R}(\boldsymbol{\phi})\right|_{\boldsymbol{\phi}=\mathbf{0}}\right)^\wedge
\end{equation}
\paragraph{Riemannian gradient descent on $\mathrm{SO}(3)$.}
We are now ready to state the Riemannian optimization in the main paper in terms of the exponential map:
\begin{equation}
\mathbf{R}_{k+1}=\mathrm{Exp}_{\mathbf{R}_k}(-\tau_k\nabla \boldsymbol{\phi}).
\end{equation}
Note that if we consider the most commonly used L2 loss $f(\mathbf{R})=\|\mathbf{R}-\mathbf{R}_{\mathrm{gt}}\|_F^2$ , where
\scriptsize
$$ \mathbf{R}=\begin{pmatrix}
a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3
\end{pmatrix} \in \mathrm{SO}(3), \text{\quad} \mathbf{R}_{\mathrm{gt}}=\begin{pmatrix}
x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3
\end{pmatrix} \in \mathrm{SO}(3),$$
\normalsize
we can get an analytical expression of $\nabla \boldsymbol{\phi}=(\nabla \phi_x,\nabla \phi_y,\nabla \phi_z)$ as follows:
\scriptsize
\begin{align}
\label{eq:wgrad}
\nabla\boldsymbol{\phi}_x &=\frac{\partial f(\mathbf{R})}{\partial \mathbf{R}} *\mathbf{R} \mathbf{x}^\wedge \notag\\
&= 2\left|\left|\begin{pmatrix}
a_1-x_1 & b_1-y_1 & c_1-z_1 \\ a_2-x_2 & b_2-y_2 & c_2-z_2 \\ a_3-x_3 & b_3-y_3 & c_3-z_3
\end{pmatrix}
\begin{pmatrix}
0 & c_1 & -b_1 \\ 0 & c_2 & -b_2 \\ 0 & c_3 & -b_3
\end{pmatrix}\right|\right|_1 \notag\\
&= 2*\sum^3_{i=1}(b_i*z_i-c_i*y_i)
\end{align}
\normalsize
Similarly, we have $ \nabla\phi_y=2*\sum^3_{i=1}(c_i*x_i-a_i*z_i)$ and $\nabla\phi_z=2*\sum^3_{i=1}(a_i*y_i-b_i*x_i)$.
\paragraph{$\tau_{converge}$ in ablation study.}
We have mentioned in Section \ref{sec:rpmg} that $\tau$ should be small at the beginning of training and be large when converging. This is because a small $\tau$ can yield $\mathbf{R}_g$ closer to $\mathbf{R}$ and greatly alleviate the reverse problem at the beginning stage of training discussed in Section \ref{sec:rpmg}. Later in training, a large $\tau$ can help us converge better. The initial $\tau$ will not influence the final results too much, and we just need to choose a reasonable value. But the final $\tau$ matters.
Right before convergence, our ideal choice for the final $\tau$ would be $\tau_{\mathrm{gt}}$.
Given that the value of $\tau_{\mathrm{gt}}$ will change according to the geodesic distance between $\mathbf{R}$ and $\mathbf{R}_{\mathrm{gt}}$, we instead choose to find a suitable constant value to act like $\tau_{\mathrm{gt}}$ when converging, which we denotes as $\tau_{converge}$.
\begin{lemma}
The final value of $\tau_{converge}$ satisfies:
\small
\begin{equation}\mathbf{R}_{\mathrm{gt}}=\lim_{<\mathbf{R},\mathbf{R}_{\mathrm{gt}}>\to0} R_\mathbf{R}(-\tau_{converge}~\grad~\mathcal{L}(f(\mathbf{R})))
\end{equation}
\normalsize
where $<\mathbf{R},\mathbf{R}_{\mathrm{gt}}>$ represents the angle between $\mathbf{R}$ and $\mathbf{R}_{\mathrm{gt}}$.
\end{lemma}
\begin{proof}
Considering the symmetry, without loss of generality, we assume that $\mathbf{R}=\mathbf{I}$, which will simplify the derivation. Based upon the conclusion in~\cref{eq:wgrad}, when we use L2 loss, we have $\nabla \boldsymbol{\phi}=(2*(z_2-y_3),2*(x_3-z_1),2*(y_1-x_2))$ and $\grad~\mathcal{L} f(\mathbf{R})=(\nabla \boldsymbol{\phi})^\wedge = 2(\mathbf{R}_{\mathrm{gt}}^\top-\mathbf{R}_{\mathrm{gt}})$. Taking the manifold logarithm of both sides, we get:
\small
\begin{equation}
\log_\mathbf{R}(\mathbf{R}_{\mathrm{gt}})=\lim_{<\mathbf{R},\mathbf{R}_{\mathrm{gt}}>\to0}-\tau_{converge} ~\grad~\mathcal{L} f(\mathbf{R})
\end{equation}
\normalsize
The solution for $\tau_{converge}$ can then be derived as follows:
\small
\begin{align}
\label{eq:tau_safe}
\tau_{converge}&=\lim_{<\mathbf{R},\mathbf{R}_{\mathrm{gt}}>\to0}-\frac{\log_\mathbf{R}(\mathbf{R}_{\mathrm{gt}})}{\grad~\mathcal{L} f(\mathbf{R})}
\notag \\ &=\lim_{\theta\to0}-\frac{(\boldsymbol{\phi}_{\mathrm{gt}})^\wedge}{2(\mathbf{R}_{\mathrm{gt}}^\top-\mathbf{R}_{\mathrm{gt}})} \notag \\
&=\lim_{\theta\to0}-\frac{(\boldsymbol{\phi}_{\mathrm{gt}})^\wedge}{2\sin\theta(((\boldsymbol{\omega}_{\mathrm{gt}})^\wedge)^\top-(\boldsymbol{\omega}_{\mathrm{gt}})^\wedge)} \notag \\
&=\lim_{\theta\to0}\frac{\theta}{4\sin\theta} \notag \\
&=\frac{1}{4}
\end{align}
\normalsize
where
\small
$(\boldsymbol{\phi}_{\mathrm{gt}})^\wedge=\log_\mathbf{I}(\mathbf{R}_{\mathrm{gt}})=\theta(\boldsymbol{\omega}_{\mathrm{gt}})^\wedge$, $\theta=<\mathbf{I},\mathbf{R}_{\mathrm{gt}}>$
\normalsize
\end{proof}
Note that though $\tau_{converge}=\frac{1}{4}$ is only true for the L2 loss, we can solve $\tau_{converge}$ for other frequently used loss formats, \textit{e.g.}, geodesic loss~\cite{peretroukhin_so3_2020}. If we use geodesic loss $\theta^2$, it can be computed that $\tau_{converge}=\frac{1}{2}$. We leave the detailed derivation to the interested readers.
\subsection{Derivations of Inverse Projection}
\label{sec:inverse_proj}
For different rotation representations, we follow the same process to find its inverse projection: we first find the inverse image space $\pi^{-1}(\mathbf{x}_g)$, then project $\mathbf{x}$ to this space resulting in $\mathbf{x}_{gp}$, and finally get our (regularized) projective manifold gradient.
\paragraph{Quaternion}
We need to solve
\begin{equation}
\mathbf{x}_{gp} = \underset{\mathbf{x}_g\in\pi_q^{-1}(\hat{\mathbf{x}}_g)}{\text{argmin}}~\|\mathbf{x}_g-\mathbf{x}\|_2^2,
\end{equation}
where $\mathbf{x}$ is the raw output of our network in \textit{ambient space} $\mathbb{R}^4$, $\hat{\mathbf{x}}_g$ is the next goal in \textit{representation manifold} $\mathcal{S}^3$, and $\mathbf{x}_g$ is the variable to optimize in \textit{ambient space} $\mathbb{R}^4$. Recall $\pi^{-1}_{q}(\hat{\mathbf{x}}_g) = \{\mathbf{x}~|~\mathbf{x} = k\hat{\mathbf{x}}_g, k\in \mathbb{R}~\text{and}~k>0\}$, and we can have
\begin{equation}
\|\mathbf{x}-\mathbf{x}_g\|_2^2=\mathbf{x}^2-2k\mathbf{x}\cdot\hat{\mathbf{x}}_g+k^2\hat{\mathbf{x}}_g^2
\end{equation}
Without considering the condition of $k>0$, We can see when $k=\frac{\mathbf{x}\cdot\hat{\mathbf{x}}_g}{\hat{\mathbf{x}}_g^2}=\mathbf{x}\cdot\hat{\mathbf{x}}_g$ the target formula reaches minimum. Note that when using a small $\tau$, the angle between $\hat{\mathbf{x}}_g$ and $\mathbf{x}$ is always very small, which means the condition of $k=\mathbf{x}\cdot\hat{\mathbf{x}}_g>0$ can be satisfied naturally. For the sake of simplicity and consistency of gradient, we ignore the limitation of $k$ no matter what value $\tau$ takes. Therefore, the inverse projection is $\mathbf{x}_{gp}=(\mathbf{x}\cdot\hat{\mathbf{x}}_g)\hat{\mathbf{x}}_g$.
\paragraph{6D representation}
We need to solve
\scriptsize
\begin{equation}
[\u_{gp},\v_{gp}] = \underset{[\u_g,\v_g]\in\pi_{6D}^{-1}([\hat{\u}_g,\hat{\v}_g])}{\text{argmin}}~(\|\u_g-\u\|_2^2+\|\v_g-\v\|_2^2)
\end{equation}
\normalsize
where $[\u,\v]$ is the raw output of network in \textit{ambient space} $\mathbb{R}^6$, $[\hat{\u}_g,\hat{\v}_g]$ is the next goal in \textit{representation manifold} $\mathcal{V}_2(\mathbb{R}^3)$ and $[\u_g,\v_g]$ is the variable to optimize in \textit{ambient space} $\mathbb{R}^6$. Recall $\pi^{-1}_{6D}([\hat{\u}_g,\hat{\v}_g]) = \{[k_1\hat{\u}_g, k_2\hat{\u}_g+k_3\hat{\v}_g]~|~k_1,k_2,k_3 \in \mathbb{R}~\text{and}~k_1, k_3>0\}$. We can see that $\u_g$ and $\v_g$ are independent, and $\u_g$ is similar to the situation of quaternion. So we only need to consider the part of $\v_g$ as below:
\begin{equation}
\|\v-\v_g\|_2^2=\v^2+k_2^2\hat{\u}_g^2+k_3^2\hat{\v}_g^2-2k_2\v\cdot\hat{\u}_g-2k_3\v\cdot\hat{\v}_g
\end{equation}
For the similar reason as quaternion, we ignore the condition of $k_3>0$ and we can see when $k_2=\v\cdot\hat{\u}_g$ and $k_3=\v\cdot\hat{\v}_g$, the target formula reaches minimum. Therefore, the inverse projection is $[\u_{gp},\v_{gp}]=[(\u\cdot\hat{\u}_g)\hat{\u}_g, (\v\cdot\hat{\u}_g)\hat{\u}_g+(\v\cdot\hat{\v}_g)\hat{\v}_g]$
\paragraph{9D representation}
For this representation, obtaining the inverse image $\pi_{9D}^{-1}$ is not so obvious. Recall $\pi_{9D}(\mathbf{x})=\mathbf{U}\Sigma'\mathbf{V}^\top$, where $\mathbf{U}$ and $\mathbf{V}$ are left and right singular vectors of $\mathbf{x}$ decomposed by SVD expressed as $\mathbf{x}=\mathbf{U}\Sigma \mathbf{V}^\top$, and $\Sigma'=\mathrm{diag}(1,1,\det(\mathbf{U}\mathbf{V}^\top))$.
\begin{lemma}
The inverse image $\pi^{-1}_{9D}(\mathbf{R}_g) = \{\mathbf{S}\mathbf{R}_g~|~\mathbf{S}=\mathbf{S}^\top\}$ satisfies that $\{\mathbf{x}_g~|~\pi_{9D}(\mathbf{x}_g)=\mathbf{R}_g\}\subset \pi^{-1}_{9D}(\mathbf{R}_g)$.
\end{lemma}
\begin{proof}
To find a suitable $\pi_{9D}^{-1}$, the most straightforward way is to only change the singular values $\Sigma_g=\mathrm{diag}(\lambda_0,\lambda_1,\lambda_2)$, where $\lambda_0,\lambda_1,\lambda_2$ can be arbitrary scalars, and recompose the $\mathbf{x}_g=\mathbf{U}\Sigma_g \mathbf{V}^\top$.
However, we argue that this simple method will fail to capture the entire set of $\{\mathbf{x}_g~|~\pi_{9D}(\mathbf{x}_g)=\mathbf{R}_g\}$, because different $\mathbf{U}'$ and $\mathbf{V}'$ can yield the same rotation $\mathbf{R}_g$. In fact, $\mathbf{U}_g$ can be arbitrary if $\mathbf{x}_g=\mathbf{U}_g\Sigma_g \mathbf{V}_g^\top$ and $\mathbf{U}_g\Sigma_g' \mathbf{V}_g^\top=\mathbf{R}_g$. Assuming $\mathbf{R}_g$ is known, we can replace $\mathbf{V}_g^\top$ by $\mathbf{R}_g$ and express $\mathbf{x}_g$ in a different way: $\mathbf{x}_g=\mathbf{U}_g\Sigma_g\frac{1}{\Sigma_g'}\mathbf{U}_g^{-1}\mathbf{R}_g$. Notice that $\mathbf{U}_g\Sigma_g\frac{1}{\Sigma_g'}\mathbf{U}_g^{-1}$ must be a symmetry matrix since $\mathbf{U}_g$ is an orthogonal matrix. Therefore, $\{\mathbf{x}_g~|~\pi_{9D}(\mathbf{x}_g)=\mathbf{R}_g\}\subseteq \pi^{-1}_{9D}(\mathbf{R}_g) = \{\mathbf{S}\mathbf{R}_g~|~\mathbf{S}=\mathbf{S}^\top\}$.
Note that such $\mathbf{x}_g\in\pi^{-1}_{9D}(\mathbf{R}_g)$ can't ensure $\pi_{9D}(\mathbf{x}_g)=\mathbf{R}_g$, because in the implementation of SVD, the order and the sign of three singular values are constrained, which is not taken into consideration. Therefore, $\{\mathbf{x}_g~|~\pi_{9D}(\mathbf{x}_g)=\mathbf{R}_g\}\neq \pi^{-1}_{9D}(\mathbf{R}_g)$.
\end{proof}
Then we need to solve
\begin{equation}
\mathbf{x}_{gp} = \underset{\mathbf{x}_g\in\pi_{9D}^{-1}(\mathbf{R}_g)}{\text{argmin}}~\|\mathbf{x}_g-\mathbf{x}\|_2^2
\end{equation}
where $\mathbf{x}$ is the raw output of our network in \textit{ambient space} $\mathbb{R}^{3\times3}$, $\hat{\mathbf{x}}_g$ is the next goal in \textit{representation manifold} $\mathrm{SO}(3)$, and $\mathbf{x}_g$ is the variable to optimize in \textit{ambient space} $\mathbb{R}^{3\times3}$.
We can further transform the objective function as below:
\begin{equation}
\|\mathbf{x}_g-\mathbf{x}\|_2^2 =\|\mathbf{S}\mathbf{R}_g-\mathbf{x}\|_2^2
=\|\mathbf{S}-\mathbf{x}\mathbf{R}_g^\top\|_2^2
\end{equation}
Now we can easily find
when $\mathbf{S}$ equals to the symmetry part of $\mathbf{x}\mathbf{R}_g^\top$, the target formula reaches minimum. Therefore, the inverse projection admits a simple form $\mathbf{x}_{gp}=\frac{\mathbf{x}\mathbf{R}_g^\top+\mathbf{R}_g \mathbf{x}^\top}{2}\mathbf{R}_g$.
\paragraph{10D representation} Recall the \textit{manifold mapping} $\pi_{10D}: \mathbb{R}^{10}~\rightarrow~\mathcal{S}^3, \pi_{10D}(\mathbf{x})=\underset{\mathbf{q}\in\mathcal{S}^3}{\min}~\mathbf{q}^\top\mathbf{A}(
\mathbf{x})\mathbf{q}$, in which
\begin{equation}\label{eq:10d}
\vspace{1mm}
\mathbf{A}(\boldsymbol{\theta})~=~
\begin{pmatrix}
\theta_1 & \theta_2 & \theta_3 & \theta_4 \\
\theta_2 & \theta_5 & \theta_6 & \theta_7 \\
\theta_3 & \theta_6 & \theta_8 & \theta_9 \\
\theta_4 & \theta_7 & \theta_9 & \theta_{10} \\
\end{pmatrix}.
\vspace{1mm}
\end{equation}
We need to solve
\begin{equation}
\mathbf{x}_{gp} = \underset{\mathbf{A}(\mathbf{x}_g)\mathbf{q}_g=\lambda\mathbf{q}_g}{\arg\min}~\|\mathbf{x}_g-\mathbf{x}\|_2^2,
\end{equation}
where $\mathbf{x}$ is the raw output of our network in \textit{ambient space} $\mathbb{R}^{10}$, $\mathbf{q}_g$ is the next goal in \textit{representation manifold} $\mathcal{S}^3$, and $\mathbf{x}_g$ is the variable to optimize in \textit{ambient space} $\mathbb{R}^{10}$. Note that $\lambda$ is also a variable to optimize. For the similar reason as before, for the sake of simplicity and consistency of analytical solution, here we also need to relax the constraint that $\lambda$ should be the smallest eigenvalue of $\mathbf{A}(\mathbf{x}_g)$.
To solve Eq. \ref{eq:10d}, we start from rewriting $\mathbf{A}(\mathbf{x}_g)\mathbf{q}_g=\lambda\mathbf{q}_g$ as
\begin{equation}
\mathbf{M}\Delta\mathbf{x}=\lambda\mathbf{q}_g-\mathbf{A}(\mathbf{x})\mathbf{q}_g,
\end{equation}
where $\Delta\mathbf{x}=\mathbf{x}_g-\mathbf{x}$ and
\scriptsize
\begin{equation}
\mathbf{M}=\begin{pmatrix}
q_1&q_2&q_3&q_4&0&0&0&0&0&0\\
0&q_1&0&0&q_2&q_3&q_4&0&0&0\\
0&0&q_1&0&0&q_2&0&q_3&q_4&0\\
0&0&0&q_1&0&0&q_2&0&q_3&q_4\\
\end{pmatrix}
\end{equation}
\normalsize
where $\mathbf{q}_g=(q_1,q_2,q_3,q_4)^\top$.
For simplicity, we denote $\lambda\mathbf{q}_g-\mathbf{A}(\mathbf{x})\mathbf{q}_g$ as $\bb$.
Once we have finished the above steps for preparation, we solve $\lambda$ and $\Delta\mathbf{x}$ for the minimal problem by two steps as below. First, we assume $\lambda$ is known and the problem becomes that given $\mathbf{M}$ and $\bb$, we need to find the best $\Delta\mathbf{x}$ to minimize $\|\Delta\mathbf{x}\|^2_2$ with the constraint $\mathbf{M}\Delta\mathbf{x}=\bb$. This is a typical quadratic optimization problem with linear equality constraints, and the analytical solution satisfies
\begin{equation}
\begin{pmatrix}\label{eq:kkt}
\mathbf{I} & \mathbf{M}^\top \\
\mathbf{M} & \mathbf{0}
\end{pmatrix}~
\begin{pmatrix}
\Delta\mathbf{x} \\
\mathbf{v}
\end{pmatrix} = \begin{pmatrix}
\mathbf{0} \\
\mathbf{b}
\end{pmatrix}
\end{equation}
\noindent where $\mathbf{v}$ is a set of Lagrange multipliers which come out of the solution alongside $\Delta\mathbf{x}$, and $\begin{pmatrix}\mathbf{I} & \mathbf{M}^\top \\ \mathbf{M} & \mathbf{0}\end{pmatrix}$ is called KKT matrix. Since this matrix has full rank almost everywhere, we can multiple the inverse of this KKT matrix in both sides of Eq. \ref{eq:kkt} and lead to the solution of $\Delta\mathbf{x}$ as below:
\begin{equation}
\begin{pmatrix}
\Delta\mathbf{x} \\
\mathbf{v}
\end{pmatrix} = \begin{pmatrix}
\mathbf{I} & \mathbf{M}^\top \\
\mathbf{M} & \mathbf{0}
\end{pmatrix}^{-1}~\begin{pmatrix}
\mathbf{0} \\
\mathbf{b}
\end{pmatrix}
\end{equation}
\noindent Recall that $\bb=\lambda\mathbf{q}_g-\mathbf{A}(\mathbf{x})\mathbf{q}_g$, therefore so far we have had the solution of $\Delta\mathbf{x}$ respect to each $\lambda$:
\begin{equation}\label{eq:kkt_solution}
\Delta\mathbf{x}=\begin{pmatrix}
\Delta\mathbf{x} \\
\mathbf{v}
\end{pmatrix}_{0:10} = \mathbf{K} (\lambda\mathbf{q}_g-\mathbf{A}(\mathbf{x})\mathbf{q}_g)=\lambda\mathbf{S}-\mathbf{T}
\end{equation}
in which $\mathbf{K}$ is the upper right part of the inverse of the KKT matrix $\mathbf{K}=\left[\begin{pmatrix}\mathbf{I} & \mathbf{M}^\top \\ \mathbf{M} & \mathbf{0}\end{pmatrix}^{-1}\right]_{10:14, 0:10}$, $\mathbf{S}=\mathbf{K}\mathbf{q}_g$ and $\mathbf{T}=\mathbf{K}\mathbf{A}(\mathbf{x})\mathbf{q}_g$.
Next, we need to optimize $\lambda$ to minimize our objective function $\|\Delta\mathbf{x}\|^2_2$. In fact, using the results of Eq. \ref{eq:kkt_solution}, $\|\Delta\mathbf{x}\|^2_2$ becomes a quadratic functions on $\lambda$, thus we can simply get the final analytical solution of $\lambda$ and $\mathbf{x}_{gp}$:
\begin{equation}
\left\{
\begin{array}{l}
\lambda= \frac{(\mathbf{S}^\top\mathbf{T}+\mathbf{T}^\top\mathbf{S})}{2\mathbf{S}^\top\mathbf{S}}\\
\mathbf{x}_{gp} = \mathbf{x} + \lambda\mathbf{S} - \mathbf{T}
\end{array}
\right.
\end{equation}
Another thing worth mentioning here is that in this special case, the \textit{representation manifold} $\mathcal{S}^3$ is no longer a subspace of the \textit{abmient space} $\mathbb{R}^{10}$, which means that we can't directly compute our regularization term $\mathbf{x}_{gp}-\mathbf{q}_g$ because $\mathbf{x}_{gp}\in\mathbb{R}^{10}$ while $\mathbf{q}_g\in\mathcal{S}^3$. However, the length vanishing problem still exists as shown in Figure \ref{fig:length}. Therefore, to compute the regularization term, we need a simple mapping to convert $\mathbf{q}_g$ to an element on $\mathbb{R}^{10}$ with stable length norm. We use the mapping $g:\mathcal{S}^3\rightarrow\mathbb{R}^{10}, g(\mathbf{q})=\mathbf{A}^{-1}(\mathbf{I}-\mathbf{q}\q^\top)$, which is proposed in \cite{peretroukhin_so3_2020}. They also proved that $\pi(g(\mathbf{q}))=\mathbf{q}$ is always true, which makes $g(\mathbf{q})$ better than simply normalizing $\mathbf{x}_{gp}$ because the latter one will suffer from the problem of opposite gradient discussed in Section \ref{sec:rpmg}.
\begin{table*}[t]
\caption{\textbf{Pose estimation from PASCAL3D+ \textit{sofa} images.} Left: a comparison of methods by 10$^\circ$ / 15$^\circ$ / 20$^\circ$ accuracy of (geodesic) errors and median errors after 60k training steps. Middle: median test error at different iterations during training. Right: test error percentiles after training completes. The legend on the right applies to both plots.}
\begin{minipage}[h]{0.28\columnwidth}
\resizebox{\columnwidth}{!}{
\begin{tabular}{lcccc}
\multirow{2}{*}{Methods}&\multicolumn{3}{c}{ Accuracy$(\%)\uparrow$}&Med$(^\circ)\downarrow$\\
& 10$^\circ$ & 15$^\circ$ & 20$^\circ$ & Err \\
\cmidrule{1-5}
Euler & 60.2& 80.9&90.6&8.3 \\
Axis-Angle & 45.0 & 70.9 & 85.1&11.0 \\
Quaternion & 34.3&60.8 & 73.5&13.2 \\
6D & 50.8& 76.7 & 89.0 & 9.9 \\
9D & 52.4 & 79.6 & 90.3&9.2 \\
9D-Inf & 70.9 & \textbf{88.0} & 93.5&\textbf{6.7} \\
10D & 50.2&77.0&89.6&9.8 \\
\midrule
RPMG-Quat & 56.6&79.6&90.9&8.9 \\
RPMG-6D & 69.6 & 86.1 &92.2&\textbf{6.7} \\
RPMG-9D & \textbf{72.5} &\textbf{88.0} &
\textbf{95.8}&\textbf{6.7} \\
RPMG-10D & 69.3 & 87.1 & 93.9 & 7.0\\
\end{tabular}}
\end{minipage}
\begin{minipage}[h]{0.28\columnwidth}
\includegraphics[width=\columnwidth]{figure/Training_curve_sofa.pdf}
\end{minipage}
\begin{minipage}[h]{0.28\columnwidth}
\includegraphics[width=\columnwidth]{figure/Percentile_sofa.pdf}
\end{minipage}
\label{tab:pascal3d_sofa}
\vspace{-1mm}
\end{table*}
\begin{table*}[t]
\caption{\textbf{Pose estimation from PASCAL3D+ \textit{bicycle} images.} We report the same metrics as Table \ref{tab:pascal3d_sofa}; see the caption there.}
\begin{minipage}[h]{0.28\columnwidth}
\resizebox{\columnwidth}{!}{
\begin{tabular}{lcccc}
\multirow{2}{*}{Methods} &\multicolumn{3}{c}{ Accuracy$(\%)\uparrow$} &Med$(^\circ)\downarrow$\\
& 10$^\circ$ & 15$^\circ$ & 20$^\circ$ & Err \\
\cmidrule{1-5}
Euler & 28.2& 48.1& 62.7& 15.7 \\
Axis-Angle & 5.3 & 8.1 & 10.1&79.7 \\
Quaternion & 20.8& 38.8 & 54.6&18.7 \\
6D & 21.8& 39.0 & 55.3 & 18.1 \\
9D & 20.6 & 37.6 & 56.9&18.0 \\
9D-Inf & 38.0 & 53.3 & 69.9&13.4 \\
10D & 23.9&42.3&56.7&17.9 \\
\midrule
RPMG-Quat & 32.3&50.0&65.6&15.0 \\
RPMG-6D & 35.4 & 57.2 &70.6&13.5 \\
RPMG-9D & 36.8&57.4 & \textbf{71.8}&\textbf{12.5} \\
RPMG-10D & \textbf{40.0} & \textbf{57.7} & 71.3 & 12.9 \\
\end{tabular}}
\end{minipage}
\begin{minipage}[h]{0.28\columnwidth}
\includegraphics[width=\columnwidth]{figure/Training_curve_bicycle.pdf}
\end{minipage}
\begin{minipage}[h]{0.28\columnwidth}
\includegraphics[width=\columnwidth]{figure/Percentile_bicycle.pdf}
\end{minipage}
\label{tab:pascal3d_bicycle}
\vspace{-1mm}
\end{table*}
\section{Projective Manifold Gradient on $S^2$}
\label{sec:supp3}
\subsection{Riemannian Optimization on $S^2$}
Our methods can also be applied for the regression of other manifolds. Taking $\mathcal{S}^2$ as an example, which is included in Experiment \ref{sec:other manifolds}, we will show the detail of how our projective manifold gradient layer works in other manifolds.
During forward, The network predicts a raw output $\mathbf{x}\in\mathbb{R}^3$, which is then mapped to $\hat{\mathbf{x}}\in\mathcal{S}^2$ through a \textit{manifold mapping} $\pi(\mathbf{x})=\mathbf{x}/\|\mathbf{x}\|$. Here we don't define the \textit{rotation mapping} and \textit{representation mapping}, and we directly compute the loss function on \textit{representation manifold} $\mathcal{S}^2$.
During backward, to apply a Riemannian optimization, we first need to know some basic concepts of $\mathcal{S}^2$. The tangent space of an arbitrary element $\hat{\mathbf{x}}\in\mathcal{S}^2$ is $\mathcal{T}_{\hat{\mathbf{x}}}\mathcal{M}$, which is a plane. And we can map a geodesic path $\v\in\mathcal{T}_{\hat{\mathbf{x}}}\mathcal{M}$ to an element on the manifold $\mathcal{S}^2$ through $\exp_{\hat{\mathbf{x}}}(\v)=\cos(\|\v\|)\hat{\mathbf{x}}+\sin(\|\v\|)\frac{\v}{\|\v\|}$, where $\|.\|$ means the ordinal Frobenius norm.
For the definition of the mapping $^\wedge$, which connects Euclidean space $\mathbb{R}^2$ and the tangent space $\mathcal{T}_{\hat{\mathbf{x}}}\mathcal{M}$, we need to first define two orthogonal axes $\hat{\mathbf{c}}_1$, $\hat{\mathbf{c}}_2$ in the tangent plane. Note that the choice of $\hat{\mathbf{c}}_1$ and $\hat{\mathbf{c}}_2$ won't influence the final result, which will be shown soon after. To simplify the derivation, we can assume ground truth unit vector $\hat{\mathbf{x}}_{gt}$ is known and choose $\hat{\mathbf{c}}_1=\frac{\mathrm{Log}_{\hat{\mathbf{x}}}(\hat{\mathbf{x}}_{gt})}{\|\mathrm{Log}_{\hat{\mathbf{x}}}(\hat{\mathbf{x}}_{gt})\|}=\frac{\hat{\mathbf{x}}_{gt}-(\hat{\mathbf{x}}_{gt}\cdot\hat{\mathbf{x}})}{\|\hat{\mathbf{x}}_{gt}-(\hat{\mathbf{x}}_{gt}\cdot\hat{\mathbf{x}})\|}$ and $\hat{\mathbf{c}}_2=\hat{\mathbf{x}}\times\hat{\mathbf{c}}_1$. Then we can say $\boldsymbol{\phi}^\wedge=\phi_1\hat{\mathbf{c}}_1+\phi_2\hat{\mathbf{c}}_2$, where $\boldsymbol{\phi}=(\phi_1, \phi_2)\in\mathbb{R}^2$. The gradient of exponential mapping with respect to $\boldsymbol{\phi}$ is
\small
\begin{align}
&\left.\frac{\partial}{\partial \phi_1}\mathrm{Exp}_{\hat{\mathbf{x}}}(\boldsymbol{\phi})\right|_{\boldsymbol{\phi}=\mathbf{0}} \notag \\
=&\left.\frac{\partial}{\partial \phi_1}(\cos(\|\phi_1\hat{\mathbf{c}}_1\|)\hat{\mathbf{x}}+
\sin(\|\phi_1\hat{\mathbf{c}}_1\|)\frac{\phi_1\hat{\mathbf{c}}_1}{\|\phi_1\hat{\mathbf{c}}_1\|})\right|_{\boldsymbol{\phi}=\mathbf{0}} \notag \\
=&~\hat{\mathbf{c}}_1
\end{align}
\normalsize
Similarly, we have $\left.\frac{\partial}{\partial \phi_2}\mathrm{Exp}_{\hat{\mathbf{x}}}(\boldsymbol{\phi})\right|_{\boldsymbol{\phi}=\mathbf{0}}=\hat{\mathbf{c}}_2$.
When using L2 loss, we can have
\small
\vspace{-1mm}
\begin{align}
\grad~\mathcal{L} f(\hat{\mathbf{x}})&=(\nabla f(\hat{\mathbf{x}}))^\wedge=(\nabla \boldsymbol{\phi})^\wedge\notag\\
&=\left(\left.\frac{\partial f(\hat{\mathbf{x}})}{\partial \hat{\mathbf{x}}}\frac{\partial}{\partial \boldsymbol{\phi}} \mathrm{Exp}_{\hat{\mathbf{x}}}(\boldsymbol{\phi})\right|_{\boldsymbol{\phi}=\mathbf{0}}\right)^\wedge \notag\\
&=((2(\hat{\mathbf{x}}-\hat{\mathbf{x}}_{gt})\hat{\mathbf{c}}_1,2(\hat{\mathbf{x}}-\hat{\mathbf{x}}_{gt})\hat{\mathbf{c}}_2))^\wedge \notag\\
&=2((\hat{\mathbf{x}}\cdot\hat{\mathbf{x}}_{gt})\hat{\mathbf{x}}-\hat{\mathbf{x}}_{gt})
\end{align}
\normalsize
Note that this expression doesn't depend on the choice of $\hat{\mathbf{c}}_1$ and $\hat{\mathbf{c}}_2$.
Similar to Eq \ref{eq:tau_safe}, we can also solve a $\tau_{converge}$
\scriptsize
\begin{equation}
\vspace{-1mm}
\tau_{converge}=\lim_{<\hat{\mathbf{x}},\hat{\mathbf{x}}_{\mathrm{gt}}>\to0}-\frac{\mathrm{Log}_{\hat{\mathbf{x}}}(\hat{\mathbf{x}}_{\mathrm{gt}})}{\grad~\mathcal{L} f(\hat{\mathbf{x}})}=\lim_{\theta\to0}\frac{\theta\hat{\mathbf{c}}_1}{2\sin\theta\hat{\mathbf{c}}_1} =\frac{1}{2}
\end{equation}
\normalsize
where $\theta=<\hat{\mathbf{x}}, \hat{\mathbf{x}}_{gt}>$. Note that in Experiment 5.4, we change the schedule of $\tau$ according to this conclusion. We increase $\tau$ from $0.1$ to $0.5$ by uniform steps.
\subsection{Inverse Projection}
Similar to quaternion, we can have $\mathbf{x}_{gp}=(\mathbf{x}\cdot\hat{\mathbf{x}}_g)\hat{\mathbf{x}}_g$. For the detail of derivation, see Section \ref{sec:inverse_proj}.
\section{Computational Cost}
\label{sec:cost}
Our method does not alter the forward pass and thus incurs no cost at test time. For backward pass at training, we observe that, before and after inserting RMPG layers, the backward time for quaternion / 6D / 9D / 10D representations, averaged among 1K iterations on a GeForce RTX 3090, changes from 4.39 / 4.48 / 4.48 / 4.53 to 4.45 / 4.43 / 4.49 / 4.63 (unit: $10^{-2}$ s), and the memory cost changes from 11449 / 11415 / 10781 / 11363 to 11457 / 11459 / 11545 / 11447 (unit: MiB). Note that the runtime is almost keep the same, as Riemannian optimization only performs an additional projection and we derive and always use analytical solutions in representation mapping, inversion and projection steps. RPMG also has a very marginal cost on the memory, as it does not introduce any weights but only a few intermediate variables.
\section{More Experiments}
\label{sec:more_exps}
\subsection{Pascal3D+}
\label{pascal3d+}
Pascal3D+ \cite{xiang2014pascal3d} is a standard benchmark for object pose estimation from real images.
We follow the same setting as in \cite{levinson2020analysis} to estimate object poses from single images. For training we discard occluded or truncated objects and augment with rendered images from \cite{renderforcnn15}. In the Table \ref{tab:pascal3d_sofa} and Table \ref{tab:pascal3d_bicycle}, we report our results on \textit{sofa} and \textit{bicycle} categories.
We use the same batch size as in \cite{levinson2020analysis}. As for the learning rate, we use the same strategy as in Experiment \ref{sec:image}. See the discussion in Section \ref{sec:img_lr}.
It can be seen that our method leads to consistent improvements to quaternion, 6D, 9D and 10D representations on both \textit{sofa} and \textit{bicycle} classes. One may be curious about why our method can only outperform 9D-inf for a margin. We think that this is because this dataset is quite challenging. The number of annotated real image for training is only around 200 for each category. Though there are a lot of synthetic images generated from \cite{renderforcnn15} for training, these images suffer from sim-to-real domain gap. Therefore, we argue that the bottleneck here is not in optimization, which makes the gains from less noise in gradient smaller(Note that 9D-inf is just a special case of our methods with $\lambda=1$ and $\tau=\tau_{gt}$). But compared to vanilla 4D/6D/9D/10D representation, our methods can still bring a great improvement.
\subsection{Using Flow Loss for Rotation Estimation from Point Clouds.}
\label{sec:flow loss}
Apart from the most widely used L2 loss, our method can also be applied to the loss of other forms, e.g. flow loss.
We mainly follow the setting in Experiment \ref{sec:pc_rotation} with \textit{airplane} point clouds dataset and the only difference is that we use flow loss $\|\mathbf{R}X-\mathbf{R}_{gt}X\|_F^2$ here, where $X$ is the complete point clouds.
Since the format of loss is changed, the previous schedule of $\tau$ is not suitable anymore, and we have to change the value of $\tau$ accordingly. Our selection skill is to first choose a $\tau$ as we like and visualize the mean geodesic distance between predicted $\mathbf{R}$ and $\mathbf{R}_g$ during training. Then we can roughly adjust $\tau$ to make the geodesic distance looked reasonable. For this experiment, we use $\tau=50$ and $\lambda=0.01$.
In Table \ref{tab:flow-loss}, we show our methods again outperform vanilla methods as well as \textbf{9D-inf}.
\begin{table}[htbp]
\centering
\caption{\textbf{Flow Loss for Rotation Estimation from Point Clouds.} All models are trained for 30K iterations. }
\resizebox{0.8\columnwidth}{!}{
\begin{tabular}{lccc}
Methods & \multicolumn{1}{c}{Mean ($^\circ$)} & \multicolumn{1}{c}{Med ($^\circ$)} & \multicolumn{1}{c}{5$^\circ$Acc ($\%$)} \\
\cmidrule{1-4}
Euler & 12.14 & 6.91 & 33.6 \\
Axis-Angle & 35.49 & 20.80 & 4.7 \\
Quaternion & 11.54 & 7.67 & 29.8 \\
6D & 14.13 & 9.41 & 23.4 \\
9D & 11.44 & 8.01 & 23.8 \\
9D-Inf & 4.07 & 3.28 & 76.7 \\
10D & 9.28 & 7.05 & 32.6 \\
\midrule
RPMG-Quat & 4.86 & 3.25 & 75.8 \\
RPMG-6D & \textbf{2.71} & \textbf{2.04} & \textbf{92.1} \\
RPMG-9D & 3.75 & 2.10 & 91.1 \\
RPMG-10D & 3.30 & 2.70 & 86.8 \\
\end{tabular}}
\label{tab:flow-loss}
\vspace{-3mm}
\end{table}
\begin{table*}[htbp]
\centering
\caption{\textbf{Camera relocalization on Cambridge Landscape dataset.} We report the \textit{median} error of translation and rotation of the best checkpoint, which is chosen by minimizing the median of rotation. We only care about the rotation error here.}
\begin{tabular}{l|cc|cc|cc|cc|cc}
\multirow{2}{*}{Methods}& \multicolumn{2}{c|}{King's College} & \multicolumn{2}{c|}{Old Hospital}& \multicolumn{2}{c|}{Shop Facade}&
\multicolumn{2}{c|}{St Mary's Church} & \multicolumn{2}{c}{Average} \\
\cmidrule{2-11}
& T($m$)& R($^\circ$) & T($m$)& R($^\circ$) & T($m$)& R($^\circ$) & T($m$)& R($^\circ$)& T($m$)& R($^\circ$) \\
\midrule
Euler&1.16&2.85&2.54&2.95&1.25&6.48&1.98&6.97&1.73&4.81\\
Axis-Angle&1.12&2.63&2.41&3.38&\textbf{0.84}&5.05&2.16&7.58&1.63&4.66\\
Quaternion&\textbf{0.98}&2.50&2.39&3.44&1.06&6.01&2.59&8.81&1.76&5.19\\
6D&1.10&2.56&2.21&3.43&1.01&5.43&\textbf{1.73}&5.82&1.51&4.31\\
9D&1.14&3.03&2.11&3.50&0.88&6.39&1.95&5.95&1.52&4.72\\
9D-Inf&\textbf{0.98}&2.32&\textbf{1.89}&3.32&1.15&6.36&1.96&6.25&\textbf{1.50}&4.56\\
10D&1.54&2.62&2.32&3.39&1.20&5.76&1.85&6.69&1.73&4.62\\
\midrule
RPMG-Quat&1.04&1.91&2.42&2.72&0.98&4.28&1.82&4.89&1.57&3.45\\
RPMG-6D&1.55&\textbf{1.70}&2.62&3.09&0.95&5.01&2.44&5.18&1.89&3.75\\
RPMG-9D&1.57&1.82&4.37&3.12&0.93&4.17&1.92&\textbf{4.69}&2.20&3.45\\
RPMG-10D & 1.30 & 1.74 & 3.21 & \textbf{2.59} & 1.10 & \textbf{3.47} &2.20&5.09 & 1.95 & \textbf{3.22} \\
\end{tabular}
\label{tab:reloc}
\end{table*}
\subsection{Camera Relocalization}
\label{sec:reloc}
The task of camera relocalization is to estimate a 6 Degree-of-Freedom camera pose (rotation and translation) from visual observations, which is a fundamental component of many computer vision and robotic applications.
In this experiment, we use all the settings (data, network, training strategy, hyperparameters, etc.) of PoseLSTM \cite{PoseLSTM17} except that we modify the rotation representations and the gradient layers. We report the results on the outdoor Cambridge Landscape dataset \cite{posenet} in Table \ref{tab:reloc}.
Notice that our RPMG layer performs the best on the rotation regression task, but not on the translation regression. We believe this results from a loss imbalance. We does not change the weights of the rotation loss and translation loss, otherwise it leads to an unfair comparison with existing results. We only care about the rotation error here.
\section{More Implementation Details}
\label{sec:imp_detail}
\subsection{Experiment \ref{sec:pc_rotation} \& \ref{sec:self-supervised} \& \ref{sec:other manifolds}}
\textbf{Data}
We generate the data from ModelNet dataset \cite{wu2015modelnet} by sampling 1024 points on the mesh surface, following the same generation method as in \cite{zhou2019continuity}. We uniformly sample M rotations for each data point and set them as the ground truth. We apply the sampled rotations on the canonical point clouds to obtain the input data.
\textbf{Network Architecture}
We use a PointNet++ MSG \cite{qi2017pointnetplusplus} backbone as our feature extractor. Our network takes input a point cloud with a resolution of 1024. It them performs three set abstractions to lower the resolution to 512, 128, and finally 1, resulting in a global feature of dimensionality 1024. The feature is finally pushed through a three-layer MLP $[1024, 512, N]$ to regress rotation, where $N$ is the dimension of the rotation representation.
\textbf{Training details} The learning rate is set to 1e-3 and decayed by 0.7 every 3k iterations. The batch size is 20. For each experiment, we train the network on one NVIDIA TITAN Xp GPU for 30k iterations.
\subsection{Experiment \ref{sec:image}}
\label{sec:img_lr}
Most of the training settings and strategies are all the same as \cite{levinson2020analysis} except learning rate. We find setting initial learning rate $lr=1e-3$ and decaying to $1e-5$ can perform much better than using $lr=1e-5$ as in \cite{levinson2020analysis}, which accounts for the inconsistency of the results of those baseline methods compared to \cite{levinson2020analysis}. We believe that the methods should be compared under hyperparameters as optimal as possible. Thus, we stick to our $lr$ schedule.
\begin{table}[htbp]
\caption{\textbf{Test error percentiles for Experiment \ref{sec:pc_rotation} \& \ref{sec:image}} Left: test error percentiles of \textit{airplane} for Experiment \ref{sec:pc_rotation} after training completes. Right: test error percentiles of \textit{chair} for Experiment 5.2 after training completes.}
\begin{minipage}[h]{0.48\columnwidth}
\includegraphics[width=\columnwidth]{figure/Percentile_full.pdf}
\end{minipage}
\begin{minipage}[h]{0.48\columnwidth}
\includegraphics[width=\columnwidth]{figure/img_percentile_chair.pdf}
\end{minipage}
\label{tab:add_fig}
\vspace{-1mm}
\end{table}
\section{Addition on Rotation Representations}
\label{sec:q_r_trans}
\paragraph{Standard mapping between rotation matrix and unit quaternion}
The \emph{rotation mapping} $\phi: \mathbf{q}\mapsto \mathbf{R}$ algebraically manipulates a unit quaternion $\mathbf{q}$ into a rotation matrix:
\scriptsize
\begin{equation}\phi(\mathbf{q})=\begin{pmatrix}
2(q_0^2+q_1^2)-1&2(q_1q_2-q_0q_3)&2(q_1q_3+q_0q_2)\\
2(q_1q_2+q_0q_3)&2(q_0^2+q_2^2)-1&2(q_2q_3-q_0q_1)\\
2(q_1q_3-q_0q_2)&2(q_2q_3+q_0q_1)&2(q_0^2+q_3^2)-1\\
\end{pmatrix}\end{equation}
\normalsize
where $\mathbf{q}=(q_0,q_1,q_2,q_3)\in\mathcal{S}^3$.
In the reverse direction, the \emph{representation mapping} $\psi(\mathbf{R})$ can be expressed as:
\small
\begin{equation}\left\{
\begin{array}{l}
q_0=\sqrt{1+R_{00}+R_{11}+R_{22}}/2 \\
q_1=(R{21}-R_{12})/(4*q_0)\\
q_2=(R_{02}-R_{20})/(4*q_0)\\
q_3=(R_{10}-R_{01})/(4*q_0)\\
\end{array}
\right.\end{equation}
\normalsize
Note that $\mathbf{q}=(q_0,q_1,q_2,q_3)$ and $-\mathbf{q}=(-q_0,-q_1,-q_2,-q_3)$ both are the valid quaternions parameterizing the same $\mathbf{R}$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,611 |
\section{Introduction}
\label{sec:main-intro}
Profusion of empirical data on real world networks has given impetus to research in mathematical models for such systems that explain the various observed statistics such as scale free degree distribution, small world properties and clustering. A range of mathematical models have been proposed both static as well as dynamic to understand the structural properties of such real world networks and their evolution over time.
One particular direction of significant research is focused on understanding the effect of choice in the evolution of random network models (see \cite{spencer2007birth} and references therein).
More precisely, suppose that at time $t=0$ we start with the empty configuration on $[n]:=\set{1,2,\ldots, n}$ vertices. At each discrete step $k=0,1,2,\ldots$, we choose two edges $(e_1(k),e_2(k))$ uniformly at random amongst all ${n\choose 2}$ edges and decide whether the graph at instant $(k+1)$, denoted as ${\bf{G}}_n(k+1)$, is ${\bf{G}}_n(k)\cup e_1(k)$ or ${\bf{G}}_n(k)\cup e_2(k)$ according to some pre-specified rule that takes into account suitable properties of the chosen edges with respect to the present configuration ${\bf{G}}_n(k)$. Speeding up time by a factor of $n$ and abusing notation, for $t\geq 0$ write, ${\bf{G}}_n(t) = {\bf{G}}_n(\lfloor nt/2 \rfloor)$.
Then the basic goal is to understand the effect of the rule governing the edge formations in the evolution of various characteristics of the network such as, the size of the largest component, the vector of sizes of all components, component complexities, etc.
Three prototypical examples to keep in mind are as follows:
\\(a) {\bf Erd\H{o}s-R\'{e}nyi random graph:} At each stage include edge $e_1$ and ignore $e_2$. This results in the classical Erd\H{o}s-R\'{e}nyi random graph evolution.
For a component $\mathcal{C}$, define $|\mathcal{C}|$ for the size (number of vertices) of the component. Well known results \cite{er-1,bollobas-rg-book} say that the critical time for the emergence of a giant component for this model is $1$, namely for $t<1$ the size of the largest component $|\mathcal{C}^{\scriptscriptstyle \bf ER}_1(t)| = O(\log{n})$ while for $t> 1$, the size of the largest component $|\mathcal{C}^{\scriptscriptstyle \bf ER}_1(t)| = \Theta(n)$.
Here $O,\Theta$ are defined in the usual manner. More precisely, given a sequence of random variables $\{\xi_n\}_{n\ge 1}$ and a function $f(n)$, we say $\xi_n = O (f)$ if there is a constant $C$ such that $\xi_n \le C f(n)$ with high probability (whp), and we say $\xi_n = \Omega(f)$ if there is a constant $C$ such that $\xi_n \ge Cf(n)$ whp. Say that $\xi_n = \Theta(f)$ if $\xi_n = O(f)$ {\bf and} $\xi_n = \Omega(f)$.
\\
(b) {\bf Bohman-Frieze (BF) process: } This was the first rigorously analyzed example of a rule that delayed the emergence of the giant component through limited choice \cite{bohman2001avoiding}. Here the rule is to use the first edge if it connects two singletons (vertices which have no connections at the present time), otherwise use the second edge. It has been shown (\cite{spencer2007birth, janson2010phase}) that there is a critical time $t_c^{\scriptscriptstyle {\bf {BF}}} \approx 1.176$ when the largest component transitions from $O(\log{n})$ to $\Theta(n)$.\\
(c) {\bf General bounded-size rules (BSR):} The BF process corresponds to a
choice rule which treats all components with size greater than one in an identical fashion.
It is a special case of the general family of models, referred to as bounded-size rules. Here one fixes $K\geq 1$ and then the rule for attachment is invariant on components of size greater than $K$. We postpone a precise description to Section \ref{sec:bsr}. General bounded-size rules were analyzed in \cite{spencer2007birth} where it was shown that there exists a (rule dependent) critical time $t_c$ such that for $t< t_c$, the largest component $|\mathcal{C}^{\scriptscriptstyle {\bf{BSR}}}_1(t)| = O(\log{n})$ when $t< t_c$ while $|\mathcal{C}^{\scriptscriptstyle {\bf{BSR}}}_1(t)| = \Theta(n)$ for $t> t_c$.
Thus as time transitions from below to above $t_c$, a giant component (of the same order as the network) emerges. Motivated by recent results on the Erd\H{o}s-R\'enyi random graph components at criticality \cite{addario2009continuum} as well as general rules such as the (unbounded-size) product rule \cite{achlioptas2009explosive}, there has been a renewed interest in understanding the precise nature of the emergence of the giant component as well as structural properties of components near $t_c$ for classes of rules which incorporate limited choice in their evolution.
Define the surplus or complexity of a component ${\bf spls}(\mathcal{C})$ as
\begin{equation}
\label{eqn:surp-def}
{\bf spls}(\mathcal{C}) = \mbox{number of edges} - (|\mathcal{C}|-1).
\end{equation}
If a component were a tree, its surplus would be zero, thus this is a measure of the deviation of the component from a tree. Write $\mathcal{C}_i(t)$ for the $i$-th largest component and $\xi_i(t) := {\bf spls} (\mathcal{C}_i(t))$ for the surplus of the component $\mathcal{C}_i(t)$. For any of the rules above and a fixed $t\geq 0$, consider the vector of component sizes and associated surplus $(|\mathcal{C}_i(t)|, \xi_i(t): i\geq 1)$. In the context of the Erd\H{o}s-R\'enyi random graph process, precise fine-scale results are known about the nature of the emergence of the giant component as time $t$ transitions through the scaling window around $t_c^{\scriptscriptstyle \bf ER}=1$. More precisely, for fixed $\lambda \in {\mathbb{R}}$ write
\[\bar{\boldsymbol{C}}^{\scriptscriptstyle {\bf ER}}(\lambda) := \left(\frac{1}{n^{2/3}}\left|\mathcal{C}_i^{\scriptscriptstyle {\bf ER}}\left(1+\frac{1}{n^{1/3}}\lambda\right)\right| :i\geq 1\right)
\mbox{ and } \bar{\boldsymbol{Y}}^{\scriptscriptstyle {\bf ER}}(\lambda) = \left (\xi_i^{\scriptscriptstyle {\bf ER}}\left(1+\frac{1}{n^{1/3}}\lambda\right) :i\geq 1\right).
\]
Then Aldous in \cite{aldous1997brownian} showed:
\begin{enumeratea}
\item The process $(\bar{\boldsymbol{C}}^{\scriptscriptstyle {\bf ER}}(\lambda): -\infty < \lambda < \infty)$ converges to a Markov process called the standard multiplicative coalescent.
\item For fixed $\lambda \in {\mathbb{R}}$, the rescaled component sizes and the corresponding surplus $(\bar{\boldsymbol{C}}^{\scriptscriptstyle {\bf ER}}(\lambda), \bar{\boldsymbol{Y}}^{\scriptscriptstyle {\bf ER}}(\lambda))$ converge jointly to a limiting random process described by excursions from zero of an inhomogeneous reflected Brownian motion $\hat{W}_\lambda$ and a counting process $\hat N_{\lambda}$ with intensity function $\hat{W}_\lambda(\cdot)$.
\end{enumeratea}
We give a precise description of these results in Section \ref{sec:mult}.
Obtaining similar results on critical asymptotics
for general inhomogeneous Markovian models such as the bounded-size rules requires new ideas. These rules lack a simple description for the dependence between edges making the direct use of the component exploration and associated random walk construction, the major workhorse in understanding random graph models at criticality (\cite{aldous1997brownian,bhamidi2009novel,bhamidi-hofstad-van,riordan2012phase,joseph2010component}), intractable. Thus it is nontrivial to identify the critical scaling window for such processes, let alone distributional asympototics for the component sizes and surplus. In the current work we develop a different machinery that allows us to identify the critical scaling window for all bounded-size rules. Furthermore, denoting the suitably scaled component sizes and surplus processes as $(\bar{\boldsymbol{C}}^{\scriptscriptstyle (n)}(\lambda), \bar{\boldsymbol{Y}}^{\scriptscriptstyle (n)}(\lambda))$, our
results describe the joint asymptotic behavior of
\[
\left((\bar{\boldsymbol{C}}^{\scriptscriptstyle (n)}(\lambda_1), \bar{\boldsymbol{Y}}^{\scriptscriptstyle (n)}(\lambda_1)), \ldots, (\bar{\boldsymbol{C}}^{\scriptscriptstyle (n)}(\lambda_m), \bar{\boldsymbol{Y}}^{\scriptscriptstyle (n)}(\lambda_m))\right)\]
for $-\infty < \lambda_1< \lambda_2< ... < \lambda_m < \infty$.
Starting point of our work is the construction of a new Markov process that is associated with the inhomogeneous reflected Brownian motion $\{\hat W_{\lambda}\}_{\lambda \in {\mathbb{R}}}$
and the associated counting process $\{\hat N_{\lambda}\}_{\lambda \in {\mathbb{R}}}$ which we refer to as the augmented multiplicative coalescent (AMC). The main result of this work
shows that AMC is the characterizing process for a new universality class that includes, in addition to critically scaled components and surplus vectors for
Erd\H{o}s-R\'enyi graphs, analogous processes for all bounded-size rules. More precisely, our contributions are as follows.
\begin{enumeratea}
\item In Theorem \ref{thm:smc-surplus} we show the existence and ``near'' Feller property of a Markov process ${\boldsymbol Z}(\lambda)$, $-\infty < \lambda < \infty$, called the augmented multiplicative coalescent, which tracks the evolution of both component sizes and surplus edges over the critical window. Aldous's standard multiplicative coalescent corresponds to the first coordinate of this process. Identifying the correct state space and topology that is suitable for obtaining the Feller property for this process turns out to be particularly delicate (see Remark \ref{prodnofell}). The (near) Feller property plays a key role in analyzing the joint distribution, at multiple time instants, of the component sizes and surplus for bounded-size rules in the critical scaling window. In proving the existence of the standard augmented process, a key role is played by Theorem \ref{theo:aldous-full-gene} which is a generalization of a result of Aldous for the component sizes of an inhomogeneous random graph, to a setting where one considers joint distributions of component sizes and surplus.
We believe that this result is of broader significance and can be used to analyze the distribution of surplus in the critical regime for various other inhomogeneous random graph models, e.g. the rank-1 inhomogeneous random graphs (\cite{bollobas-riordan-janson}).
\item In Theorems \ref{thm:suscept-funct} and \ref{thm:suscept-limit} we analyze susceptibility functions (sums of moments of component sizes) associated with a general bounded-size rule.
Spencer and Wormald \cite{spencer2007birth} showed that these susceptibility functions converge to limiting monotonically increasing deterministic functions which are finite only for $t< t_c$ and explode for $t> t_c$. Theorem \ref{thm:suscept-funct} uses a dynamic random graph process with immigration and attachment to show that these limiting functions for {\bf all bounded-size rules} have the same critical exponents as the Erd\H{o}s-R\'enyi random graph process. Theorem \ref{thm:suscept-limit} shows that the susceptibility functions are close to their deterministic analogs in a strong sense even as $t\uparrow t_c$ when the limiting functions explode.
\item The analysis of the susceptibility functions gives rise to (rule dependent) constants $\alpha, \beta > 0$ which describe the nature of the explosion of the limiting susceptibility functions as $t\uparrow t_c$. For a given bounded-size rule we consider the rescaled process $\set{{\bar\boldsymbol{Z}}^{\scriptscriptstyle (n)}(\lambda): -\infty< \lambda < \infty} $ where ${\bar \boldsymbol{Z}}^{\scriptscriptstyle (n)}(\lambda) = (\bar{\boldsymbol{C}}^{\scriptscriptstyle (n)}(\lambda),\bar{\boldsymbol{Y}}^{\scriptscriptstyle (n)}(\lambda) )$ with $\bar{\boldsymbol{C}}^{\scriptscriptstyle (n)}(\lambda)$ denoting the rescaled component sizes and $\bar{\boldsymbol{Y}}^{\scriptscriptstyle (n)}(\lambda)$ denoting the surplus of these components, namely
\[\bar{\boldsymbol{C}}^{\scriptscriptstyle (n)}(\lambda) := \left(\frac{\beta^{1/3}}{n^{2/3}}\left|\mathcal{C}_i\left(t_c+\frac{\alpha\beta^{2/3}}{n^{1/3}}\lambda\right)\right| :i\geq 1\right)
\mbox{ and } \bar{\boldsymbol{Y}}^{\scriptscriptstyle (n)}(\lambda) = \left (\xi_i\left(t_c+\frac{\alpha\beta^{2/3}}{n^{1/3}}\lambda\right) :i\geq 1\right).
\]
Using the Feller property proved in Theorem \ref{thm:smc-surplus}, Theorem \ref{theo:aldous-full-gene}, results on the susceptibility functions, and bounds on the maximal component in the barely subcritical regime from \cite{bsr-2012}, we show the convergence
of finite dimensional distribution of this process to that of the augmented multiplicative coalescent $\set{{\boldsymbol Z}(\lambda):-\infty< \lambda <\infty}$, namely
for any set of times $-\infty < \lambda_1< \lambda_2< ... < \lambda_m < \infty$,
\[
\left(\bar{\boldsymbol{Z}}^{\scriptscriptstyle (n)}(\lambda_1), \ldots, \bar{\boldsymbol{Z}}^{\scriptscriptstyle (n)}(\lambda_m)\right) \convd \left(\boldsymbol{Z}(\lambda_1), \ldots, \boldsymbol{Z}(\lambda_m)\right).
\]
The result in particular identifies the critical scaling window {\bf for all bounded-size rules} as well as the asymptotic joint distributions of component sizes and surplus for any fixed $\lambda$, implying that such rules belong to the same universality class as the Erd\H{o}s-R\'enyi random graph process. The convergence for the joint distribution of the surplus and the component sizes for multiple time points $\lambda$ in the critical scaling window is new even in the context of the Erd\H{o}s-R\'enyi random graph process.
\end{enumeratea}
The paper is organized as follows. In Section \ref{sec:main-defn} we introduce some common notation, give a precise description of bounded-size rules, and give an informal
description of the augmented multiplicative coalescent. Section \ref{sec:main-results} contains the statements of our main results. Sections \ref{sec:main-constr-mc}
and \ref{sec:main-st-ag-mc-ex} are devoted to proving the existence and near Feller property of the AMC. In particular Section \ref{sec:main-st-ag-mc-ex}
contains the proof of Theorem \ref{thm:smc-surplus}. Section \ref{sec:main-bsr-susceptibility} studies the asymptotics of the susceptibility functions associated
with general bounded-size rules and proves Theorems \ref{thm:suscept-funct} and \ref{thm:suscept-limit}.
Finally in Section \ref{sec:main-coupling} we complete the proof of Theorem \ref{thm:crit-regime}.
\section{Definitions and Notation}
\label{sec:main-defn}
\subsection{Notation}
We collect some common notation and conventions used in this work.
A graph ${\bf{G}}=\{\mathcal{V}, \mathcal{E}\}$ consists of a vertex set $\mathcal{V}$ and an edge set $\mathcal{E}$, where $\mathcal{V}$ is a subset of some type space $\mathcal{X}$.
For a finite set $A$ write $|A|$ for its cardinality. A graph ${\bf{G}}$ with no vertices and edges will be called a \textbf{null graph}.
For graphs ${\bf{G}}_1, {\bf{G}}_2$, if ${\bf{G}}_1$ is a subgraph of ${\bf{G}}_2$ we shall write this as ${\bf{G}}_1 \subset {\bf{G}}_2$.
The number of vertices in a connected component $\mathcal{C}$ of a graph ${\bf{G}}$ will be called the size of the component and will be denoted by $|\mathcal{C}|$.
Let $\mathcal{G}$ be the set of all possible graphs $(\mathcal{V}, \mathcal{E})$ on a given type space $\mathcal{X}$. When $\mathcal{V}$ is finite, we will
consider $\mathcal{G}$ to be endowed with the discrete topology and the corresponding Borel sigma field and refer to a random element of $\mathcal{G}$ as a random graph.
We use $\convp$ and $\convd$ to denote convergence in probability and in distribution respectively.
All the unspecified limits are taken as $n \to \infty$.
Given a sequence of events $\{E_n\}_{n\ge 1}$, we say $E_n$ occurs with high probability (whp) if $\mathbb{P}\{E_n\} \to 1$.
The notation $O,\Omega,\Theta$ was described in the Introduction. Furthermore, for a sequence of random variables $\{\xi_n\}_{n\ge 1}$ and a function $f(n)$, we say $\xi_n = o (f)$ if $\xi_n/f(n) \convp 0$.
For a Polish space $S$, $\mathcal{D}([0,T]:S)$ (resp. $\mathcal{D}([0,\infty):S)$) denote the space of right continuous functions with left limits (RCLL) from $[0,T]$ (resp. $[0,\infty)$) equipped with the usual Skorohod topology. For a RCLL function $f:[0,\infty) \to \mathbb{R}$,
we write $\Delta f(t) = f(t) - f(t-)$, $t >0$.
Suppose that $(S, \mathcal{S})$ is a measurable space and we are given a partial
ordering on $S$.
We say
the $S$ valued random variable $\xi$ \textbf{ stochastically dominates} $\tilde \xi$, and write $\xi \ge_d \tilde \xi$ if there exists a coupling between the two random variables on a common probability space such that $\xi^* \ge \tilde \xi^*$ a.s., where $\xi^*=_d \xi$ and $\tilde \xi^* = \tilde \xi$. For probability measures $\mu, \tilde \mu$ on $S$, we say $\mu$
stochastically dominates $\tilde \mu$, if $\xi \ge_d \tilde \xi$ where $\xi$ has distribution $\mu$ and $\tilde \xi$ has distribution $\tilde \mu$.
Two examples of $S$ relevant to this work are $\mathcal{D}([0,T]: \mathbb{R})$ and $\mathcal{D}([0,T]: \mathcal{G})$ with the natural associated partial ordering.
Given a metric space $S$, we denote by $\mathcal{B}(S)$ the Borel $\sigma$ - field on $S$ and by
$\mbox{BM}(S), C_b(S), \mathcal{P}(S)$, the space of bounded (Borel) measurable functions, continuous and bounded function, and probability measures, on $S$,
respectively. The set of nonnegative integers will be denoted by $\mathbb{N}_0$.
\subsection{Bounded-size rules (BSR)}
\label{sec:bsr}
We now define the general class of rules that will be analyzed in this paper. Much of the notation follows \cite{spencer2007birth} which provides a comprehensive analysis of the sub and supercritical regime.
Fix $K \in \mathbb{N}$ and let $\Omega_0=\set{\varpi}$ and $\Omega_K =\set{1,2,\ldots, K, \varpi}$ for $K \ge 1$, where $\varpi$ will represent components of size greater than $K$. Given a graph ${\bf{G}}$ and a vertex $v\in{\bf{G}}$, write $\mathcal{C}_v({\bf{G}})$ for the component that contains $v$. Let
\begin{equation}
\label{eqn:cv-def}
c(v) = \left\{\begin{array}{ll}
|\mathcal{C}_v({\bf{G}})| & \mbox{ if }|\mathcal{C}_v({\bf{G}})|\leq K\\
\varpi & \mbox{ if } |\mathcal{C}_v({\bf{G}})| > K.
\end{array}\right.
\end{equation}
For a quadruple of vertices $v_1, v_2, v_3, v_4$, write $\vec{v} = (v_1,v_2,v_3,v_4)$ and let $c(\vec{v}) = (c(v_1), c(v_2), c(v_3), c(v_4))$. Fix $F\subseteq \Omega_K^4$. We now define the random graph process $\{{\bf{BSR}}^{\scriptscriptstyle(n)}(k)\}_{k\ge 0}$
on the vertex set $[n]$ evolving through a $F$-bounded-size rule ($F$-BSR) as follows.
Define ${\bf{BSR}}^{\scriptscriptstyle(n)}(0) = \mathbf{0}_n$. Having defined ${\bf{BSR}}^{\scriptscriptstyle(n)}(k)$ for $k \ge 0$, ${\bf{BSR}}^{\scriptscriptstyle(n)}(k+1)$ is constructed as follows:
Choose four vertices $(v_1(k), v_2(k), v_3(k), v_4(k))$ uniformly at random amongst all possible $n^4$ vertices uniformly at random and let \[\vec{v}_k = (v_1(k), v_2(k),v_3(k), v_4(k)).\]
Denote the function $c(\vec{v})$ associated with ${\bf{BSR}}^{\scriptscriptstyle(n)}(k)$ as $c_k(\vec{v})$.
Define
\begin{equation}
\label{eqn:f-rule-def}
{\bf{BSR}}^{\scriptscriptstyle(n)}(k+1) =\left\{ \begin{array}{ll}
{\bf{BSR}}^{\scriptscriptstyle(n)}(k)\cup (v_1(k), v_2(k)) & \quad \mbox{ if } c_k(\vec{v}_k) \in F\\
{\bf{BSR}}^{\scriptscriptstyle(n)}(k)\cup (v_3(k), v_4(k)) & \quad \mbox{ otherwise. }
\end{array} \right.
\end{equation}
These rules are called bounded-size rules since they treat all components of size greater than $K$ identically. Concrete examples of such rules include
Erd\H{o}s-R\'{e}nyi random graph (here $K=0$, $F=\Omega_0^4 =\set{\varpi,\varpi,\varpi,\varpi}$) and
Bohman-Frieze process (here $K=1$, $F=\set{(1,1,\alpha, \beta): \alpha, \beta \in \Omega_1}$).
\noindent {\bf Continuous time formulation $\set{{\bf{BSR}}^{\scriptscriptstyle(n)}(t)}_{t\geq 0}$:} It will be more convenient to work in continuous time.
For every quadruple of vertices $\vec{v} = (v_1, v_2, v_3, v_4)\in [n]^4$, let $\mathcal{P}_{\vec{v}}$ be a Poisson process with rate $\frac{1}{2n^3}$, independent between quadruples.
The continuous time random graph process $\{{\bf{BSR}}^{\scriptscriptstyle(n)}(t)\}_{t\ge 0}$ is constructed recursively as follows.
We denote the function $c(v)$ [resp. $c(\vec{v})$] associated with ${\bf{BSR}}^{\scriptscriptstyle(n)}(t-)$ as $c_{t-}(v)$ [resp. $c_{t-}(\vec{v})$].
Given ${\bf{BSR}}^{\scriptscriptstyle(n)}(t-)$, and that for some
$\vec{v} \in [n]^4$, $\mathcal{P}_{\vec{v}}$ has a point at the time instant $t$, we define
\begin{equation}
\label{eqn:f-rule-def-cts}
{\bf{BSR}}^{\scriptscriptstyle(n)}(t) =\left\{ \begin{array}{ll}
{\bf{BSR}}^{\scriptscriptstyle(n)}(t-)\cup (v_1, v_2) & \quad \mbox{ if } c_{t-}(\vec{v}) \in F\\
{\bf{BSR}}^{\scriptscriptstyle(n)}(t-)\cup (v_3, v_4) & \quad \mbox{ otherwise. }
\end{array} \right.
\end{equation}
The rationale behind this scaling for the rate of the Poisson point process is that the total rate of adding edges is
\[\frac{n^4}{2n^3} = \frac{n}{2}.\]
Thus with this scaling, for the $F$-BSR rule corresponding to the Erd\H{o}s-R\'{e}nyi evolution, the giant component emerges at time $t=1$. To simplify notation, when there is no scope for confusion, we will suppress $n$ in the notation. For example, we write ${\bf{BSR}}_t:= {\bf{BSR}}^{\scriptscriptstyle(n)}(t)$.
Denote $\mathcal{C}_i^{\scriptscriptstyle (n)}(t)$ for the $i$-th largest component of ${\bf{BSR}}_t$ at time $t$.
The work of Spencer and Wormald (see \cite{spencer2007birth}) shows that for any given BSR, there exists a (model dependent) {\bf critical time} $t_c>0$ such that
for $t < t_c$, $|\mathcal{C}_1^{\scriptscriptstyle (n)}(t)|= O(\log n)$ and
for $t > t_c$, $|\mathcal{C}_1^{\scriptscriptstyle (n)}(t)| \sim f(t)n$ where $f(t) >0$.
One of the key ingredients in the proof of the above result is an analysis of the susceptibility functions: For any given time $t$ and fixed $k\geq 0$ define the
$k$-susceptibility function
\begin{equation}\label{eq:eq5.1}\mathcal{S}_k^{\scriptscriptstyle (n)}(t) \equiv \mathcal{S}_k(t) := \sum_{i\geq 1} \left|\mathcal{C}_i^{\scriptscriptstyle (n)}(t)\right |^k. \end{equation}
Then \cite{spencer2007birth} shows that
for any bounded-size rule and for every $k \ge 2$, there exists a monotonically increasing
function $s_k : [0,t_c) \to [0, \infty)$ satisfying $s_k(0)=1$ and $\lim_{t \uparrow t_c}s_k(t)=\infty$, such that
\begin{equation}
\label{eqn:suscept-defn}
\bar s_k(t) := \frac{\mathcal{S}_k(t)}{n} \convp s_k(t) \quad \forall t\in [0,t_c).
\end{equation}
Along with the size of the components, another key quantity of interest is the complexity of components.
Recall the definition of the surplus of a component from \eqref{eqn:surp-def}, and denote $\xi_i^{\scriptscriptstyle (n)}(t) :={\bf spls}(\mathcal{C}_i^{\scriptscriptstyle (n)}(t))$ for the surplus of the component $\mathcal{C}_i^{\scriptscriptstyle (n)}(t)$. We will be interested in the joint vector of ordered component sizes and corresponding surplus
\begin{equation*}
((|\mathcal{C}_i(t)|, \xi_i(t) ): i\geq 1).
\end{equation*}
\subsection{Augmented Multiplicative coalescent}
\subsubsection{The multiplicative coalescent}
\label{sec:mult}
Let $l^2 = \{x = (x_1,x_2,\ldots): \sum_i x_i^2< \infty\}$. Then $l^2$ is a separable Hilbert space with the inner product $\langle x, y\rangle = \sum_{i=1}^{\infty}x_iy_i$,
$x=(x_i), y= (y_i) \in l^2$. Let
\begin{equation}
l^2_{\downarrow} = \{(x_1,x_2,\ldots): x_1\geq x_2\geq \cdots \geq 0, \sum_i x_i^2< \infty\}.
\label{eqn:ldown}
\end{equation}
Then $l^2_{\downarrow}$ is a closed subset of $l^2$ which we equip with the metric inherited from $l^2$. In \cite{aldous1997brownian} Aldous introduced a $l^2_{\downarrow}$
valued continuous time Markov process, referred to as the {\em standard multiplicative coalescent}, that can be used to describe the asymptotic behavior
of suitably scaled component size vector in Erd\H{o}s-R\'{e}nyi random graph evolution, near criticality. Subsequently, similar results have been shown to hold
for other random graph models (see \cite{bhamidi-budhiraja-wang2011,aldous2000random} and references therein). We now give a brief description of this Markov process. \\
Fix $x=(x_i)_{i \in \mathbb{N}}$. Let $\{\xi_{i,j}, i,j \in \mathbb{N}\}$ be a collection of independent rate one Poisson processes. Given $t \ge 0$, consider the random graph with vertex set $\mathbb{N}$ in which there exist $\xi_{i,j}([0,t x_i x_j/2]) + \xi_{j,i}([0,t x_i x_j/2]) $ edges between $(i,j)$,
$1 \le i < j < \infty$, and there are $\xi_{i,i}([0,t x_i^2/2])$ self-loops with the vertex $i \in \mathbb{N}$. The volume of a component $\mathcal{C}$ of this graph is defined to be
$${\bf vol}(\mathcal{C}) :=\sum_{i\in \mathcal{C}}x_i.$$
Let $X_i(x,t)$ be the volume of the $i$-th largest (by volume) component. It can be shown that
$X(x,t) = (X_i(x,t), i \ge 1) \in l^2_{\downarrow}$, a.s. (see Lemma 20 in \cite{aldous1997brownian}).
Define
$$T_t: {\mbox{BM}}(l^2_{\downarrow}) \to {\mbox{BM}}(l^2_{\downarrow}),$$
as $T_tf(x) = \mathbb{E}(f(X(x,t)))$. It is easily checked that $(T_t)_{t\ge 0}$ satisfies the semigroup property $T_{t+s} = T_tT_s$, $s,t\ge 0$, and \cite{aldous1997brownian}
shows that $(T_t)$ is Feller, i.e. $T_t(C_b(l^2_{\downarrow})) \subset C_b(l^2_{\downarrow})$ for all $t \ge 0$. The paper \cite{aldous1997brownian} also shows that the semigroup
$(T_t)$ along with an initial distribution $\mu \in \mathcal{P}(l^2_{\downarrow})$ determines a Markov process with values in $l^2_{\downarrow}$ and RCLL sample paths. Denoting by $P^{\mu}$
the probability distribution of this Markov process on $\mathcal{D}([0,\infty): l^2_{\downarrow})$, the Feller property says that $\mu \mapsto P^{\mu}$ is a continuous map.
One special choice of initial distribution for this Markov process is particularly relevant for the study of asymptotics
of random graph models. We now describe this distribution. Let $\{W(t)\}_{t\ge 0}$ be a standard Brownian motion, and for a fixed $\lambda \in {\mathbb{R}}$, define
\[W_\lambda(t) = W(t)+\lambda t-\frac{t^2}{2},\; t \ge 0.\]
Let $\hat{W}_{\lambda}$ denote the reflected version of $W_{\lambda}$, i.e.,
\begin{equation}
\hat{W}_\lambda(t) = W_\lambda(t) - \min_{0\leq s\leq t} W_\lambda(s), \; t \ge 0.
\label{eqn:inh-ref-bm}
\end{equation}
An excursion of $\hat{W}_\lambda$ is an interval $(l,u) \subset [0,+\infty)$ such that $\hat W_\lambda(l)=\hat W_\lambda(u)=0$ and $\hat W_\lambda(t)>0$ for all $t \in (l,u)$. Define $u-l$ as the size of the excursion. Order the sizes of excursions of $\hat W_\lambda$ as
\[X^*_1(\lambda)> X^*_2(\lambda)> X^*_3(\lambda)> \cdots\] and write $\boldsymbol{X}^*(\lambda) = (X^*_i(\lambda):i\geq 1).$
Then $\boldsymbol{X}^*(\lambda)$ defines a $l^2_{\downarrow}$ valued random variable (see Lemma 25 in \cite{aldous1997brownian})
and let $\mu_{\lambda}$ be its probability distribution. Using the Feller property and asymptotic connections with
certain non-uniform random graph models, the paper \cite{aldous1997brownian} shows that $\mu_{\lambda}T_t = \mu_{\lambda+t}$, for all $\lambda \in {\mathbb{R}}$ and $t \ge 0$, where
for $\mu \in \mathcal{P}(l^2_{\downarrow})$, $\mu T_t \in \mathcal{P}(l^2_{\downarrow})$ is defined in the usual way: $\mu T_t(A) = \int T_t(1_A)(x) \mu(dx)$, $A \in \mathcal{B}(l^2_{\downarrow})$. Using this consistency
property one can determine a unique probability measure $\mu_{\mbox{\tiny{MC}}} \in \mathcal{P}(\mathcal{D}((-\infty, \infty): l^2_{\downarrow}))$ such that, denoting the canonical coordinate process
on $\mathcal{D}((-\infty, \infty): l^2_{\downarrow})$ by $\{\pi_t\}_{-\infty < t < \infty}$,
$$
\mu_{\mbox{\tiny{MC}}} \circ (\pi_{t+\cdot})^{-1} = P^{\mu_t}, \; \mbox{ for all } t \in {\mathbb{R}} ,$$
where $\pi_{t+ \cdot}$ is the process $\{\pi_{t+s}\}_{s\ge 0}$.
The measure $\mu_{\mbox{\tiny{MC}}}$ is known as the {\em standard multiplicative coalescent}.
This measure plays a central role in characterizing asymptotic distribution of component size vectors in the critical window for random graph models \cite{aldous1997brownian,aldous2000random,bhamidi-budhiraja-wang2011}.
\subsubsection{The augmented multiplicative coalescent}
\label{sec:augmented-mc}
We will now augment the above construction and introduce a measure on a larger space that can be used to describe the joint asymptotic behavior of the component size vector and the associated surplus vector, for a broad family of random graph models.
Let $\mathbb{N}^{\infty} = \{y=(y_1, \cdots) : y_i \in \mathbb{N}, \mbox{ for all } i \ge 1 \}$ and define
\[\mathbb{U}_{\downarrow} = \{ (x_i, y_i)_{i\geq 1} \in l^2_{\downarrow} \times \mathbb{N}^\infty : \sum_{i=1}^\infty x_i y_i < \infty \mbox{ and }
y_m = 0 \mbox{ whenever } x_m =0, m \ge 1\}.\]
We will view $x_i$ as the volume of the $i$-th component and $y_i$ the surplus of the $i$-th component of a graph with vertex set $\mathbb{N}$.
Writing $x = (x_i)$ and $y = (y_i)$, we will sometimes denote $(x_i, y_i)$ as $z=(x,y)$.
We equip $\mathbb{U}_{\downarrow}$ with the metric
\begin{equation}
{\bf d}_{\scriptscriptstyle \mathbb{U}}((x,y),(x',y'))=\left( \sum_{i=1}^\infty(x_i-x'_i)^2\right)^{1/2}+ \sum_{i=1}^\infty|x_iy_i-x'_iy'_i|. \label{eqn:distance}
\end{equation}
Note that one natural metric on $\mathbb{U}_{\downarrow}$, denoted as ${\bf d}_1$, is the one obtained by replacing the second term in \eqref{eqn:distance}
with
$$\sum_{i=1}^\infty \frac{|y_i-y'_i|}{2^i}\wedge 1.$$
This metric corresponds to the
topology on $\mathbb{U}_{\downarrow}$ inherited from $\ell^2 \times \mathbb{N}^{\infty}$ taking the topology generated by the inner product $\langle \cdot, \cdot\rangle$
on $\ell^2$ and the product topology on $\mathbb{N}^{\infty}$; and then considering the product topology on $\ell^2 \times \mathbb{N}^{\infty}$. Although the metric ${\bf d}_1$ in some
respects is simpler to work with, it is not a natural metric to consider for the study of the joint distribution of component size and surplus process.
Another metric (which we denote as ${\bf d}_2$) that can be considered on $\mathbb{U}_{\downarrow}$ corresponds to replacing the second term in \eqref{eqn:distance} with
${\bf d}_{vt}(\mu_z, \mu_{z'})$, where $\mu_z = \sum_{i=1}^{\infty} \delta_{z_i}$, $\mu_{z'} = \sum_{i=1}^{\infty} \delta_{z'_i}$ and ${\bf d}_{vt}$ is the metric corresponding to the
vague topology on the space of $\mathbb{N} \cup \{\infty\}$ valued locally finite measures on $(0,\infty) \times \mathbb{N}$. However this metric as well is not suitable for our
purposes.
These points are further
discussed in Remark \ref{prodnofell}.
Let $\mathbb{U}_{\downarrow}^0 = \{(x_i, y_i)_{i\geq 1}\in \mathbb{U}_{\downarrow}: \mbox{ if } x_k=x_m, k \le m, \mbox{ then } y_k \ge y_m\}$.
We now introduce the {\em augmented multiplicative coalescent} (AMC). This is a continuous time Markov process with values in $(\mathbb{U}_{\downarrow}^0, {\bf d}_{\scriptscriptstyle \mathbb{U}})$, whose dynamics can heuristically be described as follows: The process jumps at any given time instant from state $(x,y) \in \mathbb{U}_{\downarrow}^0$ to:
\begin{itemize}
\item $(x^{ij}, y^{ij})$ at rate $x_ix_j$, $i \neq j$, where $(x^{ij}, y^{ij})$ is obtained by merging components $i$ and $j$ into a component with volume
$x_i+x_j$ and surplus $y_i+y_j$ and reordering the coordinates to obtain an element in $\mathbb{U}_{\downarrow}^0$.
\item $(x, y^i)$ at rate $x_i^2/2$, $i \ge 1$, where $(x, y^i)$ is the state obtained by increasing the surplus in the $i$-th component from $y_i$ to $y_{i}+1$ and reordering the coordinates (if needed) to obtain an element in $\mathbb{U}_{\downarrow}^0$.
\end{itemize}
Whenever $z=(x,y) \in \mathbb{U}_{\downarrow}^0$ is such that $\sum_{i=1}^\infty x_i < \infty$, there is a well defined Markov process $\{\boldsymbol{Z}(z,\lambda)\}_{\lambda \ge 0}$ that corresponds to the above transition mechanism, starting at time $\lambda=0$ in the state $z$. In fact in Section \ref{sec:main-constr-mc} (see also Theorem \ref{thm:smc-surplus}) we will see, that there is a well defined Markov process $\{Z(z,\lambda)\}_{\lambda \ge 0}$ corresponding to the above dynamical description for any $z \in \mathbb{U}_{\downarrow}^0$.
Define, for $\lambda \ge 0$, $\mathcal{T}_\lambda: {\mbox{BM}}(\mathbb{U}_{\downarrow}^0) \to {\mbox{BM}}(\mathbb{U}_{\downarrow}^0)$ as
$$(\mathcal{T}_\lambda f)(z) = \mathbb{E} f(\boldsymbol{Z}(z,\lambda)).$$
As for Aldous' multiplicative coalescent, there is one particular family of distributions that plays a special role. Recall the reflected parabolic Brownian motion $\hat W_\lambda(t)$
from \eqref{eqn:inh-ref-bm}. Let $\mathcal{P}$ be a Poisson point process on $[0,\infty) \times [0,\infty)$ with intensity $\lambda_{\infty}^{\otimes2}$ (where $\lambda_{\infty}$ is the Lebesgue measure on $[0, \infty)$) independent of $\hat W_\lambda$. Let $(l_i,r_i)$ be the $i$-th largest excursion of $\hat W_{\lambda}$. Define
$$ X^*_i(\lambda) = r_i-l_i \mbox{ and } Y^*_i(\lambda) = |\mathcal{P} \cap \{(t,z):0 \le z \le \hat W_\lambda(t), l_i \le t \le r_i \}|. $$
Then $\boldsymbol{Z}^*(\lambda) = (\boldsymbol{X}^*(\lambda), \boldsymbol{Y}^*(\lambda))$ is a.s. a $\mathbb{U}_{\downarrow}^0$ valued random variable, where $\boldsymbol{X}^* = (X^*_i)_{i\ge 1}$ and
$\boldsymbol{Y}^* = (Y^*_i)_{i\ge 1}$. Let $\nu_{\lambda}$ be its probability distribution.
In Theorem \ref{thm:smc-surplus} we will show that there there exists a $\mathbb{U}_{\downarrow}^0$ valued stochastic process $(\boldsymbol{Z}(\lambda))_{ -\infty < \lambda < \infty}$
such that $\boldsymbol{Z}(\lambda)$ has probability distribution $\nu_{\lambda}$ for every $\lambda \in (-\infty, \infty)$ and for all $f \in {\mbox{BM}}(\mathbb{U}_{\downarrow}^0)$, and $ \lambda_1 < \lambda_2 $, we have
$$ \mathbb{E}[ f(\boldsymbol{Z}(\lambda_2))| \{\boldsymbol{Z}(\lambda)\}_{\lambda \le \lambda_1}] =(\mathcal{T}_{\lambda_2-\lambda_1}f)( \boldsymbol{Z}(\lambda_1) ). $$
The process $\boldsymbol{Z}$ will be referred to as the {\em standard augmented multiplicative coalescent}.
We will also show that $\{\mathcal{T}_{\lambda}\}_{\lambda \ge 0}$ is a semigroup, namely $\mathcal{T}_{\lambda_1}\circ\mathcal{T}_{\lambda_2} = \mathcal{T}_{\lambda_1+\lambda_2}$,
for $\lambda_1, \lambda_2 \ge 0$ which is {\em nearly Feller}, in the sense made precise in the statement of Theorem \ref{thm:smc-surplus}.
It will be seen that this process plays a similar role in characterizing the asymptotic joint distributions of the component size and surplus vector in the
critical window as Aldous' standard multiplicative coalescent does in the study of asymptotics of the component size vector.
\section{Results}
\label{sec:main-results}
Our first result establishes the existence of the standard augmented coalescent process.
Let $\mathbb{U}_{\downarrow}^1 = \{z=(x,y) \in \mathbb{U}_{\downarrow}^0: \sum_i x_i = \infty\}$.
\begin{Theorem}
\label{thm:smc-surplus}
There is a collection of maps $\{\mathcal{T}_t\}_{t\ge 0}$, $\mathcal{T}_t: {\mbox{BM}}(\mathbb{U}_{\downarrow}^0) \to {\mbox{BM}}(\mathbb{U}_{\downarrow}^0)$ and a $\mathbb{U}_{\downarrow}^0$ valued stochastic process
$\{{\boldsymbol Z}(\lambda)\}_{-\infty < \lambda < \infty}= \{(\boldsymbol{X}(\lambda),\boldsymbol{Y}(\lambda))\}_{-\infty < \lambda < \infty}$ such that the following hold.
\begin{enumeraten}
\item $\{\mathcal{T}_t\}$ is a semigroup: $\mathcal{T}_t \circ \mathcal{T}_s = \mathcal{T}_{t+s}$, $s, t \ge 0$.
\item $\{\mathcal{T}_t\}$ is nearly Feller: For all $t > 0$, $f \in {\mbox{BM}}(\mathbb{U}_{\downarrow}^0)$ and $\{z_n\} \subset \mathbb{U}_{\downarrow}^0$, such that
$f$ is continuous at all points in $\mathbb{U}_{\downarrow}^1$ and
$z_n \to z$ for some $z \in \mathbb{U}_{\downarrow}^1$, we have $\mathcal{T}_{t} f(z_n) \to \mathcal{T}_tf(z)$.
\item The stochastic process $\{{\boldsymbol Z}(\lambda)\}$ satisfies the Markov property with semigroup $\{\mathcal{T}_t\}$:
For all $f \in {\mbox{BM}}(\mathbb{U}_{\downarrow}^0)$, and $ \lambda_1 < \lambda_2 $, we have
$$ \mathbb{E}[ f(\boldsymbol{Z}(\lambda_2))| \{\boldsymbol{Z}(\lambda)\}_{\lambda \le \lambda_1}] =(\mathcal{T}_{\lambda_2-\lambda_1}f)( \boldsymbol{Z}(\lambda_1) ). $$
\item Marginal distribution of $\boldsymbol{Z}(\lambda)$ is characterized through the parabolic reflected Brownian motion $\hat W_{\lambda}$: For each $\lambda \in \mathbb{RR}$,
$\boldsymbol{Z}(\lambda)$ has the probability distribution $\nu_{\lambda}$.
\item If $f \in {\mbox{BM}}(\mathbb{U}_{\downarrow}^0)$ is such that $f(x,y)=g(x)$ for some $g \in {\mbox{BM}}(l^2_{\downarrow})$, then
$$ (\mathcal{T}_t f)(z) = (T_t g)(x), \;\; \forall z=(x,y) \in \mathbb{U}_{\downarrow}^0. $$
Furthermore, $\{\boldsymbol{X}(\lambda) \}_{-\infty < \lambda < \infty}$ is Aldous's standard multiplicative coalescent.
\end{enumeraten}
\end{Theorem}
A precise definition of $\mathcal{T}_t$ can be found in Section \ref{sec:main-constr-mc}. We will refer
to $\boldsymbol{Z}$ as the standard augmented multiplicative coalescent.
Theorem \ref{thm:smc-surplus} will be proved in Section \ref{sec:main-st-ag-mc-ex}.
The next few theorems deal with bounded-size rules. Throughout this work we fix $K \in \mathbb{N}_0$, $F \in \Omega_K^4$ and consider a $F$ -BSR as introduced in Section \ref{sec:bsr}.
The first two results consider the asymptotics of the susceptibility functions. Recall the deterministic functions $s_k$ from \cite{spencer2007birth} introduced above \eqref{eqn:suscept-defn}.
\begin{Theorem}[{\bf Singularity of susceptibility}]
\label{thm:suscept-funct}
There exist $\alpha, \beta \in (0, \infty)$ such that
\begin{equation} s_2(t) = (1+O(t_c-t))\frac{\alpha}{t_c-t}, \quad s_3(t) = \beta [s_2(t)]^3(1+O(t_c-t)), \label{eq:eq803}\end{equation}
as $t\uparrow t_c$.
\end{Theorem}
\begin{Theorem}{\bf (Convergence of susceptibility functions)}
\label{thm:suscept-limit}
For every $\gamma \in (1/6, 1/5)$,
\begin{align}
&\sup_{t\in [0, t_n]} \left| \frac{n^{1/3}}{\bar s_2(t)} - \frac{n^{1/3}}{s_2(t)}\right| \convp 0\\
&\sup_{t\in [0, t_n]} \left| \frac{\bar s_3(t)}{(\bar s_2(t))^3} - \frac{ s_3(t)}{(s_2(t))^3} \right| \convp 0 ,
\end{align}
where $t_n = t_c - n^{-\gamma}$.
\end{Theorem}
We now state the main result which gives the asymptotic behavior in the critical scaling window as well as merging dynamics for all bounded-size rules.
\begin{Theorem}[{\bf Bounded-size rules: Convergence at criticality}]
\label{thm:crit-regime}
Let $\alpha, \beta \in (0, \infty)$ be as in Theorem \ref{thm:suscept-funct}. For $\lambda \in {\mathbb{R}}$ define
\[\bar{\boldsymbol{C}}^{\scriptscriptstyle (n)}(\lambda) := \left(\frac{\beta^{1/3}}{n^{2/3}}\left|\mathcal{C}_i\left(t_c+\frac{\alpha\beta^{2/3}}{n^{1/3}}\lambda\right)\right| :i\geq 1\right)
\mbox{ and } \bar{\boldsymbol{Y}}^{\scriptscriptstyle (n)}(\lambda) = \left (\xi_i\left(t_c+\frac{\alpha\beta^{2/3}}{n^{1/3}}\lambda\right) :i\geq 1\right).
\]
Then $\bar{\boldsymbol{Z}}^{\scriptscriptstyle (n)} = (\bar{\boldsymbol{C}}^{\scriptscriptstyle (n)}, \bar{\boldsymbol{Y}}^{\scriptscriptstyle (n)})$ is a stochastic process with sample paths in
$\mathcal{D}((-\infty,\infty):\mathbb{U}_{\downarrow})$ and
for any set of times $-\infty < \lambda_1< \lambda_2< ... < \lambda_m < \infty$
\begin{equation}
\label{eq:eq1241}\left(\bar{\boldsymbol{Z}}^{\scriptscriptstyle (n)}(\lambda_1), \ldots, \bar{\boldsymbol{Z}}^{\scriptscriptstyle (n)}(\lambda_m)\right) \convd \left(\boldsymbol{Z}(\lambda_1), \ldots, \boldsymbol{Z}(\lambda_m)\right)\end{equation}
as $n\to\infty$, where $\boldsymbol{Z}$ is as in Theorem \ref{thm:smc-surplus}.
\end{Theorem}
\subsection{Background}
We now make some comments on the problem background and future directions.
\begin{enumeratea}
\item {\bf Critical random graphs:} Starting with the early work of Erd\H{o}s-R\'enyi \cite{er-1,er-2}, there is now a large literature on understanding phase transitions in random graph models, see e.g. \cite{bollobas-rg-book,bollobas-riordan-janson,janson-luczak-bb} and the references therein. Proving and identifying phase transitions in dynamic random graph models such as the bounded-size rule requires a relatively new set of ideas and is much more recent \cite{spencer2007birth}. The study of Erd\H{o}s-R\'enyi random graph in the critical regime was carried out
in \cite{janson1994birth,aldous1997brownian}. In particular the paper, \cite{aldous1997brownian} introduced the standard multiplicative coalescent to understand the merging dynamics of the Erd\H{o}s-R\'enyi random graph at criticality. The barely subcritical and supercritical regimes of the Bohman-Frieze process were studied respectively in \cite{bf-spencer-perkins-kang} and \cite{janson2010phase}, with the latter identifying the scaling exponents for the susceptibility functions for the special case of the Bohman-Frieze (BF) process by using the special form of the differential equations for the BF process. The current work extends this result to all bounded-size rules (Theorem \ref{thm:suscept-funct}) by viewing such processes as random graph processes with immigration and attachment (see Section \ref{sec:proof-conv-susceptibility}).
\item {\bf Unbounded-size rules: } One of the reasons for renewed interest in such models is the recent study of the product rule (\cite{achlioptas2009explosive}), where as before one chooses two edges at random and then uses the edge that minimizes the product of the components at the end points of the chosen edges. This is an example of an unbounded-size rule and simulations in \cite{achlioptas2009explosive} suggest different behavior at criticality as compared to the usual Erd\H{o}s-R\'enyi or BF random graph processes. There has been recent progress in rigorously understanding the continuity at the critical point \cite{riordan2011achlioptas} as well the subcritical regime \cite{riordan2012evolution}.
Such unbounded rules can be regarded as formal limits of $K$-bounded-size rules analyzed in the current work, as $K \to \infty$.
It would be of great interest to identify and understand the critical scaling window of such processes.
\item {\bf Related open questions:} In the context of bounded-size rules our results suggest other related questions. In particular, there has been recent progress in understanding structural properties of the component sizes of the Erd\H{o}s-R\'enyi random graph at criticality, in particular see \cite{addario2009continuum,addario2009critical} which use information about the surplus and component sizes in \cite{aldous1997brownian} to prove that the components viewed as metric spaces, converge to random fractals closely related to the continuum random tree \cite{aldous1991-crt} with shortcuts due to surplus edges. Our results strongly suggest the components in any bounded-size rule at criticality belong to the same universality class. Proving this will require substantially new ideas.
\end{enumeratea}
\subsection{Organization of the paper}
The two main results in this paper are Theorems \ref{thm:smc-surplus} and \ref{thm:crit-regime}.
In Section \ref{sec:main-constr-mc}
we introduce the semigroup
$\{\mathcal{T}_t\}_{t\ge 0}$ and, as a first step towards Theorem \ref{thm:smc-surplus}, establish in Theorem \ref{theo:welldef-feller} the existence of a $\mathbb{U}_{\downarrow}^0$ valued Markov process associated with this semigroup, starting from an arbitrary
initial value.
Then in Section \ref{sec:main-st-ag-mc-ex} we complete the proof of Theorem \ref{thm:smc-surplus}.
We then proceed to the analysis of bounded-size rules in Section \ref{sec:main-bsr-susceptibility} where we study the differential equation systems associated with the BSR process and prove Theorems \ref{thm:suscept-funct} and \ref{thm:suscept-limit}.
Finally in Section \ref{sec:main-coupling} we complete the proof of Theorem \ref{thm:crit-regime}.
\section{The augmented multiplicative coalescent}
\label{sec:main-constr-mc}
We begin by making precise the formal dynamics of the augmented multiplicative coalescent process given in Section \ref{sec:augmented-mc}.
Fix $(x,y)\in \mathbb{U}_{\downarrow}^0$. Let $\{\xi_{i,j}\}_{i,j \in \mathbb{N}}$ be a collection of i.i.d. rate one Poisson processes. Let ${\bf{G}}(z,t)$, where $z=(x,y)$, be the random graph on vertex set $\mathbb{N}$ given as follows:\\
(I) For $i \in \mathbb{N}$, there are $y_i$ self-loops to the vertex $i$. \\
(II) For $i<j \in \mathbb{N}$, there are $\xi_{i,j}([0,t x_i x_j/2]) + \xi_{j,i}([0,t x_i x_j/2])$ edges between vertices $i$ and $j$. For $i \in \mathbb{N}$, there are $\xi_{i,i}([0,t x_i^2/2])$ self-loops to the vertex $i$. \\
Let $\mathcal{F}^x_t = \sigma\{ \xi_{i,j}([0,sx_ix_j/2]): 0 \le s \le t, \; i,j \in \mathbb{N}\}$, $t\ge 0$.
Recall the volume of a component $\mathcal{C}$ is defined to be ${\bf vol}(\mathcal{C}) = \sum_{i \in \mathcal{C}} x_i$. The surplus of a finite connected graph was defined in \eqref{eqn:surp-def}.
For infinite graphs the definition requires some care.
We define the surplus for a connected graph ${\bf{G}}$ with vertex set a subset of $\mathbb{N}$ as
$$ {\bf spls}({\bf{G}}) := \lim_{k \to \infty} {\bf spls}({\bf{G}}^{\scriptscriptstyle [k]}), $$
where ${\bf{G}}^{\scriptscriptstyle [k]}$ is the {\bf induced subgraph} that has the vertex set $[k]$ (the subgraph with vertex set $[k]$ and all edges between vertices in $[k]$ that
are present in ${\bf{G}}$). It is easy to check that this definition of surplus does not depend on the labeling of the vertices. Further note that the surplus of a connected graph might be infinite with this definition.\\
Thus letting $\tilde \mathcal{C}_i(t)$ be the $i$-th largest component (in volume) in ${\bf{G}}(z,t)$,
define $X_i(z,t) : = {\bf vol}(\tilde \mathcal{C}_i(t))$ and $Y_i(z,t):= {\bf spls}(\tilde \mathcal{C}_i(t))$ to be the {\bf volume} and the {\bf surplus} of the $i$-th largest component at time $t$. In case two components have the same volume, the ordering of $(\tilde \mathcal{C}_i(t): i \ge 1)$ is taken to be such that $Y_m(z,t) \ge Y_k(z,t)$ whenever $m \le k$ and $X_m(z,t) = X_k(z,t)$.
Let $\boldsymbol{X}^z(t) := (X_i(z,t): i \ge 1 )$ and $\boldsymbol{Y}^z(t) := (Y_i(z,t): i \ge 1)$.
The paper \cite{aldous1997brownian} shows that $\boldsymbol{X}^z(t) \in l^2_{\downarrow}$ a.s. for all $t\ge 0$. The following result shows that
$\boldsymbol{Z}^z(t) = (\boldsymbol{X}^z(t), \boldsymbol{Y}^z(t)) \in \mathbb{U}_{\downarrow}^0$ a.s., for all $t$.
\begin{Theorem}
\label{theo:welldef-feller}
Fix $ z=(x,y)\in \mathbb{U}_{\downarrow}^0$ and let $(\boldsymbol{X}^z(t),\boldsymbol{Y}^z(t))_{t \ge 0}$ be the stochastic process described above, then
for any fixed $t \ge 0$, $(\boldsymbol{X}^z(t),\boldsymbol{Y}^z(t)) \in \mathbb{U}_{\downarrow}^0$.
\end{Theorem}
The above theorem will be proved in Section \ref{sec:well-defined}.
For $t\ge 0$, define $\mathcal{T}_t: {\mbox{BM}}(\mathbb{U}_{\downarrow}^0) \to {\mbox{BM}}(\mathbb{U}_{\downarrow}^0)$ as
$$\mathcal{T}_t f(z) = \mathbb{E} f(\boldsymbol{Z}^z(t)),\; z \in \mathbb{U}_{\downarrow}^0,\; f \in {\mbox{BM}}(\mathbb{U}_{\downarrow}^0).$$
The following result shows that $\{\mathcal{T}_t\}$ is a semigroup that is (nearly) Feller.
\begin{Theorem}
\label{theo:welldef-fellerb}
For $t, s \ge 0$, $\mathcal{T}_t\circ \mathcal{T}_s = \mathcal{T}_{t+s}$. For all $t > 0$, $f \in {\mbox{BM}}(\mathbb{U}_{\downarrow}^0)$ and $\{z_n\} \subset \mathbb{U}_{\downarrow}^0$, such that
$f$ is continuous at all points in $\mathbb{U}_{\downarrow}^1$ and
$z_n \to z$ for some $z \in \mathbb{U}_{\downarrow}^1$, we have $\mathcal{T}_{t} f(z_n) \to \mathcal{T}_tf(z)$.
\end{Theorem}
The above theorem will be proved in Section \ref{sec:feller-property}.
Throughout we will assume, without loss of generality, that for all $z \in \mathbb{U}_{\downarrow}^0$, $\boldsymbol{Z}^z$ is constructed using the same set of Poisson processes $\{\xi_{i,j}\}$.
This coupling of $\boldsymbol{Z}^z$ for different values of $z$ will not be noted explicitly in the statement of various results.
We begin with the following elementary lemma.
\begin{Lemma}
\label{lemma:borel-cantelli}
Let $\{\mathcal{F}_m\}_{m \in \mathbb{N}_0}$ be a filtration given on some probability space.\\
(i) Let $\{Z_m\}_{m \ge 0}$ be a $\{\mathcal{F}_m\}$ adapted sequence of nondecreasing random variables such that $Z_0=0$. Let
$\lim_{m \to \infty}Z_m=Z_\infty$. Suppose there exists a nonnegative random variable $U$ such that $U<\infty$ a.s. and
$ \sum_{m=1}^\infty \mathbb{E}[Z_m-Z_{m-1} | \mathcal{F}_{m-1} ] \le U.$ Then for any $\epsilon \in (0,1)$,
$$ \mathbb{P}\{ Z_\infty > \epsilon \} \le \frac{1+\epsilon}{\epsilon} \mathbb{E}[U \wedge 1]. $$
(ii) Let $\{A_m\}$ be a sequence of events such that $A_m \in \mathcal{F}_m$. Suppose there exists a random variable $U<\infty$ a.s. such that $\sum_{m=1}^\infty \mathbb{E}[ {{\rm 1\hspace{-0.90ex}1}}_{A_m} | \mathcal{F}_{m-1}] \le U$. Then $\mathbb{P}\{ A_m \mbox{ i.o.} \}=0$. Furthermore,
$$ \mathbb{P}\{ \cup_{m=1}^\infty A_m \} \le 2 \mathbb{E}[U \wedge 1]. $$
\end{Lemma}
{\bf Proof:} (i) Define $B_m := \sum_{i=1}^m \mathbb{E}[Z_i-Z_{i-1}| \mathcal{F}_{i-1}]$. Note that $B_m$ is nondecreasing and $\mathcal{F}_{m-1}$-measurable.
Define $\tau=\inf\{ l: B_{l+1} > 1 \}$
where the infimum over an empty set is taken to be $\infty$.
Since $B_m$ is predictable, $\tau$ is a stopping time and, for all $m$, $B_{m\wedge \tau} \le 1$.
Let $B_{\infty} = \lim_{m\to \infty} B_m$. Since $Z_{m \wedge \tau}-B_{m\wedge\tau}$ is a martingale,
$$\mathbb{E}[Z_\tau] = \lim_{m \to \infty} \mathbb{E}[Z_{m \wedge \tau}] = \lim_{m \to \infty} \mathbb{E}[B_{m \wedge \tau}] \le \lim_{m \to \infty} \mathbb{E}[B_{m} \wedge 1]=\mathbb{E}[B_\infty \wedge 1]. $$
Thus $$ \mathbb{P}\{ Z_\infty > \epsilon\} \le \mathbb{P}\{ \tau < \infty\}+ \frac{1}{\epsilon} \mathbb{E}[B_\infty \wedge 1] = \mathbb{P}\{ B_\infty > 1\}+ \frac{1}{\epsilon} \mathbb{E}[B_\infty \wedge 1] \le \frac{1+\epsilon}{\epsilon} \mathbb{E}[U \wedge 1].$$
(ii) The first statement is immediate from the Borel-Cantelli lemma (cf. Theorem 5.3.7 \cite{durrett-book}). For the second statement note that for any $\epsilon \in (0, 1)$, we have $ { \cup_{m=1}^\infty A_m } = \{\sum_{m=1}^\infty {{\rm 1\hspace{-0.90ex}1}}_{A_m} > \epsilon\} $. Now applying part (i) to $Z_m = \sum_{k=1}^m {{\rm 1\hspace{-0.90ex}1}}_{A_k}$ and taking $\epsilon \to 1$ yields the desired result. \ \ \rule{1ex}{1ex}\\
Next, we present a result from \cite{aldous1997brownian} that will be used here. We begin with some notation. For $x \in l^2_{\downarrow}$, we write $x^{\scriptscriptstyle [k]}=(x_1,...,x_k, 0, 0, ...)$ for the $k$-truncated version of $x$.
Similarly, for a sequence $x^{\scriptscriptstyle (n)}=(x_1^{\scriptscriptstyle (n)},x_2^{\scriptscriptstyle (n)},...)$ of elements in $l^2_{\downarrow}$, $x^{\scriptscriptstyle (n)[k]}$ is the $k$-truncation of $x^{\scriptscriptstyle (n)}$.
For $z = (x,y), z^{\scriptscriptstyle (n)} = (x^{\scriptscriptstyle (n)}, y^{\scriptscriptstyle (n)}) \in \mathbb{U}_{\downarrow}^0$ $z^{\scriptscriptstyle [k]}, y^{\scriptscriptstyle [k]}, z^{\scriptscriptstyle (n)[k]}, y^{\scriptscriptstyle (n)[k]}$ are defined similarly.\\
Recall the construction of ${\bf{G}}(z,t)$ described in items (I) and (II) at the beginning of the section.
We will distinguish the surplus created in $\tilde \mathcal{C}_i(t)$ by the action in item (I) and that in item (II). The former will be referred to as the type I surplus and denoted as
$\tilde Y_i(z,t)$ while the latter will be referred to as the type II surplus and denoted as $\hat Y_i(z,t) \equiv \hat Y_i(x,t)$.
More precisely,
$$ \tilde Y_i(z,t) = \sum_{j \in \tilde \mathcal{C}_i(t)} y_j \; \mbox{ and } \;\hat Y_i(z,t)=Y_i(z,t)-\tilde Y_i(z,t). $$
Also define
$$ \tilde R(z,t) := \sum_{i=1}^\infty X_i(z,t) \tilde Y_i(z,t), \;\; \hat R(x,t) \equiv \hat R(z,t) :=\sum_{i=1}^\infty X_i(z,t) \hat Y_i(z,t) $$
and
$$R(z,t):=\sum_{i=1}^\infty X_i(z,t)Y_i(z,t), \;\; S(x,t) \equiv S(z,t) := \sum_{i=1}^\infty (X_i(x,t))^2.$$
The following properties of $S$ and $\boldsymbol{X}$ have been established in \cite{aldous1997brownian}.
\begin{Theorem}
\label{theo:aldous-mc-s} [Aldous\cite{aldous1997brownian}]
(i) For every $x \in l^2_{\downarrow}$ and $t\ge 0$, we have $S(x,t)<\infty $ a.s. and $S(x^{\scriptscriptstyle [k]}, t) \uparrow S(x,t)$ as $k \to \infty$.\\
(ii) If $x^{\scriptscriptstyle (n)} \to x$ in $l^2_{\downarrow}$, then $\boldsymbol{X}(x^{\scriptscriptstyle (n)},t) \convp \boldsymbol{X}(x,t)$ in $l^2_{\downarrow}$, as $n \to \infty$. In particular, $\{S(x^{\scriptscriptstyle (n)},t)\}_{n \ge 1}$ is tight.
\end{Theorem}
\subsection{Existence of the augmented MC}
\label{sec:well-defined}
This section proves Theorem \ref{theo:welldef-feller}.
We begin by considering the type I surplus.
\begin{Proposition}
\label{lemma:211} For any $t \ge 0$ and $z \in \mathbb{U}_{\downarrow}^0$, $ \tilde R(z,t) = \sum_{i=1}^\infty X_i(z,t) \tilde Y_i(z,t) < \infty $ a.s.
\end{Proposition}
Proof of Proposition \ref{lemma:211} is given below Lemma \ref{lemma:765}. The basic idea is to bound the truncated version $\tilde R^{\scriptscriptstyle [k]}=\tilde R(z^{\scriptscriptstyle [k]},t)$ using a martingale argument, and then let $k \to \infty$. The truncation error is controlled using Lemma \ref{lemma:248} below and a suitable supermartingale is
constructed in Lemma \ref{lemma:765}.
\begin{Lemma}
\label{lemma:248}
For every $z \in \mathbb{U}_{\downarrow}^0$ and $t \ge 0$, as $k\to \infty$, $\tilde R(z^{\scriptscriptstyle [k]},t) \to \tilde R(z,t) \le \infty$ a.s.
\end{Lemma}
{\bf Proof:} Fix $t\ge 0$. Denote by $A_{ij}$ [resp. $A_{ij}^{\scriptscriptstyle [k]}$] the event that there exists a path from $i$ to $j$ in ${\bf{G}}(z,t)$ [resp. ${\bf{G}}(z^{\scriptscriptstyle [k]},t)$], with the
convention that $\mathbb{P} \set{A_{ii}} = \mathbb{P} \{ A_{ii}^{\scriptscriptstyle [k]} \}= 1$. Let
$$f_i = \sum_{j=1}^\infty y_j {{\rm 1\hspace{-0.90ex}1}}_{A_{ij}}, \;\; f_i^{\scriptscriptstyle [k]} =\sum_{j=1}^k y_j {{\rm 1\hspace{-0.90ex}1}}_{A_{ij}^{\scriptscriptstyle [k]}}.$$
Then
$$ \tilde R (z,t) = \sum_{i=1}^\infty f_i x_i, \;\; \tilde R(z^{\scriptscriptstyle [k]},t)= \sum_{i=1}^\infty f_i^{\scriptscriptstyle [k]}x_i.$$
Since $A_{ij}^{\scriptscriptstyle [k]} \uparrow A_{ij}$, we have $ f_i^{\scriptscriptstyle [k]} \uparrow f_i$. The result now follows from an application of monotone convergence theorem. \ \ \rule{1ex}{1ex}\\
\begin{Lemma}\label{lemma:765}
Suppose that $z = (x,y) = z^{\scriptscriptstyle [k]}$ for some $k \ge 1$ and that $\sum_j y_j \neq 0$. Then
$$ A_t=A(z,t)= \log \tilde R(z,t) - \int_0^t S(z,u)du$$
is a supermartingale with respect to the filtration $\mathcal{F}_t^x = \sigma\{ \xi_{i,j}([0,sx_ix_j/2]); 0 \le s \le t, \; i,j \in \mathbb{N}\}$.
\end{Lemma}
{\bf Proof:}
From the construction of $\boldsymbol{Z}(z, \cdot)$ we see that $ \tilde R(z,t)$ is a pure jump, nondecreasing process that at any time instant $t$, jumps at rate
$X_i(z,t-)X_j(z,t-)$, $1 \le i < j \le k$, with jump sizes $B_{ij}(t-) = X_i(z,t-)\tilde Y_j(z,t-)+X_j(z,t-) \tilde Y_i(z,t-)$. Consequently
$\log \tilde R(z,t)$ jumps at the same rate, with corresponding jump size
$\log ( 1 + \frac{B_{ij}(t-)}{\tilde R(z,t-)})$. From this and elementary properties of Poisson processes it follows that
$$
\log \tilde R(z,t) = \log \tilde R(z,0) + \sum_{1 \le i < j \le k} \int_0^t \log \left (1 + \frac{B_{ij}(u)}{\tilde R(z,u)}\right ) X_i(z,u)X_j(z,u) du + M(t),$$
where $M$ is a $\mathcal{F}_t^x $ martingale. Consequently, for $0 \le s < t < \infty$
\begin{equation}
\label{eq:ab619}
\log \tilde R(z,t) - \log \tilde R(z,s)= \sum_{1 \le i < j \le k} \int_s^t \log \left (1 + \frac{B_{ij}(u)}{\tilde R(z,u)}\right ) X_i(z,u)X_j(z,u) du + M(t) - M(s).
\end{equation}
Next note that, for $u \ge 0$
\begin{align*}
&\sum_{1 \le i < j \le k} \log \left (1 + \frac{B_{ij}(u)}{\tilde R(z,u)}\right ) X_i(z,u)X_j(z,u)\\
\le & \sum_{1 \le i < j \le k} \frac{B_{ij}(u)}{\tilde R(z,u)} X_i(z,u)X_j(z,u)\\
= & \sum_{1 \le i < j \le k} \frac{X_i(z,u)\tilde Y_j(z,u)+X_j(z,u) \tilde Y_i(z,u)}{\tilde R(z,u)} X_i(z,u)X_j(z,u)\\
\le & S(z,u).
\end{align*}
Using this observation in \eqref{eq:ab619} we now have
$$
\mathbb{E} \left [\log \tilde R(z,t) - \log \tilde R(z,s) \mid \mathcal{F}_s^x \right] \le \mathbb{E} \left [ \int_s^t S(z,u) du \mid \mathcal{F}_s^x \right].$$
The result follows.
\ \ \rule{1ex}{1ex}\\
{\bf Proof of Proposition \ref{lemma:211}:} Fix $z = (x,y) \in \mathbb{U}_{\downarrow}^0$. The result is trivially true if $\sum_i y_i = 0$. Assume now that $\sum_i y_i \neq 0$.
For $k \ge 1$ and $a \in (0, \infty)$, define
$T_{a}^{\scriptscriptstyle [k]} =\inf\{ s \ge 0: S(z^{\scriptscriptstyle [k]},s) \ge a \}.$
Fix $k \ge 1$ and assume without loss of generality that $\sum_{i=1}^k y_i > 0$.
Write $R^{\scriptscriptstyle [k]}(t)=R(z^{\scriptscriptstyle [k]},t)$, and $A^{\scriptscriptstyle [k]}(t)=A(z^{\scriptscriptstyle [k]},t)$ where $A$ is as in Lemma \ref{lemma:765}.
From the supermartingale property $\mathbb{E}[A^{\scriptscriptstyle [k]} (T_{a}^{\scriptscriptstyle [k]} \wedge t)] \le \mathbb{E}[A^{\scriptscriptstyle [k]}(0)]= \log \tilde R^{\scriptscriptstyle [k]}(0)$. Therefore
$$ \mathbb{E}\left[ \log \frac {\tilde R^{\scriptscriptstyle [k]}(T_{a}^{\scriptscriptstyle [k]} \wedge t)}{\tilde R^{\scriptscriptstyle [k]}(0)}\right] \le \mathbb{E}\left[ \int_0^{T_{a}^{\scriptscriptstyle [k]} \wedge t} S(z^{\scriptscriptstyle [k]}, u) du \right] \le ta. $$
Thus
\begin{align*}
\mathbb{P}\{ \tilde R^{\scriptscriptstyle [k]}(t) > m \}
\le \mathbb{P}\{ \tilde R^{\scriptscriptstyle [k]}(t) > m, T_a^{\scriptscriptstyle [k]} > t \} + \mathbb{P}\{ T_a^{\scriptscriptstyle [k]} \le t \}
\le \frac{ t a} { \log m -\log \tilde R^{\scriptscriptstyle [k]}(0)} + \mathbb{P} \{ T_a^{\scriptscriptstyle [k]} \le t\}.
\end{align*}
By Lemma \ref{lemma:248}, $\tilde R^{\scriptscriptstyle [k]}(t) \to \tilde R(z,t)$, and by Theorem \ref{theo:aldous-mc-s} (i), $S(z^{\scriptscriptstyle [k]},t) \to S(z,t)$ when $k \to \infty$.
Therefore letting $k \to \infty$ on both sides of the above inequality, we have
\begin{equation}
\mathbb{P}\{ \tilde R(z,t) > m \} \le \frac{ t a} { \log m -\log \tilde R(z,0)} + \mathbb{P} \{ S(z,t) \ge a\}. \label{eqn:282}
\end{equation}
The result now follows on first letting $m \to \infty$ and then $a \to \infty$ in the above inequality. \ \ \rule{1ex}{1ex} \\
The following result is an immediate consequence of the estimate in \eqref{eqn:282} and Theorem \ref{theo:aldous-mc-s}(ii).
\begin{Corollary}\label{corr:ab1157}
If $z^{\scriptscriptstyle (n)} \to z$ in $\mathbb{U}_{\downarrow}^0$, then for every $t\ge 0$, $\{\tilde R(z^{\scriptscriptstyle (n)},t)\}_{ n \ge 1}$ is tight.
\end{Corollary}
Next we consider the type II surplus. Let, for $x \in l^2_{\downarrow}$
$$\mathcal{G}_t^x := \sigma\{ \{\xi_{i,j}([0,sx_ix_j/2]) = 0\}:\; 0 \le s \le t,\; i,j \in \mathbb{N} \}. $$
The $\sigma$-field $\mathcal{G}_t^x$ records the information whether or not $i$ and $j$ are in the same component at time $s$, for all $i,j$ and for all $s \le t$.
In particular, components $\{\tilde \mathcal{C}_i(s), i \ge 1, s \le t\}$ can be determined from the information in $\mathcal{G}_t^x$ and consequently,
$\boldsymbol{X}(x,t)$ is $\mathcal{G}_t^x $ measurable.
\begin{Lemma}
\label{lemma:786}
(i) Fix $x \in l^2_{\downarrow}$ and $t \ge 0$. Then $\hat R(x,t) < \infty$ a.s.\\
(ii) Let $x^{\scriptscriptstyle (n)} \to x$ in $l^2_{\downarrow}$. Then the sequence $\{ \hat R(x^{\scriptscriptstyle (n)},t) \}_{n \ge 1}$ is tight.
\end{Lemma}
{\bf Proof:} Note that (i) is an immediate consequence of (ii). Consider now (ii).
For fixed $x \in l^2_{\downarrow}$ and $t \ge 0$, let $\hat \mu_i(x,t)$ denote the probability law of $\hat Y_i(x,t)$, conditioned on $\mathcal{G}_t^x$.
Then, for a.e. $\omega$, $\hat \mu_i(x,t)$ is a Poisson random variable with parameter
$$
\int_0^t \sum_{j=1}^\infty (\sum_{k,k'\in \tilde \mathcal{C}_j(s)} \frac{1}{2}x_k x_{k'} ) {{\rm 1\hspace{-0.90ex}1}}_{\{\tilde \mathcal{C}_j(s) \subset \tilde \mathcal{C}_i(t)\}} ds
= \int_0^t \frac{1}{2} \sum_{j=1}^\infty (X_j(x,s))^2 {{\rm 1\hspace{-0.90ex}1}}_{\{\tilde \mathcal{C}_j(s) \subset \tilde \mathcal{C}_i(t)\}} ds
\le \frac{t}{2} (X_i(x,t))^2,
$$
where the last inequality is a consequence of the inequality $\sum_{j:\tilde \mathcal{C}_j(s) \subset \tilde \mathcal{C}_i(t) }(X_j(x,s))^2 \le (X_i(x,t))^2$.
Therefore $\hat \mu_i(x,t)\le_d \hat \nu_i(x,t)$, a.s., where $\hat \nu_i(x,t)$ is a random probability measure on $\mathbb{N}$ such that for a.e. $\omega$,
$\hat \nu_i(x,t)$ is Poisson with parameter $\frac{t}{2} (X_i(x,t, \omega))^2 $.
A similar argument shows that the conditional distribution of $\sum_{i=1}^{\infty} \hat Y_i(x,t)$, given $\mathcal{G}_t^x$ is a.s. stochastically dominated by
a random measure on $\mathbb{N}$ that, for a.e. $\omega$ has a Poisson distribution with parameter
$\sum_{i=1}^{\infty}\frac{t}{2} (X_i(x,t, \omega))^2 = \frac{t}{2} S(x,t)$.
Also, if $x^{\scriptscriptstyle (n)}$ is a sequence converging to $x$ in $l^2_{\downarrow}$, we have that for each $n$, the conditional distribution of $\sum_{i=1}^{\infty} \hat Y_i(x^{\scriptscriptstyle (n)},t)$,
given $\mathcal{G}_t^{x^{\scriptscriptstyle (n)}}$ is a.s. stochastically dominated by a Poisson random variable with parameter $\frac{t}{2} S(x^{\scriptscriptstyle (n)},t)$.
From Theorem \ref{theo:aldous-mc-s}(ii), $\{S(x^{\scriptscriptstyle (n)},t)\}_{n\ge 1}$ is tight. Combining these facts we have that
$\{\sum_{i=1}^{\infty} \hat Y_i(x^{\scriptscriptstyle (n)},t)\}_{n\ge 1}$ is a tight family. Finally, note that
$\hat R(x^{\scriptscriptstyle (n)},t) \le X_1(x^{\scriptscriptstyle (n)},t)\left (\sum_{i=1}^{\infty} \hat Y_i(x^{\scriptscriptstyle (n)},t) \right)$. The tightness of $\{ \hat R(x^{\scriptscriptstyle (n)},t) \}_{n \ge 1}$
now follows on combining the above established tightness of $\{\sum_{i=1}^{\infty} \hat Y_i(x^{\scriptscriptstyle (n)},t)\}_{n\ge 1}$ and the tightness
of $\{X_1(x^{\scriptscriptstyle (n)},t)\}_{n\ge 1}$, where the latter is once again a consequence of Theorem \ref{theo:aldous-mc-s}(ii).
\ \ \rule{1ex}{1ex}\\
We now complete the proof of Theorem \ref{theo:welldef-feller}.\\
{\bf Proof of Theorem \ref{theo:welldef-feller}.} Fix $z=(x,y) \in \mathbb{U}_{\downarrow}^0$ and $t \ge 0$. From Lemma \ref{lemma:786} (i) $\hat R(x,t) < \infty$ a.s.
Also, from Proposition \ref{lemma:211}, $\tilde R(z,t) < \infty$ a.s. The result now follows on recalling that
$R(z,t) = \hat R(x,t) + \tilde R(z,t)$. \ \ \rule{1ex}{1ex}\\
We also record the following consequence of Lemma \ref{lemma:786} and Corollary \ref{corr:ab1157} for future use.
\begin{Corollary} \label{lemma:tight-r}
If $z^{\scriptscriptstyle (n)} \to z$ in $\mathbb{U}_{\downarrow}^0$, then
$ \{R(z^{\scriptscriptstyle (n)},t)\}_{n\ge 1} $ is tight.
\end{Corollary}
\subsection{Feller property of the augmented MC}
\label{sec:feller-property}
In this section, we will prove Theorem \ref{theo:welldef-fellerb}. In fact we will show that
if $ z^{\scriptscriptstyle (n)}= (x^{\scriptscriptstyle (n)},y^{\scriptscriptstyle (n)})$ converges to $z=(x,y)$ in $\mathbb{U}_{\downarrow}^0$, and $z \in \mathbb{U}_{\downarrow}^1$, then
\begin{equation}\label{eq:ab1200}
(\boldsymbol{X}(z^{\scriptscriptstyle(n)},t) ,\boldsymbol{Y}(z^{\scriptscriptstyle(n)},t)) \convp (\boldsymbol{X}(z,t), \boldsymbol{Y}(z,t)). \end{equation}
The following lemma is immediate from the definition of ${\bf d}_{\scriptscriptstyle \mathbb{U}}(\cdot, \cdot)$.
\begin{Lemma}
\label{lemma:basic-analysis}
Suppose $(x,y), (x^{\scriptscriptstyle (n)}, y^{\scriptscriptstyle (n)}) \in \mathbb{U}_{\downarrow}$ for $n \ge 1$. Then $\lim_{n \to \infty} {\bf d}_{\scriptscriptstyle \mathbb{U}}((x,y),(x^{\scriptscriptstyle (n)},y^{\scriptscriptstyle (n)}))=0$ if and only if the following three conditions hold:\\
(i) $\lim_{n \to \infty}\sum_{i=1}^\infty (x_i^{\scriptscriptstyle (n)}-x_i)^2 = 0$.\\
(ii) $y_i^{\scriptscriptstyle (n)}=y_i$ for $n$ sufficiently large, for all $ i \ge 1$.\\
(iii) $\lim_{n \to \infty}\sum_{i=1}^\infty x_i^{\scriptscriptstyle (n)}y_i^{\scriptscriptstyle (n)} = \sum_{i=1}^\infty x_iy_i$.
\end{Lemma}
The key ingredient in the proof is the following lemma the proof of which is given after Lemma \ref{strict}.
\begin{Lemma} \label{lemma:feller-conv-prob}
Let $ z^{\scriptscriptstyle (n)}= (x^{\scriptscriptstyle (n)},y^{\scriptscriptstyle (n)})$ converge to $z=(x,y)$ in $\mathbb{U}_{\downarrow}^0$. Suppose that $z \in \mathbb{U}_{\downarrow}^1$. Then\\
(i) $Y_i(z^{\scriptscriptstyle (n)},t) \convp Y_i(z,t)$ for all $i \ge 1$.\\
(ii) $\sum_{i=1}^\infty X_i(z^{\scriptscriptstyle (n)},t)Y_i(z^{\scriptscriptstyle (n)},t) \convp \sum_{i=1}^\infty X_i(z,t)Y_i(z,t)$.
\end{Lemma}
Proof of Theorem \ref{theo:welldef-fellerb} can now be completed as follows.\\
{\bf Proof of Theorem \ref{theo:welldef-fellerb}.}
The first part of the theorem is immediate from the construction given at the beginning of Section \ref{sec:main-constr-mc} and elementary properties of Poisson processes.
For the second part, consider $ z^{\scriptscriptstyle (n)}= (x^{\scriptscriptstyle (n)},y^{\scriptscriptstyle (n)})$, $z=(x,y)$ as in the statement of the theorem. It suffices to prove \eqref{eq:ab1200}.
From Theorem \ref{theo:aldous-mc-s}(ii), $\boldsymbol{X}(z^{\scriptscriptstyle(n)},t) \to \boldsymbol{X}(z,t)$ in probability, in $l^2_{\downarrow}$. The result now follows on combining this convergence with
the convergence in Lemma \ref{lemma:feller-conv-prob} and applying Lemma \ref{lemma:basic-analysis}. \ \ \rule{1ex}{1ex} \\
Rest of this section is devoted to the proof of Lemma \ref{lemma:feller-conv-prob}.
The key idea of the proof is as follows. Consider the induced subgraphs on the first $k$ vertices ${\bf{G}}^{\scriptscriptstyle [k]}={\bf{G}}(z^{\scriptscriptstyle [k]},t)$ and ${\bf{G}}^{\scriptscriptstyle (n)[k]}={\bf{G}}(z^{\scriptscriptstyle (n)[k]},t)$. Since there are only finite number of vertices in ${\bf{G}}^{\scriptscriptstyle [k]}$, when $n \to \infty$, ${\bf{G}}^{\scriptscriptstyle (n)[k]}$ will eventually be identical to ${\bf{G}}^{[k]}$ almost surely. The main step in the proof is to control the difference between ${\bf{G}}^{\scriptscriptstyle (n)[k]}$ and ${\bf{G}}^{\scriptscriptstyle (n)}$ when $k$ is large, uniformly for all $n$. For this we first analyze the difference between ${\bf{G}}^{\scriptscriptstyle (n)[k]}$ and ${\bf{G}}^{\scriptscriptstyle (n)[k+1]}$ in the lemma below.
Consider the set of vertices $[k+1]=\{1,2,...,k, k+1\}$, and for every $i \in [k+1]$, let vertex $i$ have label $(x_i, y_i)$ representing its size and surplus, respectively. Suppose $x_1 \ge x_2 \ge ... \ge x_{k+1}$. Fix $t > 0$. Define a random graph ${\bf{G}}^*$ on the above vertex set as follows. For $i\le k$, the number of edges, $N_i$, between $i$ and $k+1$ follows Poisson$(tx_i x_{k+1})$. In addition, there are $N_0$ = Poisson$(t x^2_{k+1}/2)$ self-loops to the vertex $k+1$. All the Poisson random variables are taken to be mutually
independent.\\
Denote $X_i$ and $Y_i$ for the component volumes and surplus of the resulting star-like graph if $i$ is the smallest labeled vertex in its component; otherwise let $X_i=Y_i=0$. A precise definition of $(X_i, Y_i)$ is as follows. Write $i \sim k+1$ if there is an edge between $i$ and $k+1$ in ${\bf{G}}^*$. By convention $(k+1) \sim (k+1)$. Let
$\mathcal{J}_k = \{i \in [k+1]: i \sim k+1\}$, and $i_0 = \min\{i: i \in \mathcal{J}_k\}$.
Then
\[
(X_i, Y_i) =\left\{
\begin{array}{cc}
\left( \sum_{i\in \mathcal{J}_k}x_i, \sum_{i\in \mathcal{J}_k}y_i\right) & \mbox{ if } i = i_0 \\ \ \\
(0,0) & \mbox{ if } i \in \mathcal{J}_k\setminus \{i_0\} \\ \ \\
(x_i,y_i) & \mbox{ if } i \in [k+1]\setminus\mathcal{J}_k.
\end{array}
\right .
\]
Define $R_k=\sum_{i=1}^k x_i y_i$, $S_k=\sum_{i=1}^k x_i^2$, $R_{k+1}=\sum_{i=1}^{k+1} X_iY_i$. Then we have the following result.
\begin{Lemma}
\label{lemma:basic-dynamic}
(i) $\mathbb{P}\{ Y_i \neq y_i \} \le tx_{k+1}y_{k+1} x_1 + t x_{k+1}^2 \left( 1 + it x_1^2 + tS_k +tR_k x_1 \right)$.\\
(ii) $ \mathbb{E}[R_{k+1}-R_k] \le x_{k+1}y_{k+1} ( 1 + t S_k) + x_{k+1}^2 (t R_k + t^2 S_k R_k + t^2 S_k x_1) + tx_{k+1}^3
(1 + 2tS_k +t^2S_k^2). $
\end{Lemma}
{\bf Proof:} (i) It is easy to see that, for $i = 1, \cdots k$,
$$
\{Y_i \neq y_i\} \subset \left( \{y_{k+1} > 0\} \cap \{i \in \mathcal{J}_k\}\right)
\cup \{N_0 \neq 0\} \cup_{j=1}^k \{N_j > 1\} \cup_{j < i} \{N_jN_i \neq 0\} \cup_{j: y_j > 0} \{N_jN_i \neq 0\}.$$
Using the observation that for a Poisson$(\lambda)$ random variable $Z$, $\mathbb{P}\{Z \ge 1\} < \lambda$ and $\mathbb{P}\{ Z \ge 2\} < \lambda^2$, we now have
that
\begin{align*}
\mathbb{P}\{ Y_i \neq y_i \}
\le& tx_i x_{k+1} \cdot y_{k+1} + \frac {t x_{k+1}^2}{2} + \sum_{j=1}^k (tx_jx_{k+1})^2 \\
+& \sum_{j=1}^{i-1} tx_jx_{k+1} \cdot tx_i x_{k+1} + \sum_{j=1}^k tx_jx_{k+1} \cdot tx_i x_{k+1} \cdot y_j.
\end{align*}
Proof is now completed on collecting all the terms and using the fact that $x_i \ge x_1$ for every $i$.\\
(ii) Note that $$X_0 =x_{k+1} + \sum_{j=1}^k x_j {{\rm 1\hspace{-0.90ex}1}}_{\{ N_j \ge 1 \}}, \; Y_0 =y_{k+1} + \sum_{j=1}^k y_{j} {{\rm 1\hspace{-0.90ex}1}}_{\{ N_j \ge 1 \}}
+ N_{0} + \sum_{j=1}^k (N_j - 1)^+ .$$ Then
\begin{align*}
R_{k+1}-R_k =& X_0Y_0 - \sum_{j \in \mathcal{J}_k} x_jy_j \\
=& x_{k+1}y_{k+1} + \sum_{j=1}^k (x_jy_{k+1}+x_{k+1}y_j){{\rm 1\hspace{-0.90ex}1}}_{\{ N_j \ge 1 \}} + \sum_{1\le j < l \le k} (x_j y_l +x_l y_j) {{\rm 1\hspace{-0.90ex}1}}_{\{ N_j \ge 1 \}}{{\rm 1\hspace{-0.90ex}1}}_{\{ N_l \ge 1 \}}\\
+& N_{0} X_0 + x_{k+1} \sum_{j=1}^k (N_j-1)^+ + \sum_{j=1}^k x_j (N_j-1)^+ \\
+& \sum_{1 \le j < l \le k}(x_j {{\rm 1\hspace{-0.90ex}1}}_{\{ N_j \ge 1 \}} (N_l-1)^+ + x_l {{\rm 1\hspace{-0.90ex}1}}_{\{ N_l \ge 1 \}} (N_j-1)^+).
\end{align*}
The result now follows on taking expectations in the above equation and using the fact that $\mathbb{E}[(N_j-1)^+] < (tx_jx_{k+1})^2 $. \ \ \rule{1ex}{1ex} \\
Recall that, by construction, $X_i(z,t) \ge X_{i+1}(z,t)$ for all $z \in \mathbb{U}_{\downarrow}$, $t\ge 0$ and $i \in \mathbb{N}$. The following lemma which is a key ingredient in the proof of Lemma \ref{lemma:feller-conv-prob} says
that if $z \in \mathbb{U}_{\downarrow}^1$, ties do not occur, a.s.
\begin{Lemma}
\label{strict}
Let $z \in \mathbb{U}_{\downarrow}^1$. Then for every $t > 0$ and $i \in \mathbb{N}$,
$X_i(z,t) > X_{i+1}(z,t)$ a.s.
\end{Lemma}
\noindent \textbf{Proof:} Fix $t > 0$. Consider the graph ${\bf{G}}(z,t)$ and write $\mathcal{C}_{x_i} \equiv \mathcal{C}_{x_i}(t)$ for the component of vertex $(x_i,y_i)$ at time $t$. It suffices to show for all $i\neq j$
\begin{equation}
\label{eqn:1257}
\mathbb{P}\set{ |\mathcal{C}_{x_i}| = |\mathcal{C}_{x_j}|, \mathcal{C}_{x_i} \ne \mathcal{C}_{x_j} } = 0.
\end{equation}
The key property we shall use is that for $z=(x,y) \in \mathbb{U}_{\downarrow}^1$, $\sum_{i=1}^\infty x_i =\infty$. Now fix $i\geq 1$. It is enough to show that $|\mathcal{C}_{x_i}|$ has no atom i.e for all $(x,y)\in \mathbb{U}_{\downarrow}^1$
\begin{equation}
\label{eqn:no-atom-one}
\mathbb{P}(|\mathcal{C}_{x_i}| =a) = 0, \qquad \mbox{ for any } a\geq 0.
\end{equation}
To see this, first note that since $|\mathcal{C}_{x_i}|< \infty$ a.s., conditional on $\mathcal{C}_{x_i}$ the vector $z^* = ((x_k,y_k): x_k\notin \mathcal{C}_{x_i}) \in \mathbb{U}_{\downarrow}^1$ almost surely. Thus on the event $x_j\notin \mathcal{C}_{x_i}$, conditional on $\mathcal{C}_{x_i}$, using \eqref{eqn:no-atom-one} with $a= |\mathcal{C}_{x_i}|$ implies that $\mathbb{P}(|\mathcal{C}_{x_j}| = |\mathcal{C}_{x_i}|\mid \mathcal{C}_{x_i}) = 0$ and this completes the proof. Thus it is enough to prove \eqref{eqn:no-atom-one}. For the rest of the argument, to ease notation let $i=1$. Let us first show the simpler assertion that the volume of direct neighbors of $x_1$ has a continuous distribution. More precisely, let $N_{i,j}(t):=\xi_{i,j}([0,tx_ix_j/2])+\xi_{j,i}([0,tx_ix_j/2])$, $1 \le i < j$, denote the number of edges between any two vertices $x_i$ and $x_j$ by time $t$. Then the volume of \emph{direct} neighbors of the vertex $x_1$ is $L := \sum_{i = 2 }^\infty x_i {{\rm 1\hspace{-0.90ex}1}}_{\set{N_{1,i}(t) \ge 1}} $ and we will first show that $L$ has no atom, namely
\begin{equation}
\label{eqn:1261}
\mathbb{P}(L = a) = 0, \qquad \mbox{ for all } a\geq 0.
\end{equation}
For any random variable $X$ define the maximum atom size of $X$ by
$${\bf atom}(X) : = \sup _{ a \in \mathbb{R}} \mathbb{P}\set{X = a}.$$
For two independent random variables $X_1$ and $X_2$ we have ${\bf atom}(X_1 + X_2) \le \min \set{ {\bf atom}(X_1), {\bf atom}(X_2) }$. For $m \ge 2$, define $L_m = \sum_{i=m}^\infty x_i {{\rm 1\hspace{-0.90ex}1}}_{\set{N_{1,i}(t) \ge 1}}$. Since $L_m$ and $L-L_m$ are independent, we have
$ {\bf atom}(L) \le {\bf atom}(L_m).$
Define the event
$$ E_m:=\set{ N_{1,i}(t) \le 1 \mbox{ for all } i \ge m }, $$
and write
$$ L^*_m(t) := \sum_{i=m}^\infty x_i N_{1,i}(t).$$
Then $L^*_m(t)$ is a pure jump Levy process with Levy measure $\nu(du) = \sum_{i=m}^\infty x_1 x_i \delta_{x_i}(du)$. By \cite{hartman1942infinitesimal}, such a Levy process has continuous marginal distribution since the Levy measure is infinite ($ \nu(0,\infty) = (\sum_{i=m}^\infty x_i) x_1 = \infty$) . Thus $L^*_m(t)$ has no atom. Next, for any $a \in \mathbb{R}$,
\begin{align*}
\mathbb{P} \set{ L_m = a}
\le& \mathbb{P} \set{ E_m^c} + \mathbb{P} \set{ E_m, L_m =a }
= \mathbb{P} \set{ E_m^c} + \mathbb{P} \set{ E_m, L^*_m(t) =a }\\
\le& \sum_{i=m}^\infty \frac{(tx_1x_i)^2}{2} + 0
= \frac{t^2 x_1^2}{2} \sum_{i=m}^\infty x_i^2.
\end{align*}
Thus ${\bf atom}(L) \le {\bf atom}(L_m) \le \frac{t^2 x_1^2}{2} \sum_{i=m}^\infty x_i^2$. Since $m$ is arbitrary, we have ${\bf atom}(L)=0$. Thus $L$ is a continuous variable, and \eqref{eqn:1261} is proved.
Let us now strengthen this to prove \eqref{eqn:no-atom-one}.
Let $\tilde {\bf{G}}$ be the subgraph of ${\bf{G}}(z,t)$ obtained by deleting the vertex $x_1$ and all related edges. Let $\tilde X_i$ be the volume of the $i$-th largest component of $\tilde {\bf{G}}$. Note that $\sum_{i=1}^\infty \tilde X_i = \sum_{i=2}^\infty x_i = \infty$ a.s. Conditional on $(\tilde X_i)_{i\geq 1}$, let $\tilde N_{1,i}$ have Poisson distribution with parameter $t x_1 \tilde X_i$. Then
\[ \mathcal{C}_{x_1} \stackrel{d}{=} x_1 + \sum_{i=1}^\infty {\tilde X}_i{\rm 1\hspace{-0.90ex}1}_{\set{\tilde N_{1,i}\geq 1}},\]
where the second term has the same form as the random variable $L$. Using \eqref{eqn:1261} completes the proof. \ \ \rule{1ex}{1ex}
We now proceed to the proof of Lemma \ref{lemma:feller-conv-prob}.
{\bf Proof of Lemma \ref{lemma:feller-conv-prob}.}
Fix $t > 0$ and $z^{\scriptscriptstyle (n)}, z$ as in the statement of the lemma. Denote
$ Y^{\scriptscriptstyle [k]} =Y(z^{\scriptscriptstyle [k]},t), \;\; Y^{\scriptscriptstyle (n)[k]}= Y(z^{\scriptscriptstyle (n)[k]},t)$.
Similarly, denote $\mathcal{C}^{\scriptscriptstyle [k]}_i$ and $\mathcal{C}_i^{\scriptscriptstyle (n)[k]}$ for the corresponding $i$-th largest component; and $X_i^{\scriptscriptstyle [k]}$ and $X_i^{\scriptscriptstyle (n)[k]}$ for their respective sizes. Also, write $X^{\scriptscriptstyle (n)} = X(x^{\scriptscriptstyle (n)},t)$ and define $Y^{\scriptscriptstyle (n)}, R^{\scriptscriptstyle (n)}, S^{\scriptscriptstyle (n)}$ similarly.
For $i \in \mathbb{N}$, define the event $E_i^{\scriptscriptstyle (n)[k]}$ as,
$$ E_i^{\scriptscriptstyle (n)[k]} := \{\omega: X_j^{\scriptscriptstyle (n)[k]}(\omega) > X_{j+1}^{\scriptscriptstyle (n)}(\omega), \mbox{ for } j=1,2,...,i \}, $$
and define $E_i^{\scriptscriptstyle [k]}$ similarly.
Then
\begin{align}
\mathbb{P}\{ Y_i^{\scriptscriptstyle (n)} \neq Y_i(t) \}
\le& \mathbb{P}\{ Y_i^{\scriptscriptstyle (n)} \neq Y_i^{\scriptscriptstyle (n)[k]} \} + \mathbb{P}\{Y_i^{\scriptscriptstyle (n)[k]} \neq Y_i^{\scriptscriptstyle [k]}\} +\mathbb{P}\{ Y_i^{\scriptscriptstyle [k]} \neq Y_i(t)\} \nonumber\\
\le& \mathbb{P}\{ Y_i^{\scriptscriptstyle (n)} \neq Y_i^{\scriptscriptstyle (n)[k]}, E_i^{\scriptscriptstyle (n)[k]} \} + \mathbb{P}\{ (E_i^{\scriptscriptstyle (n)[k]})^c \} + \mathbb{P}\{Y_i^{\scriptscriptstyle (n)[k]} \neq Y_i^{\scriptscriptstyle [k]}\} +\mathbb{P}\{ Y_i^{\scriptscriptstyle [k]} \neq Y_i(t)\}. \label{eqn:314}
\end{align}
Note that
$$
E_i^{\scriptscriptstyle (n)[k]} \subset \{\omega: \mathcal{C}_j^{\scriptscriptstyle (n)[k]}(\omega) \subset \mathcal{C}_j^{\scriptscriptstyle (n)[m]}(\omega) \subset \mathcal{C}_j^{\scriptscriptstyle (n)}(\omega), \mbox{ for all } j=1,2,...,i \mbox{ and } m \ge k\}.$$
Thus the probability of the event $\{ Y_i^{\scriptscriptstyle (n)[m+1]} \neq Y_i^{\scriptscriptstyle (n)[m]}, E_i^{\scriptscriptstyle (n)[k]} \}$, for $m \ge k$, can be estimated using Lemma \ref{lemma:basic-dynamic} (i). More precisely, let $\mathcal{F}^{\scriptscriptstyle [m]}=\sigma \{ \xi_{i,j}; {i , j \le m} \}$ for $ m \ge 1$. Then by Lemma \ref{lemma:basic-dynamic} (i),
\begin{align*}
\mathbb{P}\{ Y_i^{\scriptscriptstyle (n)[m+1]} \neq Y_i^{\scriptscriptstyle (n)[m]}, E_i^{\scriptscriptstyle (n)[k]} | \mathcal{F}^{\scriptscriptstyle [m]} \} \le& t x^{\scriptscriptstyle (n)}_{m+1}y^{\scriptscriptstyle (n)}_{m+1} X^{\scriptscriptstyle (n)[m]}_1 \\
+&
t (x^{\scriptscriptstyle (n)}_{m+1})^2 \left( 1 + it (X^{\scriptscriptstyle (n)[m]}_1)^2 + tS^{\scriptscriptstyle (n)[m]} +t R^{\scriptscriptstyle (n)[m]} X^{\scriptscriptstyle (n)[m]}_1 \right),
\end{align*}
where $S^{\scriptscriptstyle (n)[m]} = \sum_i (X_i^{\scriptscriptstyle (n)[m]})^2$ and $R^{\scriptscriptstyle (n)[m]} = \sum_i (X_i^{\scriptscriptstyle (n)[m]}Y_i^{\scriptscriptstyle (n)[m]})$.
Note that $X_1^{\scriptscriptstyle (n)[k]} \le X_1^{\scriptscriptstyle (n)}$, $R^{\scriptscriptstyle (n)[k]} \le R^{\scriptscriptstyle (n)}$ and $S^{\scriptscriptstyle (n)[k]} \le S^{\scriptscriptstyle (n)}$. Thus we have
\begin{align*}
\sum_{m=k}^\infty \mathbb{P}\{ Y_i^{\scriptscriptstyle (n)[m+1]} \neq Y_i^{\scriptscriptstyle (n)[m]}, E_i^{\scriptscriptstyle (n)[k]} | \mathcal{F}^{\scriptscriptstyle [m]} \}
\le& t
\left(\sum_{m=k+1}^\infty x^{\scriptscriptstyle (n)}_{m}y^{\scriptscriptstyle (n)}_{m}\right) X_1^{\scriptscriptstyle (n)} \\
+& t \left(\sum_{m=k+1}^\infty (x^{\scriptscriptstyle (n)}_{m})^2\right) \left( 1 + it (X_1^{\scriptscriptstyle (n)})^2 +
tS^{\scriptscriptstyle (n)} +tR^{\scriptscriptstyle (n)}X_1^{\scriptscriptstyle (n)} \right).
\end{align*}
Denote the right hand side of the above inequality as $U^{\scriptscriptstyle (n)[k]}$. Then by Lemma \ref{lemma:borel-cantelli}(ii), we have
\begin{equation}
\mathbb{P} \{ Y_i^{\scriptscriptstyle (n)} \neq Y_i^{\scriptscriptstyle (n)[k]}, E_i^{\scriptscriptstyle (n)[k]} \} = \mathbb{P} \left(
\cup_{m=k}^\infty \{ Y_i^{\scriptscriptstyle (n)[m+1]} \neq Y_i^{\scriptscriptstyle (n)[m]}, E_i^{\scriptscriptstyle (n)[k]} \}\right) \le 2 \mathbb{E}[U^{\scriptscriptstyle (n)[k]} \wedge 1] \label{eqn:879}
\end{equation}
and therefore
\begin{align}
\mathbb{P}\{ Y_i^{\scriptscriptstyle (n)} \neq Y_i(t) \}
\le 2\mathbb{E}[ U^{\scriptscriptstyle (n)[k]} \wedge 1 ] + \mathbb{P}\{ (E_i^{\scriptscriptstyle (n)[k]})^c \} + \mathbb{P}\{Y_i^{\scriptscriptstyle (n)[k]} \neq Y_i^{\scriptscriptstyle [k]}\} +\mathbb{P}\{ Y_i^{\scriptscriptstyle [k]} \neq Y_i(t)\}.
\label{eqn:879.5}
\end{align}
Next note that $X^{\scriptscriptstyle (n)}_1$, $S^{\scriptscriptstyle (n)}$ and $R^{\scriptscriptstyle (n)}$ are all tight sequences by Corollary \ref{lemma:tight-r} and Theorem \ref{theo:aldous-mc-s}(ii).
Thus $(1+ it (X_1^{\scriptscriptstyle (n)})^2 + tS^{\scriptscriptstyle (n)} + tR^{\scriptscriptstyle (n)} X_1^{\scriptscriptstyle (n)})$ is also tight. Also, since $z^{\scriptscriptstyle (n)} \to z$,
$$ \limsup_{k \to \infty} \limsup_{ n \to \infty} \sum_{i=k+1}^\infty x^{\scriptscriptstyle (n)}_i y^{\scriptscriptstyle (n)}_i =0 \mbox{ and } \limsup_{k \to \infty} \limsup_{ n \to \infty} \sum_{i=k+1}^\infty(x^{\scriptscriptstyle (n)}_i )^2 =0.$$
Combining the above observations we have that
$\limsup_{k \to \infty} \limsup_{ n \to \infty} \mathbb{P}\{U^{\scriptscriptstyle (n)[k]}>\epsilon \}=0$ for all $\epsilon > 0$. From the inequality
$$\mathbb{E} [ U^{\scriptscriptstyle (n)[k]} \wedge 1] \le \mathbb{P}\{ U^{\scriptscriptstyle (n)[k]} > \epsilon \} + \epsilon$$ we now see that
\begin{equation} \label{lemma:tight-uniformly-u} \limsup_{k \to \infty} \limsup_{n \to \infty} \mathbb{E} [ U^{\scriptscriptstyle (n)[k]} \wedge 1]=0. \end{equation}
Next, from a straightforward extension of Proposition 5 of Aldous \cite{aldous1997brownian} we have that
$(\boldsymbol{X}^{\scriptscriptstyle (n)}, X_1^{\scriptscriptstyle (n)[k]}, ..., X_i^{\scriptscriptstyle (n)[k]} ) \convd (\boldsymbol{X}(t), X_1^{\scriptscriptstyle [k]}, ..., X_i^{\scriptscriptstyle [k]} )$ in $l^2_{\downarrow} \times \mathbb{R}^i$ when $n \to \infty$, for each fixed $i$ and $k$. Combining this with Lemma \ref{strict} we now see that for fixed $i$
$$ \limsup_{k \to \infty}\limsup_{n \to \infty} \mathbb{P}\{ (E_i^{\scriptscriptstyle (n)[k]})^c \} = 0.$$
Also, for each fixed $k$
$$
\limsup_{n \to \infty} \mathbb{P}\{ Y_i^{\scriptscriptstyle (n)[k]} \neq Y_i^{[k]}\} = 0.$$
Observing that $\lim_{k \to \infty} Y_i^{\scriptscriptstyle [k]}=Y_i(t)$ and the last term in \eqref{eqn:879.5} does not depend on $n$, we have that
$$
\limsup_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\{ Y_i^{\scriptscriptstyle [k]} \neq Y_i(t)\} = 0.$$
Part (i) of the lemma now follows on combining the above observations and taking limit as $n \to \infty$ and then $k \to \infty$ in \eqref{eqn:879.5}.
We now prove part (ii) of the lemma.
Note that
$$ \liminf_{n \to \infty} R^{\scriptscriptstyle (n)} \ge \lim_{n \to \infty} R^{\scriptscriptstyle (n)[k]} = R^{[k]}. $$
With a similar argument as in Lemma \ref{lemma:248}, we have $R^{[k]} \to R(z,t)$ as $k \to \infty$. Thus sending $k \to \infty$ in the above display we have
\begin{equation}\label{eq:ab1746}\liminf_{n \to \infty} R^{\scriptscriptstyle (n)} \ge R(z,t).\end{equation}
To complete the proof, it suffices to show that
\begin{equation}
\mbox{ For any } \epsilon > 0,\, \lim_{n \to \infty} \mathbb{P}\{ R^{\scriptscriptstyle (n)} > R(z,t) + \epsilon \} =0.\label{eq:ab1928}
\end{equation}
Note that
\begin{align}
\mathbb{P}\{ R^{\scriptscriptstyle (n)} - R(z,t) > \epsilon \}
\le& \mathbb{P}\{ R^{\scriptscriptstyle (n)} - R^{\scriptscriptstyle (n)[k]} > \epsilon/2 \} + \mathbb{P}\{ R^{\scriptscriptstyle (n)[k]} - R(z,t) > \epsilon/2 \}\nonumber\\
\le& \mathbb{P}\{ R^{\scriptscriptstyle (n)} - R^{\scriptscriptstyle (n)[k]} > \epsilon/2 \} + \mathbb{P}\{ R^{\scriptscriptstyle (n)[k]} - R^{\scriptscriptstyle [k]} > \epsilon/2 \}.\label{eq:ab1956}
\end{align}
The second term on the right side above goes to zero for each fixed $k$, as $n \to \infty$. For the first term, note that
by Lemma \ref{lemma:basic-dynamic}(ii), for all $m \ge k$
\begin{align*}
\mathbb{E}[ R^{\scriptscriptstyle (n)[m+1]}- R^{\scriptscriptstyle (n)[m]} | \mathcal{F}^{\scriptscriptstyle [m]} ]
\le x^{\scriptscriptstyle (n)}_m y^{\scriptscriptstyle (n)}_m U_1^{\scriptscriptstyle (n)} + (x^{\scriptscriptstyle (n)}_m)^2 U_2^{\scriptscriptstyle (n)} + (x^{\scriptscriptstyle (n)}_{m+1})^3 U_3^{\scriptscriptstyle (n)},
\end{align*}
where $U_1^{\scriptscriptstyle (n)}= 1 + t S^{\scriptscriptstyle (n)}$, $ U_2^{\scriptscriptstyle (n)}= t R^{\scriptscriptstyle (n)} + t^2 S^{\scriptscriptstyle (n)}R^{\scriptscriptstyle (n)}+t^2 S^{\scriptscriptstyle (n)} X_1^{\scriptscriptstyle (n)}$ and $U_3^{\scriptscriptstyle (n)}= t(1 + 2t S^{\scriptscriptstyle (n)} + t^2 (S^{\scriptscriptstyle (n)})^2)$. Thus by Lemma \ref{lemma:borel-cantelli} (i),
$$ \mathbb{P}\{ R^{\scriptscriptstyle (n)} - R^{\scriptscriptstyle (n)[k]} > \epsilon \} \le (1 + 1/\epsilon) \mathbb{E}[ U^{\scriptscriptstyle (n)[k]} \wedge 1], $$
where $U^{\scriptscriptstyle (n)[k]}=( \sum_{m=k+1}^\infty x^{\scriptscriptstyle (n)}_m y^{\scriptscriptstyle (n)}_m ) U_1^{\scriptscriptstyle (n)} + (\sum_{m=k+1}^\infty(x^{\scriptscriptstyle (n)}_m)^2) U_2^{\scriptscriptstyle (n)} + (\sum_{m=k+1}^\infty(x^{\scriptscriptstyle (n)}_{m+1})^3) U_3^{\scriptscriptstyle (n)} $. Note that $U_1^{\scriptscriptstyle (n)}$, $U_2^{\scriptscriptstyle (n)}$ and $U_3^{\scriptscriptstyle (n)}$ are all tight sequences
and $z^{\scriptscriptstyle (n)} \to z$. An argument similar to the one used to prove \eqref{lemma:tight-uniformly-u} now shows that, for all $\epsilon > 0$,
$$ \limsup_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\{ R^{\scriptscriptstyle (n)} - R^{\scriptscriptstyle (n)[k]} > \epsilon \} \le \left(1 + \frac{1}{\epsilon}\right) \limsup_{k \to \infty} \limsup_{n \to \infty} \mathbb{E}[ U^{\scriptscriptstyle (n)[k]} \wedge 1] =0.$$
The statement in \eqref{eq:ab1928} now follows on using the above convergence in \eqref{eq:ab1956} and combining it with the observation below \eqref{eq:ab1956}.
This completes the proof of part (ii). \ \ \rule{1ex}{1ex}\\
\begin{Remark}
\label{prodnofell}
Lemma \ref{lemma:feller-conv-prob} is at the heart of the (near) Feller property in Theorem \ref{theo:welldef-fellerb}
which is crucial for the proof of the joint convergence in \eqref{eq:eq1241}.
The proof of the lemma reveals the reason for considering the metric ${\bf d}_{\scriptscriptstyle \mathbb{U}} $ on $\mathbb{U}_{\downarrow}$ rather than ${\bf d}_1$ or ${\bf d}_2$
The proof hinges upon the convergence of
$\sum_{m=1}^\infty x^{\scriptscriptstyle (n)}_m y^{\scriptscriptstyle (n)}_m $ to $\sum_{m=1}^\infty x_m y_m $, as $n\to \infty$, even for the proof of convergence of
$Y_i(z^{\scriptscriptstyle (n)},t) \convp Y_i(z,t)$. This suggests that the convergence in ${\bf d}_1$ or
${\bf d}_2$ is ``too weak'' to yield the desired Feller property.
\end{Remark}
\section{The standard augmented multiplicative coalescent.}
\label{sec:main-st-ag-mc-ex}
In this section we prove Theorem \ref{thm:smc-surplus}. Proposition 4 of \cite{aldous1997brownian} proves a very useful result on convergence of component size
vectors of a general family of non-uniform random graph models to the ordered excursion lengths of $\hat W _{\lambda}$. We begin in this section by extending this
result to the joint convergence of component size and component surplus vectors in $\mathbb{U}_{\downarrow}$, under a slight strengthening of the conditions assumed in
\cite{aldous1997brownian}.
Recall the excursion lengths and mark count process $\boldsymbol{Z}^*(\lambda) = (\boldsymbol{X}^*(\lambda), \boldsymbol{Y}^*(\lambda))$ defined in Section \ref{sec:augmented-mc}.
Our first
result below shows that, for fixed $\lambda \in \mathbb{R}$, $\boldsymbol{Z}^*(\lambda)$ arises as a limit of $\boldsymbol{Z}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})$ in $\mathbb{U}_{\downarrow}$ for all sequences $\{z^{\scriptscriptstyle (n)}\} \subset \mathbb{U}_{\downarrow}$ and
$q^{\scriptscriptstyle (n)}=q^{\scriptscriptstyle (n)}_\lambda \subset (0, \infty)$ that satisfy certain regularity conditions.
For $n \ge 1$, let $z^{\scriptscriptstyle (n)} = (x^{\scriptscriptstyle (n)}, y^{\scriptscriptstyle (n)}) \in \mathbb{U}_{\downarrow}^0$.
Writing $z_i^{\scriptscriptstyle (n)} = (x_i^{\scriptscriptstyle (n)}, y_i^{\scriptscriptstyle (n)})$, $i \ge 1$, define
$$x^{*\scriptscriptstyle (n)}= \sup_{i \ge 1} x^{\scriptscriptstyle (n)}_i, \; s^{\scriptscriptstyle (n)}_r = \sum_{i=1}^\infty (x_i^{\scriptscriptstyle (n) })^r, \; r \ge 1.$$
Let $\{q^{\scriptscriptstyle (n)}\}$ be a nonnegative sequence. We will suppress $(n)$ from the notation unless needed.
\begin{Theorem}
\label{theo:aldous-full-gene}
Let $z^{\scriptscriptstyle (n)} = (z_1^{\scriptscriptstyle (n)}, \cdots )\in \mathbb{U}_{\downarrow}^0$ be such that $z^{\scriptscriptstyle (n)}_i = (0,0)$ for all $i >n$.
Suppose that, as $n\to \infty$,
\begin{equation}
\label{eqn:qsigma2}
\frac{s_3}{(s_2)^3} \to 1, \quad
q- \frac{1}{s_2} \to \lambda , \quad
\frac{x^*}{s_2} \to 0,
\end{equation}
and, for some $\varsigma \in (0, \infty)$,
\begin{equation}
\label{eqn:additional-condition}
s_1 \cdot \left(\frac{x^*}{s_2}\right)^\varsigma \to 0.
\end{equation}
Further suppose that $y_i^{\scriptscriptstyle(n)} = 0$ for all $i$. Then $\boldsymbol{Z}^{\scriptscriptstyle (n)} = \boldsymbol{Z}(z^{\scriptscriptstyle (n)},q^{\scriptscriptstyle (n)})$ converges in distribution in $\mathbb{U}_{\downarrow}$ to $\boldsymbol{Z}^*(\lambda)$.
\end{Theorem}
{\bf Remark:} The convergence assumption in \eqref{eqn:qsigma2} is the same as that in Proposition 4 of \cite{aldous1997brownian}. The additional assumption in
\eqref{eqn:additional-condition} is not very stringent as will be seen in Section \ref{sec:main-coupling} when this
result is applied to a general family of bounded-size rules.
Given Theorem \ref{theo:aldous-full-gene}, the proof of Theorem \ref{thm:smc-surplus} can now be completed as follows.
{\bf Proof of Theorem \ref{thm:smc-surplus}.}
The first two parts of the theorem were shown in Theorem \ref{theo:welldef-fellerb}.
Also, part (v) of the theorem is immediate from the definition of $\{T_t\}$ in Section \ref{sec:mult}.
Recall the definition of $\nu_{\lambda}$ from
Section \ref{sec:augmented-mc}. In order to prove parts (iii)-(iv) it suffices to show that
\begin{equation}
\label{eq:ab2121}
\mbox{ for any } \lambda_1, \lambda_2 \in \mathbb{R}, \, \lambda_1 \le \lambda_2,\; \nu_{\lambda_1}\mathcal{T}_{\lambda_{2}-\lambda_1} = \nu_{\lambda_2}.
\end{equation}
Indeed, using the semigroup property of $(\mathcal{T}_{\lambda})$ and the above relation, it is straightforward to define a consistent
family of finite dimensional distributions $\mu_{\lambda_1, \cdots \lambda_k}$ on $(\mathbb{U}_{\downarrow})^{\otimes k}$, $-\infty < \lambda_1 < \lambda_2, \cdots
\lambda_k<\infty$, $k \ge 1$, such that $\mu_{\lambda} = \nu_{\lambda}$ for every $\lambda \in \mathbb{R}$. The desired result then follows from Kolmogorov's consistency
theorem.
We now prove \eqref{eq:ab2121}.
Let
$$
z^{\scriptscriptstyle (n)} = (x^{\scriptscriptstyle (n)}, y^{\scriptscriptstyle (n)}), \; x_i^{\scriptscriptstyle (n)} = n^{-2/3}, \; y_i^{\scriptscriptstyle (n)} = 0, \; i = 1, \cdots n, \; q^{\scriptscriptstyle (n)}_{\lambda_j} = \lambda_j + n^{1/3}, \; j = 1,2.$$
We set $z^{\scriptscriptstyle (n)}_i =0$ for $i > n$.
Note that with this choice of $x^{\scriptscriptstyle (n)}$, $s_1=n^{1/3}, s_2 = n^{-1/3}, s_3 = n^{-1}$ and so clearly \eqref{eqn:qsigma2} and \eqref{eqn:additional-condition} (with
any $\varsigma > 1$) are satisfied with $q = q_{\lambda_j}$, $\lambda = \lambda_j$, $j=1,2$. Thus, denoting
the distribution of $\boldsymbol{Z}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)}_{\lambda_j})$ by $\nu_{\lambda_j}^{\scriptscriptstyle (n)}$, we have by Theorem \ref{theo:aldous-full-gene} that
\begin{equation}
\label{eq:eqab2153}
\nu_{\lambda_j}^{\scriptscriptstyle (n)} \to \nu_{\lambda_j}, \mbox{ as } n \to \infty.
\end{equation}
Also, from the construction of $\boldsymbol{Z}(z,t)$ in Section \ref{sec:main-constr-mc}, it is clear that
$\nu_{\lambda_1}^{\scriptscriptstyle (n)}\mathcal{T}_{\lambda_{2}-\lambda_1} = \nu_{\lambda_2}^{\scriptscriptstyle (n)}$. The result now follows on combining the convergence in
\eqref{eq:eqab2153} with Theorem \ref{theo:welldef-fellerb} and observing that $\boldsymbol{Z}^*(\lambda) \in \mathbb{U}_{\downarrow}^1$ a.s. for every $\lambda \in \mathbb{R}$.
\ \ \rule{1ex}{1ex}\\
Rest of this section is devoted to the proof of Theorem \ref{theo:aldous-full-gene} and is organized as follows. Recall the random graph process
${\bf{G}}(z,q)$, for $z \in \mathbb{U}_{\downarrow}$, $q\ge 0$, defined at the beginning of Section \ref{sec:main-constr-mc}. In Section \ref{sec:conv-in-prod} we will
give an equivalent in law construction of ${\bf{G}}(z,q)$, from \cite{aldous1997brownian}, that defines the random graph
simultaneously with a certain breadth-first-exploration random walk. The excursions of the reflected version of this walk encode
the component sizes of the random graph while the area under the excursions gives the parameter of the Poisson distribution describing
the (conditional) law of the surplus associated with the corresponding component. Using this construction, in Theorem \ref{theo:generalized-aldous},
we will first prove a weaker result than Theorem \ref{theo:aldous-full-gene} which proves the convergence in distribution of $\boldsymbol{Z}^{\scriptscriptstyle (n)}$ to $\boldsymbol{Z}^*(\lambda)$
in $l^2_{\downarrow} \times \mathbb{N}^{\infty}$, where we consider the product topology on $\mathbb{N}^{\infty}$. This result is proved in Section \ref{sec:sec6.2}.
In Section \ref{sec:conv-d-metric} we will give the proof of Theorem \ref{theo:aldous-full-gene} using Theorem \ref{theo:generalized-aldous} and an auxiliary
tightness lemma (Lemma \ref{lemma:tightness-sup}). Finally, proof of Lemma \ref{lemma:tightness-sup} is given in Section \ref{sec:sec6.3new}.
\subsection{Breadth First Exploration Walk. }
\label{sec:conv-in-prod}
In this section, following \cite{aldous1997brownian}, we will give an equivalent in law construction of ${\bf{G}}(z,q)$ that defines the random graph
simultaneously with a certain breadth-first-exploration random walk. Given $q \in (0, \infty)$ and $z \in \mathbb{U}_{\downarrow}^0$ such that $x_i=0$ for all $i > n$ and $y_i=0$ for all $i$,
we will construct a random graph $\bar {\bf{G}}(z,q)$ that is equivalent in law to ${\bf{G}}(z,q)$, in two stages, as follows.
We begin with a graph on $[n]$ with no edges.
Let $\{\eta_{i,j}\}_{i,j \in \mathbb{N}}$ be independent Poisson point processes on $[0,\infty)$ such that $\eta_{ij}$ for $i\neq j$ has intensity
$qx_j$; and for $i=j$ has intensity $qx_i/2$.\\
{\bf Stage I: The breadth-first-search forest and associated random walk:}
Choose a vertex $v(1) \in [n]$ with $\mathbb{P}(v(1) = i)\propto x_i$. Let
$$\mathbb{I}_1 = \{ j \in [n]: j \neq v(1) \mbox{ and } \eta_{v(1),j} \cap [0, x_{v(1)}] \neq \emptyset \}.$$
Form an edge between $v(1)$ and each $j \in \mathbb{I}_1$. Let $c(1) = \# (\mathbb{I}_1)$.
Let $m_{v(1),j}$ be the first point in $\eta_{v(1),j}$ for each $j \in \mathbb{I}_1$. Order the vertices in $\mathbb{I}_1$ according to increasing
values of $m_{v(1),j}$ and label these as $v(2), \cdots v(c(1) + 1)$. Let
$$\mathcal{V}_1 =\{v(1)\}, \; \mathcal{N}_1 = \{v(2), \cdots , v(c(1) +1)\}, \; l_1 = x_{v(1)} \mbox{ and } d_1 = c(1).$$
Having defined $\mathcal{V}_{i'}$, $\mathcal{N}_{i'}$, $l_{i'}$, $d_{i'}$ and the edges up to step $i'$, with
$\mathcal{V}_{i'} = \{v(1), \cdots v(i')\}$, $\mathcal{N}_{i'} = \{ v(i'+1), v(i'+2), \cdots v(d_{i'} +1)\}$ for $1 \le i' \le i-1$, define,
if $\mathcal{N}_{i-1} \neq \emptyset$
$$
\mathbb{I}_i = \{ j \in [n]: j \not \in \mathcal{N}_{i-1}\cup\mathcal{V}_{i-1} \mbox{ and } \eta_{v(i),j} \cap [0, x_{v(i)}] \neq \emptyset \}$$
and form an edge between $v(i)$ and each $j \in \mathbb{I}_i$. Let $c(i) = |\mathbb{I}_i|$ and
let $m_{v(i),j}$ be the first point in $\eta_{v(i),j}$ for each $j \in \mathbb{I}_i$. Order the vertices in $\mathbb{I}_i$ according to increasing
values of $m_{v(i),j}$ and label these as $v(d_{i-1}+2), \cdots v(d_i + 1)$, where $d_i = d_{i-1} + c(i)$. Set
$$
l_i = l_{i-1} + x_{v(i)}, \;
\mathcal{V}_{i} = \{v(1), \cdots v(i)\}, \; \mathcal{N}_{i} = \{ v(i+1), v(i+2), \cdots v(d_{i} +1)\}.$$
In case $\mathcal{N}_{i-1} = \emptyset$, we choose $v(i)\in [n]\setminus \mathcal{V}_{i-1}$ with probability proportional to $x_j$, $j \in [n]\setminus \mathcal{V}_{i-1}$
and define $\mathbb{I}_i, c(i), d_i, l_i, \mathcal{V}_i, \mathcal{N}_i$ and the edges at step $i$ exactly as above.
This procedure terminates after exactly $n$ steps at which point we obtain a forest-like graph with no surplus edges. We will include
surplus to this graph in stage II below.
Associate with the above construction an (interpolated) random walk process $Z^{\scriptscriptstyle (n)}(\cdot)$ defined as follows. $Z^{\scriptscriptstyle (n)}(0) = 0$ and
\begin{equation}
\label{eqn:def_zn}
Z^{\scriptscriptstyle (n)}(l_{i-1}+u)=Z^{\scriptscriptstyle (n)}(l_{i-1})-u+\sum_{j \notin \mathcal{V}_i \cup \mathcal{N}_{i-1}} x_{j}{{\rm 1\hspace{-0.90ex}1}}_{\{m_{v(i),j}<u\}} \quad \mbox{for}~ 0< u< x_{v(i)},\;
i = 1, \cdots n,
\end{equation}
where by convention $l_0 =0$ and $\mathcal{N}_0 = \emptyset$. This defines $Z^{\scriptscriptstyle (n)}(t)$ for all $t \in [0, l_n)$. Define
$Z^{\scriptscriptstyle (n)}(t)=Z^{\scriptscriptstyle (n)}(l_n-)$ for all $t \ge l_n$.\\
{\bf Stage II: Construction of surplus edges:}
For each $i=1, \cdots, n$, we construct surplus edges on the graph obtained in Stage I and a point process $\mathcal{P}_x$ on $[0, l_n]$, simultaneously, as follows. \\
(i) For each $v \in \mathbb{I}_i$ and $\tau \in \eta_{v(i),v}\cap [0, x_{v(i)}] \setminus \{ m_{v(i),v} \}$, construct an edge between $v(i)$ and $v$. This corresponds to multi-edges between the two vertices $v(i)$ and $v$.\\
(ii) For each $\tau \in \eta_{v(i), v(i)} \cap [0, x_{v(i)}]$, construct an edge between $v(i)$ and itself. This corresponds to self-loops at the vertex $v(i)$.\\
(iii) For each $v(j) \in \mathcal{N}_{i-1} \setminus \{v(i)\}$ and $\tau \in \eta_{v(i), v(j)} \cap [0, x_{v(i)}]$, construct an edge between $v(i)$ and $v(j)$. This corresponds to additional edges between two vertices, $v(i)$ and $v(j)$, that were indirectly connected in stage I.\\
For each of the above cases, we also construct points for the point process $\mathcal{P}_x$ at time $l_{i-1}+\tau \in [0,l_n]$.\\
This completes the construction of the graph $\bar {\bf{G}}(z,q)$ and the random walk $Z^{\scriptscriptstyle (n)}(\cdot)$. This graph has the same law as ${\bf{G}}(z,q)$, so the associated component sizes and surplus vector denoted as $(\bar \boldsymbol{X}(z,q), \bar \boldsymbol{Y}(z,q))$ has the same law as that of $( \boldsymbol{X}(z,q), \boldsymbol{Y}(z,q))$.
Furthermore, conditioned on $Z^{\scriptscriptstyle (n)}$, $\mathcal{P}_x$ is Poisson point process on $[0, l_n]$ whose intensity we denote by $r_x(t)$.
Using the above construction we will show in next section, as a first step, a weaker result than Theorem \ref{theo:aldous-full-gene}.
\subsection{Convergence in $l^2_{\downarrow} \times \mathbb{N}^{\infty}$.}
\label{sec:sec6.2}
The following is the main result of this section.
\begin{Theorem}
\label{theo:generalized-aldous}
Let $z^{\scriptscriptstyle (n)} \in \mathbb{U}_{\downarrow}^0$ and $q^{\scriptscriptstyle (n)} \in (0,\infty)$ be sequences that satisfy the conditions in Theorem \ref{theo:aldous-full-gene}. Then
\begin{equation}
\label{eqn:fd-con-surpl}
(\boldsymbol{X}^{\scriptscriptstyle (n)},\boldsymbol{Y}^{\scriptscriptstyle (n)}) \convd (\boldsymbol{X}^*(\lambda), \boldsymbol{Y}^*(\lambda))
\end{equation}
in the space $l^2_{\downarrow} \times \mathbb{N}^\infty$ as $n\to \infty$, where we consider the product topology on $\mathbb{N}^\infty$.
\end{Theorem}
The key ingredient in the proof is the following result. With $z^{\scriptscriptstyle (n)}$ and $q^{\scriptscriptstyle (n)}$ as in the above theorem, define
$\bar \boldsymbol{X}^{\scriptscriptstyle (n)} = \bar \boldsymbol{X}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})$, $\bar \boldsymbol{Y}^{\scriptscriptstyle (n)} = \bar \boldsymbol{Y}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})$ and $r^{\scriptscriptstyle (n)}(t) = r_{x^{\scriptscriptstyle (n)}}(t)1_{[0, l_n]}(t)$, $t \ge 0$.
Denote the random walk process from Section \ref{sec:conv-in-prod} constructed using $(x^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})$ (rather than $(x,q)$), once more, by $Z^{\scriptscriptstyle (n)}(\cdot)$.
Define the rescaled process $\bar{Z}^{\scriptscriptstyle (n)}(\cdot)$ and its reflected version $\hat Z^{\scriptscriptstyle (n)} (\cdot)$ as follows
\begin{equation}
\label{eqn:reflec-def}
\bar{Z}^{\scriptscriptstyle (n)}(t) := \sqrt{\frac{s_2}{s_3}} Z^{\scriptscriptstyle (n)}(t), \quad \hat Z^{\scriptscriptstyle (n)} (t):= \bar{Z}^{\scriptscriptstyle (n)}(t) - \min_{0\leq u\leq t} \bar{Z}^{\scriptscriptstyle (n)}(u).
\end{equation}
\begin{Lemma}\label{lem:lem1325}
(i) As $n \to \infty$, the process $\bar{Z}^{\scriptscriptstyle (n)} \convd W_\lambda$ in $\mathcal{D} ([0,\infty): \mathbb{R})$.
(ii) For $n \ge 1$,
\begin{equation}
\label{eqn:rt-q-small}
\sup_{t \ge 0} \left|r^{\scriptscriptstyle (n)}(t)-q \sqrt{\frac{s_3}{s_2}} \hat Z^{\scriptscriptstyle (n)}(t) \right|
\le \frac{3}{2} q x_*.
\end{equation}
\end{Lemma}
Given Lemma \ref{lem:lem1325}, the proof of Theorem \ref{theo:generalized-aldous} can be completed as follows.\\
{\bf Proof of Theorem \ref{theo:generalized-aldous}:}
The paper \cite{aldous1997brownian} shows that the vector $\bar \boldsymbol{X}^{\scriptscriptstyle (n)}$ can be represented as the ordered sequence of excursion lengths of the
process $\hat{Z}^{\scriptscriptstyle (n)} $. Also, weak convergence of $\bar{Z}^{\scriptscriptstyle (n)}$ to $W_\lambda$ in Lemma \ref{lem:lem1325} (i) implies the convergence
of $\hat{Z}^{\scriptscriptstyle (n)} $ to $\hat W_{\lambda}$. Using these facts, Proposition 4 of \cite{aldous1997brownian} shows that
$\bar \boldsymbol{X}^{\scriptscriptstyle (n)}$ converges in distribution to the ordered excursion length sequence of $\hat W_{\lambda}$, namely $\boldsymbol{X}^*(\lambda)$, in $l^2_{\downarrow}$.
Also, conditional on $\hat{Z}^{\scriptscriptstyle (n)}$, $\mathcal{P}_x$ is a Poisson point process on $[0, \infty)$ with rate $r^{\scriptscriptstyle (n)}(t)$ and for $i \ge 1$,
$\bar Y_i^{\scriptscriptstyle (n)}$ has a Poisson distribution with parameter
$\int_{[a_i^{\scriptscriptstyle (n)}, b_i^{\scriptscriptstyle (n)}]} r^{\scriptscriptstyle (n)}(s) ds$, where $a_i^{\scriptscriptstyle (n)}, b_i^{\scriptscriptstyle (n)}$ are the left and right endpoints of the $i$-th ordered excursion
of $\hat Z^n$. From conditions in \eqref{eqn:qsigma2} it follows that $ q x^* \to 0$ and $ q \sqrt{s_3/s_2} \to 1$. Lemma \ref{lem:lem1325} (ii)
then shows that $\int_{[a_i^{\scriptscriptstyle (n)}, b_i^{\scriptscriptstyle (n)}]} r^{\scriptscriptstyle (n)}(s) ds$ converges in distribution to $\int_{[a_i, b_i]} \hat W_{\lambda}(s) ds$,
where $a_i, b_i$ are the left and right endpoints of the $i$-th ordered excursion
of $\hat W_{\lambda}$.
In fact we have the joint convergence of $\left(\hat{Z}^{\scriptscriptstyle (n)}, \left(\int_{[a_i^{\scriptscriptstyle (n)}, b_i^{\scriptscriptstyle (n)}]} r^{\scriptscriptstyle (n)}(s) ds\right)_{i\ge 1}\right)$
to $\left(\hat W_{\lambda}, \left(\int_{[a_i, b_i]} \hat W_{\lambda}(s) ds\right)_{i\ge 1}\right)$. This proves
the convergence of $(\bar\boldsymbol{X}^{\scriptscriptstyle (n)},\bar\boldsymbol{Y}^{\scriptscriptstyle (n)})$ to $(\boldsymbol{X}^*(\lambda), \boldsymbol{Y}^*(\lambda))$ in $l^2_{\downarrow} \times \mathbb{N}^\infty$.
The result follows since $(\bar \boldsymbol{X}^{\scriptscriptstyle (n)}, \bar \boldsymbol{Y}^{\scriptscriptstyle (n)})$ has the same law as $( \boldsymbol{X}^{\scriptscriptstyle (n)}, \boldsymbol{Y}^{\scriptscriptstyle (n)})$. \ \ \rule{1ex}{1ex}\\
{\bf Proof of Lemma \ref{lem:lem1325}} Part (i) was proved in Proposition 4 of \cite{aldous1997brownian}. Consider now (ii).\\
It is easy to verify that $Z^{\scriptscriptstyle (n)}$ satisfies
$$ Z^{\scriptscriptstyle (n)}(l_{i})= - \sum_{j=1}^{i} \delta_{v(j)} x_{v(j)} + \sum_{v \in \mathcal{N}_{i}} x_v,\; i = 1, \cdots n.$$
The above equation implies that for all $k \le i$, $Z^{\scriptscriptstyle (n)}(l_{k}) \ge - \sum_{j=1}^{i} \delta_{v(j)} x_{v(j)}$. In addition, taking $k_0 = \sup \set{j \le i: \delta_{v(j)}=1} $ we have $Z^{\scriptscriptstyle (n)}(l_{k_0}) = - \sum_{j=1}^{i} \delta_{v(j)} x_{v(j)}$. In particular, this implies that $\inf_{j \le i} Z^{\scriptscriptstyle (n)}(l_{j}) = - \sum_{j=1}^{i} \delta_{v(j)} x_{v(j)}$.
Also, from \eqref{eqn:def_zn} we have that for $t \in (l_{i-1},l_i]$,
$ Z^{\scriptscriptstyle (n)}(t) \ge Z^{\scriptscriptstyle (n)}(l_{i-1}) - x^*$. Consequently
\begin{equation}
\label{eqn:rw_minimum}
\left|\inf_{0 \le u \le t}Z^{\scriptscriptstyle (n)}(u) +\sum_{j=1}^{i-1} \delta_{v(j)} x_{v(j)}\right| = \left|\inf_{0 \le u \le t}Z^{\scriptscriptstyle (n)}(u) - \inf_{\set{j : l_{j} \le t}} Z^{\scriptscriptstyle (n)}(l_{j})\right| \le x^*.
\end{equation}
Let $\mathcal{N}_{i-1}=\{v(i), v(i+1), ... , v(i+l)\}$. From the above expression for $Z^{\scriptscriptstyle (n)}(l_{i})$, we have that for $t \in (l_{i-1},l_i]$
\begin{equation}
\label{eqn:rw_zn}
Z^{\scriptscriptstyle (n)}(t) = \left(- \sum_{j=1}^{i-1} \delta_{v(j)} x_{v(j)} + \sum_{j=i}^{i+l} x_{v(j) } \right) - \left(t -l_{i-1} \right) + \sum_{j \notin \mathcal{V}_i \cup \mathcal{N}_{i-1}} x_j {{\rm 1\hspace{-0.90ex}1}}_{\{m_{v(i),j}<t-l_{i-1}\}},
\end{equation}
Also, accounting for the three sources of surplus described in Stage II of the construction, one has
the following formula for $r^{\scriptscriptstyle (n)}(t)$ at time $t \in (l_{i-1},l_i]$:
$$ r^{\scriptscriptstyle (n)}(t) = q \cdot \left( \frac{x_{v(i)}}{2}+ \sum_{j=i+1}^{i+l}x_{v(j)} + \sum_{j \notin \mathcal{V}_i \cup \mathcal{N}_{i-1}} x_j {{\rm 1\hspace{-0.90ex}1}}_{\{m_{v(i),j}<t-l_{i-1}\}} \right). $$
The three terms in the above expression correspond to self-loops; edges between vertices that in stage I were only connected indirectly; and additional edges between two vertices that were directly connected in stage I. Combining the above expression with \eqref{eqn:rw_zn} and \eqref{eqn:rw_minimum}, we have
\begin{align}
\label{eqn:rate-surplus}
\left|r^{\scriptscriptstyle (n)}(t)-q \cdot \left(Z^{\scriptscriptstyle (n)}(t)-\min_{0 \le s \le t}Z^{\scriptscriptstyle (n)}(s)\right) \right|
\le q \cdot \left( \left| \inf_{0 \le s \le t}Z^{\scriptscriptstyle (n)}(s) +\sum_{j=1}^{i-1} \delta_{v(j)} x_{v(j)} \right| + \frac {x_{v(i)}}{2} \right)
\le \frac{3}{2} q x_*.
\end{align}
The result follows. \ \ \rule{1ex}{1ex}
\subsection{Proof of Theorem \ref{theo:aldous-full-gene}. }
\label{sec:conv-d-metric}
In this section we complete the proof of Theorem \ref{theo:aldous-full-gene}. The key step in the proof is the following lemma whose
proof is given in Section \ref{sec:sec6.3new}.
\begin{Lemma}
\label{lemma:tightness-sup}
Let $z^{\scriptscriptstyle (n)} \in \mathbb{U}_{\downarrow}^0$ and $q^{\scriptscriptstyle (n)} \in (0,\infty)$ be as in Theorem \ref{theo:aldous-full-gene}. Let $\hat Z^{\scriptscriptstyle (n)}$ be as
introduced in \eqref{eqn:reflec-def}. Then
$ \{\sup_{t \ge 0} \hat Z^{\scriptscriptstyle (n)}(t) \}_{n \ge 1}$ is a tight family of $\mathbb{R}_+$ valued random variables.
\end{Lemma}
\begin{Remark}
In fact one can establish a stronger statement, namely $ \sup_{u \ge t} \sup_{n \ge 1} \hat Z_u^{\scriptscriptstyle (n)} \to 0$ in probability
as $t \to \infty$. Also, although not used in this work, using very similar techniques as in the proof of Lemma \ref{lemma:tightness-sup}, it can be shown
that $ \sup_{u \ge t} \hat W_\lambda(u) $ converges a.s. to $0$, as $t \to \infty$.
\end{Remark}
{\bf Proof of Theorem \ref{theo:aldous-full-gene}.} Since $(\boldsymbol{X}^{\scriptscriptstyle (n)}, \boldsymbol{Y}^{\scriptscriptstyle (n)})$ has the same distributions as
$(\bar\boldsymbol{X}^{\scriptscriptstyle (n)}, \bar\boldsymbol{Y}^{\scriptscriptstyle (n)})$, we can equivalently consider the convergence of the latter sequence.
From Theorem \ref{theo:generalized-aldous} we have that $(\bar\boldsymbol{X}^{\scriptscriptstyle (n)}, \bar\boldsymbol{Y}^{\scriptscriptstyle (n)})$ converges to
$(\boldsymbol{X}^*(\lambda), \boldsymbol{Y}^*(\lambda))$, in distribution, in $l^2_{\downarrow} \times \mathbb{N}^{\infty}$ (with product topology on $\mathbb{N}^{\infty}$).
By appealing to Skorohod representation theorem, we can assume without loss of generality that the convergence is almost sure.
In view of Lemma \ref{lemma:basic-analysis}, it now suffices to argue that
$$ \sum_{i=1}^\infty \left|\bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i- X^*_i(\lambda) Y^*_i(\lambda)\right| \convp 0. $$
Fix $\epsilon > 0$. Then, for any $k \in \mathbb{N}$,
\begin{align}
\mathbb{P}\set{ \sum_{i=1}^{\infty} \left|\bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i- X^*_i(\lambda) Y^*_i(\lambda)\right| > \epsilon }\le &
\mathbb{P}\set{ \sum_{i=1}^k \left|\bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i- X^*_i(\lambda) Y^*_i(\lambda)\right| > \frac{\epsilon}{3} } \nonumber \\
+& \mathbb{P}\set{ \sum_{i=k+1}^\infty \bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i > \frac{\epsilon}{3}} +\mathbb{P}\set{ \sum_{i=k+1}^\infty X^{*}_i(\lambda) Y^{*}_i(\lambda) > \frac{\epsilon}{3}}. \label{eqn:1108}
\end{align}
From the convergence of $(\bar\boldsymbol{X}^{\scriptscriptstyle (n)}, \bar\boldsymbol{Y}^{\scriptscriptstyle (n)})$ to
$(\boldsymbol{X}^*(\lambda), \boldsymbol{Y}^*(\lambda))$ in $l^2_{\downarrow} \times \mathbb{N}^{\infty}$ we have that
$$
\lim_{n\to \infty} \mathbb{P}\set{ \sum_{i=1}^k \left|\bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i- X^*_i(\lambda) Y^*_i(\lambda)\right| > \frac{\epsilon}{3} } = 0.$$
Consider now the second term in \eqref{eqn:1108}. Let $E^{\scriptscriptstyle (n)}_L = \{ \sup_{t\ge 0} r_t^{\scriptscriptstyle (n)} \le L \}$. Then
$$
\mathbb{P}\set{ \sum_{i=k+1}^\infty \bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i > \frac{\epsilon}{3}}
\le \mathbb{P}\set{( E^{\scriptscriptstyle (n)}_L)^c} + \frac{3}{\epsilon} \mathbb{E} \left ({{\rm 1\hspace{-0.90ex}1}}_{E^{\scriptscriptstyle (n)}_L}\left[
\sum_{i=k+1}^\infty \bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i \wedge 1\right] \right).
$$
Let $\mathcal{G} = \sigma\{\hat Z^{\scriptscriptstyle (n)}(t): t \ge 0\}$. Since $r_t^{\scriptscriptstyle (n)}$ is $\mathcal{G}$ measurable for all $t \ge 0$, $E^{\scriptscriptstyle (n)}_L \in \mathcal{G}$.
Then
\begin{align*}
\mathbb{E} \left ({{\rm 1\hspace{-0.90ex}1}}_{E^{\scriptscriptstyle (n)}_L}\left[
\sum_{i=k+1}^\infty \bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i \wedge 1 \right]\right)
=& \mathbb{E} \left ({{\rm 1\hspace{-0.90ex}1}}_{E^{\scriptscriptstyle (n)}_L}\mathbb{E}\left[
\sum_{i=k+1}^\infty \bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i \wedge 1 \mid \mathcal{G} \right] \right)\\
\le& \mathbb{E} \left ({{\rm 1\hspace{-0.90ex}1}}_{E^{\scriptscriptstyle (n)}_L}
\left(\sum_{i=k+1}^\infty \mathbb{E}\left[\bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i \mid \mathcal{G} \right]\wedge 1 \right) \right)\\
\le& L \mathbb{E} \left[\sum_{i=k+1}^\infty (\bar X^{\scriptscriptstyle (n)}_i)^2 \wedge 1 \right],
\end{align*}
where the last inequality follows on observing that, conditionally on $\mathcal{G}$, $\bar Y^{\scriptscriptstyle (n)}_i$ has a Poisson distribution with
rate that is dominated by $\bar X^{\scriptscriptstyle (n)}_i \cdot (\sup_{t\ge 0} r_t^{\scriptscriptstyle (n)})$. Using the convergence of $\bar \boldsymbol{X}^{\scriptscriptstyle (n)}$ to $\boldsymbol{X}^*$, we now have
$$
\limsup_{n\to \infty}
\mathbb{E} \left ({{\rm 1\hspace{-0.90ex}1}}_{E^{\scriptscriptstyle (n)}_L}\left[
\sum_{i=k+1}^\infty \bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i\right] \wedge 1 \right) \le L \mathbb{E} \left[\sum_{i=k+1}^\infty ( X^*_i(\lambda))^2 \wedge 1 \right].$$
Let $\delta > 0$ be arbitrary. Using Lemma \ref{lemma:tightness-sup} and Lemma \ref{lem:lem1325} (ii) we can choose $L \in (0, \infty)$ such that
$\mathbb{P}\set{( E^{\scriptscriptstyle (n)}_L)^c} \le \delta$.
Finally, taking limit as $n \to \infty$ in \eqref{eqn:1108} we have that
\begin{align}
\limsup_{n\to \infty}\mathbb{P}\set{ \sum_{i=1}^{\infty} \left|\bar X^{\scriptscriptstyle (n)}_i \bar Y^{\scriptscriptstyle (n)}_i- X^*_i(\lambda) Y^*_i(\lambda)\right| > \epsilon }
\le & \delta + L \mathbb{E} \left[\sum_{i=k+1}^\infty ( X^*_i(\lambda))^2 \wedge 1 \right]\nonumber\\
+& \mathbb{P}\set{ \sum_{i=k+1}^\infty X^{*}_i(\lambda) Y^{*}_i(\lambda) > \frac{\epsilon}{3}}.
\end{align}
The result now follows on sending $k \to \infty$ in the above display and recalling that
$\sum_{i=1}^\infty ( X^*_i(\lambda))^2 < \infty$ and $\sum_{i=1}^\infty X^{*}_i(\lambda) Y^{*}_i(\lambda) < \infty$ a.s. and $\delta > 0$ is arbitrary. \ \ \rule{1ex}{1ex}
\subsection{Proof of Lemma \ref{lemma:tightness-sup}.}
\label{sec:sec6.3new}
In this section we prove Lemma \ref{lemma:tightness-sup}. We will only treat the case $\lambda = 0$. The general case can be treated similarly. The key step in the proof is the following proposition whose proof is given at the end of the section.
Note that
$\sup_{t \ge 0}|\bar Z^{\scriptscriptstyle (n)}(t) - \bar Z^{\scriptscriptstyle (n)}(t-)| \le x^* \sqrt{s_2/s_3} \to 0$ as $n \to \infty$.
Also, as $n \to \infty$, $qs_2 \to 1$.
Thus, without loss of generality, we will assume that
\begin{equation}\label{eq:eq1547}
\sup_{n \ge 1}\sup_{t \ge 0}|\bar Z^{\scriptscriptstyle (n)}(t) - \bar Z^{\scriptscriptstyle (n)}(t-)| \le 1, \; \sup_{n \ge 1} q^{\scriptscriptstyle (n)}s_2^{\scriptscriptstyle (n)} \le 2 .
\end{equation}
Fix $\vartheta \in (0, 1/2)$ and define $t^{* \scriptscriptstyle (n)} = \left(\frac{s_2}{x^*}\right)^{\vartheta}$.
Denote by $\{\mathcal{F}_t^{\scriptscriptstyle (n)}\}$ the filtration generated by $\{\bar Z^{\scriptscriptstyle (n)}(t)\}_{t \ge 0}$.
For ease of notation, we write $\sup_{t \in [a,b]}=\sup_{[a,b]}$. We will suppress $(n)$ in the notation, unless needed.\\
\begin{Proposition}
\label{prop:tails-bound}
There exist $\Theta \in (0, \infty)$, events $G^{\scriptscriptstyle (n)}$, increasing $\mathcal{F}_t^{\scriptscriptstyle (n)}$-stopping times $1= \sigma^{\scriptscriptstyle (n)}_0 < \sigma^{\scriptscriptstyle (n)}_1 < ... $, and
a real positive sequence $\set{\kappa_i}$ with $\sum_{i=1}^\infty \kappa_i < \infty$, such that the following hold:\\
(i) For every $i \ge 1$, $\set{\sigma_i^{\scriptscriptstyle (n)}}_{n \ge 1}$ is tight.\\
(ii) For every $i \ge 1$,
$$ \mathbb{P}\left(\set{ \sup_{[{\sigma^{\scriptscriptstyle (n)}_{i-1}},{\sigma^{\scriptscriptstyle (n)}_i}]} \hat Z^{\scriptscriptstyle (n)}(t) > 2\Theta + 1 } \cap \set{ \sigma^{\scriptscriptstyle (n)}_{i-1} < t^{* \scriptscriptstyle (n)} } \cap G^{\scriptscriptstyle (n)}\right) \le \kappa_i. $$
(iii) As $n \to \infty$, $ \mathbb{P}\set{ \sup_{[\sigma^{*\scriptscriptstyle (n)}, \infty)} \hat Z^{\scriptscriptstyle (n)}(t) > \Theta; G^{\scriptscriptstyle (n)}} \to 0$, where $\sigma^{*\scriptscriptstyle(n)}= \inf\set{ \sigma_i^{\scriptscriptstyle (n)}: \sigma_i^{\scriptscriptstyle (n)} \ge t^{*\scriptscriptstyle (n)}} $. \\
(iv) As $n \to \infty$, $\mathbb{P}(G^{\scriptscriptstyle (n)}) \to 1$.
\end{Proposition}
Given Proposition \ref{prop:tails-bound}, the proof of Lemma \ref{lemma:tightness-sup} can be completed as follows.\\ \ \\
{\bf Proof of Lemma \ref{lemma:tightness-sup}:}
Fix $\epsilon \in (0,1)$. Let $\Theta \in (0, \infty)$, $G^{\scriptscriptstyle (n)}$, $\sigma_i^{\scriptscriptstyle (n)}$, $\kappa_i$ be as in Proposition \ref{prop:tails-bound}.
Choose $i_0 > 1$ such that $\sum_{i\ge i_0} \kappa_i \le \epsilon$. Since $\set{\sigma_{i_0-1}^{\scriptscriptstyle (n)}}$ is tight,
there exists $T \in (0,\infty)$ such that $\limsup_{n \to \infty} \mathbb{P}\set{ \sigma_{i_0-1}^{\scriptscriptstyle (n)} > T } \le \epsilon$.
Thus for any $M' > 2\Theta + 1$, we have
\begin{align*} \mathbb{P}\set{\sup_{[1,\infty)}\hat Z^{\scriptscriptstyle (n)}(t) > M'} \le & \mathbb{P}\set{\sup_{[1,T]}\hat Z^{\scriptscriptstyle (n)}(t) > M'} + \mathbb{P}\set{\sigma_{i_0-1}^{\scriptscriptstyle (n)} > T } + \mathbb{P}\set{(G^{\scriptscriptstyle (n)})^c} \\
+& \mathbb{P}\set{\sup_{[\sigma^{\scriptscriptstyle (n)}_{i_0-1},\sigma^{* \scriptscriptstyle (n)}]}\hat Z^{\scriptscriptstyle (n)}(t) > 2\Theta+1; G^{\scriptscriptstyle (n)}} +
\mathbb{P}\set{ \sup_{[\sigma^{*\scriptscriptstyle (n)}, \infty)} \hat Z^{\scriptscriptstyle (n)}(t) > \Theta; G^{\scriptscriptstyle (n)}}. \end{align*}
Taking $\limsup_{n \to \infty}$ on both sides
$$ \limsup_{n \to \infty}\mathbb{P}\set{\sup_{[1,\infty)}\hat Z^{\scriptscriptstyle (n)}(t) > M'} \le \limsup_{n \to \infty}\mathbb{P}\set{\sup_{[1,T]}\hat Z^{\scriptscriptstyle (n)}(t) > M'} + \epsilon + 0 + \epsilon + 0. $$
Since $\set{\sup_{[1,T]}\hat Z^{\scriptscriptstyle (n)}(t)}_{n \ge 1}$ is tight, we have,
$$ \limsup_{M' \to \infty} \limsup_{n \to \infty}\mathbb{P}\set{\sup_{[1,\infty)}\hat Z^{\scriptscriptstyle (n)}(t) > M'} \le 2\epsilon . $$
Since $\epsilon > 0$ is arbitrary, the result follows. \ \ \rule{1ex}{1ex} \\
We now proceed to the proof of Proposition \ref{prop:tails-bound}. The following lemma is key.\\
\begin{Lemma}
\label{lemma:mn-an-prop}
There are $\{\mathcal{F}_t^{\scriptscriptstyle (n)}\}$ adapted processes $\{A^{\scriptscriptstyle (n)}(t)\}$, $\{B^{\scriptscriptstyle (n)}(t)\}$ and $\mathcal{F}_t^{\scriptscriptstyle (n)}$-martingale $\{M^{\scriptscriptstyle (n)}(t)\}$
such that\\
(i) $A^{\scriptscriptstyle (n)}(\cdot)$ is a non-increasing function of $t$, a.s. For all $t \ge 0$, $\bar Z^{\scriptscriptstyle (n)}(t) = \int_0^t A^{\scriptscriptstyle (n)}(u) du + M^{\scriptscriptstyle (n)}(t)$.\\
(ii) For $t \ge 0$, $\langle M^{\scriptscriptstyle (n)},M^{\scriptscriptstyle (n)} \rangle_t = \int_0^t B^{\scriptscriptstyle (n)}(u) du$.\\
(iii) $\sup_{n \ge 1} \sup_{u \ge 0} B^{\scriptscriptstyle (n)}(u) \le 2$.\\
(iv) With $ G^{\scriptscriptstyle (n)} = \set{ A(t) < -t/2 \mbox{ for all } t \in [1, t^{*(n)}] } $, $\mathbb{P}(G^{\scriptscriptstyle (n)}) \to 1$ as $n\to\infty$.\\
(v) For any $\alpha \in (0, \infty)$ and $t > 0$,
\begin{equation}
\mathbb{P} \set{ \sup_{ u \in [0,t]} |M^{\scriptscriptstyle (n)}(u)| > \alpha } \le 2\exp\left\{ \alpha \right\} \cdot \exp\left\{ - \alpha \log \left( 1+ \frac{\alpha}{2t} \right) \right\}. \label{eqn:1117}
\end{equation}
\end{Lemma}
{\bf Proof:} Recall the notation from Section \ref{sec:conv-in-prod}. Parts (i) and (ii) are proved in \cite{aldous1997brownian}. Furthermore, from
Lemma 11 of \cite{aldous1997brownian} it follows that, for $ t \in [l_{i-1}, l_i)$, writing $Q_2(t) = \sum_{j=1}^i (x_{v(j)})^2 $, we have
\begin{equation*}
A(t) \le \sqrt{\frac{s_2}{s_3}}(-1+q s_2 - q Q_2(t)), \;
B(t) \le q s_2 .
\end{equation*}
Part (iii) now follows on recalling from \eqref{eq:eq1547} that $qs_2 \le 2$.
To prove (iv) it suffices to show that
\begin{equation}
\label{eqn:sn-s-to-zero}
\sup_{t\leq t^*} \left|\frac{s_2}{s_3} Q_2(t) - t\right|\convp 0.
\end{equation}
To prove this we will use the estimate on Page 832, Lemma 13 of \cite{aldous1997brownian}, which says that for any fixed $\epsilon \in (0,1)$, and $ L \in (0,\infty)$
$$ \mathbb{P} \set{ \sup_{ t \in [0,L]}\left|\frac{s_2}{s_3} Q_2(t) - t \right| > \epsilon }=O\left( \frac{L^2 x^*}{s_2}+ \sqrt{\frac{L (x^*)^2 s_2}{s_3}} + \frac{L^2 s_3}{s_2^2}+ \sqrt{\frac{Ls_3}{s_2}} + \frac{s_2^2}{(1-2 Ls_2)^+}
\right). $$
Note that the first term on the right hand side determine its order when $L \to \infty$.
Taking $L = t^*$ in the above estimate we see that, since $\vartheta \in (0, 1/2)$, the expression on the right side above goes to $0$ as $n \to \infty$. This proves
\eqref{eqn:sn-s-to-zero} and thus completes the proof of (iv).
Finally, proof of (v) uses standard concentration inequalities for martingales. Indeed, recalling that the maximal jump size of
$\bar Z$, and consequently that of $M$, is bounded by $1$ and $\langle M,M \rangle_t \le 2t$, we have from Section 4.13, Theorem 5 of \cite{liptser-mart-book} that, for any fixed $\alpha>0$ and $t>0$,
$$ \mathbb{P}\{ \sup_{u \in [0,t]} |M_u| > \alpha \} \le 2 \exp \set{-\sup_{\lambda > 0}\left[\alpha \lambda - 2t \phi(\lambda)\right] },$$
where $\phi(\lambda) = (e^{\lambda }-1-\lambda ).$ A straightforward calculation shows
$$ \sup_{\lambda > 0 } [\alpha \lambda - 2t \phi(\lambda)] = \alpha \log\left(1 + \frac{\alpha}{2t}\right) - \left( \alpha - 2t \log\left(1 + \frac{\alpha } {2t}\right) \right) \ge \alpha \log\left(1 + \frac{\alpha}{2t}\right) - \alpha.$$
The result follows. \ \ \rule{1ex}{1ex}\\
The bound \eqref{eqn:1117} continues to hold if we replace $M(u)$ with $M(\tau+u)-M(\tau)$ for any finite stopping time $\tau$.
From this observation we immediately have the following corollary.
\begin{Corollary} \label{lemma:mart-concentration}
Let $M$ be as in Lemma \ref{lemma:mn-an-prop}. Then, for any finite stopping time $\tau$:\\
(i) $\mathbb{P} \set{ \sup_{ u \in [0,t]} |M(\tau+u)-M(\tau)| > \alpha } \le 2 e^{-\alpha}$, whenever $\alpha > 2(e^2-1)t$. \\
(ii) $\mathbb{P} \set{ \sup_{ u \in [0,t]} |M(\tau+u)-M(\tau)| > \alpha } \le 2 (2e/\alpha)^{\alpha} t^{\alpha}$, for all $t >0$ and $\alpha > 0$.
\end{Corollary}
Part (i) of the corollary is useful when $\alpha$ is large and
part (ii) is useful when $t$ is small.
Finally we now give the proof of Proposition \ref{prop:tails-bound}.\\
{\bf Proof of Proposition \ref{prop:tails-bound}:} From Lemma \ref{lem:lem1325} (i) we have that $\hat Z^{\scriptscriptstyle (n)}$ converges in distribution to
$\hat W_0$ (recall we assume that $\lambda = 0$) as $n \to \infty$. Let $\{\epsilon_i\}_{i \ge 1}$ be a positive real sequence bounded by $1$ and fix $\Theta \in (2, \infty)$. Choice of $\Theta$ and $\epsilon_i$ will be specified later in the proof.
Let $\sigma^{\scriptscriptstyle (n)}_0 < \tau^{\scriptscriptstyle (n)}_1 \le \sigma^{\scriptscriptstyle (n)}_1 < \tau^{\scriptscriptstyle (n)}_2 \le \sigma^{\scriptscriptstyle (n)}_2 < ... $ be a sequence of stopping times such that $\sigma^{\scriptscriptstyle (n)}_0=1$, and for $i \ge 1$,
\begin{equation}
\tau^{\scriptscriptstyle (n)}_i = \inf \{t \ge \sigma^{\scriptscriptstyle (n)}_{i-1} + \epsilon_i : \hat Z^{\scriptscriptstyle (n)}(t)\ge \Theta\} \wedge (\sigma^{\scriptscriptstyle (n)}_{i-1}+1), \;\; \sigma^{\scriptscriptstyle (n)}_i = \inf \{t \ge \tau^{\scriptscriptstyle (n)}_{i} : \hat Z^{\scriptscriptstyle (n)}(t) \le 1\}. \label{eqn:def-stopping-times}
\end{equation}
Similarly define stopping times
$1 = \bar \sigma_0 < \bar \tau_1 \le \bar \sigma_1 < \bar \tau_2 \le \bar \sigma_2 < ... $
by replacing $\hat Z^{\scriptscriptstyle (n)}$ in \eqref{eqn:def-stopping-times} with $\hat W_0$. Due to the negative quadratic drift in the definition of
$W_0$ it follows that $\bar \sigma_i < \infty$ for every $i$ and from the
weak convergence of $\hat Z^{\scriptscriptstyle (n)}$ to $\hat W_0$ it follows that $\sigma^{\scriptscriptstyle (n)}_i \to \bar \sigma_i $ and
$\tau^{\scriptscriptstyle (n)}_i \to \bar \tau_i $, in distribution, as $n \to \infty$. Here we have used the fact that if $\zeta$ denotes the first time $W_0$
hits the level $\alpha \in (0, \infty)$ then, a.s., for any $\delta > 0$, there are infinitely many crossings of the level $\alpha$ in $(\zeta, \zeta + \delta)$.
In particular we have that $\{\sigma^{\scriptscriptstyle (n)}_i\}_{n \ge 1}$ is a tight sequence, and this proves part (i) of Proposition \ref{prop:tails-bound}.\\
For the rest of the proof we suppress $(n)$ from the notation. Since the jump size of $\hat Z$ is bounded by $1$, we have that $ \sup_{[{\sigma_{i-1}},{\sigma_{i-1}+ \epsilon_i}]} \hat Z(t) \le \Theta $ implies $ \sup_{[\sigma_{i-1},\tau_i]} \hat Z(t) \le \Theta+1 $ and thus, in this case, when $t \in [\tau_i,\sigma_i]$, we have $\hat Z(t) = \hat Z(\tau_i) + ( \bar Z(t) - \bar Z(\tau_i) )\le \Theta+1 +( \bar Z(t) - \bar Z(\tau_i))$.
Let $G \equiv G^{\scriptscriptstyle (n)}$ be as in Lemma \ref{lemma:mn-an-prop} (iv) and let
$H_i = G \cap \{\sigma_{i-1} < t^*\}$, then writing $\mathbb{P} (\cdot \cap H_i)$ as $\mathbb{P}_i(\cdot)$,
\begin{align}
\mathbb{P}_i \set{ \sup_{[{\sigma_{i-1}},{\sigma_i}]} \hat Z(t) > 2\Theta + 1 }
\le& \mathbb{P}_i \set{ \sup_{[{\sigma_{i-1}},{\sigma_{i-1}+ \epsilon_i}]} \hat Z(t) > \Theta }\\
+& \mathbb{P}_i \set{ \sup_{[\tau_i, \sigma_i]} \left[\Theta + 1 + (\bar Z(t) - \bar Z(\tau_i))\right] > 2\Theta+1 }.
\end{align}
Denote the two terms on the right side by $\mathbb{T}_1$ and $\mathbb{T}_2$ respectively. Recalling that $\hat Z (\sigma_{i-1})\le 2$, we have
from the decomposition in Lemma \ref{lemma:mn-an-prop} (i) and Corollary \ref{lemma:mart-concentration}(ii) that
\begin{align}
\mathbb{T}_1 \le \mathbb{P}\set{ \sup_{[\sigma_{i-1},\sigma_{i-1}+\epsilon_i]} |M(t)-M(\sigma_{i-1})| > \frac{\Theta-2}{2} } \le C_{\frac{\Theta-2}{2}} {\epsilon_i}^{(\Theta-2)/2}, \label{eqn:1167}
\end{align}
Here, for $\alpha > 0$, $C_{\alpha} = 2 (2e/\alpha)^{\alpha}$ and we have used the fact that on $H_i$,
$A(t) \le -t/2\le 0$ for all $t \in [\sigma_{i-1}, \sigma_{i-1}+ \epsilon_i]$.
Next, let $\{\delta_i\}_{i\ge 1}$ be a sequence of positive reals bounded by $1$. Setting $d_i = \sum_{j=1}^{i-1} \epsilon_i$, we have
\begin{align}
\mathbb{T}_2
\le& \mathbb{P}_i \set{ \sup_{[\tau_i,\tau_i+\delta_i]}(\bar Z(t) - \bar Z(\tau_i)) > \Theta } + \mathbb{P}_i \set{\sup_{[\tau_i + \delta_i, \tau_i + 1]}(\bar Z(t) - \bar Z(\tau_i)) > \Theta} + \mathbb{P}\set{ \sigma_i > \tau_i+1} \nonumber \\
\le& \mathbb{P} \set{ \sup_{[\tau_i,\tau_i+\delta_i]}(M(t) - M(\tau_i)) > \Theta } +
\mathbb{P} \set{\sup_{[\tau_i + \delta_i, \tau_i + 1]}(M(t) - M(\tau_i)) > \Theta + \frac{\delta_i d_i}{2}} \nonumber\\
+& \mathbb{P}\set{ M(\tau_i + 1)-M(\tau_i) > -\Theta + \frac{d_i}{2}} \nonumber\\
\le& C_{\Theta}\delta_i^{\Theta} + 2 e^{-\delta_i d_i/2} + 2 e^{\Theta-d_i/2}, \label{eqn:1174}
\end{align}
whenever
\begin{equation}\label{eq:ins431} \min\{\delta_i d_i/2 , d_i/2 - \Theta\} > 2(e^2-1).\end{equation}
Fix $\Theta > 14$. Then $\max\set{C_{\Theta}, C_{(\Theta-2)/2}}\le 2$. We will impose additional conditions on $\Theta$ later in the proof. Combining \eqref{eqn:1167} and \eqref{eqn:1174}, we have
\begin{align}
\mathbb{P}_i \set{ \sup_{[{\sigma_{i-1}},{\sigma_i}]} \hat Z(t) > 2\Theta + 1 } \le 2 ( {\epsilon_i}^{(\Theta-2)/2} + \delta_i^{\Theta} + e^{-\delta_i d_i/2} + e^{\Theta-d_i/2}) \equiv \kappa_i. \label{eqn:basic-bound}
\end{align}
Let
$$ \epsilon_i= i^{-1/2}, \;\; d_i = \sum_{j=1}^{i-1} \epsilon_i \sim i^{1/2} , \;\; \delta_i = 1/\sqrt{d_i} \sim i^{-1/4} .$$
Then, \eqref{eq:ins431} holds for $i$ large enough, and
$$\kappa_i \sim 2 ( i^{-(\Theta-2)/4}+ i^{-\Theta/4} + e^{-i^{1/4}/2} + e^\Theta e^{-i^{1/2}/2}),$$
which, since $\Theta >14$, is summable.
This proves part (ii) of the Proposition.\\
Now we consider part (iii).
We will construct another sequence of stopping times with values in $[t^*, \infty)$, as follows.
Define
$\sigma_0^* :=\inf\set{ \sigma_i : \sigma_i \ge t^*} = \inf\set{t \ge t^*: \hat Z(t) \le 1 }$, then define $\tau^*_i, \sigma^*_i$ for $i \ge 1$ similarly as in \eqref{eqn:def-stopping-times}. Similar arguments as before give a bound as \eqref{eqn:basic-bound} with $d_i$ replaced by $t^*$, $\delta_i$ replaced by
$1/\sqrt{t^*}$, $\epsilon_i$ replaced with $1/t^*$ and $\Theta$ replaced by any $\Theta_0 > 14$. Namely,
\begin{align}
\mathbb{P} \set{ \sup_{[{\sigma_{i-1}^*},{\sigma_i^*}]} \hat Z(t) > 2\Theta_0 + 1;\; G^{\scriptscriptstyle(n)} } \le 2 ( (1/t^*)^{(\Theta_0-2)/2} + (1/\sqrt{t^*})^{\Theta_0} + e^{-\sqrt{t^*}/2 } + e^{\Theta_0-t^*/2}) . \label{eqn:basic-boundnew}
\end{align}
Here we have used the fact that
since $A(t)$ is non-increasing, on $G^{\scriptscriptstyle(n)}$, $A(t) \le -t^*/2$ for all $t \ge t^*$.\\
Recall that, by construction,
$\hat Z(t) = 0$ when $t \ge s_1$. So there exist $i_0$ such that $\tau^*_{i_0}= \infty$, in fact since $\sigma^*_i \ge \sigma^*_{i-1}+ \epsilon$, we have that $i_0 \le s_1/\epsilon$. Thus, we have from the above display that
$$ \mathbb{P}\set{ \sup_{[\sigma^{*}_0, \infty)} \hat Z(t) > 2\Theta_0+1} \le \frac{2s_1}{\epsilon} ( (1/t^*)^{(\Theta_0-2)/2} + (1/\sqrt{t^*})^{\Theta_0} + e^{-\sqrt{t^*}/2 } + e^{\Theta_0-t^*/2}).$$
Taking $\Theta > 29$, we have on setting $\Theta_0 = \frac{\Theta -1}{2}$ in the above display
$$ \mathbb{P}\set{ \sup_{[\sigma^{*}_0, \infty)} \hat Z(t) > \Theta} \le 2s_1 \left( {\left(\frac{1}{t^*}\right)}^{(\Theta-1)/4-2} + {\left(\frac{1}{t^*}\right)}^{(\Theta-1)/4-1} + \frac{1}{t^*} e^{-\sqrt{t^*}/2} + \frac{1}{t^*}e^{(\Theta-1)/2-t^*/2}\right).$$
From \eqref{eqn:additional-condition} we have that $s_1 \cdot (\frac{1}{t^*})^{\varsigma/\vartheta} \to 0$.
So if $\Theta \ge 4 (\frac{\varsigma}{\vartheta} +2) + 1$, the above expression approaches $0$ as $n \to \infty$.
The result now follows on taking $\Theta = \max\{29, 4 (\frac{\varsigma}{\vartheta} +2) + 1\}$. \ \ \rule{1ex}{1ex} \\
\section{Bounded-size rules at time $t_c- n^{-\gamma}$}
\label{sec:main-bsr-susceptibility}
Throughout Sections \ref{sec:main-bsr-susceptibility} and \ref{sec:main-coupling} we take $T = 2t_c$ which is a convenient upper bound for the time parameters of interest.
In this section we prove Theorems \ref{thm:suscept-funct} and \ref{thm:suscept-limit}.
We begin with some notation associated with BSR processes, which closely follows \cite{spencer2007birth}.
Recall from Section \ref{sec:bsr} the set $\Omega_K$ and the random graph process ${\bf{BSR}}^{\scriptscriptstyle (n)}(t)$ associated with a given $K$-BSR $F \subset \Omega_K^4$. Frequently we will suppress $n$ in the notation.
Also recall the definition of $c_t(v)$ from Section \ref{sec:bsr}.
For $i \in \Omega_K$, define
\begin{equation}
X_i(t)=|\{v \in {\bf{BSR}}_t^{\scriptscriptstyle (n)} : c_t(v)=i \}| \mbox{ and } \bar{x}_i(t)=X_i(t)/n. \label{eqn:def-x}
\end{equation}
Denote by ${\bf{BSR}}^*(t)$ the subgraph of ${\bf{BSR}}(t)$ consisting of all components of size greater than $K$, and define, for $k=1,2,3$
$$ \mathcal{S}_{k,\varpi}(t):= \sum_{\{\mathcal{C} \subset {\bf{BSR}}^*(t)\}} |\mathcal{C}|^k \mbox{ and } \bar s_{k,\varpi}(t) = \mathcal{S}_{k,\varpi}(t)/n,$$
where $\{\mathcal{C} \subset {\bf{BSR}}^*(t)\}$ denotes the collection of all components in ${\bf{BSR}}^*(t)$. For notational convenience in long formulae, we sometimes write ${\bf{BSR}}(t) = {\bf{BSR}}_t$ and similarly ${\bf{BSR}}^*(t) = {\bf{BSR}}_t^*$. Similar notation will be used throughout the paper.
Clearly
\begin{equation}
\mathcal{S}_k(t) = \mathcal{S}_{k,\varpi} + \sum_{i=1}^K i^{k-1} X_i(t), \; \bar s_k(t) = \bar s_{k,\varpi} + \sum_{i=1}^K i^{k-1} \bar x_i(t). \label{eqn:s2omega-to-s2}
\end{equation}
Also note that $\mathcal{S}_1(t) = n$ and $\mathcal{S}_{1,\varpi}(t) = X_\varpi(t)$.\\
Recall the Poisson processes $\mathcal{P}_{\vec{v}}$ introduced in Section \ref{sec:bsr}. Let
$\mathcal{F}_t = \sigma \{\mathcal{P}_{\vec{v}}(s): s \le t, \vec{v} \in [n]^4\}$.
For $T_0 \in [0,T]$ and a $\{\mathcal{F}_t\}_{0\le t < T_0}$ semimartingale $\{J(t)\}_{0 \le t < T_0}$ of the form
\begin{equation} \label{eq:semimart} dJ(t) = \alpha(t) dt + dM(t), \langle M, M \rangle_t = \int_0^t \gamma(s) ds, \end{equation}
where $M$ is a $\{\mathcal{F}_t\}$ local martingale and $\gamma$ is a progressively measurable process, we write $\alpha = {\bf d}(J)$, $M = \boldsymbol{M}(J)$
and $\gamma = \boldsymbol{v}(J)$.
{\bf Organization:} Rest of this section is organized as follows.
In Section \ref{sec:prelim-bsr}, we state a recent result on BSR models and certain deterministic maps associated with the evolution of ${\bf{BSR}}_t^*$ from \cite{bsr-2012} that will be used in this work.
In Section \ref{sec:bsr-diff-s2s3A}, we will study the asymptotics of $\bar s_{2,\varpi}$ and $\bar s_{3,\varpi}$.
In Section \ref{sec:proof-thm-alphabeta}, we will complete the proof of Theorem \ref{thm:suscept-funct}.
In Section \ref{sec:decompose-s2-s3}, we will obtain some useful semimartingale decompositions for certain functionals of $\bar s_2$ and $\bar s_3$.
In Section \ref{sec:proof-conv-susceptibility}, we will complete the proof of Theorem \ref{thm:suscept-limit}.
\subsection{Evolution of ${\bf{BSR}}^*_t$.}
\label{sec:prelim-bsr}
We begin with the following lemma from \cite{bsr-2012} (see also
\cite{spencer2007birth}).
\begin{Lemma}
\label{lemma:approx-xi}\
(a) For each $i \in \Omega_K$, there exists a continuously differentiable function $x_i:[0,T] \to [0,1]$ such that for any $\delta \in (0,1/2)$, there exist $C_1, C_2 \in (0, \infty)$ such thar
for all $n$,
\[\mathbb{P}\left( \sup_{i \in \Omega_K} \sup_{s \in [0,T]} |\bar{x}_i(t)-x_i(t) | > n^{-\delta}\right) <C_1 \exp\left( -C_2 n^{1-2\delta}\right).\]
(b) There exist polynomials $\{F^x_i({\bf{x}})\}_{i\in \Omega_K}$, ${\bf{x}}=(x_i)_{i \in \Omega_K} \in \mathbb{R}^{K+1}$, such that ${\bf{x}}(t)= (x_i(t))_{i \in \Omega_K}$ is the unique solution to the differential equations:
\begin{equation}
x_i^\prime(t)= F^x_i({\bf{x}}(t)), \;\; i \in \Omega_K, \;\; t \in [0,T] \mbox{ with initial values } {\bf{x}}(0)=(1,0,..., 0). \label{eqn:sys-dif-eqns}
\end{equation}
Furthermore, $\bar x_i$ is a $\{\mathcal{F}_t\}_{0\le t < T}$ semimartingale of the form \eqref{eq:semimart} and
$$\sup_{0\le t < T} |{\bf d}(\bar x_i)(t) - F_i^x(\bar {\bf{x}}(t)) | \le \frac{K^2}{n}.$$
Also,
for all $i \in \Omega_K$ and $t \in (0,T]$, we have $x_i(t)>0$ and $ \sum_{i\in \Omega_K} x_i(t) = 1$. \\
\end{Lemma}
Recall that ${\bf{BSR}}^*(t)$ is the subgraph of ${\bf{BSR}}(t)$ consisting of all components of size greater than $K$. The evolution of this graph is governed by three type of events:\\
{\bf Type 1 (Immigrating vertices): } This corresponds to the merger of two components of size bounded by $K$ into a component of size larger than $K$. Such an event leads to the
appearance of a new component in ${\bf{BSR}}^*(t)$ which we view as the immigration of a `vertex' into ${\bf{BSR}}^*(t)$. Denote by $n a^*_i(t)$ the rate at which a component of size $K+i$ immigrates into ${\bf{BSR}}_t^*$ at time $t$.
In \cite{bsr-2012} it is shown that there are polynomials $F_i^a({\bf{x}})$ for $1\le i\le K$ such that, with
$\bar {\bf{x}}(t) = (\bar x_i(t))_{i \in \Omega_K}$
\begin{equation}
\sup_{t \in [0,\infty) }| a^*_i(t)-F^a_i(\bar {\bf{x}}(t))| \le \frac{K}{n}. \label{eqn:error-a}
\end{equation}
We define, with ${\bf{x}}(t)$ as in Lemma \ref{lemma:approx-xi},
\begin{equation}
a_i(t) := F^a_i({\bf{x}}(t)), \; i = 1, \cdots K. \label{eqn:def-fa-a}
\end{equation}
{\bf Type 2 (Attachments):} This event corresponds to a component of size at most $K$ getting linked with some component of size larger than $K$. For $1\le i \le K$, denote by $|\mathcal{C}| c^*_i(t)$ the rate at which a component of size $i$ attaches to a component $\mathcal{C}$ in ${\bf{BSR}}^*_{t-}$.
Then (see \cite{bsr-2012}) there exist polynomials $F^c_i({\bf{x}})$ for $1\le i\le K$, such that
$c_i^*(t) = F^c_i(\bar {\bf{x}}(t))$. Define
\begin{equation}
c_i(t) := F^c_i({\bf{x}}(t)), i = 1, \cdots K. \label{eqn:def-fc-c}
\end{equation}
{\bf Type 3 (Edge formation):} This event corresponds to the addition of an edge between components in ${\bf{BSR}}^*_t$. The occurrence of this event adds one edge between two vertices in ${\bf{BSR}}^*_{t-}$, the vertex set stays unchanged, whereas the edge set
has one additional element. From \cite{bsr-2012}, there is a polynomial $F^b({\bf{x}})$ such that, defining $b^*(t) = F^b(\bar {\bf{x}}(t))$, the rate at which each pair of components $\mathcal{C}_1 \neq \mathcal{C}_2 \in {\bf{BSR}}_t^*$ merge at time $t$, equals $|\mathcal{C}_1| |\mathcal{C}_2| b^*(t)/n$.
Furthermore
\begin{equation}
b(t) := F^b({\bf{x}}(t)) \label{eqn:def-fb-b}
\end{equation}
satisfies $b(t_c) \in (0, \infty)$.
\subsection{Analysis of $\bar s_{2,\varpi}(t)$ and $\bar s_{3,\varpi}(t)$}
\label{sec:bsr-diff-s2s3A}
We begin by recalling a result from \cite{spencer2007birth}.
Define functions $F^s_{2,\varpi} : [0,1]^{K+1}\times \mathbb{R} \to \mathbb{R}$
and $F^s_{3,\varpi} : [0,1]^{K+1}\times \mathbb{R}^2 \to \mathbb{R}$
as
\begin{equation}
F^s_{2,\varpi}({\bf{x}}, s_2)
:= \sum_{j=1}^K (K+j)^2 F^a_{j}({\bf{x}}) + y_2\sum_{j=1}^K 2j F^c_{j}({\bf{x}}) + x_\varpi \sum_{j=1}^K j^2 F^c_{j}({\bf{x}}) + (s_2)^2 F^b ({\bf{x}}) ,
\label{eqn:fs2omega}
\end{equation}
for $({\bf{x}}, s_2) \in [0,1]^{K+1}\times \mathbb{R} $ and, for $({\bf{x}}, s_2, s_3) \in [0,1]^{K+1}\times \mathbb{R}^2 $
\begin{align}
F^s_{3,\varpi}({\bf{x}}, s_2, s_3)
:=& \sum_{j=1}^K (K+j)^3 F^a_{j}({\bf{x}}) + s_3 \sum_{j=1}^K 3jF^c_{j}({\bf{x}}) + 3s_2 \sum_{j=1}^K j^2 F^c_{j}({\bf{x}})\nonumber\\
+& x_\varpi \sum_{j=1}^K j^3 F^c_j({\bf{x}}) + 3s_2 s_3 F^b({\bf{x}}). \label{eqn:fs3omega}
\end{align}
\begin{Lemma}\cite{spencer2007birth}
\label{lemma:s2omega-s3omegaA}
As $n \to \infty$, $\bar s_{j,\varpi}(t) \to s_{j,\varpi}(t)$ in probability, $j=2,3$, $t \in [0, t_c)$. Furthermore, $s_{j,\varpi}$ are continuously
differentiable on $[0, t_c)$ and can be characterized as the unique solutions of the equations
\begin{align}
s'_{2,\varpi}(t) =& F^s_{2,\varpi}({\bf{x}}(t), s_{2,\varpi}(t)),\; s_{2,\varpi}(0)=0, \label{eqn:diff-s2}\\
s'_{3,\varpi}(t) =& F^s_{3,\varpi}({\bf{x}}(t), s_{2,\varpi}(t), s_{3,\varpi}(t)),\; s_{3,\varpi}(0)=0. \label{eqn:diff-s3}
\end{align}
Furthermore,
$\lim_{t \to t_c} s_{2,\varpi}(t) = \lim_{t \to t_c} s_{3,\varpi}(t) = \infty.$
\end{Lemma}
The following lemma gives additional information on the convergence of $\bar s_{j,\varpi}$ to $s_{j,\varpi}$. For $T_0 \in [0, T]$, a stochastic process
$\{\xi(t)\}_{0 \le t < T_0}$, and a nonnegative sequence $\alpha(n)$, the quantity $O_{T_0}(\xi(t)\alpha(n))$ will represent a stochastic process
$\{\eta(t)\}_{0\le t < T_0}$ such that for some $d_1 \in (0, \infty)$, $\eta(t) \le d_1 \xi(t)\alpha(n)$, for all $0 \le t < T_0$ and $n \ge 1$.
\begin{Lemma}
\label{lemma:s2omega-s3omega}
The processes $\bar s_{j,\varpi}$, $j=2,3$, are $\{\mathcal{F}_t\}_{0\le t < t_c}$ semimartingales of the form \eqref{eq:semimart} and
\begin{align*}
&|{\bf d}(\bar s_{2,\varpi})(t) - F^s_{2,\varpi}(\bar {\bf{x}}(t), \bar s_{2,\varpi}(t))| = O_{t_c}(\mathcal{S}_4(t)/n^2)\\
&|{\bf d}(\bar s_{3,\varpi})(t) - F^s_{3,\varpi}(\bar {\bf{x}}(t), \bar s_{2,\varpi}(t), \bar s_{3,\varpi}(t))|= O_{t_c}(\mathcal{S}_5(t)/n^2).
\end{align*}
\end{Lemma}
{\bf Proof:} Note that $\mathcal{S}_{2, \varpi}$ and $\mathcal{S}_{3, \varpi}$ have jumps at time instant $t$ with rates and values $ \Delta \mathcal{S}_{2, \varpi}(t)$, $\Delta \mathcal{S}_{3, \varpi}(t)$, respectively, given as follows.
\begin{itemize}
\item for each $1 \le i \le K$, with rate $n a^*_i(t)$,
$$ \Delta \mathcal{S}_{2, \varpi}(t) = (K+i)^2, \;\; \Delta \mathcal{S}_{3, \varpi}(t) = (K+i)^3.$$
\item for each $1 \le i \le K$ and $\mathcal{C} \subset {\bf{BSR}}_{t-}^*$, at rate $|\mathcal{C}| c^*_i(t)$,
$$ \Delta \mathcal{S}_{2, \varpi}(t) = 2|\mathcal{C}| i+ i^2, \;\; \Delta \mathcal{S}_{3, \varpi}(t) = 3|\mathcal{C}|^2 i + 3 |\mathcal{C}| i^2 +i^3. $$
\item for all unordered pair $\mathcal{C},\tilde \mathcal{C} \subset {\bf{BSR}}_{t-}^*$, such that $\mathcal{C} \neq \tilde \mathcal{C}$, at rate $ |\mathcal{C}| |\tilde\mathcal{C}| b^*(t)/n$,
$$ \Delta \mathcal{S}_{2, \varpi}(t) = 2|\mathcal{C}| |\tilde \mathcal{C}| , \;\; \Delta \mathcal{S}_{3, \varpi}(t) = 3|\mathcal{C}|^2|\tilde \mathcal{C}| + 3 |\mathcal{C}| |\tilde \mathcal{C}|^2. $$
\end{itemize}
Thus
\begin{align}
{\bf d}(\mathcal{S}_{2,\varpi})(t)
=& \sum_{j=1}^K (K+j)^2 n a^*_j(t) + \sum_{j=1}^K \sum_{\mathcal{C} \subset {\bf{BSR}}_t^*} (2j|\mathcal{C}| + j^2) |\mathcal{C}| c^*_j(t) + \sum_{\mathcal{C} \neq \tilde \mathcal{C} \subset {\bf{BSR}}_t^*} 2|\mathcal{C}| |\tilde \mathcal{C}| \frac{b^*(t) |\mathcal{C}| |\tilde \mathcal{C}|}{n} \nonumber \\
=& \sum_{j=1}^K (K+j)^2 n a^*_j(t) + \sum_{j=1}^K 2j c^*_j(t) \mathcal{S}_{2,\varpi}(t)
+ \sum_{j=1}^K j^2 c^*_j(t) X_\varpi(t)\nonumber\\ +& \frac{b^*(t)}{n} (\mathcal{S}_{2,\varpi}^2(t) - \mathcal{S}_{4,\varpi}(t))\nonumber\\
=& n\left(F_{2,\varpi}^s(\bar {\bf{x}}, \bar s_{2,\varpi}) + O(1/n) + O_{t_c}(S_{4,\varpi}(t)/n^2)\right) \label{eqn:2072}
\end{align}
and
\begin{align*}
&{\bf d}(\mathcal{S}_{3,\varpi})(t) \nonumber\\
=& \sum_{j=1}^K (K+j)^3 n a^*_j(t) + \sum_{j=1}^K \sum_{\mathcal{C} \subset {\bf{BSR}}_t^*} (3j|\mathcal{C}|^2 + 3 j^2 |\mathcal{C}| + j^3) |\mathcal{C}| c^*_j(t) \\
&+ \sum_{\mathcal{C} \neq \tilde \mathcal{C} \subset {\bf{BSR}}_t^*} (3|\mathcal{C}|^2 |\tilde \mathcal{C}| + 3|\mathcal{C}| |\tilde \mathcal{C}|^2 ) \frac{b^*(t) |\mathcal{C}| |\tilde \mathcal{C}|}{n} \\
=& \sum_{j=1}^K (K+j)^3 n a^*_j(t) + \sum_{j=1}^K 3j c^*_j(t) \mathcal{S}_{3,\varpi}(t)
+ \sum_{j=1}^K 3j^2 c^*_j(t) \mathcal{S}_{2,\varpi}(t) \\ +& \sum_{j=1}^K j^3 c^*_j(t) X_\varpi(t) + \frac{3b^*(t)}{n} (\mathcal{S}_{3,\varpi}(t)\mathcal{S}_{2,\varpi}(t) - \mathcal{S}_{5,\varpi}(t))\\
=& n\left(F_{3,\varpi}^s(\bar {\bf{x}}(t), \bar s_{2,\varpi}(t), \bar s_{3,\varpi}(t)) + O(1/n) + O_{t_c}(S_{4,\varpi}(t)/n^2)\right).
\end{align*}
The result follows. \ \ \rule{1ex}{1ex} \\
As an immediate consequence of \eqref{eqn:s2omega-to-s2} and the convergence of $(\bar s_k, \bar s_{k,\varpi}, \bar {\bf{x}})$
to $( s_k, s_{k,\varpi}, {\bf{x}})$
we have the following formula.
\begin{equation}
s_k(t) := s_{k,\varpi}(t) + \sum_{i=1}^K i^{k-1} x_i(t), \mbox{ for } k=2,3. \label{eqn:2134}
\end{equation}
This, along with Lemma \ref{lemma:s2omega-s3omegaA} and Lemma \ref{lemma:approx-xi}(b), yields the following differential equations for $s_2$ and $s_3$.
\begin{Lemma}
\label{lemma:diff-s2s3}
The functions $s_2,s_3$ are continuously differentiable on $[0, t_c)$ and can be characterized as the unique solutions of the following differential equations
\begin{align*}
s'_2(t) &= F^s_2({\bf{x}}(t), s_2(t)), \;\;\;\; s_2(0)=1,\\
s'_3(t) &= F^s_3({\bf{x}}(t), s_2(t) ,s_3(t)), \;\; \;\; s_3(0)=1.
\end{align*}
where the function $F^s_2(\cdot)$ and $F^s_3(\cdot)$ are defined as
\begin{align*}
F^s_2({\bf{x}},s_2) &:= F^s_{2,\varpi}\left({\bf{x}}, s_2 - \sum_{i=1}^K i x_i\right) + \sum_{i=1}^K i F^x_{i}({\bf{x}}),\\
F^s_3({\bf{x}},s_2,s_3) &:= F^s_{3,\varpi}\left({\bf{x}},\; s_2- \sum_{i=1}^K i x_i,\; s_3-\sum_{i=1}^K i^2 x_i\right) + \sum_{i=1}^K i^2 F^x_{i}({\bf{x}}).
\end{align*}
\end{Lemma}
\subsection{Proof of Theorem \ref{thm:suscept-funct}}
\label{sec:proof-thm-alphabeta}
In this section we prove Theorem \ref{thm:suscept-funct}. We begin with the following lemma which defines the two parameters $\alpha$ and $\beta $ that appear in Theorems \ref{thm:suscept-funct} and \ref{thm:crit-regime}.
Recall from Section \ref{sec:prelim-bsr} that $b(t_c) \in (0, \infty)$.
\begin{Lemma}
\label{lemma:alpha-beta}
There following two limits exist,
$$ \alpha := \lim_{t \to t_c-} (t_c-t)s_2(t), \;\; \beta := \lim_{t \to t_c-} \frac{s_3(t)}{(s_2(t))^3}.$$
Furthermore, $\alpha, \beta \in (0, \infty)$ and $\alpha = 1/{b(t_c)} $.
\end{Lemma}
{\bf Proof:} By \eqref{eqn:2134}, for $k = 2,3$, $|s_k(t)-s_{k,\varpi}(t)| \le K^{k}$. Since $s_k(t) \to \infty$, we thus have that $\lim_{t\to t_c-} s_k(t)/s_{k,\varpi}(t) = 1$.
Write $y_\varpi(t)=1/s_{2,\varpi}(t)$ and $z_\varpi(t)=y^3_\varpi(t) s_{3,\varpi}(t)$, it suffices to show that:
\begin{equation}\label{eq:eq742}
\lim_{t \to t_c-} \frac{t_c-t}{y_\varpi(t)} = \lim_{t \to t_c-} -\frac{1}{y'_\varpi(t)} = \frac{1}{b(t_c)}, \mbox{ and } \lim_{t \to t_c-} z_\varpi(t) \in (0, \infty). \end{equation}
Define $A_l(t)=\sum_{i=1}^K(K+i)^l a_i(t)$ and $C_l(t)=\sum_{i=1}^{K}i^l c_i(t)$ for $l=1,2,3$. Then by Lemma \ref{lemma:s2omega-s3omegaA}, \eqref{eqn:fs2omega} and \eqref{eqn:fs3omega}, the derivative of $y_\varpi(t)$ and $z_\varpi(t)$ can be written as follows (we omit $t$ from the notation):
\begin{align}
y_\varpi' =& -(A_2+C_2 x_\varpi) y_\varpi^2 - 2 C_1 y_\varpi - b, \label{eqn:diff-y}\\
z_\varpi' =& y_\varpi^3 \left[ A_3+3C_1s_{3,\varpi} +3C_2s_{2,\varpi}+C_3x_\varpi + 3bs_{2,\varpi}s_{3,\varpi} \right]\nonumber \\
-& 3y_\varpi^2s_{3,\varpi} \left[(A_2+C_2 x_\varpi) y_\varpi^2 + 2 C_1 y_\varpi + b\right]\nonumber\\
=& -(3y_\varpi A_2+3y_\varpi C_2x_\varpi +3C_1) z_\varpi + (y_\varpi^3A_3+ 3y_\varpi^2C_2 +y_\varpi^3C_3 x_\varpi) \nonumber\\
=& - B_1 z_\varpi + B_2,\label{eqn:diff-z}
\end{align}
where $B_1(t) = (3y_\varpi(t) A_2(t)+3y_\varpi(t) C_2(t)x_\varpi(t) +3C_1(t))$ and $B_2(t) = (y_\varpi^3(t)A_3(t)+ 3y_\varpi^2(t)C_2(t) +y_\varpi^3(t)C_3(t) x_\varpi(t)) $.
Since $\lim_{t \to t_c-} y_\varpi(t)=0$, we have $\lim_{t \to t_c-}y'_\varpi(t) = - b(t_c)$ which proves the first statement in \eqref{eq:eq742}.
Choose $t_1 \in (0,t_c)$ such that $y_{\varpi}(t), z_{\varpi}(t) \in (0,\infty)$ for all $t \in (t_1, t_c)$.
Then from \eqref{eqn:diff-z}, for all such $t$
$$ z_\varpi(t) = \int_{t_1}^{t} e^{ -\int_s^{t} B_1(u)du } B_2(s)ds + z_\varpi(t_1)e^{ -\int_{t_1}^{t} B_1(u)du } .$$
Since $B_1, B_2$ are nonnegative and $\sup_{t \in [t_1, t_c]} \{ B_1(t) + B_2(t)\} < \infty$, we have $\lim_{t \to t_c-} z_\varpi(t) \in (0, \infty)$. This completes the proof of \eqref{eq:eq742}.
The result follows.
\ \ \rule{1ex}{1ex}\\
We now complete the proof of Theorem \ref{thm:suscept-funct}.\\
{\bf Proof of Theorem \ref{thm:suscept-funct}:} Let $\alpha, \beta$ be as introduced in Lemma \ref{lemma:alpha-beta}.
From Lemma \ref{lemma:diff-s2s3} it follows that $y(t) = 1/ s_2(t)$ and $z(t) = y^3(t) s_3(t)$, for $0 \le t < t_c$, solve
the differential equations
\begin{align}
\label{eq:eq1746}
y'(t) = F^y({\bf{x}}(t),y(t)),\; z'(t) = F^z({\bf{x}}(t), y(t), z(t)), \; y(0) =z(0) = 1,
\end{align}
where
$F^y: [0,1]^{K+2} \to \mathbb{R}$ and $F^z: [0,1]^{K+2} \times \mathbb{R} \to \mathbb{R}$ are defined as
\begin{equation}
F^y({\bf{x}}, y) := -y^2 F^s_2({\bf{x}} , 1/y), \; F^z({\bf{x}}, y, z) := 3 z F^y({\bf{x}}, y)/y + y^3 F^s_3({\bf{x}}, 1/y, z/y^3), \label{eqn:def-fy}
\end{equation}
$({\bf{x}}, y, z) \in [0,1]^{K+2} \times \mathbb{R} \to \mathbb{R}$.
It is easy to check that $F^y$ and $F^z$ are polynomials in $(x_1, ...,x_K, x_\varpi, y)$ and $(x_1, ...,x_K, x_\varpi, y, z)$ respectively,
furthermore for each fixed $({\bf{x}}, y) \in [0,1]^{K+2}$
the map $z \mapsto F^x({\bf{x}}, y, z)$ is linear.
Thus \eqref{eq:eq1746} has a unique solution. Also, defining
$y(t_c) = \lim_{t\to t_c-}y(t) = \lim_{t\to t_c-}y_{\varpi}(t)$ and $z(t_c) = \lim_{t\to t_c-}z(t) = \lim_{t\to t_c-}z_{\varpi}(t)$, we see that
$y, z$ are twice continuously differentiable (from the left) at $t_c$. Furthermore, $y'(t_c-) = -\alpha^{-1}$ and
$z(t_c-) = \beta$. Thus we have
$$ y(t) = \frac{1}{\alpha} (t_c-t)(1 + O(t_c-t)), \;\; z(t) = \beta (1 + O(t_c-t)), \mbox{ as } t \uparrow t_c. $$
The result follows. \ \ \rule{1ex}{1ex}\\
\subsection{Asymptotic analysis of $\bar s_2(t)$ and $\bar s_3(t)$}
\label{sec:decompose-s2-s3}
In preparation for the proof of Theorem \ref{thm:suscept-limit}, in this section we will obtain some useful semimartingale decompositions for
$ Y(t):=\frac{1}{\bar s_2(t)}$ and $Z(t):=\frac{\bar s_3(t)}{(\bar s_2(t))^3}$. Throughout this section and next we will denote $ |\mathcal{C}_1^{\scriptscriptstyle (n)}(t)|$ as $I(t)$.
Recall the functions $F^s_2$, $F^s_3$ introduced in Lemma \ref{lemma:diff-s2s3}.
\begin{Lemma}
\label{lemma:decompose-s2-jumps}
The processes $\bar s_2$ and $\bar s_3$ are $\{\mathcal{F}_t\}_{0\le t < t_c}$ semimartingales of the form \eqref{eq:semimart} and the following equations hold.\\
\begin{enumerate}[(a)]
\item ${\bf d}(\bar s_2)(t) = F^s_2( \bar {\bf{x}}(t), \bar s_2(t)) + O_{t_c}\left(I^2(t)\bar s_2(t)/{n}\right).$ \\
\item ${\bf d}(\bar s_3)(t) = F^s_3(\bar {\bf{x}}(t), \bar s_2(t) ,\bar s_3(t)) + O_{t_c}\left( {I^3(t) \bar s_2(t)}/{n}\right).$\\
\item $\boldsymbol{v}(\bar s_2)(t) = O_{t_c}(I^2(t)\bar s_2^2(t)/n).$\\
\end{enumerate}
\end{Lemma}
\textbf{Proof: } Parts (a) and (b) are immediate from \eqref{eqn:s2omega-to-s2}, Lemma \ref{lemma:approx-xi}(b) and Lemma \ref{lemma:s2omega-s3omega}.
For part (c), recall the three types of events described in Section \ref{sec:prelim-bsr}.
For type 1, $\Delta \bar s_2(t)$ is bounded by $2K^2/n$ and the total rate of such events is bounded by $n/2$. For type 2, the attachment
of a size $j$ component, $1\le j \le K$, to a component $\mathcal{C}$ in ${\bf{BSR}}^*_{t-}$ occurs at rate $|\mathcal{C}| c^*_j(t)$ and produces a jump
$\Delta \bar s_2(t) = 2j|\mathcal{C}|/n$. For type 3, components $\mathcal{C}$ and $\tilde \mathcal{C}$ merge at rate $|\mathcal{C}| |\tilde \mathcal{C}| b^*(t)/n$ and produce a jump $ \Delta \bar s_2(t) = 2 |\mathcal{C}| |\tilde \mathcal{C}|/n$.
Thus for $t \in [0, t_c)$, $\boldsymbol{v}(\bar s_2)(t)$ can be estimated as
\begin{align*}
\boldsymbol{v}(\bar s_2)(t) \le &
\frac{n}{2} \left( \frac{2K^2 }{n}\right)^2
+ \sum_{j=1}^K \sum_{\mathcal{C} \subset {\bf{BSR}}^*_{t}} \left(\frac{2j|\mathcal{C}|}{n}\right)^2 |\mathcal{C}| c^*_j(t)
+ \sum_{\mathcal{C} \neq \tilde \mathcal{C} \subset {\bf{BSR}}^*_t} \left(\frac{2|\mathcal{C}| |\tilde \mathcal{C}|}{n}\right)^2 \frac{b^*(t) |\mathcal{C}| |\tilde \mathcal{C}|}{n} \\
\le& \frac{2K^4}{n} + \frac{4K^2 \mathcal{S}_3 }{n^2}+ \frac{4(\mathcal{S}_3)^2}{n^3} = O_{t_c}\left(\frac{I^2(t)\bar s_2^2(t)}{n}\right).
\end{align*}
This proves (c).
\ \ \rule{1ex}{1ex}\\
In the next lemma, we obtain a semimartingale decomposition for $Y$.
\begin{Lemma}
\label{lemma:decomp-s2-y}
The process $Y(t)=1/\bar s_2(t)$ is a $\{\mathcal{F}_t\}_{0\le t < t_c}$ semimartingale of the form \eqref{eq:semimart} and
(i) With $F^y(\cdot)$ as defined in \eqref{eqn:def-fy},
\begin{equation}
{\bf d}(Y)(t) = F^y(\bar {\bf{x}}(t), Y(t)) + O_{t_c}\left(\frac {I^2(t)Y(t)}{n}\right). \label{eqn:delta-y}
\end{equation}
(ii)
\begin{equation*}
\boldsymbol{v}(Y)(t) = O_{t_c}\left(\frac{ I^2(t)Y^2(t)}{n} \right).
\end{equation*}
\end{Lemma}
{\bf Proof:} Note that
\begin{equation}
\Delta Y(t) = \frac{1}{\bar s_2+\Delta \bar s_2}- \frac{1}{\bar s_2}= -\frac{\Delta \bar s_2}{\bar s_2^2}+ \frac{(\Delta \bar s_2)^2}{\bar s_2^2(\bar s_2 + \Delta \bar s_2)} =-\frac{\Delta \bar s_2}{\bar s_2^2}+ O_{t_c}\left(\frac{(\Delta \bar s_2)^2}{\bar s_2^3}\right). \label{eqn:2591}
\end{equation}
Thus by Lemma \ref{lemma:decompose-s2-jumps}(a), we have,
\begin{align*}
{\bf d}(Y)(t)
=& -\frac{1} {(\bar s_2(t))^2} {\bf d}(\bar s_2)(t) + O_{t_c}\left( \frac{1}{(\bar s_2(t))^3} \boldsymbol{v}(\bar s_2)(t)\right)\\
=& \left( -\frac{1} {(\bar s_2(t))^2} \right) \left( F^s_2( \bar {\bf{x}}(t), \bar s_2(t)) +O_{t_c}\left(\frac{I^2(t)\bar s_2(t)}{n}\right) \right) + O_{t_c}\left( \frac{1}{(\bar s_2(s))^3} \cdot \frac{I^2(t)\bar s_2^2(t)}{n} \right) \\
=& F^y(\bar {\bf{x}}(t), Y(t)) + O_{t_c}\left(\frac{I^2(t)Y(t)}{n}\right).
\end{align*}
This proves (i). For (ii), note that \eqref{eqn:2591} also implies
$ (\Delta Y(t))^2 \le \frac{(\Delta \bar s_2)^2}{\bar s_2^4}$
We then have
$$ {\bf d}(Y)(t) \le \frac{2\boldsymbol{v}(\bar s_2)(t)}{\bar s_2^4} = O_{t_c}\left(\frac{I^2(t)Y^2(t)}{n}\right).$$
The result follows. \ \ \rule{1ex}{1ex}\\
We now give a semimartingale decomposition for $Z(t)=\bar s_3(t)/(\bar s_2(t))^3$.
\begin{Lemma}
\label{lemma:decomp-s3-z}
The process $Z(t)=\bar s_3(t)/(\bar s_2(t))^3$ is a $\{\mathcal{F}_t\}_{0\le t < t_c}$ semimartingale of the form \eqref{eq:semimart} and
(i) With $F^z(\cdot)$ as defined in \eqref{eqn:def-fy},
\begin{equation*}
{\bf d}(Z)(t) = F^z (\bar {\bf{x}}(t), Y(t), Z(t)) + O_{t_c}\left(\frac{I^3(t)Y^2(t)}{n}\right).
\end{equation*}
(ii)
\begin{equation*}
\boldsymbol{v}(Z)(t) = O_{t_c}\left(\frac{ I^4(t)Y^4(t)}{n} + \frac{ I^6(t)Y^6(t)}{n}\right).
\end{equation*}
\end{Lemma}
\textbf{Proof: } Note that
\begin{equation*}
\Delta Z = Y^3 \Delta \bar s_3 + 3 Y^2 \bar s_3 \Delta Y + R(\Delta Y, \Delta \bar s_3),
\end{equation*}
where $R(\Delta Y, \Delta \bar s_3)$ is the error term which, using the observations
that $ \bar s_3 \le I \bar s_2$, $\Delta \bar s_3 \le 3 I \Delta \bar s_2$ and $|\Delta Y| \le Y^2 \Delta \bar s_2$, can be bounded as follows.
\begin{align*}
|R(\Delta Y, \Delta \bar s_3)|
\le& 3 Y^2 |\Delta Y||\Delta \bar s_3| + 3 Y \bar s_3 |\Delta Y|^2 \\
\le& 3Y^2 \cdot {Y^2 \Delta \bar s_2} \cdot {3I\Delta \bar s_2} + 3 I \cdot \left( {Y^2 \Delta \bar s_2 }\right)^2
= 12I Y^4 \cdot (\Delta \bar s_2)^2.
\end{align*}
From Lemma \ref{lemma:decompose-s2-jumps}(b), Lemma \ref{lemma:decomp-s2-y}(i) and Lemma \ref{lemma:decompose-s2-jumps}(c), we have
\begin{align*}
{\bf d}(Z)(t)
=& Y^3(t) {\bf d}(\bar s_3)(t) + 3Y^2(t) \bar s_3(t) {\bf d}(Y)(t) + O_{t_c}\left(I(t) Y^4(t)\boldsymbol{v}(\bar s_2)(t)\right)\\
=& Y^3(t) \left(F^s_3(\bar {\bf{x}}(t),\bar s_2(t), \bar s_3(t))+ O_{t_c}\left(\frac{I^3(t) \bar s_2(t)}{n}\right)\right) \\
&+3Y^2(t) \bar s_3(t) \left( F^y(\bar {\bf{x}}(t), Y(t)) + O_{t_c}\left(\frac{I^2(t) Y(t)}{n}\right)\right) + O_{t_c}\left( \frac{I^3(t)Y^2(t)}{n}\right)\\
=& F^z(\bar {\bf{x}}(t), Y(t),Z(t)) + O_{t_c}\left(\frac{I^3(t) Y^2(t)}{n}\right).
\end{align*}
This proves (i). For (ii), note that
\begin{equation*}
Y^3 |\Delta \bar s_3| + 3 Y^2\bar s_3 |\Delta Y| \le Y^3 \cdot 3I|\Delta \bar s_2|+ 3Y^2 \cdot I \bar s_2 \cdot Y^2|\Delta \bar s_2|=6Y^3I|\Delta \bar s_2|.
\end{equation*}
Thus,
$$
|\Delta Z| \le 6Y^3I|\Delta \bar s_2| + 12 I Y^4 \cdot (\Delta \bar s_2)^2.$$
Applying Lemma \ref{lemma:decompose-s2-jumps}(c) we now have,
\begin{equation*}
\boldsymbol{v}(Z)(t) = O_{t_c} \left( Y^6 I^2 \boldsymbol{v}(\bar s_2)(t)\right) + O_{t_c} \left(\frac{I^6 Y^6}{n}\right) =
O_{t_c}\left( \frac{I^4Y^4}{n} + \frac{I^6 Y^6}{n} \right).
\end{equation*}
The result follows. \ \ \rule{1ex}{1ex} \\
\subsection{Proof of Theorem \ref{thm:suscept-limit}}
\label{sec:proof-conv-susceptibility}
We begin with an upper bound on the size of the largest component at time $t \le t_n = t_c - n^{-\gamma}$ for $\gamma \in (0,1/4)$, which has been proved in \cite{bsr-2012}, and will play an important role in the proof of Theorem \ref{thm:suscept-limit}.
\begin{Theorem}[{\cite[Theorem 1.2]{bsr-2012}} {\bf Barely subcritical regime}]
\label{thm:subcrit-reg}
Fix $\gamma \in (0,1/4)$. Then there exists $C_3 \in (0, \infty)$ such that, as $n \to \infty$,
\[ \mathbb{P}\set{ I^{\scriptscriptstyle (n)}(t) \le C_3 \frac{(\log n)^4}{(t_c-t)^2},~ \forall t <t_c-n^{-\gamma} } \to 1. \]
\end{Theorem}
The next lemma is an elementary consequence of Gronwall's inequality.
\begin{Lemma}
\label{theo:conv-pure-jump}
Let $\{t_n\}$ be a sequence of positive reals such that $t_n \in [0, t_c)$ for all $n$. Suppose that $U^{\scriptscriptstyle(n)}$ is a semimartingale of the form \eqref{eq:semimart} with values in ${\mathbb{D}} \subset \mathbb{R}$.
Let $g:[0,t_c)\times {\mathbb{D}} \to {\mathbb{R}}$ be such that, for some $C_4(g) \in (0, \infty)$,
\begin{equation} \sup_{t \in [0, t_c)}|g(t,u_1)-g(t,u_2)| \le C_4(g)|u_1-u_2|, \; u_1, u_2 \in {\mathbb{D}}. \label{eq:eq2127}\end{equation}
Let $\{u(t)\}_{t \in [0,T]}$ be the unique solution of the differential equation
$$ u^\prime(t) = g(t,u(t)), \;\;\; u(0)=u_0. $$
Further suppose that there exist positive sequences:
\begin{enumerate}[(i)]
\item $\{\theta_1(n)\}$ such that, whp,
$|U^{\scriptscriptstyle(n)}(0)-u_0| \le \theta_1(n)$.
\item $\{\theta_2(n)\}$ such that, whp,
$$ \int_0^{t_n}\left| {\bf d}(U^{\scriptscriptstyle(n)})(t) - g(t,U^{\scriptscriptstyle(n)}(t))\right|dt \le \theta_2(n). $$
\item $\{\theta_3(n)\}$ such that, whp,
$\langle\boldsymbol{M}(U^{\scriptscriptstyle(n)}), \boldsymbol{M}(U^{\scriptscriptstyle(n)})\rangle_{t_n} \le \theta_3(n)$.
\end{enumerate}
Then, whp,
$$ \sup_{0\le t \le t_n}|U^{\scriptscriptstyle (n)}(t)-u(t)| \le e^{C_4(g)T}(\theta_1(n) + \theta_2(n) + \theta_4(n)), $$
where $\theta_4=\theta_4(n)$ is any sequence satisfying $ \sqrt{\theta_3(n)} = o(\theta_4(n))$.
\end{Lemma}
{\bf Proof:} We suppress $n$ from the notation unless needed. Using the Lipschitz property of $g$, we have, for all $t \in [0, t_n]$,
\begin{align*}
|U(t)-u(t)|
\le& |U(0)-u_0| + \int_0^t |{\bf d}(U)(s)-g(s,U(s))| ds + \int_0^t|g(s,U(s))-g(s,u(s))|ds + |\boldsymbol{M}(U)(t)|\\
\le& |U(0)-u_0| + \int_0^t |{\bf d}(U)(s)-g(s,U(s))| ds + |\boldsymbol{M}(U)(t)| + C_4\int_0^t |U(s)-u(s)|ds.
\end{align*}
Then by Gronwall's lemma
\begin{equation}\label{eq:eq1844}
\sup_{0 \le t \le t_n}| U(t)-u(t)| \le \left( |U(0)-u_0| + \int_0^{t_n} |{\bf d}(U)(s)-g(s,U(s))| ds + \sup_{0 \le t \le t_n}|\boldsymbol{M}(U)(t)| \right) e^{C_4 T}.
\end{equation}
Let $\tau^{\scriptscriptstyle (n)} = \inf\{ t\ge 0: \langle \boldsymbol{M}(U), \boldsymbol{M}(U) \rangle_t > \theta_3(n) \}$. By Doob's inequality
$$ \mathbb{E}[\sup_{0\le t\le t_n}|\boldsymbol{M}(U)(t \wedge \tau)|^2] \le 4 \mathbb{E}[ |\boldsymbol{M}(U)(t_n \wedge \tau)|^2] = 4 \mathbb{E} \left[ \langle \boldsymbol{M}(U), \boldsymbol{M}(U) \rangle_{t_n \wedge \tau}\right] \le 4 \theta_3(n). $$
Then for any $\theta_4(n)$ such that $ \theta_3=o((\theta_4)^2)$, we have
\begin{align*}
\mathbb{P}\{ \sup_{0 \le t \le t_n}|\boldsymbol{M}(U)(t)| > \theta_4(n) \}
\le& \mathbb{P}\{ \tau^{\scriptscriptstyle (n)} < t_n \}+ \mathbb{P}\set{ \sup_{0\le t\le t_n}|\boldsymbol{M}(U)(t \wedge \tau)| > \theta_4(n) }\\
\le& \mathbb{P}\{ \langle \boldsymbol{M}(U), \boldsymbol{M}(U) \rangle_{t_n} > \theta_3(n) \} + 4 \theta_3(n)/\theta_4^2(n) \to 0.
\end{align*}
The result now follows on using the above observation in \eqref{eq:eq1844}.
\ \ \rule{1ex}{1ex} \\
{\bf Proof of Theorem \ref{thm:suscept-limit}:} Let $y$ and $z$ be as in the proof of Theorem \ref{thm:suscept-funct}. It suffices to show
\begin{align}
\sup_{0 \le t \le t_n} \left| Y(t) - y(t)\right| n^{1/3} \convp 0 \label{eqn:conv-s2y} \\
\sup_{0 \leq t\le t_n} |Z(t)- z(t)| \convp 0. \label{eqn:conv-s3z}
\end{align}
We begin by proving the following weaker result than \eqref{eqn:conv-s2y}:
\begin{equation}
\sup_{0 \le t \le t_n}|Y(t)-y(t)| = O(n^{-1/5}), \; \mbox{whp}. \label{eqn:1142}
\end{equation}
Recalling from Theorem \ref{thm:suscept-funct} that ${\bf{x}} \mapsto F^y({\bf{x}}, y)$ is Lipschitz, uniformly in $y$, we get for some $d_1 \in (0, \infty)$
\begin{equation*}
\sup_{0 \le t \le T}|F^y(\bar {\bf{x}}(t), Y(t))-F^y({\bf{x}}(t), Y(t))| \le d_1 \sup_{i \in \Omega_K} \sup_{ 0 \le t \le T}|\bar{x}_i(t)-x_i(t)|.
\end{equation*}
From Lemma \ref{lemma:decomp-s2-y}(ii) and Lemma \ref{lemma:approx-xi}(a) we now get for some $d_2 \in (0, \infty)$, whp,
$$|{\bf d}(Y)(t) - F^y({\bf{x}}(t), Y(t))| \le d_2\left(\frac {I^2(t)Y(t)}{n} + n^{-2/5}\right), \mbox{ for all } t \in [0, t_n].$$
Thus, from Theorem \ref{thm:subcrit-reg} and recalling that $\gamma < 1/5$, we have whp,
\begin{align*}
\int_0^{t_n} |{\bf d}(Y)(t) - F^y({\bf{x}}(t), Y(t))|dt =& O \left(\int_0^{t_n} \frac{(\log n)^8}{n(t_c-t)^4}dt + n^{-2/5}\right) \\
=& O((\log n)^8n^{3\gamma-1})+O(n^{-2/5})=O(n^{-2/5}).
\end{align*}
Next, by Lemma \ref{lemma:decomp-s2-y}(ii) and using the fact $Y(t) \le 1$ for all $t \in [0, t_c)$,
\begin{align}
\langle \boldsymbol{M}(Y), \boldsymbol{M}(Y) \rangle_{t_n}
=& O\left( \int_0^{t_n} \frac{I^2(t) Y^2(t)}{n} dt\right)
= O\left( \int_0^{t_n} \frac{I^2(t)}{n} dt\right)\nonumber\\
=&
O\left( \int_0^{t_n} \frac{(\log n)^8}{n(t_c-t)^4}dt \right)
= O((\log n)^8n^{3\gamma-1}). \label{eqn:2961}
\end{align}
The statement in \eqref{eqn:1142} now follows on observing that $((\log n)^8n^{3\gamma-1})^{1/2} = o(n^{-1/5})$ and applying Lemma \ref{theo:conv-pure-jump} with ${\mathbb{D}}:=[0,1]$, $g(t,y):=F^y({\bf{x}}(t),y)$, $\theta_1=0$, $\theta_2 = n^{-2/5}$ and $\theta_3 = (\log n)^8n^{3\gamma-1}$.\\
We now strengthen the estimate in \eqref{eqn:1142} to prove \eqref{eqn:conv-s2y}. From Theorem \ref{thm:suscept-funct} it follows that
$y(t_n) = \Theta(n^{-\gamma})$. Since $\gamma < 1/5$, from \eqref{eqn:1142} we have, whp, $Y(t) \le 2 y(t)$ for all $t \le t_n$.
Thus from the first equality in \eqref{eqn:2961} and Theorem \ref{thm:suscept-funct} we get, whp,
\begin{equation*}
\langle \boldsymbol{M}(Y), \boldsymbol{M}(Y) \rangle_{t_n} = O\left( \int_0^{t_n} \frac{I^2(t) y^2(t)}{n} \right) = O\left( \int_0^{t_n} \frac{(\log n)^8}{n(t_c-t)^2}dt \right) = O((\log n)^8n^{\gamma-1}).
\end{equation*}
Since $((\log n)^8 n^{\gamma-1})^{1/2}=o(n^{-2/5})$, applying Lemma \ref{theo:conv-pure-jump} again gives
\begin{equation}
\sup_{0 \le t \le t_n}|Y(t)-y(t)| = O(n^{-2/5}), \; \mbox{whp}. \label{eqn:2971}
\end{equation}
This proves \eqref{eqn:conv-s2y}.\\
We now prove \eqref{eqn:conv-s3z}. We will apply Lemma \ref{theo:conv-pure-jump} to ${\mathbb{D}} := \mathbb{R}$ and
$
g(t,z):= F^z({\bf{x}}(t),y(t),z).
$
As noted in the proof of Theorem \ref{thm:suscept-funct}, $g$ defined as above satisfies \eqref{eq:eq2127}.
We now verify the three assumptions in Lemma \ref{theo:conv-pure-jump}.
Note that (i) is satisfied with $\theta_1 = 0$, since $Z(0)=z(0)=1$. Next, by Lemma \ref{lemma:decomp-s3-z}(ii) and the fact $Y(t) \le 2y(t)$ for $t \le t_n$, whp, we have
\begin{align*}
\langle \boldsymbol{M}(Z), \boldsymbol{M}(Z) \rangle_{t_n} =& O \left(\int_0^{t_n} \left(\frac{I^4(t)Y^4(t)}{n} + \frac{I^6(t)Y^6(t)}{n}\right) dt \right)\\
=& O\left(\int_0^{t_n}\left( \frac{ (\log n)^{16}}{n (t_c-t)^4} + \frac{ (\log n)^{24}}{n (t_c-t)^6}\right) dt\right)
= O((\log n)^{24}n^{5\gamma-1}).
\end{align*}
Since $\gamma < 1/5$, we can find $\theta_4(n) \to 0$ such that $\sqrt{(\log n)^{24}n^{5\gamma-1} }= o(\theta_4(n))$
Thus (iii) in Lemma \ref{theo:conv-pure-jump} is satisfied. Next recall from the proof of Theorem \ref{thm:suscept-funct} that $g(t,z)$ is linear in $z$. Also, $Z(t) \le I(t)$. Thus from Lemma \ref{lemma:approx-xi} and
\eqref{eqn:2971}, for some $d_3 \in (0, \infty)$ whp, for all $t \le t_n$
\begin{align*}
&|F^z(\bar {\bf{x}}(t), Y(t), Z(t)) -g(t,Z(t))| \\
\le& d_3 (1 + Z(t)) \left(\sup_{1 \le i \le K} \sup_{0 \le t \le t_n}|\bar{x}_i(t)-x_i(t)|
+ \sup_{0 \le t \le t_n}|Y(t)-y(t)|\right)
= I(t)O(n^{-2/5}).
\end{align*}
By Lemma \ref{lemma:decomp-s3-z}(i) and the above bound,
\begin{align}
\int_0^{t_n}|{\bf d}(Z)(t) - g(t,Z(t))|dt
=& O\left( \int_0^{t_n} n^{-2/5} I(t)dt \right)+O\left( \int_0^{t_n} \frac{y^2(t)I^3(t)}{n} dt \right) \nonumber\\
=& O((\log n)^4 n^{\gamma -2/5}) + O((\log n)^{12} n^{3\gamma -1}). \label{eqn:1297}
\end{align}
This verifies (ii) in Lemma \ref{theo:conv-pure-jump} with $\theta_2(n) = O((\log n)^{12} n^{3\gamma -1})$.
From Lemma \ref{theo:conv-pure-jump} we now have
\begin{equation*}
\sup_{0\le t\le t_n}|Z(t)-z(t)| \le \theta_1(n) + \theta_2(n) + \theta_4(n) = o(1).
\end{equation*}
The result follows. \ \ \rule{1ex}{1ex}\\
\section{Coupling with the multiplicative coalescent}
\label{sec:main-coupling}
In this section we prove Theorem \ref{thm:crit-regime}. Throughout this section we fix $\gamma \in (1/6,1/5)$. The basic idea of the proof is as follows. Recall $\alpha, \beta \in (0, \infty)$ from Theorem \ref{thm:suscept-funct} (see also Lemma \ref{lemma:alpha-beta}).
We begin by approximating the BSR random graph process by a process which until time $t_n := t_c - n^{-\gamma}$ is identical to
the BSR process and in the time interval $[t_n, t_c + \alpha \beta^{2/3}\frac{\lambda}{n^{1/3}}]$ evolves as an Erd\H{o}s-R\'enyi process, namely over this interval
edges between any pair of vertices appear at rate $1/\alpha n$, and self loops at any given vertex appear at rate $1/2\alpha n$.
Asymptotic behavior of this random graph is analyzed using Theorem \ref{theo:aldous-full-gene}. Theorems \ref{thm:suscept-funct}, \ref{thm:subcrit-reg}
and \ref{thm:suscept-limit} help in verifying the conditions \eqref{eqn:qsigma2} and \eqref{eqn:additional-condition} in the statement of
Theorem \ref{theo:aldous-full-gene}. We then complete the proof of Theorem \ref{thm:crit-regime} by arguing that the `difference' between the BSR process and the modified random graph process is asymptotically negligible.
Let
\begin{equation}
\label{eqn:lambdan-defn}
t_n=t_c - \alpha \beta^{2/3}\frac{\lambda_n}{n^{1/3}} \mbox{ where } \lambda_n = \frac{ n^{1/3-\gamma}}{\alpha \beta^{2/3}}.
\end{equation}
Throughout this section, for $\lambda \in \mathbb{R}$, we denote $t^{\lambda} = t_c + \alpha \beta^{2/3}{\lambda}/{n^{1/3}}$.
Recall the random graph process ${\bf{BSR}}^*(t)$ introduced in Section \ref{sec:main-bsr-susceptibility}. Denote by
$(|\mathcal{C}_{i}^{*}(t)|, \xi_{i}^{*}(t))_{i\ge 1}$ the vector of ordered component size and corresponding surplus in ${\bf{BSR}}^*(t)$ (the components
are denoted by $\mathcal{C}_{i}^{*}(t)$ ).
Let, for $\lambda \in \mathbb{R}$,
$$ \bar \boldsymbol{C}^{{\scriptscriptstyle (n)},*}(\lambda) = \left(\frac{\beta^{1/3}}{n^{2/3}}\left|\mathcal{C}_{i}^{*}(t^{\lambda})\right| :i\geq 1\right), \;\; \bar \boldsymbol{Y}^{{\scriptscriptstyle (n)},*}(\lambda) = \left(\xi_{i}^{*}(t^{\lambda}) :i\geq 1\right).$$
For $i \ge 1$, denote $\bar \boldsymbol{C}_i^{{\scriptscriptstyle (n)},*}(\lambda)$ and $\bar \boldsymbol{Y}_i^{{\scriptscriptstyle (n)},*}(\lambda)$ for the $i$-th coordinate of $\bar \boldsymbol{C}^{{\scriptscriptstyle (n)},*}(\lambda)$ and $\bar \boldsymbol{Y}^{{\scriptscriptstyle (n)},*}(\lambda)$ respectively.
Write
$\bar \boldsymbol{Y}_i^{{\scriptscriptstyle (n)},*} = \tilde \xi_{i}^{\scriptscriptstyle (n)} + \hat \xi_{i}^{\scriptscriptstyle (n)}$ where
$\tilde \xi_{i}^{\scriptscriptstyle (n)}(\lambda)$ represents the surplus in ${\bf{BSR}}^*(t^{\lambda})$ that is created before time $t_n$, namely
$$\tilde \xi_{i}^{\scriptscriptstyle (n)}(\lambda) = \sum_{j: \mathcal{C}_j^{*}(t_n) \subset \mathcal{C}_{i}^{*}(t^{\lambda})} \bar \boldsymbol{Y}_j^{{\scriptscriptstyle (n)},*}(-\lambda_n).$$
In Section \ref{sec:old-surplus} we will show that the contribution from $\tilde \xi^{\scriptscriptstyle (n)}(\lambda):= (\tilde \xi_i^{\scriptscriptstyle (n)}(\lambda): i \ge 1)$ is asymptotically negligible.
First, in Section \ref{sec:two-coupling} below we analyze the contribution from the `new surplus', i.e. $\hat \xi^{\scriptscriptstyle (n)}:= (\hat \xi_i^{\scriptscriptstyle (n)} : i \ge 1)$.
\subsection{Surplus created after time $t_n$.}
\label{sec:two-coupling}
The main result of this section is as follows. Recall the process $\boldsymbol{Z}(\lambda) = (\boldsymbol{X}(\lambda), \boldsymbol{Y}(\lambda))$ introduced in Theorem \ref{thm:smc-surplus}.
\begin{Theorem}
\label{theo:onedimc}
For every $\lambda \in \mathbb{R}$, as $n\to \infty$, $(\bar \boldsymbol{C}^{{\scriptscriptstyle (n)},*}(\lambda), \hat\xi^{\scriptscriptstyle (n)}(\lambda))$ converges in distribution, in $\mathbb{U}_{\downarrow}$, to
$({\boldsymbol{X}}(\lambda), \boldsymbol{Y}(\lambda))$.
\end{Theorem}
The basic idea in the proof of the above theorem is to argue that ${\bf{BSR}}^*(t^{\lambda})$ `lies between' two Erd\H{o}s-R\'enyi random graph processes ${\bf{G}}^{\scriptscriptstyle(n),-}(t^\lambda)$ and ${\bf{G}}^{\scriptscriptstyle(n),+}(t^\lambda)$, whp, and then apply
Theorem \ref{theo:aldous-full-gene} to each of these processes. For a graph ${\bf{G}}$, denote by $|\mathcal{C}_i({\bf{G}})|$ and $\xi_i({\bf{G}})$ the size and surplus, respectively, of the $i$-th largest component, $\mathcal{C}_i({\bf{G}})$ of graph ${\bf{G}}$.
We begin with the following lemma. Recall $\lambda_n$ from \eqref{eqn:lambdan-defn}.
\begin{Lemma}
\label{lemma:upper-lower-coupling}
There exists a construction of $\{{\bf{BSR}}^*(t)\}_{t\ge 0}$ along with two other random graph processes $\{{\bf{G}}^{\scriptscriptstyle(n),-}(t)\}_{t\ge 0}$ and $\{{\bf{G}}^{\scriptscriptstyle(n),+}(t)\}_{t\ge 0}$
such that:\\
(i) With high probability,
\begin{equation}
\label{eq:eq1136}
{\bf{G}}^{\scriptscriptstyle(n),-}(t^\lambda) \subset {\bf{BSR}}^*(t^\lambda)\subset {\bf{G}}^{\scriptscriptstyle(n),+}(t^\lambda) \qquad \mbox{ for all } \lambda \in [-\lambda_n, \lambda_n]. \end{equation}
(ii) Let for $i \ge 1$, $\bar \boldsymbol{C}^{\scriptscriptstyle(n),\mp}_i(\lambda) = \frac{\beta^{1/3}}{n^{2/3}}|\mathcal{C}_i\left({\bf{G}}^{\scriptscriptstyle(n),\mp}(t^\lambda)\right)|$
and $$\bar{\boldsymbol{Y}}^{\scriptscriptstyle (n),\mp}_{i}(\lambda) = \xi_i\left( {\bf{G}}^{\scriptscriptstyle(n),\mp}(t^\lambda)\right) - \sum_{j: \mathcal{C}_j\left({\bf{G}}^{\scriptscriptstyle(n),\mp}(t_n)\right) \subset \mathcal{C}_i\left({\bf{G}}^{\scriptscriptstyle(n),\mp}(t^{\lambda})\right)}
\xi_j\left( {\bf{G}}^{\scriptscriptstyle(n),\mp}(t_n)\right).$$
Then, for all $\lambda \in \mathbb{R}$
\begin{equation*}
(\bar \boldsymbol{C}^{\scriptscriptstyle(n),\bullet}(\lambda), \bar \boldsymbol{Y}^{\scriptscriptstyle(n),\bullet}(\lambda)) \convd (\boldsymbol{X}(\lambda), \boldsymbol{Y}(\lambda)), \; \bullet = -, +,
\end{equation*}
where $\convd$ denotes weak convergence in $\mathbb{U}_{\downarrow}$.
\end{Lemma}
We remark that $\bar{\boldsymbol{Y}}^{\scriptscriptstyle (n),\mp}(\lambda)$ represents the surplus in ${\bf{G}}^{\scriptscriptstyle(n),\mp}(t^\lambda)$ created after time instant $t_n$.
Proof of the lemma relies on the following proposition which is an immediate consequence of Theorem \ref{thm:suscept-funct}, Theorem \ref{thm:suscept-limit} and Theorem \ref{thm:subcrit-reg}.
\begin{Proposition}
\label{prop:three-conditions} There exists a $\kappa \in (0, \frac{1}{3} - \gamma)$ such that
$$ \frac{\bar s_3(t_n)}{(\bar{s}_2(t_n))^3} \convp \beta, \;\; \frac{n^{1/3}}{\bar{s}_2(t_n)}-\frac{n^{1/3-\gamma}}{\alpha} \convp 0, \;\; \frac{I^{\scriptscriptstyle (n)}(t_n)}{n^{2\gamma + \kappa}} \convp 0. $$
\end{Proposition}
We now prove Lemma \ref{lemma:upper-lower-coupling}.
\textbf{Proof of Lemma \ref{lemma:upper-lower-coupling}:}\\
We suppress $n$ in the notation for the random graph processes. Write $t_n^+:=t_c+n^{-\gamma}$. Let
${\bf{BSR}}(t)$ for $t \in [0, t_n^+]$ be constructed as in Section \ref{sec:bsr} and define ${\bf{BSR}}^*(t)$ for $t \in [0, t_n)$
as in Section \ref{sec:main-bsr-susceptibility}. Set
$$
{\bf{G}}^{\scriptscriptstyle(n),-}(t) = {\bf{G}}^{\scriptscriptstyle(n),+}(t) = {\bf{BSR}}^*(t), \mbox{ for } t \in [0, t_n).
$$
We now give the construction of these processes for
$t \in [t_n, t_n^+]$. \\
The construction is done in two rounds. In the first round, we construct processes ${\bf{G}}^{I,-}(t)$, ${\bf{BSR}}^{I,*}(t)$ and ${\bf{G}}^{I,+}(t)$ for $ t \in [t_n ,t_n^+]$ by using only the information of immigrations and attachments in ${\bf{BSR}}(t)$, while the edge formation between large components is ignored. We first construct the process $\{{\bf{BSR}}^{ I}(t)\}_{t \in [t_n, t_n^+] }$ as follows. Let ${\bf{BSR}}^{ I}(t_n) := {\bf{BSR}}(t_n)$. For $t > t_n$, ${\bf{BSR}}^{ I}(t)$ is constructed
along with and same as ${\bf{BSR}}(t)$, except for when
$$ c_{t-}(\vec{v}) \in \{ \vec{j} \in \Omega_K^4: \vec{j} \in F, j_1=j_2=\varpi \;\;\mbox{ or } \;\; \vec{j} \notin F, j_3=j_4=\varpi \},$$
in which case no edge is added to ${\bf{BSR}}^{ I}(t)$.
Let $\bar x_i(t), a_i^*(t),b^*(t), c^*_i(t)$, $1 \le i \le K$, $t \in [t_n, t_n^+]$, be the processes determined from
$\{{\bf{BSR}}(t)\}_{t\in [t_n, t_n^+] }$ as in Section \ref{sec:main-bsr-susceptibility}. These processes will be used in the second round of the construction.
Now define ${\bf{BSR}}^{I,*}(t)$ to be the subgraph that consists of all large components (components of size greater than $K$) in ${\bf{BSR}}^{I}(t)$, and then define ${\bf{G}}^{I,-}(t)$ and ${\bf{G}}^{I,+}(t)$ for $t \in [t_n, t_n^+]$ as follows:
$$ {\bf{G}}^{I,-}(t) \equiv {\bf{BSR}}^{I,*}(t_n), \mbox{ and } {\bf{G}}^{I,+}(t) \equiv {\bf{BSR}}^{I,*}(t_n^+).$$
Then
$$ {\bf{G}}^{I,-}(t) \subset {\bf{BSR}}^{I,*}(t) \subset {\bf{G}}^{I,+}(t) \mbox{ for all } t \in [t_n ,t_n^+].$$
We now proceed to the second round of the construction. Let
$$ E_n =\set{ b(t_c) -n^{-1/6} < b^*(t) < b(t_c) + n^{-1/6}, \mbox{ for all } t \in [t_n ,t_n^+] }. $$
Note that Lemma \ref{lemma:approx-xi} and \eqref{eqn:def-fb-b} implies that with probability at least $1- C_1 e^{-C_2 n^{1/5}}$,
\begin{align*}
\sup_{ t \in (t_n, t_n^+)}|b^*(t)-b(t_c)|
\le& \sup_{ t \in (t_n, t_n^+)}|b^*(t)-b(t)| + \sup_{ t \in (t_n, t_n^+)}|b(t)-b(t_c)|\\
\le& d_1 n^{-2/5} + d_2 n^{-\gamma} = o(n^{-1/6}).
\end{align*}
Thus $\mathbb{P} \set{E_n^c} \to 0$ as $n \to \infty$.
Since we only need the coupling to be good with high probability,
it suffices to construct the coupling of the three processes until the first time $t \in [t_n, t_n^+]$ when
$b^*(t) \not \in [b(t_c) -n^{-1/6}, b(t_c) + n^{-1/6}]$. Equivalently, we can assume without loss of generality that
$b^*(t) \in [b(t_c) -n^{-1/6}, b(t_c) + n^{-1/6}]$, for all $t \in [t_n, t_n^+]$, a.s.
We will construct ${\bf{G}}^{ +}(t)$, ${\bf{BSR}}^*(t)$ and ${\bf{G}}^{ -}(t)$ by adding new edges between components in the three random graph processes ${\bf{G}}^{I,-}(t)$, ${\bf{BSR}}^{I,*}(t)$ and ${\bf{G}}^{I,+}(t)$ such that, at time $t \in [t_n ,t_n^+]$ edges are added
between each pair of vertices in ${\bf{G}}^{I,-}(t)$, ${\bf{BSR}}^{I,*}(t)$ and ${\bf{G}}^{I,+}(t)$, at rates $\frac{1}{n}(b(t_c)-n^{-1/6})$, $\frac{1}{n}b^*(t)$ and $\frac{1}{n}(b(t_c)+n^{-1/6})$, respectively. The precise mechanism is as follows.
We first construct ${\bf{G}}^{+}(t)$ for $t \in (t_n,t_n^+]$ by adding edges between every pair of vertices in ${\bf{G}}^{I,+}(t)$
at the rate $\frac{1}{n}(b(t_c)+n^{-1/6})$ and creating self-loops at the rate $\frac{1}{2n}(b(t_c)+n^{-1/6})$ for each vertex in ${\bf{G}}^{I,+}(t)$.
Next, we construct ${\bf{BSR}}^*(t)$ and ${\bf{G}}^{-}(t)$ through successive thinning of ${\bf{G}}^{+}(t)$, thus obtaining the desired coupling. Let $(e_1,e_2, ...)$ be the sequence of edges that are added to ${\bf{G}}^{I,+}(t)$ to obtain ${\bf{G}}^{+}(t)$.
Let $(u_1, u_2, ...)$ be i.i.d Uniform$[0,1]$ random variables that are also independent of the random variables used to construct
${\bf{G}}^{I,-}, {\bf{BSR}}^{I,*}, {\bf{G}}^{I,+}, {\bf{G}}^{+}$.
Suppose at time $t_k$, we have ${\bf{G}}^{+}(t_k)={\bf{G}}^{+}(t_k-) \cup \{ e_k\}$, where $e_k = \{v_1,v_2\}$. We set ${\bf{BSR}}^*(t_k) = {\bf{BSR}}^*(t_k-) \cup \{ e_k\}$ if and only if
$$ v_1, v_2 \in {\bf{BSR}}^{I,*}(t_k-) \mbox{ and } u_k \le \frac{b^*(t_k)}{b(t_c) + n^{-1/6}}, $$
otherwise let ${\bf{BSR}}^*(t_k) = {\bf{BSR}}^*(t_k-)$.
This defines the process ${\bf{BSR}}^*(t)$ (with the correct probability law) such that the second inclusion in \eqref{eq:eq1136} is satisfied.
Finally, construct ${\bf{G}}^{-}(t)$ by a thinning of ${\bf{BSR}}^*(t)$
exactly as above by replacing $\frac{b^*(t_k)}{b(t_c) + n^{-1/6}}$ with $\frac{b(t_c) - n^{-1/6}}{b^*(t_k)}$.
Then ${\bf{G}}^{-}(t)$, for $t \in [t_n ,t_n^+]$ is
an Erd\H{o}s-R\'enyi type processes and the first inclusion in \eqref{eq:eq1136} is satisfied.
This completes the proof of the first part of the lemma.
We now prove (ii). Consider first the case $\bullet = -$. We will apply Theorem \ref{theo:aldous-full-gene}. With notation as in that theorem,
it follows from the Erd\H{o}s-R\'enyi dynamics of ${\bf{G}}^{\scriptscriptstyle(n),-}(t)$ that, the distribution of $(\bar \boldsymbol{C}^{\scriptscriptstyle(n),-}(\lambda), \bar \boldsymbol{Y}^{\scriptscriptstyle(n),-}(\lambda))$, conditioned on $\{\mathcal{P}_{\vec{v}}(t),\, t \le t_n \; \vec{v} \in [n]^4\}$, for each
$\lambda \in [-\lambda_n, \lambda_n]$, is same as the distribution of $\boldsymbol{Z}(z^{\scriptscriptstyle (n)}, q^{\scriptscriptstyle (n)})$, where $z^{\scriptscriptstyle (n)} = (\bar \boldsymbol{C}^{\scriptscriptstyle(n),-}(-\lambda_n), \bf{0})$, $\bf{0}$ denotes the vector $(0, 0, \cdots)$
and $q^{\scriptscriptstyle (n)}$ is determined by the equality
$$
q^{\scriptscriptstyle (n)} \bar \boldsymbol{C}^{\scriptscriptstyle(n),-}_i(-\lambda_n) \bar \boldsymbol{C}^{\scriptscriptstyle(n),-}_j(-\lambda_n)
= \frac{\alpha\beta^{2/3}}{n^{1/3}} (\lambda + \lambda_n) \frac{(b(t_c) - n^{-1/6})}{n} |\mathcal{C}_i({\bf{G}}^{\scriptscriptstyle(n),-}(t_n))| |\mathcal{C}_j({\bf{G}}^{\scriptscriptstyle(n),-}(t_n))|,$$
for $i \neq j$. Recalling that $\alpha b(t_c)=1$ it then follows that
$q^{\scriptscriptstyle (n)} = \lambda + \frac{n^{1/3-\gamma}}{\alpha\beta^{2/3}} + O(n^{1/6-\gamma}).$
We now verify the conditions of Theorem \ref{theo:aldous-full-gene}. Taking $x^{\scriptscriptstyle (n)} = \bar \boldsymbol{C}^{\scriptscriptstyle(n),-}(-\lambda_n)$ we see with,
$x^*, s_k$, $k = 1,2,3$ as in Theorem \ref{theo:aldous-full-gene},
$$ s^{\scriptscriptstyle(n)}_1 \le \beta^{1/3} n^{1/3}, \;\; s^{\scriptscriptstyle (n)}_2 = \frac{\beta^{2/3}}{n^{4/3}} \sum_{\mathcal{C} \subset {\bf{BSR}}^*(t_n)} |\mathcal{C}|^2, \;\; s^{\scriptscriptstyle (n)}_3 = \frac{\beta}{n^2} \sum_{\mathcal{C} \subset {\bf{BSR}}^*(t_n)} |\mathcal{C}|^3.$$
Recall the definition of $\bar s_k$ and $\bar s_{k,\varpi}$ from \eqref{eqn:suscept-defn} and Section \ref{sec:main-bsr-susceptibility}.
Then
$$ s_2^{\scriptscriptstyle (n)}=\frac{\beta^{2/3} \bar s_{2,\varpi}(t_n)}{n^{1/3}},\;\; s_{3}^{\scriptscriptstyle (n)} = \frac{\beta \bar{s}_{3,\varpi}(t_n)}{n}, \;\;
x^{*\scriptscriptstyle(n)}= \beta^{1/3}\frac{I(t_n)}{n^{2/3}}. $$
From the first two convergences in Proposition \ref{prop:three-conditions} and recalling that, for $k=1,2$,
$|\bar s_{k, \varpi}-\bar s_k| \le K^k$, we immediately get that the first two convergences in \eqref{eqn:qsigma2} hold.
Also,
$$\frac{x^*}{s_2} = \frac{I(t_n)}{\beta^{1/3}n^{1/3}\bar s_{2,\varpi}(t_n)} = \frac{I(t_n)}{\beta^{2/3}n^{ \gamma + 1/3}}O(1) \to 0 , \mbox{ in probability},$$
where the second equality is consequence of the second convergence in Proposition \ref{prop:three-conditions}, and the convergence of the last term follows from the third convergence in Proposition \ref{prop:three-conditions}. This proves the third convergence in \eqref{eqn:qsigma2}.
Finally we note that the convergence in \eqref{eqn:additional-condition} holds with $\varsigma = \frac{1}{1 - 3(\gamma + \kappa)}$, where $\kappa$ is as in
Proposition \ref{prop:three-conditions}, since
$$
s_1 \left(\frac{x^*}{s_2}\right)^{\varsigma} \le O(1) n^{1/3} \left (\frac{I(t_n)}{n^{\gamma + 1/3} }\right)^{\varsigma} = O(1) \left (\frac{I(t_n)}{n^{2\gamma + \kappa} }\right)^{\varsigma}
\to 0,$$
where the last equality follows from our choice of $\varsigma$
and the convergence is a consequence of Proposition \ref{prop:three-conditions}. Thus we have verified all the conditions in Theorem \ref{theo:aldous-full-gene}
and therefore we have from this result that
$(\bar \boldsymbol{C}^{\scriptscriptstyle(n),-}(\lambda), \bar \boldsymbol{Y}^{\scriptscriptstyle(n),-}(\lambda))$
converges in distribution, in $\mathbb{U}_{\downarrow}$, to $(\boldsymbol{X}^*(\lambda), \boldsymbol{Y}^*(\lambda))$ proving part (ii) of the lemma for $\bullet = -$.
To prove part (ii) of the lemma for $\bullet = +$, one needs slightly more work.
Once more we will apply Theorem \ref{theo:aldous-full-gene}.
As before, conditioned on $\{\bar \boldsymbol{C}^{\scriptscriptstyle(n),+}(\lambda_0): \lambda_0 \le -\lambda_n\}$, for
each
$\lambda \in [-\lambda_n, \lambda_n]$, the distribution of $(\bar \boldsymbol{C}^{\scriptscriptstyle(n),+}(\lambda), \bar \boldsymbol{Y}^{\scriptscriptstyle(n),+}(\lambda))$ is same as the distribution of $\boldsymbol{Z}(\bar z^{\scriptscriptstyle (n)}, \bar q^{\scriptscriptstyle (n)})$, where $\bar z^{\scriptscriptstyle (n)} = (\bar \boldsymbol{C}^{\scriptscriptstyle(n),+}(-\lambda_n), \bf{0})$
and
$\bar q^{\scriptscriptstyle (n)} = \lambda + \frac{n^{1/3-\gamma}}{\alpha\beta^{2/3}} + O(n^{1/6-\gamma}).$
Taking $x^{\scriptscriptstyle (n)} = \bar \boldsymbol{C}^{\scriptscriptstyle(n),+}(-\lambda_n)$ we see with,
$x^*, s_k$, $k = 1,2,3$ as in Theorem \ref{theo:aldous-full-gene},
$$ s^{\scriptscriptstyle(n)}_1 \le \beta^{1/3} n^{1/3}, \;\; s^{\scriptscriptstyle (n)}_2 = \frac{\beta^{2/3}}{n^{4/3}} \sum_{\mathcal{C} \subset {\bf{BSR}}^{I,*}(t_n^+)} |\mathcal{C}|^2, \;\; s^{\scriptscriptstyle (n)}_3 = \frac{\beta}{n^2} \sum_{\mathcal{C} \subset {\bf{BSR}}^{I,*}(t_n^+)} |\mathcal{C}|^3.$$
Next note that for any component $\mathcal{C} \subset {\bf{G}}^-(t_n)= {\bf{BSR}}^{I,*}(t_n) $ there is a unique component $ \mathcal{C}^+ \subset {\bf{G}}^+(t_n) ={\bf{BSR}}^{I,*}(t_n^+)$,
such that $\mathcal{C} \subset \mathcal{C}^+$. Denote by $\mathcal{C}_i$ the $i$-th largest component in ${\bf{BSR}}^{I,*}(t_n)$, and let $\mathcal{C}_i^+$
be the corresponding component in ${\bf{BSR}}^{I,*}(t_n^+)$ such that $\mathcal{C}_i \subset \mathcal{C}_i^+$. Denote by $N$ the number of immigrations that occur during $[t_n, t_n^+]$ in ${\bf{BSR}}^{I,*}$, and denote by $\{\tilde \mathcal{C}_i^+\}_{i=1}^N$ the components
in ${\bf{BSR}}^{I,*}(t_n^+)$
resulting from these immigrations.
Then
$$ s_2^{\scriptscriptstyle (n)}=\frac{\beta^{2/3} \bar s_{2}^+}{n^1/3},\;\; s_{3}^{\scriptscriptstyle (n)} = \frac{\beta \bar{s}_{3}^+}{n}, \;\;
x^{*\scriptscriptstyle(n)}= \beta^{1/3}\frac{I^+}{n^{2/3}}, $$
where
\begin{align*}
\bar s_2^+ :=& \frac{1}{n} \left( \sum_{i=1}^\infty |\mathcal{C}^+_i|^2 + \sum_{i=1}^N |\tilde \mathcal{C}_i^+|^2 \right),\\
\bar s_3^+ :=& \frac{1}{n} \left( \sum_{i=1}^\infty |\mathcal{C}^+_i|^3 + \sum_{i=1}^N |\tilde \mathcal{C}_i^+|^3 \right),\\
I^+ :=& \max\set{ \max_{i} |\mathcal{C}_i^+|, \max_{i} |\tilde \mathcal{C}_i^+|}.
\end{align*}
To complete the proof it suffices to show that the statement in Proposition \ref{prop:three-conditions} holds with
$(\bar{s}_2(t_n), \bar s_3(t_n), I^{\scriptscriptstyle (n)}(t_n))$ replaced with $(\bar s_2^+, \bar s_3^+, I^+)$. This follows from Proposition
\ref{prop:excess-vertices} given below and hence completes the proof of the lemma. \ \ \rule{1ex}{1ex} \\
\begin{Proposition}
\label{prop:excess-vertices}
With notation as in the proof of Lemma \ref{lemma:upper-lower-coupling}, as $n \to \infty$, we have
$$ I^+ = O(I), \;\; \frac{\bar s_2^+}{\bar s_2(t_n)} \convp 1, \;\; \frac{\bar s_3^+}{\bar s_3(t_n)} \convp 1,
\;\; \frac{n^{1/3}}{\bar s_2(t_n)} - \frac{n^{1/3}}{\bar s_2^+} \convp 0.$$
\end{Proposition}
{\bf Proof.} The proof is similar to that of Proposition 8.1 in \cite{bhamidi-budhiraja-wang2011} thus we only give a sketch.
Observe that the total rate of attachments is $\sum_{i=1}^K c^*_i(t) \le 1$ and each attachment has size no bigger than $K$.
Recall that $\mathcal{C}_i$ denotes the $i$-th largest component in ${\bf{BSR}}^{I,*}(t_n)$. Denote
by $V_i(t)$, $t \in [t_n, t_n^+]$, the stochastic process defining the size of the component containing $\mathcal{C}_i$
in ${\bf{BSR}}^{I,*}(t)$. Note that
$V_i(t_n) = |\mathcal{C}_i|$ and $V_i(t_n^+)=|\mathcal{C}_i^+|$. Then ${V_i(t)}/{K}$ can be stochastically dominated by a Yule process starting with $\lceil |\mathcal{C}_i|/K\rceil $ particles and birth rate $K$.
Using this and an argument similar to \cite{bhamidi-budhiraja-wang2011}, it follows that,
$$ |\mathcal{C}_i^+| - |\mathcal{C}_i| \le_d K \cdot \mbox{Negative-Binomial}(\lceil |\mathcal{C}_i|/K\rceil, e^{-2Kn^{-\gamma}}). $$
Next, note that the immigrations are of size no bigger than $2K$, and thus for the same reason, we have the bound,
$$ |\tilde \mathcal{C}_i^+| \le_d 2K + K \cdot \mbox{Negative-Binomial}(2, e^{-2Kn^{-\gamma}}).$$
Since the total number of vertices is $n$, the number of immigrations $N$ can be bounded by $n/K$.
With the above three bounds the proof of the proposition follows exactly as the proof of Proposition 8.1 in \cite{bhamidi-budhiraja-wang2011} with obvious
changes needed due to the constant $K$ that appears in the above bounds. Details are omitted. \ \ \rule{1ex}{1ex}\\
We will now use Lemma \ref{lemma:upper-lower-coupling} to complete the proof of Theorem \ref{theo:onedimc}.
We begin with the following elementary lemma.
\begin{Lemma}
\label{lemma:l1cgce}
Let $\{x^{\scriptscriptstyle (n)}_i, y^{\scriptscriptstyle (n)}_i, x_i, y_i, i \ge 1, n\ge 1\} $ be a collection of non-negative numbers such that, for each fixed $i$, as $n\to \infty$,
$x^{\scriptscriptstyle (n)}_i \to x_i$ and $y^{\scriptscriptstyle (n)}_i\to y_i$.
Also suppose that
$\sum_i x^{\scriptscriptstyle (n)}_iy^{\scriptscriptstyle (n)}_i \to \sum_i x_iy_i < \infty$ as $n \to \infty$.
Then $\sum_i|x^{\scriptscriptstyle (n)}_iy^{\scriptscriptstyle (n)}_i - x_i y_i| \to 0$ as $n\to \infty$.
\end{Lemma}
{\bf Proof.} Proof is immediate on applying Fatou's lemma, indeed
\begin{align*}
2\sum_{i} x_iy_i \le& \liminf_{n\to \infty} \sum_i (x^{\scriptscriptstyle (n)}_iy^{\scriptscriptstyle (n)}_i + x_i y_i - |x^{\scriptscriptstyle (n)}_iy^{\scriptscriptstyle (n)}_i - x_i y_i|)\\
=& 2 \sum_i x_iy_i - \limsup_{n\to \infty} \sum_i |x^{\scriptscriptstyle (n)}_iy^{\scriptscriptstyle (n)}_i - x_i y_i|.\end{align*}
\ \ \rule{1ex}{1ex}\\
The next proposition says that the inclusion in \eqref{eq:eq1136} can be strengthened to component-wise inclusion.
\begin{Proposition}
\label{prop:surplus} Fix $\lambda\in {\mathbb{R}}$ and $i_0\geq 1$. Then, as $n\to\infty$,
\[\mathbb{P}\set{\mathcal{C}_i({\bf{G}}^{\scriptscriptstyle(n), -}(t^\lambda))\subset \mathcal{C}_i({\bf{BSR}}^{*}(t^\lambda)) \subset \mathcal{C}_i({\bf{G}}^{\scriptscriptstyle(n), +}(t^\lambda))~~~ \forall~ 1\leq i\leq i_0} \to 1.\]
\end{Proposition}
{\bf Proof: }
From Lemma \ref{lemma:upper-lower-coupling} and Lemma 15 in \cite{aldous2000random} (see also Section 8.2 of \cite{bhamidi-budhiraja-wang2011} for a similar
argument), we have, as $n\to\infty$,
\begin{equation}
\label{eq:eq1154}(\bar{\boldsymbol{C}}^{\scriptscriptstyle(n),-}(\lambda),\bar{\boldsymbol{C}}^{{\scriptscriptstyle(n)},*}(\lambda), \bar{\boldsymbol{C}}^{\scriptscriptstyle(n),+}(\lambda) ) \convd (\boldsymbol{X}(\lambda), \boldsymbol{X}(\lambda), \boldsymbol{X}(\lambda)),\end{equation}
in $l^2_{\downarrow} \times l^2_{\downarrow} \times l^2_{\downarrow}$, where $\boldsymbol{X}$ is as in Theorem \ref{thm:smc-surplus}.
Define events $E_n, F_n$ as
$$ E_n=\set{ \bar \boldsymbol{C}_i^{\scriptscriptstyle (n),-}(\lambda) > \bar \boldsymbol{C}_{i+1}^{\scriptscriptstyle (n),+}(\lambda) : 1\le i \le i_0 }, F_n=\set{ {\bf{G}}^{\scriptscriptstyle (n),-}(\lambda) \subset {\bf{BSR}}^*(\lambda) \subset {\bf{G}}^{\scriptscriptstyle (n),+}(\lambda) }. $$
Then on the set $E_n \cap F_n$
$$ \mathcal{C}_i({\bf{G}}^{\scriptscriptstyle(n), -}(\lambda))\subset \mathcal{C}_i({\bf{BSR}}^{*}(\lambda)) \subset \mathcal{C}_i({\bf{G}}^{\scriptscriptstyle(n), +}(\lambda)), ~~\forall~ 1\leq i\leq i_0.$$
From Lemma \ref{lemma:upper-lower-coupling} (i) $\mathbb{P}\{ F_n^c\} \to 1$. Also
\begin{align*}
\limsup_{n} \mathbb{P}( E_n^c)
\le& \limsup_n \mathbb{P} \set{\bar \boldsymbol{C}_i^{\scriptscriptstyle (n),-}(\lambda) \le \bar \boldsymbol{C}_{i+1}^{\scriptscriptstyle (n),+}(\lambda) \mbox{ for some } 1\le i \le i_0}\\
\le& \mathbb{P} \set{ \boldsymbol{X}_i(\lambda) \le \boldsymbol{X}_{i+1}(\lambda) \mbox{ for some } 1 \le i \le i_0 }=0.
\end{align*}
This shows that $\mathbb{P}(E_n \cap F_n) \to 1$ as $n \to \infty$. The result follows.
\ \ \rule{1ex}{1ex}\\
We will also need the following elementary lemma. Proof is omitted.
\begin{Lemma}
\label{lemma:sandwich-1}
Let $ \eta ^{\scriptscriptstyle (n),-}, \eta^{\scriptscriptstyle (n), +}, \eta^*$ be real random variables such that $\eta^{\scriptscriptstyle (n),-} \le \eta^{\scriptscriptstyle (n),+}$ with high probability. Further assume $\eta^{\scriptscriptstyle (n),-} \convd \eta^*$ and $\eta^{\scriptscriptstyle (n),+} \convd \eta^*$. Then
$ \eta^{\scriptscriptstyle (n),+} - \eta^{\scriptscriptstyle (n),-} \convp 0. $
Furthermore, if $\eta^{\scriptscriptstyle (n)}$ are random variables such that $\eta^{\scriptscriptstyle (n),-} \le \eta^{\scriptscriptstyle (n)} \le \eta^{\scriptscriptstyle (n),+}$ with high probability, then
$ \eta^{\scriptscriptstyle (n)} \convd \eta^*$ and $\eta^{\scriptscriptstyle (n)} - \eta^{\scriptscriptstyle (n),-} \convp 0.$
\end{Lemma}
We now complete the proof of Theorem \ref{theo:onedimc}.\\
\textbf{Proof of Theorem \ref{theo:onedimc}:}
From Lemma \ref{lemma:upper-lower-coupling} (ii) we have that
\begin{equation} \label{eq:eq1206}
\left(\bar {\boldsymbol{C}}^{\scriptscriptstyle (n),-}(\lambda), \bar \boldsymbol{Y}^{\scriptscriptstyle (n),-}(\lambda) , \sum_{i=1}^\infty \bar {\boldsymbol{C}}^{\scriptscriptstyle (n),-}_i(\lambda)\boldsymbol{Y}_i^{\scriptscriptstyle (n),-}(\lambda)\right)
\convd \left( {\boldsymbol{X}}(\lambda), \boldsymbol{Y}(\lambda), \sum_{i=1}^\infty {\boldsymbol{X}}_i(\lambda) \boldsymbol{Y}_i(\lambda)\right),\end{equation}
in $l^2_{\downarrow} \times \mathbb{N}^{\infty} \times \mathbb{R}$, where on $\mathbb{N}^{\infty}$ we consider the product topology.
In order to prove the theorem it suffices, in view of Lemma \ref{lemma:l1cgce}, to show that
\begin{equation}
\left(\bar {\boldsymbol{C}}^{\scriptscriptstyle (n)}_*(\lambda), \hat \xi^{\scriptscriptstyle (n)}(\lambda) , \sum_{i=1}^\infty \bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda)\hat \xi_i^{\scriptscriptstyle (n)}(\lambda)\right)
\convd \left( {\boldsymbol{X}}(\lambda), \boldsymbol{Y}(\lambda), \sum_{i=1}^\infty {\boldsymbol{X}}_i(\lambda) \boldsymbol{Y}_i(\lambda)\right), \label{eqn:2576}
\end{equation}
in $l^2_{\downarrow} \times \mathbb{N}^{\infty} \times \mathbb{R}$.
From Proposition \ref{prop:surplus}, we have for any $i_0 \in \mathbb{N}$, with high probability
$$ \bar \boldsymbol{Y}_i^{\scriptscriptstyle (n),-}(\lambda) \le \hat \xi_i^{\scriptscriptstyle (n)}(\lambda) \le \bar \boldsymbol{Y}_i^{\scriptscriptstyle (n),+} \mbox{ for } 1 \le i \le i_0.$$
Also, from Lemma \ref{lemma:upper-lower-coupling} (i),
whp,
$$ \sum_{i=1}^\infty \bar {\boldsymbol{C}}^{\scriptscriptstyle (n),-}_i(\lambda)\bar\boldsymbol{Y}_i^{\scriptscriptstyle (n),-}(\lambda) \le \sum_{i=1}^\infty \bar {\boldsymbol{C}}^{\scriptscriptstyle (n)}_i(\lambda)\bar\boldsymbol{Y}_i^{\scriptscriptstyle (n)}(\lambda) \le \sum_{i=1}^\infty \bar {\boldsymbol{C}}^{\scriptscriptstyle (n),+}_i(\lambda)\bar\boldsymbol{Y}_i^{\scriptscriptstyle (n),+}(\lambda). $$
From Lemma \ref{lemma:sandwich-1} and Lemma \ref{lemma:upper-lower-coupling} (ii), we then have
$$ \left( \left |\hat \xi^{\scriptscriptstyle (n)}(\lambda) - \bar \boldsymbol{Y}^{\scriptscriptstyle (n),-}(\lambda)\right |,\;\; \sum_{i=1}^\infty \bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda)\hat \xi_i^{\scriptscriptstyle (n)}(\lambda) -\sum_{i=1}^\infty \bar {\boldsymbol{C}}^{\scriptscriptstyle (n),-}_i(\lambda)\bar \boldsymbol{Y}_i^{\scriptscriptstyle (n),-}(\lambda) \right) \convp 0, $$
in $\mathbb{N}^{\infty} \times \mathbb{R}$, where for $y = (y_1, y_2, \cdots ) \in \mathbb{Z}^{\infty}$, $|y| = (|y_1|, |y_2|, \cdots)$.
The convergence in \eqref{eqn:2576} now follows on combining \eqref{eq:eq1206} and \eqref{eq:eq1154}. The result follows. \ \ \rule{1ex}{1ex}\\
\subsection{Proof of Theorem \ref{thm:crit-regime}.}
\label{sec:old-surplus}
As a first step towards the proof we show the following convergence result for one dimensional distributions.
\begin{Theorem}
\label{theo:onedimcnew}
For every $\lambda \in \mathbb{R}$, as $n\to \infty$, $(\bar {\boldsymbol{C}}^{\scriptscriptstyle (n)}(\lambda), \bar \boldsymbol{Y}^{\scriptscriptstyle (n)}(\lambda))$ converges in distribution, in $\mathbb{U}_{\downarrow}$, to
$({\boldsymbol{X}}(\lambda), \boldsymbol{Y}(\lambda))$.
\end{Theorem}
\textbf{Proof.}
Fix $\lambda \in \mathbb{R}$. We first argue that
\begin{equation} \label{eq:eq1536}
(\bar \boldsymbol{C}^{{\scriptscriptstyle (n)},*}(\lambda), \bar \boldsymbol{Y}^{{\scriptscriptstyle (n)},*}(\lambda)) \convd ({\boldsymbol{X}}(\lambda), \boldsymbol{Y}(\lambda)), \mbox{ in }\mathbb{U}_{\downarrow}.
\end{equation}
For this, it suffices to show that
\begin{equation}\label{eq:eq1539}
\sum_{i=1}^\infty \tilde \xi_i^{\scriptscriptstyle (n)}(\lambda) \bar \boldsymbol{C}_{i}^{{\scriptscriptstyle (n)},*} (\lambda) \convp 0.
\end{equation}
Define
$$E_n=\set{ I(s) \le C_3\frac{(\log n)^4}{(t_c-s)^2} \mbox{ for } s \le t_c-n^{-\gamma} }.$$
By Theorem \ref{thm:subcrit-reg}, $\mathbb{P}\{E_n^c\} \to 0$ and $E_n \in \tilde \mathcal{F}(\lambda) = \sigma \{ |\mathcal{C}_i(s)|: i\ge 1, s \le t^{\lambda}\}$ for all $\lambda \ge -\lambda_n$.
We begin by showing that there exists $d_1 \in (0, \infty)$ such that, for all $i \in \mathbb{N}$,
\begin{equation}
\label{eq:eq1601}
\mathbb{E} \left[\tilde \xi_i^{\scriptscriptstyle (n)}(\lambda)\mid \tilde{\mathcal{F}}_\lambda\right ] 1_{E_n} \le d_1 \bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda) n^{\gamma -1/3}(\log n)^4.
\end{equation}
Note that at any time $s < t^{\lambda}$, for a component of size $\mathcal{C} \subset {\bf{BSR}}^{\scriptscriptstyle (n)}(s)$, there are at most $2 |\mathcal{C}|^2 n^2$ quadruples of vertices which may provide a surplus edge within $\mathcal{C}$. Since edges are formed at rate $2/n^3$, we have that
\begin{align*}
\mathbb{E} \left[\tilde \xi_i^{n}(\lambda)\mid \tilde{\mathcal{F}}_\lambda\right ]
\le& \int_0^{t_n} \left[\sum_{j: \mathcal{C}_j({\bf{BSR}}^{\scriptscriptstyle (n)}(s)) \subset \mathcal{C}_i({\bf{BSR}}^{*}(t^{\lambda}))}
\frac{1}{2n^3}2 n^2 |\mathcal{C}_j({\bf{BSR}}^{\scriptscriptstyle (n)}(s))|^2 \right] ds\\
\le & \frac{1}{n} |\mathcal{C}_i({\bf{BSR}}^{*}(t^{\lambda}))| \int_0^{t_n} I(s) ds.
\end{align*}
Thus, for some $d_0, d_1 \in (0, \infty)$
\begin{align*}
\mathbb{E} \left[\tilde \xi_i^{\scriptscriptstyle (n)}(\lambda)\mid \tilde{\mathcal{F}}(\lambda)\right ] 1_{E_n}
\le d_0\frac{\bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda)}{n^{1/3}} \int_0^{t_c-n^{-\gamma}} \frac{(\log n)^4}{(t_c-s)^2} ds
\le d_1
\bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda) n^{\gamma -1/3
} (\log n )^4.
\end{align*}
This proves \eqref{eq:eq1601}.
As an immediate consequence of this inequality we have that
\begin{align*}
\mathbb{E} \left [ \sum_{i}\tilde \xi_i^{\scriptscriptstyle (n)}(\lambda)\bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda) \mid \tilde\mathcal{F}(\lambda)\right]1_{E_n}
=& \sum_{i}\bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda)1_{E_n}\mathbb{E} \left [ \tilde \xi_i^{\scriptscriptstyle (n)}(\lambda) \mid \tilde\mathcal{F}(\lambda)\right] \\
\le& d_1 n^{\gamma -1/3} (\log n )^4\sum_{i} \left(\bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda) \right)^2.
\end{align*}
Observing that $\gamma - 1/3 < 0$ and, from Theorem \ref{theo:onedimc}, that $\sum_{i}\left(\bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda) \right)^2$ converges in distribution,
we have that
$$\mathbb{E} \left [ \sum_{i}\tilde \xi_i^{\scriptscriptstyle (n)}(\lambda)\bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda) \mid \tilde\mathcal{F}(\lambda)\right]1_{E_n} \convp 0.$$
Since $\mathbb{P}(E_n) \to 1$, letting $\eta^{\scriptscriptstyle (n)} = \sum_{i}\tilde \xi_i^{\scriptscriptstyle (n)}(\lambda)\bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda)$, we have that
$\mathbb{E}(\eta^{\scriptscriptstyle (n)} \mid \tilde\mathcal{F}(\lambda)) \to 0$ in probability. Convergence in \eqref{eq:eq1539} now follows
on noting that, as $n\to \infty$,
$$ \mathbb{E}[\eta^{\scriptscriptstyle (n)} \wedge 1] = E \left [ E[\eta^{\scriptscriptstyle (n)} \wedge 1 \mid \tilde \mathcal{F}(\lambda)]\right] \le \mathbb{E} \left [ \mathbb{E} [\eta^{\scriptscriptstyle (n)} \mid \tilde \mathcal{F}(\lambda)]\wedge 1\right] \to 0.$$
This proves \eqref{eq:eq1536}. Next note that
\begin{equation}
\label{eq:eq1619}
\sum_{i=1}^{\infty} |\bar {\boldsymbol{C}}^{\scriptscriptstyle (n)}_{i}(\lambda) - \bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda)|^2 \le \frac{n}{n^{4/3}}O(1) \to 0, \mbox{ as }
n \to \infty .
\end{equation}
Also,
\begin{align*}
\mathbb{E} \left[\bar \boldsymbol{Y}_i^{\scriptscriptstyle (n)}(\lambda)\mid \tilde{\mathcal{F}}(\lambda)\right ] 1_{\{|\mathcal{C}_i(t^{\lambda})| \le K\}}
\le& \left[\int_0^{t^{\lambda}} \sum_{j: \mathcal{C}_j(s) \subset \mathcal{C}_i(t^{\lambda})}
\frac{1}{2n^3}2 n^2 |\mathcal{C}_j(s)|^2 ds\right] 1_{\{|\mathcal{C}_i(t^{\lambda})| \le K\}}\\
\le & \frac{K^2}{n}.
\end{align*}
Thus, as $n \to \infty$,
$$
\mathbb{E} \sum_{i=1}^{\infty} |\bar {\boldsymbol{C}}^{\scriptscriptstyle (n)}_{i}(\lambda)\bar \boldsymbol{Y}_i^{\scriptscriptstyle (n)}(\lambda) - \bar {\boldsymbol{C}}_i^{{\scriptscriptstyle (n)},*}(\lambda)\bar \boldsymbol{Y}_{*,i}^{\scriptscriptstyle (n)}(\lambda)|
\le \frac{O(1)}{n} \mathbb{E} \left[\sum_{i=1}^{\infty}\bar {\boldsymbol{C}}^{\scriptscriptstyle (n)}_{i}(\lambda) \right] = O(n^{-2/3}) \to 0.$$
The result now follows on combining the above convergence with \eqref{eq:eq1619} and \eqref{eq:eq1536}. \ \ \rule{1ex}{1ex} \\
\begin{Remark}
\label{rem:rem1905}
The proofs of Theorems \ref{theo:onedimc} and \ref{theo:onedimcnew} in fact establish the following stronger statement:
For all $\lambda \in \mathbb{R}$,
\begin{align*}
&\left (|\bar \boldsymbol{Y}^{\scriptscriptstyle (n),-}(\lambda) - \bar \boldsymbol{Y}^{\scriptscriptstyle (n)}(\lambda)|,
\sum_{i=1}^{\infty}|\bar {\boldsymbol{C}}^{\scriptscriptstyle (n),-}_i(\lambda) - \bar {\boldsymbol{C}}^{\scriptscriptstyle (n)}_i(\lambda)|^2,\right .\\
& \left .\;\;\;\;\; \sum_{i=1}^{\infty} |\bar {\boldsymbol{C}}^{\scriptscriptstyle (n),-}_i(\lambda)
\bar \boldsymbol{Y}^{\scriptscriptstyle (n),-}_i(\lambda) - \bar {\boldsymbol{C}}^{\scriptscriptstyle (n)}_i(\lambda)
\bar \boldsymbol{Y}^{\scriptscriptstyle (n)}_i(\lambda)|\right)
\to ({\bf 0},0,0),\end{align*}
in probability, in $\mathbb{N}^{\infty}\times \mathbb{R} \times \mathbb{R}$.
\end{Remark}
{\bf Proof of Theorem \ref{thm:crit-regime}:}
For simplicity we present the proof for the case $m=2$. The general case can be treated similarly.
Fix $-\infty < \lambda_1 < \lambda_2 < \infty$.
Denote, for $\lambda \in \mathbb{R}$, $\bar \boldsymbol{Z}^{\scriptscriptstyle (n),-}(\lambda) = (\bar {\boldsymbol{C}}^{\scriptscriptstyle (n),-}(\lambda), \bar \boldsymbol{Y}^{\scriptscriptstyle (n),-}(\lambda))$.
In view of Remark \ref{rem:rem1905} it suffices to show that, as $n \to \infty$,
$$(\bar \boldsymbol{Z}^{\scriptscriptstyle (n),-}(\lambda_1),\bar \boldsymbol{Z}^{\scriptscriptstyle (n),-}(\lambda_2)) \convd (\boldsymbol{Z}(\lambda_1), \boldsymbol{Z}(\lambda_2)),$$
for which it is enough to show that for all $f_1, f_2 \in C_b(\mathbb{U}_{\downarrow}^0)$
\begin{equation}
\label{eq:eq1943}
\mathbb{E}\left [ f_1(\bar \boldsymbol{Z}^{\scriptscriptstyle (n),-}(\lambda_1))f_2(\bar \boldsymbol{Z}^{\scriptscriptstyle (n),-}(\lambda_2))\right]
\to \mathbb{E}\left [ f_1( \boldsymbol{Z}(\lambda_1))f_2( \boldsymbol{Z}(\lambda_2))\right].
\end{equation}
Note that the left side of \eqref{eq:eq1943} equals
$$ \mathbb{E}\left [ f_1(\bar \boldsymbol{Z}^{\scriptscriptstyle (n),-}(\lambda_1))\mathcal{T}_{\lambda_2-\lambda_1}f_2(\bar \boldsymbol{Z}^{\scriptscriptstyle (n),-}(\lambda_1))\right],$$
which using Theorem \ref{thm:smc-surplus} (2), Lemma \ref{lemma:upper-lower-coupling} (ii) and the fact that
$\boldsymbol{X}(\lambda) \in \mathbb{U}_{\downarrow}^1$ a.s., converges to
$$
\mathbb{E}\left [ f_1( \boldsymbol{Z}(\lambda_1))\mathcal{T}_{\lambda_2-\lambda_1}f_2( \boldsymbol{Z}(\lambda_1))\right]
= \mathbb{E}\left [ f_1( \boldsymbol{Z}(\lambda_1))f_2( \boldsymbol{Z}(\lambda_2))\right],$$
where the last equality follows from Theorem \ref{thm:smc-surplus} (3). This proves \eqref{eq:eq1943} and the result follows. \ \ \rule{1ex}{1ex} \\
\section*{Acknowledgements}
AB and XW has been supported in part by the National Science Foundation (DMS-1004418, DMS-1016441), the Army Research Office (W911NF-0-1-0080, W911NF-10-1-0158) and the US-Israel Binational Science Foundation (2008466). SB and XW have been supported in part NSF-DMS grant 1105581.
\begin{bibdiv}
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}
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ISSN={0178-8051},
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}
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}
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}
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}
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}
\bib{bsr-2012}{article}{
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author={Wang, X},
title={{Bounded size rules: The barely subcritical regime}},
date={2012},
journal={Arxiv preprint},
}
\bib{bhamidi2009novel}{article}{
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author={Van Der~Hofstad, R.},
author={van Leeuwaarden, J.S.H.},
title={{Novel scaling limits for critical inhomogeneous random graphs}},
date={2009},
journal={Arxiv preprint arXiv:0909.1472},
}
\bib{bhamidi-hofstad-van}{article}{
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author={van Leeuwaarden, Johan S.~H.},
title={Scaling limits for critical inhomogeneous random graphs with
finite third moments},
date={2010},
ISSN={1083-6489},
journal={Electron. J. Probab.},
volume={15},
pages={no. 54, 1682\ndash 1703},
review={\MR{2735378}},
}
\bib{bohman2001avoiding}{article}{
author={Bohman, T.},
author={Frieze, A.},
title={{Avoiding a giant component}},
date={2001},
journal={Random Structures and Algorithms},
volume={19},
number={1},
pages={75\ndash 85},
}
\bib{bollobas-rg-book}{book}{
author={Bollob{\'a}s, B{\'e}la},
title={Random graphs},
edition={Second},
series={Cambridge Studies in Advanced Mathematics},
publisher={Cambridge University Press},
address={Cambridge},
date={2001},
volume={73},
ISBN={0-521-80920-7; 0-521-79722-5},
review={\MR{1864966 (2002j:05132)}},
}
\bib{bollobas-riordan-janson}{article}{
author={Bollob{\'a}s, B{\'e}la},
author={Janson, Svante},
author={Riordan, Oliver},
title={The phase transition in inhomogeneous random graphs},
date={2007},
ISSN={1042-9832},
journal={Random Structures Algorithms},
volume={31},
number={1},
pages={3\ndash 122},
url={http://dx.doi.org/10.1002/rsa.20168},
review={\MR{2337396 (2008e:05124)}},
}
\bib{durrett-book}{book}{
author={Durrett, Rick},
title={Probability: theory and examples},
edition={Fourth},
series={Cambridge Series in Statistical and Probabilistic Mathematics},
publisher={Cambridge University Press},
address={Cambridge},
date={2010},
ISBN={978-0-521-76539-8},
review={\MR{2722836 (2011e:60001)}},
}
\bib{er-2}{article}{
author={Erd{\H{o}}s, P.},
author={R{\'e}nyi, A.},
title={On the evolution of random graphs},
date={1960},
journal={Magyar Tud. Akad. Mat. Kutat\'o Int. K\"ozl.},
volume={5},
pages={17\ndash 61},
review={\MR{0125031 (23 \#A2338)}},
}
\bib{er-1}{article}{
author={Erd{\H{o}}s, P.},
author={R{\'e}nyi, A.},
title={On the evolution of random graphs},
date={1961},
journal={Bull. Inst. Internat. Statist.},
volume={38},
pages={343\ndash 347},
review={\MR{0148055 (26 \#5564)}},
}
\bib{hartman1942infinitesimal}{article}{
author={Hartman, P.},
author={Wintner, A.},
title={On the infinitesimal generators of integral convolutions},
date={1942},
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volume={64},
number={1},
pages={273\ndash 298},
}
\bib{janson1994birth}{article}{
author={Janson, S.},
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author={Luczak, T.},
author={Pittel, B.},
title={{The birth of the giant component, with an introduction by the
editors}},
date={1994},
journal={Random Struct. Alg},
volume={4},
pages={231\ndash 358},
}
\bib{janson2010phase}{article}{
author={Janson, S.},
author={Spencer, J.},
title={{Phase Transitions for Modified Erdos--R{\'e}nyi Processes}},
date={2010},
journal={Arxiv preprint arXiv:1005.4494},
}
\bib{janson-luczak-bb}{book}{
author={Janson, Svante},
author={{\L}uczak, Tomasz},
author={Rucinski, Andrzej},
title={Random graphs},
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Optimization},
publisher={Wiley-Interscience, New York},
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ISBN={0-471-17541-2},
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}
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date={2010},
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}
\bib{bf-spencer-perkins-kang}{article}{
author={Kang, M.},
author={},
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date={2010},
journal={Arxiv preprint arXiv:1106.0484v1},
}
\bib{liptser-mart-book}{book}{
author={Liptser, R.~Sh.},
author={Shiryayev, A.~N.},
title={Theory of martingales},
series={Mathematics and its Applications (Soviet Series)},
publisher={Kluwer Academic Publishers Group},
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Dzhaparidze]},
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}
\bib{riordan2012phase}{article}{
author={Riordan, O.},
title={The phase transition in the configuration model},
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}
\bib{riordan2011achlioptas}{article}{
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author={Warnke, L.},
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}
\bib{riordan2012evolution}{article}{
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\bib{spencer2007birth}{article}{
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}
\end{biblist}
\end{bibdiv}
\end{document} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,226 |
{"url":"https:\/\/www3.math.tu-berlin.de\/cgi-bin\/IfM\/show_abstract.cgi?Report-04-2012.rdf.html","text":"Sie sind hier\n\n## High order asymptotic expansion for the acoustics in viscous gases close to rigid walls\n\n Source file is available as : Portable Document Format (PDF)\n\nAuthor(s) : Kersten Schmidt , Anastasia Th\u00f6ns-Zueva\n\nPreprint series of the Institute of Mathematics, Technische Universit\u00e4t Berlin\nPreprint 04-2012\n\nMSC 2000\n\n35C20 Asymptotic expansions\n35J25 Boundary value problems for second-order, elliptic equations\n41A60 Asymptotic approximations, asymptotic expansions\n35B40 Asymptotic behavior of solutions\n\nAbstract :\nWe derive a complete asymptotic expansion for the singularly perturbed problem of the acoustic wave propagation inside gases with small viscosity, this for the non-resonant case in smooth bounded domains in two dimensions. Close to rigid walls the tangential velocity shows a boundary layer of size $O(\\sqrt{\\eta})$ where $\\eta$ is the dynamic viscosity. The asymptotic expansion based on the technique of multiscale expansion is in powers of $\\sqrt{\\eta}$ and takes into account curvature effects. The terms of the velocity and pressure expansion are defined independently by partial differential equations, where the normal component of velocities or the normal derivative of the pressure, respectively, are prescribed on the boundary. The asymptotic expansion is rigorously justified with optimal error estimates.\n\nKeywords : Acoustic wave propagation, Singularly perturbed PDE, Asymptotic Expansions","date":"2019-11-13 09:10:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5369435548782349, \"perplexity\": 774.2026540855868}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496667177.24\/warc\/CC-MAIN-20191113090217-20191113114217-00295.warc.gz\"}"} | null | null |
Q: ¿Como hago repito mi codigo sin utilizar variables globales? Tengo que rehacer este codigo sin variables globales... Es una actividad evaluativa de un curso de aprendizaje que estoy tomando en python; el profesor señala que el codigo esta correcto pero que debo de plantearlo sin la necesidad de usar "global" y tampoco creo que permita que use "self" hasta el momento porque apenas es la siguiente tematica objetos. ¿Que recomiendan en este caso para que el codigo me funcione?
Intente plantear el codigo definiendo las variables dentro de la funcion pero luego la siguiente funcion me dice que la variable no existe, no se que hacer en este caso.
Aprecio los consejos, muchas gracias:
Archivo= Archivo + ".txt"
def lectura(Archivo):
lista1= []
lista2= []
lista3= []
lista4= []
listaconvacios= []
datos=open(Archivo,'r')
datos.readline()
while True:
lineas=datos.readline()
if "nan" in lineas:
listaconvacios.append(lineas)
if lineas=='':
break
else:
res=[float(num) for num in lineas.split(';')]
lista1.append(res[0])
lista2.append(res[1])
lista3.append(res[2])
lista4.append(res[3])
return lista1,lista2,lista3,lista4
lectura(Archivo)
#print('datosx', lista1)
#print('datosy', lista2)
#print('datosz', lista3)
#print('datosw', lista4)
#Columna 1 Vs Columna 2
def Grafica1(lista1,lista2,lista3,lista4):
lista1= lista1
lista2= lista2
lista3=lista3
lista4=lista4
#Columna 1 vs Columna 2
plt.plot(lista1)
plt.xlabel("Tiempo")
plt.ylabel("Cantidad")
plt.ioff()
plt.plot(lista2)
plt.ion()
plt.plot(lista2)
plt.plot(lista1, label = "Confirmados")
plt.plot(lista2, label = "Activos")
plt.legend()
#Columna 1 vs Columna 3
plt.plot(lista1)
plt.xlabel("Tiempo")
plt.ylabel("Cantidad")
plt.ioff()
plt.plot(lista3)
plt.ion()
plt.plot(lista3)
plt.plot(lista3, label = "Recuperados")
plt.legend()
#Columna 1 vs Columna 4
plt.plot(lista1)
plt.xlabel("Tiempo")
plt.ylabel("Cantidad")
plt.ioff()
plt.plot(lista4)
plt.ion()
plt.plot(lista4)
plt.plot(lista4, label = "Muertos")
plt.legend()
def Grafica2(lista1,lista2,lista3,lista4):
#Columna 1 vs Columna 2
plt.figure()
plt.plot(lista1)
plt.xlabel("Tiempo")
plt.ylabel("Cantidad")
plt.ioff()
plt.plot(lista2)
plt.ion()
plt.plot(lista2)
plt.plot(lista1, label = "Confirmados")
plt.plot(lista2, label = "Activos")
plt.legend()
#Columna 1 vs Columna 3
plt.figure()
plt.plot(lista1)
plt.xlabel("Tiempo")
plt.ylabel("Cantidad")
plt.ioff()
plt.plot(lista3)
plt.ion()
plt.plot(lista3)
plt.plot(lista1, label = "Confirmados")
plt.plot(lista3, label = "Recuperados")
plt.legend()
#Columna 1 vs Columna 4
plt.figure()
plt.plot(lista1)
plt.xlabel("Tiempo")
plt.ylabel("Cantidad")
plt.ioff()
plt.plot(lista4)
plt.ion()
plt.plot(lista4)
plt.plot(lista1, label = "Confirmados")
plt.plot(lista4, label = "Muertos")
plt.legend()
Grafica1(lista1,lista2,lista3,lista4)
#Grafica2(lista1,lista2,lista3,lista4)
A: Podrías eliminar la variable Archivo que estas usando para formar el nombre del archivo que quieres abrir. Mejor cuando quieras abrir el archivo con el método open() escribe directamente el nombre del archivo que quieres abrir. Ejemplo: datos=open("NOMBRE_DE_TU_ARCHIVO.txt,'r')
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 856 |
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University of Lynchburg
About Athletics
/ Academics / College of Business / Alumni Advisory Board
John Guagliano '90
Senior Vice President, Marketing and Communications | Minnesota United FC
John Guagliano is the senior vice president of marketing and communications at Minnesota United FC. He won numerous awards for his work in branding and marketing, including a Clio Sports award and multiple American Advertising awards, during his nine years as vice president of marketing and broadcasting for Major League Baseball's Washington Nationals. He has also worked in broadcasting and game ops for the NBA's Charlotte Hornets and NHL's Nashville Predators, winning three Emmy awards as a producer in the "Sports Program/Live" category. He has opened three sports facilities: Charlotte Arena, Nationals Park, and Allianz Field.
John received his BA in marketing from Lynchburg College and was inducted into Lynchburg's Sports Hall of Fame as a soccer player in 2006.
Greg Jackson '84
Founder and Managing Director | Spinnaker Group
Greg Jackson is founder and managing director of the Spinnaker Group, a boutique executive search firm based in Charlotte, North Carolina. Greg has 20 years of executive search experience supporting a variety of Fortune 500, private-equity, and foreign-national companies. His clients have included Aurora Capital, Blackstone Group, Boeing Corporation, JPMorgan, Capital One, and Wachovia/Wells Fargo. Greg previously served as vice president/manager of executive recruiting for Wachovia/Wells Fargo's General Banking Group, as a director with Spencer Stuart, and for 10 years as a vice president/partner at Korn/Ferry International, where he established the Supply Chain Management Center of Excellence.
Greg's earlier career experience includes serving as the assistant director of admissions at Lynchburg College. He holds a BA in English from Lynchburg.
Tim Murphy '99
Managing Director and Industry Manager, Health Care, Higher Education, and Nonprofit Commercial Banking Group | JPMorgan Chase & Co.
Tim Murphy is an industry manager in JPMorgan Chase & Co.'s health care, higher education, and nonprofit commercial banking group, based in Irvine, California. During his banking career, Tim's responsibilities have included strategy and business planning, business development, relationship management, portfolio management, credit underwriting, credit training development, and nonprofit risk analytics. Before moving to California in May 2018, Tim was based in Washington, D.C., with leadership responsibility for JPMorgan's health care, higher education and nonprofit banking teams in the mid-Atlantic and Northeast regions. Tim joined JPMorgan Chase & Co. in 2013 from Branch Banking & Trust, where he served as senior banker and nonprofit higher education industry specialist. Tim was a relationship manager with Allied Irish Bank serving the government, nonprofit, education, and health care industry segments. He currently serves on the chapter advisory board of the American Red Cross in the Desert to the Sea California Region.
Tim received his BA in business administration from Lynchburg College and is currently enrolled in leadership and management certificate programs at the University of Virginia Darden School of Business.
Cheryl Pipia '96
Head of U.S. Intermediaries Integrated Sales | T. Rowe Price
Founder and President | Mission & Movement
Founder and President | City Chik on a Mission
Cheryl Pipia is head of U.S. intermediaries integrated sales at T. Rowe Price, and founder and president of Mission & Movement, a global humanitarian company, and the nonprofit City Chik on a Mission. When she entered the corporate world, Cheryl felt pressured to participate in charity work that felt inauthentic. A volunteer trip 15 years ago to the Mother Teresa Home for the Destitute and the Dying in Lima, Peru, changed everything. Cheryl has since devoted her life to engaging in meaningful social service at home and abroad. In 2014, she founded City Chik on a Mission, a nonprofit dedicated to funding and maintaining projects that focus on education, community development, and employment in impoverished communities. Mission & Movement was the natural evolution. The sister companies merge Cheryl's corporate leadership skills with her nonprofit experience. Through sustainable programs and practices, and training and workshop opportunities, Cheryl lives her life with direction and feeling. She encourages others to tap into their emotions, identify their passions, and put them to work.
Cheryl holds a BA in art history from Lynchburg College.
Jeff Rupp '85
Founder and Managing Director | View Capital Advisors
Jeff Rupp is the founder and managing director of View Capital Advisors. He is also a member of the firm's Investment Committee, with over two decades of experience advising sophisticated investors. Jeff led the firm into a variety of alternative investments such as real estate, private equity, and operating assets. Before founding View Capital Advisors, he was managing director and regional executive for Deutsche Bank Alex. Brown in Dallas, Texas. Jeff was also a member of Deutsche Bank Alex. Brown's Private Wealth Management Executive Committee, and worked in various roles for Alex. Brown in San Francisco and Baltimore.
Jeff holds a bachelor's in business administration and economics from Lynchburg College.
Scott Shepherd '00
Senior Vice President, Corporate Partnerships | The Washington Redskins
Scott Shepherd returned to The Washington Redskins as senior vice president in January 2019. He oversees the team's Corporate Partnerships Division, as well its media properties and marketing platforms. From 2008 to 2015, he led the Redskins to record corporate partnership and special event revenues, and orchestrated landmark government relations agreements with the Commonwealth of Virginia, Loudoun County, and the City of Richmond. Before returning to the Redskins, Scott served as the chief commercial officer at Guest Services, Inc., a 102-year-old hospitality management firm headquartered in Washington, D.C. He was responsible for all revenue growth and commercial functions across the U.S. His career also includes serving in various leadership roles for the Minnesota Vikings, Palace Sports & Entertainment's Stanley Cup Champion Tampa Bay Lightning and Arena Football League Champion Tampa Bay Storm, and the Columbus Blue Jackets, a new NHL franchise at the time.
Scott graduated from Lynchburg College with a BA in business administration, He has been involved in several community initiatives, including the University of Lynchburg's College of Business Advisory Board, the Trust for the National Mall Corporate Leadership Council, the Loudoun County Chamber of Commerce Board of Directors, and the Visit Loudoun Sports Commission Executive Committee.
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Ю́лия Анато́льевна Бори́ско (; род. 28 ноября 1978 года, Львов, Украинская ССР, СССР) — украинская телеведущая, журналист.
Родилась 28 ноября 1978 года во Львове. Окончила Львовский национальный университет имени Ивана Франко.
Карьера
Первым местом работы была львовская газета «Экспресс», когда Бориско училась на третьем курсе университета.
.
В 2000 году переехала в Киев, где в период с 2000 по 2001 годы проработала журналистом ТРК «Экспресс-Информ» (специализация — финансы и налоги).
2001—2004 — журналист ТСН на канале 1+1, ведущая утреннего шоу «Завтрак с 1+1».
2005 — ведущая утренних «Фактов» на канале ICTV.
2005—2008 — ведущая «Новостей» на «Первом национальном».
2008—2020 — ведущая ночного выпуска ТСН, который выходит после 23:00. Свою работу в позднем выпуске ТСН называла «поцелуем в лоб на ночь» и считала, что программа должна содержать главные новости и нести определённый заряд позитива. Также, с сентября 2017 по август 2020 была ведущей выпуска ТСН в 16:45.
По версии журнала «Фокус», занимает 22 место в рейтинге самых успешных телевизионных ведущих Украины.
Семья
Муж — Роман Выбрановский. Дети: дочка Соломия (2003 г. р.), сын Марко (2006 г. р.). Бабушка и дедушка похоронены на Яновском кладбище.
Интересные факты
Вегетарианка, не ест мясных блюд;
любимые писатели — Герман Гессе, Джордж Оруэлл, Сергей Довлатов, Александр Мень, Экхарт Толле, Леся Украинка и Лина Костенко. Иногда перечитывает Новый Завет;
«сова», уже много лет не ложится спать раньше двух часов ночи;
домашнее животное — пёс Боно, немецкий боксёр.
Источники
Страница Юлии Бориско на сайте 1+1
Примечания
Ссылки
https://www.instagram.com/boryska_/
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Hammersley House
5-8 Warwick Street
London W1B 5LX
United Kingdom
Email: info@bluffers.com
Website: bluffers.com
Twitter: @BluffersGuide
First published 1985
This edition published 2013
Copyright © Bluffer's® 2013
Publisher: Thomas Drewry
Publishing Director: Brooke McDonald
Series Editor: David Allsop
Design and Illustration: Jim Shannon
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of Bluffer's®.
A CIP Catalogue record for this book is available from the British Library.
Bluffer's Guide®, Bluffer's® and Bluff Your Way® are registered trademarks.
ISBN:| 978-1-909365-76-6 (print)
978-1-909365-77-3 (ePub)
978-1-909365-78-0
---|---
**Contents**
Let's Get Quantum Physical
Preparing Yourself
The Quantum Realm
Matter Matters
Shocking Behaviour
GUT Instincts
What Cern is Addressing
Superstringing Us Along
Sokal's Hoax
Famous Physicists
What It All Means
Glossary
Nothing conveys the impression of substantial intellect so much as even the sketchiest knowledge of quantum physics.
LET'S GET QUANTUM PHYSICAL
Whether you are trying to explore the universe, which is very, very big, or the 'quantum realm', in which particles of light and matter are very, very small, nothing can be stated with clearer conviction than this: nobody really understands what's going on. So, if you have ever lamented your own incomprehension, let yourself off the hook immediately. Welcome to a bluffer's paradise, where no one need feel afraid, nor should anyone feel stupid.
Of course, where physicists do have knowledge and understanding, prudence dictates that astute bluffers gain some appreciation of it. This will not be too difficult. The universe is, after all, very large, and the smallest entities are extremely small – and their conduct is unusual, to say the least.
Happily, such an appreciation may be obtained without formulas, equations or fractions. It may seem to be taking the word 'appreciation' too far, but you are going to have to feign, find or affirm some sense of wonder. Nature is full of marvels at the best of times. In the quantum realm, as the great Danish physicist, Niels Bohr, used to say: if people didn't find these phenomena shocking, wild and confusing, they weren't taking it in.
And Bohr was understating it, of course. Indeed, however bizarre a description of subatomic phenomena may be, there'll always be some smarty-pants saying: 'That's underselling it'. Agree fervently and move on.
So, needless to say, overwhelming questions will continually come up. Nowadays these can be neatly batted away with the words: 'Yes, well, of course they're looking into that at Cern'. You hint at deeper knowledge by saying that Cern is an acronym for Conseil Européen pour la Recherche Nucléaire and by being specific about the actual location of the famous 'collider' – near Geneva, beneath the Franco-Swiss border. Not too far away from Goldfinger's lair in the eponymous Bond book, actually.
Mentioning Cern will become an automatic response for you, and one that falls somewhere between 'They deal with that in accounts', 'Give the ball to Brian', and 'Do you mind if I have another glass of wine?' No one should ever tire of saying, 'Yes, they're addressing that at Cern'.
Nothing conveys the impression of substantial intellect so much as even the sketchiest knowledge of quantum physics, and since the sketchiest knowledge is all anyone will ever have, never be shy of holding forth with bags of authority about subatomic particles and the quantum realm without having done any science whatsoever.
After all, what else is the act of bluffing about?
This book sets out to guide you through the main danger zones encountered in discussions about the quantum universe, and to equip you with a vocabulary and an evasive technique that will minimise the risk of being rumbled as a bluffer. It will lend you a few easy-to-learn hints and methods that will allow you to be accepted as a physics expert of rare ability and experience. But it will do more. It will give you the tools to impress legions of marvelling listeners with your knowledge and insight – without anyone discovering that before reading it you didn't know the difference between a boson and a hadron, or, indeed, a fermion or a meson, or come to that a photon, pion or gluon. Time to put a smileon, get ready to putiton and readon.
The Italian physicist Enrico Fermi said that if he could remember all the names given to particles he would have been a botanist, and if such things really do interest you, you need to get out more.
PREPARING YOURSELF
DOES SIZE MATTER?
Whichever way you approach discussions about the universe and the quantum realm, the notion of scale will somehow always be present. But remember that it should be beneath you to be impressed by size; it is for other people to reel and gasp. Scale is something with which you must appear to be comfortable and familiar.
Physicists naturally possess what bluffers must somehow acquire: the ability to drop phenomenally massive numbers into the conversation with ease, airily disparaging 'geewhizzery' while trotting out unimaginably awe-inspiring facts and figures.
Where the tendency is towards 'big', you need to appreciate that a light year measures the distance you could cover in a Julian year (that's a normal calendar year of 365.25 days), zooming at 186,283 miles per second. The nearest galaxy to earth, newly discovered, is a trillion miles away – remarkably near as galaxies go, but still pretty far, even for a commercial traveller. (And if you do go, don't talk to strangers.)
At the other extreme, you need to have some idea of what is meant by 'small'. All matter can be broken down into atoms. Atoms are small. They are smaller than affordable apartments in Manhattan, they are smaller than average portions at The Ritz, they are even smaller than the chance that a politician will give you an honest answer. The full stop at the end of this sentence will be a tiny blob of ink about a quarter of a millimetre wide which will contain close to four billion atoms.
Having taken that into account, a reminder is necessary. To a human being, or to a full stop, an atom is small. But to a subatomic particle, any one of those four billion atoms in that full stop is enormous.
At the centre of an atom is a nucleus – itself made of particles. The nucleus feels within the atom as a regulation red ball might feel in a cricket ground. The electrons that surround the atom would be akin to peanuts circling at the outskirts of the ground's car park. There are no players, spectators or refreshment-sellers around. Atoms are usually described as being made up of 99.9999999999999% 'empty space'. Without that empty space, the entire human race would fit into a space the size of a sugar cube.
Pointing to such a cube, or mentioning how many atoms there are in a grain of coffee, or Parmesan cheese, or beach sand, or, best of all, in a pretty freckle on human skin, may be a game well worth the candle for the confident bluffer. But you can take it further.
With the concept of 'touch' of any kind – innocent, compromising or downright culpable – the physics of today provides a most helpful get-out clause. No atoms of any one object can ever touch the atoms of another.
For hundreds of years, people have talked of great performers having magnetism, or public moments having electricity, but it has taken humanity more than a few millennia to realise how much electromagnetism lies within everything and how powerful it can be.
The 'empty space' in atoms, while devoid of mass – of matter, of 'stuff' – carries a tremendous electromagnetic charge. People don't fall through the ground as they walk through a town or city because the negative electromagnetic charge on the outside of the atoms in a pavement exerts a repulsive force on the negative charge of the atoms in the soles of a shoe. That is the same repulsive force in action when hammers hit nails, boxing gloves punch heads or rackets thwack tennis balls.
The charges around the atom create the illusion and even the feeling of touch. Still, however hard they are pushed, the force surrounding atoms far exceeds even the most powerful bar magnets, so atoms cannot be made to come into contact with each other. There is always that layer of separation, even around scissors and hair, food as it's digested and, therefore, between two pairs of lips.
Whatever the status of human relationships, lovers have no option but to accept the age-old adage: 'There is always space in togetherness.'
THE PARADOX OF THE BELIEF SYSTEM
Few forms of knowledge are so closely interwoven with human belief systems as studies concerning the quantum universe.
When discussing the very tiniest particles of light and matter, or wrestling with the very big universe, your belief system, and the belief systems of the people whom you will be impressing, will be powerfully challenged.
Because you are dealing with fundamental entities in nature, it will be impossible to avoid disturbing and challenging the most basic beliefs of those around the table. The mere mention of a particle will raise the issue of consciousness in snails, and how this relates to God.
Everything that anyone has ever held dear, every blind assumption, every hard-won prejudice may be horribly threatened. Clearly, this sort of thing should not be allowed; and, certainly, no one else should have to witness it.
To prepare yourself for 'deipnosophy' (the noble art of excelling at dinner parties), you need to know where you stand. Behind closed doors you will have resolved all those profound questions which, however slightly, might have upset your own world view. You will have absorbed this knowledge – and been shaken and stirred – but you will have managed to keep your cherished notions intact. Before you re-emerge to re-engage and mingle with your friends, you will have discovered how to use the very same knowledge to provide you with an opposite, more pleasing interpretation.
It will involve no effort at all to find some subatomic occurrence that has a bearing on religion, or politics, or marriage or morals, or art or azaleas, or music or marmalade, or snooker or sex. Each precious phenomenon must be seized on, every analogy squeezed until it squeaks.
TWO SCHOOLS OF THOUGHT
It has been said that there are two types of people: those who divide the world into two types of people and those who don't. Ideas about quantum mechanics can divide people in trillions of ways, but it is instructive to consider this relatively new science by considering two distinct views of the universe, two schools of thought. Here's how you can see them coming.
1. The Holists
Members of this school are, broadly speaking, holistic. Holists are interested in the whole picture, except when it comes to the letter 'w'. As time goes on, more and more holists are likely to come out of the (w)ood(w)ork.
2. The Reductionists
Those who belong to this school are interested in the parts of the picture. All they want to know is 'How small?', 'How many?', 'How wide?', etc., as if the quantum universe was a watch that could be taken apart to the smallest screw to find how it all works.
Holists are generally interested in quality. Reductionists are interested in quantity.
Holists, to paraphrase the poet William Blake, can see the universe and eternity in an atom – or even a grain of sand. They have likened the universe to an enormous ball of string. No one can see its end or its beginning, yet if you draw on the string at any part of the ball, the ball becomes tighter throughout. Try to pull one strand of it, and all of it changes. Holists say that this involves a deeper reality than mere interrelationship. For them, wherever you tweak the string, there is the entire universe because all of it changes. Everything is part of everything, every part is part of every part. There is oneness and only oneness. If it sounds like religion, it depends on how you define religion.
Modern physics, like love, or football, or being a teenager, defies language.
Reductionists concentrate on the individual atom without relating it to anything else. They think you should chop up the string ball and measure the bits.
Holists think the scientific discoveries of the twentieth century should encourage us to rethink everything. This is sometimes called a 'paradigm shift' – the altering of a model, in this case the whole model. They want to unite, to bind, to join. Reductionists don't. They like the word 'discrete', meaning 'separate from everything else'.
Reductionists think it is unnecessary to make so much of the fact that the iron at the core of planets is also contained in human haemoglobin. Holists think it's poetic and meaningful that human beings have stardust in their blood.
Holists find coincidences intensely moving; reductionists demonstrate why they aren't coincidental. Holists accept a certain floppiness in their ideas; reductionists insist on rigour.
Holists quite like the circuitous approach. A reductionist will even do a cutting gesture while explaining that space itself is curved so you will, at least, get that straight.
THE LANGUAGE BARRIER
When you come up against the limits of the quantum universe, and indeed the limits of your own understanding, you will inevitably come up against the extremes of what you can express linguistically, so to speak.
For instance, the notion that in the beginning everything was in one place and, lo, that place was nothing. No thing. It was a nothingness, so nothing like its nothingness took nothingness to untold extremes of nothingness. (Modern physics, like love, or football, or being a teenager, defies language.)
It has never been easy for anyone to find words to explain the deep mysteries of the universe. How to explain new phenomena? How to describe them? How to tell people about entities and occurrences which are so far removed from anything hitherto known about or understood that the brain starts doing backflips and wants to join the circus.
The difficulties begin with names. You will, as it happens, encounter many words in the quantum realm which end in '-on'. There are hadrons, which may be baryons or mesons, and mesons which may be pions or kaons, while muons and tauons are leptons, not to be confused with sleptons – and so on.
You may be tempted to sort them out. Don't. At the University of California at Berkeley, in an almanac which records new particles, the list was already running to more than 2,000 pages in the 1990s. The Italian physicist Enrico Fermi said if he could remember all the names given to particles he would have been a botanist, and if such things really do interest you, you need to get out more. Besides, as early as 1964, the American physicist Murray Gell-Mann and others have felt that all those differently named entities were actually combinations of a relatively few truly elementary particles. It is perhaps small wonder that the Nobel Prize-winning Gell-Mann is these days devoting himself more to the study of language than to physics.
Niels Bohr, who used to have days of intense discussion about almost every word he used, was called a bad lecturer, but often he just wanted to get the language right. He once paused for a long moment before saying, '...and', then took another very long pause before he said, '...but'. (It was he who reportedly gave a baffled French ambassador the happy and fulsome greeting: ' _Aujourd'hui!_ ') At least he conceded that there was a problem. It took him several weeks of hard thought and discussion to come up with this: 'We are somehow suspended in language.'
You could do no better yourself.
Blaming language is your parachute, your cavalry, your safety net. A reminder about linguistic difficulties will send sharp questions ricocheting back to their posers; and if or when you have dropped yourself in it, talking about the lack of suitable words can soften the horror-movie eyes around the table.
The language used by physicists is mathematics – though many of them need help with it from real mathematicians. Their best-known trick is to shut their eyes and find some Greek letters to make it seem deep and meaningful. Physics is chock-full of words which have been poached and which may once have had a sporting chance of meaning something. Most could not be more confusing if they were designed to be so, and the suspicion is that they were.
It may also be a useful gambit to cite languages other than your own, such as Algonkian spoken by the Blackfoot people of Alberta, Canada, which emphasises change and movement in nature, rather than finding names for things. For example, instead of saying, 'I saw someone long ago', they say, 'I saw him/her in the far away.' This so impressed the eminent American scientist David Bohm that he proposed that an entirely new kind of language should be developed for the quantum universe. An English one, of course.
TACTICS FOR TIGHT CORNERS
Whenever the universe becomes a topic of conversation, certain questions will always be hovering around the edges – if there are any edges (and that's one of the questions). These include:
•How did the universe begin?
•Who cares?
•If the universe had a beginning, will it have an end? Or a new series in the Autumn?
•What has 'quantum' got to do with it?
•I sort of get it, but just when I do, I lose it again. It's as if it's in the cracks of my understanding. Is that where it should be?
•Can I have a large Bloody Mary?
Some of these penetrating queries may seem to be unanswerable. And that is because they are. But defeatism is alien to science as well as to bluffing. And since the scientists who are bold enough to deal with these questions are so often conjecturing, you may as well follow suit. So try one of the following if the bluffing muse has come upon you: 'Physicists now believe that before the universe began, everything was a substance not dissimilar to bubblegum'; or, 'Last week, cosmologists declared that the universe was stacked between two bookends; now they're saying it's shaped like the upside-down torso of a man in boxer shorts.' Or, 'The smallest particle, the carryon, is even smaller than the one found last February, the dreamon, and last Tuesday's equally minuscule particle, the whatson.'
But remember to balance conviction with caution.
Never commit yourself about the outer limits of the universe or the quantum realm even with a 'probably'. Anything you utter with certainty, or declare to be 'probably true', could return to haunt you and, it can be said with confidence, probably will. If you know what's good for you, a 'possibly' is the farthest you will go.
To avoid being thought a delinquent in the light of tomorrow's new data, you need to make the lack of language work for you by:
a)saying that we are up against the limits of it;
b)frequently using the word 'possibly' as an insurance policy; and
c)capping all statements about problems that have not been solved and theories that have been proved unassailable, or the finding of the smallest particle in the universe, with the caveat 'so far'.
With some of the more difficult questions, a gentle attack may be a good defence. The 'dismissal' tactic is ideal since it conveys the impression of very deep and thorough knowledge and prevents any deviation from your chosen conversational course. The dismissal is best exemplified by the simple but salutary sentence: 'The question doesn't apply.'
For decades scientists said that there was no before. That is to say: no 'before' before the Big Bang. Everything was in one spot and that spot was nothing. Less than nothing. And then that less-than-nothing went 'pop' and the universe just 'grew'.
How could they know that? Well, they interpreted it from the mathematics.
And what was there before the 'pop'? The answer was as cold as a padlock: 'the question doesn't apply'. Everyone who was not a physicist was at that time hectored into not thinking of a 'before'. The question didn't apply. Even so much as considering a 'before' had to be avoided because there was nothing there, indeed there was no 'there' there, just as there was no 'because' because 'because' needed a 'dum-de-dum' in order to have gone from 'dum-de-dum' to 'dum-de-dee' because of 'blah-de-blah'. Which didn't happen, nor did 'happen' happen, nor did 'there', nor did 'anywhere.'
Well, time passed and slowly but surely a substantial number of influential physicists came to accept what everyone else could see was blindingly obvious.
The universe had to have a Before. After all, there must have been something 'before' the Big Bang.
There was no apology from physicists. There was not even an acknowledgement. Not one of them came out and said, 'Your splutterings were perfectly reasonable. We patronised you, in fact we were really rather rude, but of course we were talking nonsense. So sorry. You were right.'
How do scientists get away with such high-handed behaviour? The question doesn't apply.
THE QUANTUM REALM
The study of subatomic particles is called quantum mechanics. This is strange because the word 'quantum' is Latin for 'how much' or 'how great'. In this case, not a lot. And you can see within a nanosecond (a very small fraction of a second) that it is entirely alien to this subject to talk of anything 'mechanical', 'mechanistic' or 'machine-like'.
The term 'quantum' was coined in 1900 by the great physicist, Max Planck. The quantum unit was to represent a 'package of energy'. Despite the temptation to add 'on' to the end of any word to reinforce one's quantum credentials, one can only hope that neither the prolific namers of particles nor naff punsters will ever mention anything involving 'Planckton', which really won't do. Indeed bluffers should avoid all such clumsy jokes, especially the all-too-obvious 'Thick as a Planck'. (The same applies to 'Thundering' Bohr, 'Huge' Schwinger and 'Atomic' Bohm.)
THE QUANTUM MOMENT
Max Planck's professor advised him not to do physics because it was thought at that time that all its problems had been solved. Planck ignored his advice. In 1900 he found a problem to solve and, in solving it, set a million more in motion.
The problem bore the name, rather unfairly perhaps, of 'Violet Catastrophe'. You can allow a few sniggers before bringing everyone to order by adding that it is also known as the 'Ultraviolet Catastrophe'. It occurred as a result of 'black body radiation'. A black body – a black object – tends to absorb radiation when cool; but if you heat it sufficiently, it will emit radiation by going red hot. The hotter it gets, the farther up the spectrum the glow issuing from it will get, through an orange and a blue heat, all the way to ultraviolet (the sort of light that picked out teeth and specks on a jacket in the good-old golden days of the discotheque).
For a physicist, seeing something work is not the point; the point is that the mathematics of it must work. The sums identified by Planck implied that the progress through the spectrum of the differently coloured glows would be shattered in a burst of catastrophic ultraviolet radiation. But however intense the heat, no such destructive explosion occurred. Something was wrong with the calculations. So Planck did the obvious thing. He cheated. He squeezed a number into the sums: 0.000 000 000 000 000 000 000 000 00066.
Then he attempted to do that most difficult thing for a scientist: he went back to real life. But whatever he tried, he could not get rid of that embarrassing little subterfuge.
It was some time before he realised that the cheat was the whole point. The cheat was Nature's cheat. The black body did not heat, or cool, in one smooth run of energy but in little jerks or spurts. (On reflection, perhaps Quantum is a better name than Jerk or Spurt – but the question doesn't apply.)
You should call that number, with its 27 zeros before the 66, Planck's Constant.
In physics, a 'constant' is a quantity that is universal in nature and constant in time. In other words, a 'constant' occurs always and everywhere in nature. Newton's gravity and the speed of light used by Einstein are constants. Planck's constant is in that exclusive club.
For years, Planck's discovery was not taken by other physicists to be a revolution. Then when it was, it was precisely the kind of revolution he fought for the rest of his life to prove it wasn't. He had created a monster which threatened the very basis of classical science. Again and again he returned to the problem he had already solved, even making ready to undo his own work and diminish his ranking as a physicist. To remark that there never was a more reluctant springboard than Planck is a truly unintentional breach of the no-pun rule ( _see_ above).
THE MEDIUM AND THE MESSAGE
Quantum physics started off being concerned with the composition of matter, the properties of light, and the interaction between light and matter. In 1900, when Planck was playing with fire and tinkering with his maths, what he was examining was the interaction between light and matter. The young Albert Einstein explained in 1905 how photons – particles of light – could be directed at a metal and dislodge electrons – particles of matter. Bonus bluffing marks will come your way when you declare that Einstein's examinations of the photoelectric effect were indispensable when TV was invented. (You can top this with the fact that Einstein won his Nobel Prize not for the general theory of relativity, nor for his big hair, extravagant moustache or his habit of not wearing socks, but for his work on the photoelectric effect.)
Light is just one of a number of agents of transmission which dislodge, illuminate, attract, repel, bind or explode all the materials of nature in a constant dance. You will receive much gratitude and admiration for imparting the simple information that there are, basically, two main categories whereby subatomic particles may be classified:
1.particles of matter; and
2.messenger particles.
The suspicion is that for each kind of matter particle there exists a 'symmetrical' messenger particle. If so, it would create a marvellous balance of medium and message throughout the universe which you should call supersymmetry. This is a fantastic word and it can be used to great effect whenever discussing the notion of 'balance' in a universal context.
It will probably do you no lasting harm to know that particles of matter have the generic name fermions, and all messenger particles, whatever else they are called, are also known as bosons.
It is worth gaining some extra points by mentioning the 1946 talk given by the British-born physicist Paul AM Dirac. Dirac was a taciturn gent, but his lecture was noteworthy for other reasons, for in the course of it he suggested the names 'fermion' and 'boson', paying tribute to two esteemed scientific colleagues. Enrico Fermi was most unusual in that he was an excellent experimentalist as well as a top-class theoretician. Satyendra Nath Bose understood light and matter so profoundly that Einstein himself headhunted him for collaborations. Much came out of their work together, not least the hologram.
Saying that fermions are scared of commitment can be a delectable conversational gambit in mixed company.
Fermions get their mass from a unique particle, the now legendary Higgs boson; quite rightly, somewhere along the way, science writers ceased to deploy Higgs's apostrophe.
Saying that fermions are scared of commitment, individualistic and selfish, and that bosons are the outgoing, surrendering relationship particles, can be a delectable conversational gambit in mixed company.
QUANTUM MECHANICS
You can divide the early history of the development of quantum mechanics into three main periods. During the first, the discoveries of Planck and Einstein were demonstrated but not fully understood. During the second, Niels Bohr's theory of 1913 was explored; his work with atoms and light was enormously helpful without, strictly speaking, being comprehensible. It was incomplete, which is why it was not until the third period that true quantum mechanics appeared when Max Born, Werner Heisenberg, Louis de Broglie, Erwin Schrödinger, Pascual Jordan and Paul AM Dirac (among others) offered fuller descriptions of a body of physical law which not a single person on earth could understand.
There was a clear need for some outstandingly impressive mathematics which would still leave the physicists themselves thoroughly bamboozled, thereby keeping up the good work. It arrived in the form of quantum electrodynamics.
In the late 1940s, three physicists, Shin'ichiro Tomonaga, Richard Feynman and Julian Schwinger, working independently of each other, arrived at the same mathematical conclusions at the same time. (It will do you no harm to add that a fourth, the sage-like British-born Freeman Dyson, should have won a Nobel Prize for explaining quantum electrodynamics to the other three.)
When interactions between light and matter are contemplated, the possibilities are infinite; and it was the 'infinities' in calculations which sent physicists loopy. Quantum electrodynamics (QED), instead of banishing infinity from the sums, invited it to join the party. Miraculously, it solved the problem.
QED is one of the most respected scientific theories ever. As Feynman was fond of pointing out, if you imagine that the calculations deal with the distance from London to New York, they would be accurate to within the breadth of one human hair.
Such accuracy is a good thing to invoke. By 1950 quantum mechanics had become what Paul W Davies called '...the most powerful theory known to mankind' and what _New Scientist_ termed '...the most successful and wide-ranging theory devised by human ingenuity'.
There were those, however, who felt that the intricate edifice stood on quicksand. Einstein, whose work with light and electrons had opened the curtains on the whole elaborate 'quantomime', wavered between calling quantum mechanics 'incomplete' and declaring its ideas to be 'the system of delusions of an exceedingly intelligent paranoiac, concocted of incoherent elements of thought'.
Obviously, you are in the presence here of a curious blend of scientific derailment and reassurance. There will always be those who feel that confusion is entirely appropriate in God's territory; so you are absolved from all obligations to understand any of it, and need merely pick and choose from such descriptions as you deem worthy of exploitation.
But the incomprehensibility of quantum physics is a secret which you should guard with your life.
The existence of antimatter is one of the arguments put forward for the existence of parallel universes. At least, in this universe it is.
MATTER MATTERS
DOES ANY OF IT MATTER?
The greatest potential for paralysing another human being, far better than tickling, getting a submission with a step-over leg lock or administering a generous dose of ketamine, is to introduce a novice to the basic structure of matter.
All matter is made of molecules, you say, and these can be further divided into atoms. Explain that the atom was accepted as the basic building block from 430BC until 1897, so it is a twentieth-century fashion to talk blithely of 'sub'-atomic particles.
The kernel or core of the atom – the nucleus – is one ten-thousandth the size of the entire atom, and is itself composed of particles (protons and neutrons) bound together by a strong nuclear force. Protons carry a positive electrical charge. Neutrons are electrically neutral.
Whizzing round this positive-and-neutral nucleus are particles of negative electricity called electrons. (As previously noted, whenever in doubt, add an 'on' as a suffix to any word in the quantum vocabulary.)
Try to imagine, you say again, a 'shell' effect around the nucleus (like the illusion of solidity produced by a whirring top) which is formed by these electrons while zooming at a speed the equivalent of four times round the earth in one second. In fact, of course, they are whirring around a nucleus that is itself only one ten-thousandth the size of an atom.
For a clear idea of scale, thinking of the cricket ball in the cricket ground still applies. Or you might want to use the image of a half-sized speck of dust inside the Taj Mahal. Others ask people to picture the dome of St Peter's Basilica in Rome. The electron shell would be the dome itself, and within the space of the dome – more or less at the centre – the nucleus would be the size of a grain of salt. Of course, you won't forget to say that the vastness between the nucleus and the electrons is almost completely made up of empty space. If you also remember to add that within the outer shell there may be layers and layers of inner ones, like skins, then the members of the enraptured audience surrounding you will really know their onions.
It is worth emphasising that electrons are infinitesimal points or wisps, and not planet-like spheres. They do not orbit; they flit about like supercharged moths. Even the fact that particles 'spin' at the same time as they zoom and jiggle should not lead anyone to think of the earth's rotation. Apart from anything else, particle spin occurs in two directions: 'up' and 'down' (a typically idiosyncratic physicist's expression for clockwise and anti-clockwise). It was Wolfgang Pauli who showed that a single particle will insist on being alone and spinning in its own way in its own energy state. This is called the exclusion principle. If another particle joined it, it would have to have opposite spin. A third would make the others feel exceptionally crowded and would have to go off on its own or find another one with which to have opposite spin. You should limit the ways in which you let anyone say it is just like life.
The balancing of electrical charges at the subatomic level is carried on throughout the universe; all the negative charges balance out all the positive charges. To some, this may suggest that the universe all adds up to nothing. To others, it presents another example of cosmic balance.
You may care to enlarge a bit here by stating that protons are 1,836 times bigger than electrons. Scientists tend to state this precise statistic and then say that neutrons are 'slightly' bigger than protons, without saying how big slightly is. You have little choice but to follow suit, or to forget these particular facts at once.
But this isn't all. Not by a long shot.
Protons and neutrons are themselves composed of still smaller particles. Yes, afraid so. They are the notorious quarks (pronounced to rhyme with forks). They are one of the exceptions to the 'on' rule, and must never be referred to as 'quarkons'.
Protons, neutrons and the quarks within them cluster together to form the nucleus of the atom, so some nuclei are very bobbly, like little clumps of caviar, or like raspberries. Just a touch more difficult to bite into.
Billions and billions and billions of atoms make up every tangible and visible object, so considerable mileage may be had from pointing to something and stating that it is just nuclei, space, whirr-effects and spin which create the solidity that we see and touch. What seems as still as a breeze block is, in fact, in constant motion.
This is why holists speak of everything as being an illusion. (You can refute this with a well-aimed kick.)
Erwin Schrödinger suggested that the entire universe and everything in it – the yaks and yams and stars and suns, the whole caboodle – was created by a single particle on a journey of incredible rapidity and wild complexity. Schrödinger was known for taking to his bed before and after such pronouncements; he was in a place to which any of us could so easily go.
QUARKS AND THE STANDARD MODEL
Questions about subatomic particles may arise – indeed, you yourself may wish to ask them – along the lines of, 'Just a minute, no one can actually even see these little thingie-whatsits, can they?' Or, 'How do they know?' Or, 'Are drugs required?'
After all, subatomic particles are, well, subatomic: minuscule, point-like entities (variously described as clouds, tendencies, or even dreams) which leave traces among jets of particles when there are collisions in the quantum realm. Some entities may be said to put out energy. Atoms consist of energy and energy is, effectively, invisible. In short, atoms are unlikely ever to be seen; they are detected. No one had actually detected an atom until the 1980s, with the invention of the scanning tunnelling microscope which can magnify what it is looking at 100 million times.
You could proffer the information that down the decades, experimental physicists have used photomultipliers, and have developed voltage doublers, Van de Graaff generators (not the rock band), cyclotrons, Geiger counters, cloud chambers and even massive tunnels miles wide in which particles have been smashed, accelerated, collided, traced, discovered, annihilated, created, presumed or given silly names. But the impression you should be giving is that you live a life of the mind, not of toil at the coalface. You should not deign to spend more than 20 seconds on any of this. You have something in the oven.
For many years the so-called 'standard model' – the orthodox view of what happens at the subatomic level – was composed of quarks and leptons. Be wary of saying 'quarks and electrons' because electrons belong to a wider genre of particle called leptons, so while electrons are always leptons, not all leptons are electrons. Fortunately no one is ever going to ask you to name the others because this is about as much as any brain can take.
It would have been helpful if you could have just left it at that, especially since the nucleus is packed with six different kinds of quark and six varieties of lepton dancing together. That's it, you could have said. The basis of all matter. Excellent. So simple.
Except it is now suspected that quarks are made up of even smaller particles, at present called subquarks, and that the subdivisions go sideways as compulsively as they go downwards. So that there may be not just six quarks and six leptons but a whole menagerie within the Particle Zoo.
Dramatic 'particle smashing' in 1997 revealed the possibility of a strange basic entity, a 'leptoquark', which might simplify matters. To a physicist, a leptoquark is a 'chalk and cheese' or a 'sow's ear silk purse', but it has been ages since anyone expected Nature to do what she's told.
In all fairness, physicists themselves find it difficult to hold on to their sanity in this reductionist's heaven.
Quarks were originally given 'flavours' whereby, for the purposes of calculations and the kind of work they seemed to do, they could be told apart. These flavours are 'up', 'down', 'strange' (originally called 'sideways' but it became even more odd), 'charm', 'bottom' and 'top'. (The latter two were originally called 'beauty' and 'truth'.) You should thank Murray Gell-Mann (of whom more later) for these quirky quark names; they may be pretentious but they are a tad friendlier than mathematical symbols.
When quantum chromodynamics was developed in 1977, quarks were given 'colours', which have less to do with tone than flavours had to do with taste. You can get each flavour of quark in red, green or blue, and antiquarks ( _see_ below, or in another universe) in minus-red, minus-green and minus-blue.
Now, you are increasingly likely to come across books, plays, films or ballets which hitch a ride on the ideas of quantum mechanics. These tend to be vain attempts on the part of artists to appear intelligent. Amidst all of these wrong-headed and often simple-minded efforts, you may find words such as 'charm', 'colour' or 'beauty' liberally deployed and, at times, even used in titles. Those terms, like the word 'quarks', were originally nabbed by scientists trying to seem more arty. So when you are confronted with arty types trying to be more scientific – and thoroughly misunderstanding the concepts – you should scoff with boundless force. It is your bounden duty to stomp on those outpourings which are proliferating wildly and are charmless, colourless, ugly and wrong.
In all fairness, physicists themselves find it difficult to hold on to their sanity in this reductionist's heaven, and have expressed disquiet about the standard model. Its calculations have proved successful, but as Einstein might have said, 'If you were in charge, is that the way you would construct the universe?'
ANTIMATTER
All quantum mechanics has an _Alice in Wonderland_ feeling about it. And antimatter is the looking-glass world. Indeed, the existence of antimatter is one of the arguments put forward for the existence of parallel universes. At least, in this universe it is.
Antimatter is matter composed entirely of antiparticles – elementary particles that have the same mass as given particles, but opposite electrical or magnetic properties. Protons are mirrored by antiprotons, neutrons by antineutrons, and electrons – which are negatively charged – by positrons.
The existence of antimatter was first suggested by Dirac in 1930. He was only trying to help. And he was not the only person to notice during the Great Depression that there was an awful lot of negative energy around, but he had a novel way of looking at it.
Dirac was formulating a theory for the motion of electrons and produced equations that were beautiful – a major criterion for him – but they described, indeed demanded, the existence of an antiparticle. Such an entity would have the same mass as the electron but with positive instead of negative electric charge. Physicists pooh-poohed Dirac, but Dirac was one of the least poohpoohable scientists who ever lived. Sure enough, in 1931, the antiparticle of the electron was duly detected. Because of its positive charge it became known as the positron and vast numbers of positrons have been found since.
When the universe formed, equal amounts of matter and antimatter should have formed; but if that were the case, they should have annihilated each other. Now, the observable universe seems to be composed almost entirely of matter. Rather than antimatter. Or a mixture of matter and antimatter. That apparent asymmetry of matter and antimatter in the visible universe is one of the greatest unsolved problems in physics. In short, it is another question for Cern. (They're looking into it.)
When matter and antimatter meet, the mutual annihilation is even worse than the most frightening marriage you know. So if any original antiparticles are still around, they certainly can't be nearby.
Something may have happened, just after their creation, which destroyed them all. Or did they build up an antiuniverse, with antihouses, anticroissants, antilobsters, antistickytape and at least one antiyou?
All that can be said now about such an idea is 'ylbissoP'.
Space does not permit speculations here concerning the antimatter equivalents anti-anti – or pro-depressants, anti-antidandruff shampoos, anti-antiwar demonstrations, anti-auntieannes or even antimacassars.
It is a treat for physicists when particles and antiparticles collide, and, as they not-so-delicately put it, annihilate to energy. Still more magical is that nanomoment when out of pure energy, particles and antiparticles are created. It's enough to make them feel like Steven Spielberg.
Yet without the help of physicists, this dance of creation and destruction is taking place all the time – in people, phones, walls, toothpaste, rubbish bins and jelly beans. Nothing could exist if it didn't.
THE QUANTUM LEAP
When you speak of the whirring electrons inside every object, you need to bear in mind that this is what is happening at room temperature. It may be a hectic flurry to you, but to an electron it's rest. Heat them, though, and they really get going. The hotter they get, the more excited they become, and when they reach a certain pitch of excitement and heat they 'radiate'.
With radiation the electron actually jumps or is boosted through different layers around the nucleus inside the atom. This is what Bohr called the 'quantum jump', an expression that gave rise to the 'quantum leap'. You could do some interesting dinner-party dribbling with this one. First, what Bohr spotted were jumps down rather than up. (If you really need to know, an electron jumps downwards when radiation is emitted and upwards when radiation is absorbed.)
Second, the term has come to mean in everyday language a huge or sudden advance. Quantum leaps do seem to disobey a number of traditional laws of physics; and, relative to the size of the electron, they can involve impressive energy and range. But here is an opportunity for you to earn maximum points. Strictly speaking, in the scheme of things, a 'quantum leap' is a seriously small event. Indeed, you can happily startle anyone who uses the term by telling them that few events in nature are as minuscule as a quantum leap.
Billions of such leaps are involved in every light bulb, candle, barbecue, soldering iron, combustion engine, cigarette, their respective electrons all receiving and transmitting like one vast telephone exchange – with light, heat, energy and radiation instead of voices on the line.
Bohr and Heisenberg described the behaviour of these minute interacting particles as 'crazy', and Feynman – the physicists' physicist – with his customary American folksiness, called their antics 'nutty'. Today's physicists are reviving the word 'weird'.
It must not be forgotten that scientists are very restrained in their use of language. Nor must it be forgotten that many scientists have been unable to come to terms with the behaviour of subatomic particles. Nothing for them in two-and-a-half millennia has been as shocking.
Light is electromagnetic radiation. It's the kind of fact that gives the mind a lemony taste and then becomes pleasingly forgettable.
SHOCKING BEHAVIOUR
In order to captivate any gathering, you need some appreciation of the shocks received by classical scientists striding purposefully out of the nineteenth century. Such was the respect accorded at that time to anything scientific that whatever was unscientific was looked at as if it were a schnitzel that no one had ordered. Indeed, as the twentieth century began, it was believed that all the problems concerning nature and the physical world had been solved by science. Those minor problems that remained would be solved before Wednesday week, at the latest.
Wednesday week came and went. So did many very clever people, and within one quarter of a century all notions of space, time, gravity, energy and matter were changed. It was discovered that things bent, warped, scrunched and dawdled. And this was decades before rock 'n' roll. It was shown that if you went faster than the speed of light you would go back in time, and that the faster you travelled, the more distorted your body would become. Furthermore, the faster you travel, the slower your brain would run. Planes, trains and motorways are full of examples of that.
Perhaps most alarmingly, it was established that if your parents zoomed off at a speed getting on for the speed of light and stayed away for 40 earth years, they would come back younger than you. And you could get your own back. And so could they.
All these things were accepted by Einstein – after all, he proposed them. But what he could not absorb were the shocks of quantum physics. When asked why, he said: 'A good joke should never be repeated twice.' (Grammar was not Einstein's strong suit. Nor was humour. Nor, as it happens, was maths.)
Within the quantum realm, the rules of nature would be happy merely to be turned upside down. They are turned every which way, and many billions of other ways that are inconceivable to the human mind and are likely to remain so for a long, long time to come. Quantum theory challenges, at the very least, notions of causality, predictability, reversibility, continuity, locality, order, separateness (sometimes called isolability), clear definition, accurate and informative measurement, objectivity, either/or thinking and, most controversially, certainty. It threatens notions of solidity and substance, and suggests that there are absolutely no absolutes.
CAUSALITY
Causality is one of the most important pillars of science. Everyone knows that if you freeze water you will get ice and if you heat it, it will evaporate. But in quantum theory events seem not to have such well-defined causes and effects.
It all boils down to the holist/reductionist argument. Coming down on either side is dicey. Holists are convinced that an electron can jump between orbits for no discernible reason, or that a subatomic particle can come into being or disintegrate altogether without any cause. In other words, holists insist that particles do the loathsomely worst thing anybody or anything can do to a scientist: they behave spontaneously.
Reductionists share Einstein's lifelong distaste for the incorporation into science of anything 'acausal', random or 'stochastic' – a good word to use if you want to say 'subject to guesswork, or conjecture'. He said: 'I cannot believe that God plays dice with the universe.'
If you look at a glass of sparkling mineral water, Coke or, best of all, Champagne, you might begin to see the problem. To analyse or anticipate the behaviour of any individual bubble is fiendishly difficult, even if it is momentarily motionless, let alone when it suddenly decides to jiggle around, disappear or shoot off in funny directions. Quantum physicists have set themselves a task far tougher than relating a bubble to a drink, or even an atom to a bubble. Yet reductionists say they have an answer for all that fizz, zoom and kapow!
The real kapow! lies in the fact that their methods deal with particles collectively, via the distribution of probabilities within what they call a macro system, i.e., the whole picture. In other words, their solution is holistic.
Meanwhile, holists say that it's no good calculating collectively: what about the individual particle? In other words, they are thinking like reductionists.
You only need to point to this transposition of views to be protected in any discussion about the stochastic, random or spontaneous nature of subatomic particles.
You will even find a number of physicists who dismiss spontaneity – just like that. For them, the central contentious issue when it comes to causality is Time. Is causality linked to time as profoundly as the word 'consequence' is related to 'sequence'? Can an effect precede its cause? It's a question that, of course, arose from the following.
PREDICTABILITY
The remarks about causality also apply to the issue of predictability. Anyone could have seen that coming.
REVERSIBILITY
Reversibility may be understood by watching a video of some movement in nature in what is loosely called real life: a person or an animal running, or swimming, or a speeded-up film of a flower blooming. When such a scene is shown backwards it always looks odd, not lifelike, even funny, because of what is known as the irreversibility of Nature. Natural processes take place in time, and they cannot be 'run backwards'.
However, a video or film of a subatomic particle in motion could run backwards for a century or indeed, conceivably, for several millennia, without looking odd. The viewer might have clues – identifiable events, perhaps, or a peek at the label – but no one coming fresh to that video as it played would be able to identify either the forward or backward motion of it. In short, subatomic particles are somehow outside Time – occupying a gap in the very nature that they compose.
Some have been in existence since the universe began. Others have been created as recently as a second ago and might have disintegrated a relatively long time before that second was over. And there is a strong possibility that there are subatomic particles whizzing about here and now that were created in the future.
It is comforting when contemplating these phenomena to realise that even the greatest geniuses are 'up against the limits of what can be known'. This is a handy expression to use on its own or in conjunction with 'up against the barriers of language'.
The latter ploy is better if you want to imply that you do have the knowledge but it is the language that is letting you down. The former is excellent if you need to be humble. And brief.
CONTINUITY
If you were to arrive in the quantum realm, what you would first notice is the lighting. Strobe, unfortunately. It is an irritating flicker, flashing much more rapidly than in any nightclub. And the particles, like dancers in a nightclub lit by a strobe, are known to be 'here', then to be 'there', but no one knows what happens in between. Their progress is discontinuous.
In classical science it is possible to 'trace a path', 'record a progression', 'monitor a process'; but the lives of subatomic particles are mysterious flickerings, and the bursts of knowing where they are, and what they are doing, are shorter than the gaps of disappearance and unknowability. Like teenagers, really. Not all teenagers are like that, though, and teenagers do tend to grow up. These blighters are arrested at that impossibly awkward stage of not knowing what or why they are, or where they are going.
THE PARTICLE/WAVE PROBLEM
In the quantum universe there is an essential and baffling fact: subatomic particles can also be waves. This you should speak of as the 'wave/particle duality'; you should say it is the central mystery of quantum physics.
For a reductionist, the mystery concerns the so-called 'quantum amplitude': a particle is of a different size from a wave. It ought not to be one and the same. To a holist, there is no mystery; it is glorious and natural that nature's tiniest possible entity – a particle – should have connecting ripples throughout the universe.
With particles and waves you see science in a very bad light, and indeed it was in the course of a dispute about light that the major surprises suddenly appeared. Isaac Newton had stated that light was composed of particles. He called them 'corpuscles' and expected everyone to agree with him because he was a genius. They did agree with Newton until a certain Thomas Young proved that light was made up of waves. Fortunately for him, Newton had been dead since 1727 and wasn't around to argue the toss. (You can seem scholarly by saying that Young conducted his experiment in 1801 or 1805 'depending on the source'. You can seem even more scholarly if someone suggests that Young deciphered the Rosetta Stone by saying that he did find the sound value of six signs, but Champollion did most of the work and would not share the credit.)
Throughout the nineteenth century, everyone was so sure of the wave nature of light that they concentrated on finding out what was waving. As with most romantic plots, electricity and magnetism, which had started out not being suited to each other, were brought together and engaged. Michael Faraday did a lot of the basic matchmaking, then James C Maxwell showed that 'electromagnetism' travelled at the speed of light – and he married them. Light is electromagnetic radiation. It's the kind of fact that gives the mind a lemony taste and then becomes pleasingly forgettable.
While light is always electromagnetic radiation, not all electromagnetic radiation is light. The light that can be seen makes up only a small part of the kinds of electromagnetic rays that there are in the universe, which include X-rays, gamma rays, microwaves and even radio waves.
In this sorry saga so far, Newton had said that light was definitely made up of particles and Young had said that light was definitely made of waves. Maxwell, Heinrich Hertz and many others agreed: yes, light was made up of waves. That was that, then. Until Einstein, working with the photoelectric effect, stated: 'Nein, light is definitely made up of particles.' He didn't deny light's wave nature, but he could no longer deny light's particle nature either. Which was why the pioneers of quantum physics had to have another look at an old experiment – the one with the two slits, which you should refer to as the double-slit experiment.
Quote Feynman, who said that all discussions about quantum mechanics will eventually come back to the experiment 'with the two holes'. The apparatus consisted of a light, a screen into which two very thin vertical slits were cut, and a wall. In Young's test the light had clearly lapped and undulated its way through and away from the slits, because there were blocks of light and shade on the wall. Particles of light would have flown like bullets, leaving no shading, just two clear slits.
These days, the very same test shows that light is made up of both particles and waves. If you use a particle detector and test for particles you get particles. If you use a wave detector you get waves. Somehow, and no one knows how, it is the test itself that gives you the result. It is utterly alien to classical physics.
Eddington once said that on Mondays, Wednesdays and Fridays an electron is a particle and on Tuesdays and Thursdays it's a wave. (He didn't work at weekends.)
As Walter Cronkite used to say on American TV at the end of the news: 'And that's the way it is this Sunday lunchtime.'
ACCURATE AND INFORMATIVE MEASUREMENT
It is important to make it clear, in a magnanimous and sage-like way, that the measurement of subatomic particles is phenomenally precise. Scientists, especially reductionists, cannot hear that often enough.
The problem is that scientists – like all of us, only more so – need two or more measurements to work together, so that other conclusions may be drawn. If it is known that a plane is travelling at 500 miles an hour and that Rome is 1,000 miles away, then the plane should take two hours to arrive; that can be handy, or essential, to know.
In science it is generally required that data should do more than one job, and in classical science, position and momentum (by which physicists mean velocity times mass, rather than speed) go together like the Blues Brothers, Laurel and Hardy, or trains and irritating people on mobile phones.
The shock when it came to quantum physics was that it was possible to confirm the position of a particle but not its momentum; the particle was too small, too inaccessible and too capable of deflection – as always – by the instrument and the light used to trace it. All concentration was thus focused on finding the momentum. In fact, the momentum was established – with equal accuracy – but by then the particle had zoomed past and its position could not be pinpointed with anything like the same accuracy.
In any discussion about this conundrum you really need do nothing more than utter the three words: 'Heisenberg's uncertainty principle'. It is possible to be certain about one measurement, but then another would become uncertain. By the way, Heisenberg stated that just as it was impossible to ascertain simultaneously the position and the momentum of a particle, so one could not know at the same time a particle's position and its energy. It is rare for this to be pointed out and you will, or should, be highly regarded for doing so.
Inevitably, the uncertainty principle is not principally used by professional physicists for what it has to say about uncertainty. Nor for its remarks about measurements. It is simply a useful tool for some calculations. Scientists do not think that philosophy is a useful tool for anything.
The really terrifying bit of uncertainty for scientists, and indeed for all of Western society, concerns objectivity.
OBJECTIVITY
If you ever get the urge to challenge classical science, state Werner Heisenberg's finding: 'The act of observation affects that which is being observed.' And then watch.
In science, any test must work independently of the person conducting it. The worst put-down one can receive from a scientist is: 'Now you're being merely subjective' – i.e., influenced by individual feelings or opinions, and therefore irrational. Only objectivity, the collective subjective opinions of like-minded scientists, can be trusted. Like a rugby prop forward, objectivity is solid and dependable. And it won't cry.
It is from objective truth that the laws of nature are derived. And these laws hold good whether or not someone is checking on them. With objectivity, the observer cannot affect the experiment; the experiment and the observation of it are separate in every way.
But in the quantum realm it's hard enough to keep the experimental apparatus separate from the experiment itself; there is also the added problem of the physicist always getting the result being tested for. Is the human being profoundly involved? Believe it or not, you can say that physics has rather moved beyond such questions. After all, you are quite at liberty to point out that the experiment would work just as well while the physicist was having lunch in the canteen.
The physicist wouldn't see it, though, so these questions do linger. Besides, they give you the opportunity to talk about that legendary character, Schrödinger's cat.
SCHRÖDINGER'S CAT
Schrödinger's 'thought experiment' proposed that a cat be placed in a sealed box containing a small flask of cyanide gas which might (or might not) be shattered by a hammer that might (or might not) be activated by an emission from a lump of (potentially) radioactive material – and the experiment be run long enough to give the cat a 50/50 chance of survival.
The cat represents two possibilities, or probabilities. It could be alive or dead and continues to be in that state until – well, until the scientist takes a look. Until then, it is in an absurd and, it has to be said, unscientific state of live-deadness.
OBSERVATION KILLS THE CAT
This story cannot be ignored, but in fact it can be dispensed with very quickly. Apart from anything else, cat lovers never take in the details, they just want to get the cat out of the box and cuddle it. However, for some, what is important is that as the act of observation takes place, reality forks and two entire universes are simultaneously brought into being: one in which the cat is alive and scratching, and another in which the cat is dead, deceased and gone to its maker.
Indeed, many boffins go further and declare that the two possibilities represent two separate universes and that, in each of those universes, there may be other twinned possibilities, and other universes. Yes, really – many other universes, billions of other universes.
These are hard-nosed scientists, remember, who are supposed to have a grip on reality. And, by the way, for them, reality – such as it is – doesn't just fork, it fans. This offers up a potentially endless number of realities and is called the 'many-worlds interpretation' – but never say, 'Of what?' The idea of 'many worlds' was dreamed up in 1957 by the American physicist Hugh Everett; you will strike chords with the same notion, in certain circles, by speaking of 'parallel universes'. All's fair – provided you remember that ever-absolving word 'possibly'.
Schrödinger himself went back and forth wondering whether or not he agreed with quantum theory. As noted earlier, he often took to his bed. Equally often, Niels Bohr would follow him into the bedroom and talk at him some more to try to convince him that quantum events were everything and headaches were nothing. It is much regretted that the poor man had such frequent headaches and recurring bouts of illness. In a poetically just universe, he would have been allergic to cats and would have swelled up and started sneezing. Then the rest of us could have observed it – in a vast range of other universes.
LOCALITY
People have been known to scold with the words, 'I can't be in two places at the same time.' Yet it seems as if subatomic particles can. People say it automatically, like, 'How are you?' or, 'This is so civilised', as they pour the tea, or, 'I'm sorry, he's in a meeting'; they just trot it out. 'Of course, subatomic particles can be in two places at once.'
In addition to all the wacky phenomena thrown up by the double-slit experiment, there is the fact – and it is a fact – that a single particle appears to go through both of those slits. It really is as startling as the famously surreal _The New Yorker_ cartoon by Charles Addams depicting a skier apparently passing through a pine tree with single ski tracks on either side of the trunk.
The Reverend Dr John Polkinghorne KBE is the kind of person who makes hen's teeth seem plentiful: he's a theoretical physicist who is deeply religious.
Somehow, Polkinghorne's way of putting it is, so to speak, a godsend to bluffers, since he refers to 'that great and good man, Sherlock Holmes', who was fond of saying that when you have eliminated the impossible, whatever remains must have been the case, however improbable it may seem to be. Applying this Holmesian principle must lead us to the conclusion that the indivisible particle went through both slits. In terms of classical intuition this is a nonsense conclusion. In terms of quantum theory, however, it makes perfect sense.
Nothing goes to the heart of the differences between the reductionists and the holists like the question of locality. As far as the reductionists are concerned, classical science has managed more or less to accept and understand that particles of light do become waves of light and vice versa when they have to negotiate, say, the pupils of eyes, or telescope lenses or camera shutters.
And classical scientists never thought to question the fact that such events happened 'locally', at the cricket ground, or at Julia's wedding, or through Paul's window; the light negotiated a small hole, or a thick piece of glass, or a thick piece of glass and a hole, and moved on.
For the holists, there is the knowledge that Heisenberg and Bohr, when they developed quantum theory, began to be aware of something vaguely mystical about it all. Bohr incorporated the Taoist symbol of oneness into his coat of arms. Cult books about quantum physics published in the 1980s created considerable excitement among Buddhists, vegetarians and ageing hippies: 'I mean, don't look for why, man, don't look back, don't explain, know what I'm saying? It just like happens, man. Spontaneously. Why won't my van start?'
This weirdness, or non-locality, is still controversial, and therefore a winner for you. Besides, physicists do their best to shrug off, dismiss or wriggle round any questions that have the faintest whiff of mysticism, so when they take such a subject seriously and address it, so should you.
ENTANGLEMENT
If you want to seem especially knowledgeable and professional when quantum mechanics is discussed these days, you will find few words that are used more frequently than 'entanglement'.
There are some entrancing ways of discussing entanglement; an important thought underlying them is Einstein's conviction that no interactions over any distance could take place faster than the speed of light. Even in his day, though, it did seem as if interactions between particles took place with suspicious simultaneity. Bohr's argument was that separated particles were still part of a single totality, even if one was on earth and the other on another planet.
To challenge this, Einstein and two younger colleagues, Boris Podolsky and Nathan Rosen, dreamed up a thought experiment, the EPR experiment.
E, P and R believed that they could show quantum mechanics to be 'incomplete'. They suggested an imaginary collision between two particles, which then fly off separately to some distance from each other. Then, by knowing the position of the smash and by probing one of the particles immediately afterwards, all the data about position and momentum concerning both particles could be calculated. Of course, this examination would have to be extremely quick. It would have to happen in a flicker of time. But it would be worth it to contradict Heisenberg and, indeed, to nullify all the shocks wrought by quantum mechanics and save classical science.
There would only be one way in which the EPR experiment could ever be wrong: if there were some kind of connection or message between particles that travelled faster than light (i.e., at a speed that was 'superluminal'). In that event, it would not be possible to probe one particle without simultaneously probing the other. Such a notion was worse than disturbing to Einstein; it made scientists look like fairground psychics, gamblers or dabblers in voodoo. He called it 'spooky and absurd'.
Einstein died. Bohr died. Then, in 1964, John Bell at Cern in Switzerland published some challenging calculations, which you should call Bell's Theorem. Bell had actually begun work in the hope of proving Einstein and his young assistants right, but his theorem opened the door to Einstein's worst nightmare. Repeated experiments, especially during the late 1990s, confirmed that once two particles have had any interaction, they do somehow remain linked as parts of the same indivisible system. Researchers seem perpetually on the brink of putting Bell's Theorem beyond question - especially, in this case, at institutions away from Cern. Where they're also looking into it.
You can expect discussions about the connection between twins, the collective unconscious, TV ratings – any number of subjects can be swept into this gigantic dustpan.
If entanglement doesn't bring out the storyteller in you, nothing will. You'll find your own way of telling this, but your tale requires anything that comes in pairs, one left and one right (therefore not earrings), such as shoes. Einstein used gloves. The Timothy McSweeney website explored the glove analogy too. As with all tales, you will adapt it to suit your style. Be sure to make it clear that your story is describing occurrences that are 'a bit like' phenomena in nature.
You are telling a magical tale. You put a right-hand glove in one box and a left-hand glove in the other. Wrap them if you like. Then, go on a journey, taking one of the boxes, so that the gloves are far away from each other; go even to the edge of the universe.
Then open your box. Inside it you will find a right- or left-hand glove. At that moment you will be able to see whether it is a right- or left-hand glove. And you'll instantly know from that information whether the other glove, in the other box, some distance away, is a left- or right-hand glove.
The two gloves in this story are said, by physicists, to be 'entangled'. One can know certain things from the one by looking at the other.
But now you can notch up the enchantment. Involve a lover. Imagine that you and your lover wrap and box separately a left-hand glove that is bright pink and a right-hand glove that is dark green.
You and your lover travel far from each other, even to opposite boundaries of the universe. When you reach your destination, you open your box and unwrap your glove. A trillion, trillion, trillion gloves swirl around you: left and right, different shapes and sizes, different colours. You reach up into the cloud and pick one glove. It is dark green and it fits your right hand perfectly.
And, because the clouds of gloves are entangled, because they're connected in some spooky but nonetheless real way, you know for certain that on the other side of the universe, at the very moment you're flexing your fingers in your glove, admiring the back and front of your hand, a bright-pink glove has just been slipped onto your lover's left hand. That really is how entanglement works in the quantum realm.
Many people love Einstein as if he were their puppy.
Now, however smug it feels to have proved Einstein wrong and to show how spooky the cosmos really can be, you should not allow this to show. Firstly, it would be unseemly. Secondly, you would be hated for it; many people love Einstein as if he were their puppy. Thirdly and most importantly, the questions that Einstein posed and the doubts that he expressed have not completely gone away. You should never stop stating, gently, that quantum mechanics is incomplete. A situation that, needless to say, is being addressed daily – constantly – at Cern.
ORDER
It is worth noting, for conversations in which others do not mind getting technical, that all electrons are identical to each other, all protons are identical to each other, and all neutrons are identical to each other. In other words, three basic entities, by combining to form atoms, which combine again to form molecules, combine and combine again to build everything in the universe from bratwurst to battleships, from waffles to warthogs, from neck braces to New York. But it will also by now be clear that in every cubic millimetre of space there are billions and billions of subatomic particles which are capable of behaving at times in astonishing and unpredictable ways for no rhyme or reason. These repeatedly, spontaneously and 'nuttily' change their physical qualities, their whereabouts, their energies and their flight paths and, small as they are, seem to have the entire universe in tow. It is quite difficult to put them in neat piles, label them and file them away.
CLEAR DEFINITION
The point never stops being made, unfortunately, that particles that can clearly be defined as particles can also be found to have none of the qualities that are recognised as particle qualities; at such times they have all the qualities of waves and must be defined as waves. This is not a new idea, and yet the shock waves from it still do not seem to have subsided.
It challenges classical science, because clarity of definition is a cornerstone of science. It is almost the definition of it.
SEPARATENESS
Particles are often defined as 'discrete' (ibid.). This is a good word to use. 'Discrete' means 'individually distinct', 'separate'. Holists believe that each particle tugs at and is tugged by the universe. Reductionists cannot quite budge from their stand of separateness and splendid isolation.
Whether you support one or the other (or neither), you need to concede that it is possible to study and conduct experiments with individual particles. Indeed, the ability to send a single electron whizzing through silicon is the very basis of the entire computer industry. (This fact should get good reactions. If people are receptive, it is good drama to add that computers, transistor radios, digital systems in TV, sound systems and even – pointing around the table at a sequence of wrists – digital watches, could not have been developed without quantum physics. Or you may prefer doing it the other way around, making computers the big finish.)
But, when trying to study subatomic particles, it is impossible to subject them to traditional, classical scientific examination, because particles relate to each other and to equipment itself with such determination. It is sometimes stated that these particles are not experimentally 'isolable'. Neither is it possible to separate off any of the shocks that quantum theory has given to physics, to classical science and to civilisation – challenges that arrive from every direction, at all times, simultaneously.
It might seem obvious that this should be so, but Einstein didn't like it – at all.
TITANS AT BREAKFAST
In 1927, during the Solvay Conference in Brussels, held in the Metropole Hotel, Bohr and Einstein conducted a debate which you might choose to call the greatest intellectual encounter since Newton's famous clash with Gottfried Leibniz. Using the word 'famous' when people are unlikely to have heard of something is an unkind but delicious ploy. It helps the story along if you mention that they conducted their arguments outside the business of the conference, and always at breakfast time.
Einstein was in effect defending the Newtonian universe against all those shocks with which you are now familiar.
Bohr narrowly won the first encounter, in 1927, by gently insisting that subatomic particles cannot avoid being affected by the very equipment with which they are examined. It might go against Einstein's deepest convictions, but obviously there are particles in the instruments. Surely they must affect each other.
The second bout took place in 1930, during that year's Solvay Conference, in the same breakfast room of the same hotel.
Einstein had made up a 'thought experiment' which concentrated, subtly, on a tiny particle's tiny energy. As they finished their croissants, it looked as if Bohr had lost.
But at yet another breakfast, Bohr proved Einstein incorrect. Bohr was able to show that the equipment would have to be affected and this time did so triumphantly and dramatically, referring to gravitational fields. Put simply, Bohr defeated Einstein by using Einstein's own theory of relativity.
Scientists might suggest that Bohr was convincing the great man that quantum theory is not inconsistent with relativity. Bluffers may want to accept that kind view. Hard hearts may wish to push the point that, for once, Einstein was wrong and lost by scoring an own goal.
Niels Bohr and Einstein kept up their argument for three and a half decades. Even after Einstein died, Bohr would take issue with him as if he were still there, as a warm-up. The last thing Bohr did before his death in 1962 was to draw a diagram of that thought experiment proposed 35 years earlier in Belgium by Einstein.
He was probably just reassuring himself one last time.
EITHER/OR THINKING
All the new shocks and questions have not shaken the fundamental difference between holists and reductionists. Reductionists say 'either/or', holists say 'both/and'. Both can be either right or wrong or, indeed, both right _and_ wrong. This applies to both 'either/or' and/or 'both/and'.
CERTAINTY
Obviously, if definitions, accuracy and predictability are at risk, certainty will be in trouble, too. Note:
1.The uncertainty principle refers to the phenomenon of uncertainty with regard to the position and momentum of a particle. It is not saying that nothing is to be trusted anywhere anymore.
2.Classical science is still, for the most part, accurate and certain. You need to say this, even if you do feel moved to add that reductionists need to accept some doubt, some spontaneity, some life.
3.All possible or probable jokes about the uncertainty principle have been made and made and made again. You can be certain of that.
According to the dreadful old joke, W and Z particles beg the Higgs boson to come to the cathedral on Sunday because without it they can't have Mass.
GUT INSTINCTS
In the quantum universe the end of physics will be found. And thereby, the solution to everything.
While it is clear that there should be a grand unifying theory (GUT), and there have been any number of attempts to prove that there is, all such comprehensive proposals remain satisfyingly loose.
There is no reason why you should not join the legions of super-ambitious physicists in the search for this theory of everything (TOE) which Einstein dreamed of finding. (In Einstein's day, a GUT or TOE would have been called a UFT, for unified field theory, because it had been discovered that four kinds of 'fields' are permeating the physical universe, and that, sportingly, they allow four 'forces' to frolic there.)
In the 1950s Heisenberg said that he and his friend Wolfgang Pauli had just about got there. Pauli, however, drew a cartoon of an artist, brush in hand in front of a canvas on which there were a few lines and squiggles. The caption read: 'I can paint like Titian; only a few details are missing.'
This should inspire you to prepare and cook your own GUT. You are ready to do so.
First, catch your particles of matter. These can be difficult to clean and separate, so just cram them in the pot and persuade as many as you can to stay.
If you thought particles of matter were unwieldy, wait until you try dealing with messenger particles.
Then get hold of your messenger particles. Be warned: if you thought particles of matter were unwieldy, wait until you try dealing with these.
You will find them in the forces of nature which have been marinating at least since the Big Bang. In ascending order of their powers of attraction or repulsion the forces are:
1.gravity;
2.the weak nuclear force;
3.electromagnetism; and
4.the strong nuclear force, which you should always refer to as 'the strong force' – the immense tug at the core of atoms. Nothing in the universe binds like it. Its particles are called gluons. (It is the breaking of this binding that makes nuclear weapons wreak havoc, and the harnessing of it that creates nuclear energy.)
By the time you have stewed, minced, mashed, stirred, whisked and added a pinch of salt, you would hope that they would all be reduced – after straining – to a single force. But be prepared for disappointment. A GUT has never been achieved before (nor has it been grand or unified) for the simple but persistent reason that gravity has been left out and even the other three have not yet been united.
DIPPING A TOE
The major stumbling block is that subatomic particles have always seemed free of gravity, flying about with as much abandon as international businessmen. More to the point, the maths makes utter nonsense.
You can show how difficult this problem is by saying that Einstein laboured on it for decades, trying to unite gravity with the other three forces, so that quantum physics could coexist with his general theory of relativity, and subatomic particles would make sense within the universe as a whole.
Uniting the weak nuclear force with electromagnetism was only achieved some 20 years after his death. Since electromagnetism has a vastly longer range, operates more symmetrically and is 10 billion times stronger than the weak nuclear force, you should regard the discovery that they are the same fundamental force as spectacular, and call the theory that proves it the 'electroweak theory'. Particles that carry this combined force are called, inventively, W, Z+ and Z-, which excites reductionists and no one else.
Two of the men who came up with this were the physicists Steven Weinberg and Sheldon Glashow, who had been friends since their days at the Bronx High School of Science. Still, they worked more or less independently – which excites holists. Also labouring apart but simultaneously was Abdus Salam, who remains without honour in his own country, Pakistan, because his family belonged to the wrong community.
So much for what has been done. For the rest, the belief is that the more billions that are spent on them, the more answers they will find. A superconducting supercollider in the Texan desert, for instance, cost the Americans mega-dollars (think billions, then think double that) before Congress pulled the plug when 14 of the intended 54 miles had been built. Nobody mentions it any more. It would have been nearly three times as powerful as the Large Hadron Collider at Cern.
Nor can discoveries always be as newsworthy as the Higgs boson. By the way, a minor bluffing point: Peter Higgs may live in Edinburgh but he is not a Scot. He was born in Tyneside and most of his early years were spent in Bristol, where he attended Dirac's old school, Cotham Grammar.
Electroweak particles (W and Z to their friends) are puzzlingly dense. According to the dreadful old joke, W and Z particles beg the Higgs boson to come to the cathedral on Sunday because without it they can't have Mass. And indeed, with impressive creativity, those entities are somehow, perhaps miraculously, endowed with mass, which is why the Higgs boson was called 'the God particle' for a short while.
On 4 July 2012, experimental physicists at Cern announced that they had found a particle which very jolly nearly was, at the very least, Higgs-like. It was then that the world came to know of the expression 'Sigma-5'. Sigma is a Greek letter. Sigma 1 to 4 do represent the close-but-nocigar results, but this is the world of science, remember, so 3.6 sigma means 99.9% certain; 5 sigma means 99.99995% certain. There is a distinct possibility, however, that it will be adapted to mean 'pretty sure'. In answer to questions about used cars, damp courses and contraception, stand by for the firm, less than trustworthy response, 'Sigma-5'.
In 1993, William Waldegrave, then the UK government's science minister, awarded bottles of Champagne to physicists who could explain the Higgs boson and why money should be spent looking for it. The winning answer was in the form of a metaphor involving Margaret Thatcher, the recently departed prime minister, making her way through a cocktail party. As political hangers-on cluster around she finds it harder to move across the room – acquiring mass due to the 'field' of fans, with each fan acting like a single Higgs boson.
Some scientists have tried to put across the image of particles as flour being bound together, but they appear to have confused the name 'Higgs' for the word 'eggs'. Another overworked explanation involves a tray of sugar or treacle and several ping-pong balls. Such a tray would represent what is known as a Higgs field and the particles within that field, Higgs bosons, give mass to the balls as they make their way along. You can trump such explanations by stating loftily: 'Higgs himself loathes such metaphors.'
You can add that he especially dislikes the idea that a Higgs field slows things down. However treacly the environment may be, particles power through it. Peter Higgs is in his eighties, but still rather youthful, and he's possibly defensive about pace of movement.
If you're feeling less than charitable, you can add that Higgs was not by any means the only – nor even the first – person to conceive of these mass-giving phenomena. So you can trump the media and even the man himself by calling this entity the 'Englert-Brout-Higgs-Guralnik-Hagen-Kibble' field.
It mustn't be forgotten that the Higgs boson is even more magical than the acorns from which oak trees grow. It is a messenger particle and yet it gives mass to other particles. For what it's worth, Higgs himself sees the Higgs field and the Higgs boson as refracting – shining, as it were. Don't forget that light is made up of bosons. And so the Higgs boson might be described as giving off a blue light that colours everything blue. And just as plants use light to grow, tiny particles gain mass from that light, that message.
Another tactic is to duck or dodge the details and simply say that the Higgs boson gives mass to particles and thereby matter to everything from people to planets.
WHAT CERN IS ADDRESSING
A few questions (more than a few questions) are being looked into, you can say with quiet confidence. The following are some examples:
•What's with the Higgs boson?
•What's with matter and antimatter?
•Is the standard model substandard or hunky-dory?
•Do all known particles have supersymmetric partners? Or are they only good for one-night stands?
•Are there extra dimensions? Are Grandad's glasses there?
•Are there additional sources of quark-flavour mixing? If so, why won't Waitrose stock them?
•Are electromagnetism, the weak nuclear force and the strong nuclear force different manifestations of one universal, unified force? And is it with us?
•Why is the fourth fundamental force, gravity, so many orders of magnitude weaker than the other three fundamental forces? Can we finally be cleverer than Einstein?
•What are superstrings and has somebody tidied them away in a drawer?
•How humble should we be about our ignorance of so much of the universe? Not humble at all? Exactly.
One of the most controversial projects at Cern involved the neutrino. Enrico Fermi suggested this word (Italian for 'little neutral one') for particles that have no electrical charge, are subject to the weak nuclear force and are capable of passing through a block of lead which stretches from here to Jupiter. What role they may play in the picture of the universe as a whole is still up for grabs – or not, as the case may be. Pauli offered a bottle of Champagne for anyone who could prove that they existed. He paid up.
During 2011, experimentalists at Cern announced repeatedly that they had created many bunches of neutrinos and had sent them through 730km (454 miles) of rock to a giant detector at the INFN-Gran Sasso laboratory in Italy. They claimed that 15,000 separate measurements, spread out over three years, showed that the neutrinos arrived 60 billionths of a second faster than light would have, travelling unimpeded over the same distance.
Einstein had insisted that nothing can travel faster than the speed of light.
Perhaps the neutrinos had ignored the universe's absolute speed limit, like so many drivers before them, once they'd crossed the border into Italy.
Then in March 2012, in a repeat experiment, the neutrinos went fast, but not that fast. The timepieces were checked and found faulty, thus rather embarrassing the Swiss who are known to pride themselves on efficient timekeeping, although it was an Italian who led the research and who, almost at the speed of light, was asked to resign. Generally speaking, though, Cern is thriving, firm in the conviction that the day's not far off when the world will finally have an all-encompassing theory of everything and an equation that can be printed on a T-shirt. The novelist Martin Amis has suggested that such revelations lie 'five Einsteins away'. You, too, should develop a way of spouting smart-sounding stuff like that.
You are well within your rights to be somewhat sniffy about superstrings and M-theory, in the interests of science – and in a bitchily even-handed way.
SUPERSTRINGING US ALONG
Supporters of superstrings and M-theory believe that the ultimate truths are coming closer, courtesy of Cern, and that they have one or two Einsteins on the case. String theory and M-theory have been known to bring physicists the closest some will ever get to visible manifestations of emotion. As with all GUTs, TOEs and UFTs, they will answer questions about the beginning of time, the origin of the universe, whether there are multidimensional universes and the best way to make a cup of tea.
The idea is that the fruits of centuries of enthusiastic investigations – the totality of information about the universe – will be encapsulated in a single equation; and that if the right equipment was available, it could be seen that the ultimate fundamental entities are minuscule filaments, vibrating strings or small membranes. (When you say 'small', know that those strings look up at an atom as an atom looks up at the entire solar system.)
Superstrings are believed to thrum like the strings of a musical instrument. Instead of notes, the strings are supposed to produce streams of particles, and thus all of creation is somehow tuned into existence.
Michael Green and John H Schwarz had worked out string theory by the late 1960s and over the next decade and a half, they managed to sort out such mathematical inconsistencies as were obvious at the time. In 1984, before the days of email, they FedExed a letter about their precious strings to Edward Witten, a brilliant young physicist and an exceptional mathematician at the prestigious Institute for Advanced Study in Princeton. Witten was smitten. He was already an influential fellow and wires all around began to buzz about strings; they have buzzed unceasingly since.
For Witten and cohorts of other researchers, calculations have a better chance of attaching to vibrating thread-like entities than to minuscule point-like particles, which send their numbers into the dreaded world of infinities. Also, it is a great scientific insight that superstrings carry not just the strong force, but all forces. So superstring theory doesn't just propose that gravity is included. It requires it.
M-theory stands for mother of all strings, magic, mystery or membrane theory – though some of late have called it madness. It proposes that strings exist in various dimensions, as membranes or webs. The membranes whose dimensions exist as no dimension are called 'zerobrane'; the ones that have string-like dimensions are called 'one-brane'; those with dimensions like a bubble are called 'two-brane' and so forth. At one stage it was half-seriously suggested that the word 'p' for particle might establish a category called p-brane.
Imagine the kudos you can get when talking about extra dimensions. People will believe you can actually perceive them. You can't. They exist in the sums and, the theory goes, they exist (in effect) non-existently in real life.
Hypothetical, multidimensional regions are named Hilbert spaces after a great mathematician, David Hilbert. Anyone mentioning Hilbert spaces sounds impressively like a professional physicist; and to appear even more knowledgeable, it is worth lamenting the fact that they are not named after the mathematician John von Neumann (pronounced Noymun).
Von Neumann's master stroke in 1929 was to adapt Hilbert's ideas, providing physicists with spaces that have an infinity of dimensions. Pictures of Hilbert spaces tend to look like the tangle of proliferating pipes outside the Pompidou Centre in Paris – happily without the white-faced mimes and also without someone selling T-shirts saying 'My physicist friend went to a Hilbert space and all I got was a T-shirt saying, "My physicist friend went to a Hilbert Space and all I got was a T-shirt saying 'My Professor went to a Hilbert Space...etc.'
You are well within your rights to be somewhat sniffy about superstrings and M-theory, in the interest of science – and in a bitchily even-handed way.
You should be able to comment on a book by the American physicist Lee Smolin entitled _The Trouble with Physics_. He may as well have called it _The Trouble with Physicists,_ but you definitely shouldn't go there.
It used to be said that testing string theory would require 150 years and a microscope the size of a galaxy. Now, things are even worse. String theory cannot be tested at all, nor proved, nor falsified. So a growing number of brave scientists say it is hurting science. Too many hypotheses, too much hype.
Initially, string theory was mathematically elegant, even beautiful. Now, after decades of manipulating and mangling the figures, neither the numbers nor the universes they are seeking to represent are attractive, not even to motherly types.
Critics of string theory argue forcefully that its proponents have ladled into their bowl an indigestible gruel containing an extra six or seven or 26 dimensions which no one could ever see, trillions and trillions of other parallel universes which no one could ever visit, tunnels through space and time, and other creepy-crawly ideas taking string theory from being a theory of everything to being a theory, quite possibly, of nothing.
You're quite at liberty to use the expressions 'superstring theory' and 'string theory' interchangeably and you can certainly use the plural form. Without necessarily dismissing string theory entirely – a dangerous ploy – you might gently point out that, because of all those hypothetical universes and far-flung notions, string theories have proliferated to such an extent that there are now effectively more string theories than there are atoms in the unknown universe. Where calculations pertaining to string theory can be judged – and there are very few of them for science's liking – one prediction made by string theorists is so inaccurate that it would be like saying that a baseball is the size of the sun.
Tough as it may seem, you should be even-handed here and point out that at such a scale, dramatic deviations in the mathematics always occur. String theory is an entirely new science and by definition it breaks all the rules; the connections among those hypothetical strings are different, deep down, and so distant from the world around us that completely different approaches are required.
One prediction made by string theorists is so inaccurate that it would be like saying that a baseball is the size of the sun.
It is worth betting on more breakthroughs. It has had some astonishing successes and so an entirely new approach is required to understand it. There will be some spectacular surprises and if they could be described now, they wouldn't be surprises, would they?
Never forget that a number of geniuses support string theory. Gell-Mann called it the only game in town – some time ago, admittedly. Still, Green, Schwarz and Witten are not slouches and should not be opposed lightly. Top-class scientists tend not to work on failed ideas; no guns are pointed at them and there is no barbed wire between fields of physics. Real, true, heroic science involves discovery, wonder and being prepared to accept what seems unimaginable. It's not that string theorists love the scientific method last; they just love science more.
If you're brave enough to do so, you could hold fast to the many strong objections to string theory. Physics departments that look to other theories don't get funding; and yet string theory might have shattered the hopes and dreams of thousands of brilliant scientists and dedicated people in those departments. And there are powerful witnesses against these ideas, including no less a figure than Richard Feynman who remained disapproving and unconvinced. Also unimpressed is the renowned Nobel-Prize winner Sheldon Glashow, who penned some doggerel verse, enjoining us all to heed good advice, lest we all become smitten: 'The book is not finished, the last word is not Witten.'
SOKAL'S HOAX
There is a great deal of satisfaction to be had from this story because it goes to the heart of the Two Cultures discussion. You can maintain your superiority throughout your telling of it, giving yourself great boffin-like authority, if you show how ridiculous humanities numpties can be.
In 1994 a New York physics professor, Alan D Sokal, sent a paper to a hitherto respected cultural studies journal, _Social Text._ His piece, which was published in 1996, was deliberately full of scientific howlers. None of the academics who read the article had sufficient knowledge to correct his work and it was published. He showed that even lefties like himself would accept a paper that agreed with their own politics and would make it public, if it had the imprimatur of science.
Sokal made up a rambling title: 'Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity'. The piece itself was chock-full of statements that went way beyond the merely bonkers. Sokal talked of the 'external world, whose properties are independent of any individual human being and indeed of humanity as a whole'. Even first-year physics students would roar with laughter at his claim that 'the [Pi] of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity'. He pushed the gobbledygook on and on into Drivelsville: '...and the putative observer becomes fatally de-centered, disconnected from any epistemic link to a space-time point that can no longer be defined by geometry alone'.
Sokal followed up his paper with a book called _Fashionable Nonsense_ , written with the Belgian theoretical physicist Jean Bricmont. Here, the targets were authors in France, a veritable pantheon of contemporary French 'thinkers' whom you can safely and comfortably debunk. These swaggering figures include: Jean Baudrillard, Gilles Deleuze, Félix Guattari, Luce Irigaray, Julia Kristeva, Jacques Lacan, Bruno Latour and Jean-François Lyotard.
Lacan really did state that the erect male member is equal to the square root of minus one.
You should concede that they couldn't pin anything on the philosopher Jacques Derrida, who wrote a critique called 'Sokal and Bricmont Aren't Serious,' first published in the French newspaper _Le Monde._ He commented in passing that the Sokal hoax was rather 'sad [grave]', not only because Alan Sokal's name would now be linked primarily to a hoax, not to science, but also because the chance to reflect seriously on the issue had been lost to a broad public forum that deserved better.
Notwithstanding the lack of seriousness, each and every one of the overblown figures identified by Sokal and Bricmont were condemned in the most embarrassing way possible, by simply quoting verbatim what they themselves had written. They had tried to make themselves seem clever and to bolster any number of wayward world views and prejudices, but all those kings and queens were all together in the altogether.
Julia Kristeva really did write that 'poetic language can be theorised in terms of the cardinality of the continuum', Baudrillard did assert that 'modern war takes place in a non-Euclidean space', and Latour honestly did muse about what Einstein had learned from him.
These words really were written by notionally sentient beings: 'In the Euclidean space of history, the shortest path between two points is the straight line, the line of Progress and Democracy', as well as 'Is E=mc2 a sexed equation?' Lacan really did state that the erect male member is equal to the square root of minus one.
Clearly, like particles gaining mass from the Higgs boson, those who cannot measure up intellectually to physicists are trying to hoover some scientific credibility into themselves, while at the same time disparaging science for being too much of a bully. The suggestion is that science is an oppressive force. But truth does not oppress anyone. Science at its best liberates people. It is lies that oppress us.
You have a golden opportunity to ensure that science does not win hands down because of this scandal. First of all, in order to get his article published, Alan Sokal was quite happy to exploit his own reputation and authority, so his experiment was not 100% pure.
Secondly, thus far no one has dared to write such a thing, but it's a fair bet that a carefully composed and suitably zany theory in a certain scientific field would get peer reviewed and published.
The time is ripe for someone to write a barking-mad paper, as crazy as Sokal's, involving string theory.
FAMOUS PHYSICISTS
Whenever the conversation gets a bit sticky, it is well worth your while to have some snappy facts about the People Who Made the New Physics. This is especially helpful if no one else has heard of them and you can blind everyone with scientists.
JOHN BARDEEN (1908–1991)
John Bardeen is the only person in history to have received two Nobel Prizes in physics. Bardeen should be far more famous than any number of loud-mouthed scientific hype-meisters, but 'Whispering John' simply did his work and never bothered with boasting. At Bell Labs in New Jersey, the golf-crazy Bardeen and Walter Brattain, with contributions from William Shockley, succeeded in creating the first transistor.
**Maximum bluffing value (MBV)** In 1956, just before his first Nobel ceremony for that work, Bardeen had to borrow a shirt, his own being deemed unsuitable for a major occasion. To make things worse, when King Gustav of Sweden presented the award, he scolded Whispering John for leaving his children behind on such an important occasion.
In 1972, Bardeen won his second Nobel Prize for being in the group that pioneered work in the field of superconductivity. That time he brought the kids.
DAVID BOHM (1917–1992)
Recording chats with the Dalai Lama and Krishnamurti, Bohm was one of the holiest of the holists, and you need to be aware of him. He was sure that there is an underlying unity to all physical, psychological and spiritual experiences. This unity is as yet indivisible; he spoke often of 'hidden variables' – a notion that is gaining popularity among physicists.
For him, waves and particles were not always evident but were always present, the waves somehow guiding the particles. Bohm's calculations in the early 1950s led to later proofs of subatomic non-local quality, but then he lived a non-local life in the USA, the UK, South America and the Middle East.
**MBV** Bohm spent long periods of time living among the Blackfoot people in what reductionists call Canada. On an entirely different note, Einstein and Bohm, despite their age difference, were known to go out on double dates. (The best-known story of their love lives involves two sisters.)
NIELS BOHR (1885–1962)
A family man, happily married for 50 years to Margrethe. They had six sons. One died in childhood; one died in a boating accident; the other four went on to success in medicine, engineering, law and physics. The lawyer played hockey for Denmark and the physicist, Aage, won the Nobel Prize in 1975.
Bohr led the group of physicists who developed one of the greatest theories of all time – at his home. Heisenberg, Schrödinger, Fermi and many others came to stay with him and his family in Copenhagen, wore down the floor at either end of the ping-pong table, wrote letters home, munched on sandwiches and ran around with his laughing children.
Bohr, who was a member of the Danish underground during the Second World War, had a portentous walk with his German friend and colleague Heisenberg in 1941. No one knows exactly what was said, except that Bohr was extremely shocked and the friendship with Heisenberg ended there and then.
Possibly Heisenberg told Bohr of the Germans' plans for an atom bomb. Within months Bohr had contacted the Americans and was whisked to the USA. As a direct result of Bohr's alarm, work on the atom bomb was stepped up at Los Alamos.
**MBV** Bohr had an unfortunate way of speaking. Einstein's comment about Bohr's lack of ability to allow words to come out of his mouth properly was: 'He utters his opinions like one perpetually groping and never like one who believes himself to be in possession of the definite truth.' A fellow physicist once drew a cartoon of Bohr talking to a friend who is bound and gagged. Bohr is saying 'Please, please may I get a word in?'
MAX BORN (1882–1970)
Known as 'The Probability Man', the Polish-born Born was already pretty nifty with the old physics before he made his vast contribution to quantum theory by sorting out the mathematical formulations and practicalities of the ideas of Heisenberg and Schrödinger. You should say that it is his technique – the Born approximation – which is used by working physicists far more than the philosophical speculations of Heisenberg and Uncertainty.
You should also be scandalised that Born did not win the Nobel Prize until the day before he turned 72, and even then had to share it. It's worth mentioning that Einstein's famous statement about quantum mechanics – 'I cannot believe that God would play dice with the universe' – was made in a letter to his great friend Born.
**MBV** Born was Olivia Newton-John's grandfather.
LOUIS-VICTOR DE BROGLIE (1892–1987)
De Broglie (pronounced de Broy) devised a simple mathematical relationship connecting the wave and particle properties of matter.
Einstein had proposed the possibility that there were such things as matter waves in 1905, and in fact it was Einstein's enthusiastic response to de Broglie's thesis that made his name – another instance of Einstein's support for quantum mechanics, despite his loathing of the claims that were made for it.
**MBV** De Broglie's great-great-grandfather was a French aristocrat who died on the guillotine during the French Revolution's 'Reign of Terror'.
PAUL ADRIEN MAURICE DIRAC (1902–1984)
The word associated with Dirac is 'antimatter' which he was the first to suggest might exist. Of Anglo-Swiss parentage and born in Bristol, Dirac was Feynman's hero. He makes many physicists quite misty-eyed because he believed that calculations and theories should be endowed with beauty. His approach to mathematics was always elegant and simple. So was his approach to life.
**MBV** Dirac hated wasting words. Once asked if he took sugar, he answered 'Yes' and was surprised to be asked: 'How many lumps?' If he had wanted more than one lump, he as a mathematician would have specified the number.
FREEMAN JOHN DYSON (1923–)
Nobel-laureate Steven Weinberg has said that the Nobel committee 'fleeced' Dyson, who has won numerous scientific awards but has never won a Nobel Prize – although if there were prizes for infectious chortling he would be a shoo-in. Dyson was the first person to appreciate the power of Feynman diagrams – apart from Feynman himself, of course – and to present Feynman's work in a form that could be understood by other physicists. Dyson received a lifetime appointment at the Institute for Advanced Study in Princeton along with heartfelt thanks from J Robert Oppenheimer 'for proving me wrong'.
Two of Dyson's six children, Esther and George, write with intelligence and perception about modern technology, and Dyson himself has never ceased to comment on atoms, bytes and cells. Many see him as a climate change denier. Some say he simply insists on proper evidence-based arguments, as any hard-nosed scientist should.
**MBV** People notice that Dyson, an Englishman who has lived in the USA for over 50 years, speaks with a slight German accent. His wife is German.
PAUL EHRENFEST (1880–1933)
Einstein's great friend, Ehrenfest had tears rolling down his cheeks during Einstein's debate with Bohr. He forcefully reminded Einstein not to be as rigidly against quantum mechanics as people had been against relativity. Yet he himself called the quantum theorists 'Klugscheisser' – clever excrement.
Many have felt overwhelmed by quantum mechanics, but Ehrenfest's was an extreme case.
**MBV** The letter he sent before he committed suicide began: 'My dear friends Bohr, Einstein [and others]. In recent years it has become ever more difficult for me to follow developments [in physics] with understanding. After trying, ever more enervated and torn, I have finally given up in DESPERATION.'
ALBERT EINSTEIN (1879–1955)
Einstein was married twice (to Mileva and to his cousin, Elsa). His first child with Mileva was born before their marriage, and had to be given away because they were too poor to provide for it. His second son spent most of his life in a mental hospital.
Einstein worked alone. People who knew him well said that he was somehow always 'apart'. He said: 'I am not much good with people...I feel the insignificance of the individual and it makes me happy.' It is not surprising that he kept insisting that subatomic particles had to be thought of as separate entities – 'discrete'.
**MBV** Einstein died in the USA, far from his place of birth in Germany. Reporters jostling around the nurse who had been the only person present when he died were told that, yes, the great sage had spoken as he took his last breath. What did he say? 'I'm sorry,' said the nurse, 'I don't speak German.'
ENRICO FERMI (1901–1954)
Born in Rome, Fermi was rare in that he actually did his own experimental dirty work, even in the USA. The Fermi Award for innovation is still much coveted by physicists. Fermi built the first nuclear reactor.
**MBV** When he established the first chain reaction, Fermi made a phone call to a military colleague, saying calmly and in code, 'The Italian navigator has landed in the new world.' New world indeed.
RICHARD FEYNMAN (1918–1988)
Feynman's work on light and matter perfected what is possibly the most accurate theory ever developed in science. His famous diagrams show quantum interaction. You should always stress that although there is so much probability in quantum mechanics, there is also accuracy and precision.
Feynman (it rhymes with lineman) was a hugely popular figure, as much for his personality as for his mathematical wizardry. His diagrams were controversial because no one knew how he managed to derive them, but few are willing to argue for long about a drawing that saves hundreds of pages of algebra.
**MBV** A brilliant lecturer, Feynman enjoyed doing sums on paper napkins at nightclubs and was an accomplished picker of locks which, if you are dealing with atomic secrets and one of your best friends is the spy Klaus Fuchs (who gave US atomic secrets to the USSR), is a pretty risky business. Feynman had his kookier moments, such as his refusal, for some months, to brush his teeth or to wash his hands after urinating because of his ideas about germs. Late in life he became famous all over again for explaining why the Challenger space shuttle disaster happened: sealing rings became less resilient and were subject to failures at ice-cold temperatures in space.
MURRAY GELL-MANN (1929–)
Gell-Mann had an office across the way from Feynman at the California Institute of Technology in Pasadena in the 1950s, and was judged the more urbane of the two. He sprinkled his scientific work with literary and classical references, and ordered particles into arrangements or families which he called the eightfold way in honour of the Buddha.
Gell-Mann himself is not known for his tranquility. Indeed, he is a renowned curmudgeon. He hasn't begun to consider the notion that fools could be suffered and would find it unthinkable that anyone might do so gladly.
**MBV** Gell-Mann's most famous contribution is the 'quark' which he both discovered and named – insisting that it be pronounced 'quork', to rhyme with pork, after a line in _Finnegans Wake_ by James Joyce: 'Three quarks for Muster Mark!' (Quarks, like sailors in old musicals, go around in threes. But understanding Mark's relevance to the whole thing needs more enthusiasm than anyone can muster.)
STEPHEN HAWKING (1942–)
Hawking (together with Sir Roger Penrose) proved that the beginning of the universe was a 'singularity', a mathematical point of infinite density, the explosion of which was the Big Bang. Hawking fathered a subject whereby, instead of wrestling to bring all the forces together, the quantum-mechanical implications for gravity alone are studied. It's called quantum cosmology.
**MBV** Hawking loathes the idea of parallel universes. He once said, 'When I hear the words "Schrödinger's cat", I reach for my gun.'
WERNER HEISENBERG (1901–1976)
Famous for his uncertainty principle (published around his 26th birthday), Heisenberg is also notorious for having gained some advancement during the Hitler years in Germany. But no one quite understands what his role was because plans for a German atom bomb were abandoned by order of the Führer.
**MBV** Heisenberg's musings about the observer and the observed have been exploited far beyond his intentions, and you should scoff at the cavalier use of the word 'uncertainty', which is not an excuse to avoid marriage or filling in tax returns.
PETER WARE HIGGS (1929–)
It was with others that Higgs proposed the mechanism by which particles are endowed with mass by interacting with a field, which is carried by bosons. So it is truly remarkable – and an indication of the way science and our culture operates – that, famously, the mechanism is called the Higgs mechanism, the field is called the Higgs field, and the bosons are called Higgs bosons.
**MBV** His father was a sound man at the BBC.
LISE MEITNER (1878–1968)
Meitner was an Austrian, later Swedish, physicist who worked on radioactivity and nuclear physics. Meitner was part of the team that discovered nuclear fission, an achievement for which her colleague Otto Hahn was awarded the Nobel Prize. Meitner is often mentioned as one of the most glaring examples of women's scientific achievement overlooked by the Nobel committee.
**MBV** Meitner was taught by the legendary nineteenth-century physicist Ludwig Boltzmann and then by Max Planck, who until then had rejected any women wanting to attend his lectures. After one year, Meitner became Planck's assistant. Otto Hahn was also working for Planck.
WOLFGANG PAULI (1900–1958)
At the age of 20, Pauli wrote a 200-page encyclopaedia entry on the theory of relativity. It was his ideas that led to the discovery of the neutrino, but he is best known for his exclusion principle.
Pauli was excluded many times from bars for being pixillated. Even when sober, he had no trouble speaking his mind. To one student he said, 'Ach, so young and already you are unknown'; to another, 'That isn't even wrong'. He even put Einstein down for not seeing the difference between mathematics and physics. It was to Pauli that Bohr made the legendary remark: 'Your theory is crazy, but it's not crazy enough to be true.'
**MBV** Pauli, an Austrian, was also fascinated by subatomic particles and consciousness, collaborating for some time with psychologist Carl Jung, whose patient he was for a while. Their association may not have proved much, but it probably made them feel better.
Pauli, along with Feynman, had an obsession with the number 137, which is important in physics and in the ancient wisdom of the Kabbalah – the real thing (not Madonna's idea of Dolce and Kabbalah). The universe is 13.7 billion years old, the number 137 refers to the absorption of light by matter and – if you use a little mathematical magic – unites the electron charge, the speed of light and Planck's constant. In ancient tradition, 137 unites wisdom and prophecy as well as the spiritual and the material. When Pauli was taken ill in Zürich with pancreatic cancer in 1958, he was convinced his time had come when he was put in hospital room number 137. He was right.
Planck could never come to terms with quantum mechanics, of which he himself was a pioneer.
MAX PLANCK (1858–1947)
Famous for Planck's constant: the energy of a light wave is always proportional to its frequency. The mathematically minded marvel at the strange relationship between Planck's constant and Heisenberg's uncertainty principle. By multiplying Heisenberg's two uncertainties you actually get Planck's constant. Truth is stranger than fiction.
Planck's constant is represented by h, which has the value of 6.63 times 10 to the power of minus 34 Js (joule second). It is always 6.63 x 10 to the power of minus 34 Js, and the energy of a photon is h – or 6.63 times 10 to the power of minus 34 Js multiplied by f, where f is the frequency of the wave. It should be clear by now why jokes about being thick as a Planck are not such a great idea.
He could never come to terms with quantum mechanics, of which he himself was a pioneer.
**MBV** Planck's son Erwin was executed in 1945 for attempting to assassinate Hitler.
ERNEST RUTHERFORD (1871–1937)
Rutherford was a New Zealand-born physicist who sang 'Onward Christian Soldiers' loudly and out of tune all day and every day for the whole of his life. His loudness was legendary and his own researchers manufactured an electrical sign in their lab which read, 'TALK SOFTLY PLEASE'. Rutherford set down the groundwork for the development of nuclear physics by discovering the alpha particle, the nucleus and the proton. He was also wise enough to employ Niels Bohr, as well as Hans Geiger, who developed the Geiger counter as a result of sitting in the dark and totting up flashes of radiation.
**MBV** Rutherford contributed to the atom bomb by speaking so loudly and firmly against it that he infuriated a young Hungarian physicist called Leó Szilárd. Within the decade Szilárd was a leader of the Manhattan Project.
Rutherford used to say, 'In science there is only physics; everything else is stamp collecting.' In 1908, when he won the Nobel Prize, it was for chemistry.
ERWIN SCHRÖDINGER (1887–1961)
A brilliant physicist who stands out because of his extraordinary intellectual versatility, Schrödinger's numerous contributions to science include an extremely useful wave equation, and a handy and profound book about quantum physics and genetic structure.
**MBV** In 1944 Schrödinger published a landmark book, _What Is Life?_ Both James D Watson and Francis Crick, codiscoverers of the structure of DNA, credited Schrödinger's book with presenting an early theoretical description of how the storage of genetic information would work. The great physicist had become a pioneering molecular biologist.
Such a love of nature shouldn't lead you to think that, despite the fact that he is most famous for a dratted cat, he would actually own one.
WHAT IT ALL MEANS
Apart from the obvious fun that can be derived from believing impossible things for as much as half an hour a day, with six of them before breakfast, quantum mechanics is a big part of everyday life. It deals with the most basic of basic stuff. It is behind every chemical reaction, every biological and medical miracle. It underlies all of existence in some way.
And therein lies the controversy. For the holists, what is so powerful is the discovery of these fundamental interrelationships which permeate the entire universe and make it a cohesive (w)hole. They feel that Western civilisation should finally resign its obsession with dividing, compartmentalising and separating – an obsession that has governed intellectual activity since the ancient Greeks, and especially since Aristotle, whom they regard as the chief villain of the pieces.
They say it's time to stop dismembering and begin remembering. They hold divisive thinking responsible for all the world's splitting, clefting and rending; they say it's what causes revolutions, riots and wars.
The holists latch on to statements like that made by John Wheeler, an American physics professor who worked with some of the greats, including Bohr. Wheeler stated: 'Nothing is more important about quantum physics than this: it has destroyed the concept of the world as "sitting out there".' Wheeler was very firm: this is no longer 'a universe for observation'. It is a universe that requires, demands and expects participation.
Reductionists don't buy that. Imagine a performer in a theatre asking if anyone hates audience participation. A reductionist would fall into the trap and say, 'Yes! Me!'
THE CUTTING EDGE
There are two fashionable topics at the cutting edge, or even the outer limits, whose popularity has mushroomed as a result of quantum mechanics. The first is mysticism, the second is consciousness.
1. Mysticism
To reductionists, this means anything from wearing a saffron robe to having a fondness for breathing too deeply. To them there is no worse word.
Supreme caution must be exercised with mysticism in the presence of any member of the scientific establishment, unless you are tired of life.
2. Consciousness
The fact that it is considered to be no more than a possibility that consciousness is connected with phenomena in the quantum realm should not prevent you from pontificating about it. Sir Roger Penrose, sometime colleague of Stephen Hawking, goes farther. For him, it's a 'definite possibility'.
Evidence for such a convinced 'maybe' lies in the swiftness with which the human brain makes choices. For instance, sifting all the visual alternatives as the eye focuses on a single printed word takes the brain less than a tenth of a second; a standard computer would have to start calculating before time and the universe began, so you wouldn't want to be stuck behind one of them on a roundabout.
To reductionists, mysticism means anything from wearing a saffron robe to having a fondness for breathing too deeply.
But speculation about the act of observation takes the issue far beyond the ordinary miracle of the brain's astounding speed, beyond settling the fate of Schrödinger's cat. It takes it towards affecting reality itself, and even into the clutches of Bishop Berkeley's assertion in the eighteenth century that mind creates matter.
With the double-slit experiment, are the particles and waves merely sensitive to screens and other equipment? Or are they picking up messages from the physicist's brain?
John von Neumann, biologist George Wald and physicists David Bohm and Arthur Eddington all made statements to the effect that the universe is mind-stuff, but it is difficult to get out of one's head for long enough to prove it, even in this day and age. What's more, no scientist would risk making statements to that effect in an actual paper.
More and more, though, in popular science magazines, writers are venturing to suggest that a watched pot might well boil – and boil faster – if your mind asks it to, while the fridge defrosts, the furniture rearranges itself and the loft gets converted.
The softer option is to say that particles of matter and particles of mind may come into being together, but that in any case our self-awareness and what seems to be some kind of consciousness at the quantum level are in profound communication.
Consciousness is also non-local. No one can say where the mind is, or how far the effects of thought can reach. The anthropic principles (both strong and weak – _see_ 'Glossary') suggest that in Homo sapiens there are intellectual capacities that are there for a reason, that somehow human beings are obliged to help the universe through the next stage – although it is unclear whether this means giving it homework, or getting stoned with it and talking all night.
You can be supremely engaging and effective, without being too technical, by telling people about the messenger particles in the brain and how profoundly they reinforce the interconnectedness of everything.
Messenger particles – which by now you will be able to call 'bosons' without flinching – are the strongest, most basic forces of relationship in nature. If anything relates more fervently, it hasn't yet been found. Indeed, within the brain, all the forces are active, including gravity, which is nothing less than the force that holds the entire universe together.
Perhaps there are such things as thought particles. And perhaps they know that this is being supposed. Perhaps they supposed it first. Perhaps the supposing happened simultaneously. Just a thought.
TO BE CONTINUED...
If questions in the quantum realm go to the root of everything and the answers that physicists seek will reveal the secret to Life, the Universe and Everything, then it is easy to see why, whenever you raise the subject, you are firing the starting gun for a conversational marathon.
The quantum universe has different rules about endings and beginnings. Many physicists believe that the Big Bang was a quantum event, the explosion of a single particle out of which the universe ballooned.
Whether the universe is contracting or expanding – crunching down to a single particle or creating from each particle a potential universe – there is nothing to stop you proposing that the universe, which is said to have begun with a Big Bang, could eventually evolve, dissolve and resolve into a Little Whimper. How could that ever happen?
The question doesn't apply.
There's no point in pretending that you know everything about the quantum universe – nobody does – but if you've got this far and absorbed at least a modicum of the information and advice contained within these pages, then you will almost certainly know more than 99% of the rest of the human race about what it is, why it is, how it is, where it is, and why nobody has much idea about what's going on.
What you now do with this information is up to you, but here's a suggestion: be confident about your newfound knowledge, see how far it takes you, but above all have fun using it. You are now a bona fide expert in the art of bluffing about a subject that is so impenetrably complicated that the world's finest minds are still confounded by what it all means.
And remember: when in doubt, simply answer: 'They're looking into it at Cern.'
**Think you're ready to shine with your knowledge of the quantum universe? Test it first with our quiz atbluffers.com.**
**GLOSSARY**
**Anthropic principle** The theory that the universe has evolved so that human beings could come along. Because of this they have a purpose and their minds matter. The strong anthropic principle states that people matter a lot. The weak anthropic principle states that human beings matter quite a lot, especially if they understand the EPR paradox.
**Antimatter** Matter composed entirely of antiparticles.
**Atom** The smallest unit of any material which still retains the characteristics of that material. Obviously, if you go into the subatomic realm, anything can happen and probably will. The word 'atom' comes from the Greek meaning 'that which cannot be cut or split'. Right. And the Titanic was that which could not be sunk.
**Big Bang** Generally accepted notion that Time had a beginning, and an incredible amount of action during the opening credits.
**Billion** Big number which should be bandied about with abandon. The British billion is traditionally a million million. In physics, the American billion is used: a thousand million. Piffling in either case.
**Constant** A comforting predictable factor. The speed of light is a constant. So is the constant discovered by Max Planck. Useful in science and in life: buses are always late, they don't write songs like that any more, and money is never going to grow on trees.
**Dark Energy** A purely hypothetical form of energy. It is said to permeate all of space. Dark energy was put forward as an idea in the 1990s, when it seemed as if the expansion of the universe was not slowing because of gravity but accelerating because of, well, something. It has been ventured that roughly 68% of the universe is made up of it.
**Dark Matter** A purely hypothetical form of matter. Makes up about 27% of the universe. Add that to the 68% of the universe that is dark energy and you will see that we understand less than 5% of the universe. That man Rumsfeld with his 'known knowns' and 'unknown unknowns' was onto something after all.
**Electromagnetism** A weak force. Ultraviolet rays from the sun are electromagnetic. So are X-rays, microwaves in ovens and radio waves.
**Elementary particle** A particle without any internal structure which is therefore considered to be one of the basic building blocks of matter. In other words, the smallest thing there is. So far.
**Energy** Oomph, pizazz, vigour. In physics, as in life, it also means the 'capacity to do the work'.
**Gluons** Mega-strong messenger particles which bind everything together in the nucleus. Far more binding than marriage vows, contracts or even eggs.
**God** Among physicists, even though they work at the boundaries of the cosmos, please, please don't mention God. If God were actually to arrive during one of your discussions, berate Him for providing so little evidence.
**Gravity** The binding force of the universe, central to relativity, left out of the quantum dance. If anyone doubts gravity's existence, point to the ageing process.
**GUT** Grand unifying theory. A theory of everything which unites all the forces, the search for which goes on and on. On that, at least, physicists are united.
**Laser light amplification by stimulated emission of radiation** The emission is stimulated from an excited source. (Don't even think of making jokes about this.)
**Mechanics** Branch of applied maths dealing with motion and tendencies to motion. (Someone's got to do it.)
**Molecule** Two or more atoms in a 'bound system'. There are 92 naturally occurring substances (called elements) and another 17 man-made ones (so far), the atoms of which can combine to form molecules which ultimately form everything from guacamole to Guatemala.
**Nucleus** The dense core of an atom, made up of protons and neutrons. Nuclear energy and the nuclear bomb refer to this kind of nucleus. The nuclear family is so called because it, too, is a basic unit; to point out that families split, experience fallout or blow up is in very bad taste.
**Particle** Small part (sometimes the least possible part) of something. It is the central mystery of quantum mechanics that a particle sometimes behaves like a wave ( _see_ below).
**Probability** Bluffers need to know that a number of physicists prefer notions of probability to the use of the word 'uncertainty'. Physics graduates are much sought-after for jobs in the financial markets because they know about probabilities, waywardness and wildness from quantum mechanics. These prized whizz-kids are called 'quants'.
**Quantum** If you don't know now, you never will. But that doesn't mean you won't have a happy life.
**Superstring theory** Theory that minuscule vibrating strings will be, for certain, the smallest possible anythings in nature. A neat solution to bring the threads together? Reductionists are already sharpening their scissors.
**Time** Still not found in the quantum realm. Past and future particles just go jiggling and jiggling and jiggling along.
**Wave** A 'disturbance' like a ripple, a shiver or a vibration. It should be obvious to anyone – other than a physicist that of course a tiny particle could somehow connect to the outer reaches of the universe.
**ABOUT THE AUTHOR**
**JACK KLAFF** is a writer, performer and occasional academic. His first film was _Star Wars._ His first TV show featured Morecambe and Wise. His London stage debut, involving wild sex scenes, led to a season with the Royal Shakespeare Company. He has won awards for his work in all media; now that's bluffing on an astronomical scale. He has held four visiting professorships at Princeton University and was for three years Professor of the Public Understanding of Science at Starlab in Brussels. That's bluffing on an unfathomable scale. Jack has worked closely with Intelligence Squared, for whom he has interviewed a number of scientists, including aged Nobel-winning physicists, all of whom – unbidden – do brilliant, incomprehensible impressions of Niels Bohr. Among Jack's unusual performing gigs have been a talk at the European Parliament and his theatrical trilogy about male/female relationships – performed in Saudi Arabia. His next book, _Does Science Have all the Answers?_ could be even shorter than this one, containing just the word 'No'.
**NEW EDITIONS**
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\title{On fixed points of rational self-maps of complex projective plane}
\author{S. Ivashkovich}
\date{\today}
\address{
Universit\'e de Lille-1, UFR de Math\'ematiques, 59655 Villeneuve
d'Ascq, France} \email{ivachkov@math.univ-lille1.fr}
\address{IAPMM Nat. Acad. Sci. Ukraine
Lviv, Naukova 3b, 79601 Ukraine}
\subjclass{Primary - 37F10, Secondary - 32D20, 32H04}
\keywords{Fixed point, rational map, meromorphic map.}
\begin{document}
\begin{abstract}
For any given natural $d\geqslant 1$ we provide examples of rational self-maps of
complex projective plane ${\mathbb P}^2$ of degree $d$ without
(holomorphic) fixed points. This makes a contrast with the situation
in one dimension. We also prove that the set of fixed point free rational
self-maps of ${\mathbb P}^2$ is closed (modulo ``degenerate'' maps) in some natural
topology on the space of rational self-maps of ${\mathbb P}^2$.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\newsect[1]{Introduction} Along this note ${\mathbb P}^2$ stands for the complex
projective plane as well as ${\mathbb P}^1$ for the projective line. A
meromorphic self-map $f:{\mathbb P}^2\to{\mathbb P}^2$ can be viewed as a
holomorphic map $f:{\mathbb P}^2\setminus \{ \text{ isolated points } \}\to
{\mathbb P}^2$. Then the graph of $f$ extends to an analytic subvariety of
the product ${\mathbb P}^2\times{\mathbb P}^2$, this extension will be denoted as
$\Gamma_f$ and called the graph of the meromorphic mapping $f$. The
graph $\Gamma_f$ is then an algebraic subvariety by Chow's theorem
and consequently $f$ itself can be defined by a pair of rational
functions, \hskip1pt plus1pt{\sl i.e.\/,\ \hskip1pt plus1pt} $f$ is rational.
\smallskip A {\slshape \sffamily meromorphic} fixed point of $f$ is a point
$p\in{\mathbb P}^2$ such that $p\in f[p]$. Here by $f[p]$ one means the full
image of $p$ by $f$:
\[
f[p]\mathrel{\raise0.3pt\hbox{\rm:}{=}} \Gamma_f\cap \Big(\{p\}\times {\mathbb P}^2\Big),
\]
\hskip1pt plus1pt{\sl i.e.\/,\ \hskip1pt plus1pt} $f[p]$ can be a complex curve (such $p$ is, by definition, a
point of indeterminacy of $f$). By obvious homological reasons
$\Gamma_f$ intersects the diagonal $D$ in ${\mathbb P}^2\times{\mathbb P}^2$.
Therefore meromorphic fixed points for any $f:{\mathbb P}^2\to{\mathbb P}^2$ always
exist.
\smallskip In this paper we shall consider only {\slshape \sffamily holomorphic}
fixed points: a point $p\in{\mathbb P}^2$ is said to be a {\slshape \sffamily
holomorphic} fixed point of $f$, if $f$ is holomorphic in a
neighborhood of $p$ and $f(p)=p$. In what follows holomorphic fixed
points shall be called simply - fixed points. Let $X$ be a compact
complex manifold and $f:X\to X$ a meromorphic self-map. The
topological degree of $f$ is the number of preimages of a generic
point. The goal of this note is to prove the following:
\begin{thm}
\label{fixed} For any given integer $d\geqslant 1$ there exist rational
self-maps $f:{\mathbb P}^2\to{\mathbb P}^2$ of degree $d$ without holomorphic fixed
points.
\end{thm}
\smallskip One of the reasons for the interest in fixed points of
meromorphic maps lies in the attempt to understand what should be an
analog of a Lefschetz Fixed Point Formula in meromorphic case, see
ex. \cite{B} and \cite{BK}.
\medskip In Section 4 we define a natural topology on the space
${\cal M} ({\mathbb P}^2)$ of meromorphic self-maps of the complex projective
plane. Denote by $FFix ({\mathbb P}^2)$ the set of fixed point free
meromorphic self-maps of ${\mathbb P}^2$. By $DFix(X)$ the subset of ${\cal M}
({\mathbb P}^2)$ which consists of the maps with a {\slshape \sffamily curve} of fixed
points (\hskip1pt plus1pt{\sl i.e.\/,\ \hskip1pt plus1pt} in some sense they are degenerate maps). $DFix(X)$ is a
closed subset of ${\cal M} ({\mathbb P}^2)$. We prove the following:
\begin{thm}
\label{closed} The set $FFix({\mathbb P}^2)\cup DFix({\mathbb P}^2)$ is closed in
${\cal M} ({\mathbb P}^2)$.
\end{thm}
\noindent I.e., a sequence of fixed point free meromorphic mapping
converge either to a fixed point free meromorphic map, or to a map
with a curve of fixed points. Notice that by the Theorem \ref{fixed}
the set $FFix({\mathbb P}^2)\cup DFix({\mathbb P}^2)$ is a proper subset of ${\cal M}
({\mathbb P}^2)$.
\medskip\noindent{\slshape \sffamily Acknowledgement.} 1. I heard of the question
whether any rational self-map of ${\mathbb P}^2$ has a holomorphic fixed
point for the first time in the talk of J.-E. Fornaess on the
``Colloque en l'honneur de P. Dolbeault'' in Paris, June 1992.
\smallskip\noindent 2. I would like also to give my thanks to the
organizers and participants of the Conference ``Dynamique et
G\'eom\'etrie Complexe'' in Luminy, June 2009, where this note was
basically done, for the encouraging atmosphere and stimulating
discussions. I am especially grateful to E. Bedford for sending me
his Lecture Notes \cite{B}.
\newsect[sect2]{Examples in the product of projective lines}
We shall work first with ${\mathbb P}^1\times{\mathbb P}^1$ as a model of the
complex projective plane, because in this manifold the geometric
picture is particularly clear and formulas are simpler. Then we
shall transfer our examples to ${\mathbb P}^2$.
\newprg[prg2.1]{Equation for the fixed points}
From what was said above in the Introduction it is clear that in
order to produce mappings without fixed points one needs to produce
mappings $f$ such that $\Gamma_f$ intersects the diagonal $D$ in
$\big({\mathbb P}^1\times{\mathbb P}^1\big)\times \big({\mathbb P}^1\times{\mathbb P}^1\big)$ only
over the points of indeterminacy of $f$.
\smallskip
Every rational self-map $f:{\mathbb P}^1\times{\mathbb P}^1\to
{\mathbb P}^1\times{\mathbb P}^1$ can be written as $f=(f_1,f_2)$, where
$f_j(z_1,z_2)$, $j=1,2$, are rational functions of two complex
variables. A point $p=(z_1,z_2)\in{\mathbb P}^1\times{\mathbb P}^1$ is a fixed point
for $f$ if and only if both $f_1$ and $f_2$ are holomorphic in a
neighborhood of $p$ and
\begin{equation}
\eqqno(fix-pnt)
\begin{cases}
f_1(z_1,z_2) = z_1,\cr
f_2(z_1,z_2) = z_2.
\end{cases}
\end{equation}
When writing \eqqref(fix-pnt) we mean always affine coordinates. If
a fixed point $p$ has one or both of its coordinates equal to
$\infty$ then one should take an appropriate affine coordinates in a
neighborhood of $p$ and appropriately rewrite $\eqqref(fix-pnt)$.
But holomorphicity here means holomorphicity {\slshape \sffamily with values in
${\mathbb P}^1$}.
\begin{figure}[h]
\centering
\includegraphics[width=2.5in]{fixd.eps}
\caption{Curves $C_1$ and $C_2$ are defined by equation \eqqref(fix-pnt).
Points of intersection of these curves are the fixed points of our map unless
they are indeterminacy points at the same time. Points, marked by asterisks,
are indeterminacy points. Therefore, on this picture $a$ is a fixed point,
but $b$, $c$ and $d$ are not.}
\label{fix-fig}
\end{figure}
\smallskip
Set $C_1\mathrel{\raise0.3pt\hbox{\rm:}{=}}\{ (z_1,z_2)\in{\mathbb P}^1\times{\mathbb P}^1: f_1(z_1,z_2)=z_1\}$
and $C_2\mathrel{\raise0.3pt\hbox{\rm:}{=}}\{ (z_1,z_2)\in{\mathbb P}^1\times{\mathbb P}^1: f_2(z_1,z_2)=z_2\}$.
These are complex curves and their intersection $C_1\cap C_2$
contains all fixed points of $f$. More precisely, if by $I(f)$ we
denote the indeterminacy set of $f$ then $Fix(f) = (C_1\cap
C_2)\setminus I(f)$. Here $Fix(f)$ stands for the set of
(holomorphic) fixed points of $f$. Remark, finally, that a point
$p=(z_1,z_2)$ is an indeterminacy point of $f$ if it is an
indeterminacy point of at least one of $f_1$ or $f_2$.
\medskip
Let us try to find rational functions $f_2$ such that for
$f_1(z_1,z_2) = \frac{z_2}{z_1}$ the rational map $f=(f_1,f_2)\in
Rat({\mathbb P}^1\times{\mathbb P}^1)$ has no fixed points. Write $f_2(z_1,z_2) =
\frac{P(z_1,z_2)}{Q(z_1,z_2)}$. Then $C_1=\{z_2=z_1^2\}$ and
$C_2=\{P(z_1,z_2)=z_2Q(z_1,z_2)\}$. Therefore
\begin{equation}
\eqqno(z-z2)
Fix(f) = \{ (z,z^2)\in{\mathbb P}^1\times{\mathbb P}^1: P(z,z^2)=z^2Q(z,z^2)\}
\setminus I(f).
\end{equation}
Remark that in order for $Fix(f)$ to be an empty set any solution
$\lambda$ of \eqqref(z-z2) should be either $0$ or $\infty$ or,
otherwise, $\lambda$ should be the root of both polynomials
$P(z,z^2)$ and $Q(z,z^2)$. More precisely the following is true:
\begin{lem}
\label{lem-z-z2} Let $P$ and $Q$ be relatively prime and suppose
that every non-zero root of $P(z,z^2)-z^2Q(z,z^2)$ is the root of
both $P(z,z^2)$ and $Q(z,z^2)$. Then the map
\[
f : (z_1,z_2) \to \Big( \frac{z_2}{z_1},\frac{P(z_1,z_2)}{Q(z_1,z_2)}\Big)
\]
has no fixed points in ${\mathbb P}^1\times{\mathbb P}^1$.
\end{lem}
\state Proof. \2 Suppose that $p=(z_1,z_2)$ is a fixed point of $f$. Then
$p\not= (0,0),(\infty,\infty)$ and $z_2=z_1^2$, \hskip1pt plus1pt{\sl i.e.\/,\ \hskip1pt plus1pt} $p=(z,z^2)$ for
some non-zero complex number $z$. In addition we have that
$P(z_1,z_2)=z_2Q(z_1,z_2)$ and this implies that
$P(z,z^2)-z^2Q(z,z^2)=0$. Therefore $z$ is the root of both
$P(z,z^2)$ and $Q(z,z^2)$ by the assumption of the Lemma. But that
means that our point $p=(z,z^2)$ belong to the zero divisor of both
$P$ and $Q$. Since they are relatively prime, the point $p$ is an
indeterminacy point of $f_2=\frac{P}{Q}$.
\smallskip\ \ \hfill\hbox to .1pt{}\hfill\hbox to .1pt{}\hfill $\square$\par
\begin{rema} \rm
Two polynomials $P(z_1,z_2)$ and $Q(z_1,z_2)$ are relatively prime
if their zero divisors do not have common component. This is easily
checked when $P$ and $Q$ are simple enough.
\end{rema}
\newprg[prg2.2]{Examples}
Now let us give a list of examples following from Lemma \ref{lem-z-z2}.
Let's start, for the sake of clarity, with low degrees.
\begin{exmp} \rm
\label{exmp1}
Consider the map
\begin{equation}
f(z_1,z_2) = \Big( \frac{z_2}{z_1} , \frac{z_1^2-1}{z_2-1}\Big).
\end{equation}
\end{exmp}
In this case $P(z_1,z_2) = z_1^2-1$ and $Q(z_1,z_2) = z_2-1$. $P$
and $Q$ are obviously relatively prime. Moreover, polynomials
$P(z,z^2)= z^2-1$ and $Q(z,z^2)= z^2-1$ do satisfy the condition of
Lemma \ref{lem-z-z2}. Really: $z^2-1-z^2(z^2-1)= -(z^2-1)^2$.
Therefore this map has no fixed points. The degree of $f$ is $2$,
the indeterminacy points are: $(0,0), (\infty , \infty), (\pm 1,1)$.
\begin{exmp} \rm
\label{exmp2}
Consider the map
\begin{equation}
f(z_1,z_2) = \Big( \frac{z_2}{z_1} , \frac{(z_1-1)(z_1-2)(3z_1-2)}{z_2-3z_1 +2}\Big).
\end{equation}
\end{exmp}
In this case $P(z_1,z_2) = (z_1-1)(z_1-2)(3z_1-2)$ and
$Q(z_1,z_2) = z_2 - 3z_1 +2$. Therefore $P(z,z^2) = (z-1)(z-2)(3z-2)$,
$Q(z,z^2) = z^2 - 3z +2 = (z-1)(z-2)$ and $P(z,z^2)-z^2Q(z,z^2) =
(z-1)(z-2)(3z-2-z^2)=-(z-1)^2(z-2)^2$. Condition of
Lemma \ref{lem-z-z2} is again satisfied. The degree of $f$ is $3$, the
indeterminacy points are: $(0,0), (\infty , \infty),
(1,1), (2,4)$ and $(\frac{2}{3}, 0)$.
\medskip Now let us give examples in all degrees.
\begin{exmp} {\slshape \sffamily (Even degrees).} \rm
\label{exmp3}
Let $(a_1,b_1),...,(a_d,b_d)$ be such pairs of complex numbers that quadratic
polynomials $(z-a_1)(z-b_1),...,(z-a_d)(z-b_d)$
are pairvise relatively prime. Consider the rational map
\begin{equation}
f(z_1,z_2) = \Big( \frac{z_2}{z_1} , \frac{(z_1^2-1)\prod_{j=1}^d(z_1-a_j)
(z_1-b_j)}{\prod_{j=1}^d[z_2-(a_j+b_j)z_1 + a_jb_j]}\Big).
\end{equation}
Polynomials $P(z_1,z_2) = (z_1^2-1)\prod_{j=1}^d(z_1-a_j)(z_1-b_j)$ and
$Q(z_1,z_2) = \prod_{j=1}^d[z_2-(a_j+b_j)z_1 + a_jb_j]$
are obviously relatively prime (think about their zero divisors). To
check Lemma \ref{lem-z-z2} write
\[
P(z,z^2) - z^2Q(z,z^2) = (z^2-1)\prod_{j=1}^d(z-a_j)(z-b_j) -
z^2\prod_{j=1}^d[z^2-(a_j+b_j)z + a_jb_j] =
\]
\[
= \prod_{j=1}^d(z-a_j)(z-b_j)(z^2-1-z^2) = - \prod_{j=1}^d(z-a_j)(z-b_j).
\]
Therefore the condition of Lemma \ref{lem-z-z2} is satisfied. The degree
of $f$ is $2d+2$.
\end{exmp}
\begin{exmp} {\slshape \sffamily (Odd degrees).} \rm
\label{exmp4}
Let $(a_1,b_1),...,(a_d,b_d)$ be such as in Example \ref{exmp3} and, in
addition, such that quadratic polynomials $(z-a_1)(z-b_1),...,(z-a_d)(z-b_d)$
are relatively prime with $(z-1)(z-2)$. Consider the rational map
\begin{equation}
f(z_1,z_2) = \Big( \frac{z_2}{z_1} , \frac{(z_1-1)(z_1-2)(3z_1-2)
\prod_{j=1}^d(z_1-a_j)(z_1-b_j)}
{(z_2-3z_1 +2)\prod_{j=1}^d[z_2-(a_j+b_j)z_1 + a_jb_j]}\Big).
\end{equation}
Polynomials $P(z_1,z_2) = (z_1-1)(z_1-2)(3z_1-2)\prod_{j=1}^d(z_1-a_j)(z_1-b_j)$
and $Q(z_1,z_2) = \prod_{j=1}^d[z_2-(a_j+b_j)z_1 + a_jb_j]$
are again relatively prime and
\[
P(z,z^2) - z^2Q(z,z^2) = (z-1)(z-2)(3z-2-z^2)\prod_{j=1}^d(z-a_j)(z-b_j) =
\]
\[
= - (z-1)^2(z-2)^2\prod_{j=1}^d(z-a_j)(z-b_j),
\]
where $P(z) = (z-1)(z-2)(3z-2)\prod_{j=1}^d(z-a_j)(z-b_j)$ and $Q(z,z^2) =
(z-1)(z-2)\prod_{j=1}^d(z-a_j)(z-b_j)$.
Therefore the condition of Lemma \ref{lem-z-z2} is again satisfied. The
degree of $f$ is $2d+3$.
\end{exmp}
\newprg[prg2.3]{More examples}
One can produce in a similar way other examples. For example one may
take as $f_1(z_1,z_2) = \frac{z_2}{z_1^k}$ and then look for
$f_2(z_1,z_2) = \frac{P(z_1,z_2)}{Q(z_1,z_2)}$ with the condition
that $P$ and $Q$ are relatively prime and every non-zero root of
$P(z,z^{k+1}) - z^{k+1}Q(z,z^{k+1})$ should be a root of both
$P(z,z^{k+1})$ and $Q(z,z^{k+1})$. With such $f_2$ the map
$f=(f_1,f_2)$ will not have fixed points. Its degree will be at
least $k \text{ plus degree of } f_2 \text{ in } z_2$.
\begin{exmp} \rm
Consider the following map
\[
f : (z_1,z_2) \to \Big(\frac{z_2}{z_1^k}, \frac{z_1^{k+1} -1}{z_2-1}\Big).
\]
Then
\[
P(z,z^{k+1}) - z^{k+1}Q(z,z^{k+1}) = -(z^{k+1}-1)^2
\]
and therefore $f$ has no fixed points. Its degree is $k+1$.
\end{exmp}
\begin{exmp} \rm
One can start with $f_1(z_1,z_2)= \frac{z_2^{k}}{z_1^{k-1}}$. Then
the condition for a coprime $P, Q$ to define a map
$f=(f_1,\frac{P}{Q})$ without fixed points is this: for every
$k$-root of $1$ (denote it by $\zeta_l$, $l=1,...,k$), every
non-zero $z$, which satisfies $P(z,\zeta_l z) - \zeta_l zQ(z,\zeta_l
z)=0$, should also be the root of both $P(z,\zeta_l z)$ and
$Q(z,\zeta_l z)$. The following map will do the job:
\[
f: (z_1,z_2) \to \Big(\frac{z_2^{k}}{z_1^{k-1}}, \frac{z_1^k - 1}{z_2^k - 1}\Big).
\]
\end{exmp}
\newsect[sect3]{Transfer to the complex projective plane}
Now let us explain how to translate our examples from
${\mathbb P}^1\times{\mathbb P}^1$ to ${\mathbb P}^2$. This task is not completely obvious
because of rationality (\hskip1pt plus1pt{\sl i.e.\/,\ \hskip1pt plus1pt} non-holomorphicity) of this transfer.
Fix some point $p=(p_1,p_2)\in{\mathbb P}^1\times{\mathbb P}^1$ and denote as
$l_1\mathrel{\raise0.3pt\hbox{\rm:}{=}} \{p_1\}\times{\mathbb P}^1$ and $l_2\mathrel{\raise0.3pt\hbox{\rm:}{=}} {\mathbb P}^1\times \{p_2\}$ the
``vertical'' and ``horizontal'' lines passing through $p$. Blow up
$X\mathrel{\raise0.3pt\hbox{\rm:}{=}}{\mathbb P}^1\times{\mathbb P}^1$ at $p$ and then blow down the strict
transforms of $l_1$ and $l_2$. The obtained surface we denote as
$\hat X_p$. This $\hat X_p$ is biholomorphic to ${\mathbb P}^2$. If $f:X\to
X$ is a rational self-map of $X$ then it naturally lifts to a
rational self-map $\hat f_p:\hat X_p\to\hat X_p$ of $\hat X_p$.
\begin{prop}
\label{transfert} Let $f:{\mathbb P}^1\times{\mathbb P}^1\to{\mathbb P}^1\times{\mathbb P}^1$ be a
rational map without fixed points. Take a regular point point
$p=(p_1,p_2)$ of $f$ such that:
\smallskip
{\sl i)} } \def\slip{{\sl i)} all $\{q_1,...,q_d\}\mathrel{\raise0.3pt\hbox{\rm:}{=}} f^{-1}(p)$ and $s\mathrel{\raise0.3pt\hbox{\rm:}{=}} f(p)$ are also
regular points of $f$;
{\sl i$\!$i)} } \def\sliip{{\sl i$\!$i)} $l_1$ and $l_2$ do not intersect neither $\{s,q_1,...,q_d\}$
no $I(f)$ and are not contracted by $f$.
\noindent Then $\hat f_p$ has no fixed points to.
\end{prop}
\state Proof. \2 Denote by $E_p$ the exceptional divisor of the blow-up, by
$X_p$ the obtained surface and by $\pi_0 : X_p\to X$ the
corresponding blow-down map which sends $E_p$ to $\{p\}$. Lift $f$
to $X_p$ and denote by $f_p:X_p\to X_p$ the lifted map, \hskip1pt plus1pt{\sl i.e.\/,\ \hskip1pt plus1pt}
$f_p\mathrel{\raise0.3pt\hbox{\rm:}{=}} \pi_0^{-1}\circ f\circ \pi_0$.
\medskip\noindent{\slshape \sffamily Claim 1. {\it $f_p$ has no fixed points.}}
By $q_j$ denote the preimages of $q_j$ for $j=1,...,d$ and by $s$
that of $s$. Likewise denote by $I(f)$ the indeterminacy set of $f$
as well as its preimage under $\pi_0$. Then $I(f_p) = \{I(f);
q_1,..., q_d\}$. Really, $\pi_0$ is biholomorphic near every point
from $I(f)$ and therefore it remains an indeterminacy point also for
$f_p$. As for, say $q_1$, the map $f$ sends it to $p$ and
$\pi_0^{-1}$ blows it up. So $q_1$ becomes to be an indeterminacy
point of $f_p$. For any other point $r\in X_p$ both $\pi_0$ is
regular at $r$ and $f$ is regular at $\pi_0(r)$ and
$f(\pi_0(r))\not=p$. Therefore $\pi_0^{-1}$ is regular at
$f(\pi_0(r))$.
\smallskip Suppose $r\in X_p$ is a fixed point for $f_p$. Then
$r\not\in \{I(f); q_1,..., q_d\}$. If in addition $r\not\in E_p$
then both $r$ and $f_p(r)$ belong to the domain where $\pi_0$ is
biholomorphic. Therefore $f_p(r)=r$ would imply $f(\pi_0(r)) =
\pi_0(r)$ and this is not the case. The only case left is $r\in
E_p$. But hen $\pi_0$ sends $r$ to $p$ and $f$ further to $s\not=p$.
Finally $\pi^{-1}(s)\not\in E_p$ and we are done.
\medskip Let $\pi_1:X_p\to X_1$ be the blow-down of $l_1$ and let
$f_1\mathrel{\raise0.3pt\hbox{\rm:}{=}} \pi_1\circ f_p\circ \pi_1^{-1}$ be the pulled down map.
Set $s_1=\pi_1(l_1)$.
\medskip\noindent{\slshape \sffamily Claim 2. {\it $f_1$ has no fixed points
as well.}} Start form $s_1$, which is the only ``new'' point in
$X_1$. It is an indeterminacy point for $f_1$. Really,
$\pi_1^{-1}(s_1)=l_1$ and $f_p(l_1)$ is not a point because $f(l_1)$
is not a point by the assumption. At the same time
$f_p(l_1)\not\subset E_p$, because again $f(l_1) = \pi_0(f(l_1))$
should not be a point. Therefore $s_1$ is not a fixed point of
$f_1$.
Take any other point $r\in X_1$. $\pi_1$ is biholomorphic near $r$
and therefore would $r$ be a fixed point for $f_1$ the point
$\pi_1^{-1}(r)$ would be a fixed point for $f_p$. Which cannot
happen according to the Case 1.
\medskip Let $\pi_2:X_1\to X_2$ be the blow-down of $l_2$ and let
$f_2\mathrel{\raise0.3pt\hbox{\rm:}{=}} \pi_2\circ f_1\circ \pi_2^{-1}$ be the pulled down map.
Set $s_2=\pi_2(l_2)$.
\medskip\noindent{\slshape \sffamily Claim 3. {\it $f_2$ has no fixed points to.}}
The proof is the same as for Claim 2. But $X_2\equiv {\mathbb C}{\mathbb P}^2$ and
we are done.
\smallskip\ \ \hfill\hbox to .1pt{}\hfill\hbox to .1pt{}\hfill $\square$\par
\medskip The relevance of this Proposition to our task is clear: for
a given map $f$ a generic choice of $p$ will satisfy conditions
({\sl i)} } \def\slip{{\sl i)}, ({\sl i$\!$i)} } \def\sliip{{\sl i$\!$i)} of Proposition \ref{transfert} and therefore the
lifted map $\hat f_p$ will be also without fixed points if such was
$f$.
\newsect[sect4]{Closure of the set of fixed point free rational
maps}
In Theorem \ref{fixed} we proved that the set of fixed point free
rational self-maps of ${\mathbb P}^2$ is not empty. Now we are going to
prove that it is also closed modulo degenerate maps.
\newprg[prgCLOS.topol]{Topology on the space of rational maps}
Possible notions of convergence on the space of meromorphic mappings
${\cal M} (X,Y)$ between complex manifolds (or, spaces) are discussed
in \cite{I}. For us in this paper an appropriate one is the
following. Let $\{ f_n \}$ be a sequence of meromorphic maps from
complex manifold into $X$ a complex manifold $Y$.
\begin{defi}
\label{conv} We say that $f_n$ {\slshape \sffamily strongly converges} on
compacts in $X$ to a meromorphic map $f : X \to Y$ if for any
compact $K\Subset X$
\begin{equation}
{\cal H}- \mathop{\sf{lim}}_{n\rightarrow \infty}\Gamma_{f_n}\cap (K\times Y) =
\Gamma_f\cap (K\times Y).
\end{equation}
\end{defi}
\noindent Here by ${\cal H}-\mathop{\sf{lim}} $ we denote the limit in the Hausdorff
metric, supposing that both $X$ and $Y$ are equipped with some
Hermitian metrics. Remark that this notion of convergency doesn't
depend on a particular choice of metrics. We say that $f_n$ {\slshape \sffamily
converge weakly}, if their graphs converge i Hausdorff metric, see
\cite{I} for more details about relation between strongly and weakly
converging sequences of meromorphic mappings.
\smallskip The definition of strong convergence agrees well with the
usual notion of convergence of holomorphic mappings. Namely, in
\cite{I} we proved the following:
\begin{thm} {\slshape \sffamily (Rouch\'e principle).} Let a sequence of meromorphic
mappings $\{ f_n\} $ between normal complex spaces $X$ and $Y$
converge strongly on compacts in $X$ to a meromorphic map $f$. Then:
\smallskip
{\sl i)} } \def\slip{{\sl i)} If $f$ is holomorphic then for any relatively compact open
subset $X_1\Subset X$ all restrictions $f_n\mid_{X_1}$ are holomorphic
for $n$ big enough, and $f_n \longrightarrow f$ on compacts in $X$
in the usual sense (of holomorphic mappings).
{\sl i$\!$i)} } \def\sliip{{\sl i$\!$i)} If $f_n$ are holomorphic then $f$ is also holomorphic and
$f_n \longrightarrow f$ on compacts in $X$.
\end{thm}
\newprg[prgCLOS.proof]{Proof of Theorem \ref{closed}}
In the case of compact $X$ the notion of strong convergence (and
corresponding topology on the space ${\cal M} (X)$) is also well
adapted for understanding the structure of the space of meromorphic
self-maps of $X$. We shall not discuss here the general case any
more, but just turn to our case $X={\mathbb P}^2$ (or, $X={\mathbb P}^1\times
{\mathbb P}^1$). The space ${\cal M} ({\mathbb P}^2)$ naturally splits into a double
sequence $\{{\cal M}_{d,s}\}_{d=1,s=0}^{\infty}$ of subspaces, indexed
by the degree $d$ and the ``skew-degree'' $s$. Each ${\cal M}_{d,s}$ is
infinite dimensional. If we equip ${\mathbb P}^2$ with the Fubini-Studi
metric form $\omega$ of total volume one then
\begin{equation}
\eqqno(integrals)
d = \int\limits_{{\mathbb P}^2}(f^*\omega)^2 \qquad \text{ and } \qquad s =
\int\limits_{{\mathbb P}^2}\omega\wedge f^*\omega .
\end{equation}
The volume of the graph of $f$ will be $\sf{Vol} (\Gamma_f) = d+1+2s$.
Now if $\{f_n\}\subset{\cal M}_{d,s}$, then by Bishop compactness
theorem we know that some subsequence $\{\Gamma_{f_{n_k}}\}$
converge to an analytic set $\Gamma$. This $\Gamma$ naturally
decomposes as the union of compact components: $\Gamma =
\Gamma_f\cup \Gamma_{\delta}\cup \Gamma_s$, where $\Gamma_f$ is a
graph of some meromorphic $f$, $\Gamma_{\delta}$ projects to points
and $\Gamma_s$ to curves under the natural projection
$\pi_1:{\mathbb P}^2\times{\mathbb P}^2\to {\mathbb P}^2$ onto the source. Now $\sf{Vol}
(\Gamma_{\delta})$ contributes to the first integral in
\eqqref(integrals), \hskip1pt plus1pt{\sl i.e.\/,\ \hskip1pt plus1pt} to the degree and $\sf{Vol} (\Gamma_{s})$ to the
second, \hskip1pt plus1pt{\sl i.e.\/,\ \hskip1pt plus1pt} to the skew-degree. Therefore if the Hausdorff limit of
$\Gamma_{f_{n_k}}$ is different from $\Gamma_f$ our subsequence
diverge from ${\cal M}_{d,s}$. Summing up we conclude that each
${\cal M}_{d,s}$ is closed in the strong topology we introduced (and is
relatively compact in the weak one).
\medskip Now let us prove the Theorem \ref{closed} from the Introduction.
Denote by $Fix ({\mathbb P}^2)$ the set of rational self-mappings of ${\mathbb P}^2$
admitting at least one fixed point. Let $f_n$ have no fixed points
and converge strongly to $f$. First of all we remark that Theorem 2
from \cite{I} implies that a strongly converging sequence has
bounded volume. From this fact and from what was said above we see
that $f_n$ for $n>>1$ and $f$ belong to the same component
${\cal M}_{d,s}$. If $f\in Fix({\mathbb P}^2)\setminus DFix({\mathbb P}^2)$ then
$\Gamma_f$ intersects the diagonal $D$ by an isolated point ${\bf
p}=(p,p)$. But then $f_n$, converging to $f$, should be holomorphic
in a neighborhood of $p$ and converge to $f$ as holomorphic
mappings by Rouch\'e's Principle. Therefore they should have their
graphs intersect $D$ by the standard version of Rouch\'e theorem for
vector functions. Contradiction.
\medskip Let us end up with an explicit example when a sequence without
fixed points converge to a map with a curve of fixed points.
\begin{exmp} \rm Take $\theta_n$ to be irrational modulo $2\pi$ and
converging to zero. Take the following self-maps of ${\mathbb P}^2$:
$f_n:[z_0:z_1:z_2] \to \Big[z_0z_1:z_0z_2:e^{i\theta_n}z_1z_2\Big]$.
Then it is straightforward to check that $f_n$ have no fixed
points. But the limit map $f_n:[z_0:z_1:z_2] \to \Big[z_0z_1:z_0z_2:
z_1z_2\Big]$ has the curve $\{ z_2z_0=z_1^2\}$ of fixed points.
\end{exmp}
\ifx\undefined\leavevmode\hbox to3em{\hrulefill}\,
\newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
\def\entry#1#2#3#4\par{\bibitem[#1]{#1}
{\textsc{#2 }}{\sl{#3} }#4\par\vskip2pt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,071 |
Q: Calculate percentile for every column in a data frame in R I have a data set of 3 categorial columns and 40 columns with numerical values. I want to calculate the 90th percentile for each of the 40 numerical columns separetly.
Take this data frame as a reproducible example:
fruit = c("apple","orange","banana","berry") #1st col
ID = c(123,3453,4563,3235) #2nd col
price1 = c(3,5,10,20) #3rd col
price2 = c(5,7,9,2) #4th col
price3 = c(4,1,11,8) #5th col
df = data.frame(fruit,ID,price1,price2,price3) #combine into a dataframe
I want to do something like: calc_percentile = quantile(df[,3:5], probs = 0.90)
The output I'm looking for would be:
# Column 90thPercentile
# price1 17
# price2 8.4
# price3 10.1
Doing this one by one is not practical given that I have 40 columns. Your help is appreciated!
A: stack(lapply(df[3:5], quantile, prob = 0.9, names = FALSE))
# values ind
#1 17.0 price1
#2 8.4 price2
#3 10.1 price3
A: Using dplyr and tidyr:
df %>%
summarise_at(3:5, ~ quantile(., probs = 0.9)) %>%
gather("Column", "90thPercentile")
Column 90thPercentile
1 price1 17.0
2 price2 8.4
3 price3 10.1
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,118 |
\section{Introduction}
The knowledge of correlations is important and interesting for any
system. Looking from the practical point of view correlation means
additional information: if two quantities are correlated,
the knowledge of one of them implies certain information about the other one. In
physical systems correlations usually indicate interactions
between parts of the system. The prototypical example is given by the
Ising spin system where the nearest-neighbors interactions induce
long-range correlations leading to a phase transition.
The situation in random graphs is somewhat different. It is known that
for random geometries even in the absence of any explicit terms
inducing interactions between vertices their degree may be
correlated \cite{bs,b1,b2,b3,Satorras2001,msz,pn,bo,kim}. Moreover, those
correlations are long-range, i.e., they fall off as some power of
distance \cite{bs,b1,b2,b3}. They are generated by model
constraints rather than by explicit interactions. It should be also
stressed that the distance dependent correlation functions in the
ensemble of random graphs are much more complicated objects than their
fixed lattice counterparts \cite{bs,b2}. To see that let us take some
generic correlation function on random graphs
\begin{equation}
\av{\sum_{i,j}A({q_i}) B({q_j})\,\delta_{l,|i-j|}},
\end{equation}
where $q_i$ denotes the {\em degree} of the vertex $i$, i.e., the
number of branches emerging from it. $A$ and $B$ are some arbitrary
functions depending on the vertex degree and $|i-j|$ is the graph
(geodesic) distance between vertices $i$ and $j$. For random geometry
it makes no sense in general to choose two fixed points---that is why
we sum over all the pairs of points on the graph. The graph distance
is the length of the shortest path between those two vertices and as
such it is dependent on the whole graph. That means that the above
expression is not a two-point function but a highly nonlocal
object. That is a fundamental difference between random and fixed
geometries.
In this paper, we study in detail correlations between degrees
of vertices as a function of distance. We consider an ensemble of
all labeled trees ${\mathcal T}(V)$ with a fixed number of vertices $V$, on
which we define the probability measure
\begin{equation}\label{eq:PT}
P(T)\equiv\Omega_V^{-1}\frac{1}{V!}\prod_{i\in T}w_{q_i}.
\end{equation}
$\Omega_V$ denotes the partition function of this ensemble
(normalization factor) and $q_i$ is the degree of vertex $i$; $w_{q}$'s $(q>0)$ are some
non-negative numbers (weights).
This is a maximal-entropy
ensemble with a given degree distribution
(see Appendix). An important property of the measure
\eqref{eq:PT} is that it factorizes into a product of one-point
measures, so it does not introduce any explicit
correlations. This means that any observed correlations arise from the
fact that we consider a specific set of graphs and not from the
measure itself.
We show that the connected degree-degree correlations
are not zero and fall off with the distance as $l^{-2}$
\begin{eqnarray}\label{eq:pbcontree}
\bar\pi^{con}_{q,r}(l)&=& \bar\pi_{q,r}(l)-\bar\pi_q(l)\bar\pi_r(l)\nonumber\\
&=&-\frac{(q-2)(r-2)}{\left[2+ (\av{q^2}-4)(l-1)\right]^2}\,\pi_q\pi_r.
\end{eqnarray}
Here $\bar\pi_{q,r}(l)$ is the joint probability that two vertices
distance $l$ apart will have degrees $q$ and $r$, respectively. Those
correlations are disassortative. The average degree of the distance $l$
neighbors of a vertex with degree $q$ decreases
\begin{equation}\label{eq:kl}
\bar k_l(q)=
2+\frac{\av{q^2}-4}{q+(\av{q^2}-4)(l-1)}.
\end{equation}
For $l=1$ this reduces to the results obtained in Ref. \cite{kim}. In the
following sections we provide the detailed definitions of the
quantities introduced above and derive those results. We will also
discuss what happens for the scale-free trees when $\av{q^2}$
diverges.
The paper is organized as follows: Sec. \ref{sec:correlations}
introduces some basic definitions concerning correlations in random
trees. Then we derive the vertex degree distribution using the field
theory approach in Sec. \ref{sec:trees} and proceed on in
Sec. \ref{sec:correlationstrees} calculating the distance dependent
correlation functions. Two examples of Erd\"{o}s-R\'{e}nyi and
scale-free trees are given in Secs. \ref{sec:erdos} and \ref{sec:sf},
respectively. In the following Sec. \ref{sec:simulations} the results
for scale-free trees are verified using Monte Carlo (MC) simulations.
Final discussion and summary of our results are given in
Sec.~\ref{sec:summary}.
\section{Correlations}
\label{sec:correlations}
For each graph we introduce
\begin{equation}
n_{q,r}(l)\equiv\sum_{i,j\in G}\delta_{q_i,q}\delta_{q_j,r}\delta_{|i-j|,l},
\end{equation}
which is the number of pairs of points with degrees $q$ and $r$ separated by the distance $l$. We
define two further quantities: the number of pairs at the distance $l$
with one end point of specified degree
\begin{equation}
n_q(l)\equiv\sum_{r}n_{q,r},
\end{equation}
and the number of all pairs of vertices at the distance~$l$
\begin{equation}
n(l)\equiv\sum_{q,r}n_{q,r}(l).
\end{equation}
If we want to define the joint probability $\pi_{q,r}(l)$ we have two
obvious choices. The first one is
\begin{equation}\label{eq:piquenched}
\pi_{q,r}(l)\equiv\av{\frac{n_{q,r}(l)}{n(l)}}_{n(l)\neq0},
\end{equation}
where the subscript denotes that we restrict the average to the ensemble of
graphs for which $n(l)$ is not zero. The second possibility is to use
\begin{equation}
\bar\pi_{q,r}(l)\equiv\frac{\av{n_{q,r}(l)}}{\av{n(l)}}.
\end{equation}
In Ref.~\cite{bo} we have argued that the first {\em quenched} definition
is more natural in the context of random graphs. However, it is much
more difficult to work with. In this paper we will assume that the
ensemble of generic trees is {\em self-averaging} and the two above
definitions are equivalent. For a more detailed discussion of this issue we refer
to \cite{bo}.
Similarly we define
\begin{equation}
\bar\pi_{q}(l)\equiv\frac{\av{n_{q}(l)}}{\av{n(l)}},
\end{equation}
and the connected two point probability
\begin{equation}\label{eq:pbcon}
\bar\pi^{con}_{q,r}(l)\equiv\bar\pi_{q,r}(l)-\bar\pi_q(l)\,\bar\pi_r(l).
\end{equation}
We further define the connected correlation function \cite{bs,b1,b2}
\begin{equation}\label{eq:corrf}
\pi^{con}_{\bar q,\bar r}(l)\equiv\sum_{q,r}q \, r \,\bar\pi^{con}_{q,r}(l).
\end{equation}
Finally we define average degree of the vertices at the distance $l$
from a vertex of degree $q$ as follows:
\begin{equation}\label{eq:def-kl}
\bar k_l(q)\equiv\frac{\av{n_{q,\bar{r}}(l)}}{\av{n_q(l)}},
\end{equation}
where
\begin{equation}
n_{q,\bar{r}}(l)\equiv\sum_{r} r \, n_{q,r}.
\end{equation}
\section{Generic random trees}
\label{sec:trees}
We consider an ensemble of all labeled trees with the probability
measure \eqref{eq:PT}. The partition function $\Omega_V$ is defined
as the sum of the weights of all the trees in the ensemble
\begin{equation}\label{eq:weight}
\Omega_V\equiv\frac{1}{V!}\sum_{T\in\mathcal{T}(V)}\prod_{i\in T}w_{q_i}.
\end{equation}
The partition function of the corresponding grand-canonical ensemble
is defined by the discrete Laplace transform
\begin{equation}\label{eq:omega-grand}
\Omega(\mu)=\sum_{V=1}^\infty e^{-\mu V} \Omega_V.
\end{equation}
We will use the
field theory approach to calculate it \cite{jk}. We define the function
\begin{equation}\label{eq:W}
W(\mu)\equiv\int\text{d}\phi
\exp \left[ N \left(-\frac{1}{2}\phi^2+e^{-\mu}\sum_{q=0}\frac{w_{q}}{q!}\phi^q\right)\right].
\end{equation}
Its formal perturbative expansion in $e^{-\mu}$ will generate Feynman's diagrams
with desired weights and symmetry factors (for an
introduction see any textbook on field theory, e.g.,
Refs.~\cite{bdfn} and \cite{ift} or Ref.~\cite{graphs}). This expansion will,
however, contain all the graphs including those which are not
connected or contain loops. We can restrict the expansion to
connected graphs only by considering the function $\log
W(\mu)$. To obtain just the tree graphs we will use the expansion in
$N^{-1}$. According to Feyman's rules for the expression \eqref{eq:W}
each edge in the graph introduces a factor $N^{-1}$ and each vertex a
factor $N$ which together contribute $N^{-E+V}$, where $E$ is the
number of edges in the graph. If $L$ is the number of independent
loops in the graph then $E-V = L-1$, so the first term of the $N^{-1}$
expansion will group graphs with no loops, the second one graphs with
one loop, and so on.
That means that the contribution of tree graphs is given by the first
term in the saddle-point approximation.
The saddle-point equation is
\begin{equation}
\frac{\text{d}}{\text{d}\phi}\left(-\frac{1}{2}\phi^2+e^{-\mu}\sum_{q=0}^\infty \frac{w_{q}}{q!}\phi^q\right)=0.
\end{equation}
We will denote by $Z(\mu)$ the
solution of the above equation and rewrite it as
\begin{equation}\label{eq:rec}
Z(\mu)= e^{-\mu}\sum_{q=1}^\infty \frac{w_{q}}{(q-1)!}Z^{q-1}=
e^{-\mu}\frac{F\left(Z(\mu)\right)}{Z(\mu)},
\end{equation}
where
\begin{equation}\label{eq:F}
F(Z)\equiv\sum_{q=1}^\infty \frac{w_{q}}{(q-1)!}Z^q.
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{fig1}
\end{center}
\caption{\label{fig:Z}Graphical representation of Eq.~\eqref{eq:rec}.
Each gray bubble corresponds to the sum over planted trees given by
the partition function $Z(\mu)$. Branches without a vertex (small empty circle) at one end
denote a stem.}
\end{figure}
Inserting Eq.~\eqref{eq:rec} into Eq.~\eqref{eq:W} and taking the
logarithm to keep only connected graphs we obtain
\begin{equation}
\begin{split}
\label{eq:Omega}
\Omega(\mu)&=
e^{-\mu}\sum_{q=0}^\infty \frac{w_{q}}{q!}Z^q(\mu)-\frac{1}{2}Z^2(\mu).
\end{split}
\end{equation}
It is easy to check that
\begin{equation}
e^{-\mu}Z(\mu)=\frac{\partial\Omega(\mu)}{\partial w_{1}}.
\end{equation}
Figure~\ref{fig:Z} shows a graphical interpretation of $Z(\mu)$: it
is the partition function of the ensemble of {\em planted}
trees \footnote{The trees considered in Refs.
\cite{adfo} and \cite{bb} were planar. Here we do not impose such a
restriction. The only difference is the appearance of the $1/(q-1)!$
factor. This is due to the fact that now we are free to permute
branches emerging from the vertex.}. Planted trees are the trees
with a {\em stem} attached to one of the vertices.
Its properties and resulting critical
behavior were calculated in Refs.~\cite{adfo,bb,jk}.
\begin{figure}[!b]
\begin{center}
\includegraphics[width=7.5cm]{fig2}
\end{center}
\caption{\label{fig:singularity}Graphical representation of Eq.~\eqref{eq:rec-2} and the singularity of $Z(\mu)$.}
\end{figure}
The model has two main phases. In the so called {\em generic} or {\em
tree} phase the the function $F(Z)/Z^2$ has a minimum inside its
domain (see Fig.~\ref{fig:singularity}) and so Eq.~\eqref{eq:rec}, which can be rewritten as
\begin{equation}\label{eq:rec-2}
e^{\mu}=\frac{F(Z)}{Z^2},
\end{equation}
does not have any solution for $\mu<\mu_0$. The
function $Z(\mu)$ has a singularity at $\mu_0$ given by the condition for the minimum
\begin{equation}\label{eq:Z0}
2F(Z_0)=Z_0F'(Z_0),\quad\text{and}\quad \mu_0=\log \frac{F(Z_0)}{Z^2_0}.
\end{equation}
At this singularity the partition function behaves like
\begin{equation}\label{eq:treeexp}
Z(\mu)\approx Z_0-Z_1\sqrt{\mu-\mu_0}+Z_2(\mu-\mu_0),
\end{equation}
regardless of the form of the weights $w_{q}$. In this contribution
we will limit our self to this phase only. Inserting the expansion
\eqref{eq:treeexp} into Eq. \eqref{eq:rec} and expanding to the order
$\Delta\mu\equiv \mu-\mu_0$ ($Z_2$ cancels in the resulting equation)
we obtain
\begin{equation}\label{eq:Z1}
\frac{Z_0^2}{Z^2_1}=\frac{1}{2}\frac{F''(Z_0)Z_0^2}{F(Z_0)}-1.
\end{equation}
\begin{figure}[!t]
\begin{center}
\includegraphics[height=35mm]{fig3}
\end{center}
\caption{\label{fig:omega_p}Graphical representation of the partition
function $\Omega(q,\mu)$ given by Eq.~\eqref{eq:omegaq} for $q=4$.
Each gray bubble corresponds to the sum over planted trees given by
the partition function $Z(\mu)$; the smaller black circle represents
the root which contributes a weight factor $e^{-\mu}w_4$; the
additional $\frac{1}{4!}$ factor comes from the fact that the
relative position
of branches in the compound tree is irrelevant. }
\end{figure}
The vertex degree distribution of this model was calculated using the
correspondence with the balls in boxes model in Ref.~\cite{bbj}. Here
we rederive it using a different method which can be easily extended
to the case of two-point correlations studied in Ref.~\cite{b3}. Let
us denote by $\Omega(q;\mu)$ the partition function of the rooted
grand-canonical ensemble of trees with the condition that the degree
of the root is $q$. Then
\begin{equation}\begin{split}
\label{eq:omegaq}
\Omega(q;\mu)=
w_{q}\frac{\partial\Omega(\mu)}{\partialw(q)}=e^{-\mu}\frac{w_{q}}{q!}Z^q(\mu).
\end{split}
\end{equation}
The graphical interpretation of this equation is shown in
Fig.~\ref{fig:omega_p}. The sought degree distribution is
proportional to the canonical partition function $\Omega_V(q)$. Inserting
the expansion \eqref{eq:treeexp} into Eq. \eqref{eq:omegaq} we obtain
\begin{eqnarray}
\Omega(q;\mu)&\approx&e^{\mu_c}e^{\Delta\mu}\frac{w_q}{q!}Z_0^{q}\left(1-q\frac{Z_1}{Z_0}\sqrt{\Delta\mu}\right)\nonumber\\
&\approx&e^{\mu_c}e^{\Delta\mu}\frac{w_q}{q!}Z_0^{q}\exp\left(
-q\frac{Z_1}{Z_0}\sqrt{\Delta\mu}\right).
\end{eqnarray}
The last expression has a known inverse Laplace transform
\begin{equation}\label{eq:lap}
e^{-a \sqrt{\Delta\mu}}\; \xleftrightarrow[]{\text{Lap.}} \;
\frac{1}{2\sqrt{\pi}}\,\frac{a}{V^{\frac{3}{2}}}\;e^{-\frac{a^2}{4V}},
\end{equation}
so finally
keeping only the first terms in the $V^{-1}$ expansion
and fixing the normalization we obtain the formula
\begin{equation}\label{eq:piq}
\pi(q)=\frac{1}{F(Z_0)}\;\frac{w_{q}Z_0^{q}}{(q-1)!}.
\end{equation}
Using the above formula we can give an interpretation of the
right-hand side of Eq.~\eqref{eq:Z1}
\begin{eqnarray}
\frac{1}{2}\frac{F''(Z_0)Z_0^2}{F(Z_0)}-1&=&
\frac{1}{2}\frac{\sum_{q=1} \frac{q(q-1)}{(q-1)!}w_q Z_0^q}{F(Z_0)}-1\nonumber\\
&=&\frac{1}{2}\sum_{q=1} q(q-1)\pi(q)-1\nonumber\\
&=&\frac{1}{2}\left(\av{q^2}-4\right).
\end{eqnarray}
Here we have used the fact that on trees the average degree $\av{q}=2$ (in
the large $V$ limit). Please note that
\begin{equation}
\av{q^2}-4=\av{(q-2)^2}\ge 0.
\end{equation}
The $\av{q^2}$ is equal to $4$ only in the $\omega_1\rightarrow 0$ limit.
\subsection{Correlations in generic random trees}
\label{sec:correlationstrees}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=7cm]{fig4}
\end{center}
\caption{\label{fig:omega_pql}Graphical representation of the
partition function $\Omega_l(q,r;\mu)$ given by
Eq. \eqref{eq:omega_pql} for $q=4$ and $r=3$. Gray bubbles
correspond to the partition function $Z(\mu)$ and the smaller black
circles mark the vertices with degrees $q$ and $r$; double circles
represent the $l-1$ vertices along the path connecting them in which
we sum over all possible insertions of the $Z(\mu)$ function (see
Fig. \ref{fig:node}).}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=7cm]{fig5}
\end{center}
\caption{\label{fig:node}Double circles denote the summation over all possible insertions of the $Z(\mu)$ function (gray bubbles).}
\end{figure}
We proceed as in the previous section but this time we introduce a
partition function $\Omega_l(q,r;\mu)$ of all the trees with two
points marked, such that the points are at the distance $l$ and have
degrees $q$ and $r$, respectively. Because we are considering the
trees there is exactly one path linking the two marked vertices (see
Fig.~\ref{fig:omega_pql}). As in the previous section we can express the
partition function $\Omega_l(q,r;\mu)$ by $Z(\mu)$ \cite{b2,adj}
\begin{eqnarray}\label{eq:omega_pql}
\Omega_l(q,r;\mu)&=&\frac{e^{-\mu}w_{q}}{(q-1)!}\;Z^{q-1}(\mu)\frac{e^{-\mu}w_r}{(r-1)!}\;Z^{r-1}(\mu)\nonumber\\
&\times& \left[e^{-\mu}\sum_{k=2}\frac{w_{k}}{(k-2)!}\;Z^{k-2}\right]^{l-1}.
\end{eqnarray}
The last term comes from the vertices along the path for which we have
to sum up all the possible insertions of the $Z(\mu)$ function (see
Fig.~\ref{fig:node}). This summation can be done in the following way:
\begin{eqnarray}
e^{-\mu}\sum_{q=2}\frac{w_{q}}{(q-2)!}\;Z^{q-2}&=&
e^{-\mu}\frac{\partial}{\partial Z}\sum_{q=1}\frac{w_{q}}{(q-1)!}\;Z^{q-1}\nonumber\\
&=&e^{-\mu}\frac{\partial}{\partial Z}\frac{F(Z)}{Z}.
\end{eqnarray}
Differentiating Eq.~\eqref{eq:rec} with respect to $\mu$ we come to the relation
\begin{equation}
Z'(\mu) \left(1 - e^{-\mu}\frac{\partial }{\partial Z}\frac{F(Z)}{Z}\right)= -Z.
\end{equation}
Using it we finally obtain
\begin{equation}\begin{split}
e^{-\mu}\sum_{q=2}\frac{w_{q}}{(q-2)!}\;Z^{q-2}
=
1+\frac{Z(\mu)}{Z'(\mu)}.
\end{split}
\end{equation}
Inserting into Eq.~\eqref{eq:omega_pql} first the above formula and then the expansion \eqref{eq:treeexp} we get
\begin{eqnarray}
&&\Omega_l(p,q;\mu)\approx\frac{w_q}{(q-1)!}\,\frac{w_r}{(r-1)!}\;Z_0^{q+r-2}\nonumber\\
&&\times\left(1-\frac{Z_1}{Z_0}\sqrt{\Delta\mu}\right)^{q+r-2}\left(1-2\frac{Z_0}{Z_1}\sqrt{\Delta\mu}\right)^{l-1}.
\end{eqnarray}
This can be further approximated by
\begin{eqnarray}
\Omega_l(p,q;\mu)&\approx& \frac{w_q}{(q-1)!}\,\frac{w_r}{(r-1)!}\;Z_0^{q+r-2}\nonumber\\
&\times& e^{-\left[\frac{Z_1}{Z_0}(q+r-2) + 2\frac{Z_0}{Z_1}(l-1)\right]\sqrt{\Delta\mu}},
\end{eqnarray}
and using Eq. \eqref{eq:lap} we obtain to the leading order in $V$
\begin{eqnarray}
\av{n_{qr}}\propto\Omega_l(p,q;V)&\approx&\frac{w_q}{(q-1)!}\,\frac{w_r}{(r-1)!}\;Z_0^{q+r-2}\nonumber\\
&\times& \left[\frac{Z_1}{Z_0}(q+r-2) + 2\frac{Z_0}{Z_1}(l-1)\right].\nonumber\\
\end{eqnarray}
Finally, we get
\begin{align}
\bar\pi_{q,r}(l)=&\,\pi_q \pi_r \frac{ (q+r-2) +\left(\av{q^2}-4\right)(l-1)} {2 + \left(\av{q^2}-4\right)(l-1)}, \label{eq:piqrl}\\
\bar\pi_{q}(l)= &\,\pi_q \frac{ q + \left(\av{q^2}-4\right)(l-1)} {2 + \left(\av{q^2}-4\right)(l-1)}.\label{eq:piql}
\end{align}
Inserting this into Eqs. \eqref{eq:pbcon} and \eqref{eq:def-kl} we
obtain the results \eqref{eq:pbcontree} and \eqref{eq:kl}. Summing up
Eq.~\eqref{eq:pbcontree} over $q$ and $r$ we get the
connected correlation function \eqref{eq:corrf}
\begin{equation}
\begin{split}
\bar\pi^{con}_{\bar q,\bar r}(l)=-\frac{\left[\av{q^2}-4\right]^2}{\left[2+ (\av{q^2}-4)(l-1)\right]^2}.
\end{split}
\end{equation}
\subsection{Example 1}
\label{sec:erdos}
In the first example we put $w_{q}=1$, so all the trees in the
ensemble have the same weight. In this case $F(Z)=Z e^{Z}$. The
solution of Eq. \eqref{eq:Z0} is $Z_0=1$ from which
follows:
\begin{equation}
\pi_q=\frac{1}{e}\frac{1}{(q-1)!},\quad
\av{q^2}=\sum_{q=1}^\infty\frac{1}{e}\frac{q^2}{(q-1)!}=5.
\end{equation}
leading to
\begin{align}
\bar{\pi}_q(l)&=\pi_q\frac{q+l-1}{1+l},\\
\bar\pi^{con}_{q,r}(l)&=-\frac{1}{e^2}\frac{(q-2)(r-2)}{(l+1)^2}
\frac{1}{(q-1)!}\frac{1}{(r-1)!},
\end{align}
and
\begin{equation}
\bar \pi^{con}_{\bar q, \bar r}(l)=-\frac{1}{(l+1)^2},\quad \bar k_{l}(q)=2+\frac{1}{q+l-1}.
\end{equation}
\subsection{Example 2: Scale-free trees}
\label{sec:sf}
In this example we choose $w_{q}=q^{-\beta}(q-1)!$ which corresponds
to the planar graphs studied in Refs.~\cite{adfo} and \cite{bb}. Then $F(Z)$ is
given by the polylogarithm function $\operatorname{Li}_\beta(Z)$
\begin{equation}
F(Z) = \sum_{q=1}^\infty \frac{Z^q}{q^\beta} \equiv \operatorname{Li}_\beta(Z).
\end{equation}
and Eq.~\eqref{eq:Z0} takes the form
$2\operatorname{Li}_\beta(Z_0)=\operatorname{Li}_{\beta-1}(Z_0)$. It has the solution for
$\beta<\beta_C$ with $\beta_C\approx2.4788$ given by
$2\zeta(\beta_C)=\zeta(\beta_C-1)$. At the critical value of
$\beta=\beta_C$ the partition function no longer scales as in Eq.~\eqref{eq:treeexp}
and in principle we cannot use the Laplace transform Eq.~\eqref{eq:lap} anymore.
However, as shown in Ref.~\cite{bepw} the large $V$
behavior is not changed and we expect our formula to hold in the
large $V$ limit. From Eq.~\eqref{eq:piq} we read-off the degree
distribution
\begin{equation}\label{eq:piscale}
\pi(q)=\frac{q^{-\beta}Z_0^{q-1}}{F(Z_0)}.
\end{equation}
At the critical value of $\beta$,
$Z_0=1$ and the vertex degree distribution is scale free. The average
\begin{equation}
\av{q^2}=2+\frac{1}{\operatorname{Li}_\beta(Z_0)}\left[\operatorname{Li}_{\beta-2}(Z_0)-\operatorname{Li}_{\beta-1}(Z_0)\right],
\end{equation}
diverges as $\beta\rightarrow\beta_C$. Formula
\eqref{eq:pbcontree} leads for $l>1$ to
\begin{equation} \label{eq:piconsf}
\lim_{V\rightarrow\infty}\bar\pi^{con}_{q,r}(l)=0.
\end{equation}
This would imply that the correlations vanish in the large $V$ limit. However, this
limit \eqref{eq:piconsf} is not uniform. It is easy to check that the {\em integrated} correlation functions do not disappear
\begin{equation} \label{eq:pisf}
\lim_{V\rightarrow\infty}
\bar \pi^{con}_{\bar q,\bar r}(l)=-\frac{1}{(l-1)^2},
\end{equation}
and
\begin{equation}\label{eq:klsf}
\lim_{V\rightarrow\infty}\bar k_l(q)=2+\frac{1}{l-1}.
\end{equation}
Please note that the above results are universal and valid for any kind of
scale-free trees with $\beta<3$.
\subsection{Monte Carlo simulations}
\label{sec:simulations}
The results obtained in the previous sections are valid only in the strict
$V\rightarrow\infty$ limit and it is clear that for finite $V$
our formulas will not hold for any $l$. Defining the
average distance on a graph
\begin{equation}
\av{l}=\sum_l l\frac{n(l)}{V^2},
\end{equation}
we may expect that the formulas will be valid only for $l\ll \av{l}$.
The scaling of $\av{l}$ with the graph size depends on the Hausdorff's
dimension $d_H$
\begin{equation}
\av{l}\sim V^{{1}/{d_H}}.
\end{equation}
For generic trees considered here $d_H=1/2$. For scale-free
trees considered in example 2 we expect
\begin{equation}
d_H=\frac{1}{\gamma}, \qquad
\gamma=\frac{\beta_C-2}{\beta_C-1},
\end{equation}
which gives $d_H\approx 3$. In the case of scale-free trees the volume dependence
manifests itself by the cutoff in the degree distribution $\pi_q$ as well \cite{bck}.
Expecting finite size effects to be more severe in the
scale-free trees, we checked the $V$ dependence performing
MC simulations of the ensemble described in the
example~2. We have used an algorithm similar to ``baby-universe
surgery'' \cite{abbjp}. The basic move consisted of picking up an edge
at random and cutting it. Then the
smaller of the two resulting trees was grafted on some vertex of the bigger one. The most
time consuming part of the algorithm was to find which tree was
smaller. To save time the two trees were traversed simultaneously until one of them was filled
completely. Additionally, to pick the attachment point from the bigger tree efficiently, the vertices of the trees were marked during the traversal.
This move was supplemented with moves consisting of
cutting up leaf nodes and attaching them to some other parts of the
tree. This was much faster as it did not require traversing the tree.
However, the autocorrelation time for such moves alone was
much higher, especially for the scale-free trees. Because those trees
are at the phase transition between the generic and the crumpled
phase \cite{bb} the autocorrelation time is high even for the tree
grafting algorithm.
\begin{figure}[!t]
\includegraphics[height=7cm]{fig6}
\caption{\label{fig:kl} Average degree of distance $l$ neighbors $k_l(q)$ for $l=2$ to $6$ and trees with
64~000 (empty symbols) and 128~000 (filled symbols) vertices. Each symbol denotes
different $l$; straight lines are the predictions given by Eq.~\eqref{eq:klsf}.}
\end{figure}
\begin{figure}[!t]
\includegraphics[height=7cm]{fig7}
\caption{\label{fig:conn} Connected correlation function $\pi_{\bar q,\bar
r}^{con}(l)$ for scale-free trees with 64~000 (empty circles) and
128~000 (filled diamonds) vertices. The solid line represents the prediction given by Eq.~\eqref{eq:pisf}.}
\end{figure}
We have simulated trees of the size up to 128~000 vertices. To verify
to which extent the ensemble is self-averaging we measured the
quenched quantities which we then compared to our predictions. In
Fig.~\ref{fig:kl} we have plotted the measured
\begin{equation}
k_l(q)=\av{\frac{n_{q,\bar r}(l)}{n_q(l)}},
\end{equation}
as a function of $q$ for various values of $l$. Please note the large
finite size effects for $l=2$. This is to be expected: For finite $V$ the
$\av{q^2}$ is also finite and actually grows slowly with
$V$ \cite{bck}. For larger $l$ the agreement with our results is quite good.
Figure~\ref{fig:conn} shows the quenched correlation function
\begin{equation}
\pi^{con}_{\bar q,\bar r}(l)=
\av{\frac{n_{\bar q,\bar r}(l)}{n(l)}}-
\av{\frac{n_{\bar q}(l)}{n(l)}}^2,
\end{equation}
which is also well reproduced by our results.
\section{Summary and Discussion}
\label{sec:summary}
The appearance of the long-range correlations in generic trees is
puzzling. Usually we expect the powerlike (scale-free) behavior to
be manifested in systems at the criticality. The trees studied here
apart from the scale-free example are, however, not critical. The
free-energy density can be calculated in the infinite volume limit
and remains an analytic function of the weights $\omega_q$
\cite{bb,bbj}. It has been also shown that the critical behavior in
random trees is not associated with the diverging correlation length
\cite{bbju}. The correlations described here are thus of the
structural and not of dynamical origin. A possible mechanism
explaining it was proposed in Refs. \cite{b1} and \cite{bo}: in connected graphs
vertices of degree one must have neighbors of degree greater than
one. It remains, however, to be understood how this effect can be
propagated to larger distances.
\acknowledgments
The authors thank Z.~Burda for valuable discussion. This paper was
partially supported by EU grants No. MTKD-CT-2004-517186 (COCOS) and
No. MRNT-CT-2004-005616 (ENRAGE). P.B. thanks the Service de Physique
Th\'eorique, CEA/Saclay for the kind hospitality during his stay.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,593 |
Жак Савойски-Немур (; * 12 октомври 1531, Абатство "Волюизан" в днешния френски департамент Йон, † 15 юни 1585, имение "Касин-Шастелие" близо до Монкалиери, Савойско херцогство) е втори херцог на Немур (1533 – 1585), граф и от 1564 г. херцог на Женева, маркиз на Сен Сорлен, губернатор на Лион, генерал-полковник от Френската кавалерия.
Той е важно лице във френския двор. По време на Италианските войни и Религиозните войни се бие в Кралската армия. Известен със своя чар и елегантност, Жак е замесен в няколко любовни авантюри.
Произход
Роден е на 12 октомври 1531 г. като син на първия херцог на Немур Филип Савойски-Немур (* 1490 † 25 октомври 1533), граф апанажист на Женева, и съпругата му Шарлот Орлеанска-Лонгвил (* 1512 † 1549). Дядо му по бащина линия е савойският херцог Филип ІІ, а по майчина – херцог Луи Орлеански-Лонгвил, херцог на Лонгвил.
Има една сестра:
Жана Савойска-Немур (* 1532 † 1568), отт 1555 г. съпруга на Никола дьо Лотарингски, граф на Водемон и регент на херцогствата. Тя е грижовна свекърва за децата от първия брак на съпруга си и по-специално за принцеса Луиз дьо Лорен-Водемон, бъдеща кралица на Франция през 1575 г..
Има и двама полубратя и една полусестра от извънбрачна връзка на баща му.
Биография
Принц на служба на Франция
15-годишният Жак е представен на френския крал Франсоа I. Години по-късно, след като неуспешно предизвиква братовчед си Леонор Орлеански-Лонгвил за наследяването на Княжество Ньошател, той се отличава при обсадата на Ланс през 1552 г. На следващата година, научавайки за намеренията на Карл V за град Мец, той участва интелигентно в обсадата на Мец. След това служи във Фландрия и Италия до примирието във Восел през 1556 г. Той е издигнат в генерал-полковник на леката кавалерия. През 1558 г., по време на обсадата на Тионвил, командва леката кавалерия като полковник.
Бие се на дуел с Маркиза на Пискер по неизвестна причина. Когато крал Анри II е смъртоносно ранен по време на турнир на 30 юни 1559 г., Жак е един от неговите поддръжници. Няколко дни по-късно е посланик на херцог Еманил-Филиберт Савойски по време на брака му с Маргьорит дьо Валоа, дъщеря на Франсоа I и Клод Френска, който се състои в Париж на 10 юли 1559 г.
Става рицар на Ордена на Свети Михаил на 7 декември 1561 г. Продължава да се отличава по време на Религиозните войни срещу протестантите. Освен всичко друго Жак допринася за превземането на Бурж и на два пъти побеждава Барон дез Адре. Той командва швейцарците, които връщат Шарл IX обратно в Париж, когото калвинистите искат да отвлекат в Монсо ле Мьо. През 1562 - 1563 г., в Дофине, замествайки Антоан дьо Бурбон-Вандом , той търпи поражение пред Лион, държан от Жан V дьо Партене от името на партията на хугенотите. Договорът за мир от 19 март 1563 г. помирява двете страни за известно време.
Той става 11-ият губернатор на Лион с Оверн, Бурбоне и Марш на 27 декември 1562 г., регистриран на 4 юли 1564 г.
След като участва в битката при Сен Дени през 1567 г., е обвинен в противопоставяне на навлизането на войските, които херцогът на Дьо Пон изпраща при победените. Той се проваля в тази експедиция по вина на Херцог д'Омал, негов съперник, и се оттегля в своето Женевско херцогство, където се посвещава на култивирането на науките и изкуствата. Графство Женева е издигнато в херцогство през 1564 г.
Принц-прелъстител
Френската историографка Жаклин Буше описва Жак Савойски-Немур като "велик господар, начетен, спортен, културен и светски, известен с таланта си на прелъстител".
През лятото на 1559 г. той е сериозен ухажор на английската кралица Елизабет I. Бракът се приема за даденост във френския двор, но обичайното нежелание на кралицата и Франсоа II поради войната в Шотландия оставят този проект за съюз без последващи действия. За да му позволи да посети кралицата, крал Анри II го назначава за извънреден посланик. След смъртта на краля Жак продължава да подготвя пътуването си до Лондон (юли-септември 1559 г.). Това няма последствия, главно поради франкофобията на английския двор и натиска на съветниците на кралицата да го ангажират да прогони французите от Шотландия. Друга причина би могла да бъде любовта му към Анна д'Есте, която откриваме описана в романа "Принцесата на Клев" (1678) от Мадам дьо Ла Файет, където Жак Савойски отказва да тръгне заради любовта си към споменатата принцеса.
Той е обект на няколкогодишен съдебен процес, заведен срещу него от дама с висше потекло – Франсоаз дьо Роан, която забременява от него и той ѝ обещава брак. Тя ражда от него Анри Женевски. Жак обаче се жени на 5 май 1566 г. за Анна д'Есте от Дом Есте, вдовица на лотарингския херцог Франсоа I дьо Гиз, дъщеря на херцога на Ферара Ерколе II д'Есте и съпругата му Рене Френска, дъщеря на френския крал Луи XII и Анна Бретонска. По бащина линия тя е внучка на Алфонсо I д'Есте и Лукреция Борджия. Кралицата на Навара Жана д'Албре, роднина на Дом Роан, напразно се опитва да предотврати този брак. Произтичащият процес отравя живота на Жак и забавлява двора по време на всички религиозни войни. Завършва при крал Анри III с решение, дължащо се на математика Франсоа Виете, което удовлетворява всички страни.
Смърт
Жак Савойски-Немур умира от подагра на 18 юни 1585 г. на 53-годишна възраст в имението си Касин-Шастелие близо до Монкалиери (Пиемонт). Франсоа Мюние посочва, че е умира на 15 или 18 юни 1585 г. и тази първа дата е посочена по-специално от генеалогичния сайт Foundation for Medieval Genealogy.
Тялото му е откарано в Анси, столицата на Женевското херцогство, където тялото му е погребано в семейната църква "Нотр Дам дьо Лиес". Наследен е от сина си Шарл-Еманюел.
Брак и потомство
∞ 29 април 1566 в Сен Мор де Фосе за Анна д'Есте (1531 † 1607), вдовица на Франсоа I, херцог на Гиз, и дъщеря на Ерколе II д'Есте, херцог на Ферара, и Рене Френска, принцеса от Франция от Орлеанския клон на династията Валоа и херцогиня на Шартър. Имат двама сина и една дъщеря:
Шарл-Еманюел II (* 1567 † 1595), принц на Женева, херцог на Немур (1585 – 1595), неомъжен и бездетен.
Маргьорит-Мари (* 1569 † 1572)
Анри I (* 1572 † 1632), херцог на Немур (1595 – 1632), ∞ 1618 за Ан Лотарингска, херцогиня на Омал (* 1600, † 1638), от която има трима сина.
Еманюел-Филибер
Има две извънбрачни деца.
От Франсоаз дьо Роан (* ок. 1540, † 1 декември 1591, Бовоа сюр Мер) от Дом Роан, господарка на Ла Гарнаш, херцогиня на Луден, има един син:
Анри Женевски или Немурски (* 1557 † 1596), известен като "Принц на Женева", неженен. Има един извънбрачен син.
От Жана Дюпре има един син:
Жак, наречен Жак II, Жак Савойски или Жак Младши († 1595), абат на Талоар и Антремон.
Вижте също
Савойска династия
Савоя-Немур
Списък на херцозите на Немур
Литература
Jean-Hippolyte Mariéjol, Charles-Emmanuel de Savoie duc de Nemours Gouverneur du Lyonnais, Beaujolais et Forez (1567 – 1595), Paris: Librairie Hachette, 1938. In-8 (124 x 17 cm), XVI – 287 p.
Louis Charles Dezobry et Théodore Bachelet, Dictionnaire de Biographie et d'Histoire, Paris, 1863.
Max Bruchet, Étude biographique sur Jacques de Savoie, duc de Génevois-Nemours, suivie de son Instruction et discours sur le faict du gouvernement, 1582.
La Savoie au coeur de l"Europe du XVI siècle
Бележки
Външни препратки
Archives de Savoie
Généalogie de la maison de Savoie
Италианска аристокрация
Херцози на Немур
Графове на Женева
Савойска династия
Мъже
Италианци | {
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Henoch-Schönlein purpura (HSP) is a common autoimmune vasculitis in childhood. The detailed pathogenesis of HSP is still unclear, whereas several types of predisposing factors have been proved to be the initial step. The objectives of present study were to analyze the distribution of predisposing factors, association of the predisposing factors with clinical manifestations and HSP relapse/recurrence. 1200 children with HSP were recruited between January 2015 and December 2017. We reviewed their laboratory tests and medical histories associated with HSP onset. The annual incidence of HSP was 8.13–9.17 per 100 000 in Anhui province. HSP occurred more commonly in spring and winter than in summer with an obvious west-to-east gradient. Cutaneous purpura was the most prevalent manifestation (100%), followed by arthritis/arthralgias (43.67%), abdominal pain (40.17%) and renal involvement (18.17%). On admission, series of potential infections were identified in 611 patients (50.92%). The histories of allergy, injury, surgery, vaccination and tick bite were declared by 231 patients (19.25%), 15 patients (1.25%), 12 patients (1.00%), 4 patients (0.33%) and 3 patients (0.25%), respectively. However, predisposing factors could not be identified in 521 children with HSP (43.42%) yet. 123 cases (10.25%) relapsed or recurred more than one time; the mean number was 2.92, and the mean interval was 11.4 weeks. The infection is the most frequent predisposing factor regardless of clinical phenotypes and relapse/recurrence, whereas the clinical manifestations exhibit an obvious heterogenicity according to different predisposing factors.
Contributors PH conceptualized and designed the study, coordinated and supervised data collection, and critically reviewed the manuscript for important intellectual content. YX, JJW and YFW drafted the manuscript. YW, FFL and SS collected data and carried out the initial analyses. All authors approved the final manuscript as submitted and agree to be accountable for all aspects of the work.
Funding This study was supported by the National Natural Science Foundation of China (No. 81570637, 81000306) and the New Technology Project of the First Affiliated Hospital, Anhui Medical University (2014-01).
Ethics approval Approval for this research was acquired from the Medical Ethic Committee of the First Affiliated Hospital of Anhui Medical University. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,640 |
News Australia 3CR 855AM Community Radio Cait Kelly Carly Baque Em Castle
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Is it best that our food is Local and Organic or Big and Conventional? Our view is "Both, and.." We don't come to the table with a bias, except that good farming like good food comes in all shapes and sizes. Farm to Table Talk explores issues and the growing interest in the story of how and where the food on our tables is produced, processed and marketed. The host, Rodger Wasson is a food and agriculture veteran. Although he was the first of his family to leave the grain and livestock farm a ...
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"redpajama_set_name": "RedPajamaCommonCrawl"
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ENV['RAILS_ENV'] ||= 'test'
if ["manual", "travis"].include?(ENV["RUN_COVERAGE"])
require 'simplecov'
require 'codeclimate-test-reporter' if ENV["RUN_COVERAGE"] == "travis"
SimpleCov.add_filter 'vendor/'
SimpleCov.add_filter 'app/mailers/mail_preview.rb'
if ENV["RUN_COVERAGE"] == "travis"
SimpleCov.formatters = []
SimpleCov.start CodeClimate::TestReporter.configuration.profile
else
SimpleCov.start 'rails'
end
end
# This must be AFTER SimpleCov is required, or it won't work
require_relative '../config/environment'
require 'rails/test_help'
require "strip_attributes/matchers"
require "minitest/reporters"
require "valid_attribute"
require "factory_bot_rails"
require "sidekiq/testing"
require "webmock/minitest"
require_relative './matchers'
Minitest::Reporters.use!
def sample_file(filename = "sample_pdf.pdf")
File.new("test/fixtures/#{filename}")
end
# Clean up Paperclip uploads generated during tests
Minitest.after_run do
FileUtils.rm_rf(Dir["#{Rails.root}/public/system/"])
end
class ActiveSupport::TestCase
extend StripAttributes::Matchers
include ValidAttribute::Method
include FactoryBot::Syntax::Methods
extend HackathonManager::Shoulda::Matchers
# Setup all fixtures in test/fixtures/*.yml for all tests in alphabetical order.
fixtures :all
# Add more helper methods to be used by all tests here...
end
class ActionController::TestCase
include Devise::Test::ControllerHelpers
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,715 |
\section{Introduction}
The two body problem is one the most important and difficult
challenges in general relativity. As a particular case, the head-on
collision of two black holes has recently gained new insights from
several techniques, namely, numerical\cite{ncsa},
perturbative\cite{pullin}, post-Newtonian\cite{will} and
semi-analytical methods\cite{our1}. These in turn concentrated their
initial efforts in studying the collision of two Schwarzschild black
holes. A semi-analytical approach developed \cite{our1} to treat the
Schwarzschild case was based on an exact solution of Einstein's field
equations. A particular case of this solution describes a geometry
that can be interpreted as a static axisymmetric spacetime with two
black holes plus a conical singularity between them. It is then
possible to obtain the ``force'' of attraction between the holes and
their ``acceleration''. These expressions were used to calculate the
rate of emission and the total amount of gravitational wave energy
released in the process. The remarkable agreement between these
results and those of the numerical treatment seems to indicate that
the static acceleration contribution is very important for the head-on
collision problem. This conjecture is emphasized by the result that,
in the context of perturbation theory, the gravitational wave emission
of boosted perturbed black holes is very similar to that of the static
case\cite{cook}.
The previous success with the semi-analytical approach strongly
motivated the present development of a similar treatment of a toy
model for the head-on collision problem of two Kerr black holes. We
start with an exact solution of Einstein's field equations that can be
interpreted as a stationary axisymmetric spacetime with two collinear
Kerr black holes and a singularity on the axis of symmetry. This
solution has some free parameters that determine the nature of the
singularity. This singularity has been shown to be either a spinning
rod between the holes or a pair of spinning strings extending from
each hole to infinity, or both in the most general case\cite
{letoliv}. For each of these cases, one can obtain a stationary
``force'' of attraction and the corresponding ``acceleration'' from
which we calculate the rate of emission and the total gravitational
wave energy released in the collision when we ``cut'' the strut or the
strings and let the holes collide. It is important to point out that
in the Kerr case there is also a spin-spin interaction between the
holes\cite{wald}, which is always a repulsive contribution to the
``attractive force''.
The plan of this paper is as follows: In section II we discuss the
``stationary'' force between the holes with axisymmetric spins. In
section III the equations of motion for the holes are obtained. In
section IV we derive an expression for the rate of emission of
gravitational radiation and calculate the total output of
gravitational wave energy released for several initial
configurations. These results are summarized in one table. Section V
is dedicated to the possibility of the holes' spin variation during
the collision process and section VI to our conclusions.
\section{The two body Interaction force}
The line element of a stationary axisymmetric spacetime is given by
the Lewis metric \cite{papapetrou}
\begin{equation}
ds^2=\exp (2\psi )\left( dt-w\;d\phi \right) ^2-\exp (-2\psi )\left[
\exp (2\nu )\left( dr^2+dz^2\right) +r^2d\phi ^2\right] \label{ds2}
\end{equation}
in cilindrical coordinates $0\leq r<\infty $, $-\infty <z<\infty $ and
$ 0\leq \phi <2\pi $ . The metric functions $\psi =\psi (r,z)$, $\nu
=\nu (r,z) $ and $w=w(r,z)$ satisfy the vacuum Einstein's field
equations:
\begin{eqnarray}
\psi _{rr}+\psi _r/r+\psi _{zz} &=&-
{\textstyle {\exp (4\psi ) \over 2r^2}}
\left( w_r^2+w_z^2\right) \;, \label{psieq} \\
w_{rr}-w_r/r+w_{zz} &=&-
{\textstyle {4 \over r}}
\left( w_r\psi _r+w_z\psi _z\right) \;, \label{omegaeq}
\end{eqnarray}
\begin{eqnarray}
\nu _r &=&r\left( \psi _r^2-\psi _z^2\right) -
{\textstyle {\exp (4\psi ) \over 2r}}
\left( w_r^2+w_z^2\right) \;, \label{nureq} \\
\nu _z &=&2r\psi _r\psi _z-
{\textstyle {\exp (4\psi ) \over 2r}}
w_rw_z\;. \label{nuzeq}
\end{eqnarray}
The metric function $\nu $ is obtained, up to a constant, by
quadrature once the solutions $\psi $ and $w$ for the non-linear
coupled system of equations (\ref{psieq}) and (\ref{omegaeq}) are
known. These equations are also the integrability conditions for the
equations (\ref{nureq}) and (\ref{nuzeq}) and they can also be
obtained from the action $\int \left[ \left( \psi _r^2+\psi
_z^2\right) -
{\textstyle {\exp (4\psi ) \over 4r^2}}
\left( w_r^2+w_z^2\right) \right] rdrd\phi dz$ \cite{carmeli}.
Asymptotically $\psi $ plays the role of a Newtonian potential and
$w/4$ of a body's angular momentum per radial distance. $w$ is related
to the usual twist potential $\chi $ by $r\;\chi _r=w_z$ and $r\;\chi
_z=-w_r$ .
Solutions of the system of equations (\ref{psieq})--(\ref{omegaeq})
have been obtained using B\"{a}cklund transformation and inverse
scattering techniques \cite{BelinskyZhakarov}. In general these
solutions have singularities. We point out that the choice of the
integration constants in the solution is a crucial step towards the
interpretation of the nature of the singularity and consequently of
the resulting spacetime geometry. In this paper we are going to deal
with two possible configurations. One that corresponds to two
collinear Kerr black holes with a twisted spinning strut between them
and the other case, that of a couple of spinning strings extending
from each of the holes to infinity.
The strut and string's energy momentum vanishes everywhere except at
the axis. Furthermore, their effective mass is null and their angular
momentum per unit length is given by $w\exp (\psi )/4$ at $r=0$
\cite{cosmicstrings} \cite{letoliv}. Let $m_1$ , $m_2$ be the masses
and $a_1$, $a_2$ be the angular momenta per unit mass of the holes
(hereafter called simply spins), respectively. It has shown that the
compression force on the strut is given by \cite{letoliv}
\begin{equation}
F_{\text{strut}}=-\frac{m_1m_2}{d^2-(m_1+m_2)^2+(a_1-a_2)^2}\;,
\label{force1}
\end{equation}
where $d$ is the coordinate distance between the center of the
holes. We assume that this is also the ``stationary force of
attraction'' between the holes if the strut is removed. In order to
get rid off the semi-infinity strings the spacetime has the constraint
$\frac{a_1}{a_2}=\frac{m_1}{m_2}$ so it is applicable to the parallel
spin case only.
The anti-parallel spin case can be treated using the tension force on
the string as the ``stationary force of attraction'' between the holes
if both strings are cut. The force of attraction is given by
\begin{equation}
F_{\text{strings}}=-\frac{m_1m_2}{d^2-(m_1-m_2)^2+(a_1-a_2)^2}
\label{force2}
\end{equation}
which is opposit to the tension on the strings.
Both forces are singular at a finite separation $d$ , nevertheless our
computation stops before that. Recall that $\sigma _k\equiv
\sqrt{m_k^2-a_k^2}$ , $m_k\geq a_k$ , $k=1,2$ is the coordinate
distance between the center of each isolated hole and its infinite
redshift surface along the axis. In this paper we consider
$d^2>(m_1+m_2)^2-(a_1-a_2)^2>\left( \sigma _1+\sigma _2\right) ^2$.
Recall also that these coordinates distances correspond asymptotically
to the distances measured by an observer at infinity.
Note that the spin-spin interaction reduces the attraction intensity
unless the holes are co-rotating $\left( a_1=a_2 \right)$. If the
holes have anti-parallel spins ( $a_1=-a_2$), the spin-spin repulsion
has a maximum\cite{letoliv}. But it gives always a repulsive
contribution to the force. For large distances the spin-spin repulsion
is $m_1m_2(a_1-a_2)^2/d^4$ .
\section{Equations of motion}
We now investigate the time evolution of a two-body system interacting
through the force (\ref{force1}) applicable only to the parallel spins
black holes. We assume that Newton's equations of motion are adequate
to describe the black holes motion and that their spins and the masses
are constants. We then have a Newtonian two-body problem that can be
transformed into a central force problem with a fixed center of
mass. Let $M=m_1+m_2$ and $\mu =m_1m_2/M$ be the total and reduced
mass of the system, respectively. Let $ a_R=\left| a_1-a_2\right| $
be the relative spin of the holes. Their relative acceleration is
given by
\begin{equation}
\frac{d^2z}{dt^2}=-\frac M{z^2-M^2+a_R^2}\;. \label{accel1}
\end{equation}
Let $\overline{z}=z/M$ , $a=a_R/M$ and $t\rightarrow t/M$. Recall that
the spin of a hole is constrained to be less than its mass, so $a\leq
1$.
The first integral of equation (\ref{accel1}) is
\begin{equation}
{\textstyle {1 \over 2}}
\stackrel{.}{\overline{z}}^2+V=\varepsilon \;, \label{fint}
\end{equation}
where $\varepsilon $ is the total energy per unit of reduced mass, $
\stackrel{.}{\overline{z}}$ $=d\overline{z}/dt$ and the potential $V$
is
\begin{eqnarray}
V(\overline{z},a)=-\frac 1{2\sigma }\ln \left[
\frac{\overline{z}+\sigma }{ \overline{z}-\sigma }\right] \;,
\label{potential}
\end{eqnarray}
where $\sigma =\sqrt{1-a^2}$ .
We now investigate the time evolution of a two-body system interacting
through the force given in (\ref{force2}) applicable to the
anti-parallel spins black holes. We remark that this space-time is not
asymptotically flat because of the semi-infinity strings. Even so we
assume that Newton's equations of motion are adequate to describe the
black holes motion and that the spins and the masses of the holes are
constants.
Their relative acceleration is given by
\begin{equation}
\frac{d^2z}{dt^2}=-\frac{m_1+m_2}{z^2-\left( m_1-m_2\right)
^2+a_R^2}\;.
\label{accel2}
\end{equation}
Let $\overline{z}=\overline{z}/M$ , $a=a_R/M$ , $\delta \equiv \left(
m_1-m_2\right) /M$ and $t\rightarrow t/M$. The first integral of
equation (\ref{accel2}) is similar to (\ref{fint}) with the potential
\begin{equation}
V(\overline{z},a,\delta )=\left\{
\begin{array}{cc}
-\frac 1{2\beta }\ln \left[ \frac{\overline{z}+\beta
}{\overline{z}-\beta } \right] & \text{for }\delta ^2>a^2 \\ -\frac
1{\overline{z}} & \text{for }\delta ^2=a^2 \\ -\frac 1{\left| \beta
\right| }\left( \frac \pi 2-\arctan \left( \frac{ \overline{z}}{\left|
\beta \right| }\right) \right) & \text{for }\delta ^2<a^2
\end{array}
\right. \label{pot2}
\end{equation}
where $\beta =\sqrt{\delta ^2-a^2}$ .
Therefore we can treat both, parallel and antiparallel spins cases,
using the potential (\ref{pot2}) since $\sigma =\beta $ if we set
$\delta =1$ . If $\delta ^2=a^2$ the potential becomes the Newtonian.
\section{Emission of gravitational radiation}
For the computation of the gravitational radiation luminosity (rate of
emission of gravitational wave energy) we use the standard quadrupole
formula \cite{quadrupoleformula}. In our model this is a good first
approximation to get an estimate of the total amount of gravitational
radiation during the head-on collision. The formula takes an average
over several wave-lenghts and we integrate the luminosity along the
whole collision path up to a point where the holes coalesce.
For a two particle system separated
by a distance $\overline{z}$ the luminosity is given by
\begin{equation}
\frac{dE_g}{dt}=
{\textstyle {6 \over 45}}
\mu ^2\left( \stackrel{\cdot \cdot \cdot }{\overline{z}^2}\right)^2
\;.
\label{lum}
\end{equation}
Our main concern is the energy emitted along the collision process.
Therefore we need the rate of emission of gravitational wave energy
per unit length. For a head-on collision of two bodies with equation
of motion (\ref {fint}) the quadrupole formula (\ref{lum}) yields
\begin{equation}
\frac{dE_g}{d\overline{z}}=\frac{dE_g}{dt}\frac
1{\stackrel{.}{\overline{z}}} =\left\{
\begin{array}{cc}
{\textstyle {8 \over 15}}
\mu ^2\frac{\left( \overline{z}^2-3\beta ^2\right) ^2}{\left(
\overline{z} ^2-\beta ^2\right) ^4}\sqrt{2\varepsilon +\frac 1\beta
\ln \left[ \frac{ \overline{z}+\beta }{\overline{z}-\beta }\right] } &
\text{for }\delta ^2>a^2\;, \\
{\textstyle {8 \over 15}}
\mu ^2\frac 1{\overline{z}^4}\sqrt{2\varepsilon +\frac
2{\overline{z}}} & \text{for }\delta ^2=a^2\;, \\
{\textstyle {8 \over 15}}
\mu ^2\frac{\left( \overline{z}^2+3\left| \beta \right| ^2\right)
^2}{\left( \overline{z}^2+\left| \beta \right| ^2\right)
^4}\sqrt{2\varepsilon +\frac 2{ \left| \beta \right| }\left( \frac \pi
2-\arctan \left( \frac{\overline{z}}{ \left| \beta \right| }\right)
\right) } & \text{for }\delta ^2<a^2\;.
\end{array}
\right. \label{dedr}
\end{equation}
Note that, for $\delta ^2>a^2$ , as $\overline{z}$ is decreasing in
the collision process,$\frac{dE_g}{d\overline{z}}$ reaches a maximum
at $ \overline{z}\approx \sqrt{5}\beta $ , then it decreases to zero
at $ \overline{z}=\sqrt{3}\beta $ and finally it diverges at
$\overline{z}=\beta $ . The behavior of the rate of emission of
gravitational radiation as a function of the distance between the
holes is similar to the Schwarzschild case\cite{our1}. The main
difference is the fact that the two Kerr black holes horizons touch
each other at $\overline{z}=\left( \sigma _1+\sigma _2\right) /M<\beta
<\sigma $ and at this point $\frac{dE_g}{d\overline{z}}$ is finite,
whereas it diverges at the horizon's touching point in the
Schwarzschild case. In any case, as before, the description of the
head-on collision problem by this approach is valid only for
$\overline{z}\geq \overline{z}_{\text{final} }\equiv \beta /\tanh
\left( \frac \beta 2\right) $, the point where the holes would reach
the speed of light if released at the rest from infinity.
For the $\delta ^2<a^2$ case, as $\overline{z}$ is decreasing
$\frac{dE_g}{d \overline{z}}$ reaches a maximum at $\overline{z}=0$
and it is finite everywhere. Again the description is valid only for
$\overline{z}\geq \overline{z}_{\text{final}}\equiv \left| \beta
\right| /\tan \left( \frac{ \left| \beta \right| }2\right) $ due to
the speed limit.
The total amount of gravitational wave energy emitted is then given by
\[
\Delta
E_g(\overline{z}_{\text{initial}},\overline{z}_{\text{final}})=-\int_{
\overline{z}_{\text{initial}}}^{\overline{z}_{\text{final}}}\frac{dE_g}{d
\overline{z}}d\overline{z}\;,
\]
where $\overline{z}_{\text{initial}}$ is the point of release from
rest. We make a restriction to the cases $-1/2<\varepsilon \leq 0$ ,
which corresponds to $ \overline{z}_{\text{final}}\leq
\overline{z}_{\text{initial}}<\infty $. Our results are summarized in
the table below, where the total amount of gravitational wave energy
is given as a function of $\varepsilon $ and $ \beta $ in units
of$\;10^{-3}\times \mu ^2/M$.
The efficiency of the emission of gravitational waves decreases with
$\beta $ for $\delta ^2>a^2$ , increases with $\left| \beta \right| $
for $\delta ^2<a^2$ and decreases with $\varepsilon $ . Note that the
anti-parallel case can emit as much as 3\% of the rest energy $\mu
^2/M$. We see for example that the collision of equal masses extremal
($m=a$) black holes with anti-parallel spins ($\beta =i$) can release
at least 3 times more gravitational wave energy than the similar
collision with parallel spins ($ \beta =1$).
\section{Time variation of spins}
Let us now turn to the possibility of time variation of the holes'
spins. Since both strut or strings attached to the holes have angular
momentum per unit length, they may affect the angular momentum of the
holes along the collision path. It would be a mutual Lense-Thiring
effect or the dragging of each other's inertial frame. The
gravitational radiation does not carry out angular momentum in the
quadrupole approximation due to the axisymetry \cite{will}. Therefore
it is reasonable to expect angular momentum conservation in the
configuration.
In the parallel case the strut's total angular momentum is given by
\cite{letoliv}
\[
J_{\text{strut}}=-\frac{2m_2\,a_1\,\left( \,\,d+\,m_2+m_1\right)
\left( d-\sigma _1-\sigma _2\right)
}{d^2-(m_1-m_2)^2+(a_1-a_2)^2}\;\;.
\]
The constraint $\frac{a_1}{a_2}=\frac{m_1}{m_2}$ has been used. Let the
total angular momentum be $\ell \equiv J_{\text{strut}}+m_1a_1+m_2a_2$ . Let
us assume that $\ell $ is conserved during the collision process. Then $a_1$
and $a_2$ have to change with the separation distance $d$. Let
$\alpha \equiv a_1/m_1=$ $a_2/m_2$, $\delta \equiv m_2-m_1$ and set
$m_1+m_2$ to one. In order to keep $\ell $ constant, $\alpha $ must be a
real root of
\begin{equation}
\sqrt{1-\alpha ^2}\alpha =\left[ \frac{\overline{z}}{\left( \overline{z}
+1\right) }\frac{\left( \delta ^2+2\overline{z}\delta ^2-1\right) }{\left(
1-\delta ^2\right) }-\frac{\delta ^2\left( 1+\delta ^2\right) }{\left(
\overline{z}+1\right) \left( 1-\delta ^2\right) }\left( 1-\alpha ^2\right)
\right] \alpha -2\ell \frac{\left( \overline{z}^2-\delta ^2\left( 1-\alpha
^2\right) \right) }{\left( \overline{z}+1\right) \left( 1-\delta ^2\right) }
\label{eq0}
\end{equation}
For a given initial condition, $\ell $ and $\delta $ are known, then
$\alpha $ is the root of (\ref{eq0}) that changes with $\overline{z}$
from its initial value continuously. We find that $\alpha $ slightly
decreases as the holes get close to each other for most of the initial
conditions. For example, considering identical holes $\left(\delta
=0\right)$ and setting $\ell =0$ we find $\alpha ^2=\left( 2
\overline{z}+1\right) /\left( \overline{z}+1\right) ^2$ . In Fig 1 whe
show the change of $\alpha $ as a function of the separation
$\overline{z}$ for $\eta =1/3$ with initial value $\alpha =1/3$ at
$\overline{z}=40$.
\input psbox
\begin{figure}[tbh]
\mbox{\psbox{fig1.eps}}
\caption{Change of $\alpha $ as a function of the separation $d$ for
$\eta =1/3$ with initial value $\alpha =1/3$ at $d=40$.}
\end{figure}
We shall now consider the torque of the semi-infinite strings on the
holes. The strings total angular momenta are not
finite. Nevertheless, the rate of their change may be computed as
follows
\begin{eqnarray}
S_2\;\stackrel{\cdot }{\overline{z}_2} &=&m_2\stackrel{\cdot }{a_2}
\label{spin1} \\
S_1\;\stackrel{\cdot }{\overline{z}_1} &=&m_1\stackrel{\cdot }{a_1}
\label{spin2}
\end{eqnarray}
where $S_k$ , $k=1,2$ are the strings spins given by \cite{letoliv}
\begin{eqnarray*}
S_1 &=&S_{1;2}+\frac{2m_1a_2\,d}{\Delta ^2} \\
S_2 &=&S_{1;2}+\frac{2m_2a_1\,d}{\Delta ^2} \\
S_{1;2} &\equiv &-\frac{2m_1m_2a_2}{\Delta ^4}\,\left( m_2^2-\left(
m_1+d\right) ^2\right) +a_1\;\,\left( m_1-\left( m_2+d\right) ^2\right)
-\left( a_1-a_2\right) \left( a_1^2-a_2^2\right)
\end{eqnarray*}
where $\Delta ^2=d^2-(m_1+m_2)^2+(a_1-a_2)^2$ . That is, we assume
that each string exchanges angular momentum with its attached black
hole. We use a fixed center of mass reference system in which
$z_2-z_1=d$ and $ m_1z_1+m_2z_2=0$ and a further requirement that the
holes angular momentum $\ell _H\equiv m_1a_1+m_2a_2$ is kept constant
along the collision path. Then the equations (\ref{spin1}-\ref{spin2})
are equivalent to the algebraic equations
\begin{eqnarray}
m_1S_2-m_2S_1 &=&0 \label{spin3} \\
m_1a_1+m_2a_2 &=&\ell _H \label{spin4}
\end{eqnarray}
Thus, $a_1\left( d\right) =a_1\left( d_{\text{initial}}\right)
+A\left( d\right) -A\left( d_{\text{initial}}\right) $ where $A\left(
d\right) $ is the solution of (\ref{spin3}-\ref{spin4}), that is, it
is the real root of
\begin{eqnarray*}
P(A) &\equiv &4\left( \,\delta ^2-1\right) A^3-4l\left( \delta +2\delta
^2-3d\right) A^2 \\
&&+\left[ 4l^2\left( 2\delta +\delta ^2-3d\right) +d\left( d^2-1\right)
\left( 2\delta -1\right) +\left( d+1\right) ^2\left( \delta ^4-\delta
^2\left( d^2-1\right) -2\delta ^3\right) \right] A \\
&&-l\left( \left( d+1\right) \left( \delta -1\right) ^2\left( \delta
^2+d\delta -d^2+d\right) +4l^2\left( \delta -d\right) \right) =0
\end{eqnarray*}
Here again we set $m_1+m_2$ to one and $\delta \equiv m_2-m_1$. We
find that the holes spins decreases as the holes get close to each
other. In Fig. 2 we show the change of spins during the collision
process.
\begin{figure}[tbh]
\mbox{\psbox{fig2.eps}}
\caption{Change of spins during the collision process. The settings
are the same of Figure 1 except for the initial values $a_1=-1/3$ and
$a_2=1/6$}
\end{figure}
The change in the spins are small during most of the collision
process, so we neglet the change in the gravitational radiation output
presented in the Table I.
In both cases, strut and string's, we see the transfer of angular
momentum from the defects to the holes. One then expects some exchange
of angular momentum between the holes along the collision path.
\section{Conclusions}
We developed a semi-analytical approach of treating the head-on
collision problem of two Kerr black holes. This is based on an exact
solution of Einstein's field equations that can be interpreted as the
configuration of two Kerr black holes with some singularities on the
axis from which we obtain the attraction force between the holes. We
use the force and Newtonian equations of motion to find the dynamics
of the holes. We then calculate the amount of gravitational radiation
released in the collision using the quadrupole radiation formula. We
also discuss the possible exchange of the holes spins.
It is well known that the efficiency of conversion of rest mass into
gravitational radiation in the head-on collision problem is extremely
small for Schwarzschild black holes. We expect higher efficiency for
Kerr black holes. If we consider two identical spinning holes far
apart and compare their initial energies with the final energy of a
single Kerr hole, assuming hole's angular momentum conservation, the
maximum radiation efficiency is 29\% for parallel spins whereas the
maximum ranges between 29\% and 50\% for antiparallel
spins\cite{Yorketal}. Our results show that the process is more
efficient for the case of Kerr black holes than for the Schwarzschild
black holes, but still small. The Kerr holes emit at most 3\% of the
rest energy $ \mu ^2/M$ where $\mu $ and $M$ are the reduced and total
mass respectively whereas the Schwarzschild holes emit no more than
1\% \cite{our1}. Recall that $\mu/M \leq 1/4$, so that the maximum
radiation efficiency we found is less than $ 0.2\% $ of the total rest
energy $M$.
According to the area theorems for coalescing black holes the
antiparallel case can radiate more energy than the parallel
case\cite{HawkingGibbons}. Our approach agree with the area
theorems. See the first (last) columns of Table I for the
gravitational wave output of parallel (antiparallel) holes in
collision.
We also found that the spin-spin interaction gives a repulsive
contribution to the force for both parallel and antiparallel spins. In
contradistinction, according to Maxwell theory, two magnetic dipoles
repel (attract) each other and, according to some authors, a spinning
test particle is attracted to (repelled from) a large rotating body if
they are aligned in parallel (antiparallel)\cite{wald}.
We then addressed the possible exchange of spins between the
holes. With simple and reasonable assumptions we conclude that the
holes may spin down or spin up depending on the initial conditions and
on the constant parameters of the configuration. The effect is small
for most of the collision path. When the holes are close enough the
spins may then change up to 100\%. It is known that the spin-spin
force is of post-post-Newtonian magnitude \cite{ThorneHartle}.
Further investigations must be done towards the clarification of this
spin exchange effect.
\section{Acknowledgments}
M.E. Ara\'{u}jo and S.R. Oliveira thank FAPDF for financial support
and IMECC- Unicamp and Departamento de Matem\'{a}tica-UnB for their
hospitality during the preparation of this work. M.E. Ara\'{u}jo and
P.S. Letelier thank CNPq for research grants. P.S. Letelier and
S.R. Oliveira also acknowledge FAPESP for financal support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,811 |
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See our other Electronics Calculators. Cohen's d = (M2 - M1) \u2044 SDpooled. 45 + 1. 8 30","date":"2021-05-12 01:03:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.46657314896583557, \"perplexity\": 2486.9392783386606}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991693.14\/warc\/CC-MAIN-20210512004850-20210512034850-00304.warc.gz\"}"} | null | null |
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Ministry of Defence and Cabinet Office: Central Intelligence Machinery: Joint Intelligence Sub-Committee Later Committee: Minutes (JIC Series). Joint Intelligence Committee. Meetings 70-139 > 1. Review of the Situation Round the Soviet and Satellite Perimeter and other Sensitive Areas; 2. J.I.C. Weekly Intelligence Review (Europe) for S.H.A.P.E.; 3. Likelihood of the Soviet Union Negotiating a German Peace Treaty; 4. The Effect on Israel of the Association of Egypt with the Middle East Command; 5. Review of Assessments Made since January 1947. By the J.I.C. On Communists Intentions; 6. Review of the Middle East and North Africa; 7. Possible Extension of the War in the Far East; 8. Intelligence from Sweden on the Soviet Union; 9. Visit of Director General, Security Service, to New Zealand; 10. Intelligence for N.A.T.O.; 11. S.H.A.P.E. Request for Intelligence on Soviet Air Defence Capabilities; 12.
Document Title 1. Review of the Situation Round the Soviet and Satellite Perimeter and other Sensitive Areas; 2. J.I.C. Weekly Intelligence Review (Europe) for S.H.A.P.E.; 3. Likelihood of the Soviet Union Negotiating a German Peace Treaty; 4. The Effect on Israel of the Association of Egypt with the Middle East Command; 5. Review of Assessments Made since January 1947. By the J.I.C. On Communists Intentions; 6. Review of the Middle East and North Africa; 7. Possible Extension of the War in the Far East; 8. Intelligence from Sweden on the Soviet Union; 9. Visit of Director General, Security Service, to New Zealand; 10. Intelligence for N.A.T.O.; 11. S.H.A.P.E. Request for Intelligence on Soviet Air Defence Capabilities; 12.
Reference CAB 159/10
Document Date 20 September 1951
Conflicts Cold War
Themes Military Intelligence and Operations, Foreign Policy and International Relations, Intelligence Organisation and Administration
Regions Africa, Atlantic, East Asia, Europe, Middle East, North America, Pacific
Countries Bulgaria, Burma, China, Egypt, France, Germany, Indochina, Israel, Korea, New Zealand, Persia, Poland, Soviet Union, Sweden, United States of America, West Germany, Yugoslavia
Document Type Meeting Minutes, Reports
Organisations British Joint Services Mission, Chiefs of Staff, Commonwealth Relations Office, Foreign Office, Joint Intelligence Bureau, Joint Intelligence Staff, Joint Planning Staff, Ministry of Defence, Security Service, United Nations, Joint Intelligence Sub-Committee
People Anthony Buzzard, John Gardiner, Stewart Menzies, Neill Ogilvie-Forbes, Patrick Reilly, Eric Searight, Percy Sillitoe, Kenneth Strong, Edward Travis
Notes Sections of this document have been retained under Section 3(4) of the Public Records Act, 1958.
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Chinese foreign policy, Chinese army, troop movements, military dispositions, military supplies, military operations, Chinese communists, Hong Kong, colonies, empire, First Indochina War, colonial independence movements, military strength, assessing enemy strength, volunteers, artillery, equipment, Soviet army, Burmese politics, Chinese Nationalists, Arab world, political pressure for peace, information sharing, intelligence reports, alliances, treaties, peace proposals, occupied Germany, Anglo-Egyptian relations, Anglo-Israeli relations, diplomacy, foreign policy, predicting enemy intentions, Allied strategy, strategy, intelligence channels, Swedish foreign policy, intelligence gathering, intelligence services responsibilities, defence, anti-aircraft defence, intelligence requirements, weapons development, Soviet-occupied Europe, Soviet navy, warships, Polish armed forces, manpower, Bulgarian armed forces, diplomatic disputes, border disputes, press reports, Yugoslav foreign policy
1. Director, H.C.H.Q.; 2. Intelligence from Japan; 3. Threat to Hong Kong; 4. Likelihood of the Soviet Union Initiating a German Peace Treaty; 5. The Effect on Israel of the Association of Egypt with Middle East Command; 6. World Wide Intelligence Organisation in War; 7.; 8. Future Developments in the Middle East; 9. Review of the Situation Round the Soviet and Satellite Perimeter; 10. Joint Intelligence Committee Weekly Intelligence Review (Europe) for S.H.A.P.E.; 11. N.A.T.O. National Intelligence Staffs Conference; 12. Intelligence for N.A.T.O.; 13. Intelligence for N.A.T.O. - Lecture at N.A.T.O Defence College; 14. Intelligence from Korea
1. Exchange of Information with Yugoslavia; 2. Intelligence for S.H.A.P.E.; 3. Military Implications of the Entry of British Forces into Persian Territory; 4. Invasion of Te United Kingdom, 1951-1952; 5. Liaison Between United States and United Kingdom Intelligence Organisations; 6. Study by the E.M.M.O. Regional Planning Group of the Association of Turkey and Greece with N.A.T.O. Military Planning; 7. Review of the Situation Round the Soviet and Satellite Perimeter; 8. Information on Formosa; 9. Acquisition of Merchant Shipping by Soviet and Satellite Countries; 10. J.I.C. Weekly Intelligence Review (Europe) for S.H.A.P.E.
1. Present State of Tension between India and Pakistan; 2. Situation on the Sino-Burmese Frontier; 3. Intelligence for S.H.A.P.E.; 4. Military and Economic Strength of the Soviet Union; 5. Chinese Communist Intentions in the Light of their Experiences during the First Ear of War in Korea; 6. Soviet Research and Development; 7. Review of the Situation Round the Soviet and Satellite Perimeter; 8. J.I.C. Weekly Intelligence Review (Europe) for S.H.A.P.E.
1. Intelligence for N.A.T.O.; 2. Intelligence from Sweden on the Soviet Union; 3. Reducing the Enemy Threat to the U.K.; 4. Tripartite Intelligence Conference on South East Asia; 5. Meetings during Week Ending 26th October; 6. Review of the Situation Round the Soviet and Satellite Perimeter and other Sensitive Areas; 7. Joint Intelligence Committee Weekly Intelligence Review (Europe) for S.H.A.P.E.
1. TASS Agency Radio Monitoring Station; 2. Moscow Publication "News"; 3. Potential Threat to the British Position in Egypt; 4. Chinese Communist Intentions in the Light of their Experiences during the First Year of War in Korea; 5. Intelligence for Planning Purposes; 6. Review on the Situation Round the Soviet and Satellite Perimeter; 7. J.I.C. Weekly Intelligence Review (Europe) for S.H.A.P.E.
1. Military and Economic Strength of the Soviet Union; 2. Review of the State of Our Intelligence and Measures Recommended to Improve It; 3. Future of Intelligence Division - Germany; 4. Review of the Situation Round the Soviet and Satellite Perimeter; 5. J.I.C. Weekly Intelligence Review (Europe) for S.H.A.P.E.; 6. Intelligence for S.H.A.P.E.; 7. Meetings During the Whitsun Period
1. Review of the Situation Round the Soviet and Satellite Perimeter; 2. J.I.C. Weekly Intelligence Review (Europe) for S.H.A.P.E.; 3. Probable Soviet Capabilities and Strategy in the Event of Total War in 1952; 4. Intelligence for S.H.A.P.E.; 5. Intelligence Division, Germany; 6. Basic Intelligence Appreciation for Dutch and Belgian Units in Northern Army; 7. Chinese Communist Intentions in the Light of Their Experience During the First Yeat of War in Korea; 8. Major R.D.Wilson
1. Invasion of the United Kingdom; 2. Scale and Nature of Air Attack on the United Kingdom 1951-1954; 3. Chinese Communist Intentions in the Light of Operations in Korea; 4. Review of the Situation round the Soviet and Satellite Perimeter; 5. J.I.C. Weekly Intelligence Review (Europe) for S.H.A.P.E.; 6.; 7. Intelligence for S.H.A.P.E.
1. Developments in Soviet Policy and Tactics; 2.; 3. TASS Agency Radio Monitoring Station; 4. Soviet Service Attache Representation in the United Kingdom; 5. Soviet Air Force Activity in the (Soviet) Far East; 6. Review of the Situation Round the Soviet and Satellite Perimeter; 7. Weekly Appreciation of Indications of Aggessive Moves by the Communists in the F.E.; 8. J.I.C. Weekly Intelligence Review (Europe) for S.H.A.P.E.; 9. Encouragement of Defectors
1. Meeting with the Chiefs of Staff; 2. "Science for Peace"; 3. "Intelligence Digest-A Review of World Affairs; 4. Communist Sponsored Youth Festival in Sheffield; 5. Measures to Improve Our Intelligence; 6. Intelligence Guidance for the Standing Group - S.G. 176; 7. Meeting with the Directors of Plans; 8. Review of the Situation Round the Soviet and Satellite Perimeter and Other Sensitive Areas; 9. Relations Between Service Attaches and N.A.T.O. Subordinate Commands
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Harold Adrian Linstone (15 June 1924 – 8 July 2016) was a German-American mathematician, consultant, futurist and University Professor Emeritus of Systems Science at Portland State University and a specialist in applied mathematics.
Biography
Harold Linstone was a naturalized citizen of the United States born in Hamburg, Germany in 1924. He received an M.A. from Columbia University and a PhD from the University of Southern California, both in mathematics.
Linstone worked for twenty-two years in industry, which included positions at Hughes Aircraft and Lockheed Corporation since 1963, where he was Associate Director of Corporate Planning - Systems Analysis since 1968. He has been a consultant to many organizations, including the US House of Representatives, State of Alaska oil Spill Commission, Alberta Economic Development Commission, and UN Asian-Pacific center for Technology Transfer, as well as corporations such as IBM and United Airlines.
Later he worked as University Professor of Systems Science at Portland State University, where from 1970 to 1977 he served as director of its Systems Science PhD Program and Futures Research Institute. He has served as visiting Professor at the University of Rome, the University of Washington, and Kiel University in West Germany.
Harold Linstone was editor-in-chief of the professional journal "Technological Forecasting and Social Change", which he founded in 1969, and which is now in its 56th volume. In 1993 to 1994 he served as president of the International Society for the Systems Sciences. In 2003 he won the World Future Society's Distinguished Service Award.
He died on 8 July 2016 in Pasadena, California.
Work
The Delphi Method, 1975
According to Linstone and Murray Turoff (1975) the concept underlying the Delphi method is developed in defense research by the Rand Corporation sponsored by the US Air Force, which started in the early 1950s. The original goal of the research project was "obtain the most reliable consensus of opinion of a group of experts ... by a series of intensive questionnaires interspersed with controlled opinion feedback."
The most noted outcomes were published in the 1962 memorandum of the Rand Corporation, entitled "An experimental application of the Delphi method to the use of experts" by Norman Dalkey and Olaf Helmer, republished under the same title in Management science in 1963. The research had started a decade earlier, and was published earlier in the RAND Memorandum, entitled "The Use of Experts for the Estimation of Bombing Requirements." It concerned the application of "expert opinion to the selection, from the point of view of a Soviet strategic planner, of an optimal U. S. industrial target system and to the estimation of the number of A-bombs required to reduce the munitions output by a prescribed amount."
Linstone and Turoff (1975) further explained that "it is interesting to note that the alternative method of handling this problem at that time would have involved a very extensive and costly data-collection process and the programming and execution of computer models of a size almost prohibitive on the computers available in the early fifties. Even if this alternative approach had been taken, a great many subjective estimates on Soviet intelligence and policies would still have dominated the results of the model. Therefore, the original justifications for this first Delphi study are still valid for many Delphi applications today, when accurate information is unavailable or expensive to obtain, or evaluation models require subjective inputs to the point where they become the dominating parameters. A good example of this is in the "health care" evaluation area, which currently has a number of Delphi practitioners."
Multiple Perspectives for Decision Making, 1984
The 1984 book Multiple Perspectives for Decision Making, again co-authored with Ian Mitroff, presented a multiple perspective approach for decision making. This work was based on ideas of Graham T. Allison, published in his Essence of Decision: Explaining the Cuban Missile Crisis from 1971. Linstone (1999) explained:
Allison had seen that his analysis and modeling for corporate decision making only took into account some of the factors vital in the corporate decision process and Allison's work examined the missile crisis from three different points of view, rational actor, organizational process, and bureaucratic politics. Each provided insights not obtainable with the others.
Combined with his own experience in the aerospace industry, Linstone & Mitroff distinguished three types of perspectives for decision making. At first the Technical Perspectives (T), with the characteristics:
Problems are simplified by abstraction, idealization, and isolation from the real world around us. There is the implicit assumption that the processes of reduction and simplification permit "solution" of problems.
Data and models comprise the basic building blocks of inquiry. Logic and rationality as well as objectivity are likewise presupposed. Order, structure, and quantification are sought wherever possible. Observation and model building, experimentation and analysis are usually aimed at improving predictive capability. Validation of hypotheses and replicability of observations and experiments are expected. The attainment of elegant models and best or optimal solutions is particularly prized.
Second The Organizational Perspectives (O), which "focuses on process rather than product, on action rather than problem-solving. The critical questions are 'does something need to be done, and if so, what?' and 'who needs to do it and how?' rather than 'what is the optimal solution?' There must be a recognition that top-down imposition of solutions may well fail if there is no 'bottom-up' support."
And third the Personal Perspectives (P), which "views the world through a unique individual. It sweeps in aspects that relate individuals to the system and are not captured by technical and organizational perspectives."
Later Linstone further developed this approach to decision making, and presented it in his 1999 Decision Making for Technology Executives: Using Multiple Perspectives to Improve Performance.
Publications
Books published by Linstone:
1975. The Delphi Method. Edited with Murray Turoff. Addison-Wesley. (online)
1976. Futures Research: New Directions. Edited with W. H. Clive Simmonds. Mass. : Addison-Wesley Pub. Co.
1977. Technological Substitution: forecasting techniques and applications. Edited with Devendra Sahal. New York : Elsevier Pub. Co.
1984. Multiple Perspectives for Decision Making : bridging the gap between analysis and action. North-Holland : Elsevier Science Pub. Co.
1993. The Unbounded Mind : breaking the chains of traditional business thinking. With Ian Mitroff. New York : Oxford University Press.
1994. The Challenge of the 21st Century: managing technology and ourselves in a shrinking world. With Ian I. Mitroff . Albany : State University of New York Press.
1999. Decision making for technology executives : using multiple perspectives to improved performance. Boston : Artech House.
References
External links
Harold A. Linstone on www.isss.org.
The Delphi Method: Techniques and Applications online.
1924 births
2016 deaths
20th-century American mathematicians
21st-century American mathematicians
American systems scientists
Portland State University faculty
German emigrants to the United States
University of Southern California alumni
Presidents of the International Society for the Systems Sciences | {
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\section{Introduction}\label{sec:intro}
In mechanics, variable-mass systems are systems which have mass that does not remain constant with respect to time. In such systems, Newton's second law of motion cannot directly be applied because it is valid for constant mass systems only. Instead, a body whose mass $m$ varies with time can be described by rearranging Newton's second law and adding a term to account for the momentum carried by mass entering or leaving the system \cite{Plastino,Alonso}.
Due to some conceptual difficulties, this topic is not commonly addressed in basic physics courses. So it may be pedagogically interesting to propose new approaches to the topic for students of science and engineering at the undergraduate level.
In this respect, the study of the motion of a single oscillator as its mass
varies can serve as a rich topic of discussion in a physics classroom. What happens to the equilibrium position, the amplitude, the period and frequency of a single leaking oscillator? The answer to this question is of pedagogical interest.
In this article, we discuss the proper form of Newton's second law when applied to a single-degree of freedom system with a time-dependent mass. Using theoretical considerations, we present a description of the dynamics of the single one-dimensional oscillator with the mass modeled as a quadratic function of time. The obtained equation of motion is solved by using a suitable numerical procedure for given initial conditions.
The work is mainly addressed to college students and teachers. The study of this topic requires acquaintance with basic concepts of calculus and physics at basic level.
\section{Newton's second law for variable-mass systems}\label{sec:sec2}
Consider a particle of mass $m$ which is moving with velocity $v$. By definition its linear momentum is $p=mv$. According to Newton's second law, the change of the linear momentum $p$ in time is determined by $F = \frac{dp}{dt}= \frac{d}{dt} (m v)$, where $F$ is the net force acting on the particle. If the particle mass is a constant (mass does not depend on time), the Newton's second law entails $F = m \frac{dv}{dt}=ma$. The last equation represents Newton's second law commonly presented in textbooks. This form is particularly useful in obtaining the equation of motion of a constant mass particle.
In addition, Newton's second law is invariant under the Galilean transformations, which are defined by $x^{\prime}=x-ut$, and $v^{\prime} = v-u$, where $u$ is the velocity of the primed inertial frame of reference relative to the unprimed inertial frame. Applying the Galilean transformations to Newton's second law we get simply $F^{\prime }=F$. That is, all inertial observers must measure the same force regardless the relative speed among them.
But, what if the the particle mass is not constant in time? Can we use in this case Newton's second law forms $F = m \frac{dv}{dt}$ or even $F = \frac{d}{dt} (m v)$? The answer is no. To see that, remember that the invariance of Newton's second law enforces that the equation of motion in the primed frame of reference must retain the same form, that is:
\begin{equation}
F^{\prime} = \frac{d}{dt} (m v^{\prime}). \label{cmass33c}
\end{equation}
The time derivative in the right-hand side of (\ref{cmass33c}) yields
\begin{equation}
F^{\prime} = m \frac{dv}{dt} + \frac{dm}{dt} \left( v - u \right),
\end{equation}
and hence
\begin{equation}
F^{\prime} = \frac{d}{dt}\left( mv \right) - \frac{dm}{dt} u \ne F .
\end{equation}
So, Newton's second law in the form $F = \frac{d}{dt} (m v)$ is not Galilean invariant when the particle mass is time dependent. In order to properly obtain the equation of motion, we have to apply the principle of conservation of linear momentum for the entire system, which is the basic principle behind the Newton's second law. Thus, consider a single-degree of freedom system with a time-varying mass $m(t)$, as illustrated in Figure \ref{fig:100}. The system (the body labeled 1 in the figure) moves with velocity $v$ at the time $t$. The particle of mass $\Delta m$ (labeled 2 in the figure) and mean velocity $w$ collides with the system during a time interval $\Delta t$, imparting mass to the system. Assuming that the mass of the entire system is conserved during the collision, the new mass and the new velocity of the original system increase to $m + \Delta m$ and to $v + \Delta v$, respectively. The linear momentum of the system at the time $t$ is thus given by $ p(t) = mv + \left( \Delta m \right) w$, while the new linear momentum at the time $t+\Delta t$ reads $p(t+\Delta t) = (m + \Delta m)(v +\Delta v)$. Hence, the change in the total linear momentum is
\begin{equation}
\frac{\Delta p}{\Delta t} = m \frac{\Delta v}{\Delta t} + \frac{\Delta m}{\Delta t} \Delta v - \frac{\Delta m}{\Delta t} \left(w - v\right) . \label{varp1a}
\end{equation}
\begin{figure}[htpb]
\centering
\includegraphics[width=.22\textheight]{Figure_1.eps}
\vspace*{-.1cm}
\caption{(Color online) - The particle of mass $\Delta m$ and velocity $w$ collides with a particle of mass $m$ and gets stuck in it. After the process, the new particle of mass $m + \Delta m$ moves with velocity $v + \Delta v $. \label{fig:100}}
\end{figure}
Taking the limit $\Delta t\rightarrow 0$, $\Delta m\rightarrow 0$, and $\Delta v \rightarrow 0$ in equation (\ref{varp1a}), one arrives to \cite{Plastino}
\begin{equation}
F = \frac{dp}{dt}=m\frac{dv}{dt} - \frac{dm}{dt}\left(w - v\right), \label{dpdt1a}
\end{equation}%
where $F$ is the external force acting on the system, and $w - v$ is the velocity of the incoming mass with respect to the centre of mass. Equation (\ref{dpdt1a}) can be put in the form
\begin{equation}
m\frac{dv}{dt}= F + \frac{dm}{dt} \left(w - v\right) , \label{eqmotion1a2}
\end{equation}
Analogously, for $\frac{dm}{dt}<0$ (system losing mass) we would obtain
\begin{equation}
m\frac{dv}{dt}= F - \frac{dm}{dt} \left(w - v\right) . \label{eqmotion1a2b}
\end{equation}
Equations (\ref{eqmotion1a2}) and (\ref{eqmotion1a2b}) describe the motion of a time-varying mass particle, and represent the proper extension of Newton's second law. The term $\frac{dm}{dt} (w-v) $ in the right-hand side should be interpreted as a real force acting on the particle, apart from the external force $F$. Also notice that equation (\ref{dpdt1a}) may be put in the form
\begin{equation}
F = \frac{d}{dt} (m v) - \frac{dm}{dt} w , \label{eqmotion1a3}
\end{equation}
which means that equation (\ref{eqmotion1a3}) recovers the form $F = \frac{d}{dt} (m v)$ in the particular case $w=0$. It is easy to prove that equation (\ref{eqmotion1a3}) is invariant under Galilean transformation.
\section{Modeling a leaking oscillator}\label{sec:sec4}
In order to model the variable-mass oscillator, consider a leaking bucket of water which is attached to a spring, as illustrated in Figure {\ref{fig:200}}. The water exits out the bucket through a small hole at the bottom. Assume that the mass loss of water and the motion of the oscillator are along a line (the $z$-axis). In this situation, and ignoring friction, the system is subjected to the action of three different forces, namely, the elastic force exerted by the spring, the weight of the oscillator, and the force exerted by the leaking water. In accordance with equation (\ref{eqmotion1a2b}), the dynamical behaviour of the system is governed by the equation of motion
\begin{equation}
m \dot v = -\frac{d m}{d t} (w-v) - kz - mg, \label{eqmotion2}
\end{equation}
where $z(t)$ is the displacement of the centre of mass measured from the initial equilibrium position; $w$ is the mean velocity at which the water leaves the system; $v=\dot z$ is the velocity of the oscillator; $k$ is the stiffness coefficient of the linear restoring force; and $g$ is the acceleration of gravity.
\vspace*{.2cm}
\begin{figure}[htpb]
\centering
\includegraphics[width=.22\textheight]{Figure_2.eps}
\vspace*{-.1cm}
\caption{(Color online) - Oscillator with a variable mass. A bucket filled with water is attached to a spring. The water flows out through a small hole in the bottom of the bucket. \label{fig:200}}
\end{figure}
For the conditions $w=v=0$, $\frac{dm}{dt}=0$, and $\dot v =0$, at the time $t=0$, we obtain the initial equilibrium position
\begin{equation}
z_{0}=-\frac{m(0) g}{k} , \label{zinit}
\end{equation}
where $m(0)$ is the initial mass of the oscillator. If $m$ is constant, the system would oscillate around the equilibrium position $z_{0}$.
The mass of water has a quadratic dependence on the time (see appendix for details of calculation) which is given by
\begin{equation}
m_w(t) = m_w(0) \left(1-f t\sqrt{\frac{g}{2h_{0}}}\right)^{2}, \label{masst}
\end{equation}
where $m_w(0)$ is the initial mass of water, $f=\frac{a}{A}$ is the ratio between the cross-sectional area $a$ of the hole, and the cross-sectional area $A$ of the column of water, and $h_{0}$ is the initial height of the column of water. The mass of the oscillator is given by the summation of the mass of the bucket $m_b$, and the time-varying mass of water $m_w(t)$.
Assuming the leaking of water occurs at a very low rate, one can neglect the effect of the first term on the right side of equation (\ref{eqmotion2}). In this approach the equation of motion reads
\begin{equation}
\dot v = - \frac{k}{m_b + m_w}z - g. \label{61}
\end{equation}
According to equation (\ref{masst}), the bucket of water is completely empty after the elapsed time given by
\begin{equation}
\tau=\frac{1}{f}\sqrt{\frac{2h_{0}}{g}} . \label{60}
\end{equation}
After the elapsed time $\tau$, the oscillations are governed by the equation of motion
\begin{equation}
\dot v = -\frac{k}{m_b}z- g, \, \, \, \, \, \, t\geq\tau . \label{62}
\end{equation}
\section{Numerical solution}\label{sec:sec5}
Equation (\ref{62}) represents a simple harmonic motion, whose exact solution is known and is given by a sinusoidal function. However, the exact solution of equation (\ref{61}) is beyond the scope of the present article. That does not mean we have to give up the equation (\ref{61}). It is possible to deal with approximate numerical solutions, which can be discussed in standard physics class. With this aim, we use a simple numerical method which can be implemented, for example, in electronic calculators or even by using an Excel spreadsheet.
We start with a discrete set of equidistant instants of time $\{ t_i \}$, where $t_i=t_0 +i h$. The time step $h$ and the starting time $t_0$ are suitable chosen in order to provide results as accurate as possible. From equation (\ref{61}), we see that the time evolution of $v(t)$, the value of $\dot v$, depends on the current value $z(t)$ and the mass of water $m_w(t)$. With a sufficiently small time step $h$ the values of $v(t)$ and $z(t)$ will vary approximately linearly between two consecutive instants of time $t_i$ and $t_{i+1}$. So, we may calculate the value $v(t_{i+1})$ from $v(t_{i})$ and the values of $z(t_i)$ and $m_w(t_i)$, setting $\dot v \approx (v_{i+1} -v_i)/h$. Analogously, we may compute the value of $z(t_{i+1})$ from $z(t_{i})$ and $v(t_i)$, by setting now $\dot z \approx (z_{i+1} -z_i)/h$.
So, for the time interval $0\leq t\leq\tau$, we first compute $v(t)$ at the time step $t_{i+1} \doteq t_i + h$:
\begin{equation}
v_{i+1}=v_{i} - h \left(\frac{k}{m_b + m_w(t_i)} z(t_i) + g \right) , \label{z11}
\end{equation}
and thus one computes the value of $z(t)$
\begin{equation}
z_{i+1} = z_i + hv_{i}, \label{az11}
\end{equation}
with $z_{i+1} = z(t_{i+1})$, $v_{i+1} = v(t_{i+1})$, and $m_{i} = m_w(t_i)$, as given by equation (\ref{masst}).
Thus, the output of each loop are the values $z_{i+1}$ and $v_{i+1}$, provided the values of $v_{i}$, $z_{i}$, and $m_{i}$ of the previous loop.
In order to compute the dynamical evolution for $t_i>\tau$, replace equation (\ref{z11}) by
\begin{equation}
v_{i+1} = v_i - h \left( \frac{k}{m_b}z_i + g \right) , \label{62bn}
\end{equation}
according to (\ref{62}).
The approach of numerically solving ordinary differential equations outlined above is known as Euler's method. Eventually, more sophisticated methods, as the family of Runge-Kutta methods, are extensions of these basic ideas.
\section{Numerical results}\label{sec:sec6}
In this section we present results obtained for the case of a bucket of mass $m_b = 1.0$ kg, filled with the initial mass of water of $m_w(0) = 10$ kg with a column of initial height $h_0 = 0.5$ m. The bucket of water is attached to the spring of stiffness coefficient $k=100$ N/m. The mass of the oscillator is given at every time by $m(t) = m_b + m_w(t)$, $m_w(t)$ being the time-varying mass of water.
The algorithm for solving the equations (\ref{61}) and (\ref{62}) comprises the following steps. First, assign initial values to all variables: the starting time $t_0 = 0$; the initial position $z=z(0)$; the initial velocity $v(0)$; the initial mass of water $m_w(0)$; the initial height of the water column $h_0$; the value of the ratio between the cross-sectional areas $f=\frac{a}{A}$. Assign values to constants $g$, $k$, and the mass of the bucket $m_b$. Now, compute the velocity $v_{i+1}$ given by equation (\ref{z11}) valid for $0\leq t \leq \tau$, or by (\ref{62bn}) for $t_i > \tau$. Then, compute the position $z_{i+1}$, given by equation (\ref{az11}). At each step the elapsed time is incremented by the time step $h$.
\begin{figure}[htpb]
\centering
\includegraphics[width=.60\textheight]{Figure_3.eps}
\vspace*{-.2cm}
\caption{(Color online) - Screen shot of the spreadsheet implementation of the numerical procedure described in Section \ref {sec:sec5} and detailed above. The columns contain the sequences $t_i$, $z_i$, $v_i$, $a_i$, $m_w(t_i)$ and the $z-z_0$, with $z_0$ given by equation (\ref{zinit}). Subsequent values of $z_{i+1}$ and $v_{i+1}$ are computed from cell entries in rows ($i-1$). The plot depicts the obtained results for $z(t)-z_0$. The starting values of $z$, $v$, and $m_w$ are $0$ m, $0$ m$\cdot$s$^{-1}$ and $10$ kg, respectively. The used values of the other parameters are $g=9.8$ m$\cdot$s$^{-2}$, $k=100.0$ kg$\cdot$s$^{-1}$, $m_b = 1.0$ kg (mass of the bucket), and $h = 0.05$ s (the time step). \label{fig:30}}
\end{figure}
We start with the initial condition $z(0)=0$ and $v(0)=0$. Thus, the change of the dynamical state of the system is purely caused by the change in mass of the oscillator with time. We also compute the kinetic energy, $T=mv^2/2$, the elastic potential energy, $U_k=k z^2/2$, and the gravitational potential energy, $W = m g (z-z_0)$. The mechanical energy of the system is thus given by the summation $E= T + U_k + W$.
For the value of the ratio $f=a/A=0.01$, for example, the water takes tens of seconds to exit the bucket completely. So, this number does not demand a huge computational time, providing very accurate numerical results. In the simulations presented here, we adopt the value $h = 0.05$ s for the time step. With this value, we need nicely a few hundred loops to carry out the simulation.
The screen shot shown in Figure \ref{fig:30} illustrates the spreadsheet implementation of the numerical integration of equations (\ref{61}) and (\ref{62}) using the Euler method. We have implemented the essentially same algorithm in a Fortran compiler, and some obtained results are shown in the figures below.
\begin{figure}[htpb]
\centering
\includegraphics[width=.30\textheight]{Figure_4.eps}
\vspace*{-.1cm}
\caption{(Color online) - Position as a function of time for $f=0.01$. The used values of the other parameters are $g=9.8$ m$\cdot$s$^{-2}$, $k=100.0$ kg$\cdot$s$^{-1}$, $m_b = 1.0$ kg (mass of the bucket), $m_w(0) = 10.0$ kg (initial mass of water), $h(0) = 0.5$ m (initial height of the water column), and $h = 0.05$ s (the time step). \label{fig:40}}
\end{figure}
\begin{figure}[htpb]
\centering
\includegraphics[width=.30\textheight]{Figure_5.eps}
\vspace*{-.1cm}
\caption{(Color online) - Energy as a function of time for $f=0.01$. The values of the other parameters are the same used in the previous figure. \label{fig:50}}
\end{figure}
Figure \ref{fig:40} depicts the position of the oscillator as a function of time for the adopted values of the model parameters outlined in the caption of the figure. Notice as the ''instantaneous'' equilibrium position of the oscillator moves upward while the water within the bucket flows out. The oscillations are obviously caused by the action of the restoring force, as the mass of the oscillator decreases. In special, one notices that the amplitude of the oscillations decreases, while the frequency increases as the mass of the oscillator decreases. The final equilibrium position, around which the system oscillates for $t>\tau$, can be computed as $z_{eq}=\frac{m_w(0)g}{k}$, which in the present case for $m_w(0)=10$ kg and $k=100$ N$/$m gives the value $0.98$ m.
Figure \ref{fig:50} shows the behaviour of the energy of the oscillator as a function of time for the same set of values of the parameters used in Figure \ref{fig:40}. We conclude that part of energy of the system is dissipated due to loss of mass, that is, the leaking oscillator behaves like a damped oscillator.
\section{Conclusions}\label{sec:sec8}
In this work, we discuss the appropriate form of Newton's second law applied to single-degree of freedom systems with a time-variable mass. We present a set of equations which are used to model the dynamics of a one-dimensional oscillator with a time-varying mass. The dependence of the mass on the time is taken into account, by means of a simple modeling (the leaking bucket of water) where the mass of the oscillator has a quadratic dependence on time.
The resulting equation of motion is numerically solved in terms of the improved Euler method, and some results for chosen values of the model parameters have been presented and discussed in the text.
According to the results obtained by the numerical simulations, the system shows a typical oscillatory behaviour with "amplitude" and "frequency" which vary as the water leaves the bucket. At the end, there remains only the bucket that oscillates like a one-dimensional harmonic oscillator with constant amplitude and frequency.
At this point, we point out that the quadratic dependence of mass on time is only a motivator for the leaking oscillator problem, treated here as a purely theoretical problem. Therefore, this result should be considered within its appropriate limitations. Probably, when the bucket is moving, going up and down with the oscillations, the flow rate through the hole could be seen to change as well, deviating slightly from the results obtained here. In other words, we are ignoring the fact that the bucket, as well as the water within it, are accelerating frames. However, we can admit that the quadratic dependence of mass must work as a reasonable approximation in the case the loss of water occurs at a very low rate, and the oscillating bucket experiences smooth motions as investigated in this article. As a suggestion, the model could be investigated experimentally using a motor and a leaking bucket of water, for example, in order to validate or not the assumptions made for the present model for the quadratic dependence of mass.
This study, despite its simplicity, is intended to be used as a useful approach for students get acquainted with the physics of systems with time-varying mass at the undergraduate level.
\bigskip
\bigskip
\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,647 |
Missed the New York Times essay contest on meat-eating? Well, you've got till midnight tonight to read the entries and vote for your favorite. The official judges — Michael Pollan, Mark Bittman, and the like — will pronounce their verdict on May 6. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,381 |
Chen Dong (; born 3 May 1978, in Jilin) is a Chinese football player. He last sided with Dalian Transcendence as goalkeeping coach.
Club career
Chen Dong started his professional career in 1998 playing for the Dalian Wanda where for several seasons with the club he played understudy to Han Wenhai until Han left before the beginning of the 2001 league season. Finally established as the team's first choice goalkeeper he would continue Dalian's dominance within China and win the 2001 league title. His reign was short lived and the emergence of An Qi saw Chen drop back down as to second choice the following season, unable to re-establish himself as the regular first choice goalkeeper. Despite winning several further league titles Chen received very little playing time and in the 2003 league season he would go on loan to Sichuan Guancheng and then Dalian Changbo in the 2004 season where he would gain much more playing time and establish himself as the first choice goalkeeper at both of these clubs. In the 2005 season he would return to Dalian, which had changed its name to Dalian Shide to become their first choice goalkeeper once more and go on to win another league title with them. He remained a consistent performer for Dalian until the 2008 league season when he was criticised for his poor performances with several other players and was dropped for Sun Shoubo, the club kept on to him until the end of the 2009 league season when his contract expired and he was released.
International career
Chen Dong would make his senior international debut in a friendly on February 16, 2003 against Estonia in a 1-0 victory. In total he made three appearances for China, the last was against Thailand in a friendly on the 15 March 2008 coming on as a substitute for Zong Lei, which ended 3-3.
Honours
Dalian Wanda
Chinese Jia-A League/Chinese Super League: 1998, 2000, 2001, 2002, 2005
Chinese FA Cup: 2001, 2005
References
External links
1978 births
Living people
Footballers from Changchun
Association football goalkeepers
Chinese footballers
China international footballers
Dalian Shide F.C. players
Sichuan Guancheng players
Chinese Super League players
China League One players
Asian Games medalists in football
Footballers at the 1998 Asian Games
Asian Games bronze medalists for China
Medalists at the 1998 Asian Games | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,127 |
{"url":"https:\/\/answers.ros.org\/answers\/51939\/revisions\/","text":"# Revision history [back]\n\nHey,\n\nI couldn't get this working either. I looked through the source and it seems that there is a bug in the graspitGUI.cpp code for loading plugins on the command line. The problem is here starting at line 155 in graspitGUI.cpp:\n\n\/\/look for plugins of the form plugin:name in the arguments\nfor (int i=1; i<argc; i++) {\nQString arg(argv[i]);\nif (arg.section(',',0,0)==\"plugin\") {\nQString libName = arg.section(',',1,1);\nstd::cout << \"Processing arguments \\n \";","date":"2021-09-17 11:26:08","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\": 0.348720908164978, \"perplexity\": 6574.545091010034}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780055632.65\/warc\/CC-MAIN-20210917090202-20210917120202-00377.warc.gz\"}"} | null | null |
Q: TPKeyboardAvoiding issue I'm using the TPKeyboardAvoiding project and in doing so have an issue with the size of the scroll view.
The scrolling works great but for some reason it's not using the full screen, despite being set to in the storyboard.
https://github.com/michaeltyson/TPKeyboardAvoiding
WOW that turned out to be a large image! Anyway, it nicely shows the cut off at the top.
Has any one had a similar issue? How was it resolved?
A: Are you using autolayouts? I had these problems with autolayouts. Disabling them makes it work.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
Microtubules (hereafter abbreviated as MTs) are filamentous macro-polymers
built from tubulin dimers. They are one of the components of the cytoskeleton
of all eukaryotic cells. They play a number of roles ranging from providing
mechanical stability to the cell, serving as transport pathways enabling
linear transport of versicular cargo by motor proteins, and providing forces
for the positioning of organelles and other intracellular components (for a
general overview see \cite{Alberts2002}). Perhaps their most striking function
appears during mitosis, where they form the mitotic spindle, the machinery for
positioning and separating the duplicated chromosomes prior to cell division.
They owe their functional plasticity to an intrinsic so-called `dynamical
instability' mechanism \cite{Mitchison1984} that causes individual MTs to
stochastically alternate between growing and shrinking states. By controlling
this dynamical process, through MT associated proteins (MAPS) that nucleate
new MTs or selectively stabilize or destabilize them by locally or globally
changing the rates with which they switch between dynamical states, cells are
able to reconfigure MT assemblies on timescales as fast as a few minutes. The
canonical model to describe MT dynamics was developed in the early 1990's by
Dogterom and Leibler \cite{Dogterom1993}. This model showed that isolated MTs,
depending on their dynamical parameters, can either be in a regime of bounded
growth leading to an exponential length distribution in a steady state, or in
a regime of unbounded growth in which the MT length on average increases
linearly in time. Of course, MTs `live' within the confines of a finite size
cell, whose dimensions ($\sim10\mu m$) are comparable to the observed lengths
of MTs. Interactions between MTs and boundaries, be it the cell cortex or the
surface of other intracellular compartments, are therefore important. Indeed,
a number of MT functions depends critically on these interactions: examples
are nuclear positioning in yeast \cite{Tran2001}, spindle positioning in
\emph{C. elegans }\cite{Grill2003}, and the orientation of the cortical MT
array in plant cells \cite{Ambrose2011}.
In spite of this clear relevance, it appears that a systematic approach to the
theory of MTs interacting with boundaries is lacking from the literature. The
one problem of this type which did receive substantial attention, is the
search-and-capture mechanism by which MTs are thought to find the condensed
chromosomes prior to mitosis \cite{Holy1994}\cite{Holy1997}\cite{Wollman2005
\cite{Gopalakrishnan2011}, which involves estimating the mean first-passage
time of a MT to hit a limited size target at a distance from its nucleation
point. However, although these works in fact do contain some of the basic
features of the MT-boundary problem, it is mostly hidden (literally in the
case of Ref. \cite{Holy1997}, actually an unpublished thesis) under the
specifics of the intended application. Moreover, these works also rely heavily
on `forward' techniques involving the time evolution of the full probability
density for reaching a given state from specified initial conditions. Although
this approach obviously yields a full solution of the problem, for
passage-time problems, which effectively require integrating over final
states, the full probability density is in a sense a form of \textquotedblleft
overkill\textquotedblright. The treatment of this type of problems can in fact
be simplified considerably by using `backward' techniques, as is
e.g.\ elegantly illustrated for diffusion problems in Redner's monograph
\cite{Redner2001}. In the present work we show how this approach can be used
from the ground up to solve the problem of a MT interacting with a boundary.
The outline of the paper is as follows: In Section \ref{sec:meanl} we
introduce the dynamical model of a MT interacting with a boundary, solve for
its steady state properties, such as the average length, and choose an
appropriate set of non-dimensionalized parameters and variables. In Section
\ref{sec:MFPT} we turn to the analysis of the mean first-passage time to the
boundary, formally solving this in terms of a small set of splitting
probabilities and conditioned mean first-passages times, which are
subsequently determined explicitly through the solution of appropriate linear
boundary value problems. We then use some biological data on MT dynamics to
estimate order of magnitudes for the quantities involved. Finally, in Section
\ref{sec:GG} we revisit the recent search-and-capture model discussed by
Gopalakrishnan and Govindan \cite{Gopalakrishnan2011}, and show how it is
compactly solved using the techniques introduced. We then finish with a number
of concluding remarks and two technical appendices.
\section{Mean length\label{sec:meanl}}
\subsection{Dynamical model}
The standard two-state dynamical instability model describes MTs with length
$l$ that are either growing with velocity $v_{+}$ or shrinking with velocity
$v_{-}$ and can switch between the growing and the shrinking state (a
\emph{catastrophe}) with a constant rate $r_{+}$ and between the shrinking and
the growing state (a \emph{rescue}) with a constant rate $r_{-}$. Collectively
we call these two states the \emph{active} states, and denote the
corresponding state space by $A$. We extend this model by two more states: a
\emph{nucleation} state $N$, in which a MT enters upon shrinking back to zero
length, and from which it can be (re)nucleated into a zero-length growing
state at a constant rate $r_{n}$, and a boundary state $B$, in which a MT
enters upon hitting a boundary at a distance $L$ from the nucleation point,
and which it leaves in a shrinking state at a rate $r_{b}$. Formally the state
space of this extended model is therefore given by $\Omega=N\cup A\cup B$. We
illustrate the model and its state space in Figure~\ref{fig:barrier}.
\begin{figure}[ptb]
\includegraphics[width=\textwidth]{barrier}\caption{Schematic of our model
showing the three components of the state space: The nucleation state $N$ from
which MTs are nucleated at rate $r_{n}$. The active state $A$, in which MTs
either grow or shrink, with velocities $v_{+}$ and $v_{-}$, and switch between
these two states with rates $r_{+}$ and $r_{-}$, and a barrier state $B$ from
which MTs exit at a rate $r_{b}$.
\label{fig:barrier
\end{figure}It should be noted that, from a biophysical point of view, the
model is of course an idealization. In reality a growing MT impinging on a
boundary will generate forces. These forces will affect the growth velocity
and the propensity to switch to the shrinking state, so that the latter is no
longer a simple Poisson process \cite{Janson2003}. Moreover, these forces may
also deform the boundary, lead to buckling of the MT, or cause it to slide
along the boundary (for a review see \cite{Dogterom2005}). These additional
complexities, however, will have limited impact on the results to be presented
here, as long as the residence time at the boundary is small compared to the
time to traverse the distance between nucleation point and boundary, which is
certainly the case for effectively reflective boundaries for which $r_{b}\gg
r_{+}$, and we choose to ignore them here.
The dynamics of the model described above defines a time-homogeneous Markov
process on the full state space. It is, however, technically convenient to
split the dynamics on the `active' part of the state space $A$ from that on
the two `waiting' states $N$ and $B$, and deal with the communication between
these different states through boundary conditions. We therefore first define
the probability densities (per unit length) $m_{s}\left( l,t|\omega_{0
,t_{0}\right) $ for an active MT to have length $l$ and be in state $s=+,-$
at time $t$, given that it was in some state $\omega_{0}$ at time $t_{0}<t$.
These densities satisfy the evolution equation
\begin{align}
\frac{\partial}{\partial t}m_{+}\left( l,t|\omega_{0},t_{0}\right) &
=-v_{+}\frac{\partial}{\partial l}m_{+}\left( l,t|\omega_{0},t_{0}\right)
-r_{+}m_{+}\left( l,t|\omega_{0},t_{0}\right) +r_{-}m_{-}\left(
l,t|\omega_{0},t_{0}\right) ,\label{eq:actp}\\
\frac{\partial}{\partial t}m_{-}\left( l,t|\omega_{0},t_{0}\right) &
=v_{-}\frac{\partial}{\partial l}m_{-}\left( l,t|\omega_{0},t_{0}\right)
-r_{-}m_{-}\left( l,t|\omega_{0},t_{0}\right) +r_{+}m_{+}\left(
l,t|\omega_{0},t_{0}\right) . \label{eq:actm
\end{align}
Likewise we define the probability $M_{n}\left( t|\omega_{0},t_{0}\right) $
for the MT to be in the nucleation state at time $t$, given that it was in
some state $\omega_{0}\in\Omega$ at time $t_{0}<t$. This probability satisfies
the evolution equatio
\begin{equation}
\frac{\partial}{\partial t}M_{n}\left( t|\omega_{0},t_{0}\right) =v_{-
m_{-}\left( 0,t|\omega_{0},t_{0}\right) -r_{n}M_{n}\left( t|\omega
_{0},t_{0}\right) . \label{eq:0
\end{equation}
The probability $M_{b}\left( t|\omega_{0},t_{0}\right) $ for the MT to be in
the boundary state at time $t$ in turn satisfie
\begin{equation}
\frac{\partial}{\partial t}M_{b}\left( t|\omega_{0},t_{0}\right) =v_{+
m_{+}\left( L,t|\omega_{0},t_{0}\right) -r_{b}M_{b}\left( t|\omega
_{0},t_{0}\right)
\end{equation}
This system of equations is closed by the boundary condition
\begin{align}
v_{+}m_{+}\left( 0,t|\omega_{0},t_{0}\right) & =r_{n}M_{n}\left(
t|\omega_{0},t_{0}\right) ,\\
v_{-}m_{-}\left( L,t|\omega_{0},t_{0}\right) & =r_{b}M_{b}\left(
t|\omega_{0},t_{0}\right) .
\end{align}
By construction the dynamics on the full state space conserves probability,
and indeed if we define the total probabilit
\begin{align}
M\left( t|\omega_{0},t_{0}\right) & =M_{n}\left( t|\omega_{0
,t_{0}\right) +M_{a}\left( t|\omega_{0},t_{0}\right) +M_{b}\left(
t|\omega_{0},t_{0}\right) \nonumber\\
& =M_{n}\left( t|\omega_{0},t_{0}\right) +\int_{0}^{\infty}dl\,\left\{
m_{+}\left( l,t|\omega_{0},t_{0}\right) +m_{-}\left( l,t|\omega_{0
,t_{0}\right) \right\} +M_{b}\left( t|\omega_{0},t_{0}\right)
\label{eq:norm
\end{align}
the
\begin{equation}
\frac{\partial}{\partial t}M\left( t|\omega_{0},t_{0}\right) =0,
\end{equation}
allowing us to set $M\left( t|\omega_{0},t_{0}\right) =1$.
\subsection{Steady state behaviour}
\label{sec:SS} In steady state the probabilities do not depend on time nor on
initial conditions, allowing us to write the evolution equations as a set of
balance equation
\begin{align}
v_{+}\frac{d}{dl}m_{+}\left( l\right) & =-r_{+}m_{+}\left( l\right)
+r_{-}m_{-}\left( l\right) ,\label{eq:SSp}\\
-v_{-}\frac{d}{dl}m_{-}\left( l\right) & =-r_{-}m_{-}\left( l\right)
+r_{+}m_{+}\left( l\right) ,\label{eq:SSm}\\
v_{-}m_{-}\left( 0\right) & =r_{n}M_{n},\label{eq:SSn}\\
v_{+}m_{+}\left( L\right) & =r_{b}M_{b}, \label{eq:SSb
\end{align}
to be supplemented by the boundary condition
\begin{align}
v_{+}m_{+}\left( 0\right) & =r_{n}M_{n},\label{eq:SSBC0}\\
v_{-}m_{-}\left( L\right) & =r_{b}M_{b}. \label{eq:SSBCL
\end{align}
Adding Eqs.~(\ref{eq:SSp}) and (\ref{eq:SSm}) yield
\begin{equation}
\frac{d}{dl}\left\{ v_{+}m_{+}\left( l\right) -v_{-}m_{-}\left( l\right)
\right\} =0.
\end{equation}
Combining either Eqs.~(\ref{eq:SSn}) and (\ref{eq:SSBC0}), or
Eqs.~(\ref{eq:SSb}) and (\ref{eq:SSBCL}), shows that the constant of
integration vanishes, and henc
\begin{equation}
v_{+}m_{+}\left( l\right) =v_{-}m_{-}\left( l\right) . \label{eq:SSbal
\end{equation}
This identity is now used to eliminate $m_{-}\left( l\right) $ from
(\ref{eq:SSp}) from which we then readily find that for $l\leq L$
\begin{align}
m_{+}\left( l\right) & =\frac{r_{n}M_{n}}{v_{+}}e^{-l/\bar{l}},\\
m_{-}\left( l\right) & =\frac{r_{n}M_{n}}{v_{-}}e^{-l/\bar{l}},
\end{align}
where the lengt
\begin{equation}
\bar{l}=\left( \frac{r_{+}}{v_{+}}-\frac{r_{-}}{v_{-}}\right) ^{-1
\end{equation}
is of course only positive when $r_{+}v_{-}-r_{-}v_{+}>0$, the so called
bounded-growth regime, and represents the steady-state average length of an
active MT in the absence of the boundary. Although in the presence of
boundaries one can also consider the unbounded growth regime $r_{+}v_{-
-r_{-}v_{+}<0$, as was done e.g.\ in \cite{Govindan2004} for the case where
the boundary is fully reflecting, we will not do so here, and focus
exclusively on the bounded growth case. The dependence of the probability that
the MT is in the boundary state can, from (\ref{eq:SSb}), be shown to obey the
following relationshi
\begin{equation}
M_{b}=\frac{r_{n}}{r_{b}}e^{-\frac{L}{\bar{l}}}M_{n
\end{equation}
The normalization condition (\ref{eq:norm}) can then be used to determine the
probabilities for an MT to be in a nucleation state, an active state and a
boundary state respectively
\begin{align}
M_{n} & =\frac{1}{1+r_{n}\bar{l}\left( \frac{1}{v_{+}}+\frac{1}{v_{-
}\right) \left( 1-e^{-L/\bar{l}}\right) +\frac{r_{n}}{r_{b}}e^{-L/\bar{l}
},\label{eq:MdaveB}\\
M_{a} & =\frac{r_{n}\bar{l}\left( \frac{1}{v_{+}}+\frac{1}{v_{-}}\right)
\left( 1-e^{-L/\bar{l}}\right) }{1+r_{n}\bar{l}\left( \frac{1}{v_{+}
+\frac{1}{v_{-}}\right) \left( 1-e^{-L/\bar{l}}\right) +\frac{r_{n}}{r_{b
}e^{-L/\bar{l}}},\\
M_{b} & =\frac{\frac{r_{n}}{r_{b}}e^{-L/\bar{l}}}{1+r_{n}\bar{l}\left(
\frac{1}{v_{+}}+\frac{1}{v_{-}}\right) \left( 1-e^{-L/\bar{l}}\right)
+\frac{r_{n}}{r_{b}}e^{-L/\bar{l}}
\end{align}
Taking the limit $L\rightarrow\infty$ we find $M_{a}\propto r_{n}\bar
{l}\left( \frac{1}{v_{+}}+\frac{1}{v_{-}}\right) $. As density $\propto$
nucleation rate $\times$ lifetime, this suggests that the tim
\begin{equation}
\bar{t}=\bar{l}\left( \frac{1}{v_{+}}+\frac{1}{v_{-}}\right) =\frac
{v_{+}+v_{-}}{r_{+}v_{-}-r_{-}v_{+}} \label{eq:tmean
\end{equation}
is the expected lifetime of a, otherwise unconstrained, length-zero newly
nucleated MT, a result indeed first derived by Rubin \cite{Rubin1988}.
We now define the mean length of the active MTs
\begin{equation}
\left\langle l\right\rangle _{a}=\frac{1}{M_{a}}\int_{0}^{L}dl\,l\left\{
m_{+}\left( l\right) +m_{-}\left( l\right) \right\} =\bar{l}\frac{\left(
1-\left( 1+\frac{L}{\bar{l}}\right) e^{-L/\bar{l}}\right) }{\left(
1-e^{-L/\bar{l}}\right) } \label{eq:laveB
\end{equation}
The time-averaged length of the MT over the full ensemble, is then simpl
\begin{equation}
\left\langle l\right\rangle =M_{a}\left\langle l\right\rangle _{a}+M_{b}L
\end{equation}
We can readily check the limits $\left\langle l\right\rangle _{a
\rightarrow\bar{l}$ when $L\rightarrow\infty$, and $\left\langle
l\right\rangle _{a}\simeq\frac{1}{2}L$ for $L\rightarrow0$. The latter limit
can be understood by considering that for very small $L$ the growing and
shrinking traversal times $L/v_{+}$ and $L/v_{-}$ become small with respect to
the mean time between between catastrophes, $1/r_{+}$, and rescues, $1/r_{-}$,
respectively, so that the MT is deterministically \textquotedblleft
bouncing\textquotedblright\ between the endpoints $l=0$ and $l=L$.
\subsection{Dimensional analysis}
\label{sec:dimen}We have deliberately deferred the dimensional analysis of the
system up to this point to allow the results of the steady state solution to
guide us to a natural choice of the units of length and time. In view of
(\ref{eq:MdaveB}) and (\ref{eq:laveB}), we choose $\bar{l}$ (\ref{eq:tmean})
as the unit length, and $\bar{t}$ as the unit of time. For completeness sake,
we can also introduce the unit of speed
\begin{equation}
\frac{1}{\bar{v}}\equiv\frac{\bar{t}}{\bar{l}}=\frac{1}{v_{+}}+\frac{1}{v_{-}
\end{equation}
By convention we will adopt the Greek alphabet do denote dimensionless
quantities. We can now introduce the dimensionless \emph{parameters
\begin{table}[ptb
\begin{tabular}
[c]{|l|l|l|}\hline
Symbol & Definition & Meaning\\\hline\hline
$\nu_{+}$ & $\frac{v_{+}}{\bar{v}}$ & growth speed\\\hline
$\nu_{-}$ & $\frac{v_{-}}{\bar{v}}$ & shrinkage speed\\\hline
$\rho_{+}$ & $r_{+}\bar{t}$ & catastrophe rate\\\hline
$\rho_{-}$ & $r_{-}\bar{t}$ & rescue rate\\\hline
$\rho_{n}$ & $r_{n}\bar{t}$ & nucleation rate\\\hline
$\rho_{b}$ & $r_{b}\bar{t}$ & barrier unbinding rate\\\hline
$\Lambda$ & $\frac{L}{\bar{l}}$ & distance to barrier\\\hline
\end{tabular}
\caption{Dimensionless parameters of the MT dynamical model
\label{tab:dparameters
\end{table}Note that this assignment, which has the clear advantage of
maximizing the interpretability of the non-dimensional equations, does have
the disadvantage of leaving dependencies among the parameters, as by
constructio
\begin{align}
\frac{1}{\nu_{+}}+\frac{1}{\nu_{-}} & =1,\label{eq:meanl}\\
\frac{\rho_{+}}{\nu_{+}}-\frac{\rho_{-}}{\nu_{-}} & =1.
\end{align}
To denote the \emph{independent variables} of time and length we write $\tau$
and $\lambda$ respectively. Finally, the densities, as our \emph{dependent
variables,} are denoted by $\mu_{s}\left( \lambda,\tau|\omega_{0},\tau
_{0}\right) =\bar{l}m_{s}\left( \lambda\bar{l},\tau\bar{t}|\omega_{0
,\tau_{0}\bar{t}\right) $.
Using these notations the steady-state results of the previous section can be
summarized a
\begin{align}
M_{n} & =\frac{1}{1+\rho_{n}\left( 1-e^{-\Lambda}\right) +\frac{\rho_{n
}{\rho_{b}}e^{-\Lambda}},\\
\left\langle \lambda\right\rangle _{a} & =\frac{\left( 1-\left(
1+\Lambda\right) e^{-\Lambda}\right) }{\left( 1-e^{-\Lambda}\right) }.
\end{align}
For future reference we will also rewrite the evolution equations in a more
compact notation. To do so we treat pairs of functions $(\varphi_{+
(\lambda,\tau),\varphi_{-}(\lambda,\tau))$, defined on the growing and
shrinking parts of the state space respectively, as a single vector valued
function $\varphi_{s}(\lambda,\tau),s=+,-$. This allows us to write
\begin{equation}
\frac{\partial}{\partial\tau}\mu_{s}(\lambda,\tau|\omega_{0},\tau_{0
)=\sum_{s\prime}G_{s,s^{\prime}}^{\ast}[\mu_{s^{\prime}}(\lambda,\tau
|\omega_{0},\tau_{0})],
\end{equation}
where $G^{\ast}$ is the operator matrix
\begin{equation}
G_{s,s^{\prime}}^{\ast}=\left(
\begin{array}
[c]{cc
-\nu_{+}\frac{\partial}{\partial\lambda}-\rho_{+} & \rho_{-}\\
\rho_{+} & \nu_{-}\frac{\partial}{\partial\lambda}-\rho_{-
\end{array}
\right) . \label{eq:Gadj
\end{equation}
The fact that we use the notation $G^{\ast}$, signifying the Hermitean
conjugate of the \emph{generator} $G$ of the Markov process, is conventional
when discussing the \emph{forward} Kolmogorov equation, which is the formal
term for the evolution equation for the probability densities
\cite{Williams1979}.
\section{The Mean-First Passage Time\label{sec:MFPT}}
We approach the problem of calculating the Mean-First Passage Time (MFPT) for
a microtubule to hit the boundary at a distance $L$ in three steps. We first
provide a formal solution to the problem in terms of suitably chosen set of
survival (and ruin) probabilities. We then calculate the static splitting
probabilities that describe the relative weights of the direct and indirect
paths of reaching the boundary, and finally determine the conditional MFPTs
corresponding to these sets of paths.
\subsection{Formal solution}
\label{sec:formal}We first define the \emph{survival set}, the subset of state
space excluding the boundary state, i.e. $\Omega_{\Lambda}=\Omega/B=N\cup A$.
Our goal is to determine the survival probability of the process in this set
starting at $\tau=0$ from an arbitrary active state with length $\lambda
<\Lambda$, which we denote by $S_{\Omega_{\Lambda}}\left( \tau|\lambda
,s\right) $. We will deconstruct this survival probability with the aid of a
number additional probabilities defined as follows: $S_{A}\left( \tau
|\lambda,s\right) $, the probability of surviving in the active part of the
state space, i.e. of not having passed either $\Lambda$ or shrunk back to zero
length from the initial condition, and the conditional \emph{ruin}
probabilities $R_{A}^{B}\left( \tau|\lambda,s\right) $ and $R_{A}^{N}\left(
\tau|\lambda,s\right) $, being the probabilities to have exited into the
boundary state at $\lambda=\Lambda$ or the nucleation state at $\lambda=0$ at
time $\tau$ respectively, \emph{without} leaving the active state at any prior
moment. Finally, we define the survival probability to remain in the
nucleation state, which, because nucleation is a simple Poisson process, is
given by $S_{N}\left( \tau|N\right) =\exp\left( -\rho_{n}\tau\right) $.
Each of these survival or ruin probabilities has a corresponding waiting time
distribution, symbolically given by $\sigma\left( \tau|\omega\right)
=-\frac{\partial}{\partial\tau}S\left( \tau|\omega\right) $ or
$\sigma\left( \tau|\omega\right) =\frac{\partial}{\partial\tau}R\left(
\tau|\omega\right) $.
We now note the following two identitie
\begin{align}
S_{\Omega_{\Lambda}}\left( \tau|\lambda,s\right) & =S_{A}\left(
\tau|\lambda,s\right) +\int_{0}^{\tau}d\tau^{\prime}\sigma_{A}^{N}\left(
\tau^{\prime}|\lambda,s\right) S_{\Omega_{\Lambda}}\left( \tau-\tau^{\prime
}|N\right) \label{eq:surva}\\
S_{\Omega_{\Lambda}}\left( \tau|N\right) & =S_{N}\left( \tau|N\right)
+\int_{0}^{\tau}d\tau^{\prime}\sigma_{N}\left( \tau^{\prime}|N\right)
S_{\Omega_{\Lambda}}\left( \tau-\tau^{\prime}|0,+\right) \label{eq:survd
\end{align}
The first captures the fact that starting from an active state the MT survives
either by remaining active and not reaching $\Lambda$, or by shrinking back to
zero at some intermediate time and then surviving from the nucleation state.
The second states that a MT in the nucleation state survives either by
remaining in this state, or being nucleated into a growing one at an
intermediate time and then surviving from the zero-length growing state. As
survival in the active state means not exiting either at length $\Lambda$ into
the $B$ state or at length $0$ into the $N$ state we have that
\begin{equation}
S_{A}\left( \tau|\lambda,s\right) =1-R_{A}^{B}\left( \tau|\lambda,s\right)
-R_{A}^{N}\left( \tau|\lambda,s\right)
\end{equation}
As is clear from the steady state solution a MT will always leave the active
state for large enough time ($M_{n}>0$ independent of the initial conditions)
so that $S_{A}\left( \infty|\lambda,s\right) =0$. The ultimate conditional
ruin probabilities $R_{A}^{B}\left( \infty|\lambda,s\right) $ and $R_{A
^{N}\left( \infty|\lambda,s\right) $ are usually, and aptly, called
\emph{splitting probabilities}, as the total ruin probability is `split'
betwen them $R_{A}^{B}\left( \infty|\lambda,s\right) +R_{A}^{N}\left(
\infty|\lambda,s\right) =1$. We can thus rewrite identity (\ref{eq:surva}) a
\begin{multline}
S_{\Omega_{\Lambda}}\left( \tau|\lambda,s\right) =\left( R_{A}^{B}\left(
\infty|\lambda,s\right) -R_{A}^{B}\left( \tau|\lambda,s\right) \right)
+\left( R_{A}^{N}\left( \infty|\lambda,s\right) -R_{A}^{N}\left(
\tau|\lambda,s\right) \right) \label{eq:survtot}\\
+\int_{0}^{\tau}d\tau^{\prime}\sigma_{A}^{N}\left( \tau^{\prime
|\lambda,s\right) S_{\Omega_{\Lambda}}\left( \tau-\tau^{\prime}|N\right)
\end{multline}
We can now define
\begin{equation}
T_{\Omega_{\Lambda}}\left( \lambda,s\right) =\int_{0}^{\infty
d\tau\,S_{\Omega_{\Lambda}}\left( \tau|\lambda,s\right) =\int_{0}^{\infty
}d\tau\,\tau\,\sigma_{\Omega_{\Lambda}}\left( \tau|\lambda,s\right)
\end{equation}
as the MFPT for the process to pass length $\Lambda$ starting from the active
state $\left( \lambda,s\right) $. Integrating the first two terms on the
righthand side of (\ref{eq:survtot}) over time yields
\begin{multline}
\int_{0}^{\infty}d\tau\,\left( R_{A}^{B}\left( \infty|\lambda,s\right)
-R_{A}^{B}\left( \tau|\lambda,s\right) \right) =\int_{0}^{\infty
d\tau\,\tau\,\sigma_{A}^{B}\left( \tau|\lambda,s\right) \label{eq:Tcond}\\
=R_{A}^{B}\left( \infty|\lambda,s\right) \frac{\int_{0}^{\infty}d\tau
\,\tau\,\sigma_{A}^{B}\left( \tau|\lambda,s\right) }{\int_{0}^{\infty
d\tau\,\sigma_{A}^{B}\left( \tau|\lambda,s\right) }\equiv R_{A}^{B}\left(
\infty|\lambda,s\right) T_{A}^{B}\left( \lambda,s\right)
\end{multline}
which introduces the conditional MFPT of the process to exit at $\Lambda$
without ever shrinking to $0$, and similarl
\begin{equation}
\int_{0}^{\infty}d\tau\,\left( R_{A}^{N}\left( \infty|\lambda,s\right)
-R_{A}^{N}\left( \tau|\lambda,s\right) \right) =R_{A}^{N}\left(
\infty|\lambda,s\right) T_{A}^{N}\left( \lambda,s\right)
\end{equation}
Integrating the final term give
\begin{multline}
\int_{0}^{\infty}d\tau\,\int_{0}^{\tau}d\tau^{\prime}\sigma_{A}^{N}\left(
\tau^{\prime}|\lambda,s\right) S_{\Omega_{\Lambda}}\left( \tau-\tau^{\prime
}|N\right) =\nonumber\\
\int_{0}^{\infty}d\tau^{\prime}\,\sigma_{A}^{N}\left( \tau^{\prime
|\lambda,s\right) \int_{\tau^{\prime}}^{\infty}d\tau S_{\Omega_{\Lambda
}\left( \tau-\tau^{\prime}|N\right) =R_{A}^{N}\left( \infty|\lambda
,s\right) T_{\Omega_{\Lambda}}\left( N\right)
\end{multline}
The MFPT from the nucleation state is readily obtained from (\ref{eq:survd})
and yield
\begin{equation}
T_{\Omega_{\Lambda}}\left( N\right) =T_{N}\left( N\right) +T_{\Omega
_{\Lambda}}\left( 0,+\right) =\frac{1}{\rho_{n}}+T_{\Omega_{\Lambda}}\left(
0,+\right)
\end{equation}
where we have used that exiting the nucleation state is sure i.e.
$S_{N}\left( \infty|N\right) =0$. Collecting all these results then yield
\begin{equation}
T_{\Omega_{\Lambda}}\left( \lambda,s\right) =R_{A}^{B}\left( \infty
|\lambda,s\right) T_{A}^{B}\left( \lambda,s\right) +R_{A}^{N}\left(
\infty|\lambda,s\right) \left\{ T_{A}^{N}\left( \lambda,s\right) +\frac
{1}{\rho_{n}}+T_{\Omega_{\Lambda}}\left( 0,+\right) \right\}
\label{eq:MFTP
\end{equation}
The interpretation of this result is clear: Starting from $\left(
\lambda,s\right) $ the MT either exits directly at $\lambda=\Lambda$, which
happens with probability $R_{A}^{B}\left( \infty|\lambda,s\right) $ and (on
average) takes time $T_{A}^{B}\left( \lambda,s\right) $, or the MT first
shrinks back to $\lambda=0$ (with probability $R_{A}^{N}\left( \infty
|\lambda,s\right) $) which takes a time $T_{A}^{N}\left( \lambda,s\right)
$, and then has to wait a time $\frac{1}{\rho_{n}}$ to be renucleated, after
which it takes time $T_{\Omega_{\Lambda}}\left( 0,+\right) $ to reach
$\lambda=\Lambda$ for the first time.We illustrate this result schematically
in Figure~\ref{fig:MFPT}. \begin{figure}[ptb]
\includegraphics[height=5cm]{MFPT}\caption{Schematic showing how the
unconditional MFPT (grey arrow) for a growing MT to reach the boundary from
starting length $\lambda$ splits into two conditional MFPTs: The first
associated with all direct paths (black arrows) from $\lambda$ to the boundary
at $\Lambda$. The second associated with all paths that shrink to zero length
without reaching the boundary, spend time in the nucleation state (wiiglly
line) and then tries again unconditionally growing from length $0$.
\label{fig:MFPT
\end{figure}
If we now consider a MT starting at $\lambda=0$ and in the growing state $s=+$
, we can selfconsistenly solve for the MFPT\ $T_{\Omega_{\Lambda}}\left(
0,+\right) $, which is our main resul
\begin{equation}
T_{\Omega_{\Lambda}}\left( 0,+\right) =T_{A}^{B}\left( 0,+\right)
+\frac{R_{A}^{N}\left( \infty|0,+\right) }{R_{A}^{B}\left( \infty
|0,+\right) }\left\{ T_{A}^{N}\left( 0,+\right) +\frac{1}{\rho_{n
}\right\} . \label{eq:MFPT0p
\end{equation}
This latter result is in fact sufficient to solve the general problem, because
in order to reach the boundary at $\Lambda$ from length $0$ the MT first has
to pass through each intermediate length $\lambda<\Lambda$, taking time
$T_{\Omega_{\lambda}}\left( 0,+\right) $ (note the subscript $\lambda$ to
$\Omega$ here!$)$ and then reach $\Lambda$ from there, i.e.
\begin{equation}
T_{\Omega_{\Lambda}}\left( 0,+\right) =T_{\Omega_{\lambda}}\left(
0,+\right) +T_{\Omega_{\Lambda}}\left( \lambda,+\right) ,
\end{equation}
showing that the MFPT for a MT starting in the growing state at an arbitrary
length can be expressed fully in terms of MFPTs starting from the zero-length state.
Although due to the fundamental asymmetry of the problem, there is no
corresponding simple rule for the shrinking case, the following argument shows
how we can leverage the results of the growing case to obtain a fairly compact
representation. We first introduce the survival probability of the shrinking
state with respect to rescues, which is simply given by $S_{-}\left(
\tau|-\right) =\exp\left( -\rho_{-}\tau\right) $. If no rescue occurs, the
shrinking MT will hit zero length at the deterministic time $\tau_{-}\left(
\lambda\right) =\lambda/\nu_{-}$, so tha
\begin{multline}
S_{\Omega_{\Lambda}}\left( \tau|\lambda,-\right) =\left( 1-H\left(
\tau-\tau_{-}\left( \lambda\right) \right) \right) \left\{ S_{-}\left(
\tau|-\right) +\int_{0}^{\tau}d\tau^{\prime}\,\sigma_{-}\left( \tau^{\prime
}|-\right) S_{\Omega_{\Lambda}}\left( \tau-\tau^{\prime}|\lambda-\nu_{-
\tau^{\prime},+\right) \right\} \\
H\left( \tau-\tau_{-}\left( \lambda\right) \right) \left\{ \int_{0
^{\tau_{-}\left( \lambda\right) }d\tau^{\prime}\,\sigma_{-}\left(
\tau^{\prime}|-\right) S_{\Omega_{\Lambda}}\left( \tau-\tau^{\prime
|\lambda-\nu_{-}\tau^{\prime},+\right) +S_{-}\left( \tau_{-}\left(
\lambda\right) |-\right) S_{\Omega_{\Lambda}}\left( \tau-\tau_{-}\left(
\lambda\right) |N\right) \right\}
\end{multline}
where $H\left( x\right) $ is the standard Heavyside function. Integrating
over all time yields the desired resul
\begin{align}
T_{\Omega_{\Lambda}}\left( \lambda,-\right) & =\frac{1}{\rho_{-}}\left(
1-e^{-\rho_{-}\tau_{-}\left( \lambda\right) }\right) +\rho_{-}\int
_{0}^{\tau_{-}\left( \lambda\right) }d\tau\,e^{-\rho_{-}\tau}T_{\Omega
_{\Lambda}}\left( \lambda-\nu_{-}\tau,+\right) +e^{-\rho_{-}\tau_{-}\left(
\lambda\right) }T_{\Omega_{\Lambda}}\left( N\right) \nonumber\\
& =\left( 1-e^{-\rho_{-}\tau_{-}\left( \lambda\right) }\right) \left\{
\frac{1}{\rho_{-}}+\frac{\int_{0}^{\tau_{-}\left( \lambda\right)
d\tau\,\sigma_{-}\left( \tau|-\right) T_{\Omega_{\Lambda}}\left(
\lambda-\nu_{-}\tau,+\right) }{\int_{0}^{\tau_{-}\left( \lambda\right)
}d\tau\,\sigma_{-}\left( \tau|-\right) }\right\} +e^{-\rho_{-}\tau
_{-}\left( \lambda\right) }T_{\Omega_{\Lambda}}\left( N\right) \nonumber\\
& =R_{-}\left( \tau_{-}\left( \lambda\right) |-\right) \left\{ \frac
{1}{\rho_{-}}+\left\langle T_{\Omega_{\Lambda}}\left( \lambda-\nu_{-
\tau,+\right) \right\rangle _{\left( \lambda,-\right) }\right\}
+S_{-}\left( \tau_{-}\left( \lambda\right) |-\right) \left\{ \frac
{1}{\rho_{n}}+T_{\Omega_{\Lambda}}\left( 0,+\right) \right\}
\end{align}
where $\left\langle T_{\Omega_{L}}\left( \lambda-\nu_{-}\tau,+\right)
\right\rangle _{\left( \lambda,-\right) }$ is the average MFPT of an MT that
starts in a growing state after a single rescue from a shrinking state
originally at length $\lambda$ at time $0$, provided this happens before the
shrinking state hits zero length.
\subsection{The splitting probabilities}
\label{sec:split}To calculate the splitting probabilities $R_{A}^{N}\left(
\infty|\lambda,s\right) $ and $R_{A}^{B}\left( \infty|\lambda,s\right) $,
we first recall from the theory of Markov processes that expectation values of
future events seen as functions of the initial time and state satisfy the
\emph{backward} Kolmogorov equation \cite{Williams1979}. Specifically, any
ruin probability $R_{K}^{C}\left( T|\tau,\lambda,s\right) $, where $K$ is
some subset of $\Omega$, and $C$ a conditioning event, satisfie
\begin{equation}
\frac{\partial}{\partial\tau}R_{K}^{C}\left( T|\tau,\lambda,s\right)
=-\frac{\partial}{\partial T}R_{K}^{C}\left( T|\tau,\lambda,s\right)
=-\sum_{s^{\prime}}G_{s,s^{\prime}}\left[ R_{K}^{C}\left( T|\tau
,\lambda,s^{\prime}\right) \right] \label{eq:back
\end{equation}
where the generator $G_{s,s^{\prime}}$ is the Hermitian conjugate of the
operator (\ref{eq:Gadj}), i.e
\begin{equation}
G_{s,s^{\prime}}=\left(
\begin{array}
[c]{cc
\nu_{+}\frac{\partial}{\partial\lambda}-\rho_{+} & \rho_{+}\\
\rho_{-} & -\nu_{-}\frac{\partial}{\partial\lambda}-\rho_{-
\end{array}
\right) \label{eq:G
\end{equation}
Since our process is time homogeneous, we can of course take the initial time
to be $\tau=0$. Letting our final time $T\rightarrow\infty$, we see that
$\frac{\partial}{\partial\tau}R_{K}^{C}\left( \infty|\tau,\lambda,s\right)
=\frac{\partial}{\partial\tau}R_{K}^{C}\left( \infty|0,\lambda,s\right) =0$,
so that splitting probabilities satisfy
\begin{equation}
\sum_{s^{\prime}}G_{s,s^{\prime}}\left[ R_{K}^{C}\left( \infty
|\lambda,s^{\prime}\right) \right] =0,
\end{equation}
and are said to be \emph{harmonic.}
\bigskip For convenience sake we now drop the explicit mention of the final
time and set $R_{A}^{N}\left( \lambda,s\right) =R_{A}^{N}\left(
\infty|\lambda,s\right) $ and $R_{A}^{B}\left( \lambda,s\right) =R_{A
^{B}\left( \infty|\lambda,s\right) $. We consider the latter splitting
probabilities first, and write out (\ref{eq:G}) to obtain
\begin{align}
\nu_{+}\frac{\partial}{\partial\lambda}R_{A}^{B}\left( \lambda,+\right)
-\rho_{+}R_{A}^{B}\left( \lambda,+\right) +\rho_{+}R_{A}^{B}\left(
\lambda,-\right) & =0\\
-\nu_{-}\frac{\partial}{\partial\lambda}R_{A}^{B}\left( \lambda,-\right)
-\rho_{-}R_{A}^{B}\left( \lambda,-\right) +\rho_{-}R_{A}^{B}\left(
\lambda,+\right) & =0
\end{align}
with the obvious boundary conditions $R_{A}^{B}\left( \Lambda,+\right) =1$,
$R_{A}^{B}\left( 0,-\right) =0$. At this point it is useful to define the
mean forward and backward run-length
\begin{align}
\lambda_{+} & =\frac{\nu_{+}}{\rho_{+}}\\
\lambda_{-} & =\frac{\nu_{-}}{\rho_{-}
\end{align}
The first equation allows us to eliminate $R_{A}^{B}\left( \lambda,-\right)
\begin{equation}
R_{A}^{B}\left( \lambda,-\right) =R_{A}^{B}\left( \lambda,+\right)
-\lambda_{+}\frac{\partial}{\partial\lambda}R_{A}^{B}\left( \lambda,+\right)
\label{eq:mprel
\end{equation}
Insertion into the second then yields a second order equatio
\begin{equation}
\lambda_{+}\lambda_{-}\frac{\partial^{2}}{\partial\lambda^{2}}R_{A}^{B}\left(
\lambda,+\right) -\left( \lambda_{-}-\lambda_{+}\right) \frac{\partial
}{\partial\lambda}R_{A}^{B}\left( \lambda,+\right) =0
\end{equation}
or equivalentl
\begin{equation}
\frac{\partial^{2}}{\partial\lambda^{2}}R_{A}^{B}\left( \lambda,+\right)
-\frac{\partial}{\partial\lambda}R_{A}^{B}\left( \lambda,+\right) =0
\end{equation}
as (\ref{eq:meanl}) implies
\begin{equation}
\frac{1}{\lambda_{+}}-\frac{1}{\lambda_{-}}=1
\end{equation}
We then obtain the following solution
\begin{align}
R_{A}^{B}\left( \lambda,+\right) & =\frac{e^{\lambda}-\left(
1-\lambda_{+}\right) }{e^{\Lambda}-\left( 1-\lambda_{+}\right) }\\
R_{A}^{B}\left( \lambda,-\right) & =\frac{\left( e^{\lambda}-1\right)
\left( 1-\lambda_{+}\right) }{e^{\Lambda}-\left( 1-\lambda_{+}\right)
\end{align}
In a fully similar manner the corresponding quantities $R_{A}^{N}\left(
\lambda,s\right) $ are also readily determine
\begin{align}
R_{A}^{N}\left( \lambda,+\right) & =\frac{e^{\Lambda}-e^{\lambda
}{e^{\Lambda}-\left( 1-\lambda_{+}\right) }\\
R_{A}^{N}\left( \lambda,-\right) & =\frac{e^{\Lambda}-\left(
1-\lambda_{+}\right) e^{\lambda}}{e^{\Lambda}-\left( 1-\lambda_{+}\right)
\end{align}
One checks that these forms satisfy the a priori requirements $R_{A
^{B}\left( \lambda,+\right) +R_{A}^{N}\left( \lambda,+\right) =R_{A
^{B}\left( \lambda,-\right) +R_{A}^{N}\left( \lambda,-\right) =1$, which
follow from the fact that the ultimate ruin of an MT on a finite length
interval is sure. These splitting probabilities were also derived in
\cite{Holy1997} by considering Laplace transforms of recurrence relations
satisfied by the probability density.
\subsection{The conditional MFPTs}
\label{sec:CMFPT}With the splitting probabilities determined, we can directly
calculate the conditional MFPTs by solving a time integrated form of the
backward equation. Indeed, integrating (\ref{eq:back}) over the final time $T$
, and recalling that $R_{K}^{C}\left( \tau|\tau,\lambda,s\right) =0$, yields
as a first ste
\begin{equation}
-\int_{\tau}^{\infty}dT\frac{\partial}{\partial T}R_{K}^{C}\left(
T|\tau,\lambda,s\right) =-R_{K}^{C}\left( \infty|\lambda,s\right)
=-\int_{\tau}^{\infty}dT\sum_{s^{\prime}}G_{s,s^{\prime}}\left[ \,R_{K
^{C}\left( T|\tau,\lambda,s^{\prime}\right) \right] . \label{eq:BackR
\end{equation}
At this point we would like to interchange the integration and the operation
of $G_{s,s^{\prime}}$, but as $R_{K}^{C}\left( T|\tau,\lambda,s^{\prime
}\right) $ tends to a constant for $T\rightarrow\infty$ this is not directly
possible. However, we can use the fact that the splitting probabilities are
harmonic, i.e $G_{s,s^{\prime}}\left[ R_{K}^{C}\left( \infty|\lambda
,s^{\prime}\right) \right] =0$, and the linearity $G_{s,s^{\prime}}$ to
obtain the identity
\begin{equation}
-G_{s,s^{\prime}}\left[ \,R_{K}^{C}\left( T|\tau,\lambda,s^{\prime}\right)
\right] =G_{s,s^{\prime}}\left[ R_{K}^{C}\left( \infty|\lambda,s^{\prime
}\right) -R_{K}^{C}\left( T|\tau,\lambda,s^{\prime}\right) \right] .
\end{equation}
Substitution of this identity into Eq.~(\ref{eq:BackR}) yields an integrable
argument exactly of the form previously encountered in Eq.~(\ref{eq:Tcond}),
so tha
\begin{multline}
-\int_{\tau}^{\infty}dT\sum_{s^{\prime}}G_{s,s^{\prime}}\left[ \,R_{K
^{C}\left( T|\tau,\lambda,s^{\prime}\right) \right] =\int_{\tau}^{\infty
}dT\sum_{s^{\prime}}G_{s,s^{\prime}}\left[ R_{K}^{C}\left( \infty
|\lambda,s^{\prime}\right) -R_{K}^{C}\left( T|\tau,\lambda,s^{\prime
}\right) \right] \\
=\sum_{s^{\prime}}G_{s,s^{\prime}}\left[ \int_{\tau}^{\infty}dT\left\{
R_{K}^{C}\left( \infty|\lambda,s^{\prime}\right) -R_{K}^{C}\left(
T|\tau,\lambda,s^{\prime}\right) \right\} \right] =\sum_{s^{\prime
}G_{s,s^{\prime}}\left[ R_{K}^{C}\left( \infty|\lambda,s^{\prime}\right)
T_{K}^{C}\left( \lambda,s^{\prime}\right) \right] \label{eq:BackR2
\end{multline}
Combining, Eqs.~(\ref{eq:BackR}) and (\ref{eq:BackR2}), yields the sought
after relatio
\begin{equation}
\sum_{s^{\prime}}G_{s,s^{\prime}}\left[ R_{K}^{C}\left( \infty
|\lambda,s^{\prime}\right) T_{K}^{C}\left( \lambda,s^{\prime}\right)
\right] =-R_{K}^{C}\left( \infty|\lambda,s\right) , \label{eq:TcondG
\end{equation}
which together with appropriate boundary conditions yields a closed form
equation for the conditional MFPTs $T_{K}^{C}\left( \lambda,s^{\prime
}\right) $.
We now apply Eq.~(\ref{eq:TcondG}) to our problem, starting with the case of
exiting at $\Lambda$ we hav
\begin{align}
\lambda_{+}\frac{\partial}{\partial\lambda}R_{A}^{B}\left( \lambda,+\right)
T_{A}^{B}\left( \lambda,+\right) -R_{A}^{B}\left( \lambda,+\right)
T_{A}^{B}\left( \lambda,+\right) +R_{A}^{B}\left( \lambda,-\right)
T_{A}^{B}\left( \lambda,-\right) & =-\frac{1}{\rho_{+}}R_{A}^{B}\left(
\lambda,+\right) \\
-\lambda_{-}\frac{\partial}{\partial\lambda}R_{A}^{B}\left( \lambda,-\right)
T_{A}^{B}\left( \lambda,-\right) -R_{A}^{B}\left( \lambda,-\right)
T_{A}^{B}\left( \lambda,-\right) +R_{A}^{B}\left( \lambda,+\right)
T_{A}^{B}\left( \lambda,+\right) & =-\frac{1}{\rho_{-}}R_{A}^{B}\left(
\lambda,-\right)
\end{align}
with boundary conditions $R_{A}^{B}\left( \Lambda,+\right) T_{A}^{B}\left(
\Lambda,+\right) =R_{A}^{B}\left( 0,-\right) T_{A}^{B}\left( 0,-\right)
=0$. Eliminating $R_{A}^{B}\left( \lambda,-\right) T_{A}^{B}\left(
\lambda,-\right) $ and introducing the shorthand $\Theta_{A}^{B}\left(
\lambda,+\right) =R_{A}^{B}\left( \lambda,+\right) T_{A}^{B}\left(
\lambda,+\right) $ we find the following inhomogeneous second order equatio
\begin{multline}
\frac{\partial^{2}}{\partial\lambda^{2}}\Theta_{A}^{B}\left( \lambda
,+\right) -\frac{\partial}{\partial\lambda}\Theta_{A}^{B}\left(
\lambda,+\right) =\label{eq:CMFPTp}\\
-\frac{1}{\lambda_{+}\rho_{+}}\frac{\partial}{\partial\lambda}R_{A}^{B}\left(
\lambda,+\right) -\frac{1}{\lambda_{+}\lambda_{-}}\left( \frac{1}{\rho_{+
}R_{A}^{B}\left( \lambda,+\right) +\frac{1}{\rho_{-}}R_{A}^{B}\left(
\lambda,-\right) \right) \equiv A_{A}^{B}\left( \lambda,+\right)
\end{multline}
Using Eq.~(\ref{eq:mprel}) we can eliminate $R_{A}^{B}\left( \lambda
,-\right) $ in the inhomogeneous term, so tha
\begin{equation}
A_{A}^{B}\left( \lambda,+\right) =\left( \frac{1}{\lambda_{-}\rho_{-
}-\frac{1}{\lambda_{+}\rho_{+}}\right) \frac{\partial}{\partial\lambda
R_{A}^{B}\left( \lambda,+\right) -\frac{1}{\lambda_{+}\lambda_{-}}\left(
\frac{1}{\rho_{+}}+\frac{1}{\rho_{-}}\right) R_{A}^{B}\left( \lambda
,+\right)
\end{equation}
The boundary equations for the resulting equation ar
\begin{align}
\Theta_{A}^{B}\left( \Lambda,+\right) & =0\label{eq:CMFPTbc1}\\
\lambda_{+}\frac{\partial}{\partial\lambda}\Theta_{A}^{B}\left( 0,+\right)
-\Theta_{A}^{B}\left( 0,+\right) & =-\frac{1}{\rho_{+}}R_{A}^{B}\left(
0,+\right) \label{eq:CMFPTbc2
\end{align}
The equation for the conditional MFPT for exiting at $0$ is the formally the
same as (\ref{eq:CMFPTp}), together with boundary conditions
(\ref{eq:CMFPTbc1}) and (\ref{eq:CMFPTbc2}), but with $\Theta_{A}^{N}\left(
\lambda,+\right) $ and $R_{A}^{N}\left( \lambda,+\right) $ substituted for
$\Theta_{A}^{B}\left( \lambda,+\right) $ and $R_{A}^{B}\left(
\lambda,+\right) $ respectively. These inhomogeneous linear second order
equations are readily solved, and we present the resultant rather unwieldy
expressions in the appendix. As an explicit check we consider the probability
of a growing MT to shrink back to the origin, in the absence of the boundary
\begin{equation}
\lim_{\Lambda\rightarrow\infty}\Theta_{A}^{N}\left( \lambda,+\right)
=\lim_{\Lambda\rightarrow\infty}T_{A}^{N}\left( \lambda,+\right) \equiv
T\left( \lambda,+\right) =1+\lambda\frac{\left( \rho_{+}+\rho_{-}\right)
}{\left( \rho_{+}\nu_{-}-\rho_{-}\nu_{+}\right)
\end{equation}
Upon redimensionalizing, this expression is identical to the one derived
earlier by Bicout from the full time and space-dependent survival probability
\cite{Bicout1997}. As an aside, we note that the result $T(0,+)=1$ shows that
the time-scale we have adopted is indeed that of the origin return time of an
unconstrained MT, as already stated in Section \ref{sec:SS}.
\subsection{Application to biological data}
In order to get a feel what the results derived above mean in real-world
terms, we apply them to two sets of fairly well characterized kinetic
parameters for MTs, one derived from observations on fission yeast
\cite{Janson2007}, and one on interphase Tobacco Bright Yellow-2 plant culture
cells \cite{Vos2004}. These data sets are summarized in Table
\ref{tab:bparameters}. \begin{table}[ptb
\begin{tabular}
[c]{|l|c|c|}\hline
Parameter & Yeast & Plant\\\hline\hline
Growth speed( $\mu$m/min) & $2.4$ & $4.8$\\\hline
Shrinking speed ($\mu$m/min) & $9.6$ & $9.6$\\\hline
Catastrophe rate (/min) & $0.3$ & $0.28$\\\hline
Rescue rate (/min) & $-$ & $0.42$\\\hline
Nucleation rate (/min) & $0.15$ & $0.15^{\dagger}$\\\hline
Mean length (no boundary) ( $\mu$m) & $8$ & $68.5$\\\hline
Expected lifetime (no boundary) (min) & $4.17$ & $21.43$\\\hline
\end{tabular}
\caption{Table of MT dynamical parameters for yeast and interphase plant
cells. {\footnotesize {$\dagger$: In the absence of available data we take
this number equal to that of yeast.} }
\label{tab:bparameters
\end{table}
We now confront these two types of MTs with boundaries located at $5\mu m$,
smaller than both mean lengths in the absence of boundaries and comparable to
half the length of a fission yeast cell, $20\mu m$ double the mean length for
the yeast MT and still significantly smaller than that of the Tobacco BY-2
MTs, and $100\mu m$ of the order of typical lengthscale of a Tobacco BY-2
cell. We first consider the splitting probabilities $R_{A}^{B}\left(
l,+\right) $ and $R_{A}^{N}\left( l,+\right) $,which we plot in Figure
\ref{fig:split}. \begin{figure}[ptb]
\includegraphics[width=\textwidth]{splittingprobabilities}\caption{Splitting
probabilities $R_{A}^{B}(l,+)$ (red line) and $R_{A}^{N}(l,+)$ (blue line) as
a function of the initial length $l$ for different values of the distance
$L$.
\label{fig:split
\end{figure}We see that for the smallest boundary distance $L=5\mu m$ both for
yeast and plant MTs the probability to reach the boundary from zero length is
already appreciable, and increases roughly linearly with starting length,
consistent with it being dominated by uninterrupted growth. As we increase the
boundary distance, the probabilties depend more strongly non-linearly on the
starting length. This is most striking for the case of yeast at $L=100\mu m$,
where $R_{A}^{B}\left( l,+\right) $ is essentially $0$ until the starting
length is within the natural length $\bar{l}\approx8\mu m$ from the boundary.
Next, we turn to the conditional MFPTs $T_{A}^{B}\left( l,+\right) $ and
$T_{A}^{N}\left( l,+\right) $. Here, we first need to take \ a little care,
as for yeast\ the rescue probability vanishes ($r_{-}=0)$, so that the
backward runlength $l_{-}$ is ill-defined. One can of course go through the
procedure in Section \ref{sec:CMFPT} again setting $\rho_{-}=0$ at the outset
in Eq.~(\ref{eq:G}). However, in this case the conditional MFPTs are also
readily determined from first principle
\begin{align}
r_{-} & =0:T_{A}^{B}\left( l,+\right) =\frac{\left( L-l\right) }{v_{+
}\label{eq:cMFTPYa}\\
r_{-} & =0:T_{A}^{N}\left( l,+\right) =\frac{\int_{0}^{\frac{\left(
L-l\right) }{v_{+}}}dt\,e^{-r_{+}t}\left\{ t+\frac{\left( l+v_{+}t\right)
}{v_{-}}\right\} }{\int_{0}^{\frac{\left( L-l\right) }{v_{+}}
dt\,e^{-r_{+}t}}=\frac{l}{v_{-}}+\bar{t}\frac{\left( 1-e^{-\frac{L-l}{\bar
{l}}}\left( 1+\frac{L-l}{\bar{l}}\right) \right) }{1-e^{-\frac{L-l}{\bar
{l}}}}\label{eq:cMFTPYb
\end{align}
where (\ref{eq:cMFTPYa}) follows because a non-rescuable MT can only reach the
boundary without first shrinking away by growing towards it deterministically,
and (\ref{eq:cMFTPYb}) is obtained by averaging (i) the time to experience a
catastrophe before reaching the boundary plus (ii) the time to shrink to zero
length from that moment on over the ensemble of histories that do not reach
the boundary. These two approaches indeed give the same results, which serves
as another independent check on the general formalism. Figure \ref{fig:cMFPT}
shows the resulting passage times. \begin{figure}[ptb]
\includegraphics[width=\textwidth]{cMFPTs}\caption{Conditional MFPTs
$T_{A}^{B}\left( l,+\right) $ (red line) and $T_{A}^{N}\left( l,+\right) $
(blue line) as a function of the initial length $l$ for different values of
the distance $L$.
\label{fig:cMFPT
\end{figure}A perhaps at first sight puzzling feature of these results is the
decrease of $T_{A}^{N}\left( l,+\right) $ for increasing starting length
$l$, which is evident for the yeast case. This, however, is a direct
consequence of the conditioning on shrinking back without reaching the
boundary: If the starting length is within the forward runlength $l_{+}$ from
the boundary, a conditioned MT must rapidly undergo a catastrophe after which
it deterministically shrinks back to zero length. The conditioned return time
(the second term on the far right hand side of Eq.~(\ref{eq:cMFTPYb})) is
therefore a strongly non-linearly decreasing function of the distance to the
boundary, whereas the time to deterministically shrink back from the starting
length $l/v_{-}$ only increases linearly with length.
Finally, in Table \ref{tab:MFPT} we give the MFPT $T_{\Omega_{L}}\left(
0,+\right) $ for reaching the barrier starting from zero length in the growing
state as calculated from Eq.~(\ref{eq:MFPT0p}).
\begin{table}[ptb
\begin{tabular}
[c]{|l|l|l|}\hline
$L$ & Yeast & Plant\\\hline\hline
$5\mu m$ & $8.89$ & $3.29$\\\hline
$20\mu m$ & $119.1$ & $17.23$\\\hline
$100\mu m$ & $2.9\times10^{6}$ & $266.55$\\\hline
\end{tabular}
\caption{Table of the MFPT $T_{\Omega_{L}}(0,+)$ (in min) as a function of the
distance $L$.
\label{tab:MFPT
\end{table}We see that for a yeast MT the largest boundary distance $L=100\mu
m$ is effectively unbridgeable, and that even the plant MT needs on average
$\approx10$ times its natural lifetime $\bar{t}\approx20\min$ to first reach
the boundary, although it is only $\approx1.5$ times its natural length of
$\bar{l}\approx65\mu m$.
\section{The Gopalakrishnan-Govindan search-and-capture model}
\label{sec:GG}The first-passage-time model Gopalakrishnan and Govindan
recently introduced (\cite{Gopalakrishnan2011}, hereafter referred to as GG)
considers the problem of the `capture' of a chromosome by a MT 'searching' for
it. It has the following ingredients: The MT is nucleated at a rate $r_{n}$
from a centrosome in an arbitrary direction into a cone with solid angle
opening of $\Delta\Omega$. The centrosome is located at a distance $d$ from
the chromosome, which has a cross-sectional area $a$ and therefore subtends a
solid angle $\Delta\Omega_{c}=a/d^{2}$ as seen from the centrosome. The
probability of being nucleated into a direction in which the target can
possibly be hit is therefore given by $p_{c}=\Delta\Omega_{c}/\Delta\Omega$.
When the MT is nucleated outside of the `\emph{target cone}', it can
potentially grow until it hits a cell boundary located at a distance we will
call $D$ from the centrosome. At this boundary, the MT is initially stalled,
but experiences an increased catastrophe rate $r_{b}>r_{+}$.
We will now revisit this model, using the formalism derived in the previous
sections. The state space of this model is conveniently represented by
$\Omega=N\cup A_{b}\cup B\cup A_{c}\cup C$. Here, as before, $N$ is the
nucleation state, $A_{b}$ are the active states with lengths in the interval
$[0,D]$ in the directions that do not `see' the target, $B$ the state of being
at the cell boundary, $A_{c}$ the active states with lengths in the interval
$[0,d]$ and directions within the target cone, and, finally, $C$ the state of
being on the target chromosome. We non-dimensionalize using the same
prescription as in Section \ref{sec:dimen}, denoting the additional parameters
needed by $\Delta=D/\bar{l}$ and $\delta=d/\bar{l}$. Using the results of
Section~\ref{sec:formal} we can immediately write down an expression for the
search-time starting from the nucleation stat
\begin{multline}
T_{\Omega/C}\left( N\right) =\frac{1}{\rho_{n}}+\left( 1-p_{c}\right)
\left\{ T_{A_{b}\cup B}\left( \left( 0,+\right) _{A_{b}}\right)
+T_{\Omega/C}\left( N\right) \right\} \label{eq:Tc}\\
+p_{c}\left\{ R_{A_{c}}^{C}\left( \left( 0,+\right) _{A_{c}}\right)
T_{A_{c}}^{C}\left( \left( 0,+\right) _{A_{c}}\right) +R_{A_{c}
^{N}\left( \left( 0,+\right) _{A_{c}}\right) \left\{ T_{A_{c}}^{N}\left(
\left( 0,+\right) _{A_{c}}\right) +T_{\Omega/C}\left( N\right) \right\}
\right\}
\end{multline}
The logic of this equation is simple. Starting from the nucleation state the
MT (on average) waits $1/\rho_{n}$ before being nucleation. With probability
$\left( 1-p_{c}\right) $ the nucleation will be in a direction that can not
hit the target. In that case the MT will spend the origin-return time
$T_{A_{b}\cup B}\left( \left( 0,+\right) _{A_{b}}\right) $ in this part of
state space before shrinking back to zero length and starting again from the
nucleation state. With probability $p_{c}$ the initial nucleation is inside
the target cone. In that case the MT either hits the target, without first
shrinking back to zero-length, with probability $R_{A_{c}}^{C}\left( \left(
0,+\right) _{A_{c}}\right) $ taking time $T_{A_{c}}^{C}\left( \left(
0,+\right) _{A_{c}}\right) $, or, with probability $R_{A_{c}}^{N}\left(
\left( 0,+\right) _{A_{c}}\right) =1-R_{A_{c}}^{C}\left( \left(
0,+\right) _{A_{c}}\right) $ shrinking back to zero-length before hitting
the target, taking time $T_{A_{c}}^{N}\left( \left( 0,+\right) _{A_{c
}\right) $ and then trying again from the nucleation state. This process is
illustrated in Figure~\ref{fig:GG}. \begin{figure}[ptb]
\includegraphics[width=\textwidth]{GG}\caption{Schematic illustration of the
structure of the search process in the Gopalakrishnan-Govindan
search-and-capture model. From the nucleation state the MT must either perform
(with probability $1-p_{c}$) a fruitless search (gray arrows) in the
directions where it can interact with the cell boundary $A_{b}$, or (with
probability $p_{c}$) it is nucleated in the right direction and then either
directly traverses $A_{c}$ to reach the chromosome $C$ or shrinks back to zero
length without reaching the target and must try again.
\label{fig:GG
\end{figure}
Extracting $T_{\Omega/C}\left( N\right) $ from the relation (\ref{eq:Tc})
yield
\begin{multline}
p_{c}R_{A_{c}}^{C}\left( \left( 0,+\right) _{A_{c}}\right) T_{\Omega
/C}\left( N\right) =\frac{1}{\rho_{+}}+\left( 1-p_{c}\right) T_{A_{b}\cup
B}\left( \left( 0,+\right) _{A_{b}}\right) \\
+p_{c}\left\{ R_{A_{c}}^{C}\left( \left( 0,+\right) _{A_{c}}\right)
T_{A_{c}}^{C}\left( \left( 0,+\right) _{A_{c}}\right) +R_{A_{c}
^{N}\left( \left( 0,+\right) _{A_{c}}\right) T_{A_{c}}^{N}\left( \left(
0,+\right) _{A_{c}}\right) \right\} \label{eq:Tsearch
\end{multline}
As we show in Appendix \ref{sec:GGapp}, this is, apart from the changed
notation, precisely the result derived by GG (their Eq.~(34) in Appendix A)
from an explicit sum-over-histories argument. We also note the calculation of
the fruitless search time, $T_{A_{b}\cup B}\left( \left( 0,+\right)
_{A_{b}}\right) $, in the directions not containing the target, is also
readily simplified using the methods presented her
\begin{multline}
T_{A_{b}\cup B}\left( \left( 0,+\right) _{A_{b}}\right) =R_{A_{b}
^{N}\left( \left( 0,+\right) _{A_{b}}\right) T_{A_{b}}^{N}\left( \left(
0,+\right) _{A_{b}}\right) \\
+R_{A_{b}}^{B}\left( \left( 0,+\right) _{A_{b}}\right) \left\{ T_{A_{b
}^{B}\left( \left( 0,+\right) _{A_{b}}\right) +\frac{1}{\rho_{b}
+T_{A_{b}\cup B}\left( \left( \Delta,-\right) _{A_{b}}\right) \right\} ,
\end{multline}
where in tur
\begin{align}
T_{A_{b}\cup B}\left( \left( \Delta,-\right) _{A_{b}}\right) &
=R_{A_{b}}^{N}\left( \left( \Delta,-\right) _{A_{b}}\right) T_{A_{b}
^{N}\left( \left( \Delta,-\right) _{A_{b}}\right) \\
& +R_{A_{b}}^{B}\left( \left( \Delta,-\right) _{A_{b}}\right) \left\{
T_{A_{b}}^{B}\left( \left( \Delta,-\right) _{A_{b}}\right) +\frac{1
{\rho_{b}}+T_{A_{b}\cup B}\left( \left( \Delta,-\right) _{A_{b}}\right)
\right\} .\nonumber
\end{align}
The latter expression allows $T_{A_{b}\cup B}\left( \left( \Delta,-\right)
_{A_{b}}\right) $ to be expressed solely of splitting probabilities and
conditional MFPTs. While GG use an ingenious symmetry argument interpreting a
shrinking MT as a growing `anti'-MT to calculate these latter quantities, we
point out that they can also be obtained in a straightforward manner from the
differential equations presented in Sections \ref{sec:split} and
\ref{sec:CMFPT}.
\section{Conclusions and outlook}
Our aim was to present a structured approach to the problem of MTs interacting
with boundaries. To this end we relied exclusively on `backward' techniques,
focussing on survival probabilities and their associated boundary value
problems. The upshot of this approach is that it allows one to decompose a
complex MFPT problem \emph{a priori} into closed form self-consistency problem
involving a small set of relevant splitting probabilities and conditional
MFPTs that readily follow from a proper disjoint decomposition of the state
space. The utility of this approach is illustrated by its application to the
Gopala\-krishnan-\-Govindan model, where the key decomposition of the search
time in terms of the time spent fruitlessly searching in the wrong directions,
waiting in the nucleation state and finally reaching the target is the
starting point of the calculation, rather than, as in
\cite{Gopalakrishnan2011}, the result of collecting the results of
intermediate steps in the calculation. We hope that the technique presented
will serve as a convenient starting point for future applications to current
problems in microtubule cytoskeleton organization.
\acknowledgments This work is part of the research program of the
\textquotedblleft Stichting voor Fundamenteel Onderzoek der Materie
(FOM)\textquotedblright, which is financially supported by the
\textquotedblleft Nederlandse Organisatie voor Wetenschappelijk Onderzoek
(NWO)\textquotedblright.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,630 |
\section{Introduction} Due to the better performance in many situations than existing classical iterative algorithms, randomized iterative algorithms for solving a linear system of equations
\beq\label{lin} {\bf Ax=b},\quad \mbf A\in\mbbr^{m\times n},\quad \mbf b\in\mbbr^m,\eeq have attracted much attention recently; see, for example, \cite{strohmer2009rando,leventhal2010rando,needell2010rando,eldar2011accel,zouzias2013rando,needell2014paved,dumitrescu2015relat,ma2015conve,gower2015rando,liu2016accel,needell2016stoch,hefny2017rows} and the references therein. In this paper, we consider the randomized Kaczmarz (RK) algorithm \cite{strohmer2009rando}, the randomized Gauss-Seidel (RGS) algorithm \cite{leventhal2010rando}, the randomized extended Kaczmarz (REK) algorithm \cite{zouzias2013rando}, and the randomized extended Gauss-Seidel (REGS) algorithm \cite{ma2015conve}.
Let $\mbf A^\dag$ denote the Moore-Penrose pseudoinverse \cite{ben2003gener} of $\bf A$. We summarize the convergence of RK, RGS, REK, and REGS in expectation to the Moore-Penrose pseudoinverse solution $\mbf A^\dag\mbf b$ for all types of linear systems in Table \ref{t1}.
\begin{table}[htp]
\caption{Summary of the convergence of {\rm RK}, {\rm RGS}, {\rm REK}, and {\rm REGS} in expectation to the Moore-Penrose pseudoinverse solution $\mbf A^\dag\mbf b$ for all types of linear systems: {\rm Y} means the algorithm is convergent and {\rm N} means not.} \label{t1}
\begin{center}
\begin{tabular}{c|c|c|c|c|c} \toprule
linear system (\ref{lin}) & $\rank(\mbf A)$ & RK & RGS &REK & REGS \\
\hline
consistent & $=n$ & Y & Y & Y & Y \\
consistent & $<n$ & Y & N & Y & Y \\
inconsistent & $=n$ & N & Y & Y & Y \\
inconsistent & $<n$ & N & N & Y & Y \\ \bottomrule
\end{tabular}
\end{center}
\end{table}
{\it Main contributions}. We show that REK is essentially an RK-RK approach (see Remark \ref{RKRK}) and that REGS is essentially an RGS-RK approach (see Remark \ref{RGSRK}). We present refined upper bounds for the convergence of REK and REGS. These bounds hold for all types of linear systems (consistent or inconsistent, overdetermined or underdetermined, $\mbf A$ has full column rank or not) and are attainable. In addition, we point out that the proof for Theorem 4.1 of \cite{ma2015conve} is incomplete and we resolve this issue.
{\it Organization of the paper}. In the rest of this section, we give some notation and preliminaries. In section 2, we review the randomized Kaczmarz algorithm and the randomized extended Kaczmarz algorithm. We present a slightly different variant of REK and prove its convergence. In section 3, we review the randomized Gauss-Seidel algorithm and the randomized extended Gauss-Seidel algorithm. We show that the convergence analysis for REGS of \cite{ma2015conve} is incomplete. We present a mathematically equivalent variant of REGS and prove its convergence. Numerical examples are given in section 4 to illustrate the theoretical results. We present brief concluding remarks in section 5.
{\it Notation and preliminaries}. For any random variable $\bm\xi$, let $\mbbe\bem\bm\xi\eem$ denote its expectation. For an integer $m\geq 1$, let $[m]:=\{1,2,3,\ldots,m\}$. Throughout the paper all vectors are assumed to be column vectors. For any vector $\mbf u\in\mbbr^m$, we use $\bf u^\rmt$, $u_i$, and $\|\mbf u\|_2$ to denote the transpose, the $i$th entry, and the Euclidean norm of $\mbf u$, respectively. We use ${\bf e}_j$ to denote the $j$th column of the identity matrix $\mbf I$ whose order is clear from the context. For any matrix $\mbf A\in\mbbr^{m\times n}$, we use $\mbf A^\rmt$, $\|\mbf A\|_\rmf$, $\rank(\mbf A)$, $\ran(\mbf A)$, $\nul(\mbf A)$, $\sigma_{1}(\mbf A)$, and $\sigma_{r}(\mbf A)$ to denote the transpose, the Frobenius norm, the rank, the column space, the nullspace, the largest singular value, and the smallest nonzero singular value of $\mbf A$, respectively.
We denote the columns and rows of $\bf A$ by $\{\mbf a_j\}_{j=1}^n$ and $\{\wt{\mbf a}_i^\rmt\}_{i=1}^m$, respectively. That is to say, $${\bf A}=\bem {\bf a}_1 & {\bf a}_2 &\cdots& {\bf a}_n\eem,\quad {\bf A}^\rmt=\bem\wt{\bf a}_1 & \wt{\bf a}_2 & \cdots & \wt{\bf a}_m \eem.$$ All the convergence results depend on the positive number $\rho$ defined as $$\rho:=1-\frac{\sigma_{r}^2(\bf A)}{\|\mbf A\|_\rmf^2}.$$ The following lemmas will be used extensively in this paper. Their proofs are straightforward.
\lemma\label{leq} Let $\mbf A$ be any nonzero real matrix. For every $\mbf u\in\ran(\mbf A)$, it holds \beqs\label{est1}\mbf u^\rmt\l(\mbf I-\frac{\bf AA^\rmt}{\|\mbf A\|_\rmf^2}\r)\mbf u\leq\rho\|\mbf u\|_2^2.\eeqs The equality holds if $\sigma_1(\mbf A)=\sigma_r(\mbf A)$, i.e., all the nonzero singular values of $\mbf A$ are the same.
\endlemma
\lemma\label{proj} Let $\mbf a$ be any nonzero vector. Then $$\l(\frac{\mbf a\mbf a^\rmt}{\|\mbf a\|_2^2}\r)^2=\frac{\mbf a\mbf a^\rmt}{\|\mbf a\|_2^2},\quad \l(\mbf I-\frac{\mbf a\mbf a^\rmt}{\|\mbf a\|_2^2}\r)^2=\mbf I-\frac{\mbf a\mbf a^\rmt}{\|\mbf a\|_2^2}.$$
\endlemma
\section{Randomized Kaczmarz and its extension}
Strohmer and Vershynin \cite{strohmer2009rando} proposed the following randomized Kaczmarz algorithm (Algorithm 1).
\begin{table}[htp]
\begin{center}
\begin{tabular*}{115mm}{l}
\toprule {\bf Algorithm 1}. Randomized Kaczmarz \cite{strohmer2009rando} for $\bf Ax=b$
\\ \hline \vspace{-3mm}\\
\qquad Initialize $\mbf x^0\in\mbbr^n$ \\
\qquad {\bf for} $k=1,2,\ldots$ {\bf do}\\
\qquad\qquad Pick $i\in[m]$ with probability $\|\wt{\bf a}_i\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp{\bf x}^k={\bf x}^{k-1}-\frac{\wt{\bf a}_i^\rmt{\bf x}^{k-1}-b_i}{\|\wt{\bf a}_i\|_2^2}\wt{\bf a}_i$\\
\bottomrule
\end{tabular*}
\end{center}
\end{table}
If $\bf Ax=b$ is consistent, Zouzias and Freris \cite[Theorem 3.4]{zouzias2013rando} proved that RK with initial guess $\mbf x^0\in\ran(\mbf A^\rmt)$ generates $\mbf x^k$ which converges linearly in expectation to the Moore-Penrose pseudoinverse solution ${\bf A^\dag b}$: $$\mbbe\bem\|{\bf x}^k-{\bf A^\dag b} \|_2^2\eem\leq\rho^k\|{\bf x}^0-{\bf A^\dag b}\|_2^2.$$ By the same approach as used in the proof of Theorem 3.2 of \cite{zouzias2013rando}, we can prove the following theorem, which will be used to prove the refined upper bound for the convergence of REK.
\theorem\label{estzk} Let ${\bf A}\in\mbbr^{m\times n}$ and ${\bf b}\in\mbbr^m$. Let ${\bf z}^k$ denote the $k$th iterate of {\rm RK} applied to ${\bf A^\rmt z=0}$ with initial guess ${\bf z}^0\in{\bf b}+\ran({\bf A})$. In exact arithmetic, it holds \beqs\label{bound1}\mbbe\bem\|{\bf z}^k-{(\bf I-AA^\dag)b}\|_2^2\eem\leq\rho^k\|\mbf z^0- {(\mbf I-\bf AA^\dag) b} \|_2^2.\eeqs
\endtheorem
\proof
The iteration is $${\bf z}^k={\bf z}^{k-1}-\frac{{\bf a}_j^\rmt{\bf z}^{k-1}}{\|{\bf a}_j\|_2^2}{\bf a}_j.$$ By $\mbf a_j^\rmt{\bf(I- AA^\dag) b}=0$ (since ${\bf A^\rmt (I- AA^\dag) b=0}$), we have \beqa
{\bf z}^k-{\bf (I- AA^\dag) b}&=&{\bf z}^{k-1}-{\bf (I- AA^\dag) b}-\frac{{\bf a}_j^\rmt{\bf z}^{k-1}-\mbf a_j^\rmt{\bf(I- AA^\dag) b}}{\|{\bf a}_j\|_2^2}{\bf a}_j\\ \nn &=&{\bf z}^{k-1}-{\bf (I- AA^\dag) b}-\frac{{\bf a}_j^\rmt({\bf z}^{k-1}-{\bf(I- AA^\dag) b})}{\|{\bf a}_j\|_2^2}{\bf a}_j\\\nn&=&\l({\bf I}-\frac{{\bf a}_j{\bf a}_j^\rmt}{\|{\bf a}_j\|_2^2}\r)({\bf z}^{k-1}-{\bf(I- AA^\dag) b}).\eeqas
By ${\bf z}^0\in{\bf b}+\ran({\bf A})$ and ${\bf AA^\dag b}\in\ran({\bf A})$, we have ${\bf z}^0-{\bf(I- AA^\dag) b}\in\ran(\bf A)$. Then it is easy to show that ${\bf z}^k-{\bf(I- AA^\dag) b}\in\ran(\bf A)$ by induction. Let $\mbbe_{k-1}\bem\cdot\eem$ denote the conditional expectation conditioned on the first $k-1$ iterations of RK. It follows that \beqas&&\mbbe_{k-1}\bem\|{\bf z}^k-{\bf(I- AA^\dag) b}\|_2^2\eem\\&=&\mbbe_{k-1}\bem ({\bf z}^{k}-{\bf(I- AA^\dag) b})^\rmt({\bf z}^{k}-{\bf(I- AA^\dag) b})\eem\\&=&\mbbe_{k-1}\bem({\bf z}^{k-1}-{\bf(I- AA^\dag) b})^\rmt\dsp\l({\bf I}-\frac{{\bf a}_j{\bf a}_j^\rmt}{\|{\bf a}_j\|_2^2}\r)^2({\bf z}^{k-1}-{\bf(I- AA^\dag) b})\ee
\\&=&\mbbe_{k-1}\bem({\bf z}^{k-1}-{\bf(I- AA^\dag) b})^\rmt\dsp\l({\bf I}-\frac{{\bf a}_j{\bf a}_j^\rmt}{\|{\bf a}_j\|_2^2}\r)({\bf z}^{k-1}-{\bf(I- AA^\dag) b})\eem
\\ &=&({\bf z}^{k-1}-{\bf(I- AA^\dag) b})^\rmt\l({\bf I}-\frac{\bf AA^\rmt}{\|{\bf A}\|_\rmf^2}\r)({\bf z}^{k-1}-{\bf(I- AA^\dag) b})\\&\leq&\rho\|{\bf z}^{k-1}-{\bf(I- AA^\dag) b}\|_2^2. \quad (\mbox{by Lemma \ref{leq}})\eeqas Taking expectation gives $$\mbbe\bem\|{\bf z}^k-{\bf(I- AA^\dag) b}\|_2^2\eem\leq \rho\mbbe\bem\|{\bf z}^{k-1}-{\bf(I- AA^\dag) b}\|_2^2\eem.$$ Unrolling the recurrence yields the result. \endproof
If $\bf Ax=b$ is inconsistent, Needell \cite{needell2010rando} and Zouzias and Freris \cite{zouzias2013rando} showed that RK does not converge to $\mbf A^\dag\mbf b$. To resolve this problem, Zouzias and Freris \cite{zouzias2013rando} proposed the following randomized extended Kaczmarz algorithm (here we call it REK-ZF, see Algorithm 2). They proved the convergence bound \beq\label{rekb} \mbbe\bem\|{\bf x}^k-\mbf A^\dag\mbf b\|_2^2\eem\leq\rho^{\lf k/2\rf}(1+2\sigma_{1}^2(\mbf A)/\sigma_{r}^2(\mbf A))\|\mbf A^\dag\mbf b\|_2^2.\eeq
\begin{table}[htp]
\begin{center}
\begin{tabular*}{115mm}{l}
\toprule {\bf Algorithm 2}. REK-ZF \cite{zouzias2013rando}
\\ \hline \vspace{-3mm}\\
\qquad Initialize $\mbf x^0\in\ran({\bf A}^\rmt)$ and ${\bf z}^0\in\mbf b+\ran(\mbf A)$ \\
\qquad {\bf for} $k=1,2,\ldots$ {\bf do}\\
\qquad\qquad Pick $j\in[n]$ with probability $\|{\bf a}_j\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp{\bf z}^k={\bf z}^{k-1}-\frac{{\bf a}_j^\rmt{\bf z}^{k-1}}{\|{\bf a}_j\|_2^2}{\bf a}_j$\\
\qquad\qquad Pick $i\in[m]$ with probability $\|\wt{\bf a}_i\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp{\bf x}^k={\bf x}^{k-1}-\frac{\wt{\bf a}_i^\rmt{\bf x}^{k-1}-b_i+z_i^{k-1}}{\|\wt{\bf a}_i\|_2^2}\wt{\bf a}_i$\\
\bottomrule
\end{tabular*}
\end{center}
\end{table}
\remark\label{RKRK} The original randomized extended Kaczmarz algorithm of {\rm\cite{zouzias2013rando}} uses $\mbf x^0=\mbf 0$ and ${\bf z}^0=\mbf b$. Essentially, {\rm REK-ZF} is an {\rm RK-RK} approach: $\mbf z^k$ is the $k$th iterate of {\rm RK} applied to $\bf A^\rmt z=0$ with initial guess $\mbf z^0$, and $\mbf x^k$ is one step {\rm RK} update for the linear system ${\bf Ax=b-z}^{k-1}$ from $\mbf x^{k-1}$.
\endremark
Next we present a slightly different randomized extended Kaczmarz algorithm (we call it REK-S, see Algorithm 3) which generates $\mbf x^k$ by one step RK update for the linear system ${\bf Ax=b-z}^k$ (used in REK-S) instead of ${\bf Ax=b-z}^{k-1}$ (used in REK-ZF) from $\mbf x^{k-1}$.
\begin{table}[htp]
\begin{center}
\begin{tabular*}{115mm}{l}
\toprule {\bf Algorithm 3}. REK-S
\\ \hline \vspace{-3mm}\\
\qquad Initialize $\mbf x^0\in\ran(\mbf A^\rmt)$ and ${\bf z}^0\in{\bf b}+\ran({\bf A})$ \\
\qquad {\bf for} $k=1,2,\ldots$ {\bf do}\\
\qquad\qquad Pick $j\in[n]$ with probability $\|{\bf a}_j\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp{\bf z}^k={\bf z}^{k-1}-\frac{{\bf a}_j^\rmt{\bf z}^{k-1}}{\|{\bf a}_j\|_2^2}{\bf a}_j$\\
\qquad\qquad Pick $i\in[m]$ with probability $\|\wt{\bf a}_i\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp{\bf x}^k={\bf x}^{k-1}-\frac{\wt{\bf a}_i^\rmt{\bf x}^{k-1}-b_i+z_i^k}{\|\wt{\bf a}_i\|_2^2}\wt{\bf a}_i$\\
\bottomrule
\end{tabular*}
\end{center}
\end{table}
In the following theorem, we present the convergence bound for REK-S, which is obviously better than the bound (\ref{rekb}). Actually, our bound is attainable (see Remark \ref{sharp1}). We note that a refined convergence bound for REK-ZF can be obtained by the same approach.
\theorem\label{main1} Let ${\bf A}\in\mbbr^{m\times n}$ and ${\bf b}\in\mbbr^m$. Let ${\bf x}^k$ denote the $k$th iterate of {\rm REK-S} with $\mbf x^0\in\ran(\mbf A^\rmt)$ and $\mbf z^0\in\mbf b+\ran(\mbf A)$. In exact arithmetic, it holds \beq\label{bound2}\mbbe\bem\|{\bf x}^k-{\bf A^\dag b}\|_2^2\eem\leq\rho^k\|{\bf x}^0-{\bf A^\dag b}\|_2^2+\frac{\rho^k(1-\rho^k)}{\sigma_{r}^2(\mbf A)}\|{\bf z}^0-{\bf (I-AA^\dag) b}\|_2^2.\eeq
\endtheorem
\proof Let $$\wh{\mbf x}^k={\bf x}^{k-1}-\frac{\wt{\bf a}_i^\rmt{\bf x}^{k-1}-b_i+\mbf e_i^\rmt{\bf (I-AA^\dag) b}}{\|\wt{\bf a}_i\|_2^2}\wt{\bf a}_i.$$ We have \beqas\wh{\mbf x}^k-{\bf A^\dag b}&=&{\bf x}^{k-1}-{\bf A^\dag b}-\frac{\wt{\bf a}_i^\rmt{\bf x}^{k-1}-\mbf e_i^\rmt{\bf AA^\dag b}}{\|\wt{\bf a}_i\|_2^2}\wt{\bf a}_i\\&=&{\bf x}^{k-1}-{\bf A^\dag b}-\frac{\wt{\bf a}_i^\rmt{\bf x}^{k-1}-\wt{\bf a}_i^\rmt{\bf A^\dag b}}{\|\wt{\bf a}_i\|_2^2}\wt{\bf a}_i\\&=&\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)({\bf x}^{k-1}-{\bf A^\dag b})\eeqas and $${\mbf x}^k-\wh{\mbf x}^k=\frac{\mbf e_i^\rmt({\bf (I-AA^\dag) b}-\mbf z^k)}{\|\wt{\bf a}_i\|_2^2}\wt{\bf a}_i.$$ By the orthogonality $(\wh{\mbf x}^k-{\bf A^\dag b})^\rmt({\mbf x}^k-\wh{\mbf x}^k)=0$ (which is obvious from the above two equations), we have \beq\label{ksum}\|{\mbf x}^k-{\bf A^\dag b}\|_2^2=\|{\mbf x}^k-\wh{\mbf x}^k\|_2^2+\|\wh{\mbf x}^k-{\bf A^\dag b}\|_2^2.\eeq Let $\mbbe_{k-1}\bem\cdot\eem$ denote the conditional expectation conditioned on the first $k-1$ iterations of REK-S. That is, $$\mbbe_{k-1}\bem\cdot\eem=\mbbe\bem\cdot|j_1,i_1,j_2,i_2,\ldots,j_{k-1},i_{k-1}\eem,$$ where $j_l$ is the $l$th column chosen and $i_l$ is the $l$th row chosen. We denote the conditional expectation conditioned on the first $k-1$ iterations and the $k$th column chosen as $$\mbbe_{k-1}^i\bem\cdot\eem=\mbbe\bem\cdot|j_1,i_1,j_2,i_2,\ldots,j_{k-1},i_{k-1},j_k\eem.$$ Similarly, we denote the conditional expectation conditioned on the first $k-1$ iterations and the $k$th row chosen as $$\mbbe_{k-1}^j\bem\cdot\eem=\mbbe\bem\cdot|j_1,i_1,j_2,i_2,\ldots,j_{k-1},i_{k-1},i_k\eem.$$ Then by the law of total expectation we have $$\mbbe_{k-1}\bem\cdot\eem=\mbbe_{k-1}^j\bem\mbbe_{k-1}^i\bem\cdot\eem\eem.$$
It follows from \beqas\mbbe_{k-1}\bem\|{\mbf x}^k-\wh{\mbf x}^k\|_2^2\eem&=&\mbbe_{k-1}\bem\dsp\frac{(\mbf e_i^\rmt({\bf (I-AA^\dag) b}-\mbf z^k))^2}{\|\wt{\bf a}_i\|_2^2}\eem\\ &=&\mbbe_{k-1}^j\bem\mbbe_{k-1}^i\bem\dsp\frac{(\mbf e_i^\rmt({\bf (I-AA^\dag) b}-\mbf z^k))^2}{\|\wt{\bf a}_i\|_2^2}\eem\eem\\&=&\mbbe_{k-1}^j\bem\dsp\frac{\|\mbf z^k-{\bf (I-AA^\dag) b}\|_2^2}{\|\mbf A\|_\rmf^2}\eem\\&=&\frac{1}{\|\mbf A\|_\rmf^2}\mbbe_{k-1}\bem\|\mbf z^k-{\bf (I-AA^\dag) b}\|_2^2\eem\eeqas that \beqa\nn\mbbe\bem\|{\mbf x}^k-\wh{\mbf x}^k\|_2^2\eem&=&\frac{1}{\|\mbf A\|_\rmf^2}\mbbe\bem\|\mbf z^k-{\bf (I-AA^\dag) b}\|_2^2\eem\\\label{ksum1}&\leq&\frac{\rho^k}{\|\mbf A\|_\rmf^2}\|\mbf z^0-{\bf (I-AA^\dag) b}\|_2^2. \quad (\mbox{by Theorem \ref{estzk}})\eeqa By $\mbf x^0 \in\ran(\mbf A^\rmt)$ and ${\bf A^\dag b}\in\ran(\mbf A^\rmt)$, we have $\mbf x^0-{\bf A^\dag b}\in\ran(\mbf A^\rmt)$. Then it is easy to show that $\mbf x^k-{\bf A^\dag b}\in\ran(\mbf A^\rmt)$ by induction. It follows from \beqas\mbbe_{k-1}\bem\|\wh{\mbf x}^k-{\bf A^\dag b}\|_2^2\eem&=&\mbbe_{k-1}\bem(\wh{\mbf x}^k-{\bf A^\dag b})^\rmt(\wh{\mbf x}^k-{\bf A^\dag b})\eem\\&=&\mbbe_{k-1}\bem\dsp({\bf x}^{k-1}-{\bf A^\dag b})^\rmt\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)^2({\bf x}^{k-1}-{\bf A^\dag b})\eem\\&=&\mbbe_{k-1}\bem\dsp({\bf x}^{k-1}-{\bf A^\dag b})^\rmt\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)({\bf x}^{k-1}-{\bf A^\dag b})\eem\\&=&\dsp({\bf x}^{k-1}-{\bf A^\dag b})^\rmt\l(\mbf I-\frac{\bf A^\rmt A}{\|{\bf A}\|_\rmf^2}\r)({\bf x}^{k-1}-{\bf A^\dag b})\\&\leq&\rho\|{\bf x}^{k-1}-{\bf A^\dag b}\|_2^2\quad (\mbox{by Lemma \ref{leq}})\eeqas that \beq\label{ksum2}\mbbe\bem\|\wh{\mbf x}^k-{\bf A^\dag b}\|_2^2\eem\leq\rho\mbbe\bem\|{\bf x}^{k-1}-{\bf A^\dag b}\|_2^2\eem.\eeq Combining (\ref{ksum}), (\ref{ksum1}), and (\ref{ksum2}) yields \beqas\mbbe\bem[\|{\mbf x}^k-{\bf A^\dag b}\|_2^2\eem&=&\mbbe\bem\|{\mbf x}^k-\wh{\mbf x}^k\|_2^2\eem+\mbbe\bem\|\wh{\mbf x}^k-{\bf A^\dag b}\|_2^2\eem\\&\leq&\frac{\rho^k}{\|\mbf A\|_\rmf^2}\|\mbf z^0-{\bf (I-AA^\dag) b}\|_2^2+\rho\mbbe\bem\|{\bf x}^{k-1}-{\bf A^\dag b}\|_2^2\eem\\&\leq&\cdots\leq\rho^k\|{\bf x}^0-{\bf A^\dag b}\|_2^2+\frac{\rho^k}{\|\mbf A\|_\rmf^2}\|\mbf z^0-{\bf (I-AA^\dag) b}\|_2^2\sum_{l=0}^{k-1}\rho^l\\&=&\rho^k\|{\bf x}^0-{\bf A^\dag b}\|_2^2+\frac{\rho^k}{\|\mbf A\|_\rmf^2}\|\mbf z^0-{\bf (I-AA^\dag) b}\|_2^2\frac{1-\rho^k}{1-\rho}\\&=&\rho^k\|{\bf x}^0-{\bf A^\dag b}\|_2^2+\frac{\rho^k(1-\rho^k)}{\sigma_{r}^2(\mbf A)}\|\mbf z^0-{\bf (I-AA^\dag) b}\|_2^2.\eeqas
This completes the proof. \endproof
\remark The vector $\wh{\mbf x}^k$ used in the proof is actually one step {\rm RK} update for the linear system ${\bf Ax=AA^\dag b}$ from $\mbf x^{k-1}$.
\endremark
\remark\label{sharp1} By Lemma {\rm 1}, if $\sigma_1(\mbf A)=\sigma_r(\mbf A)$, then all the inequalities in the proofs of Theorems {\rm \ref{estzk}} and {\rm \ref{main1}} become equalities.
\endremark
\section{Randomized Gauss-Seidel and its extension}
Leventhal and Lewis \cite{leventhal2010rando} proposed the following randomized Gauss-Seidel (RGS) algorithm (Algorithm 4, also called the randomized coordinate descent algorithm).
\begin{table}[htp]
\begin{center}
\begin{tabular*}{115mm}{l}
\toprule {\bf Algorithm 4}. Randomized Gauss-Seidel \cite{leventhal2010rando}
\\ \hline \vspace{-3mm}\\
\qquad Initialize $\mbf x^0\in\mbbr^n$ \\
\qquad {\bf for} $k=1,2,\ldots$ {\bf do}\\
\qquad\qquad Pick $j\in[n]$ with probability $\|{\bf a}_j\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp{\bf x}^k={\bf x}^{k-1}-\frac{{\bf a}_j^\rmt({\bf Ax}^{k-1}-{\bf b})}{\|{\bf a}_j\|_2^2}{\bf e}_j$\\
\bottomrule
\end{tabular*}
\end{center}
\end{table}
The following theorem is a restatement of Lemma 4.2 of \cite{ma2015conve} and will be used to prove the refined bound for REGS. Here we provide a proof for completeness.
\theorem\label{AAdag} Let ${\bf A}\in\mbbr^{m\times n}$ and ${\bf b}\in\mbbr^m$. Let ${\bf x}^k$ denote the $k$th iterate of {\rm RGS} applied to ${\bf Ax=b}$ with arbitrary ${\bf x}^0\in\mbbr^n$. In exact arithmetic, it holds $$\mbbe\bem\|{\bf Ax}^{k}-{\bf AA^\dag b}\|_2^2\eem\leq \rho^k\|{\bf Ax}^0-{\bf AA^\dag b}\|_2^2.$$
\endtheorem
\proof
By ${\bf A^\rmt b}={\bf A^\rmt AA}^\dag{\bf b}$, we have \beqas {\bf x}^k-{\bf A^\dag b}&=&{\bf x}^{k-1}-{\bf A^\dag b}-\frac{{\bf a}_j^\rmt({\bf Ax}^{k-1}-{\bf b})}{\|{\bf a}_j\|^2_2}{\bf e}_j\\&=&{\bf x}^{k-1}-{\bf A^\dag b}- \frac{{\bf e}_j^\rmt({\bf A^\rmt Ax}^{k-1}-{\bf A^\rmt b})}{\|{\bf a}_j\|^2_2}{\bf e}_j\\&=&{\bf x}^{k-1}-{\bf A^\dag b}- \frac{{\bf e}_j^\rmt{\bf A^\rmt A}({\bf x}^{k-1}-{\bf A^\dag b})}{\|{\bf a}_j\|^2_2}{\bf e}_j\\&=&\l({\bf I}-\frac{{\bf e}_j{\bf e}_j^\rmt{\bf A^\rmt A}}{\|{\bf a}_j\|^2_2}\r)({\bf x}^{k-1}-{\bf A^\dag b}),\eeqas which yields $$ {\bf Ax}^k-{\bf AA^\dag b}=\l({\bf I}-\frac{{\bf a}_j{\bf a}_j^\rmt}{\|{\bf a}_j\|^2_2}\r)({\bf Ax}^{k-1}-{\bf AA^\dag b}).$$
It follows that \beqas&&\mbbe_{k-1}\bem\|{\bf Ax}^k-{\bf AA^\dag b}\|_2^2\eem=\mbbe_{k-1}\bem({\bf Ax}^k-{\bf AA^\dag b})^\rmt({\bf Ax}^k-{\bf AA^\dag b})\eem\\&=&\mbbe_{k-1}\bem({\bf Ax}^{k-1}-{\bf AA^\dag b})^\rmt\dsp\l({\bf I}-\frac{{\bf a}_j{\bf a}_j^\rmt}{\|{\bf a}_j\|^2_2}\r)^2({\bf Ax}^{k-1}-{\bf AA^\dag b})\eem\\&=&\mbbe_{k-1}\bem({\bf Ax}^{k-1}-{\bf AA^\dag b})^\rmt\dsp\l({\bf I}-\frac{{\bf a}_j{\bf a}_j^\rmt}{\|{\bf a}_j\|^2_2}\r)({\bf Ax}^{k-1}-{\bf AA^\dag b})\eem\\&=&({\bf Ax}^{k-1}-{\bf AA^\dag b})^\rmt\dsp\l({\bf I}-\frac{\bf AA^\rmt}{\|{\bf A}\|^2_\rmf}\r)({\bf Ax}^{k-1}-{\bf AA^\dag b})\\&\leq&\rho\|{\bf Ax}^{k-1}-{\bf AA^\dag b}\|_2^2.\quad (\mbox{by Lemma \ref{leq}}) \eeqas Taking expectation gives $$\mbbe\bem\|{\bf Ax}^k-{\bf AA^\dag b}\|_2^2\eem\leq\rho\mbbe\bem\|{\bf Ax}^{k-1}-{\bf AA^\dag b}\|_2^2\eem.$$ Unrolling the recurrence yields the result.
\endproof
If $\bf A$ has full column rank, Theorem \ref{AAdag} implies that $\mbf x^k$ converges linearly in expectation to $\bf A^\dag b$. If $\bf A$ does not have full column rank, RGS fails to converge (see \cite[section 3.3]{ma2015conve}). Ma, Needell, and Ramdas \cite{ma2015conve} proposed the following randomized extended Gauss-Seidel algorithm (we call it REGS-MNR, see Algorithm 5) to resolve this problem.
\begin{table}[htp]
\begin{center}
\begin{tabular*}{115mm}{l}
\toprule {\bf Algorithm 5}. REGS-MNR \cite{ma2015conve}
\\ \hline \vspace{-3mm}\\
\qquad Initialize $\mbf x^0\in\mbbr^n$ and $\mbf z^0\in\mbf x^0+\ran(\mbf A^\rmt)$ \\
\qquad {\bf for} $k=1,2,\ldots$ {\bf do}\\
\qquad\qquad Pick $j\in[n]$ with probability $\|{\bf a}_j\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp{\bf x}^k={\bf x}^{k-1}-\frac{{\bf a}_j^\rmt({\bf Ax}^{k-1}-{\bf b})}{\|{\bf a}_j\|_2^2}{\bf e}_j$\\
\qquad\qquad Pick $i\in[m]$ with probability $\|\wt{\bf a}_i\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp\mbf P_i=\mbf I-{\wt{\bf a}_i\wt{\bf a}_i^\rmt}/{\|\wt{\bf a}_i\|^2_2}$\\
\qquad\qquad Set $\dsp{\bf z}^k=\mbf P_i({\bf z}^{k-1}+\mbf x^k-\mbf x^{k-1})$\\
\qquad Output $\mbf x^t-\mbf z^t$ at some step $t$ as the estimated solution\\
\bottomrule
\end{tabular*}
\end{center}
\end{table}
\remark The original randomized extended Gauss-Seidel algorithm of {\rm\cite{zouzias2013rando}} uses $\mbf x^0=\mbf 0$ and ${\bf z}^0=\mbf 0$. Here, we use $\mbf x^0\in\mbbr^n$ and $\mbf z^0\in\mbf x^0+\ran(\mbf A^\rmt)$.
\endremark
Ma, Needell, and Ramdas proved that REGS-MNR converges linearly in expectation to the least norm solution in the case that $\mbf A$ has full row rank and $m<n$. They provided the convergence bound (see \cite[Theorem 4.1]{ma2015conve} for details) \beq\label{regsb}\mbbe\bem\|\mbf x^k-\mbf z^k-\mbf A^\dag\mbf b\|_2^2\eem\leq\rho^k\|\mbf A^\dag\mbf b\|_2^2+\frac{2\rho^{\lf k/2\rf}}{\sigma_r^2(\mbf A)}\|{\bf AA^\dag b}\|_2^2.\eeq
Their proof (see \cite[Page 1600, lines 10-11]{ma2015conve}) uses Fact 1 of \cite[Page 1598]{ma2015conve} (which is that for any $\mbf P_i$ as in Algorithm 5, $\mbbe\bem\|\mbf P_i\mbf v\|_2^2\eem\leq\rho\|\mbf v\|_2^2$ for any vector $\mbf v\in\mbbr^n$) to show that $$\mbbe\bem\|\mbf P_i(\mbf x^{k-1}-\mbf z^{k-1}-\mbf A^\dag\mbf b)\|_2^2\eem\leq\rho\|\mbf x^{k-1}-\mbf z^{k-1}-\mbf A^\dag\mbf b\|_2^2.$$
However, Fact 1 of \cite[Page 1598]{ma2015conve} is not true because for any nonzero vector $\mbf v\in\nul(\bf A)$, we have $$\mbbe\bem\|\mbf P_i\mbf v\|_2^2\eem=\mbbe\bem\mbf v^\rmt\mbf P_i\mbf v\eem=\mbf v^\rmt\l(\mbf I-\frac{\bf A^\rmt A}{\|\mbf A\|_\rmf^2}\r)\mbf v=\|\mbf v\|_2^2.$$
Therefore, the proof is incomplete. This issue can be resolved easily. Actually, by Lemma \ref{leq}, the above inequality still holds because $\mbf x^{k-1}-\mbf z^{k-1}-\mbf A^\dag\mbf b\in\ran(\mbf A^\rmt)$, which can be proved by induction.
\begin{table}[htp]
\begin{center}
\begin{tabular*}{115mm}{l}
\toprule {\bf Algorithm 6}. REGS-E
\\ \hline \vspace{-3mm}\\
\qquad Initialize $\mbf x^0\in\mbbr^n$ and $\mbf z^0\in\ran(\mbf A^\rmt)$\\
\qquad {\bf for} $k=1,2,\ldots$ {\bf do}\\
\qquad\qquad Pick $j\in[n]$ with probability $\|{\bf a}_j\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp{\bf x}^k={\bf x}^{k-1}-\frac{{\bf a}_j^\rmt({\bf Ax}^{k-1}-{\bf b})}{\|{\bf a}_j\|_2^2}{\bf e}_j$\\
\qquad\qquad Pick $i\in[m]$ with probability $\|\wt{\bf a}_i\|_2^2/\|{\bf A}\|_\rmf^2$\\
\qquad\qquad Set $\dsp{\bf z}^k={\bf z}^{k-1}-\frac{\wt{\bf a}_i^\rmt({\bf z}^{k-1}-{\bf x}^k)}{\|\wt{\bf a}_i\|^2_2}\wt{\bf a}_i$\\
\bottomrule
\end{tabular*}
\end{center}
\end{table}
Next we study the convergence of REGS for a general linear system (consistent or inconsistent, full rank or rank-deficient). For the convenience of discussion, we present the following randomized extended Gauss-Seidel algorithm (we call it REGS-E, see Algorithm 6) which is mathematically equivalent to REGS-MNR. Actually, in exact arithmetic, the vector $\mbf z^k$ in REGS-E is equal to the vector $\mbf x^k-\mbf z^k$ in REGS-MNR.
\remark\label{RGSRK} Essentially, {\rm REGS-E} is an {\rm RGS-RK} approach: $\mbf x^k$ is the $k$th iterate of {\rm RGS} and $\mbf z^k$ is one step {\rm RK} update for the linear system ${\bf Az=Ax}^k$ from $\mbf z^{k-1}$
\endremark
In the following theorem, we show that the vector ${\bf z}^k$ in REGS-E converges linearly in expectation to $\bf A^\dag b$. Our proof is almost the same as that of \cite[Theorem 4.1]{ma2015conve} but avoiding their mistake. The new convergence bound is attainable (see Remark \ref{sharp2}) and obviously better than the bound $(\ref{regsb})$.
\theorem\label{main2} Let ${\bf A}\in\mbbr^{m\times n}$ and ${\bf b}\in\mbbr^m$. Let ${\bf z}^k$ denote the $k$th iterate of {\rm REGS-E} with arbitrary $\mbf x^0\in\mbbr^n$ and $\mbf z^0\in\ran(\mbf A^\rmt)$. In exact arithmetic, it holds \beq\label{bound3}\mbbe\bem\|{\bf z}^k-{\bf A^\dag b}\|_2^2\eem\leq\rho^k\|{\bf z}^0-{\bf A^\dag b}\|_2^2+\frac{\rho^k(1-\rho^k)}{\sigma_{r}^2(\mbf A)}\|{\bf Ax}^0-{\bf AA^\dag b}\|_2^2.\eeq
\endtheorem
\proof By $\mbf z^0\in\ran(\mbf A^\rmt)$ and $\mbf A^\dag\mbf b\in\ran(\mbf A^\rmt)$, we have $\mbf z^0-\mbf A^\dag\mbf b\in\ran(\mbf A^\rmt)$. Then it is easy to show that $\mbf z^k-\mbf A^\dag\mbf b\in\ran(\mbf A^\rmt)$ by induction.
We now analyze the norm of $\mbf z^k-{\bf A^\dag b}$. Note that \beqas\mbf z^k-{\bf A^\dag b}&=&{\bf z}^{k-1}-\frac{\wt{\bf a}_i^\rmt({\bf z}^{k-1}-{\bf x}^k)}{\|\wt{\bf a}_i\|^2_2}\wt{\bf a}_i-{\bf A^\dag b}\\&=&\dsp\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r){\bf z}^{k-1}+\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\mbf x^k-{\bf A^\dag b}\\&=&\dsp\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)({\bf z}^{k-1}-{\bf A^\dag b})+\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}(\mbf x^k-{\bf A^\dag b}).\eeqas It follows from the orthogonality, i.e., $$(\mbf x^k-{\bf A^\dag b})^\rmt\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)({\bf z}^{k-1}-{\bf A^\dag b})=0,$$ that \beqa\label{sum}\|\mbf z^k-{\bf A^\dag b}\|_2^2&=&\l\|\dsp\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)({\bf z}^{k-1}-{\bf A^\dag b})\r\|_2^2+\l\|\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}(\mbf x^k-{\bf A^\dag b})\r\|_2^2
\eeqa
It follows from \beqas&&\mbbe_{k-1}\bem\l\|\dsp\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)({\bf z}^{k-1}-{\bf A^\dag b})\r\|_2^2\eem\\&=&\mbbe_{k-1}\bem({\bf z}^{k-1}-{\bf A^\dag b})^\rmt\dsp\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)^2({\bf z}^{k-1}-{\bf A^\dag b})\eem\\&=&\mbbe_{k-1}\bem({\bf z}^{k-1}-{\bf A^\dag b})^\rmt\dsp\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)({\bf z}^{k-1}-{\bf A^\dag b})\ee
\\&=&({\bf z}^{k-1}-{\bf A^\dag b})^\rmt\dsp\l(\mbf I-\frac{\bf A^\rmt A}{\|\mbf A\|_\rmf^2}\r)({\bf z}^{k-1}-{\bf A^\dag b})\\&\leq&\rho\|{\bf z}^{k-1}-{\bf A^\dag b}\|_2^2\quad (\mbox{by Lemma \ref{leq}})\eeqas that \beq\label{sum1}\mbbe\bem\l\|\dsp\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)({\bf z}^{k-1}-{\bf A^\dag b})\r\|_2^2\eem\leq\rho\mbbe\bem\|{\bf z}^{k-1}-{\bf A^\dag b}\|_2^2\eem.\eeq
It follows from \beqas&&\mbbe_{k-1}\bem\dsp\l\|\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}(\mbf x^k-{\bf A^\dag b})\r\|_2^2\eem\\&=&\mbbe_{k-1}\bem\dsp(\mbf x^k-{\bf A^\dag b})^\rmt\l(\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)^2(\mbf x^k-{\bf A^\dag b})\eem\\&=&\mbbe_{k-1}^j\bem\mbbe_{k-1}^i\bem\dsp(\mbf x^k-{\bf A^\dag b})^\rmt\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}(\mbf x^k-{\bf A^\dag b})\eem\ee
\\&=&\mbbe_{k-1}^j\bem\dsp(\mbf x^k-{\bf A^\dag b})^\rmt\frac{\bf A^\rmt A}{\|\mbf A\|_\rmf^2}(\mbf x^k-{\bf A^\dag b})\eem\\&=&\frac{1}{\|\mbf A\|_\rmf^2}\mbbe_{k-1}\bem\|{\bf Ax}^k-{\bf AA^\dag b}\|_2^2\eem\eeqas that \beqa\nn\mbbe\bem\dsp\l\|\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}(\mbf x^k-{\bf A^\dag b})\r\|_2^2\eem&=&\frac{1}{\|\mbf A\|_\rmf^2}\mbbe\bem\|{\bf Ax}^k-{\bf AA^\dag b}\|_2^2\eem\\\label{sum2}&\leq&\frac{\rho^k}{\|\mbf A\|_\rmf^2}\|{\bf Ax}^0-{\bf AA^\dag b}\|_2^2. \quad(\mbox{by Theorem \ref{AAdag}})\eeqa
Combining (\ref{sum}), (\ref{sum1}), and (\ref{sum2}) yields \beqas\mbbe\bem\|\mbf z^k-{\bf A^\dag b}\|_2^2\eem&=&\mbbe\bem\l\|\dsp\l(\mbf I-\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}\r)({\bf z}^{k-1}-{\bf A^\dag b})\r\|_2^2\eem+\mbbe\bem\l\|\dsp\frac{\wt{\bf a}_i\wt{\bf a}_i^\rmt}{\|\wt{\bf a}_i\|_2^2}(\mbf x^k-{\bf A^\dag b})\r\|_2^2\eem\\&\leq&\rho\mbbe\bem\|{\bf z}^{k-1}-{\bf A^\dag b}\|_2^2\eem+\frac{\rho^k}{\|\mbf A\|_\rmf^2}\|{\bf Ax}^0-{\bf AA^\dag b}\|_2^2\\&\leq&\cdots\leq\rho^k\|{\bf z}^0-{\bf A^\dag b}\|_2^2+\frac{\rho^k}{\|\mbf A\|_\rmf^2}\|{\bf Ax}^0-{\bf AA^\dag b}\|_2^2\sum_{l=0}^{k-1}\rho^l\\&=&\rho^k\|{\bf z}^0-{\bf A^\dag b}\|_2^2+\frac{\rho^k}{\|\mbf A\|_\rmf^2}\|{\bf Ax}^0-{\bf AA^\dag b}\|_2^2\frac{1-\rho^k}{1-\rho}\\&=&\rho^k\|{\bf z}^0-{\bf A^\dag b}\|_2^2+\frac{\rho^k(1-\rho^k)}{\sigma_{r}^2(\mbf A)}\|{\bf Ax}^0-{\bf AA^\dag b}\|_2^2.\eeqas This completes the proof.
\endproof
\remark\label{sharp2} By Lemma {\rm 1}, if $\sigma_1(\mbf A)=\sigma_r(\mbf A)$, then all the inequalities in the proofs of Theorems {\rm \ref{AAdag}} and {\rm \ref{main2}} become equalities.
\endremark
\section{Numerical results} We compare the bounds (\ref{rekb}), (\ref{bound2}), (\ref{regsb}), and (\ref{bound3}) via a set of small examples. For given $m$, $n$, $r=\rank(\mbf A)$, $\sigma_1(\mbf A)$, and $\sigma_r(\mbf A)$, we construct a matrix $\bf A$ by $\bf A=UD V^\rmt$, where $\mbf U\in\mbbr^{m\times r}$ and $\mbf V\in\mbbr^{n\times r}$. Entries of $\mbf U$ and $\mbf V$ are generated from a standard normal distribution and then columns are orthonormalized. The matrix $\mbf D$ is an $r\times r$ diagonal matrix whose first $r-2$ diagonal entries are uniformly distributed numbers in $[\sigma_r(\mbf A), \sigma_1(\mbf A)]$, and the last two diagonal entries are $\sigma_r(\mbf A)$ and $\sigma_1(\mbf A)$.
We consider four cases: (i) $\bf Ax=b$ is consistent and $\rank(\mbf A)=n$; (ii) $\bf Ax=b$ is consistent and $\rank(\mbf A)<n$; (iii) $\bf Ax=b$ is inconsistent and $\rank(\mbf A)=n$; (iv) $\bf Ax=b$ is inconsistent and $\rank(\mbf A)<n$. To construct a consistent linear system, we set $\bf b=Ax$ where $\bf x$ is a vector with entries generated from a standard normal distribution.
To construct an inconsistent linear system, we set $\bf b=Ax+r$ where $\bf x$ is a vector with entries generated from a standard normal distribution and the residual $\mbf r\in\nul(\mbf A^\rmt)$. Note that one can obtain such a vector $\mbf r$ by the MATLAB function {\tt null}.
\begin{figure}[!htpb]
\centerline{\epsfig{figure=cfk.eps,height=2.1in}\epsfig{figure=cfgs.eps,height=2.1in}}
\caption{The error $\|\mbf x^k-\mbf A^\dag\mbf b\|_2^2$ for {\rm REK-S (}left{\rm)} and the error $\|\mbf z^k-\mbf A^\dag\mbf b\|_2^2$ for {\rm REGS-E (}right{\rm)} on a consistent linear system with full column rank $\bf A$: $m=500$, $n=250$, $r=250$, $\sigma_1(\mbf A)=1.25$, and $\sigma_r(\mbf A)=1$.} \label{fig1}
\end{figure}
\begin{figure}[!htpb]
\centerline{\epsfig{figure=cnk.eps,height=2.1in}\epsfig{figure=cngs.eps,height=2.1in}}
\caption{The error $\|\mbf x^k-\mbf A^\dag\mbf b\|_2^2$ for {\rm REK-S (}left{\rm)} and the error $\|\mbf z^k-\mbf A^\dag\mbf b\|_2^2$ for {\rm REGS-E (}right{\rm)} on a consistent linear system with rank-deficient $\bf A$: $m=500$, $n=250$, $r=150$, $\sigma_1(\mbf A)=1.5$, and $\sigma_r(\mbf A)=1$.}\label{fig2}
\end{figure}
\begin{figure}[!htpb]
\centerline{\epsfig{figure=ifk.eps,height=2.1in}\epsfig{figure=ifgs.eps,height=2.1in}}
\caption{The error $\|\mbf x^k-\mbf A^\dag\mbf b\|_2^2$ for {\rm REK-S (}left{\rm)} and the error $\|\mbf z^k-\mbf A^\dag\mbf b\|_2^2$ for {\rm REGS-E (}right{\rm)} on an inconsistent linear system with full column rank $\bf A$: $m=500$, $n=250$, $r=250$, $\sigma_1(\mbf A)=1.75$, and $\sigma_r(\mbf A)=1$.}\label{fig3}
\end{figure}
\begin{figure}[!htpb]
\centerline{\epsfig{figure=ink.eps,height=2.1in}\epsfig{figure=ings.eps,height=2.1in}}
\caption{The error $\|\mbf x^k-\mbf A^\dag\mbf b\|_2^2$ for {\rm REK-S (}left{\rm)} and the error $\|\mbf z^k-\mbf A^\dag\mbf b\|_2^2$ for {\rm REGS-E (}right{\rm)} on an inconsistent linear system with rank-deficient $\bf A$: $m=500$, $n=250$, $r=150$, $\sigma_1(\mbf A)=2$, and $\sigma_r(\mbf A)=1$.}\label{fig4}
\end{figure}
In Figures \ref{fig1}-\ref{fig4}, we plot the error $\|\mbf x^k-\mbf A^\dag\mbf b\|_2^2$ for REK-S with $\mbf x^0=\mbf 0$ and $\mbf z^0=\mbf b$ and the error $\|\mbf z^k-\mbf A^\dag\mbf b\|_2^2$ for REGS-E with $\mbf x^0=\mbf 0$ and $\mbf z^0=\mbf 0$ for the four cases, respectively. For each case, we average the error over 20 trials for the same problem. For all cases, our bounds (\ref{bound2}) and (\ref{bound3}) are much better than the existing bounds (\ref{rekb}) and (\ref{regsb}).
\section{Conclusion} We have shown that REK is essentially an RK-RK approach and that REGS is essentially an RGS-RK approach. We have proposed refined upper bounds for the convergence of both algorithms. These upper bounds are attained for the case that all nonzero singular values of $\bf A$ are the same. Our convergence analysis applies to all types of linear systems.
The acceleration technique such as that used in \cite{liu2016accel} is being considered.
| {
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{"url":"https:\/\/cscherrer.github.io\/","text":"## Confusion Confusion\n\nHarder Than it Needs to Be Say you've just fit a (two-class) machine learning classifier, and you'd like to judge how it's doing. This starts out simple: Reality is yes or no, and you predict yes or no. Your model will make some mistakes, which you'd like to characterize. So you go to Wikipedia, and see this: There's a lot of \"divide this sum by that sum\", without much connection to why we're doing that, or how to interpret the result. [Read More]\n\n## Soss.jl: Design Plans for Spring 2019\n\nIf you've followed my work recently, you've probably heard of my probabilistic programming system Soss.jl. I recently had the pleasure of presenting these ideas at PyData Miami: [N.B. Above is supposed to be an embedded copy of my slides from PyData Miami. I can see it from Chrome, but not Firefox. Very weird. ] In April I'll begin another \"passion quarter\" (essentially a sabbatical) and hope to really push this work forward. [Read More]\n\n## Julia for Probabilistic Metaprogramming\n\nSince around 2010, I've been involved with using and developing probabilistic programming languages. So when I learn about new language, one of my first questions is whether it's a good fit for this kind of development. In this post, I'll talk a bit about working in this area with Julia, to motivate my Soss project. Domain-Specific Languages At a high level, a probabilistic programming languages is a kind of domain-specific language, or DSL. [Read More]\n\n## A Prelude to Pyro\n\nLately I've been exploring Pyro, a recent development in probabilistic programming from Uber AI Labs. It's an exciting development that has a huge potential for large-scale applications. In any technical writing, it's common (at least for me) to realize I need to add some introductory material before moving on. In writing about Pyro, this happened quite a bit, to the point that it warranted this post as a kind of warm-up. [Read More]\n\n## Bayesian Optimal Pricing, Part 2\n\nThis is Part 2 in a series on Bayesian optimal pricing. Part 1 is here. Introduction In Part 1 we used PyMC3 to build a Bayesian model for sales. By the end we had this result: A common advantage of Bayesian analysis is the understanding it gives us of the distribution of a given result. For example, we very easily analyze a sample from the posterior distribution of profit for a given price. [Read More]\n\n## Bayesian Optimal Pricing, Part 1\n\nPricing is a common problem faced by businesses, and one that can be addressed effectively by Bayesian statistical methods. We'll step through a simple example and build the background necessary to extend get involved with this approach. Let's start with some hypothetical data. A small company has tried a few different price points (say, one week each) and recorded the demand at each price. We'll abstract away some economic issues in order to focus on the statistical approach. [Read More]\n\n## The Bias-Variance Decomposition\n\nSay there's some experiment that generates noisy data. You and I each go through the process independently, and model the results. Would the resulting models be exactly the same? Well no, of course not. That's the whole problem with noise. Instead, we'll usually end up with something like this (for a quadratic fit): The idea is that we'd like to find an approximation to $$f(x)$$, but we can never observe this function directly. [Read More]\n\n## Bayesian Changepoint Detection with PyMC3\n\nA client comes to you with this problem: The coal company I work for is trying to make mining safer. We made some change around 1900 that seemed to improve things, but the records are all archived. Tracking down such old records can be expensive, and it would help a lot if we could narrow the search. Can you tell us what year we should focus on? Also, it would really help to know this is a real effect, and not just due to random variability - we don't want to waste resources digging up the records if there's not really anything there. [Read More]","date":"2019-03-25 18:17:46","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\": 0.5185138583183289, \"perplexity\": 939.8352787407813}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-13\/segments\/1552912204086.87\/warc\/CC-MAIN-20190325174034-20190325200034-00415.warc.gz\"}"} | null | null |
Amandla är ett musikalbum från 1989 av Miles Davis. Här blandas elektroniska instrument med levande musiker.
Amandla är det tredje och sista albumet i Davis samarbete med producenten och basisten Marcus Miller. De två tidigare är Tutu (1986) and Music from Siesta (1987).
Låten Mr. Pastorius är en hyllning till den bortgångne jazzbasisten Jaco Pastorius.
Låtlista
Låtarna är skrivna av Marcus Miller om inget annat anges.
Catembe – 5:38
Cobra (George Duke) – 5:16
Big Time – 5:41
Hannibal – 5:51
Jo-Jo – 4:52
Amandla – 5:21
Jilli (John Bingham) – 5:06
Mr. Pastorius – 5:42
Medverkande
Miles Davis – trumpet
Marcus Miller – bas (spår 1–8), basklarinett (spår 2–4, 7), keyboard (spår 1–8), sopransax (spår 3), gitarr (spår 4, 7)
Kenny Garrett – sopransax (spår 2), altsax (spår 1, 3–7)
Rick Margitza – tenorsax (spår 5)
Michael Landau – gitarr (spår 2)
Foley – gitarr (spår 3, 4, 7)
Jean-Paul Bourelly – gitarr (spår 3, 5)
Billy Patterson – wah-wah gitarr (spår 7)
John Bigham – gitarr (spår 7), keyboard (spår 7)
George Duke – keyboard (spår 2), synthesizer (spår 2)
Joey DeFrancesco – keyboard (spår 2)
Jason Miles – synthesizer-programmering (spår 8)
Joe Sample – piano (spår 6)
Ricky Wellman – trummor (spår 3, 7)
Omar Hakim – trummor (spår 4, 6)
Al Foster – trummor (spår 8)
Don Alias – slagverk (spår 1, 3, 6)
Mino Cinelu – slagverk (spår 1)
Paulinho Da Costa – slagverk (spår 4–5)
Bashiri Johnson – slagverk (spår 6)
Listplaceringar
Källor
Musikalbum 1989
Album av Miles Davis | {
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Ingrid Nestle (born 22 December 1977) is a German politician of Alliance 90/The Greens who has been a member of the Bundestag, the German parliament, from 2009 to 2012 and since 2017.
Early life and education
Nestle was born in Schwäbisch Gmünd. From 1998 to 2003 she studied energy management and environmental management at the University of Flensburg. Her 2012 dissertation in economics was titled "The costs of climate change in the agricultural sector – A comparison of two calculation approaches" She now lives in Elmshorn.
Political career
Nestle became a member of the Green party in 2001.
From 2012 to 2017, Nestle served as state secretary under minister Robert Habeck in the state government of Minister-President Torsten Albig of Schleswig-Holstein.
Nestle has been a member of the German Bundestag since the 2017 elections, representing the Steinburg – Dithmarschen South districts. In parliament, she has since been serving on the Committee on Economic Affairs and Energy.
In the negotiations to form a so-called traffic light coalition of the Social Democratic Party (SPD), the Green Party and the Free Democratic Party (FDP) following the 2021 federal elections, Nestle was part of her party's delegation in the working group on climate protection and energy policy, co-chaired by Matthias Miersch, Oliver Krischer and Lukas Köhler.
Other activities
Federal Network Agency for Electricity, Gas, Telecommunications, Post and Railway (BNetzA), Member of the Advisory Board (since 2018)
References
External links
1977 births
Living people
People from Schwäbisch Gmünd
Members of the Bundestag for Schleswig-Holstein
Members of the Bundestag 2021–2025
Members of the Bundestag 2017–2021
Members of the Bundestag 2009–2013
Members of the Bundestag for Alliance 90/The Greens | {
"redpajama_set_name": "RedPajamaWikipedia"
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A Woman, Her Men, and Her Futon is a 1992 drama film directed, written and produced by Michael Sibay and starring Jennifer Rubin, Lance Edwards and Grant Show.
Background
The film stars Jennifer Rubin as Helen, Lance Edwards as Donald and Grant Show as Randy. Others in the film included Michael Cerveris as Paul, Robert Lipton as Max and Delaune Michel as Gail. Richard Gordon had played a character named Jimmy but all the scenes involving the character were cut from the final release.
Originally, both Marisa Tomei and Julianne Moore auditioned for the lead role of Helen.
Following the film's original laser-disc release in 1992 and its American VHS release in 1997, the film was released on DVD in the US only, in 2001, which remains in-print to date.
The film's original tagline read "How many men, how many times, how many ways?" The film's DVD tagline reads "She's got all the right stuff, but all the wrong men..."
According to Independent Feature Project Filmmaker, Volume 1, the film was warmly received at the Houston festival and was to screen at Stockholm in the fall. Overseas Film Group had also picked up foreign rights, already selling the film in a number of territories.
Plot
Helen, a divorced, attractive twenty-something Los Angeles office worker, has just broken up with her possessive boyfriend Paul. Living beyond her means, Helen soon loses her car and her apartment, and has to move in with her friend Donald, an aspiring screenwriter. Helen helps Donald with his screenplay, while secretly writing her own. Donald introduces Helen to Max, a producer who takes an interest in her and her screenplay. Feeling cornered by Paul, Max and Donald, who also wants a relationship with her, Helen has a series of casual affairs. These flings and her past relationships end up in her screenplay, which she is successful in selling. The story ends as Helen, now a published screenwriter, moves out of Donald's house, and drives away with her futon in tow.
Cast
Jennifer Rubin as Helen
Lance Edwards as Donald
Grant Show as Randy
Michael Cerveris as Paul
Robert Lipton as Max
Delaune Michel as Gail
Richard Gordon as Jimmy (scenes deleted)
Jennifer Zuniga as Waitress #1
Kathryn Atwood as Waitress #2
Gary Cusano as Apartment Manager
Kirsten Hall as Sales Woman
Reception
Upon release, Kevin Thomas of the Los Angeles Times described the film as "thoughtful" and "engaging", adding: "In his feature film debut, Sibay is wonderful with actors and dialogue but needs to tighten his pacing and, in long verbal stretches, to learn how to avoid occasional tedium. These, however, are typical first-film flaws, easily forgivable in the light of Sibay's overall accomplishment." Bob Strauss of the Los Angeles Daily News wrote: "A Woman, Her Men and Her Futon aspires to be an up-to-the-minute report from the front of the war between the sexes. Some of the film's particulars feel a tad dated, and the basic conflict, of course, is as old as Adam and Eve. But despite its familiarity and predictable, self-pitying tone, Futon generates a remarkable degree of behavioral credibility. Thanks to a solid, multilayered performance by Rubin, Sibay concocts a spare, sexy movie, a tale of old-fashioned romantic confusion that often seems fresh."
TV Guide gave the film two out of four stars and described the film as a "slow-moving drama", which "drags in spots, but [is] well acted and has good production values". The reviewer added: "It also intelligently addresses the issues of manipulation, frustration and false hope that take place within relationships." After the film was shown at the Stockholm International Film Festival, movie critic Susanne Ljung wrote: "This film should be obligatory pepping for all young women! Women's shaky way to independence, both sexual and professional, is all too seldom portrayed as sensitively and penetratingly as in this film."
The Dutch VPRO Cinema awarded the film three and a half stars out of five and wrote: "An independent production in the tradition of Sex, Lies & Videotape, [but] without this level being reached. The parallel between Helen and the film project being talked about is a bit too obvious." Joe Leydon of Variety (magazine) stated: "Provocative title and a few steamy scenes are the only conceivable selling points for Sibay's A Woman, Her Men and Her Futon. Small-budget pic is by turns laughably stilted and sophomorically self-referential as a drama about L.A. scriptwriter wannabes, their sexual hang-ups and their mind games." In their DVD & Video Guide 2004, authors Mick Martin and Marsha Porter gave an unfavorable review: "Pointless film about a shallow woman who uses men for sex, money, and her career. She's so insipid and unsympathetic that it's painful to watch."
Iotis Erlewine of AllMovie gave the film a two out of five star rating. The book VideoHound's Golden Movie Retriever gave the film two out of five stars.'''
References
External links
1992 films
1992 drama films
1990s English-language films
American drama films
Films scored by Joel Goldsmith
Films about screenwriters
Films set in Los Angeles
1990s American films | {
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\section{Fluctuations for ergodic sums}\label{sec:1}
The central limit theorem (CLT) for Gibbs measures was first observed by Bowen in \cite{Bow75}, although this was in fact preceded by earlier work on this subject (see \cite{Den89} for more details). Since then a vast amount of research on the CLT in dynamical systems has appeared.
In most cases this law is shown to hold for H\"older continuous obervables or functions of bounded variation. This is due to the fact that, in the majority of cases, one can show a sufficiently rapid (summable) decay of correlations which entails the CLT by Gordin's martingale-coboundary decomposition (\cite{Gor69}).
However, all but one of the early attempts to prove such laws in dynamics used mixing concepts which also lead to the Gaussian limit law in cases of functions with $2+\delta$ moments, which are sufficiently well approximated by an underlying probabilistic mixing structure (cf. \cite{Bra07} for basic definitions of probabilistic mixing conditions). This method allows formulating CLT's for non-continuous functions which merely need to be sufficiently well approximable (in $L^2$, or in probability) (see \cite{DenKel83} for an example of such an approach).
This mixing structure is simple when the transfer operator has a spectral gap (see \cite{Rou83} for a first result of this type), but it is not of the type considered in probability.
Extensions of the spectral gap and of the mixing methods have appeared, allowing the treatment of intermittent maps (see \cite{AarDenUrb93} for one of the first results) via tower constructions (inducing on Darling-Kac sets, Schweiger's jump transformation, or recently Young towers) as well as maps with slow decay of correlations.
We cite \cite{Gou04, GouMel14, HayNicVaiZha13, Liv95, Tho14, Tho15} for an incomplete list of some recent articles, including an example of a CLT for general arrays, as is presented here.
Our methods differ from the probabilistic mixing approach (see \cite{Bra07, Dou94, IbrLin71}) as well as from the transfer operator method (see \cite{HenHer01} for a general approach).
We build upon the specification properties on periodic points, which were first introduced by Bowen (see \cite{Bow71}).
The use of periodic points has many conveniences, for example any invariant measure can be approximated by a distribution on the periodic points (see \cite{Sig70}) and periodic points can be used towards a numerical understanding of the dynamics (see \cite{DenDuaMcC09}). But it is worth noting that our results do not depend heavily on the existence of periodic points. Specification properties using other sets of points are likely to work in the same way.
In particular, when choosing weighted point measures to approximate equilibrium states which are subject to specification, it is possible to study the fluctuation of ergodic sums under equilibrium measures; this applies especially to maps of the interval with a Markov structure.
The classical CLTs in dynamics study the fluctuation of partial sums of H\"older functions around their means globally using a global mixing concept (including spectral gaps), and there does not seem to be any other significant source of fluctuation. Our results show that local randomness can also contribute to fluctuation. Hence it is important to investigate the sources of the fluctuations. We suggest a decomposition into \textit{global} and \textit{local} fluctuations of Ces\'aro averages and study the CLTs in this context. This description still allows for an unknown source of fluctuation which may arise from the non-uniform distribution of periodic orbits.
To set the stage, we choose to consider dynamical systems $(X,T)$ exhibiting either \textit{global} or \textit{local} specification. The specification is called global if the concatenation of any number of orbit pieces of a given length can be shadowed by a single periodic orbit, provided sufficient time (referred to as a \textit{gap}) is allowed to migrate from the end of one piece of orbit to the next (see Definition~\ref{def:spec}). The specification is called {local} when orbits that are close enough at their initial and terminal time periods can by shadowed by periodic points (see Definition~\ref{def:specloc}).
Thanks to the relation between periodic orbits given by the specification properties, one can pick out certain sets of periodic points and compare uniform measures on them with product measures to capture an independence-like structure (assuming that there are only finitely many periodic points of any given period).
We call these \textit{globally or locally $\epsilon$-independent sets} (see definitions~\ref{def:independence} and \ref{def:locidp}).
This {global} structure allows for the study of the distribution of Birkhoff sums with gaps, i.e. incomplete time series, whereas the {local} version allows one to infer the distribution of the Birkhoff sums from the readings taken only at certain locations in the phase space.
In the global scenario, $\epsilon$-independence structure is constructed on the orbit pieces outside the gaps. To obtain a limit law it would be necessary to require certain condition on the observables over the gaps. We call this type of condition a \textit{gap condition}:
{the sum of the variances along the gaps is negligible when compared to the sum of the variances along the orbit pieces}. After neglecting all the gaps, an \textit{oscillation condition} on the observables ensures us to treat the measures on $\epsilon$-independent sets like product measures: {the first and second moments of the oscillation of the observable along the orbit pieces is negligible when compared to the sum of the variances.} Our first main theorem is a Lindeberg-type CLT for systems with global specification property (see Theorem~\ref{theo:1}).
\begin{theoremA}
Consider a dynamical system $(X, T)$ with \emph{global} specification. Then along a sequence of $\epsilon_l$-independent sets $\mathcal P_{l}$ the following holds:
given observables $h_l$ satisfying the oscillation condition~\eqref{eq:4.2-0} and the gap condition~\eqref{eq:4.2-1}, if
the Lindeberg condition~\eqref{eq:epslind} holds with respect to the uniform measure $\nu_{\mathcal P_{l}}$ on $\mathcal P_{l}$, then the central limit theorem holds:
$$ \lim_{l\to\infty} \nu_{\mathcal P_{l}}\left(\bigg\{x\in X: \sum_{j=0}^l\Big(h_l(T^jx)-\mathbb{E}_{\nu_{\mathcal P_{l}}}(h_l)\Big)\leqslant ts_l\bigg\}\right) =\frac 1{\sqrt{2\pi}} \int_{-\infty}^t e^{-u^2/2} du$$
where $s_l$ denotes a suitably-defined total variance. Additionally, the reverse holds true under a uniform oscillation condition~\eqref{eq:4.2-00}.
\end{theoremA}
In the course of proving this theorem, we prove CLT for the more general \textit{dynamical arrays} (equivalently defined in~\cite[Definition 5.1]{DenSenZha17}) instead of Birkhoff sums.
\begin{definition}\label{def:arraygeral}[Dynamical Array]
For each $l\in\mathbb N$ and $k_l\in \mathbb N$ consider two increasing sequences of positive integers
$\{a_{l,i}\}_{1\leqslant i\leqslant k_l}$ and $\{b_{l,i}\}_{1\leqslant i\leqslant k_l}$ with
$$
0\leqslant a_{l,1}\leqslant b_{l,1}<a_{l,2}\leqslant b_{l,2}<\ldots< a_{l, k_l}\leqslant b_{l, k_l}
$$
and real-valued functions $h_{l,i}:X\to \mathbb R$ where $1\leqslant i\leqslant k_l$.
A \emph{dynamical array} is a sequence $\{H_{l,i}\}_{ 1\leqslant i\leqslant k_l,\, l\in \mathbb N}$ of real valued functions of the form
$$ H_{l,i} = \sum_{j=a_{l,i}}^{b_{l,i}} h_{l,i}\circ T^{j} \qquad 1\leqslant i\leqslant k_l.$$
We refer to the intervals $a_{l,i+1}-b_{l,i}$ as the \textit{gaps} of the dynamical array.
\end{definition}
\noindent We will only consider the case $b_{l,i}-a_{l,i}=n_l$ and $a_{l,i+1}-b_{l,i}=M_l$ for all $1\leqslant i\leqslant k_l$, whereas similar results can be obtained for the more general case.
While Theorem A holds for the uniform measure supported on $\epsilon$-independent set, one can find a weighted measure supported on the set of all periodic points for which a Lindeberg-type CLT holds (see Corollary~\ref{cor:1}).
On the other hand, although the $\epsilon$-independent sets only contain incomplete periodic points, there is rich enough structure so that the entropy of any weak limit of uniform measures on these sets equals the topological entropy in an expansive system (see Theorem~\ref{theo:weak-convergence}).
We apply Theorem~A to show that it suffices to take the Birkhoff average over the full orbit of a typical periodic point to approximate the integrals of continuous functions with respect to the measure of maximal entropy (see Theorem~\ref{theo:3}).
\begin{theoremB}
Consider an expansive dynamical system with \emph{global} specification. Assume some mild additional conditions on the constants $N_l=k_l(n_l+M_l)$ defining the array and $\epsilon_l$-independent set $\mathcal P_l$, one has for a Lipschitz function $h$ and any $\eta>0$
$$\lim_{l\to\infty}\nu_{\mathcal P_{l}}\left(\bigg\{
\bigg|\frac 1{N_l}
\sum_{j=0}^{N_l}\Big(h(T^jx)-\mathbb{E}_{\nu_{\mathcal P_{l}}}(h)\Big)
\bigg|
\leqslant k_l^{-\frac 1 2+\eta}\bigg\}\right) = 1.$$
Moreover, if $(X,T)$ admits a unique measure of maximal entropy, then for random sequences of periodic points $p_l\in\mathcal{P}_l$, the uniform distributions over the orbit of $p_l$ converge to the measure of maximal entropy.
\end{theoremB}
In the {local} scenario, local $\epsilon$-independence structure is constructed at pre-defined locations. One would not need a gap condition but an oscillation condition is still necessary. In place of Theorem A we obtain a local Lindeberg-type CLT which studies the fluctuation of partial sums around local means (see Theorem~\ref{theo:2}).
\begin{theoremC}
Consider a dynamical system with \emph{local} specification. Then along a sequence of locally $\epsilon_l$-independent sets $\mathcal P_{l}$ the following holds:
given a dynamical array satisfying the oscillation condition, the Lindeberg condition
holds with respect to the uniform distribution $\nu_{\mathcal P_{l}}$ on $\mathcal P_{l}$, if and only if the array
is $\nu_{\mathcal{P}_l}$-asymptotically negligible and the CLT holds.
\end{theoremC}
As local specification implies global specification in a topologically mixing system, one can study the fluctuation of partial sums with respect to the measure of maximal entropy by local $\epsilon$-independence structure.
We obtain the following result (see Theorem~\ref{theo:8.2}).
\begin{theoremD}
Let $(X,T)$ be an expansive and topologically mixing dynamical system with the \emph{local} specification property. Then, with respect to the unique measure of maximal entropy $\mu$, the class of wildly oscillating functions in $L^3(\mu)$ (see Definition~\ref{def:5.2}) satisfying the moment condition~\eqref{eq:9.30} and with integrable local variance~\eqref{eq:9.32} belongs to the partial domain of attraction of a mixed normal distribution, i.e. a subsequence of properly centered and normed partial sums converges weakly to a mixed normal distribution.
\end{theoremD}
Since we only assume continuity for the transformation $T$, we will prove our results for positively expansive systems in this paper. Proofs are analogous for expansive homeomorphisms.
Our results clarify the amount of randomness present in a dynamical system which is due to its periodic orbit structure. From a dynamical viewpoint such theorems are important for various reasons: first, they allow one to derive CLTs for non-standard functions and for limits of invariant discrete probabilities, such as equilibrium states. In our context we only consider measures of maximal entropy but our results should also apply to more general equilibrium states. This applies in particular to Anosov diffeomorphisms and subshifts of finite type.
Secondly, such theorems provide methods for data and numerical analysis of time series. Indeed this analysis can be carried out through descriptive statistics based on dynamical arrays and their asymptotic normality. From the point of view of applications (data and numerical analysis) it is also important to study how the variance of the dynamical arrays is determined by the periodic point structure.
\textbf{Structure of the paper}
In Section \ref{sec:2} we collect basic definitions and notations.
We recall the notions of local and global specification used in the paper and illustrate them with a few examples which include
Bowen's Axiom-A$^*$-homeomorphisms, hyperbolic rational map of $\mathbb{S}^2$ and topological Markov chains with restricted entries.
In Section~\ref{sec:globalCLT} we define the $\epsilon$-independence structure in systems with global specification and prove the CLT Theorem~A. In Section~\ref{sec:localCLT} we define the local $\epsilon$-independence in systems with local specification and prove the CLT Theorem~C. In Section~\ref{sec:mme} we apply both concepts of $\epsilon$-independence structure to study the fluctuation problem with respect to the measure of maximal entropy. We show that the uniform measures on $\epsilon$-independent sets converge weakly to the measure of maximal entropy if the latter is unique and $T$ restricted to periodic points is a homeomorphism, and prove Theorems B and D.
In Section~\ref{sec:decomposition} a decomposition of the variance is described.
\section{Notations and definitions}\label{sec:2}
Consider a continuous transformation $T:X\to X$ of a compact metric space $(X, d)$.
Denote the sets of periodic points by
$$
P_n:=\{x\in X\colon T^n x=x\} \quad\mbox{ and }\quad P:=\bigcup_{n\in\mathbb N} P_n
$$
and denote
$$
B_\epsilon(x):=\{y\in X\colon d(x,y)<\epsilon\},\quad
d_n(x,y):=\sup_{0\leqslant k\leqslant n-1} d(T^kx, T^ky),$$
and
$$B^n_\epsilon(x):=\{y\in X\colon d_n(x,y)<\epsilon\},$$
$$B_{\epsilon}^{n}(A):=\{y\in X: d(T^iy, T^i A)<\epsilon, 0\leqslant i\leqslant n-1\} \text{ for } A\subset X.$$
A set $E\subset X$ is
\emph{$(n,\epsilon)$-separated} if $d_n(x,y)>\epsilon$ for all $x\neq y \in E$ and
\emph{$(n, \epsilon)$-spanning for $Y$} if $Y\subset \bigcup_{x\in E}B^n_\epsilon(x)$.
The map $T$ is said to be
\emph{$\epsilon^*$-positively expansive}, if for any $x\ne y\in X$ there exists $n\in\mathbb N$ such that
$$ d(T^nx, T^ny)>\epsilon^*.$$
If $T$ is invertible, one considers expansiveness rather than positive expansiveness with $\mathbb N$ replaced by $\mathbb Z$ and the results in this paper can be proved analogously.
Suppose throughout the paper that $P_n$ is finite for every $n$. This is true, for instance, for positively expansive maps.
For any finite subset $Z\subset X$, denote its cardinality by $|Z|$ and the uniform probability on $Z$ by $\nu_Z$, i.e.
$$\nu_Z(W):= \frac {|W\cap Z|}{|Z|} = \frac 1{|Z|}\sum_{z\in Z} \mathbbm 1_W(z)\qquad W\subset X.$$ For a real valued function $h$ on $X$, denote by $\mathbb E_Z(h)$ its expectation with respect to $\nu_Z$ and by $\sigma_Z^2(h)$ the variance when they exist.
Recall that the \emph{Birkhoff sums} are given by, for $n,m\in\mathbb N$,
$$ S_m^n h(x):=\sum_{i=m}^{m+n-1} h(T^i x),\quad\mbox{ and }\quad S^n h(x):= S_0^n h (x).$$
The \emph{oscillations} of $h: Y\to\mathbb R$, $Y$ being $P$ or $X$, around $x\in Y$ are given by
$$ \omega_m^n(h,\epsilon, x) :=\sup\Big\{ |S_m^n h (x)- S^n h(y)|\colon y\in B_\epsilon^{n}(T^m x)\cap Y\Big\}$$
and $\omega^n(h,\epsilon, x):=\omega_0^n(h,\epsilon, x)$.
The notation $f\lesssim g$ means that $f\leqslant Cg$ for some constant $C$. Denote the distribution function of the standard normal distribution by $\mathcal N(t)$.
\subsection{Specification}\label{sec:spec}
Specification property was introduced by Bowen in \cite{Bow71} (see also \cite[(21.1)]{DenGriSig76}), from which were derived many related definitions in the literature (see e.g. \cite{KwiLacOpr16}) \footnote{We thank W. Cordeiro for helpful discussions on this subject}. Each definition may lead to a CLT similar to the one proven here.
The notions of specification in this paper are defined as follows:
\begin{definition}\label{def:spec}
[Global Specification]
The dynamical system $(X,T)$ has the (global) specification property
if for every $\epsilon>0$ there exists $M(\epsilon)\in \mathbb N$ such that: for any $M\geqslant M(\epsilon)$,
$x_1,...,x_k\in X$, $k\in \mathbb{N}$ and $n\in \mathbb{N}$ there exists a periodic point $p\in P_{k(n+M)}$ with
$$T^{(i-1)(n+M)}p\in B^n_\epsilon(x_i)\qquad i=1,...,k.$$
\end{definition}
\begin{definition}\label{def:specloc}
[Local Specification]
The dynamical system $(X,T)$ has the local specification property
if for any $\epsilon>0$ there exist $\delta=\delta(\epsilon)>0$ and $N(\epsilon)\in \mathbb N$ such that for any $x_1,...,x_k\in X$, $k\in\mathbb N$ and $n\geqslant N(\epsilon)$ with
$$ d(T^{n}x_i, x_{i+1}) <\delta\qquad i=1,...,k\mbox{ and }x_{k+1}=x_1$$
there exists a periodic point $p\in P_{kn}$ with
$$T^{(i-1)n}p\in B^n_\epsilon(x_i)\qquad i=1,...,k.$$
\end{definition}
For topologically mixing maps the local specification property implies the global specification (see e.g. \cite{Hasselblatt2017a}), often simply referred to as \emph{specification}. However, in absence of topological mixing, both notions are distinct.
Indeed, global specification implies that the map $T$ is topologically mixing. This does not need to be the case for maps with the local specification. On the other hand, Example 3 in Section \ref{sec:examples} shows that global specification does not imply local specification.
In order to simplify calculations and notations, our definition of global specification requires equal length for all the stretches, still it is possible to connect any two stretches of different lengths.
\begin{proposition}\label{prop:weakspec}
Suppose $(X,T)$ satisfies the global specification property. Then for every $\epsilon>0$, any $n_1, n_2\in\mathbb N$, $M_1, M_2\geqslant \tilde M(\epsilon)=M(\epsilon/3)$ and $x_1, x_2\in X$, there exists a periodic point $p\in P_{n_1+M_1+n_2+M_2}$ such that
$$p\in B_{\epsilon}^{n_1}(x_1)\text{~ and ~} T^{(n_1+M_1)}p\in B_\epsilon^{n_2}(x_2)\qquad i=1,2.$$
\end{proposition}
\begin{proof}
First we note that it suffices to show the statement for any $M_1=M_2\geqslant \tilde M(\epsilon)$, since $n_1$ and $n_2$ can be adjusted so that $M_1=M_2$. Given any $n_1, n_2\in\mathbb N$, $M\geqslant \tilde M(\epsilon)$ and $x_1, x_2\in X$, suppose $n_1\leqslant n_2$ wlog. By the specification property, there exists $p_1\in P_{n_1+M}$ such that $$p_1\in B_{\epsilon/3}^{n_1}(x_1).$$
Choose $k\in\mathbb N$ such that $k(n_1+M)\geqslant n_2-n_1.$ Use the specification property again to obtain $p_2\in P_{2(n_2+M)}$ such that $$p_2\in B_{\epsilon/3}^{n_2}(T^{k(n_1+M)-n_2+n_1}p_1)\text{~ and ~} T^{n_2+M}p_2\in B_{\epsilon/3}^{n_2}(x_2).$$ In particular, noting that $T^{k(n_1+M)}p_1=p_1$, the first inclusion implies
$$T^{n_2-n_1}p_2\in B_{\epsilon/3}^{n_2}(p_1)\subset B_{\epsilon/3}^{n_1}(p_1).$$
Now the specification property implies the existence of some $p\in P_{n_1+n_2+2M}$ with $$p\in B_{\epsilon/3}^{n_1+n_2+M}(T^{n_2-n_1}p_2),$$
in particular, $$p\in B_{\epsilon/3}^{n_1}(T^{n_2-n_1}p_2)\text{~ and ~} T^{n_1+M}p\in B_{\epsilon/3}^{n_2}(T^{n_2+M}p_2).$$
Therefore $p\in B_\epsilon^{n_1}(x_1)$ and $T^{n_1+M}p\in B_\epsilon^{n_2}(x_2).$
\end{proof}
\subsection{Examples of maps with specification} \label{sec:examples}
We now provide a few examples for which
these specification properties hold.
\subsubsection*{Example 1}
An \emph{Axiom A$^*$ homeomorphism} (c.f. \cite[pg 378]{Bow71}) is a homeomorphism $T:X\to X$ of a compact metric space $X$ which satisfies the following properties:
\begin{enumerate}
\item [(A1)]Periodic points are dense in $X$.
\item [(A2)]
For every $\epsilon>0$ there exists $\delta>0$ such that
$$d(x,y)<\delta\quad\Longrightarrow\quad W_\epsilon^u(x)\cap W_\epsilon^s(y)\ne \emptyset$$
where
$$ W_\epsilon^s(x):=\{y\in X: d(T^n(x), T^n(y))<\epsilon\quad \forall n\geqslant 0\}$$
and
$$ W_\epsilon^u(x):=\{y\in X: d(T^{-n}(x), T^{-n}(y))<\epsilon \quad\forall n\geqslant 0\}.$$
\item [(A3)]
There exist $\eta>0$, $c\geqslant 1$ and $0<\lambda<1$ such that for all $x\in X$ and $n\geqslant 0$
$$ d(T^n x,T^n y)< c\lambda^n d(x,y)\qquad y\in W_{\eta}^s(x)$$
and
$$ d(T^{-n}x,T^{-n}y)< c\lambda^n d(x,y)\qquad y\in W_{\eta}^u(x).$$
\end{enumerate}
Axiom A$^*$ homeomorphisms satisfy the local specification property. Topologically mixing Axiom A$^*$ homeomorphisms satisfy both the local and the global specification properties.
\subsubsection*{Example 2}
Consider a dynamical system $(X,T)$ which possesses a Markov partition $\alpha$ of sets $A$ which are contained in the closure of their interior. Then $(X,T)$ is a continuous factor of a subshift of finite type $(Z,S)$. The periodic points in $Z$ of period $n$ correspond to sets in the refinement $\alpha_0^{n-1}$ and thus define measures on the $\sigma$-field generated by $\alpha_0^{n-1}$. Thus statements about measures on periodic points in $(Z,S)$ correspond to statements on natural measures on $(X,T)$ equipped with finite $\sigma$-fields. The results presented in this article can therefore be applied to this type of dynamical systems.
As an example, consider a hyperbolic rational map $T$ of the Riemannian sphere $\mathbb{S}^2$.
Ma\~n\'e~\cite{Man85} proved that $T$ is semi-conjugated to a
Bernoulli shift
by a finite-to-one factor map which is one-to-one almost everywhere.
Hence, the map $T$ on the completed Julia set equipped with the sequence topology trivially satisfies the local specification property.
\subsubsection*{Example 3}
A simple example of a system satisfying the global specification property which is not an Axiom A$^*$ homeomorphism is given by the subshift $\Sigma\subset \{1,...,N\}^{\mathbb Z}$ obtained by excluding a set of blocks from the full shift on $\{1,...,N\}^{\mathbb Z}$ as follows (see \cite[(17.1)]{DenGriSig76}).
For any $n\in\mathbb N$ pick two sequences $[p(n)]\neq [q(n)]$ of length $n$ from $\{1,..., N-1\}^{n}$. Exclude from the set of admissible words the cylinders of the form $\{[p(n)Nq(n)N]:n\in \mathbb N\}$. Consider the usual distance defined by $d(x,y):= r^k$ where $k=\min\{ |l|: x_{l}\ne y_{l} \ \mbox{\rm or}\ x_{-l}\ne y_{-l}\}$ for some $0<r<1$ and denote the shift transformation by $\sigma$. The subshift $\Sigma$ does not have the local specification property. To see this, use the blocks $[p(n)Nq(n)]$ and $[q(n)Np(n)]$ to generate periodic points $x(n)$ and $y(n)$ of period $2n+1$. Then for any $\delta>0$ and $n>-\log_{1/r}\delta$,
$$d(\sigma^{n+1}x(n),y(n))=r^n<\delta\quad\mbox{ and }\quad d(\sigma^{n+1}y(n),x(n))=r^n<\delta.$$
But if $p\in P_{4n+2}$ satisfies $d(\sigma^j p, \sigma^j x(n))<r/2$ and $d(\sigma^{n+1+j} p, \sigma^j y(n))<r/2$ for all $j=0,...,n$, then $p$ contains the excluded block $[p(n)Nq(n)N]$, a contradiction. It can be shown that such systems have the global specification property.
\section{A CLT under global specification}\label{sec:globalCLT}
The global specification property allows us to single out sets of periodic points which exhibit independent structure in the measure theoretical sense, and are spread out over the whole phase space.
\begin{definition}[$\epsilon$-Independence]\label{def:independence}
Let $\epsilon>0$ and $k,n, M\in\mathbb N$. A set ${\mathcal P}\subset P_{k(n+M)}$ is
$\epsilon$-independent if there exist a subset $E\subset P_{n+M}$
which $(n, 3\epsilon)$-spans $P_{k(n+M)}$ and a bijection $\Phi: E^k\to {\mathcal P}$ such that for any ${\bf x}=(x_1,...,x_{k})\in E^k$ and $1\leqslant i\leqslant k$,
\begin{equation}\label{eq:epsind}T^{(i-1)(n+M)}(\Phi({\bf x}))\in B^n_{\epsilon}(x_{i}).\end{equation}
The set $F:=E^k$ will be called the product set of $\mathcal P$.
\end{definition}
Recall that in the definition of global specification (Definition \ref{def:spec}) any given $\epsilon>0$ defines a $M(\epsilon)\in\mathbb N$.
\begin{proposition}\label{prop:indstr}
Let $(X,T)$ satisfy the (global) specification property.
Then for any $\epsilon>0$, $k,n\in \mathbb N$ and $M\geqslant M(\epsilon)$ there exists a $\epsilon$-independent set $\mathcal P\subset P_{k(n+M)}$.
\end{proposition}
\begin{proof}
Choose a maximal $(n,2\epsilon)$-separated set $E\subset P_{n+M}$, then $E$ $(n,2\epsilon)$-spans $P_{n+M}$ by maximality.
For every $p\in P_{k(n+M)}$ there exists $x\in P_{n+M}\cap B^n_\epsilon(p)$ by specification.
Hence $E$ also $(n,3\epsilon)$-spans $P_{k(n+M)}$.
Again due to specification, for every ${\bf x}=(x_1,\ldots, x_k)\in E^k$, there exists $p\in P_{k(n+M)}$ such that $T^{(i-1)(n+M)}(p)\in B^n_{\epsilon}(x_i)$ for all $i=1,...,k$. This correspondence defines a map
$\Phi:E^k\mapsto P_{k(n+M)}$.
$\Phi$ is injective since $\Phi({\bf x})=\Phi({\bf y})$ implies $x_i\in B^n_{2\epsilon}(y_i)$ for all $i=1,...,k$. As $E$ is $(n,2\epsilon)$-separated, this implies $x_i=y_i$.
Hence ${\mathcal P}:=\Phi(E^k)$ is $\epsilon$-independent.
\end{proof}
We can use this structure to prove a Lindeberg type CLT for Birkhoff sums with respect to the uniform measure on $\epsilon$-independent sets. To this end, we approximate the measure on the $\epsilon$-independent sets by the uniform measure on the product sets.
Recall that for any finite $Z\subset X$, $h\in L^2(\nu_Z)$ and any $\eta> 0$ the \emph{Lindeberg function} is given by
$$ L_Z(h, \eta):= \int \big(h(z)-\mathbb E_Z(h)\big)^2 \mathbbm 1_{\{|h(z)-\mathbb E_Z(h)| >\eta\}}(z)d\nu_Z,$$
and that $\sigma_Z^2$ stands for the variance with respect to the uniform measure $\nu_Z$.
\begin{proposition}\label{prop:indclt} For every $l\in\mathbb N$ consider a finite subset $E_l\subset X$ and the Cartesian product set $F_l=E_l^{k_l}$ for some $k_l\in\mathbb N$. For each $1\leqslant i\leqslant k_l$ let $G_{l,i}:F_l \to\mathbb R$ be a function which depends only on the $i$-th component:
$$G_{l,i}(x_1,\ldots,x_{k_l}) =G_{l,i}(x_i),\quad \forall (x_1,\ldots,x_{k_l})\in F_l.$$
Set $\hat{s}^2_{l}=\sum_{i=1}^{k_l} \sigma_{F_l}^2(G_{l,i}).$
Then the Lindeberg condition holds, i.e.
\begin{equation}\label{eq:prop1}
\lim_{l\to \infty}\frac 1 {\hat{s}_{l}^2}\sum_{i=1}^{k_l} L_{F_l}(G_{l,i}, \eta\hat s_l) =0, \quad \forall \eta>0
\end{equation}
if and only if the array is asymptotically negligible, i.e.
\begin{equation}\label{eq:propngl}
\lim_{l\to\infty}\max_{1\leqslant i\leqslant k_l}\nu_{F_l}(\{|G_{l,i}-\mathbb E_{F_l}(G_{l,i})|\geqslant \eta\hat s_l\})=0, \quad \forall\eta>0
\end{equation}
and the CLT holds, i.e.
\begin{equation}\label{eq:propclt}
\lim_{l\to\infty}\nu_{F_l}
\left(\bigg\{
\sum_{i=1}^{k_l}\left( G_{l,i} - \mathbb E_{F_l}(G_{l,i})\right)\leqslant t \hat{s}_{l}
\bigg\}\right)
=\mathcal{N}(t), \quad \forall t\in\mathbb R.
\end{equation}
\end{proposition}
\begin{proof}
Recall Lindeberg's CLT for independent random variables (see e.g. \cite[Theorem 15.43]{Klenke2014}): for an independent array of random variables, the Lindeberg condition holds if and only if the array is asymptotically negligible and the CLT holds. Then the proposition follows immediately since the functions $G_{l,i}$ form an independent array on $(F_l, \nu_{F_l})$.
\end{proof}
To realize the previously mentioned approximation, we require the observables to have controlled oscillations along the $\epsilon$-independent sets (see~\eqref{eq:4.2-0}) and to have negligible variance over the gaps (see~\eqref{eq:4.2-1}).
\begin{theorem}\label{theo:1}
Let $\{\epsilon_l>0, M_l, k_l, n_l\in \mathbb N\}_{l\in\mathbb N}$ be sequences of numbers, with $k_l\xrightarrow{l\to\infty}\infty$. Consider a sequence of $\epsilon_l$-independent sets $\mathcal P_{l}\subset P_{k_l(n_l+M_l)}$
and observables $h_l:P\to \mathbb R$ satisfying that
\begin{equation}\label{eq:4.2-0}
\lim_{l\to\infty}\,\frac{1}{s_l^j} \sum_{i=1}^{k_l}
\int \left(\omega_{a_i}^{n_l}(h_l,4\epsilon_l,p)\right)^j d\nu_{\mathcal P_{l}}(p)=0,\quad j=1,2,
\end{equation}
where $a_{i}=(i-1)(n_l+M_l)$ and
$s_{l}^2=\sum_{i=1}^{k} \sigma^2_{\mathcal P_{l}}(S_{a_{i}}^{n_l} h_l)$, and
that
\begin{equation}\label{eq:4.2-1}
\lim_{l\to\infty}\, \frac1{s^2_l}\ \sigma^2_{\mathcal P_{l}}\left(\sum_{i=1}^{k_l} S_{a_{i}+n_l}^{M_l} h_l\right)=0.
\end{equation}
Then the Lindeberg condition
\begin{equation}\label{eq:epslind}
\lim_{l\to\infty}\frac 1{s_l^2} \sum_{i=1}^{k_l}L_{\mathcal P_l}(S_{a_i}^{n_l} h_l, \eta s_l) =0, \quad \forall \eta>0
\end{equation}
implies the CLT: for every $t\in\mathbb R$
\begin{equation}\label{eq:epsclt}
\lim_{l\to\infty}\nu_{\mathcal P_{l}}
\left(\Big\{
S^{k_l(n_l+M_l)} h_l-\mathbb{E}_{\mathcal{P}_{l}}(S^{k_l(n_l+M_l)} h_l)
\leqslant t s_l
\Big\}\right)=\mathcal{N}(t).\end{equation}
If
\begin{equation}\label{eq:4.2-00}
\lim_{l\to\infty}\,\frac{1}{\sigma^2_{\mathcal P_{l}}(S^{n_l} h_l)}
\int \left(\omega^{n_l}(h_l,2\epsilon_l,p)\right)^2d\nu_{\mathcal P_{l}}(p)=0
\end{equation}
holds additionally then the Lindeberg condition is also necessary.
\end{theorem}
\begin{proof}
First notice that condition~\eqref{eq:4.2-1} on the gaps implies that we can replace $S^{k_l(n_l+M_l)}h_l$ by the dynamical array $\sum_{i=1}^{k_l} S_{a_{i}}^{n_l} h_l$ without affecting the limit distribution.
Define an array $\{G_{l,i}\}$ on $F_l$, the product set of $\mathcal P_l$, by setting $$G_{l,i}(x_1,\ldots, x_{k_l}):=S^{n_l}h_l(x_i)$$ and let $\hat{s}^2_l=\sum_{i=1}^{k_l}\sigma^2_{F_l}(G_{l,i})$.
We show that $\frac{1}{s_l}\sum_{i=1}^{k_l}S_{a_i}^{n_l}h_{l}$ on $(\mathcal P_l, \nu_{\mathcal P_l})$ has the same limit distribution as $\frac{1}{\hat s_l}\sum_{i=1}^{k_l}G_{l,i}$ on $(F_l, \nu_{F_l})$, hence \eqref{eq:epsclt} is equivalent to \eqref{eq:propclt}.
In fact because \begin{equation}\label{eq:epsindaprox}
|G_i(\Phi^{-1}p)-S^{n_l}_{a_i}h_l(p)|\leqslant \omega_{a_i}^{n_l}(h_l,\epsilon_l, p),
\end{equation}
we have
for any $t>0$
$$
\nu_{\mathcal P_l}\left(\Big\{|\sum_{i=1}^{k_l}(G_{l,i}\circ \Phi^{-1}-S_{a_i}^{n_l} h_l|>ts_l\Big\}\right)\leqslant \frac{1}{t s_l}\sum_{i=1}^{k_l}\int \omega_{a_i}^{n_l}(h_l,\epsilon_l,p)d\nu_{\mathcal P_l}.
$$
Then as $l$ tends to $\infty$ the upper bound tends to $0$ by \eqref{eq:4.2-0}. Similar calculation shows that $$\lim_{\l\to\infty}{\hat{s}_l}/{s_l}=1.$$
Next we show that \eqref{eq:epslind} is equivalent to \eqref{eq:prop1}, consequently the first part of the theorem follows from Proposition \ref{prop:indclt}.
For any $\eta>0$, if $|G_{l,i}(\Phi^{-1}p)-\mathbb E_{F_l}(G_{l,i})|>\eta \hat{s}_l$ then by \eqref{eq:epsindaprox} either
$$|S_{a_i}^{n_l} h_l(p)-\mathbb E_{\mathcal P_l}(S_{a_i}^{n_l}h_l)|>\frac{\eta}3 s_l
$$
or else with
$$
\omega_{a_i}^{n_l}(h_l,\epsilon_l,p)>\frac{\eta}3 s_l
$$
since when $l$ is large enough \eqref{eq:4.2-0} yields
$$\int \omega_{a_i}^{n_l}(h_l,\epsilon_l,p)d\nu_{\mathcal P_l}\leqslant \eta \hat{s}_l-\frac{2\eta}{3} s_l.$$
This implies the equivalence of the Lindeberg conditions because
$$
L_{F_l}(G_{l,i},\eta \hat{s}_l)\lesssim L_{\mathcal P_l}(S^{n_l}_{a_i} h_l, \frac\eta 3 s_l) + \int (\omega_{a_i}^{n_l}(h_l,\epsilon_l, p))^2 d\nu_{\mathcal P_l}
$$
and \eqref{eq:4.2-0} and that the roles of $G_{l,i}$ and $S_{a_i}^{n_l}h_l$ can be switched.
Conversely we need to verify \eqref{eq:propngl}. We will show that the additional oscillation condition \eqref{eq:4.2-00} implies
\begin{equation}\label{eq:4.2-3}
\lim_{l\to\infty}\frac{\sigma^2_{\mathcal P_{l}}(S^{n_l}_{a_{i}}h_l)}{\sigma^2_{\mathcal P_{l}}(S^{n_l} h_l)} =1, \quad \text{ uniformly in } 1\leqslant i\leqslant k_l.
\end{equation}
Recall that there exists a bijection $\Phi$ between $\epsilon$-independent set $\mathcal P_l$ and its product set $F_l:=E_l^{k_l}$. Let $p\in \mathcal P_{l}$ and $\Phi^{-1}(p)=(x_1,\ldots, x_{k_l})\in F_l.$
Fix $i$, let $q_i=q_i(p):=\Phi(x_i, x_{i+1},...,x_{k_l}, x_1,...,x_{i-1})\in\mathcal P_{l}$ where $x_j:=x_{j \!\!\!\mod k_l}$ for $j\geqslant k_l$. Then for all $1\leqslant j\leqslant k_l$
$$T^{a_{j}}q_i\in B_\epsilon^{n_l}(x_{i+j-1}), \text{ hence } T^{a_{j}}q_i\in B_{2\epsilon}^{n_l}(T^{a_{i+j-1}}p).$$
Since $\Phi$ is bijective, so is the map $p\mapsto q_i(p)$ on $\mathcal P_{l}$. Therefore
\begin{align*}
&\quad |\sigma^2_{\mathcal P_{l}}(S_{a_{i}}^{n_l} h_l)-\sigma^2_{\mathcal P_l}(S^{n_l}h_l)|\notag\\
&=\frac{1}{|\mathcal P_{l}|} \cdot\left|\sum_{p\in \mathcal P_{l}}\Big(S^{n_l}_{a_i}h_l(p)-\mathbb E_{\mathcal P_{l}}(S_{a_i}^{n_l}h_l)\Big)^2-\sum_{p\in \mathcal P_{l}}\Big(S^{n_l}h_l(q_i)-\mathbb E_{\mathcal P_{l}}(S^{n_l}h_l)\Big)^2\right|\\
&\leqslant \frac{1}{|\mathcal P_{l}|} \cdot\sum_{p\in\mathcal P_{l}}\bigg(\Delta_p^2+2\Delta_p\cdot \big|S^{n_l}h_l(q_i)-\mathbb E_{\mathcal P_{l}}(S^{n_l}h_l)\big|\bigg)\notag\\
&\leqslant \int \Delta_p^2 \,d\nu_{\mathcal P_{l}}+2\Big(\int \Delta_p^2\,d\nu_{\mathcal P_{l}}\Big)^{1/2}\cdot\sigma_{\mathcal P_{l}}(S^{n_l}h_l)
\end{align*}
where
\begin{align*}
\Delta_p&:=\big|(S^{n_l}_{a_{i}}h(p)- S^{n_l}h_l(q_i))-(\mathbb E_{\mathcal P_{l}}(S_{a_i}^{n_l} h_l)-\mathbb E_{\mathcal P_{l}}(S^{n_l}h_l))\big|\\
&\leqslant \omega^{n_l}(h_l, 2\epsilon_l, q_i)+ \int \omega^{n_l}(h_l, 2\epsilon_l, p)\, d\nu_{\mathcal P_{l}}(p).
\end{align*}
It follows that
$$\left|\frac{\sigma^2_{\mathcal P_{l}}(S^{n_l}_{a_i}h_l)}{\sigma^2_{\mathcal P_{l}}(S^{n_l}h_l)}-1\right|\lesssim \frac{\int (\omega^{n_l}(h_l,2\epsilon_l, p))^2 \nu_{\mathcal P_{l}}}{\sigma^2_{\mathcal P_{l}}(S^{n_l}h_l)}+\frac{(\int (\omega^{n_l}(h_l,2\epsilon_l, p))^2 \nu_{\mathcal P_{l}})^{1/2}}{\sigma_{\mathcal P_{l}}(S^{n_l}h_l)}$$
and hence \eqref{eq:4.2-3}.
Therefore $\{\frac{1}{\hat s_l}(G_{l,i}-\mathbb E_{F_l}(G_{l,i}))\}$ is asymptotically negligible, since
$$\nu_{F_l}(\{|G_{l,i}-\mathbb E_{F_l}(G_{l,i})|\geqslant \eta\hat s_l\})\leqslant \frac{\sigma^2_{F_l}(G_{l,i})}{\eta^2\hat s_l^2}\lesssim \frac{\sigma^2_{\mathcal P_l}(S_{a_i}^{n_l}h_l)}{\eta^2 s_l^2}\lesssim \frac{1}{\eta^2k_l}\xrightarrow{l\to\infty} 0.$$
\end{proof}
Note that a $\epsilon$-independent set $\mathcal P\subset P_{k(n+M)}$ is generally a proper subset of $P_{k(n+M)}$. Nevertheless we still can obtain much information in the limit of the uniform measures on $\mathcal P$ as if in the case of $P_{k(n+M)}$ (see Theorem \ref{theo:weak-convergence}).
On the other hand, we consider here a weighted measure with support $P_{k(n+M)}$ coming naturally from $\mathcal P$ and show that a Lindeberg CLT for this weighted measure can be deduced from the CLT for the uniform measure.
For every $p\in \mathcal P$ and its counterpart in the product set $(x_1,\ldots,x_k)=\Phi^{-1}(p)\in E^k$, let
$$Q(p):=\{q\in P_{k(n+M)}: d_n(T^{(i-1)(n+M)}q, x_i)<3\epsilon, \forall 1\leqslant i\leqslant k\}.$$
Because $E$ is $(n,3\epsilon)$-spanning for $P_{k(n+M)}$ it follows that
$$P_{k(n+M)}\subset\bigcup_{p\in\mathcal P_{\epsilon}}Q(p)$$
and for every $p\in \mathcal P$ and $q \in Q(p)$, one has
\begin{equation}\label{eq:qpeps}
d_n(T^{(i-1)(n+M)} p, T^{(i-1)(n+M)} q )< 4\epsilon, \quad \forall 1\leqslant i\leqslant k.
\end{equation}
However, a point $q\in P_{k(n+M)}$ may belong to multiple $Q(p)$. To account for this multiplicity define a weighted probability measure
on $P_{k(n+M)}$ by
$$\nu^w(A):=\sum_{q\in P_{k(n+M)}}w(q)\,\mathbbm 1_A(q)$$
where
$$w(q):=\frac1{|\mathcal P|}\sum_{\{p\colon q\in Q(p)\}}\frac1{|Q(p)|}.$$
\begin{corollary}\label{cor:1}
Consider $\epsilon_l$-independent sets $\mathcal P_l\subset P_{k_l(n_l+M_l)}$ and observables $h_l$ satisfying the oscillation condition \eqref{eq:4.2-0} and that
$$\lim_{l\to\infty}\frac{k_l^2M_l^2\|h_l\|^2_\infty}{s_l^2}=0.$$
Then the Lindeberg condition \eqref{eq:epslind} implies the CLT with respect to the weighted measure $\nu_l^w$
$$\lim_{l\to\infty}\nu^w_l\left(\Big\{
S^{k_l(n_l+M_l)}h_l
- \mathbb E_{\nu^w_l} (S^{k_l(n_l+M_l)}h_l) \leqslant t s_l\Big\}\right)=\mathcal{N}(t).$$
\end{corollary}
\begin{proof}
First note that \eqref{eq:4.2-1} is satisfied, because
$$\frac{\sigma^2_{\mathcal P_{l}}(\sum_{i=1}^{k_l} S_{a_{i}+n_l}^{M_l} h_l)}{s^2_l} \lesssim \frac{k_l^2M_l^2\|h_l\|_\infty^2}{s_l^2}\xrightarrow{l\to\infty} 0$$
by the assumption, as well as that $$\lim_{l\to\infty}\frac1{s^2_l}\sigma^2_{\nu_l^w}\left(\sum_{i=1}^{k_l} S_{a_{i}+n_l}^{M_l} h_l\right)=0.$$
So it suffices to show that the dynamical array $\{\frac{1}{s_l}\sum_{i=1}^{k_l}S_{a_i}^{n_l} h_l\}$ has the same limit distribution with respect to $\nu_{\mathcal P_l}$ and to $\nu_l^w$ when centered accordingly.
Note that for every pair of $p\in \mathcal P_{l}$ and $q \in Q_l(p)$, it follows from \eqref{eq:qpeps} that
$$
\left| \sum_{i=1}^{k_l} \left(S^{n_l}_{a_i} h_l(p) - S^{n_l}_{a_i} h_l(q)\right) \right|\leqslant \sum_{i=1}^{k_l} \omega_{a_i}^{n_l}(h_l,4\epsilon_l, p)=:\Omega_l(p).
$$
Denote
\begin{eqnarray*}
&&U_l(t):=\left\{q\in P_{k(n+M)}\colon \sum_{i=1}^{k_l} \left(S_{a_i}^{n_l} h_l (q)-\mathbb{E}_{\nu^w_{l}}(S^{n_l}_{a_i} h_l)\right)\leqslant t s_{l}\right\}\\
&&V^+_l(t) :=\left\{p\in \mathcal P_{l}\colon \sum_{i=1}^{k_l} \left(S^{n_l}_{a_i} h_l(p)-\mathbb E_{\mathcal P_{l}}(S^{n_l}_{a_i} h_l)\right)\leqslant t s_{l} + \Omega_l(p)+\mathbb{E}_{\mathcal P_{l}}(\Omega_l)\right\}\\
&&V^-_l(t) :=\left\{p\in \mathcal P_{l}\colon \sum_{i=1}^{k_l} \left(S^{n_l}_{a_i} h_l(p)-\mathbb E_{\mathcal P_{l}}(S^{n_l}_{a_i} h_l)\right)\leqslant t s_{l} - \Omega_l(p)-\mathbb{E}_{\mathcal P_l}(\Omega_l)\right\}.
\end{eqnarray*}
Then
$$\bigcup_{q\in U_l(t)}\{p\in\mathcal P_{l}: q\in Q_l(p)\}\subset V^+_l(t),
\quad \bigcup_{p\in V^-_{l}(t)} Q_l(p)\subset U_l(t)$$
and thus
\begin{align*}
\nu^w_{l}( U_l(t))&=\frac 1{|\mathcal P_l|} \sum_{q\in U_l(t)}\sum_{\{p\colon q\in
Q_l(p)\}}\frac{1}{|Q_l(p)|}\\
&\leqslant \frac{1}{|\mathcal P_{l}|}{\sum_{p\in V^+_l(t)}}\sum_{q\in Q_l(p)}\frac{1}{|Q_l(p)|}=\nu_{\mathcal P_{l}}(V^+_l(t)).
\end{align*}
Similarly $\nu^w_{l}( U_l(t))\geqslant \nu_{\mathcal P_{l}}(V_l^-(t)).$
Observe that because of \eqref{eq:4.2-0} $\frac 1{s_l}\Omega_l$ converges to $0$ in probability $\nu_{\mathcal P_l}$.
So $\displaystyle\lim_{l\to\infty}\nu_{\mathcal P_{l}}(V_l^+(t))=\lim_{l\to\infty}\nu_{\mathcal P_{l}}(V_l^-(t))=\mathcal N(t)$ by Theorem \ref{theo:1}, and hence $\displaystyle\lim_{l\to\infty}\nu^w_l( U_l(t))=\mathcal N(t)$ as desired.
\end{proof}
\section{A CLT under local specification}\label{sec:localCLT}
Compared to the global specification property, the local specification property allows us to single out sets of periodic points with independence structure in a local scenario in which the positions of certain orbits are specified a priori.
\begin{definition}[Local $\epsilon$-Independence]\label{def:locidp}
Let $\epsilon>0$ and $\mathcal U$ be a family of open sets. Let $A\in \bigvee_{i=0}^{k-1} T^{-in}\mathcal U$ for some $k, n\in\mathbb N$. A subset $\mathcal P\subset P$
is locally $\epsilon$-independent with respect to $A$
if there exist $E_i\subset T^{(i-1)n}A$, $1\leqslant i\leqslant k$,
and a bijection
$\Phi$ from $F:=\prod_{i=1}^{k} E_i$ to ${\mathcal P}$ such that for any ${\bf x}=(x_1,...,x_{k})\in F$ and $1\leqslant i\leqslant k$.
\begin{equation}\label{eq:6.2}
T^{(i-1)n}(\Phi({\bf x}))\in B^n_{\epsilon}(x_{i}).
\end{equation}
The set $F$ will be called the product set of $\mathcal P$.
\end{definition}
Recall that in the definition of local specification (Definition \ref{def:specloc}) any given $\epsilon>0$ defines a $N(\epsilon)\in\mathbb N$ and a $\delta(\epsilon)>0$.
\begin{proposition}\label{prop:locidp}
Let $(X,T)$ satisfy the local specification property. Then for any $\epsilon>0$, $k\in \mathbb N$, $n\geqslant N(\epsilon)$ and any family $\mathcal U$ of open subsets of diameter at most $\delta(\epsilon)$ and $A\in\bigvee_{i=0}^{k-1} T^{-in}\mathcal U$,
there exists a locally $\epsilon$-independent set $\mathcal P\subset P_m$ with respect to $A$,
where $m\geqslant kn$ is any given multiple of $n$, and such that
if in addition the system is $\epsilon^*$-expansive and $\epsilon<\epsilon^*/3$ then $$A\cap P_m\subset \mathcal P\subset B_{\epsilon}^{m}(A)\cap P_m.$$
\end{proposition}
\begin{proof}
For each $1\leqslant i\leqslant k-1$ choose a maximal $(n,2\epsilon)$-separated set $E_i\subset T^{(i-1)}A$ and a maximal $(m-(k-1)n, 2\epsilon)$-separated set $E_k\subset T^{(k-1)n}A\cap T^{-m+(k-1)n}A$. For any $(x_1,\ldots,x_k)\in F=\prod_{i=1}^{k} E_i$, by the local specification property there exists $p\in P_m$ such that $$T^{(i-1)n}p\in B_{\epsilon}^n(x_i) \qquad 1\leqslant i\leqslant k-1$$ and $$T^{(k-1)n}p\in B_{\epsilon}^{m-(k-1)n}(x_k).$$
It defines a map $\Phi$ from $F$ to $P_m$. This map is injective as $E_i$ is $(n,2\epsilon)$-separated for each $1\leqslant i\leqslant k$, hence its image, denoted by $\mathcal P$, is a locally $\epsilon$-independent set with respect to $A$.
Clearly $\mathcal P\subset B_\epsilon^m(A)\cap P_m$.
Suppose now the system is $\epsilon^*$-expansive and $\epsilon<\epsilon^*/3$. For any $q\in A\cap P_m$, due to maximality of $E_i$, there exists $\mathbf y\in F$ such that $d_n(y_i, T^{(i-1)n}q)<2\epsilon, 1\leqslant i\leqslant k-1$ and $d_{m-(k-1)n}(y_k, T^{(k-1)n}q)<2\epsilon$. Therefore expansiveness implies that $q=\Phi(\mathbf y)\in\mathcal P$.
\end{proof}
The local $\epsilon$-independence structure also entails a Lindeberg type CLT.
\begin{theorem}\label{theo:2}
Let $\{\epsilon_l>0, k_l, n_l\in\mathbb N\}_{l\in\mathbb N}$ be sequences of numbers. Let $\mathcal U_l$ be a family of open sets and $A_l\in \bigvee_{i=0}^{k_l-1}T^{-in_l}\mathcal U_l$. Consider a sequence of locally $\epsilon_l$-independent set $\mathcal P_l$ with respect to $A_l$
and observables $h_{l,i}:P\to\mathbb R$ satisfying that
$$\lim_{l\to\infty}\frac 1{s_l^j} \sum_{i=1}^{k_l} \int \left(\omega_{a_i}^{n_l}(h_{l,i}, \epsilon_l, p)\right)^j d\nu_{\mathcal P_l}(p)=0, \quad j=1,2,$$
where $a_i=(i-1)n_l$ and
$s_l^2=\sum_{i=1}^{k_l} \sigma^2_{\mathcal P_l}(S^{n_l}_{a_i}h_{l,i})$.
Then the Lindeberg condition holds, i.e.
$$\lim_{l\to\infty}\frac 1{s_l^2} \sum_{i=1}^{k_l}
L_{\mathcal P_l}(S^{n_l}_{a_i}h_{l,i}, \eta s_l)=0,\qquad\forall\eta>0$$
if and only if
the array $\big\{\frac{1}{s_l}\left(S^{n_l}_{a_i}h_{l,i}-\mathbb E_{\mathcal P_l}(S^{n_l}_{a_i}h_{l,i})\right)\big\}_{1\leqslant i\leqslant k_l, l\in\mathbb N}$ is $\nu_{\mathcal P_l}$-asymptotically negligible, i.e.
$$\lim_{l\to\infty}\max_{1\leqslant i\leqslant k_l}\nu_{\mathcal P_l}\left(\left\{\left|S^{n_l}_{a_i}h_{l,i}-\mathbb E_{\mathcal P_l}(S^{n_l}_{a_i}h_{l,i})\right|\geqslant \eta s_l\right\}\right)=0,\qquad\forall\eta>0$$
and the CLT holds, i.e.
for every $t\in \mathbb R$
$$ \lim_{l\to\infty} \nu_{\mathcal P_l} \left(\bigg\{ \sum_{i=1}^{k_l} \left(S^{n_l}_{a_i}h_{l,i}-\mathbb E_{\mathcal P_l}(S^{n_l}_{a_i}h_{l,i})\right)\leqslant ts_l\bigg\}\right) = \mathcal N(t).$$
\end{theorem}
\begin{proof}
Like the proof of Theorem \ref{theo:1}, define an array $\{G_{l,i}\}$ on $F_l$, the product set of $\mathcal P_l$, by setting
\begin{equation*}
G_{l,i}(x_1,...,x_{k_l}):= S^{n_l} h_{l,i}(x_i), \quad 1\leqslant i\leqslant k_l
\end{equation*}
and let $\hat{s}_l^2=\sum_{i=1}^{k_l} \sigma^2_{F_l}(G_{l,i})$.
Due to the structure of locally independent sets, specifically by \eqref{eq:6.2}, for any $p\in\mathcal P_l$ one has similar to \eqref{eq:epsindaprox}
$$\left|G_{l,i}(\Phi^{-1}p)-S^{n_l}_{a_i} h_{l,i}(p)\right|\leqslant \omega^{n_l}_{a_i}(h_{l,i}, \epsilon_l, p).$$
Then one can show that each statement for the dynamical array $\{\frac{1}{s_l}(S_{a_i}^{n_l}h_{l,i}-\mathbb E_{\mathcal P_l}(S^{n_l}_{a_i}h_{l,i}))\}$ in the theorem is equivalent to the statement for the array $\{\frac{1}{\hat s_l}(G_{l,i}-\mathbb E_{F_l}(G_{l,i}))\}$ in Proposition \ref{prop:indclt}, much in the same way as illustrated in Theorem \ref{theo:1}.
\end{proof}
\section{Applications: Measures of maximal entropy}\label{sec:mme}
In this section we provide application to system with a unique measure of maximal entropy, since such a measure is the weak-* limit of the equidistributed Dirac measures on periodic points. However, we believe our methods apply in more generality.
Note that in a positively expansive system satisfying global specification property the measure of maximal entropy is unique (by adapting Bowen's proof \cite{Bow71, Bow74} in the homeomorphism case or by taking the natural extension \cite{Rue92}).
Under certain conditions the measure of maximal entropy is in fact the limit of the equidistributed Dirac measures on the $\epsilon$-independent sets.
This is the case if, for instance:
\begin{equation}\label{eq:pphomeo}
T|_{P_n} \text{ is a homeomorphism for every } n. \tag{*}\footnote{This is automatic if $T$ is a homeomorphism.}
\end{equation}
Recall that in the global specification property the length of the gap $M(\epsilon)$ depends on $\epsilon>0$ (see Definition~\ref{def:spec} and Proposition~\ref{prop:weakspec}). In this section, for a sequence $\{\epsilon_l\}_{l\in\mathbb N}$, $M(\epsilon_l)$ will be abbreviated to $M_l$.
\begin{theorem}\label{theo:weak-convergence}
Let $(X,T)$ be a $\epsilon^*$-positively expansive system satisfying global specification property. Let $\mu$ be the unique measure of maximal entropy. Assume condition \eqref{eq:pphomeo}.
Given any $0<\epsilon<\epsilon^*/8$, then for any sequence of integers $k_l\in\mathbb N$, $n_l\xrightarrow{l\to\infty}\infty$ and
$\epsilon$-independent sets $\mathcal P_l\subset P_{k_l(n_l+M(\epsilon))}$, $$\nu_{\mathcal P_l}\xRightarrow{l\to\infty}\mu.$$
\end{theorem}
\begin{proof}
Note that a positively expansive transformation has finite entropy. We first prove that
any weak accumulation measure $\nu$ is $T$-invariant.
Let wlog $\nu=\text{w-}\lim_{l\to\infty}\nu_{\mathcal{P}_l}$. Recall $E_l\subset P_{k_l(n_l+M)}$ from Definition~\ref{def:independence}
with the bijection $\Phi_l: E_l^{k_l}\to \mathcal P_l$. In the rest of the proof we will write $k,n$ for $k_l, n_l$ for simplicity, where the dependence is self-evident.
Given $p\in\mathcal P_l$, since $E_l$ $(n,3\epsilon)$-spans $P_{k(n+M)}$ and $Tp\in P_{k(n+M)}$, there exists $\mathbf{y}=(y_1,\ldots,y_{k})\in E_l^k$ such that for any $1\leqslant i\leqslant k$
$$ T^{(i-1)(n+M)}\left(Tp\right)\in B^n_{3\epsilon}(y_i).$$
Let $q(p):=\Phi_l(\mathbf{y})\in\mathcal P_l$, then
$T^{(i-1)(n+M)}(q(p))\in B_\epsilon^n(y_i)$,
hence
$$d_n(Tp, q(p))\leqslant d_n(Tp, y_1)+d_n(y_1, q(p))<4\epsilon<\epsilon^*.$$
Since $T$ is $\epsilon^*$-positively expansive and $n\xrightarrow{l\to\infty}\infty$,
$$ \lim_{l\to \infty} d(Tp, q(p))=0$$
uniformly in $p$, that is, for any $\delta>0$ there exists $\ell(\delta)\in\mathbb N$ such that for any $l>\ell(\delta)$ and any $p\in\mathcal P_l$, $d(Tp, q(p))<\delta$.
Moreover, the map $p\mapsto q(p)$ is injective:
for $p, p'\in \mathcal P_l\subset P_{k(n+M)}$, if $q(p)=q(p')$ then they correspond to the same $\mathbf y\in E_l^k$, hence for all $1\leqslant i\leqslant k$
$$d_n(T^{(i-1)(n+M)}Tp, T^{(i-1)(n+M)}Tp')< 6\epsilon<\epsilon^*.$$
Therefore uniformly
$$\lim_{l\to\infty} d(T^{(i-1)(n+M)}Tp, T^{(i-1)(n+M)}Tp')=0.$$
With condition \eqref{eq:pphomeo}
it implies that for large $l$ and all $1\leqslant i\leqslant k$
$$d_M(T^{i(n+M)+n}Tp, T^{i(n+M)+n}Tp')<\epsilon^*.$$
Thus orbits of $p$ and $p'$ are always $\epsilon^*$-close and $p=p'$ by expansiveness.
Finally, for any $\delta>0$, any Lipschitz function $f$ and $l>\ell(\delta)$,
$$\left| \int f(Tp) \,d\nu_{\mathcal P_l}(p) - \int f(p) \,d\nu_{\mathcal P_l}(p)\right| \leqslant
\frac{1}{|\mathcal P_l|}\sum_{p\in \mathcal P_l}
|f(Tp)-f(q(p))| \leqslant \delta\cdot Lip(f).$$
Letting $l\to\infty$ proves the $T$-invariance of $\nu$.
We now prove that the metric entropy $h(\nu)$ of any weak accumulation point $\nu$ of the sequence $\nu_{\mathcal{P}_l}$ agrees with the topological entropy $h(T)$.
Let $\alpha$ be a partition of $X$ into Borel sets $A\in\alpha$ of diameter $<\epsilon$ with $\nu(\partial A)=0$, where $\partial A$ denotes the boundary of $A$.
For any $m\geqslant 1$ and $A\in \bigvee_{i=0}^{m-1} T^{-i}\alpha$ one has
$$
\nu(A)=\lim_{l\to\infty} \nu_{\mathcal P_l}(A)=\lim_{l\to\infty}
\frac{|\mathcal P_l\cap A|}{|\mathcal P_l|}.$$
Denote the maximal cardinality of all $(m, \epsilon^*)$-separated sets of $X$ by $r_{m,\epsilon^*}(X)$. We claim there exists $C=C(\epsilon)>0$ such that
\begin{equation}
\label{claim:upper-bound}
\lim_{l\to\infty}
\frac{|\mathcal P_l\cap A|}{|\mathcal P_l|}\leqslant
\frac{C}{r_{m,\epsilon^*}(X)}.
\end{equation}
Assuming this claim it follows that
$$ \frac 1m H_\nu\left(\bigvee_{i=0}^{m-1} T^{-i}\alpha\right) \geqslant \frac 1m \log \left(\frac{ r_{m,\epsilon^*}(X)}{C}\right)$$
and, letting $m\to\infty$ yields
$$ h_\nu(T,\alpha)\geqslant h(T).$$
Since $(X,T)$ is $\epsilon^*$-positively expansive, partitions by sets of diameter $<\epsilon^*$ are generating and thus $h_\nu(T)=h_\nu(T,\alpha)\geqslant h(T)$.
The reversed inequality is well-known.
We now prove claim~\eqref{claim:upper-bound}. Assume $l$ large such that $n>m+2M-1$ and consider $p\in\mathcal P_l\cap A$. Let $(x_1, \ldots,x_{k})=\Phi_l^{-1}(p)$, then $p\in B_\epsilon^n(x_1)$ and $\displaystyle{\sup_{0\leqslant i\leqslant m-1}diam(T^iA)<\epsilon}$, hence
$$
x_1\in \hat B^m_{2\epsilon}(A):=\{x:d_m(x,y)<2\epsilon, \forall y\in A\}.
$$
It follows that
$$|\mathcal P_l\cap A|\leqslant |E_l\cap \hat B^m_{2\epsilon}(A)|\cdot |E_l|^{k-1}.$$
Given a $(m,\epsilon^*)$-separated set $R$ of maximal cardinality, define a map
$$\kappa: R\times \{E_l\cap \hat B_{2\epsilon}^m(A)\} \to E_l$$
for which the number of preimages is bounded by a constant $C(\epsilon)$ so that
$$ r_{m,\epsilon^*}(X)\cdot |E_l\cap \hat B^m_{2\epsilon}(A)| \leqslant C |E_l|,$$
which in turn implies claim~\eqref{claim:upper-bound}.
Define $\kappa$ as follows. By
Proposition~\ref{prop:weakspec} for any $(z,x)\in R\times \{E_l\cap \hat B_{2\epsilon}^m(A)\}$, and since $n>m+2M-1$, there exists $p\in P_{n+M}$ with
$$p\in B_{\epsilon}^m(z) \quad \text{ and }\quad
T^{m+M}p\in B_\epsilon^{n-(m+M)}\left(T^{m+M}x\right).$$
Since $E_l$ is $(n,3\epsilon)$-spanning for $P_{k(n+M)}$ there exists $y\in E_l\cap B_{3\epsilon}^n(p)$.
The map $\kappa$ is defined by choosing $\kappa(z,x):=y$.
To prove the uniform bound on the number of pre-images of $\kappa$, assume
$\kappa(z,x)=\kappa(z',x')=y$ for some $y\in E_l$.
Denote by $p, p'\in P_{n+M}$ the associated periodic points. It follows that
$$ d_n(p,p')< 6\epsilon\quad\mbox{ and }\quad
d_m(z,z')< 8\epsilon<\epsilon^*.$$
But since $z, z'\in R$ and $R$ is $(m,\epsilon^*)$-separated, it follows that $z=z'$.
Moreover, since $x,x'\in \hat B_{2\epsilon}^m(A)$
we have
$$ d_m(x,x')< 4\epsilon<\epsilon^*$$
and also
$$ d_{n-(m+M)}(T^{m+M}x, T^{m+M}x')<8\epsilon<\epsilon^*.$$
Therefore if $x\ne x'\in E_l\subset P_{n+M}$, there exists $t\in [m,m+M]\cup[n,n+M]$ with
$$ d(T^t(x), T^t(x'))>\epsilon^*.$$
Let $c$ denote the minimal amount of balls of radius $<\epsilon^*/2$ necessary to cover $X$. Then for fixed $x\in E_l\cap \hat B_{2\epsilon}^m(A)$ there are at most $c^{2M}$ points $x'$ mapping to the same image under $\kappa$. Hence setting
$ C:= c^{2M}$ proves the statement.
\end{proof}
Moreover, applying a diagonal argument to the previous theorem yields:
\begin{corollary} \label{cor:wkcvg}
Let $(X,T)$ be a positively expansive system satisfying the global specification property. Let $\mu$ be the unique measure of maximal entropy. Assume condition \eqref{eq:pphomeo}. Then, given sequences $\{\epsilon_l>0, k_l\in\mathbb N\}_{l\in\mathbb N}$ with $\epsilon_l\to 0$ and $k_l\to\infty$, there exists $n_l^*\to\infty$ such that for any sequence $\{n_l\}$ with $n_l> n^*_l$ and any $\epsilon_l$-independent sets $\mathcal P_l\subset P_{k_l(n_l+M_l)}$, $\nu_{\mathcal P_l}\Rightarrow \mu$.
\end{corollary}
\begin{theorem}\label{theo:3}
Let $(X,T)$ be a positively expansive system satisfying the global specification property. Let $\mu$ be the unique measure of maximal entropy. Assume condition \eqref{eq:pphomeo}. Let $\{\epsilon_l>0, k_l, n_l\in\mathbb N\}_{l\in\mathbb N}$ be sequences such that $\epsilon_l\to 0, k_l^{1-\eta_l^*}\to\infty, \epsilon_lk_l^{(1+\eta_l^*)/2}\to 0, n_l>n^*_l$ and ${k_l^{1+\eta_l^*} M_l^2}/{n_l^2}\to 0$ as $l\to\infty$ for some $\eta_l^*\to1^-$.
Consider a sequence of $\epsilon_l$-independent sets ${\mathcal P}_{l}\subset P_{k_l(n_l+M_l)}$ and a Lipschitz function $h$.
Then
$$
\lim_{l\to\infty}\nu_{\mathcal P_{l}}
\left(\bigg\{
\frac{1}{k_l(n_l+M_l)}\big(S^{k_l(n_l+M_l)}h -\mathbb{E}_{\mathcal P_{l}}(S^{k_l(n_l+M_l)}h)\big)
\leqslant k_l^{-\frac12+\eta} \bigg\}\right)=1.
$$ for any $\eta>0$.
Moreover, if
$$\sum_{l\in\mathbb N} \frac{1}{\sqrt{k_l}}\ \cdot\, \sup_{\|h\|_{\rm Lip}\leqslant 1}\ \frac {\mathbb E_ {{\mathcal P}_{l}} (S^{n_l}h)^4}{\sigma^4_{{\mathcal P}_{l}}(S^{n_l}h)} <\infty,$$
then one has
$$
\frac 1{k_l(n_l+M_l)} \sum_{i=0}^{k_l(n_l+M_l)-1}\delta_{T^i(p_{l})}
\xRightarrow{l\to\infty}\mu
$$
for $\nu$ a.e. sequence $\displaystyle{\{p_{l}\}_{l\in\mathbb N}\in \prod_{l\in\mathbb N}{\mathcal P}_{l}}$
where $\nu$ denotes the product measure of $\nu_{{\mathcal P}_{l}}$ on $P^{\mathbb N}$.
\end{theorem}
\begin{proof}
Let $a_{i}=(i-1)(n_l+M)$ and
$s_{l}^2=\sum_{i=1}^{k_l} \sigma^2_{\mathcal P_{l}}(S_{a_{i}}^{n_l} h)$
$$I:=\{ l\in\mathbb N: s_l^2 \geqslant {k_l}^{1-\eta_l^*} n_l^2\}.$$
Assume $I$ has infinitely many elements. We verify conditions in Theorem \ref{theo:1} for $l\in I$.
Since $h$ is Lipschitz, one has for $j=1,2$ that
$$\frac1{s_l^j}\sum_{i=1}^{k_l}
\int \left(\omega_{a_i}^{n_l}(h,4\epsilon_l,p)\right)^j d\nu_{\mathcal P_{l}}\lesssim \frac{k_l(\epsilon_l n_l)^j}{(k_l^{1-\eta_l^*} n_l^2)^{j/2}}=k_l^{1-j/2+\eta_l^*j/2}\epsilon_l^{j}\xrightarrow{l\to\infty} 0.$$
and
$$\frac1{s^2_l}\ \sigma^2_{\mathcal P_{l}}\left(\sum_{i=1}^{k_l} S_{a_{i}+n_l}^{M_l} h\right)\lesssim \frac{k_l^2M_l^2\|h\|_\infty^2}{k_l^{1-\eta_l^*} n_l^2}\xrightarrow{l\to\infty}0$$
by assumptions. Also since
$${|S_{a_i}^{n_l} h(p)-\mathbb E_{\mathcal P_l}(S_{a_i}^{n_l} h)|}/{s_l}\lesssim n_l/s_l\leqslant 1/k_l^{1/2-\eta_l^*/2}\xrightarrow{l\to\infty}0,$$
$$\lim_{l\to\infty}\frac 1{s_l^2} \sum_{i=1}^{k_l}L_{\mathcal P_l}(S_{a_i}^{n_l} h_l, \eta s_l)=0\quad \forall \eta>0.$$
Hence Theorem \ref{theo:1} implies that for any subsequence $l\in I$ and any $\eta>0$
\begin{align}
&\quad \lim_{l\to\infty}\nu_{\mathcal P_l}\left(\bigg\{\frac{1}{k_l(n_l+M_l)}\big(S^{k_l(n_l+M_l)}h -\mathbb{E}_{\mathcal P_{l}}(S^{k_l(n_l+M_l)}h)\big)
\leqslant k_l^{-1/2+\eta}\bigg\}\right)\label{eq:4.2-5}\\
&=\lim_{l\to\infty} \mathcal N(k_l^{1/2+\eta}(n_l+M_l)/s_l)=1 \notag
\end{align}
since $s_l^2\lesssim k_l n_l^2$, and hence
$k_l^{1/2+\eta}(n_l+M_l)/s_l\gtrsim k_l^{1/2+\eta}(n_l+M_l)/(k_l^{1/2}n_l)\xrightarrow{l\to\infty}\infty.$
On the other hand, for any subsequence $l\not\in I$, where $\eta>(1-\eta^*_l)/2$
\begin{align*}
&\quad \nu_{\mathcal P_l}\left(\bigg\{\big| \frac 1 {k_l(n_l+M_l)} (S^{k_l(n_l+M_l)}h -\mathbb E_{{\mathcal P}_{l}}(S^{k_l(n_l+M_l)}h))\big |\geqslant k_l^{-1/2+\eta}\bigg\}\right)\\
&\lesssim \frac{k_l^2M_l^2+k_ls_l^2} {k_l^{1+2\eta}(n_l+M_l)^2}\leqslant \frac{k_l^{1-2\eta}M_l^2}{n_l^2}+k_l^{1-\eta_l^*-2\eta}\xrightarrow{l\to\infty} 0.
\end{align*}
This establishes the first part of the theorem.
For the second part, notice that the Berry-Esseen theorem applied to the $k$ blocks in the proof of Theorem \ref{theo:1} yields that the set of points satisfying (\ref{eq:4.2-5}) is of order $k_l^{-1/2}$, hence the set of $p$ not satisfying (\ref{eq:4.2-5}) has measure zero by Borel-Cantelli lemma. This implies the second part of the theorem.
\end{proof}
Since the measure of maximal entropy is often mixing it seems to be suggesting that the central limit theorem can only be derived through measures on periodic orbits if these reflect the mixing properties of the measure of maximal entropy. This seems to be difficult (if not at all impossible in general) using the global specification property. This is supported by the fact that periodic points do not have a natural filtration of $\sigma$-fields. Thus, an application of the method in Section \ref{sec:globalCLT} provides a different approach to this problem. At this point we are not discussing this further; instead,
we will derive a CLT of a somewhat different type.
Assume that $(X,T)$ satisfies local specification property with a unique measure of maximal entropy $\mu$.
In a topologically mixing system with local specification property, one can show in parallel to Theorem \ref{theo:weak-convergence} that $\mu$ is the weak limit of $\nu_{P_n}$, the uniform measure on periodic points.
The following proposition is one from a variety of others which can be proved along the lines.
We denote by
$\mu_Y$ the induced measure of $\mu$ on $Y$ and call an open, non-empty set $B$ an $\mu$-set if $\mu(\partial B)=0$.
\begin{proposition}\label{prop:9.1} Let $(X,T)$ be a $\epsilon^*$-positively expansive and topologically mixing system satisfying the local specification property. Let $\mu$ be the unique measure of maximal entropy.
Consider $\{\mathcal U_l\}_{l\in\mathbb{N}}$ a family of pairwise disjoint open $\mu$-sets of diameter less than $\delta(\epsilon^*/3)$ and let $\alpha_l:= \bigvee_{i=0}^{k_l-1} T^{-in_l}\mathcal U_l$, where
$k_l, n_l\in\mathbb N$ and $n_l\geqslant N(\epsilon^*/3)$ (see Definition~\ref{def:specloc}).
Assume
\begin{equation}\label{eq:9.2}
\mu\left(\bigcup_{U\in\mathcal U_l} U\right) = 1- o\left(\frac1{k_l}\right),
\end{equation}
\begin{equation}\label{eq:9.2b}
\lim_{l\to\infty}\sup_{A\in \alpha_l}\limsup_{j\in\mathbb N}\frac {\mu(B_{\epsilon^*}^j(A)\cap P_j)}{\mu(A\cap P_j)}= 1.
\end{equation}
\noindent
Consider a family $\{h_l\}_{l\in\mathbb{N}}$ of real valued functions in $L^3(\mu)$ such that
\begin{enumerate}
\item for $a_i=(i-1)n_l$ and $s_l^2=\max_{A\in \alpha_l} \sigma^2_{\mu_A} (S^{k_l n_l} h_l)\to\infty$
\begin{equation}\label{eq:9.1}
\lim_{l\to\infty}\frac 1{s_l^2} \max_{A\in\alpha_l}\sum_{1\leqslant i\leqslant k_l} \int_A (\omega_{a_i}^{n_l}(h_l, \epsilon^*, p))^2 d\mu_A(p)=0,
\end{equation}
\item $\exists K\in \mathbb R$ such that $\forall l\in\mathbb N, A \in \alpha_l$ and $1\leqslant i \leqslant k_l$
\begin{equation}\label{eq:9.3}
\mathbb E_{\mu_A}\left(|S^{n_l}_{a_i}h_l - \mathbb E_{\mu_A} (S_{a_i}^{n_l}h_l)|^3\right) \leqslant K {\sigma}^3_{\mu_A}(S_{a_i}^{n_l}h_l),
\end{equation}
\item for some measurable function $\sigma_x$
\begin{equation}\label{eq:9.5}
\frac1{s_l^2}\sum_{A\in\alpha_l}\mathbbm 1_A(x)\cdot \sigma^2_{\mu_A}(S^{k_l n_l}h_l)
\xrightarrow{L^1(\mu)}\sigma_x.
\end{equation}
\end{enumerate}
Then for any $t\in\mathbb R$
$$\lim_{l\to\infty}\mu\big(\big\{ S^{k_l n_l}h_l- \sum_{A\in \alpha_l} \mathbbm 1_A \cdot \mathbb E_{\mu_A} (S^{k_l n_l}h_l) \leqslant t s_l\big\}\big)=\int_X \mathcal N(t/\sigma_x) d\mu(x),$$
where for $\sigma_x=0$, $\mathcal N(t/\sigma_x)= \mathbbm 1_{\{s>0\}}(t)$
is the distribution function of the Dirac measure in $0$.
\end{proposition}
\begin{proof}
Notice that since by (\ref{eq:9.2})
$$ \mu \left(X\setminus\bigcup_{A\in \alpha_l} A\right)\leqslant \sum_{i=1}^{k_l} \left(1-\mu(T^{-a_i}(\bigcup_{U\in\mathcal U_l}U))\right) \to 0,$$
we can consider the distribution restricted to $\cup_{A\in\alpha_l} A$.
We first estimate
\begin{equation}\label{eq:9.6}
\mu_A\big(\big\{ S^{k_l n_l}h_l(x)- \mathbb E_{\mu_A} (S^{k_l n_l}h_l)\leqslant t s_{l,A}\big\}\big) - \mathcal N(t)
\end{equation}
for $A\in\alpha_l$, where
$$s_{l,A}^2= \sum_{i=1}^{k_l} \sigma^2_{\mu_A}(S^{n_l}_{a_i}h_l).$$
Approximating $h_l$ by a $\mu$-a.e. continuous function and then $\mu$ by the uniform distribution on all periodic points in ${P}_j$ for sufficiently large $j$, it is sufficient to estimate the expression (\ref{eq:9.6}) with $\mu_A$ replaced by $\nu_{P_{j}\cap A}$. We also may assume that $j$ is so large that we can approximate
\begin{eqnarray*}
&& \sigma^2_{\mu_A} (S_{a_i}^{n_l}h_l); \qquad \mathbb E_{\mu_A}\left(|S^{n_l}_{a_i}h_{l}- \mathbb E_{\mu_A} (S_{a_i}^{n_l}h_l)|^3\right), \ 1\leqslant i\leqslant k_l
\end{eqnarray*}
by the corresponding expressions under $\nu_{{P}_j\cap A}$ in such a way that it does not affect the limiting distribution and that the assumptions still hold for $\nu_{P_j\cap A}$.
The problem now reduces to proving the statement for the dynamical array $$\{ S^{n_l}_{a_i} h_l- \mathbb E_{P_j\cap A}(S^{n_l}_{a_i} h_{l})\}_{1\leqslant i\leqslant k_l}$$ on the probability space $(X, \nu_{{P}_j\cap A})$.
By Proposition \ref{prop:locidp} there exists a locally $\epsilon^*/3$-independent set $\mathcal P_l\subset P_j$ with respect to $A$ such that $ A\cap P_j \subset\mathcal P_l\subset B_{\epsilon^*}^j(A)\cap P_j$ and that $\mathcal P_l$ is bijective with a product set $F_l\subset X^{k_l}$ by $\Phi: F_l\to \mathcal P_l$, which approximates $\mathcal P_l$ as in \eqref{eq:6.2}.
By (\ref{eq:9.2b}) and approximating $\mu$ by $\nu_{P_j}$ we may assume that
$$ 1+o(1)\geqslant \frac{\nu_{P_j}(B_{\epsilon^*}^j(A))}{\nu_{P_j}(A)}\geqslant \frac{|F_l|}{|P_j\cap A|}\geqslant 1.$$
Together with condition \eqref{eq:9.1}, it follows that
we may exchange the measure $\nu_{P_j\cap A}$ for the product measure $\nu_{F_l}$.
With $\nu_{F_l}$ we can apply the central limit theorem with rate estimate (see \cite{Pet75}, Chapter V, Section 2, Theorem 3). Since the quotient of the third centered moment of $S^{n_l}_{a_i} h_l\circ \Phi$ divided by its variance raised to the $3/2$ moment stays bounded by (\ref{eq:9.3}) it follows that
$$\left|\nu_{F_l} (\{S^{k_l n_l} h_l\circ \Phi - \mathbb E_{F_l}(S^{k_l n_l} h_{l}\circ \Phi)\leqslant t s_{{l},A}\})- \mathcal N(t)\right|
= O(1/\sqrt{k_l}),$$
where the estimate is uniform over $\mathcal P_l$ and $A\in\alpha_l$.
The same estimate holds therefore for $\nu_{P_j\cap A}$, hence
\begin{eqnarray*}
&& \nu_{P_j}\left(\bigg\{x\in \bigcup_{A\in \alpha_l} A: S^{k_ln_l} h_l(x)- \sum_{A\in \alpha_l} \mathbbm 1_A(x) \cdot \mathbb E_{\mu_A} (S^{k_ln_l}h_l)\leqslant t s_{l}\bigg\}\right) \\
&=& \sum_{A\in\alpha_l} \nu_{P_j\cap A}(\{S^{k_ln_l} h_l - \mathbb E_{\mu_A} (S^{k_ln_l}h_l)\leqslant t s_{l}\}) \nu_{P_j}(A)+o(1)\\
&=& \sum_{A\in\alpha_l} \mathcal N(ts_{l}/s_{{l},A}) \nu_{P_j}(A)+o(1)\\
&=& \int \mathcal N(t/\sigma_x) d\nu_{P_j}(x)+o(1),
\end{eqnarray*}
since $ s_{{l},A}/s_{l}$ converges to $\sigma_x$ in $L^1(\mu)$ by assumption (\ref{eq:9.5}). By the choice of $P_j$, the conclusion holds as well for $\mu$.
\end{proof}
The method of proof of Proposition \ref{prop:9.1} has many applications. Here we use the proposition to obtain convergence to a mixture of normal distributions for a special class of square integrable functions, which we call wildly oscillating.
We call a finite partition $\beta$ of $X$ a $\nu$-partition if for each $B\in \beta$, $B\subset\overline{\mbox{int} B}$ and $B$ is a $\nu$-set.
\begin{definition}\label{def:5.2} Let $(X, T, \nu)$ be a probability preserving dynamical system. Suppose
for every $l\in\mathbb N$, $\beta_l$ is a $\nu$-partition of diameter less than $\epsilon_l$ with $\epsilon_l\to0$ and $W_l\geqslant n_l$ are natural numbers. A square integrable function $h:X\to \mathbb R$ is wildly oscillating with data $(\beta_l, \epsilon_l, W_l, n_l)_{l\in \mathbb N}$, if
with $k_l:=[W_l/n_l]$, $\alpha_l:=\bigvee_{i=0}^{k_l-1}T^{-i n_l} \beta_{l}$ and $s_l^2:=\max_{A\in\alpha_l} \sigma^2_{\nu_A}(S^{k_l n_l}h)$,
\begin{enumerate}
\item
$$s_l\to\infty \quad \text{ and }\quad \frac{1}{s_l^2}\max_{1\leqslant r< n_l}\max_{A\in\alpha_l} \sigma^2_{\nu_A}(S^r h)\to 0,$$
\item
$$\lim_{l\to\infty}\frac 1{s_l^2} \max_{A\in\alpha_l}\sum_{i=1}^{k_l} \int_A \left(\omega_{(i-1)n_l}^{n_l}(h, \epsilon_l, p)\right)^2 d\nu_A(p)=0.$$
\end{enumerate}
\end{definition}
\begin{theorem}\label{theo:8.2} Let $(X,T)$ be a positively expansive and topologically mixing dynamical system with the local specification property.
Let $\mu$ be the unique measure of maximal entropy. Let $h :X\to \mathbb R$ be a wildly oscillating function in $L^3(\mu)$ with data $(\beta_l,\epsilon_l, W_l, n_l=N(\epsilon_l))_{l\in \mathbb N}$. Assume that with notations above
\begin{enumerate}
\item $\exists K>0$
such that $\forall l\in \mathbb N, A\in \alpha_l, 1\leqslant i\leqslant k_l$
\begin{equation}\label{eq:9.30}
\mathbb E_{\mu_A}(| S_{(i-1)n_l}^{n_l} h-\mathbb E_{\mu_A}(S^{n_l}_{(i-1)n_l}h)|^3)\leqslant K (\sigma^2_{\mu_A}(S^{n_l}_{(i-1)n_l} h))^{3/2},
\end{equation}
\item
\begin{equation}\label{eq:9.32}
\frac1{s_l^2}\sum_{A\in\alpha_l}\mathbbm 1_A\cdot \sigma^2_{\mu_A}(S^{W_l}h) \text{ converges in } L^1.
\end{equation}
\end{enumerate}
Then with respect to $\mu$
\begin{equation}\label{eq:9.36}
\frac 1{s_l} \left(S^{W_l}h- W_l \int hd\mu\right)
\end{equation}
converges weakly to a mixture of centered normal distributions as $l\to \infty$.
\end{theorem}
\begin{proof} First note that by the assumptions Theorem \ref{theo:weak-convergence} applies so that the measure $\mu$ of maximal entropy is approximable by periodic point measures. The proof now follows applying Proposition \ref{prop:9.1}.
\end{proof}
\begin{remark} (1) Condition (\ref{eq:9.32}) of the above theorem may be replaced by the assumption that
\begin{equation}\label{eq:9.40}
\frac 1{s_l} \left(\mathbb E_\mu(S^{W_l} h|\alpha_l)- W_l \int h d\mu\right)
\end{equation}
converges to some normal distribution. Moreover, if
$$\frac{1}{s_l^2}{\sigma^2_\mu(S^{W_l} h|\alpha_l)}\to 1,$$
then \eqref{eq:9.36} converges to the standard normal distribution.
(2) It seems to be a difficult problem to show asymptotic normality for the functions in (\ref{eq:9.40}). One needs to prove a CLT under Lindeberg conditions for locally constant functions. If the dynamics is Gibbs-Markov (\cite{AarDen01}) the CLT in \cite{DenSenZha17} is not applicable, since the Lipschitz norm of the conditional expectation $\mathbb E_\mu(\cdot|\alpha_l)$ generally is of exponential growth as $l\to \infty$. In this case it seems to be more promising to use concepts of probabilistic mixing.
(3) The conditions of Theorem \ref{theo:8.2} are natural for functions for which one wants to study local fluctuation of ergodic sums. It seems to be quite possible to weaken condition (\ref{eq:9.30}) since a uniform speed of convergence in the Berry-Esseen type theorem might be more than sufficient to get the result.
(4) Nontrivial examples of functions which satisfy the assumptions of the theorem
may be constructed as in \cite{BurDen87}.
\end{remark}
\section{The decomposition theorem for fluctuations}\label{sec:decomposition}
Here we are considering a positively expansive and topologically mixing dynamical system $(X,T)$ with the local specification property. As a consequence, $(X,T)$ also fulfills the global specification property, and hence the fluctuation of an ergodic sum has at least two competing sources of randomness demonstrated by Theorems \ref{theo:1} and \ref{theo:2}. Both effects may be present for a given ergodic sum. The global specification determines a global CLT while the local one determines many local CLT's determined by sequences of open sets.
The effect with the fastest growing variances dominates the CLT. If variances have asymptotic equivalent growth rates the resulting limit distribution will be a mixture of Gaussian distributions. In Section \ref{sec:mme} we present such an application.
As a result one needs to study the behavior of the variance of ergodic sums and to decompose it according to different sources of fluctuation. This is the content of this section.
Let $\epsilon>0$, $\mathcal U$ be a finite collection of open sets and $k,n \in \mathbb N$. Let $\alpha=\bigvee_{i=0}^{k-1}T^{-in}\mathcal U$. Recall that any locally $\epsilon$-independent set $\mathcal P\subset P_{kn}$ is associated to a product set $F_{\mathcal P}\subset X^k$ by a bijection $\Phi$.
\begin{definition}\label{def:var} The variation by periodic orbits over $\mathcal U$ of a function $h:X\to \mathbb R$ is defined by
$$ \mbox{\rm Var}_{\text{\rm per}}(h)= \min\left\{ \sum_{\mathbf x\in F_{\mathcal P}}\sum_{i=1}^k \left(h(T^{(i-1)n}\Phi(\mathbf x))- h(x_i)\right)^2\right\},$$
where the minimum is taken over all possible choices of $F_{\mathcal P}$ as given above.
\end{definition}
We denote by $\Pi(\mathcal U)$ the collection of all $F_{\mathcal P}=\prod_{i=1}^k E_i$ where the minimum in Definition \ref{def:var} is attained.
\begin{definition}\label{def:local}
The local variation over $\mathcal U$ of a function $h:X\to \mathbb R$ is defined as
$$ \mbox{\rm Var}_{\text{\rm loc}}(h)= \min\left\{\sum_{i=1}^k \sum_{x_i\in E_i}\left( h(x_i)- \mathbb E_{E_i}(h)\right)^2: F_{\mathcal P}\in \Pi(\mathcal U)\right\}.$$
\end{definition}
We say that $F_{\mathcal P}\in\Pi(\mathcal U)$ is CLT-admissible (for a function $h$) if it minimizes the expression in Definition \ref{def:local} (and \ref{def:var} as well).
\begin{definition}\label{def:Holder}
The H\"older variation over $\mathcal U$ and $F_{\mathcal P}\in \Pi(\mathcal U)$ of a function $h:X\to \mathbb R$ is defined by
$$ \mbox{\rm Var}_{\text{\rm H\"ol}}= |F_{\mathcal P}|\sum_{i=1}^k \left(\mathbb E_{E_i}(h)- \mathbb E_{P_{kn}}(h)\right)^2.$$
\end{definition}
\begin{definition}\label{def:global} The total variation over $\mathcal U$ of a function $h:X\to\mathbb R$ is defined as
$$ \mbox{\rm Var}_{\text{\rm tot}} = \sum_{p\in \mathcal P} \sum_{i=1}^k \left(h(T^{(i-1)n}p) - \mathbb E_{P_{kn}}(h)\right)^2.$$
\end{definition}
\begin{theorem}\label{theo:7.5} Let $h:X\to \mathbb R$ be a function. Then for any CLT-admissible $F_{\mathcal P}\in\Pi(\mathcal U)$ we have
$$ \mbox{\rm Var}_{\text{\rm tot}} = \mbox{\rm Var}_{\text{\rm per}} +\mbox{\rm Var}_{\text{\rm loc}}+ \mbox{\rm Var}_{\text{\rm H\"ol}} + 2 \mbox{\rm Cov}_{\text{\rm per},\text{\rm loc}+ \text{\rm H\"ol}},$$
where
$$\mbox{\rm Cov}_{\text{\rm per},\text{\rm loc}+ \text{\rm H\"ol}}= \sum_{p\in \mathcal P} \sum_{i=1}^k \left(h(T^{(i-1)n}(p)) - h(x_i(p))\right)\left(h(x_i(p))- \mathbb E_{P_{kn}}(h)\right).$$
Moreover,
$$ \mbox{\rm Cov}_{\text{\rm per},\text{\rm loc}+ \text{\rm H\"ol}} \leqslant \sqrt{ \mbox{\rm Var}_{\text{\rm per}}(\mbox{\rm Var}_{\text{\rm loc}}+\mbox{\rm Var}_{\text{\rm H\"ol}})}.$$
\end{theorem}
\begin{proof}
\begin{align*}
\mbox{\rm Var}_{\text{\rm tot}} &= \sum_{p\in \mathcal P}\sum_{i=1}^k \left( h(T^{(i-1)n}(p)) -
h(x_i(p)) +h(x_i(p))- \mathbb E_{E_i}(h)\right.\\
&\left. \quad + \mathbb E_{E_i}(h) - \mathbb E_{P_{kn}}(h)\right)^2\\
&= \mbox{\rm Var}_{\text{\rm per}}+\mbox{\rm Var}_{\text{\rm loc}}+\mbox{\rm Var}_{\text{\rm H\"ol}}\\
&\quad +2\sum_{p\in \mathcal P} \sum_{i=1}^k \left(h(T^{(i-1)n}(p)) - h(x_i(p))\right)\left(h(x_i(p))- \mathbb E_{P_{kn}}(h)\right)\\
&\quad + 2 \sum_{p\in \mathcal P} \sum_{i=1}^k \left(h(x_i(p))- \mathbb E_{E_i}(h)\right)\left(\mathbb E_{E_i}(h)- \mathbb E_{P_{kn}}(h)\right).
\end{align*}
Since the fifth summand vanishes the theorem follows.
\end{proof}
\begin{remark} A similar decomposition of the total variation can be obtained for systems with the global specification property.
\end{remark}
\section*{Acknowledgments} The research was supported by {\it n\'umero 158/2012 de Pe\-squisador Visitante Especial de CAPES}. M.D and X.Z also thank the Center of Mathematical Sciences of Huazhong University of Science and Technology for the support when the paper was finalized. S.S was supported by the CNPq. X.Z was supported by PNPD/CAPES and then by Fapesp grant \#2018/15088-4.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 333 |
Grace Mc Callum (Cambridge, 30 de octubre de 2002) es una deportista estadounidense que compite en gimnasia artística.
Participó en los Juegos Olímpicos de Tokio 2020, obteniendo una medalla de plata en la prueba por equipos (junto con Simone Biles, Jordan Chiles y Sunisa Lee).
Ganó dos medallas de oro en el Campeonato Mundial de Gimnasia Artística, en los años 2018 y 2019, ambas en el concurso por equipos.
Palmarés internacional
Referencias
Gimnastas artísticos de Estados Unidos
Gimnastas de Estados Unidos en los Juegos Olímpicos de Tokio 2020
Medallistas olímpicos de plata de gimnasia
Medallistas olímpicos de plata de Estados Unidos
Medallistas olímpicos de plata de Tokio 2020
Campeones mundiales de gimnasia artística
Nacidos en Cambridge (Minnesota) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,217 |
Q: How to execute SQL statement in Pipeline script of Jenkins? I have created on pipeline and I want to execute one sql query in it. I have written following statements(Only two Lines of code, no imports/Class etc.) and throws error while executing it.
import groovy.sql.Sql
def sql = Sql.newInstance("jdbc:mysql://myIP:3306/dbName", "uname","password", "com.mysql.jdbc.Driver")
sql.execute "select count(*) from TableName"
I am getting this error
org.jenkinsci.plugins.scriptsecurity.sandbox.RejectedAccessException: Scripts not permitted to use staticMethod groovy.sql.Sql newInstance java.lang.String java.lang.String java.lang.String java.lang.String
at org.jenkinsci.plugins.scriptsecurity.sandbox.whitelists.StaticWhitelist.rejectStaticMethod(StaticWhitelist.java:174)
at org.jenkinsci.plugins.scriptsecurity.sandbox.groovy.SandboxInterceptor.onStaticCall(SandboxInterceptor.java:142)
at org.kohsuke.groovy.sandbox.impl.Checker$2.call(Checker.java:180)
at org.kohsuke.groovy.sandbox.impl.Checker.checkedStaticCall(Checker.java:177)
at org.kohsuke.groovy.sandbox.impl.Checker.checkedCall(Checker.java:91)
at com.cloudbees.groovy.cps.sandbox.SandboxInvoker.methodCall(SandboxInvoker.java:15)
at WorkflowScript.run(WorkflowScript:3)
at ___cps.transform___(Native Method)
at com.cloudbees.groovy.cps.impl.ContinuationGroup.methodCall(ContinuationGroup.java:55)
at com.cloudbees.groovy.cps.impl.FunctionCallBlock$ContinuationImpl.dispatchOrArg(FunctionCallBlock.java:106)
at com.cloudbees.groovy.cps.impl.FunctionCallBlock$ContinuationImpl.fixArg(FunctionCallBlock.java:79)
at sun.reflect.GeneratedMethodAccessor841.invoke(Unknown Source)
at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
at java.lang.reflect.Method.invoke(Method.java:606)
at com.cloudbees.groovy.cps.impl.ContinuationPtr$ContinuationImpl.receive(ContinuationPtr.java:72)
at com.cloudbees.groovy.cps.impl.ConstantBlock.eval(ConstantBlock.java:21)
at com.cloudbees.groovy.cps.Next.step(Next.java:58)
at com.cloudbees.groovy.cps.Continuable.run0(Continuable.java:154)
at org.jenkinsci.plugins.workflow.cps.SandboxContinuable.access$001(SandboxContinuable.java:19)
at org.jenkinsci.plugins.workflow.cps.SandboxContinuable$1.call(SandboxContinuable.java:33)
at org.jenkinsci.plugins.workflow.cps.SandboxContinuable$1.call(SandboxContinuable.java:30)
at org.jenkinsci.plugins.scriptsecurity.sandbox.groovy.GroovySandbox.runInSandbox(GroovySandbox.java:108)
at org.jenkinsci.plugins.workflow.cps.SandboxContinuable.run0(SandboxContinuable.java:30)
at org.jenkinsci.plugins.workflow.cps.CpsThread.runNextChunk(CpsThread.java:164)
at org.jenkinsci.plugins.workflow.cps.CpsThreadGroup.run(CpsThreadGroup.java:277)
at org.jenkinsci.plugins.workflow.cps.CpsThreadGroup.access$000(CpsThreadGroup.java:77)
at org.jenkinsci.plugins.workflow.cps.CpsThreadGroup$2.call(CpsThreadGroup.java:186)
at org.jenkinsci.plugins.workflow.cps.CpsThreadGroup$2.call(CpsThreadGroup.java:184)
at org.jenkinsci.plugins.workflow.cps.CpsVmExecutorService$2.call(CpsVmExecutorService.java:47)
at java.util.concurrent.FutureTask.run(FutureTask.java:262)
at hudson.remoting.SingleLaneExecutorService$1.run(SingleLaneExecutorService.java:112)
at jenkins.util.ContextResettingExecutorService$1.run(ContextResettingExecutorService.java:28)
at java.util.concurrent.Executors$RunnableAdapter.call(Executors.java:471)
at java.util.concurrent.FutureTask.run(FutureTask.java:262)
at java.util.concurrent.ThreadPoolExecutor.runWorker(ThreadPoolExecutor.java:1145)
at java.util.concurrent.ThreadPoolExecutor$Worker.run(ThreadPoolExecutor.java:615)
at java.lang.Thread.run(Thread.java:745)
Finished: FAILURE
Pls help. Thanks in Advance.
A: From the error it looks like you are running the script in sandbox mode, in which case there are a lot of limitation what you can execute in the script. For example the following wont work neither:
echo binding.hasVariable("test")
The things you can do:
*
*If you are not hosting the Jenkins yourself you cannot do much from the pipeline script itself, you have to create a separate process to do the select you want to do and pass back the result to the pipeline script. You can do this by adding a groovy script to the slave and executing it from the pipeline, for example.
All scenarios below assume that you are hosing Jenkins yourself
*
*If you are using Pipeline script definition you have the option to run the script not in sandbox mode (uncheck Use Groovy Sandbox in the job configuration page). In this case if you are an admin it should work just fine. If you are not an admin follow the advice below
*If you are using Pipeline script from SCM definition, then the script will be executed in sandbox mode and you will also encounter the error you posted. Then an admin needs to go to Manage Jenkins » In-process Script Approval and approve the method call that was denied (just press the Approve button)
Note: The above was tested on Jenkins version: 2.7.1
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,517 |
The Crossroad Publishing Company and Jade Music are proud to present:
**THE RULE OF BENEDICT**
A Spirituality for the 21st Century
Joan Chittister, O.S.B.
A classic text with commentary, images, and sound.
To download the selection of accompanying Gregorian chants, please go to: www.milanrecords.com/crossroadruleofbenedict
_The Crossroad Publishing Company_ offers a 200-year global family tradition of publishing on spiritual living and religious thought. Offering books by authoritative and award-winning authors, our catalog includes original editions, monographs, commentary, reflections, prayers, guides, and spiritual fiction. We promote reading as a time-tested discipline for focus and enlightenment. We help authors shape, clarify, write, and effectively promote their ideas. We conceive, create, and distribute books. Our expertise and passion is to provide healthy spiritual nourishment in written form.
_Jade Music,_ the specialty division of Milan Records, has a proven dedication to releasing quality classical and spiritual music for more than 20 years. It established itself as the premiere record label of the Abbey of Santo Domingo de Silos, Spain. Jade Music's catalog includes works by world-renowned choirs of Saint-Wandrille, Notre-Dame de Ganagobie, and Saint-Madeleine du Barroux, among others. In 2007, the record label distinguished itself by releasing the soundtrack to the highly acclaimed documentary film _Into Great Silence_ by Philip Gröning. More recently, Jade Music released two Gregorian chant albums by the Norbertine Fathers of Orange County, California.
For this edition, numerous people have shared their talents and ideas, and we gratefully acknowledge Sr. Chittister and the Benedictine Sisters of Erie, Pennsylvania who have been most gracious during the course of our cooperation. We thank especially:
John Farina, initial idea and acquisition of this book
George Foster, cover design
Eve Vaterlaus, layout and image composition
Scribe, Inc., copyediting, proofreading, and file management
Versa Press, Inc., book production
Concept and idea for this edition, art research and acquisition, and project management by The Crossroad Publishing Company.
This printing: December 2016
The Crossroad Publishing Company
www.CrossroadPublishing.com
Copyright © 1992, 2010 by Joan Chittister
All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written permission of The Crossroad Publishing Company.
The compilation and composition of the artwork and music accompanying The Rule of Benedict is an original work owned by The Crossroad Publishing Company, and may not in any way be copied, used with other texts or excerpted.
The stylized crossed letter C logo is a registered trademark of The Crossroad Publishing Company.
In continuation of our 200-year tradition of independent publishing, The Crossroad Publishing Company proudly offers a variety of books with strong, original voices and diverse perspectives. The viewpoints expressed in our books are not necessarily those of The Crossroad Publishing Company, any of its imprints, or of its employees. No claims are made or responsibility assumed for any health or other benefit.
Originally published in 1992 in slightly different form and without art and music.
Library of Congress Cataloging-in-Publication Data
Chittister, Joan.
The rule of Benedict : insights for the ages / Joan D. Chittister.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-8245-2594-1 (alk. paper)
1. Benedict, Saint, Abbot of Monte Cassino. Regula. 2. Monasticism and religious orders—Rules. I. Title.
BX3004.Z5C35 2010
255'.106—dc2
2010019403
Printed in The United States of America
Books published by The Crossroad Publishing Company may be purchased at special quantity discount rates for classes and institutional use. For information, please e-mail info@CrossroadPublishing.com.
CONTENTS
AN INVITATION
INTRODUCTION
PROLOGUE
1. THE KINDS OF MONASTICS
2. QUALITIES OF THE ABBOT OR PRIORESS
3. SUMMONING THE COMMUNITY FOR COUNSEL
4. THE TOOLS FOR GOOD WORKS
5. OBEDIENCE
6. RESTRAINT OF SPEECH
7. HUMILITY
8. THE DIVINE OFFICE AT NIGHT
9. THE NUMBER OF PSALMS AT THE NIGHT OFFICE
10. THE ARRANGEMENT OF THE NIGHT OFFICE IN SUMMER
11. THE CELEBRATION OF VIGILS ON SUNDAY
12. THE CELEBRATION OF THE SOLEMNITY OF LAUDS
13. THE CELEBRATION OF LAUDS ON ORDINARY DAYS
14. THE CELEBRATION OF VIGILS ON THE ANNIVERSARIES OF SAINTS
15. THE TIMES FOR SAYING ALLELUIA
16. THE CELEBRATION OF THE DIVINE OFFICE DURING THE DAY
17. THE NUMBER OF PSALMS TO BE SUNG AT THESE HOURS
18. THE ORDER OF THE PSALMODY
19. THE DISCIPLINE OF PSALMODY
20. REVERENCE IN PRAYER
21. THE DEANS OF THE MONASTERY
22. THE SLEEPING ARRANGEMENTS OF MONASTICS
23. EXCOMMUNICATION FOR FAULTS
24. DEGREES OF EXCOMMUNICATION
25. SERIOUS FAULTS
26. UNAUTHORIZED ASSOCIATION WITH THE EXCOMMUNICATED
27. THE CONCERN OF THE ABBOT AND PRIORESS FOR THE EXCOMMUNICATED
28. THOSE WHO REFUSE TO AMEND AFTER FREQUENT REPROOFS
29. READMISSION OF MEMBERS WHO LEAVE THE MONASTERY
30. THE MANNER OF REPROVING THE YOUNG
31. QUALIFICATIONS OF THE MONASTERY CELLARER
32. THE TOOLS AND GOODS OF THE MONASTERY
33. MONASTICS AND PRIVATE OWNERSHIP
34. DISTRIBUTION OF GOODS ACCORDING TO NEED
35. KITCHEN SERVERS OF THE WEEK
36. THE SICK
37. THE ELDERLY AND THE YOUNG
38. THE READER FOR THE WEEK
39. THE PROPER AMOUNT OF FOOD
40. THE PROPER AMOUNT OF DRINK
41. THE TIMES FOR MEALS
42. SILENCE AFTER COMPLINE
43. TARDINESS AT THE OPUS DEI OR AT TABLE
44. SATISFACTION BY THE EXCOMMUNICATED
45. MISTAKES IN THE ORATORY
46. FAULTS COMMITTED IN OTHER MATTERS
47. ANNOUNCING THE HOURS FOR THE OPUS DEI
48. THE DAILY MANUAL LABOR
49. THE OBSERVANCE OF LENT
50. MEMBERS WORKING AT A DISTANCE OR TRAVELING
51. MEMBERS ON A SHORT JOURNEY
52. THE ORATORY OF THE MONASTERY
53. THE RECEPTION OF GUESTS
54. LETTERS OR GIFTS
55. CLOTHING AND FOOTWEAR
56. THE PRIORESS'S OR ABBOT'S TABLE
57. THE ARTISANS OF THE MONASTERY
58. THE PROCEDURE FOR RECEIVING MEMBERS
59. THE OFFERING OF CHILDREN BY NOBLES OR BY THE POOR
60. THE ADMISSION OF PRIESTS TO THE MONASTERY
61. THE RECEPTION OF VISITING MONASTICS
62. THE PRIESTS OF THE MONASTERY
63. COMMUNITY RANK
64. THE ELECTION OF A PRIORESS OR ABBOT
65. THE PRIOR AND SUBPRIORESS OF THE MONASTERY
66. THE PORTER OF THE MONASTERY
67. MEMBERS SENT ON A JOURNEY
68. ASSIGNMENT OF IMPOSSIBLE TASKS
69. THE PRESUMPTION OF DEFENDING ANOTHER IN THE MONASTERY
70. THE PRESUMPTION OF STRIKING ANOTHER MONASTIC AT WILL
71. MUTUAL OBEDIENCE
72. THE GOOD ZEAL OF MONASTICS
73. THIS RULE ONLY A BEGINNING OF PERFECTION
ACKNOWLEDGMENTS
_This book is dedicated to all the past prioresses of my own community, who interpreted the Rule for us in every age of the community's history, to Sister Phyllis Schleicher, the prioress of the community, whose insight and insistence make this work possible now, and in particular to those prioresses who made the Rule real and holy to me in the various periods of my own life:_
Sister Sylvester Groner, O.S.B., 1946–1958
Sister Alice Schierberl, O.S.B., 1958–1964
Sister Mary Margaret Kraus, O.S.B., 1964–1978
AN INVITATION
The Rule of Benedict is a spiritual guide, rare by virtue of its ancient origins, valued for its continuing meaningfulness in every century since. It has weathered every period of Western history since the fall of the Roman Empire and been a dynamic source of light and energy to each. Surely someone ought to ask, How is it that anything can last that long and still be considered viable in ages so distant from its own? Someone ought to care why it is that this way of life has been found to be both holy and helpful, whatever the social changes of the era, whatever the pitfalls of the time. Someone ought to wonder, then, in what directions, if any, would this Rule point our own lives in a period in which every system in the Western world—government, economics, family, social values, and human relationships—is once again in flux?
On the one hand, this is a book about a document and a lifestyle that is over 1,500 years old but which in every era becomes increasingly more important as we fumble and stumble our way toward fullness of life in a world whose foundations are shaking.
On the other hand, this is also a book being written for a culture whose mantra is "progress" and whose character is change. We do not, as a people, often set out to preserve our past as have cultures before us.
Generations and cultures before us, for instance, made walls, houses, carriages and furniture to defy time forever. We have become a throwaway society. Everything our world creates, geared to maintaining a manufacturing economy, is timed to become useless: the tires on our cars, the heating elements in our microwaves, the motherboards in our televisions and phones and computers and MP3 players that connect us to the world around us. They all wear out on schedule; they are all made to run down in time for us to buy the next version of them. However much any of them cost, they live measured lives. They are built to be thrown away long before the equipment they power has outlived its usefulness.
We have, in fact, become a culture that conditions people to wait for what's coming next. So we go through life, tiring easily both of what was and what is, living in expectation of what is to come. We think of what went before us as "old fashioned."
Or to put it another way, this highly technological culture of ours has learned to look more to the future for answers to the great questions of life than it does to the past. The version to come, we know—we have been trained to expect—will be more vigorous, more effective than anything of its type that has gone before it, even though we know that it rarely ever is.
We forget that, new as the future may be, its value will depend entirely on what we bring to it ourselves. It will depend fundamentally for its character, its value, on what happens to us as we grow into it. Its quality will depend more on what is in us when we get there than on what is in it, however new.
In a consumer society, in a society whose economy is based on planned obsolescence for its financial stability, this movement from one continual "upgrade" of a thing to the next has managed to blur for us the difference between what is passing and what must perdure. Social systems may change, for instance, but the justice that cements them must endure. As mobility increases, the nature of personal relationships is changing but the ability to live well, to love rightly, must endure. Our knowledge of the cosmos and evolution is challenging some of our standard spiritual truisms but the search for God and the discernment of the Godly life must endure if society itself is to endure.
Which is where this book comes in.
The fact is that not everything has failed us over the centuries. There are things that remain from one culture to another with substance enough, with pith enough, to lead us through the dark days and difficult questions and questionable social systems of our own times. The insights of great thinkers, the model of great figures, the wisdom of the great spiritual traditions of all times root us in the marrow of the past while they go on pointing a way through the challenges of an evolving future.
This commentary, then, looks with respect and amazement at a document and a lifestyle that has been part of Western development for over 1,500 years and under which thousands of people around the world purport to live even now. It asks the question, What meaning, if any, can this Rule possibly have for average people of our own day who grapple daily with a culture awash in the transitory and the tenuous, in superficiality and confusion?
The answers to those questions are fairly simple ones: in the first place, the Rule of Benedict is not historical literature, it is wisdom literature. In the second place, it teaches what this world, what every culture, needs most. Especially, perhaps, in our own time.
Wisdom literature endures precisely because it is not the history of a particular people, it is not the codification of the ethical mores of a single culture, it is not the teachings of science, it is not, in fact, devoted to the presentation of any particular body of knowledge. Wisdom literature takes as its subject matter the meaning and manner of achieving the well-lived life. It deals with the spiritual, the ascetic, the Divine, and the nature of virtue. Its concerns lie in the meaning of holiness and the fundaments of happiness.
Wisdom literature is common to every great tradition. It lifts the spiritual life from the legal to the mystical, from theology to spirituality, from a study of the nature of religion to the depths of the personal spiritual life. In Hinduism, the basic outlines of the spiritual life are found in texts like the Upanishads; in Buddhism, in the Dhammapada; in Judaism, in the books of Proverbs and Job, of Ecclesiastes and Wisdom; in Islam, in the writings of the Sufis; and in Christianity, in the Apothegmata or the writings of the Desert Monastics. Out of wells of wisdom like these have sprung the teachings of the great mystics and spiritual directors of every age and every tradition. It will not be surprising, then, to find that there are bits of wisdom from all of these sources included in this commentary, in this presentation of the Rule of Benedict, a work of ancient Western wisdom literature, where the concerns of every tradition meet.
It is this concentration on meaning and purpose in life that—in those systems in which time proves true—endures, outlasts, lives on through the ages. It is this wisdom that drew people to Benedictinism in the sixth century when the Roman Empire had lost its center. It is this wisdom that drew people to monastic centers in the Dark Ages when Benedictine monasteries provided the only communal, civic, and social systems that existed. It is respect for this wisdom that made Benedictine monasteries the spiritual center of every village, every major city in medieval Europe. And it is this wisdom—this deep, heartful human presentation of the core of the spiritual life—that draws thousands upon thousands of people to this Rule yet today.
More than that, perhaps, the foundations of the Benedictine way of life that this Rule preserves are based on the very foundations that the modern world most lacks but, at the same time, most needs.
To a world fragmented by transience and distance, the Rule of Benedict stresses the need and nature of real community.
To a world dry to the core with secularism, the Rule of Benedict brings the rhythm and ointment of prayer. Prayer in the monastic tradition is always psalm centered—always the cry of the universal human soul down one age and up the other. It is also always in tune with the turning of the liturgical year and the tender, haunting, mystical chants of a praying church everywhere.
To a world that has, to the peril of both, severed human life from the creation that sustains it, the Rule of Benedict brings a new respect for the seasons of life and the stewardship of the world.
To a world torn apart by random and state violence, the Rule of Benedict brings a life based on the equality and reverence that a world in search of peace requires.
To a world where arrogance separates the developed from the "underdeveloped" by assuming that one has the right to the basics of life while the other must exist on less, the Rule of Benedict requires the development of the kind of humility that makes none of us subject to the whims of the rest of us.
To a world where people work for money, the Rule of Benedict requires that we work to continue the will of God for all of creation.
To a world where leisure has been reduced to aimlessness, the Rule of Benedict provides a sense of contemplation, the fruits of which reflection enable us to see the world as God sees the world.
Indeed, Benedictine spirituality is the spirituality for the twenty-first century.
The basic contentions of this book, then, are clearly two: first, that Benedictine spirituality deals with the issues facing us now—stewardship, relationships, authority, community, balance, work, simplicity, prayer, and spiritual and psychological development. Its strength, therefore, is that it is both fresh and ancient, current and tried at the same time. Second, its currency lies in the fact that Benedictine spirituality offers more a way of life and an attitude of mind than it does a set of religious prescriptions. The Benedictine way of life, after all, is credited with having saved Christian Europe from the ravages of the Dark Ages. In an age bent again on its own destruction, the world could be well served by asking how so simple a system could possibly have contributed so complex a thing as that.
The Rule of Benedict is not a treatise in systematic theology. Its logic is the sagacity of daily life lived in Christ and lived well. This commentary simply takes the work as it is, a paragraph at a time, and attempts to discover under the crusts of language and time, the concept being treated there and its meaning to us now. It attempts to explain, for instance, why Benedict dealt with the role and functions of a porter or doorkeeper of a monastery at all and what that might have to say to the way we ourselves deal with the world outside us.
Benedict of Nursia emerges from that rare stream of thinkers who lived out of a single tradition but from the perspective of universal and fundamental insights into life. In the Rule we profit from all of them.
The real function of the book is to make an ancient document accessible to a modern reading public who are searching, often in vain, for some spiritual framework around which to organize their own lives in a period when regular public devotions are largely a thing of the past and the overarching questions of life are more pressing than ever. I was particularly concerned that women find a welcome in this text, since they have lived the Rule for as many centuries as men. Thus, I have edited the Rule to read in universal language rather than in the usual male form. The purpose is to demonstrate clearly not only that spirituality is neither a male prerogative nor the norm of its existence but rather that the Rule has guided the lives of both women and men from its inception. The subject matter of the book is Benedictine spirituality, a spirituality for all humankind.
Benedict of Nursia was born in the year 480. As a student in Rome, he tired of the decadent culture around him and left it to live a simple spiritual life as a hermit in the countryside of Subiaco, about thirty miles away. It wasn't long, however, before he was discovered both by the people of the area and disciples who were themselves looking for a more meaningful way of life. Out of these associations sprang the monastic life that would eventually cover Europe.
We do not know much more than that about the life of Benedict of Nursia but we know enough about the history of the times to know in what ways his Rule departed from it, gave the world fresh eyes and called people to live life with a new heart. The Dialogues of Gregory, in the biographical style of the day, give us, in a set of monastic parables about Benedict, an outline of his spiritual qualities and a look into his personality and leadership gifts. Other than that, it is the perdurance of his Rule itself and the contributions of Benedictinism to all the periods of Western history that speak most clearly to us about the purpose and impact of the life outlined here in the midst of a superficial world.
In our own day there are over 1,500 communities of roughly 30,000 Benedictine and Cistercian men and women around the world who live under this Rule. In addition to the professed monastics who follow the Benedictine way of life, however, there are innumerable laypersons around the globe who also find in the Rule a guide and a ground for their own lives in the middle of a chaotic and challenging world. With that in mind, the text is divided into dated reading segments to allow for the three readings that Benedict prescribes for students of the Rule in chapter 58. It is for these people, primarily, and for people like them that this book has been written in order to keep accessible a text that has been a lifeline for many across the ages.
Clearly, this is a life we do not dare to throw away with all the other disposables we have been trained to discard in our time if life as we know it is to be as rich in the future as it has been in the past.
These reflections, in fact, come out of more than fifty years of my own personal experience of monastic life, twelve of which I spent as prioress of a Benedictine monastery and eight as president of a Benedictine federation of twenty-three autonomous communities.
Thanks to the quality of this Rule, that life has had both dynamism and depth for me. It has plumbed both my humanity and my spirituality. It has brought both psychological growth and spiritual insight. It is as worth living now as it was those long years ago when, at a very early age, I began the living of it. It is a life I would not only recommend to others but would beg women and men everywhere to consider in all its forms, both ancient and new, not only as life giving to the individuals who choose it, but also as gift to the culture wise enough to seek it.
My hope is that those who read this book, and use "this little Rule for beginners," may also find new hope and new meaning in this century and so themselves become a light to centuries to come.
INTRODUCTION
The Rule of Benedict, ancient as it is, has a very subtle power and a very serious problem, as well: it is extremely simple to read. There is nothing convoluted about it, nothing metaphysical. On the contrary. The Rule of Benedict is direct; it is clear; it is a relatively uncomplicated text that uses simple language to make simple references to simple things that have meaning even now after 1,500 years. As a result, it is difficult to miss what is being said in it. There is little wonder it has lasted so long.
At the same time, because it is so unvarnished, so uncomplicated in its structure, so simple in its concepts, it is also fairly easy to discount its concern for early sixth-century agendas and fairly difficult to recognize its continuing value. It's an essentially straightforward, clean-spoken document, true, but not always very relevant, it seems, to twenty-first century culture and lifestyles. To readers who have inherited the mysticism of the Middle Ages, the treatises of the scholastic philosophers, and the theology texts of centuries of church life, it is almost incomprehensible that this brief document, almost 1,500-years old, is now enshrined as one of the greatest spiritual handbooks of all time. Volumes have been written about it but the small, unassuming text itself is almost bound to be disappointing to a culture that likes things to sound impressive and to look slick.
What is it, then, that the Rule of Benedict says to the sixth century that gives it not only the right but the need to be heard by the twenty-first century as well? What is it about the Rule of Benedict that stays both authentic and necessary century after century after century in culture after culture after culture?
The answer surely lies more in the ideas with which it concerns itself and the attitudes it sets out to form than in the particulars it prescribed for the people who were reading it in early Europe.
The Rule of Benedict is not concerned with a single time and place, a single view of church, a single set of devotions or a single ministry. The Rule of Benedict is concerned with life: what it's about, what it demands, how to live it. And it has not failed a single generation.
The Prologue to the rule is its cornerstone and its gauntlet. Read this, the rule says, and if this is not what you're about, do not read on.
PROLOGUE
_Jan. 1 – May 2 – Sept. 1_
_Listen carefully, my child, to my instructions, and attend to them with the ear of your heart. This is advice from one who loves you; welcome it and faithfully put it into practice. The labor of obedience will bring you back to God from whom you had drifted through the sloth of disobedience. This message of mine is for you, then, if you are ready to give up your own will, once and for all, and armed with the strong and noble weapons of obedience to do battle for Jesus, the Christ._
Life is a teacher of universal truths. That may be the reason why the religious readings of so many nations speak of the same situations and fasten on the same insights. The Rule of Benedict, too, is a wisdom literature that sounds life's themes. It deals with answers to the great questions of the human condition: the presence of God, the foundation of relationships, the nature of self-development, the place of purpose. To the wise, it seems, life is not a series of events to be controlled. Life is a way of walking through the universe whole and holy.
This first paragraph of the Rule of Benedict brings into instant focus the basis for being able to do that.
Benedict says, "Listen." Pay attention to the instructions in this Rule and attend to the important things in life. Let nothing go by without being open to being nourished by the inner meaning of that event in life. There is an Oriental proverb that teaches, "Take from death before it takes from thee." If we do not live life consciously, in other words, we may not be living at all.
The Prologue is asking us to do the same thing. If we want to have a spiritual life, we will have to concentrate on doing so. Spirituality does not come by breathing. It comes by listening to this Rule and to its insights into life "with the ear of the heart," with feeling, with more than an academic interest.
One part of spirituality, then, is learning to be aware of what is going on around us and allowing ourselves to feel its effects. If we live in an environment of corporate greed or personal violence, we can't grow from it spiritually until we allow ourselves to recognize it. The other part of spirituality, the Prologue makes quite clear, is learning to hear what God wants in any given situation and being quick to respond to that, to "welcome it and faithfully put it into practice." To see the greed or sense the violence without asking what the Gospel expects in such a situation is not spirituality. It is piety at best.
Most important of all, perhaps, is the Prologue's insistence that this Rule is not being written by a spiritual taskmaster who will bully us or beat us down in a counterfeit claim to growing us up but by someone who loves us and will, if we allow it, carry us along to fullness of life. It is an announcement of profound importance. No one grows simply by doing what someone else forces us to do. We begin to grow when we finally want to grow. All the rigid fathers and demanding mothers and disapproving teachers in the world cannot make up for our own decision to become what we can by doing what we must.
In this very first paragraph of the Rule, Benedict is setting out the importance of not allowing ourselves to become our own guides, our own gods. Obedience, Benedict says—the willingness to listen for the voice of God in life—is what will wrench us out of the limitations of our own landscape. We are being called to something outside of ourselves, something greater than ourselves, something beyond ourselves. We will need someone to show us the way: the Christ, a loving spiritual model, this Rule.
_First of all, every time you begin a good work, you must pray to God most earnestly to bring it to perfection. In God's goodness, we are already counted as God's own, and therefore we should never grieve the Holy One by our evil actions. With the goodgifts which are in us, we must obey God at all times, that God may never become the angry parent who disinherits us, nor the dreaded one, enraged by our sins, who punishes us forever as worthless servants for refusing to follow the way to glory._
The person who prays for the presence of God is, ironically, already in the presence of God. The person who seeks God has already found God to some extent. "We are already counted as God's own," the Rule reminds us. Benedict knows this and clearly wants us to know it as well. A dull, mundane life stays a dull, mundane life, no matter how intent we become on developing spiritually. No amount of churchgoing will change that. What attention to the spiritual life does change is our appreciation for the presence of God in our dull, mundane lives. We come to realize that we did not find God; God finally got our attention. The spiritual life is a grace with which we must cooperate, not a prize to be captured or a trophy to be won.
But, the Rule implies, we have been given a grace that is volatile. To feel it and ignore it, to receive it but reject it, the paragraph suggests, is to be in a worse situation than if we had never paid any attention to the spiritual life at all. For disregard of God's good gifts, Benedict says, for refusing to use the resources we have for the upbuilding of the reign of God, for beginning what we do not intend to complete, the price is high. We are disinherited. We lose what is ours for the taking. We miss out on the life we are meant to have. We are dealt with, not as children of the owner who know instinctively that they are meant to grow into new and deeper levels of relationship here, but as hired help in the house, as people who look like they are part of the family but who never reap its real benefits or know its real nature. In failing to respond to God everywhere God is around us, we may lose the power of God that is in us.
The words were not idle metaphors in sixth-century Italy.
To be a member of a Roman family, the family whose structures Benedict understood, was to be under the religious, financial, and disciplinary power of the father until he died, whatever the age of the children. To be disinherited by the father was to be stranded in a culture in which paid employment was looked down upon. To be punished by him was to lose the security of family, outside of which there was no security at all. To lose relationship with the father was then, literally, to lose one's life.
And who has not known the truth of it? Who of us has not been failed by all the other things besides God—money, status, security, work, people—that we have clung to and been disappointed by in our cleaving? Whose life has not been warped by a series of twisted hopes, the roots of which were sunk in the shale of false promises and empty treasures that could not satisfy? Benedict is begging us here to realize that God is the only lifeline that life guarantees us. We have been loved to life by God, and now we must love God back with our whole lives or forever live a living death.
_Jan. 2 – May 3 – Sept. 2_
_Let us get up then, at long last, for the Scriptures rouse us when they say, "It is high time for us to arise from sleep" (Rom. 13:11). Let us open our eyes to the light that comes from God, and our ears to the voice from the heavens that every day calls out this charge: "If you hear God's voice today, do not harden your hearts" (Ps. 95:8). And again: "Youthat have ears to hear, listen to what the Spirit says to the churches" (Rev. 2:7). And what does the Spirit say? "Come and listen to me; I will teach you to reverence God" (Ps. 34:12). "Run while you have the light of life, that the darkness of death may not overtake you" (John 12:35)._
The paragraph is an insistent one, full of intensity, full of urgency. We put off so much in life—visiting relatives, writing letters, going back to school, finding a new job. But one thing stays with us always, present whether pursued or not, and that is the call to the center of ourselves where the God we are seeking is seeking us. Benedict says, Listen today. Start now. Begin immediately to direct your life to that small, clear voice within.
In this paragraph Benedict makes his first of the multiple allusions to Scripture that emerge in the Rule time and time again to the point that a reader gets the idea that the Rule is simply a chain of scriptural quotations. The particular passages cited are important, of course, and give emphasis to the point of the excerpt. In these first references, for instance, Benedict reminds us that life is short, that we don't have time to waste time, that some things are significant in life and some things are not. We all have to ask ourselves what time it is in our own lives. We each have to begin to consider the eternal weight of what we are spending life doing. We have to start someday to wonder if we have spent our lives on gold or dross.
But as important as the content of the scriptural quotations themselves is the very message of their presence: the life laid out in this Rule is a life based on the gospel of Jesus Christ. It is not the prescriptions of a private guru. It is an immersion in the gospel life so intense that we never forget for a moment what we are really about. We don't just stumble through life from one pious exercise to another, hoping that in the end everything will be all right. We don't surfeit on this life, even the spiritual systems of it, and forget the life to come. No, we run toward the light, not with our hair shirts in hand but with the Scriptures in hand, responsible to the presence of God in every moment and sure that life is only beginning when it ends.
_Jan. 3 – May 4 – Sept. 3_
_Seeking workers in a multitude of people, God calls out and says again: "Is there anyone here who yearns for life and desires to see good days?" (Ps. 34:13). If you hear this and your answer is "I do,"God then directs these words to you: if you desire true and eternal life, "keep your tongue free from vicious talk and your lips from all deceit; turn away from evil and do good; let peace be your quest and aim" (Ps. 34:14–15). Once you have done this, my "eyes will be upon you and my ears will listen for your prayers; and even before you ask me, I will say" to you: "Here I am" (Isa. 58:9). What is more delightful than this voice of the Holy One calling to us? See how God's love shows us the way of life. Clothed then with faith and the performance of good works, let us set out on this way, with the gospel for our guide, that we may deserve to see the Holy One "who has called us to the eternal presence" (1 Thess. 2:12)._
In Benedict's mind, apparently, the spiritual life is not a collection of asceticisms; it is a way of being in the world that is open to God and open to others. We struggle, of course, with temptations to separate the two. It is so easy to tell ourselves that we overlooked the needs of others because we were attending to the needs of God. It is so easy to go to church instead of going to a friend whose depression depresses us. It is so easy to want silence rather than the demands of the children. It is so much easier to read a book about religion than it is to listen to a husband talk about his job or a wife talk about her loneliness. It is so much easier to practice the privatized religion of prayers and penances than it is to make fools out of ourselves for the Christian religion of globalism and peace. Deep, deep spiritual traditions everywhere, however, reject those rationalizations: "Is there life after death?" a disciple once asked a Holy One. And the Holy One answered, "The great spiritual question of life is not, 'Is there life after death?' The great spiritual question is, 'Is there life before death?'" Benedict obviously believes that life lived fully is life lived on two planes: attention to God and attention to the good of the other.
The godly are those, this paragraph says, who never talk destructively about another person—in anger, in spite, in vengefulness—and who can be counted on to bring an open heart to a closed and clawing world.
The godly know when the world they live in has them on a slippery slope away from the good, the true, and the holy, and they refuse to be part of the decline. What's more striking, they set out to counter it. It is not enough, Benedict implies, simply to distance ourselves from the bad. It is not enough, for instance, to refuse to slander others; we must rebuild their reputations. It is not enough to disapprove of toxic waste; we must do something to save the globe. It is not enough to care for the poor; we must do something to stop the poverty. We must be people who bring creation to life. "Once you have done this," the Rule reminds us, "my eyes will be upon you and my ears will listen for your prayers." Once you have done these things, you will be in the presence of God.
Finally, as far as Benedict is concerned, the spiritual life depends on our being peaceful peacemakers.
Agitation drives out consciousness of God. When we're driven by agitation, consumed by fretting, we become immersed in our own agenda, and it is always exaggerated. We get caught up in things that, in the final analysis, simply don't count, in things that pass away, in things that are concerned with living comfortably rather than with living well. We go to pieces over crying children and broken machines and the length of stoplights at intersections. We lose touch with the center of things.
At the same time, a kind of passive tranquility is not the aim of Benedictine life. The call of this spirituality is to be gentle ourselves and to bring nonviolence in our wake. It is an amazing position for a sixth-century document to take in a violent world. There is no Armageddon theology here, no call to a pitched battle between good and evil in a world that subscribed to dualism and divided life into things of the spirit and things of the flesh.
In this rule of life, violence is simply discounted. Violence doesn't work. Not political violence, not social violence, not physical violence, not even the violence that we do to ourselves in the name of religion. Wars haven't worked. Classism hasn't worked. Fanaticism hasn't worked. Benedictinism, on the other hand, simply does not have as its goal either to beat the body down or to vanquish the world. Benedictinism simply sets out to gentle a universe riddled with violence by being a peaceful voice for peace in a world that thinks that everything—international relations, child rearing, economic development, even everything in the spiritual life—is accomplished by force.
Benedictinism is a call to live in the world not only without weapons raised against the other but also by doing good. The passage implies clearly that those who make God's creation their enemy simply do not "deserve to see the Holy One."
It is a strong passage clothed in words long dulled by repetition.
_Jan. 4 – May 5 – Sept. 4_
_If we wish to dwell in God's tent, we will never arrive unless we run there by doing good deeds. But let us ask with the prophet: "Who will dwell in your tent, O God; who will find rest upon your holy mountain?" (Ps. 15:1). After this question, then, let us listen well to what God says in reply, for we are shown the way to God's tent. "Those who walk without blemish and are just in all dealings; who speak truth from the heart and have not practiced deceit; who have not wronged another in any way, not listened to slanders against a neighbor" (Ps. 15:2–3). They have foiled the evil one at every turn, flinging both the devil and these wicked promptings far from sight. While these temptations were still "young, the just caught hold of them and dashed them against Christ" (Ps. 15:4, 137:9). These people reverence God, and do not become elated over their good deeds; they judge it is God's strength, not their own, that brings about the good in them. "They praise" (Ps. 15:4) the Holy One working in them, and say with the prophet: "Not to us, O God, not to us give the glory, but to your name alone" (Ps. 115:1)._
Two themes emerge very strongly here. In case the meaning of the earlier paragraphs has escaped us, Benedict repeats them.
Justice, honesty, and compassion are the marks of those who dwell with God in life, he insists. Then he reminds us again that we are not able to achieve God's grace without God's help. If we do good for the poor, it is because God has given us the courage to do good. If we speak truth in the face of lies, it is because God has given us a taste for the truth. If we uphold the rights of women and men alike, it is because God has given us eyes to see the wonders of all creation. We are not a power unto ourselves.
The two ideas may seem innocent enough today, but at the time at which Benedict wrote them they would both have had great social impact.
In the first place, physical asceticism had become the mark of the truly holy. The Fathers and Mothers of the Desert, living a spirituality that was the dominant form of religious life prior to the emergence of communal monasticism, had been known and revered for the frugality, discipline, and asceticism of their lives. They lived in the desert as solitaries. They ate little. They prayed night and day. They deprived their bodies to enrich their souls. They struggled against the temptations of the flesh and fled the world. Theirs was a privatized version of religious development not unlike those theologies that still thrive on measuring personal penances and using religion as personal massage rather than on making the world look the way God would want it to look. Benedict, then, introduces very early in the Rule the notion of responsibility for the human community as the benchmark of those who "dwell in God's tent," know God on earth, live on a higher plane than the mass of humanity around them. The really holy, the ones who touch God, Benedict maintains, are those who live well with those around them. They are just, they are upright, they are kind. The ecology of humankind is safe with them.
In the second place, Benedict puts to rest the position of the wandering monk Pelagius, who taught in the fifth century that human beings were inherently good and capable of achieving God's great presence on the strength of their own merits. Benedict wants "good deeds" but he does not want pride. We do what we do in life, even holy things, the Prologue teaches, not because we are so good but because God is so good and enables us to rise above the misery of ourselves. Even the spiritual life can become an arrogant trap if we do not realize that the spiritual life is not a game that is won by the development of spiritual skills. The spiritual life is simply the God-life already at work in us.
An obligation to human community and a dependence on God, then, become the cornerstones of Benedictine life.
_Jan. 5 – May 6 – Sept. 5_
_In just this way Paul the apostle refused to take credit for the power of his preaching. He declared: "By God's grace I am what I am" (1 Cor. 15:10). And again Paul said: "They who boast should make their boast in God" (2 Cor. 10:17). That is why it is said in the Gospel: "Whoever hears these words of mine and does them is like a wise person who built a house upon rock; the floods came and the winds blew and beat against the house, but it did not fall: it was founded on rock" (Matt. 7:24–25)._
Clearly, for Benedict, God is not something to be achieved; God is a presence to be responded to but to whom without that presence we cannot respond. God isn't something for which spiritual athletes compete or someone that secret spiritual formulas expose. God is the breath we breathe. It is thanks to God that we have any idea of God at all. God is not a mathematical formula that we discover by dint of our superior intelligence or our moral valor. God is the reason that we can reach God. It is to this ever-present Presence that the Rule of Benedict directs us. It is to God already in our lives that Benedict turns our minds. The Hasidim tell the story of the preacher who preached over and over, "Put God into your life; put God into your life." But the holy rabbi of the village said, "Our task is not to put God into our lives. God is already there. Our task is simply to realize that."
The words of the Rule are as fresh on this point as the day they were written. The fact is that we still compartmentalize God. We tell ourselves that we are working on reaching the spiritual life by saying prayers and doing penances and making pilgrimages and giving things up. And we keep score: so many daily Masses, so many rosaries, so many fast days, so many spiritual books read, so many conferences attended equal so many steps toward the acquisition of God. The Rule of Benedict sets us straight. God is with us, for the taking, but not for any spiritual payment, only for realizing what we already have.
God is neither cajoled nor captured, the Rule makes plain. God is in the here and now in Benedictine spirituality. It is we who are not. It is we who are trapped in the past, angry at what formed us, or fixated on a future that is free from pain or totally under our control. But God is in our present, waiting for us there.
**_With this conclusion, God waits for us daily to translate into action, as we should, these holy teachings. Therefore our life span has been lengthened by way of a truce, that we may amend our misdeeds. As the apostle says: "Do you not know that the patience of God is leading you to repent?" (Rom. 2:4). And indeed God assures us in love: "I do not wish the death of sinners, but that they turn back to me and live" (Ezek. 33:11)._**
"Life is only lent to us," a Jewish proverb instructs, and the Rule of Benedict explains further "by way of a truce." Long life, in other words, is given for the gift of insight: to give us time to understand life and to profit from its lessons and to learn from its failures and to use its moments well and make sense out of its chaos. That, perhaps, is why we expect the elderly to be wise. That, perhaps, is why we look back over the years of our own lives and wonder what happened to the person we were before we began to see more than ourselves. The problem is that there is a lot of life that dulls the senses. Too much money can make us poor. Too much food can make us slow. Too much partying can make us dull. Only the spiritual life enervates the senses completely. All life takes on a new dimension once we begin to see it as spiritual people. The bad does not destroy us and the good gives us new breath because we are always aware that everything is more than it is. The family is not just a routine relationship; it is our sanctification. Work is not just a job; it is our exercise in miracle making. Prayer is not just quiet time; it is an invitation to grow. We begin to find God where we could not see God before, not as a panacea or an anesthetic, not as a cheap release from the problems of life, but as another measure of life's meaning for us.
Clearly, living life well is the nature of repentance. To begin to see life as life should be and to live it that way ourselves is to enable creation to go on creating in us.
_Jan. 6 – May 7 – Sept. 6_
_Now that we have asked God who will dwell in the holy tent, we have heard the instruction for dwelling in it, but only if we fulfill the obligations of those who live there. We must, then, prepare our hearts and bodies for the battle of holy obedience to God's instructions. What is not possible to us by nature, let us ask the Holy One to supply by the help of grace. If we wish to reach eternal life, evenas we avoid the torments of hell, then—while there is still time, while we are in this body and have time to accomplish all these things by the light of life—we must run and do now what will profit us forever._
There is a poignancy in this paragraph that is little associated with great spiritual documents. First, Benedict stresses again that we are not alone in our undertaking to live above the dregs of life. What is "not possible to us by nature," we must "beg for by grace," he says. This is an enterprise between two spirits, in other words, God's and our own. We will fail often, but God will not fail us and we must not stop.
"God," the elder said, "is closer to sinners than to saints."
"But how can that be?" the eager disciple asked.
And the elder explained, "God in heaven holds each person by a string. When we sin, we cut the string. Then God ties it up again, making a knot—bringing the sinner a little closer. Again and again sins cut the string—and with each knot God keeps drawing the sinner closer and closer."
Even our weaknesses take us to God if we let them.
It is a very liberating thought: we are not capable of what we are about to do but we are not doing it alone and we are not doing it without purpose. God is with us, holding us up so that the reign of God may be made plain in us and become hope to others. If we can become peacemakers, if we can control our need to control, if we can distinguish between our wants and our needs, then anybody can.
_Jan. 7 – May 8 – Sept. 7_
_Therefore we intend to establish a school for God's service. In drawing up its regulations, we hope to set down nothing harsh, nothing burdensome. The good of all concerned, however, may prompt us to a little strictness in order to amend faults and to safeguard love. Do not be daunted immediately by fear and run away from the road that leads to salvation. It is bound to be narrow at the outset. But as we progress in this way of life and in faith, we shall run on the path of God's commandments, our hearts overflowing with the inexpressible delight of love. Never swerving from God's instructions, then, but faithfully observing God's teaching in the monastery until death, we shall through patience share in the sufferings of Christ that we may deserve also to share in the eternal presence. Amen._
The spiritual life is not something that is gotten for the wishing or assumed by affectation. The spiritual life takes discipline. It is something to be learned, to be internalized. It's not a set of daily exercises; it's a way of life, an attitude of mind, an orientation of soul. And it is gotten by being schooled until no rules are necessary.
Among the ancients there is a story told that confirms this insight to this day:
"What action shall I perform to attain God?" the disciple asked the elder.
"If you wish to attain God," the elder said, "there are two things you must know. The first is that all efforts to attain God are of no avail."
"And the second?" the disciple insisted.
"The second is that you must act as if you did not know the first," the elder said.
Clearly, great pursuers of the spiritual life know that the secret of the spiritual life is to live it until it becomes real.
The difference between Benedict and other spiritual masters of his time lay in the fact that Benedict believed that the spiritual life was not an exercise in spiritual gymnastics. It was to be nothing "harsh or burdensome." And it was not a private process. It was to be done in community with others. It was to be a "school" dedicated to "the good of all concerned." It was to be lived with "patience."
The private preserves of the spiritual life are far from dead, however. It is so much easier to go to daily Mass and feel good about it than it is to serve soup at a soup kitchen. It is so much more comfortable to say bedtime prayers than it is to speak peace in a warring world. It is so much more satisfying to contribute to the building of a new church than it is to advocate fair trade over free trade. It is so much more heroic to fast than it is to be patient with a noisy neighbor. It is so much easier to give the handshake of peace in church than it is to speak gently in the family. And yet one without the other is surely fraud if life with God in community is truly of the essence of real spiritual growth.
The messages of the Prologue are clear: life is very short. To get the most out of it, we must begin to attend to its spiritual dimensions without which life is only half lived. Holiness is in the now, but we go through life only half conscious of it, asleep or intent on being someplace other than where we are. We need to open our eyes and see things as they exist around us: what is valuable and what is not, what enriches and what does not, what is of God and what is not. It may be the neighborhood we live in rather than the neighborhood we want that will really make human beings out of us. It may be the job we have rather than the position we are selling our souls to get that will finally liberate us from ourselves. It may be what we do rather than the prayers we pray that will finally be the measure of our sanctity.
God is calling us to more than the material level of life and God is waiting to bring us to it. All we have to do is to live well with others and live totally in God. All we have to do is to learn to listen to the voice of God in life. And we have to do it heart, soul, and body. The spiritual life demands all of us.
CHAPTER 1
THE KINDS OF MONASTICS
_Jan. 8 – May 9 – Sept. 8_
_There are clearly four kinds of monastics. First, there are the cenobites, that is to say, those who belong to a monastery, where they serve under a rule and an abbot or prioress._
In this chapter, Benedict describes each of the four main classes of religious life that were common at the time of his writing. The effects of the descriptions and definitions are apparent. He is for all intents and purposes telling us the characteristics that he values most in spiritual development and emphasizing the qualities that in his opinion are most important to spiritual growth.
In one brief sentence, then, Benedict describes the life of the cenobite. Cenobites are the seekers of the spiritual life who live in a monastery—live with others—and are not a law unto themselves. Holiness, he argues, is not something that happens in a vacuum. It has something to do with the way we live our community lives and our family lives and our public lives as well as the way we say our prayers. The life-needs of other people affect the life of the truly spiritual person and they hear the voice of God in that.
Cenobites, too, live "under a rule." Meaningless spiritual exercises may not be a Benedictine trait but arbitrariness or whim are not part of Benedict's prescription for holiness either. Monastic spirituality depends on direction. It is a rule of life. Self-control, purpose, and discipline give aim to what might otherwise deteriorate into a kind of pseudoreligious life meant more for public show than for personal growth. It is so comforting to multiply the practices of the church in our life and so inconvenient to have to meet the responsibilities of the communities in which we live.
But the spiritual life is not a taste for spiritual consolations. The spiritual life is a commitment to faith where we would prefer certainty. It depends on readiness. It demands constancy. It flourishes in awareness.
The ancients say that once upon a time a disciple asked the elder, "Holy One, is there anything I can do to make myself enlightened?"
And the Holy One answered, "As little as you can do to make the sun rise in the morning."
"Then of what use," the surprised disciple asked, "are the spiritual exercises you prescribe?"
"To make sure," the elder said, "that you are not asleep when the sun begins to rise."
The Rule prescribes directions that will keep us, like the mythical disciple, awake until what we live, lives in us.
Then, Benedict says, the cenobite lives under an abbot or prioress, someone who will mediate past and future for us, call us to see where we have come from and where we are going, confront us with the call to the demands of living fully in the now when we might be most likely to abandon our own best ideals for the sake of the easy and the selfish. It is a basic Christian call. Everyone in life lives under someone and something. Adulthood is not a matter of becoming completely independent of the people who lay claim to our lives. Adulthood is a matter of being completely open to the insights that come to us from our superiors and our spouses, our children and our friends, so that we can become more than we can even begin to imagine for ourselves.
The cenobite, like most of the people of the world, works out the way to God by walking with others. In monastic spirituality, there is no escape from life, only a chance to confront it, day after day in all its sanctifying tedium and blessed boredom and glorious agitation in the communities of which we are a part at any given moment of our lives.
_Second, there are the anchorites or hermits, who have come through the test of living in a monastery for a long time, and have passed beyond the first fervor of monastic life. Thanks to the help and guidance of many, they are now trained to fight against evil. They have built up their strength and go from the battle line in the ranks of their members to the single combat of the desert. Self-reliant now, without the support of another, they are ready with God's help to grapple single-handed with the vices of body and mind._
If any paragraph in the Rule dispels the popular notion of spirituality, surely this is it. Modern society has the idea that if you want to live a truly spiritual life, you have to leave life as we know it and go away by yourself and "contemplate," and that if you do, you will get holy. It is a fascinating although misleading thought. The Rule of Benedict says that if you want to be holy, stay where you are in the human community and learn from it. Learn patience. Learn wisdom. Learn unselfishness. Learn love. Then, if you want to go away from it all, then and only then will you be ready to do it alone.
There is, of course, an anchorite lurking in each of us who wants to get away from it all, who finds the tasks of dailiness devastating, who looks for God in clouds and candlelight. Perhaps the most powerful point of this paragraph is that it was written by someone who had himself set out to live the spiritual life as a hermit and then discovered, apparently, that living life alone is nowhere near as searing of our souls as living it with others. It is one thing to plan my own day well with all its balance and its quiet and its contemplative exercises. It is entirely another rank of holiness to let my children and my superiors and my elderly parents and the needs of the poor do it for me.
_Third, there are sarabaites, the most detestable kind of monastics, who with no experience to guide them, no rule to try them as "gold is tried in a furnace" (Prov. 27:21), have a character as soft as lead. Still loyal to the world by their actions, they clearly lie to God by their signs of religion. Two or three together, or even alone, without a shepherd, they pen themselves up in their own sheepfolds, not God's. Their law is what they like to do, whatever strikes their fancy. Anything they believe in and choose, they call holy; anything they dislike, they consider forbidden._
There's passion in the Rule of Benedict, lots of it, and sarabaites come in for a good share. Benedict calls this sort of "spirituality" detestable.
Anchorites separate themselves from a community in order to concentrate their energies and strengthen their virtues apart from the distractions of everyday life. They are seasoned seekers who want to center their lives in God alone, naïvely perhaps but sincerely nevertheless.
Sarabaites separated themselves also. Before the codification of religious law, people could assume a habit without formal training or approval. Sarabaites presented themselves as religious but separated themselves from a disciplined life and spiritual guidance and serious purpose in order to concentrate their energies on themselves. They called themselves religious, but they were the worst of all things religious. They were unauthentic. They pretended to be what they were not.
They lived lives of moderate commitment, chaste and even simple to a point, but they listened to no one's wisdom but their own. They were soft.
Perhaps the real importance of the paragraph for today is to remind ourselves that it's not all that uncommon for people of all eras to use religion to make themselves comfortable. It is a sense of personal goodness that they want, not a sense of gospel challenge. They are tired of being challenged. They want some proof that they've arrived at a spiritual height that gives consolation in this life and the promise of security in the next. There comes a time in life for everyone where the effort of it all begins to seem too much, when the temptation to settle down and nestle in becomes reasonable.
After years of trying to achieve a degree of spiritual depth with little result, after a lifetime of uphill efforts with little to show for it, the lure is to let it be, to stop where we are, to coast. We begin to make peace with tepidity. We begin to do what it takes to get by but little that it takes to get on with the spiritual life. We do the exercises but we cease to "listen with the heart." We do the externals—the churchgoing and church giving—and we call ourselves religious, but we have long since failed to care. A sense of self-sacrifice dies in us and we obey only the desires and the demands within us.
_Fourth and finally, there are the monastics called gyrovagues, who spend their entire lives drifting from region to region, staying as guests for three or four days in different monasteries. Always on the move, they never settle down, and are slaves to their own wills and gross appetites. In every way they areworse than sarabaites. It is better to keep silent than to speak of all these and their disgraceful way of life. Let us pass them by, then, and with the help of God, proceed to draw up a plan for the strong kind, the cenobites._
The gyrovagues, whom Benedict rejected out of hand, actually had a noble beginning. Founded to follow the Christ "who had nowhere to lay his head," the earliest gyrovagues threw themselves on the providence of God, having nothing, owning nothing, amassing nothing. Originally, therefore, a sign of faith and simplicity to the Christian community, gyrovagues soon became a sign of indolence and dissipation.
Gyrovagues went from community to community, living off the charity of working monks, begging from the people, dependent on the almsgiving of others. But they never stayed anyplace long enough to do any work themselves or to be called to accountability by the community. As admirable as their call to total poverty may have been in the beginning, it began to be their own particular brand of self-centeredness. They took from every group they visited but they gave little or nothing back to the communities or families that supported them. Gyrovagues abound in religious groups: they talk high virtue and demand it from everybody but themselves. They know how to shop for a parish but they do little to build one. They live off a community but they are never available when the work of maintaining it is necessary. They are committed to morality in the curriculum of grade schools but completely unmoved by the lack of morality in government ethics. Gyrovagues were an extreme and undisciplined kind of monastic and Benedict decried them, not so much because of their ideals surely as because of their lack of direction and good work.
Benedict's reference to the gyrovagues teaches a good lesson yet today. Extremes in anything, he implies, even in religion, are dangerous. When we go to excess in one dimension of life, the unbalance in something else destroys us. Work, for instance, is good but not at the expense of family. Love is good but not at the expense of work.
Too much of a good thing can creep into life very easily and become our rationalization for avoiding everything else. Achievement becomes more important than family. Prayer becomes more important than work. Religious exercises become more important than personal responsibilities. There is a little gyrovague in us all.
The _Tao Te Ching,_ the Chinese Book of the Way, an ancient manual on the art of living that is the most widely translated book in world literature after the Bible, says on the same subject:
Fill your bowl to the brim
and it will spill.
Keep sharpening your knife
and it will blunt.
Chase after money and security
and your heart will never unclench.
Care about people's approval
and you will be their prisoner.
Do your work, then step back.
The only path to serenity.
CHAPTER 2
QUALITIES OF THE ABBOT OR PRIORESS
_Jan. 9 – May 10 – Sept. 9_
_To be worthy of the task of governing a monastery, the prioress or abbot must always remember what the title signifies and act accordingly. They are believed to hold the place of Christ in the monastery. Therefore, a prioress or abbot must never teach or decree or command anything that would deviate from God's instructions. On the contrary, everything they teach and command should, like the leaven of divine justice, permeate the minds of the community._
The social revolution of the Rule starts in this paragraph on authority. This will be a different kind of life than the sixth-century Roman ever saw. The head of the monastery will not be a chief or a queen or a feudal lord. The superior of a monastery of Benedictines will be a Christ figure, simple, unassuming, immersed in God, loving of the marginal, doer of the gospel, beacon to the strong.
Once you begin to understand that, you begin to understand the whole new type of authority that the Rule models for a world gone wild with power. You begin to understand that it is not the laws of the mighty that will govern this group. It is the law of God that will preempt all other considerations.
Like Christ, this leader does not lead with brute force. This leader understands the leavening process. This leader, called appropriately abbot or abbess or prioress, is a spiritual parent, a catalyst for the spiritual and psychological growth of the individual monastic, not a border guard or a warden. This leader is not a parent who terrorizes a child into submission; this leader believes in the best and gives people the opportunities to make the mistakes that lead to growth.
The prioress and abbot provide an environment that confronts the monastic with the presence of God, that shows them the Way. After that it is up to the monastic to let the practices of the community and the rhythm of the prayer life work their way until the piercing good of God rises in them like yeast in bread.
"If you meet the Buddha on the road," the Zen master teaches the disciple, "kill him." Don't let any human being become the measure of your life, the Zen implies. Eliminate whatever you would be tempted to idolize, no matter how worthy the object. The role of the spiritual leader, in other words, is not to make martinets out of people; it is to lead them to spiritual adulthood where they themselves make the kind of choices that give life depth and quality. Like the teacher of Zen, Benedict does not make the superior of the monastery the ultimate norm of life. Pleasing the abbot is not what monastic life is all about. Becoming what the abbess or prioress thinks you should be is not the goal of monasticism. Following the leader is not the end for which we're made; finding God is. Benedict makes the superior of his monasteries a lover of people, a leader who can persuade a person to the heights, show them the mountain and let them go.
In our own culture, becoming someone important, climbing the corporate and ecclesiastical ladder has so often meant pleasing the person at the top rather than doing what conscience demands or the situation requires. That kind of leadership is for its own sake. It makes the guru, rather than the gospel, the norm of life. That kind of obedience puts the business before the soul. That kind of authority is not monastic and it is not spiritual. That kind of authority so often leads to the satisfaction of the system more than to the development of the person and the coming of the reign of God. That kind of authority breeds scandals and cover-ups in the face of a tradition that holds up for public emulation Joan of Arc and Thomas More, whose obedience was always to a much higher law than that of countries or institutions.
_Jan. 10 – May 11 – Sept. 10_
_Let the prioress and abbot always remember that at the judgment of God, not only their teaching but also the community's obedience will come under scrutiny. The prioress and abbot must, therefore, be aware that the shepherd will bear the blame wherever the owner of the household finds that the sheep have yielded no profit. Still, if they have faithfully shepherded a restive and disobedient flock, always striving to cure their unhealthy ways, it will be otherwise: the shepherd will be acquitted at God's judgment. Then, like the prophet, they may say to God: "I have not hidden your justice in my heart; I have proclaimed your truth and your salvation (Ps. 40:11) but they spurned and rejected me" (Isa. 1:2; Ezek. 20:27).Then at last the sheep that have rebelled against their care will be punished by the overwhelming power of death._
Benedict puts a great deal of responsibility on the shoulders of people in authority, but not all of it. Abbots and prioresses are to teach, to proclaim, but the community's responsibility is to listen and to respond.
Benedict wants a community that is led, but not driven.
The concept is clear: people are not acquitted of the responsibility for their own souls. Personal decisions are still decisions, personal judgments are still judgments, free will is still free will. Being in a family does not relieve a child of the responsibility to grow up. The function of twenty-one-year-olds is not to do life's tasks as their parents told them to when they were six years old. The function of twenty-one-year-olds is simply to do the same tasks well and to take accountability themselves for having done them.
Perhaps the most important result of a model of authority like this is the environment it creates. The monastery is not a royal court, a military barracks, or a detention home. The role of leadership is not to make lackeys or foot soldiers or broken children out of adult Christians.
The purpose of Benedictine spirituality is to gather equally committed adults for a journey through earthen darkness to the dazzling light that already flames in each of us, but in a hidden place left to each of us to find.
The Rule's model of leadership and authority, then, is a paradigm for any relationship, husband and wife, parent and child, supervisor and employee. The function of authority is not to control the other; it is to guide and to challenge and to enable the other. Benedictine authority is a commitment to that, a promise of that.
A midrash on Genesis points out that "God prefers your deeds to your ancestors' virtues." We are not here simply to follow someone else. Being part of something good does not automatically make us good. What we do with our own lives is the measure of their value. We are here to learn to take ourselves in hand.
_Jan. 11 – May 12 – Sept. 11_
_Furthermore, those who receive the name of prioress or abbot are to lead the community by a twofold teaching: they must point out to the monastics all that is good and holy more by example than by words, proposing God's commandments to a receptive community with words, but demonstrating God's instructions to the stubborn and the dull by a living example. Again, if they teach the community that something is not to be done, then neither must they do it, "lest after preaching to others, they themselves be found reprobate" (1 Cor. 9:27) and God some day call to them in their sin: "How is it that you repeat my just commands and mouth my covenant when you hate discipline and toss my words behind you?" (Ps. 50:16–17). And also this: "How is it that you can see a splinter in another's eye, and never notice the plank in your own?" (Matt. 7:3)._
The _Tao Te Ching_ says,
We join spokes together in a wheel
but it is the center hole
that makes the wagon move.
Benedict says that those who hold authority in a community are not to be above the group, they are to be the centers of it, the norm of it, the movers of it. They themselves are to mirror its values. Their job is not simply to give orders. Their job is to live out the ideals. It is an authority far removed from office elitism or pompous hierarchy or high-handed parenting.
Benedict calls a community to obedience, yes, but he does not call it to servitude. He does not call people to conformity for the sake of conformity. That's where modern concepts of blind obedience and the monastic concept of cenobitic obedience are so distinct from one another. Blind obedience demands that underlings comply with authority without thought of consequences. Cenobitic obedience insists that equals must bring a thoughtful concern for what is best for everyone before they ask anything of consequence.
Autocrats and militarists and spiritual charlatans and abusive parents and corporate moguls want the people under them to obey laws from which their exalted positions hold them exempt. Benedict says that the only authentic call for obedience comes from those who themselves demonstrate the value of the law.
The point is that what we do not live we do not have a right to require, and that for two reasons: first, because it is a hollow call to insist that others do what we do not do ourselves and, second, because it requires for the sake of requiring something rather than for the merit of the requirement itself. To hold people under us to a law that we ourselves have no intention of respecting is to make a mockery of what we ask. Employees whom we require to work because we will not, children who are told to avoid what they see us doing with impunity, citizens who must do what they see us declaring exempt for ourselves do learn from us. They learn that law is useless and that we are frauds and that power protects only the powerful. Benedict is saying that if the laws are good, then people will be able to see that in the lawgiver.
But Benedict is saying even more than this. Benedict is saying that the function of spiritual leadership is not to intimidate people into submission by fear or guilt. The function of spiritual leadership is to show in our own lives the beauty that oozes out of those who live the spiritual life to its fullness. The function of spiritual leadership is to enshrine what a good life can be.
The abbot and prioress are to make of themselves the light that guides and the crystal that rings true. Otherwise, why should anyone else attempt the Way at all? "Love work and hate lordship," the Hasidim teach their rabbis. It is Benedict's teaching, too.
_Jan. 12 – May 13 – Sept. 12_
_The prioress or abbot should avoid all favoritism in the monastery. They are not to love one more than another unless they find someone better in good works and obedience. One born free is not to be given higher rank than one born a slave who becomes a monastic, except for some other good reason. But the prioress and abbot are free, if they see fit, to change anyone's rank as justice demands. Ordinarily, all are to keep to their regular places, because "whether slave or free, we are all one in Christ" (Gal. 3:28; Eph. 6:8) and share equally in the service of the one God, for "God shows no partiality among persons" (Rom. 2:11). Only in this are we distinguished in God's sight: if we are found better than others in good works and in humility. Therefore, the prioress and abbot are to show equal love to everyone and apply the same discipline to all according to their merits._
If Benedict of Nursia was anything, he was not a pious romantic. He knew the gospel and he knew life and he set out to bring the two together.
In one paragraph of this chapter, he shapes a completely new philosophy of authority, in another paragraph he hints at a different philosophy of religious life, and in this one he rejects, out of hand, the common social structures of the period. In his communities, slave and free are equal, as the Gospels demand.
This is the Jesus life. What is insane in the streets is common coin here. What is madness to politicians is life breath here. What is unheard of in nice company is taken for granted here. Here people are ranked in the order in which they came to the group—not by education, not by money, not by social status, but simply according to the moment they came to Christ. There is, as a result, no rank at all and this is very disconcerting to a world that loves uniforms and titles and knowing people who are in _Who's Who._
But do not be misled. Benedict is a realist, not a feckless libertarian. There are differences among us and he recognizes those. There is a kind of natural hierarchy of gifts. Some of us are business people and some of us are not. Some of us are musicians and some are not. Some of us are leaders and some are not. The question is not whether or not some of us should be put over others of us. The question is how we get there and why we're put there.
Here Benedict draws another sharp contrast with life as we know it. The monastic life, the spiritual life, is not a life dedicated to climbing and clawing to the top. The monastic mind is not set on politicking or groveling. Abbots and prioresses, good leaders anywhere, are not in the business of forming kitchen cabinets or caucuses.
No, favoritism and intrigue are not the mint of the monastic mind-set; commitment is.
Benedict doesn't just want a business manager who can make money for the monastery. He doesn't want workers for their productivity only. He doesn't take for leaders simply those who know how to control a group or build a business. Whom Benedict wants appointed to positions of responsibility are people who are distinguished "in good works and obedience," in "good works and humility." It is a model for leadership in those places where profit and power and the party line take precedence over what the business or the diocese or the social service agency proclaims it is about.
He does not want people in positions simply to get a job done. He wants people in positions who embody why we bother to do the job at all. He wants holy listeners who care about the effect of what they do on everybody else.
Imagine a world that was run by holy listeners.
_Jan. 13 – May 14 – Sept. 13_
_In their teachings, the prioress or abbot should always observe the apostle's recommendation in which it is said: "Use argument, appeal, reproof" (2 Tim. 4:2). This means that they must vary with circumstances, threatening and coaxing by turns, at times stern, at times devoted and tender. With the undisciplined and restless, they will use firm argument; with the obedient and docile and patient, they will appeal for greater virtue; but as for the negligent and disdainful, we charge the abbot or prioress to use reproof and rebuke. They should not gloss over the sins of those who err, but cut them out while theycan, as soon as they begin to sprout, remembering the fate of Eli, priest of Shiloh (1 Sam. 2:11–4:18). For the upright and perceptive, the first and second warnings should be verbal; but those who are evil or stubborn, arrogant or disobedient, can be curbed only by blows or some other physical punishment at the first offense. It is written, "The fool cannot be corrected with words" (Prov. 29:19), and again, "Strike your children with a rod and you will free their souls from death" (Prov. 23:14)._
To "vary with the circumstances" may be the genius of the entire Rule of Benedict. It is undoubtedly clear here.
The Rule of Benedict does not turn people into interchangeable parts. Benedict makes it quite plain: people don't all learn the same way; they don't all grow the same way; they can't all be dealt with the same way. Those concepts, of course, have become commonplace in a culture that is based on individualism. But they were not commonplace as recently as the 1950s. Historically, there has been a more acceptable way for just about everything: a more acceptable way to pray; a more acceptable way to celebrate the Mass; a more acceptable way to think; a more acceptable way to live. Not everyone did it, of course, but everyone had very clear criteria by which to judge the social fit of everyone else.
Personalism is a constant throughout the Rule of Benedict. Here, in a chapter on the abbot or prioress, you would certainly expect at least to find a clear call for order, if not for perfection and discipline and conformity. There is no room in Benedictine spirituality, though, for bloodless relationships between people in authority and the people for whom they have responsibility. Benedictine authority is expected to have meaning. It is to be anchored in the needs and personality of the other person. For the prioress or abbot or parent or supervisor, it is an exhausting task to treat every individual in their care as an individual but nothing else is worth their time. It is easy to intimidate the stubborn with power. It is simple to ignore the mediocre. It is possible to leave the docile on their own and hope for the best.
In the Rule, though, the function of the leader is to call each individual to become more tomorrow than they were today. The point of the paragraph is not how the calling is to be done, with firmness or tenderness or persuasion or discipline. The theories on that subject change from period to period. Some types respond to one approach, some respond better to another. The point here is simply that the calling is to be done. The person who accepts a position of responsibility and milks it of its comforts but leaves the persons in a group no more spiritually stirred than when they began, no more alive in Christ than when they started, no more aflame with the gospel than when they first held it in their hands, is more to be criticized than the fruitless group itself. It was Eli, Benedict points out, the father who did not correct his sinful sons, whom God indicted, not the sons alone.
_Jan. 14 – May 15 – Sept. 14_
_The prioress and abbot must always remember what they are and remember what they are called, aware that more will be expected of one to whom more has been entrusted. They must know what a difficult and demanding burden they have undertaken: directing souls and serving a variety of temperaments, coaxing, reproving, and encouraging them as appropriate. They must so accommodate and adapt themselves to each one's character and intelligence that they will not only keep the flock entrusted to their care from dwindling, but will rejoice in the increase of a good flock._
There are some interesting distinctions made in this paragraph. The abbot and prioress are to remember what they are and what they are called. What they and every other leader are is painfully clear: they are people just like everybody else in the monastery. They are not royalty. They are not potentates. They are only people who also struggle and fail just like the people they lead.
But what they are and what they are called—abbot, abbess, spiritual father, spiritual mother—are not unrelated. They are not called to be either lawgivers or camp counselors. They are not expected to be either rigid moralists or group activity directors. They are to be directors of souls who serve the group by "coaxing, reproving, and encouraging" it—by prodding and pressing and persuading it—to struggle as they have struggled to grow in depth, in sincerity, and in holiness, to grow despite weaknesses, to grow beyond weaknesses.
Abbots or prioresses of Benedictine monasteries, then, parents and supervisors and officials and bishops everywhere who set out to live a Benedictine spirituality, are to keep clearly in mind their own weak souls and dark minds and fragile hearts when they touch the souls and minds and hearts of others.
But there is another side to the question as well. It is not easy for honest people who hold their own failures in their praying hands to question behavior in anyone else. "There but for the grace of God go I," John Bradford said at the sight of the condemned on their way to execution. Aware of what I myself am capable of doing, on the one hand, how can I possibly censure or disparage or reprimand or reproach anyone else? On the other hand, Benedict reminds us, how can those who know that conversion is possible, who have been called to midwife the spiritual life, for this generation and the next, do less?
The Hasidim tell a story that abbots and prioresses, mothers and fathers, teachers and directors may understand best. Certainly Benedict did.
When in his sixtieth year after the death of the Kotzker, the Gerer accepted election as leader of the Kotzker Hasidim, the rabbi said, "I should ask myself: 'Why have I deserved to become the leader of thousands of good people?' I know that I am not more learned or more pious than others. The only reason why I accept the appointment is because so many good and true people have proclaimed me to be their leader. We find that a cattlebreeder in Palestine during the days when the Temple stood was enjoined by our Torah (Lev. 27:32) to drive newborn cattle or sheep into an enclosure in single file. When they went to the enclosure, they were all of the same station, but when over the tenth one the owner pronounced the words: 'consecrated unto the Lord,' it was set aside for holier purposes. In the same fashion when the Jews pronounce some to be holier than their fellows, they become in truth consecrated persons."
Once chosen, it is their weakness itself that becomes the anchor, the insight, the humility, and the gift of an abbot or prioress, a pope or a priest, a parent or a director. But only if they themselves embrace it. It is a lesson for leaders everywhere who either fear to lead because they know their own weaknesses or who lead defensively because they fear that others know their weaknesses. It is a lesson for parents who remember their own troubles as children. It is a lesson for husbands and wives who cannot own the weaknesses that plague their marriage. We must each strive for the ideal and we must encourage others to strive with us, not because we ourselves are not weak but because knowing our own weaknesses and admitting them we can with great confidence teach trust in the God who watches with patience our puny efforts and our foolish failures.
_Jan. 15 – May 16 – Sept. 15_
_Above all, they must not show too great a concern for the fleeting and temporal things of this world, neglecting or treating lightly the welfare of those entrusted to them. Rather, they should keep in mind that they have undertaken the care of souls for whom they must give an account. That they may not plead lack of resources as an excuse, they are to rememberwhat is written: "Seek first the reign and justice of God, and all these things will be given you as well" (Matt. 6:33), and again, "Those who reverence the Holy One lack nothing" (Ps. 34:10)._
In an age of great institutions and unending development campaigns, Benedict makes a statement in this paragraph that stretches the modern mind to the extremity of disbelief. Benedict instructs the abbot and prioress to be more concerned about the spiritual needs of the monastery than its physical ones. You have to wonder how long a group like that will last. You also have to wonder whether a monastery that is not like that should last at all. The implications are profound.
A monastery does not have to be wealthy, Benedict implies, a monastery does not have to be large, a monastery does not have to be popular. What a monastery must be, without doubt and without fail, is holy. The role of the abbot or prioress, therefore, is not to concentrate on the physical development of the community, on the "fleeting and temporal things of this world." The role of the abbot or prioress is to direct their energies to bringing the community to the white heat of the spiritual life, after which no challenge is too great and no effort is too much because we know for certain that "those who reverence the Holy One lack nothing."
In monastic spirituality, then, leadership is not intent on making things right; leadership is intent on making life right. The number of families who have succumbed to the notion that giving their children everything that money can buy assures their happiness need this insight from monastic spirituality. The number of business people who have put their entire lives into developing their businesses instead of their quality of life need this insight from monastic spirituality. The number of young people who have learned to believe that success depends on having it all may need this monastic lesson in life. The Rule of Benedict teaches us that nothing, not even a monastery, is worth the loss of the development of the important things in life, the spiritual things in life.
_The prioress and abbot must know that anyone undertaking the charge of souls must be ready to account for them. Whatever the number of members they have in their care, let them realize that on judgment day they will surely have to submit a reckoning to God for all their souls—and indeed for their own as well. In this way, while always fearful of the future examination of the shepherd about the sheep entrusted to them and careful about the state of others' accounts, they become concerned also abouttheir own, and while helping others to amend by their warnings, they achieve the amendment of their own faults._
The word here is clear: abbots and prioresses are responsible for the community, yes, but they are responsible for the quality and integrity of their own lives as well. Being an abbot or prioress, a president or corporate tycoon does not put people above the law or outside the law. On the contrary, it may instead create a double burden. In being concerned for the spiritual well-being of others, the caretaker will have to be alert to the demands it makes on her own life. Any leader knows the litany of emotional responses: anger with those who resist, frustration with things that can't be changed, disappointment with things that showed promise but never came to fruit, hurt because of rejection by the people you tried to love, grief over the failure of projects that you counted on to succeed—all tax the soul of a leader. "Thought breaks the heart," the Africans say. Thought also robs the leader of confidence and energy and trust. Despite it all, though, Benedict counsels leaders against the sin of resignation, despair, depression, and false hope. Monastic spirituality teaches us that everything we want to do will not succeed, but monastic spirituality also teaches us that we are never to stop trying. We are never to give in to the lesser in life. We are never to lose hope in God's mercy.
People looking for a spirituality of leadership have substance in this chapter for years of thought. Benedict's leaders are to birth souls of steel and light; they are to lead the group but not drive it; they are to live the life they lead; they are to love indiscriminately; they are to favor the good, not to favor the favorites; they are to call the community to the height and depth and breadth of the spiritual life; they are to remember and rejoice in their own weaknesses in order to deal tenderly with the weaknesses of others; they are to attend more to the spiritual than to the physical aspects of community life; and, finally, they are to save their own souls in the process, to be human beings themselves, to grow in life themselves.
In this chapter, monasteries become the image of a world where leadership exists for the people it leads and not for itself. It is a model for businesses and families and institutions that would change the world. It is also a model for leaders who become so consumed in leadership that they themselves forget what it means to live a rich and holy life.
CHAPTER 3
SUMMONING THE COMMUNITY FOR COUNSEL
_Jan. 16 – May 17 – Sept. 16_
_As often as anything important is to be done in the monastery, the prioress or abbot shall call the whole community together and explain what the business is; and after hearing the advice of the members, let them ponder it and follow what they judge the wiser course. The reason why we have said all should be called for counsel is that the Spirit oftenreveals what is better to the younger. The community members, for their part, are to express their opinions with all humility, and not presume to defend their own views obstinately. The decision is rather the prioress's or the abbot's to make, so that when the abbot or prioress of the community has determined what is more prudent, all must obey. Nevertheless, just as it is proper for disciples to obey their teacher, so it is becoming for the teacher to settle everything with foresight and fairness._
An African proverb says, "You do not teach the paths of the forest to an old gorilla." Experience counts. Wisdom is simply its distillation. Abbots may be abbots and prioresses may be prioresses but the community was there long before them, and the community will remain long after they have gone as well. To ignore the counsel of a group, then, is to proceed at risk.
But Benedict knows about more than the value of experience. Benedict knows about the presence and power of God. And Benedict knows that there is a spark of the divine in all of us. The function of an abbot or prioress, of leaders and spouses everywhere, is not so much to know the truth as it is to be able to espy it and to recognize it in the other when they hear it. Calling the community for counsel is Benedict's contribution to the theology of the Holy Spirit.
In the monastic community, this common search for truth is pitched at a delicate balance. The abbot and prioress are clearly not dictators, but the community is not a voting bloc either. They are each to speak their truth, to share the perspective from which they see a situation, to raise their questions and to open their hearts, with honesty and with trust. The prioress and abbot are to listen carefully for what they could not find in their own souls and to make a decision only when they can come to peace with it, weighing both the community's concerns and the heart they have for carrying the decision through.
"Foresight and fairness" are essentials for leaders who lead out of a sense of Benedictine spirituality. The decision is all theirs and they will answer for it in conscience and in consequences. They must not make it lightly, and they must take all of its effects into consideration. The emphasis in this paragraph is clearly on results rather than on power. It is easy to gain power. It is difficult to use it without being seduced by it. The Rule of Benedict reminds us that whatever authority we hold, we hold it for the good of the entire group, not for our own sense of self.
_Jan. 17 – May 18 – Sept. 17_
_Accordingly in every instance, all are to follow the teaching of the rule, and no one shall rashly deviate from it. In the monastery, monastics are not to follow their own heart's desire, nor shall they presume to contend with the prioress or abbot defiantly, or outside the monastery. Should any presume to do so, let them be subjected to the discipline of the rule. Moreover, the prioress or abbot must themselves reverence God and keep the rule in everything they do; they can be sure beyond any doubt that they will have to give an account of all their judgments to God, the most just of judges._
_If less important business of the monastery is to be transacted, the prioress and abbot shall take counsel with the elders only, as it is written: "Do everything with counsel and you will not be sorry afterward" (Sir. 32:24)._
Benedictine monasticism is life lived within the circuit of four guy wires: the gospel, the teachings of its abbots and prioresses, the experience of the community, and the Rule of Benedict itself.
The gospel gives meaning and purpose to the community.
The teaching of its abbots and prioresses gives depth and direction to the community. The experience of the community, spoken by its members in community chapter meetings, gives truth to the community. But it is the Rule of Benedict that gives the long arm of essential definition and character to the community.
Each of us, monastic or not, deals with the same elements in life. We are all bound to the gospel, under leadership of some kind, faced with the dictates of tradition or the cautions of experience and in need of a direction. Monastic spirituality offers enduring principles and attitudes far beyond whatever culture embodies them. Once embraced, they guide our way through whatever the psychological fads or religious practices or social philosophies of the time that offer comfort but lack staying power. "All are to follow the teaching of the Rule," Benedict, the great abbot, teaches, "and no one shall rashly deviate from it." Adapt the Rule, yes. Abandon the Rule, no.
The fact is that it is in the Rule itself that the principles and values of Benedictine spirituality are stored and maintained. No matter how far a group goes in its attempts to be relevant to the modern world, it keeps one foot in an ancient one at all times. It is this world that pulls it back, time and time again, to the tried and true, to the really real, to a Beyond beyond ourselves. It is to these enduring principles that every age looks, not to the customs or practices that intend to embody them from one age to another.
CHAPTER 4
THE TOOLS FOR GOOD WORKS
_Jan. 18 – May 19 – Sept. 18_
_First of all, "love God with your whole heart, your whole soul and all your strength, and love your neighbor as yourself" (Matt. 22:37–39; Mark 12:30–31; Luke 10:27). Then the following: "You are not to kill, not to commit adultery; you are not to steal nor to covet" (Rom. 13:9); "you are not to bear false witness" (Matt. 19:18; Mark 10:19; Luke 18:20). "You must honor everyone" (1 Pet. 2:17), and "never do to another what you do not want done to yourself" (Tob. 4:16; Matt. 7:12; Luke 6:31)._
At first glance, this opening paragraph on the instruments of the spiritual art seems to be a relatively standard and basic reference to a biblical description of the holy life. And that seems sound. The trouble is that it also seems strange.
The surprise is that Benedict does not call us first to prayer or sacrifice or devotions or asceticisms. This is, after all, a contemplative lifestyle. It is at the same time, however, a communal lifestyle for "that most valiant kind of monastic heart," who sets out to find the holy in the human. The call to contemplation here is the call not simply to see Christ in the other but to treat the other as Christ. Benedict calls us first to justice: love God, love the other, do no harm to anyone.
_Renounce yourself in order to follow Christ (Matt. 16:24; Luke 9:23); discipline your body (1 Cor. 9:27); "do not pamper yourself, but love fasting." You must relieve the lot of the poor, "clothe the naked, visit the sick" (Matt. 25:36), and bury the dead. Go to help the troubled and console the sorrowing._
First, Benedict instructs the monastic to keep the commandments. Then, in this next paragraph, the Rule requires the keeping of the corporal works of mercy. Benedictine monasticism is, apparently, not an escape from life. This spirituality is life lived with an eye on those for whom life is a terrible burden. "Do not pamper yourself," the Rule insists. "Relieve the lot of the poor."
The monastic heart is not just to be a good heart. The monastic heart is to be good for something. It is to be engaged in the great Christian enterprise of acting for others in the place of God.
_Jan. 19 – May 20 – Sept. 19_
_Your way of acting should be different from the world's way; the love of Christ must come before all else. You are not to act in anger or nurse a grudge.Rid your heart of all deceit. Never give a hollow greeting of peace or turn away when someone needs your love. Bind yourself to no oath lest it prove false, but speak the truth with heart and tongue._
The end of Benedictine spirituality is to develop a transparent personality. Dissimulation, half answers, vindictive attitudes, a false presentation of self are all barbs in the soul of the monastic. Holiness, this ancient Rule says to a culture that has made crafty packaging high art, has something to do with being who we say we are, claiming our truths, opening our hearts, giving ourselves to the other pure and unglossed. Shakespeare's Hamlet (act 1, sc. 5) noted once that "a man can smile and smile and be a villain." Benedict is intent on developing people who are what they seem to be.
_"Do not repay one bad turn with another" (1 Thess. 5:15; 1 Pet. 3:9). Do not injure anyone, but bear injuries patiently. "Love your enemies" (Matt. 5:44; Luke 6:27). If people curse you, do not curse them back but bless them instead. "Endure persecution for the sake of justice" (Matt. 5:10)._
A peacemaker's paragraph, this one confronts us with the gospel stripped and unadorned. Nonviolence, it says, is the center of the monastic life. It doesn't talk about conflict resolution; it says don't begin the conflict. It doesn't talk about communication barriers; it says, stay gentle even with those who are not gentle with you. It doesn't talk about winning; it talks about loving.
Most of all, perhaps, it offers us no false hope that all these attempts will really change anything. No, it says instead that we must be prepared to bear whatever blows it takes for the sake of justice, quietly, gently, even lovingly, with never a blow in return.
A story from the Far East recounts that a vicious general plundered the countryside and terrorized the villagers. He was, they said, particularly cruel to the monks of the place, whom he despised.
One day, at the end of his most recent assault, he was informed by one of his officers that, fearing him, all the people had already fled the town, with the exception of one monk who had remained in his monastery going about the order of the day.
The general was infuriated at the audacity of the monk and sent the soldiers to drag him to his tent.
"Do you not know who I am?" he roared at the monk. "I am he who can run you through with a sword and never bat an eyelash."
But the monk replied quietly, "And do you not know who I am? I am he who can let you run me through with a sword and never bat an eyelash."
Nonviolence plunges the monastic into the core of Christianity and allows for no rationalizations. Monastic spirituality is Christianity to the hilt. It calls for national policies that take the poor into first account; it calls for a work life that does not bully underlings or undercut the competition; it calls for families that talk to one another tenderly; it calls for a foreign policy not based on force. Violence has simply no place in the monastic heart.
_"You must not be proud, nor be given to wine" (Titus 1:7; 1 Tim. 3:3). Refrain from too much eating or sleeping, and "from laziness" (Rom. 12:11). Do not grumble or speak ill of others._
The _Tao Te Ching_ phrases it as follows,
Be content with what you have;
rejoice in the way things are,
When you realize there is nothing lacking
the whole world belongs to you.
Benedict reminds us, too, that physical control and spiritual perspective are linked: pride and gluttony and laziness are of a piece. We expect too much, we consume too much, and we contribute too little. We give ourselves over to ourselves. We become engorged with ourselves and, as a result, there is no room left for the strippeddown, stark, and simple furniture of the soul.
_Jan. 20 – May 21 – Sept. 20_
_Place your hope in God alone. If you notice something good in yourself, give credit to God, not to yourself, but be certain that the evil you commit is always your own and yours to acknowledge._
Grace and goodness come from God, the Rule insists. We are not the sole authors of our own story. What does come from us, though, are the decisions we make in the face of the graces we receive. We can either respond to each life grace and become what we might be in every situation, whatever the effort, or we can reject the impulses that the magnet in us called goodness brings in favor of being less than we ought to be.
It is those decisions that we must bend our lives to better.
_Live in fear of the day of judgment and have a great horror of hell. Yearn for everlasting life with holy desire. Day by day remind yourself that you are going to die. Hour by hour keep careful watch over all you do, aware that God's gaze is upon you, wherever you may be. As soon as wrongful thoughts come into your heart, dash them against Christ and disclose them to your spiritual guide. Guard your lips from harmful or deceptive speech._
Motives for the spiritual life change as we change, grow as we grow. At earlier stages it is the fear of punishment that controls passions not yet spent. At a more developed stage, it is the desire for ceaseless life that impels us. At another point, it is the shattering awareness of our own mortality that brings us to brave the thought of a life beyond life and its claim on us.
Whatever the motive, Benedict reminds us that the consciousness of God's presence, behind us, within us, in front of us demands a change of heart, a change of attention from us. From now on we must think differently and tell a different truth.
_Prefer moderation in speech and speak no foolish chatter, nothing just to provoke laughter; do not love immoderate or boisterous laughter._
A Jewish proverb reads, "Not every heart that laughs is cheerful," and Ben Sirach taught in Ecclesiasticus 21:20, "Fools raise their voices when they laugh, but the wise smile quietly."
Unlike a culture that passionately pursues unmitigated and undisciplined bliss, Benedict wants moderation, balance, control in everything. Life, he knows, is more than one long party. He wants a spirituality in which people are happy but not boisterously unaware of life in all its aspects, responsive but thoughtful, personable but serious. He wants us to keep everything in perspective. Benedict warns us over and over again in the Rule not to be overtaken, consumed, swept up, swallowed by anything because, no matter how good the thing that absorbs us, we lose other goods in life because of our total lack of discipline about a single part of it.
The Talmud says, "The Torah may be likened to two paths, one of fire, the other of snow. Turn in one direction, and you die of heat; turn to the other and you die of the cold. What should you do? Walk in the middle" (Hagigah 2:1).
_Jan. 21 – May 22 – Sept. 21_
_Listen readily to holy reading, and devote yourself often to prayer. Every day with tears and sighs confess your past sins to God in prayer and change from these evil ways in the future._
A willingness to be formed is the basis of formation. Anything else is fraud. People cannot be beaten into sanctity. They can only be beaten into submission. No, Benedict says, you can't get the spiritual life by waiting for it. You have to reach for it. Read things that gild your soul. Turn your mind to prayer, to a conscious response to the God present here and now. Remember who you are.
The ancients considered the gift of tears a sign of God's great favor. If we could be always sorry for what we have done to distort life in the past then perhaps we could be safeguarded against distorting it in the future. Regret is a gift long gone in contemporary culture but critically needed perhaps. In this society, guilt has disappeared and sorrow is labeled unhealthy. As a people, then, we separate one action from another in such a way that patterns escape us and pitfalls elude us. We simply stumble on, from one event to the next, unaware of the dangers in it for us, uncaring of our past behaviors, unfeeling of the calluses on our hearts.
Life, Benedict implies, is a tapestry woven daily from yesterday's threads. The colors don't change, only the shapes we give them. Without the past to guide us, the future itself may succumb to it.
_"Do not gratify the promptings of the flesh" (Gal. 5:16); hate the urgings of self-will. Obey the orders of the prioress and abbot unreservedly, even if their own conduct—which God forbid—be at odds with what they say. Remember the teachings of the Holy One: "Do what they say, not what they do" (Matt. 23:3)._
There are two ways to live in the world—as if we were connected to it like a leaf to a tree or as if we were a universe unto ourselves. Obedience, faithful listening, is essential to the choice. A Benedictine sense of obedience is not designed to diminish a person. It is designed to connect us to the rest of the human race. If we have the discipline to curb our own caprice, we can develop the self-control it takes to listen to the wisdom of another when our own insights are limited. The fact is that there are few right ways to do a thing; there are only other ways of doing a thing. To be open to the way of those who have already gone the ground before us is potentially soul saving. That is the function of Benedictine obedience and that is a tool of the spiritual art. It shows us in others ways to goodness that otherwise we might miss of ourselves.
_Do not aspire to be called holy before you really are, but first be holy that you may more truly be called so. Live by God's commandments every day; treasure chastity, harbor neither hatred nor jealousy of anyone, and do nothing out of envy. Do not love quarreling; shun arrogance. Respect the elders and love the young. Pray for your enemies out of love for Christ. If you have a dispute with someone, make peace with that person before the sun goes down._
The seduction of embarking on a spiritual life is that people can be fooled into believing that wanting it is doing it. They begin to believe that by traveling they have arrived. Worse, perhaps, they begin to allow others to think that by traveling they have arrived. They mistake the idea for the thing and perpetuate the idea.
Benedict knew better. He knew that the secret of the holy life was not so much a holy reputation as it was a holy attitude toward all of creation: reverence for God, reverence for the body, reverence for the other who is younger and unimportant, or older and useless now, or in opposition to us and an irritant now.
Benedict wants us to guard against a notion of superiority that will, in our most honest moments, only discourage us with ourselves.
_And finally, never lose hope in God's mercy._
What Benedict wants is simply that we keep trying. Failures and all. Pain and all. Fear and all. The God of mercy knows what we are and revels in weakness that tries.
_These, then, are the tools of the spiritual craft. When we have used them without ceasing day and night and have returned them on the day of judgment our wages will be the reward God has promised: "What the eye has not seen nor the ear heard, God has prepared for those who love" (1 Cor. 2:9)._
These tools of the spiritual life—justice, peacemaking, respect for all creation, trust in God—are the work of a lifetime. Each one of them represents the unearthed jewel that is left in us to mine. Each of them represents the gem that we can be. Benedict says that in the dark days of the spiritual life, when we have failed ourselves miserably, we must remember the God who walks with us on the journey to our best selves and cling without end to the God who fails us never.
_The workshop where we are to toil faithfully at all these tasks is the enclosure of the monastery and stability in the community._
The spiritual life for Benedict of Nursia is not an errant idea. It is not something we do without thought, without concentration, without direction, without help. Monastic spirituality is a spirituality of love. It is a way of life, not a series of ascetical exercises. It takes persistence. It takes dedication. It takes a listening commitment to the human community. It asks a great deal more of us than a series of pious formulas. It asks for an attitude of mind and a style of life and way of relating that takes me out of myself into the mind of God for humanity.
CHAPTER 5
OBEDIENCE
_Jan. 22 – May 23 – Sept. 22_
_The first step of humility is unhesitating obedience, which comes naturally to those who cherish Christ above all. Because of the holy service they have professed, or because of dread of hell and for the glory of everlasting life, they carry out the ordersof the prioress or abbot as promptly as if the command came directly from God. The Holy One says of people like this: "No sooner did they hear than they obeyed me" (Ps. 18:45); again, God tells teachers: "Whoever listens to you, listens to me" (Luke 10:16). Such people as these immediately put aside their own concerns, abandon their own will, and lay down whatever they have in hand, leaving it unfinished. With the ready step of obedience, they follow the voice of authority in their actions. Almost at the same moment, then, as the teacher gives the instruction the disciple quickly puts it into practice out of reverence for God; and both actions together are swiftly completed as one._
There is an urgency in the Rule of Benedict. The hallmark of obedience for Benedict, in fact, is immediacy. Monasticism is a process, true, but it is lived out in a million little ways day after day. Most of all, perhaps, it is lived out in obedience, the ability to hear the voice of God in one another—in the members of the community, both old and young; in the person we married and all of whose aphorisms we know by now; in underlings and children; in old parents and boring in-laws. This voice of God in the demands of community life is not something to be dallied with or contended with or speculated about or debated.
The necessary question, of course, is how is it that a rule that purports to deal with the spiritual life can possibly put so much stock in the human dimensions of community. Obedience to God is imperative, yes, but so much emphasis on obedience to a prioress or abbot, to leaders whose mundane lives are as limited as our own, almost seems to make a mockery of the very concept. If this is a life centered in the call of God, then why so much attention to the human?
The answer, of course, is that the human is the only place we can really be sure that God is. It is so easy to love the God we do not see but it is so much more sanctifying to serve the God we learn to see in others.
The self-giving of real obedience is very clear to Benedict. When we follow the voice of the ones who call us to higher service, we put down our own concerns, allow ourselves to be led by the sights of another, treat our own best interests with a relaxed grasp. We empty ourselves out so that the presence of God can come in, tangible and present and divinely human.
_It is love that impels them to pursue everlasting life; therefore, they are eager to take the narrow road of which God says: "Narrow is the road that leads to life" (Matt. 7:14). They no longer live by their own judgment, giving in to their whims and appetites; rather they walk according to another's decisions and directions, choosing to live in monasteries and to have a prioress or abbot over them. Monastics of this resolve unquestionably conform to the saying of Christ: "I have come not to do my own will, but the will of the One who sent me" (John 6:38)._
Two ideas permeate the Rule of Benedict: love and wisdom. Love is the motive; wisdom is the goal and the Way. Two great loves, love of God and love of the other, impel us to look outside ourselves and learn from those outside of ourselves where we really are in life. When we love something besides ourselves and when we listen to someone besides ourselves we have glimmers of growth to guide us.
That's why the Rule alone is not enough. The Rule is a luminaria, a lighted path, a clear direction. The presence of a prioress and abbot, of spiritual guides and spiritual giants in our lives, the living interpreters of a living spirituality and Way of Life, holds us up during the hard times in life. These living, breathing, loving vessels of the best in the spiritual life act as antidotes to our confusions and selfishness and pain when we are least able to make clear decisions. They act as corrections when we of all people would be least satisfied with ourselves. They become the compasses when we are veering off course, not because we do not want to see but because our sight is blinded now by age or stress or fatigue. They become the track when our hearts stray or our lives hurt.
What Benedict is saying, obviously, is that there is no going through life alone. Each of us needs a wisdom figure to walk the Way with us as well as a rule to route us. The Rule is clearly not enough.
"Why do you need teachers?" the visitor asked a disciple.
"Because," the disciple answered, "if water must be heated it needs a vessel between the fire and itself."
Abbots and prioresses, good leaders and teachers, fine parents and mentors, tender husbands and gentle wives, good friends and quality administrators, who listen to us as much as we listen to them, are there to help us bear the heat of life that shapes us, not to escape it.
_Jan. 23 – May 24 – Sept. 23_
_This very obedience, however, will be acceptable to God and agreeable to people only if compliance with what is commanded is not cringing or sluggish or halfhearted, but free from any grumbling or any reaction of unwillingness. For the obedience shown to an abbot or prioress is given to God, who has said: "Whoever listens to you, listens to me" (Luke 10:16). Furthermore, the disciples' obedience mustbe given gladly, for "God loves a cheerful giver" (2 Cor. 9:7). If disciples obey grudgingly and grumble, not only aloud but also in their hearts, then, even though the order is carried out, their actions will not be accepted with favor by God, who sees that they are grumbling in their hearts. These disciples will have no reward for service of this kind; on the contrary, they will incur punishment for grumbling, unless they change for the better and make amends._
If there is one determinant of monastic spirituality, this is surely it: you must want it. You must give yourself to it wholeheartedly. You must enter into it with hope and surety. You must not kick and kick and kick against the goad.
It is so easy to begin the spiritual life with a light heart and then, one day, drowning in the sea that is ourselves, refuse to go another step without having to be dragged. We ignore the teachings or demean the teachings. We ignore the prioress or criticize the abbot. We defy the teachers to teach.
We do what we are told, of course. We come to the meetings or keep the schedule or go through the motions of being part of the community or part of the family or part of the staff, but there is no truth in us and we weigh the group down with our complainings. We become a living lamentation. We become a lump of spiritual cement around the neck of the group.
This, Benedict says, is not obedience. This is only compliance, and compliance kills, both us and the community whose one heart is fractured by those who hold theirs back. Real obedience depends on wanting to listen to the voice of God in the human community, not wanting to be forced to do what we refuse to grow from.
CHAPTER 6
RESTRAINT OF SPEECH
_Jan. 24 – May 25 – Sept. 24_
_Let us follow the prophet's counsel: "I said, I have resolved to keep watch over my ways that I may never sin with my tongue. I was silent and was humbled, and I refrained even from good words" (Ps. 39:2–3). Here the prophet indicates that there are times when good words are to be left unsaid out of esteem for silence. For all the more reason, then, should evil speech be curbed so that punishment for sin may be avoided. Indeed, so important is silence that permission to speak should seldom be granted even to mature disciples, no matter how good or holy or constructive their talk, because it is written: "In a flood of words you will not avoid sin" (Prov. 10:19); and elsewhere, "The tongue holds the key to life and death" (Prov. 18:21). Speaking and teaching are the teacher's task; the disciple is to be silent and listen._
_Therefore, any requests to an abbot or prioress should be made with all humility and respectful submission. We absolutely condemn in all places any vulgarity and gossip and talk leading to laughter, and we do not permit a disciple to engage in words of that kind._
Silence is a cornerstone of Benedictine life and spiritual development, but the goal of monastic silence is not nontalking. The goal of monastic silence, and monastic speech, is respect for others, a sense of place, a spirit of peace. The Rule does not call for absolute silence; it calls for thoughtful talk. This chapter provides the principles upon which this "guard upon the tongue" is based. Silence for its own selfish, insulating sake, silence that is passiveaggressive, silence that is insensitive to the present needs of the other is not Benedictine silence.
Benedictine spirituality forms us to listen always for the voice of God. When my own noise is what drowns that word out, the spiritual life becomes a sham. Benedictine spirituality forms us to know our place in the world. When we refuse to give place to others, when we consume all the space of our worlds with our own sounds and our own truths and our own wisdom and our own ideas, there is no room for anyone else's ideas. When a person debates contentiously with anyone, let alone with the teachers and the guides of their life, the ego becomes a majority of one and there is no one left from whom to learn. But Benedictine spirituality is a builder of human community. When talk is unrestrained, when gossip becomes the food of the soul, then the destruction of others can't be far behind. When talk is loud and boisterous, when we make light of everything, when nothing is spared the raillery of a joke, the seriousness of all of life is at stake and our spirits wither from a lack of beauty and substance.
Make no doubt about it, the ability to listen to another, to sit silently in the presence of God, to give sober heed, and to ponder is the nucleus of Benedictine spirituality. It may, in fact, be what is most missing in a new century saturated with information but short on gospel reflection. The Word we seek is speaking in the silence within us. Blocking it out with the static of nonsense day in and day out, relinquishing the spirit of silence, numbs the Benedictine heart in a noise-polluted world.
The ancients wrote,
Once upon a time a disciple asked the elder, "How shall I experience my oneness with creation?"
And the elder answered, "By listening."
The disciple pressed the point: "But how am I to listen?"
And the elder taught, "Become an ear that pays attention to every single thing the universe is saying. The moment you hear something you yourself are saying, stop."
CHAPTER 7
HUMILITY
_Jan. 25 – May 26 – Sept. 25_
_Sisters and Brothers, divine Scripture calls to us saying: "Whoever exalt themselves shall be humbled, and whoever humble themselves shall be exalted" (Luke 14:11, 18:14). In saying this, therefore, it shows us that every exaltation is a kind of pride, which the prophet indicates has been shunned, saying: "O God, my heart is not exalted; my eyes are not lifted up and I have not walked in the ways of the great nor gone after marvels beyond me" (Ps. 131:1). And why? "If I had not a humble spirit, but were exalted instead, then you would treat me like a weaned child on its mother's lap" (Ps. 131:2)._
If the modern age has lost anything that needs to be rediscovered, if the Western world has denied anything that needs to be owned, if individuals have rejected anything that needs to be professed again, if the preservation of the globe in the twenty-first century requires anything of the past at all, it may well be the commitment of the Rule of Benedict to humility.
The Roman Empire in which Benedict of Nursia wrote his alternative rule of life was a civilization in a decline not unlike our own. The economy was deteriorating; the helpless were being destroyed by the warlike; the rich lived on the backs of the poor; the powerful few made decisions that profited them but plunged the powerless many into continual chaos; the empire expended more and more of its resources on militarism designed to maintain a system that, strained from within and threatened from without, was already long dead.
It is an environment like that into which Benedict of Nursia flung a rule for privileged Roman citizens calling for humility, a proper sense of self in a universe of wonders. When we make ourselves God, no one in the world is safe in our presence. Humility, in other words, is the basis for right relationships in life.
Later centuries distorted the notion and confused the concept of humility with lack of self-esteem and substituted the warped and useless practice of humiliations for the idea of humility. Eventually the thought of humility was rejected out of hand, and we have been left as a civilization to stew in the consequences of our arrogance.
Benedict's Magna Carta of humility directs us to begin the spiritual life by knowing our place in the universe, our connectedness, our dependence on God for the little greatness we have. Anything else, he says, is to find ourselves in the position of "a weaned child on its mother's lap," cut off from nourishment, puny, helpless—however grandiose our images of ourselves—and left without the resources necessary to grow in the Spirit of God. No infant child is independent of its mother, weaned or not. No spiritual maturity can be achieved independent of a sense of God's role in our development.
_Accordingly, if we want to reach the highest summit of humility, if we desire to attain speedily thatexaltation in heaven to which we climb by the humility of this present life, then by our ascending actions we must set up that ladder on which Jacob in a dream saw "angels descending and ascending" (Gen. 28:12). Without doubt, this descent and ascent can signify only that we descend by exaltation and ascend by humility. Now the ladder erected is our life on earth, and if we humble our hearts God will raise it to heaven. We may call our body and soul the sides of this ladder, into which our divine vocation has fitted the various steps of humility and discipline as we ascend._
Jacob's ladder is a recurring image of spiritual progress in classic spiritual literature, as clear in meaning to its time as the concept of the spiritual journey, for instance, would be to a later age. It connected heaven and earth. It was the process by which the soul saw and touched and climbed and clung to the presence of God in life, whose angels "descended and ascended" in an attempt to bring God down and raise us up. That ladder, that precariously balanced pathway to the invisible God, Benedict says, is the integration of body and soul. One without the other, it seems, will not do. Dualism is a hoax.
Just as false, though, is the idea that "getting ahead" and "being on top" are marks of real human achievement. Benedict says that in the spiritual life up is down and down is up: "We descend by exaltation and we ascend by humility." The goals and values of the spiritual life, in other words, are just plain different from the goals and values we've been taught by the world around us. Winning, owning, having, consuming, and controlling are not the high posts of the spiritual life. And this is the basis for social revolution in the modern world.
_Jan. 26 – May 27 – Sept. 26_
_The first step of humility, then, is that we keep "the reverence of God always before our eyes" (Ps. 36:2) and never forget it. We must constantly remember everything God has commanded, keeping in mind that all who despise God will burn in hell for their sins, and all who reverence God have everlasting life awaiting them. While we guard ourselves at every moment from sins and vices of thought or tongue, of hand or foot, of self-will or bodily desire, let us recall that we are always seen by God in the heavens, that our actions everywhere are in God's sight and are reported by angels at every hour._
The very consciousness of God in time is central to Benedict's perception of the spiritual life. Benedict's position is both shocking and simple: being sinless is not enough. Being steeped in the mind of God is most important. While we restrain ourselves from harsh speech and bad actions and demands of the flesh and pride of soul, what is most vital to the fanning of the spiritual fire is to become aware that the God we seek is aware of us. Sanctity, in other words, is not a matter of moral athletics. Sanctity is a conscious relationship with the conscious but invisible God. The theology is an enlivening and liberating one: it is not a matter, the posture implies, of our becoming good enough to gain the God who is somewhere outside of us. It is a matter of gaining the God within, the love of Whom impels us to good.
_Jan. 27 – May 28 – Sept. 27_
_The prophet indicates this to us, showing that our thoughts are always present to God, saying: "God searches hearts and minds" (Ps. 7:10); and again: "The Holy One knows our thoughts" (Ps. 94:11); likewise, "From afar you know my thoughts" (Ps. 139:3); and, "My thoughts shall give you praise" (Ps. 76:11). That we may take care to avoid sinful thoughts, we must always say to ourselves: "I shall be blameless in God's sight if I guard myself from my own wickedness" (Ps. 18:24)._
Benedict, whose whole way of life is steeped in the psalms, relies heavily on the psalms here to prove God's probing presence to the individual soul. God, Benedict says quite clearly, is within us to be realized, not outside of us to be stumbled upon. It is not a game of hide-and-seek we play in the spiritual life. It is simply a matter of opening our eyes to the light that drives out the darkness within us.
"How does a person seek union with God?" the seeker asked.
"The harder you seek," the teacher said, "the more distance you create between God and you."
"So what does one do about the distance?"
"Understand that it isn't there," the teacher said.
"Does that mean that God and I are one?" the seeker said.
"Not one. Not two."
"How is that possible?" the seeker asked.
"The sun and its light, the ocean and the wave, the singer and the song. Not one. Not two."
_Jan. 28 – May 29 – Sept. 28_
_Truly, we are forbidden to do our own will, for Scripture tells us: "Turn away from your desires" (Sir. 18:30). And in prayer too we ask that God's "will be done" in us (Matt. 6:10). We are rightlytaught not to do our own will, since we dread what Scripture says: "There are ways which some call right that in the end plunge into the depths of hell" (Prov. 16:25). Moreover, we fear what is said of those who ignore this: "They are corrupt and have become depraved in their desires" (Ps. 14:1)._
_As for the desires of the body, we must believe that God is always with us, for "All my desires are known to you" (Ps. 38:10), as the prophet tells God. We must then be on guard against any base desire, because death is stationed near the gateway of pleasure. For this reason Scripture warns us, "Pursue not your lusts" (Sir. 18:30)._
Benedict makes two points clearly: First, we are capable of choosing for God in life. We are not trapped by an essential weakness that makes God knowable but not possible. Second, we are more than the body. Choosing God means having to concentrate on nourishing the soul rather than on sating the flesh, not because the flesh is bad but because the flesh is not enough to make the human fully human. To give ourselves entirely to the pleasures of the body may close us to beauties known only to the soul.
Humility lies in knowing who we are and what our lives are meant to garner. The irony of humility is that, if we have it, we know we are made for greatness, we are made for God.
_Jan. 29 – May 30 – Sept. 29_
_Accordingly, if "the eyes of God are watching the good and the wicked" (Prov. 15:3), if at all times "the Holy One looks down from the heavens on us to see whether we understand and seek God" (Ps. 14:2); and if every day the angels assigned to us report our deeds to God day and night, then we must be vigilant every hour or, as the prophet says in the psalm, God may observe us "falling" at some time into evil and "so made worthless" (Ps. 14:3). After sparing us for a while because God is loving and waits for us to improve, we may be told later, "This you did, and I said nothing" (Ps. 50:21)._
The God-life, Benedict is telling us, is a never-ending, unremitting, totally absorbing enterprise. God is intent on it; so must we be. The Hebrew poet Moses Ibn Ezra writes, "Those who persist in knocking will succeed in entering." Benedict thinks no less. It is not perfection that leads us to God; it is perseverance.
_Jan. 30 – May 31 – Sept. 30_
_The second step of humility is that we love not our own will nor take pleasure in the satisfaction of our desires; rather we shall imitate by our actions that saying of Christ's: "I have come not to do my own will, but the will of the One who sent me" (John 6:38). Similarly we read, "Consent merits punishment; constraint wins a crown."_
The first rung of the ladder of the spiritual life is to recognize that God is God, that nothing else can be permitted to consume us or satisfy us, that we must reach out for God before we can even begin to live the God-life. We must come to understand that we are not our own destinies.
The second rung of the spiritual life follows naturally: if God is my center and my end, then I must accept the will of God, knowing that in it lies the fullness of life for me, however obscure. The question, of course, is, How do we recognize the will of God? How do we tell the will of God from our own? How do we know when to resist the tide and confront the opposition and when to embrace the pain and accept the bitterness because "God wills it for us." The answer lies in the fact that the Jesus who said, "I have come not to do my own will but the will of the One who sent me" is also the Jesus who prayed in Gethsemane, "Let this chalice pass from me." The will of God for us is what remains of a situation after we try without stint and pray without ceasing to change it.
_Jan. 31 – June 1 – Oct. 1_
_The third step of humility is that we submit to the prioress or abbot in all obedience for the love of God, imitating Jesus Christ, of whom the apostle says: "Christ became obedient even to death" (Phil. 2:8)._
It is so simple, so simplistic, to argue that we live for the God we do not see when we reject the obligations we do see. Benedictine spirituality does not allow for the fantasy. Benedict argues that the third rung on the ladder of humility is the ability to submit ourselves to the wisdom of another. We are not the last word, the final answer, the clearest insight into anything. We have one word among many to contribute to the mosaic of life, one answer of many answers, one insight out of multiple perspectives. Humility lies in learning to listen to the words, directions, and insights of the one who is a voice of Christ for me now. To stubbornly resist the challenges of people who have a right to lay claim to us and an obligation to do good by us—parents, spouses, teachers, supervisors—is a dangerous excursion into arrogance and a denial of the very relationships that are the stuff of which our sanctity is made.
Rungs one and two call for contemplative consciousness. Rung three brings us face to face with our struggle for power. It makes us face an authority outside of ourselves. But once I am able to do that, then there is no end to how high I might rise, how deep I might grow.
_Feb. 1 – June 2 – Oct. 2_
_The fourth step of humility is that in this obedience under difficult, unfavorable, or even unjust conditions, our hearts quietly embrace suffering and endure it without weakening or seeking escape. For Scripture has it: "Anyone who perseveres to the end will be saved" (Matt. 10:22), and again, "Be brave of heart and rely on God" (Ps. 27:14). Another passage shows how the faithful must endure everything, even contradiction, for the sake of the Holy One, saying in the person of those who suffer, "For yoursake we are put to death continually; we are regarded as sheep marked for slaughter" (Rom. 8:36; Ps. 44:22). They are so confident in their expectation of reward from God that they continue joyfully and say, "But in all this we overcome because of Christ who so greatly loved us" (Rom. 8:37). Elsewhere Scripture says: "O God, you have tested us, you have tried us as silver is tried by fire; you have led us into a snare, you have placed afflictions on our backs" (Ps. 66:10–11). Then, to show that we ought to be under a prioress or an abbot, it adds: "You have placed others over our heads" (Ps. 66:12)._
One thing about Benedict of Nursia: he is not a romantic. It is so easy to say, "Let God be the center of your life; do God's will; see God's will in the will of others for you." It is outrageous to say, even under the best of conditions, that it will be easy. We cling to our own ways like snails to sea walls, inching along through life, hiding within ourselves, unconscious even of the nourishing power of the sea that is seeking to sweep us into wider worlds.
And all of that when the words that control us are good for us. What about when they are not? Benedict admits the situation. There are times when the words of those over us will not be good for us.
The fourth step on the spiritual ladder, Benedict says, is the ability to persevere, even in the face of downright contradiction because it is more right to be open to the Word of God through others and have our enterprises fail sometimes than to be our own guide and have things turn out right.
It is more right to be able to deal with the difficult things of life and grow from them than it is to have things work out well all the time and learn nothing from them at all.
This is the degree of humility that calls for emotional stability, for holding on when things do not go our way, for withstanding the storms of life rather than having to flail and flail against the wind and, as a result, lose the opportunity to control ourselves when there is nothing else in life that we can control.
_In truth, those who are patient amid hardships and unjust treatment are fulfilling God's command: "When struck on one cheek, they turn the other; when deprived of their coat, they offer their cloakalso; when pressed into service for one mile, they go two" (Matt. 5:39–41). With the apostle Paul, they bear with "false companions, endure persecution, and bless those who curse them" (2 Cor. 11:26; 1 Cor. 4:12)._
To bear bad things, evil things, well is for Benedict a mark of humility, a mark of Christian maturity. It is a dour and difficult notion for the modern Christian to accept. The goal of the twenty-first century is to cure all diseases, order all inefficiency, topple all obstacles, end all stress, and prescribe immediate panaceas. We wait for nothing and put up with little and abide less and react with fury at irritations. We are a people without patience. We do not tolerate process. We cannot stomach delay. Persist. Persevere. Endure, Benedict says. It is good for the soul to temper it. God does not come on hoofbeats of mercury through streets of gold. God is in the dregs of our lives. That's why it takes humility to find God where God is not expected to be.
_Feb. 2 – June 3 – Oct. 3_
_The fifth step of humility is that we do not conceal from the abbot or prioress any sinful thoughts entering our hearts, or any wrongs committed in secret, but rather confess them humbly. Concerning this, Scripture exhorts us: "Make known your way to the Holy One and hope in God" (Ps. 37:5). And again, "Confess to the Holy One, for goodness and mercy endure forever" (Ps. 106:1; Ps. 118:1). So too the prophet: "To you I have acknowledged myoffense; my faults I have not concealed. I have said: Against myself I will report my faults to you, and you have forgiven the wickedness of my heart" (Ps. 32:5)._
The fifth rung of the ladder of humility is an unadorned and disarming one: self-revelation, Benedict says, is necessary to growth. Going through the motions of religion is simply not sufficient. No, the Benedictine heart, the spiritual heart, is a heart that has exposed itself and all its weaknesses and all its pain and all its struggles to the One who has the insight, the discernment, the care to call us out of our worst selves to the heights to which we aspire.
The struggles we hide, psychologists tell us, are the struggles that consume us. Benedict's instruction, centuries before an entire body of research arose to confirm it, is that we must cease to wear our masks, stop pretending to be perfect, and accept the graces of growth that can come to us from the wise and gentle hearts of people of quality around us.
Humility such as this gives us energy to face the world. Once we ourselves admit what we are, what other criticism can possibly demean us or undo us or diminish us? Once we know who we are, all the delusions of grandeur, all the righteousness that's in us dies and we come to peace with the world.
_Feb. 3 – June 4 – Oct. 4_
_The sixth step of humility is that we are content with the lowest and most menial treatment, and regard ourselves as a poor and worthless worker inwhatever task we are given, saying with the prophet: "I am insignificant and ignorant, no better than a beast before you, yet I am with you always" (Ps. 73:22–23)._
In a classless society status is snatched in normally harmless but corrosive little ways. We are a people who like embossed business cards and monogrammed leather briefcases and invitations to public events. We spend money we don't have to buy flat-screen TVs and electronic books that read to us instead of having to read a book for ourselves. We go into debt to buy at the right stores and live on the right street and go to the right schools. We call ourselves failures if we can't exchange last year's models for this year's styles. We measure our successes by the degree to which they outspan the successes of the neighbors. We have lost a sense of "enoughness."
Benedict tells us that it is bad for the soul to have to have more than the necessary, that it gluts us, that it protects us in Plexiglas from the normal, the natural. Benedict says that the goal of life is not to amass things but to get the most out of whatever little we have. Benedict tells us to quit climbing. If we can learn to love life where we are, in what we have, then we will have room in our souls for what life alone does not have to offer.
The _Tao Te Ching_ teaches, "Free from desire, you realize the mystery. Caught in desire, you see only the manifestations."
_Feb. 4 – June 5 – Oct. 5_
_The seventh step of humility is that we not only admit with our tongues but are also convinced in our hearts that we are inferior to all and of less value, humbling ourselves and saying with the prophet: "I am truly a worm, not even human, scorned and despised by all" (Ps. 22:7). "I was exalted, then I was humbled and overwhelmed with confusion" (Ps. 88:16). And again, "It is a blessing that you have humbled me so that I can learn your commandments" (Ps. 119:71, 73)._
At one stage of life, the temptation is to think that no human being alive could ever really believe themselves to be "inferior to all and of less value." At a later stage in life you begin to understand that secretly everybody thinks exactly that and that's why we deny it with such angst to ourselves and such unfairness to others. We set out systematically to hide the truth of it by clutching at money and degrees and positions and power and exhaust ourselves in the attempt to look better than we fear we really are.
The only difference between that stage of life and this degree of humility is that in the seventh degree of humility Benedict wants us to realize that accepting our essential smallness and embracing it frees us from the need to lie, even to ourselves, about our frailties. More than that, it liberates us to respect, revere, and deal gently with others who have been unfortunate enough to have their own smallnesses come obscenely to light.
Aware of our own meager virtues, conscious of our own massive failures despite all our great efforts, all our fine desires, we have in this degree of humility, this acceptance of ourselves, the chance to understand the failures of others. We have here the opportunity to become kind.
_Feb. 5 – June 6 – Oct. 6_
_The eighth step of humility is that we do only what is endorsed by the common rule of the monastery and the example set by the prioress or abbot._
"It is better to ask the way ten times than to take the wrong road once," a Jewish proverb reads. The eighth degree of humility tells us to stay in the stream of life, to learn from what has been learned before us, to value the truths taught by others, to seek out wisdom and enshrine it in our hearts. The eighth degree of humility tells us to attach ourselves to teachers so that we do not make the mistake of becoming our own blind guides.
It is so simple to become a law unto ourselves. The problem with it is that it leaves us little chance to be carried by others. It takes a great deal of time to learn all the secrets of life by ourselves. It makes it impossible for us to come to know what our own lights have no power to signal. It leaves us dumb, undeveloped, and awash in a naked arrogance that blocks our minds, cripples our souls, and makes us unfit for the relationships that should enrich us beyond our merit and despite our limitations.
Our living communities have a great deal to teach us. All we need is respect for experience and the comforting kind of faith that it takes to do what we cannot now see to be valuable, but presume to be holy because we see the holiness that it has produced in those who have gone before us in the family and the church.
_Feb. 6 – June 7 – Oct. 7_
_The ninth step of humility is that we control our tongues and remain silent, not speaking unless asked a question, for Scripture warns, "In a flood of words you will not avoid sinning" (Prov. 10:19), and "A talkative person goes about aimlessly on earth" (Ps. 140:12)._
When arrogance erupts anywhere, it erupts invariably in speech. Our opinions become the rule. Our ideas become the goal. Our judgments become the norm. Our word becomes the last word, the only word. To be the last one into a conversation, instead of the first, is an unheard-of assault on our egos. Benedict says, over and over, listen, learn, be open to the other. That is the ground of humility. And humility is the ground of growth and graced relationships on earth. Humility is what makes the powerful accessible to the powerless. Humility is what allows poor nations a demand on rich ones. Humility is what enables the learned to learn from the wise.
_Feb. 7 – June 8 – Oct. 8_
_The tenth step of humility is that we are not given to ready laughter, for it is written: "Only fools raise their voices in laughter" (Sir. 21:23)._
Humor and laughter are not necessarily the same thing. Humor permits us to see into life from a fresh and gracious perspective. We learn to take ourselves more lightly in the presence of good humor. Humor gives us the strength to bear what cannot be changed and the sight to see the human under the pompous. Laughter, on the other hand, is an expression of emotion commonly inveighed against in the best finishing schools and the upper classes of society for centuries. Laughter was considered vulgar, crude, cheap, a loud demonstration of a lack of self-control.
In the tenth degree of humility, Benedict does not forbid humor. On the contrary, Benedict is insisting that we take our humor very seriously. Everything we laugh at is not funny. Some things we laugh at are, in fact, tragic and need to be confronted. Ethnic jokes are not funny. Sexist jokes are not funny. The handicaps of suffering people are not funny. Pornography and pomposity and shrieking, mindless noise is not funny. Derision is not funny, sneers and sarcasm and snide remarks, no matter how witty, how pointed, how clever, how cutting, are not funny. They are cruel. The humble person never uses speech to grind another person to dust. The humble person cultivates a soul in which everyone is safe. A humble person handles the presence of the other with soft hands, a velvet heart, and an unveiled mind.
_Feb. 8 – June 9 – Oct. 9_
_The eleventh step of humility is that we speak gently and without laughter, seriously and with becoming modesty, briefly and reasonably, but without raising our voices, as it is written: "The wise are known by few words."_
Humility, Benedict teaches, treads tenderly upon the life around it. When we know our place in the universe, we can afford to value the place of others. We need them, in fact, to make up what is wanting in us. We stand in the face of others without having to take up all the space. We don't have to dominate conversations or consume all the time or call all the attention to ourselves. There is room, humility knows, for all of us in life. We are each an ember of the mind of God and we are each sent to illumine the other through the dark places of life to sanctuaries of truth and peace where God can be God for us because we have relieved ourselves of the ordeal of being god ourselves. We can simply unfold ourselves and become.
The _Tao Te Ching_ teaches the following:
The best people are like water,
They benefit all things,
And do not compete with them.
They settle in low places,
One with nature, one with Tao.
"Settling in low places," being gentle with others and soft in our comments and kind in our hearts and calm in our responses, never heckling, never smothering the other with noise or derision is an aspect of Benedictine spirituality that the world might well afford to revisit.
_Feb. 9 – June 10 – Oct. 10_
_The twelfth step of humility is that we always manifest humility in our bearing no less than in our hearts, so that it is evident at the Opus Dei, in the oratory, the monastery or the garden, on a journey or in the field, or anywhere else. Whether sitting, walking, or standing, our heads must be bowed and our eyes cast down. Judging ourselves always guilty on account of our sins, we should consider that we are already at the fearful judgment, and constantly say in our hearts what the publican in the Gospel said with downcast eyes: "I am a sinner, not worthy to look up to the heavens" (Luke 18:13). And with the prophet: "I am bowed down and humbled in every way" (Ps. 38:7–9; Ps. 119:107)._
This paragraph is, at first reading, a very difficult excursion into the tension between the apparent and the real. Bowing and scraping have long since gone out of style. What is to be made today of a dictum that prescribes bowed heads and downcast eyes in a culture given to straight-shouldered, steady-eyed self-esteem?
Benedict is telling us that true humility is simply a measure of the self that is taken without exaggerated approval or exaggerated guilt. Humility is the ability to know ourselves as God knows us and to know that it is the little we are that is precisely our claim on God. Humility is, then, the foundation for our relationship with God, our connectedness to others, our acceptance of ourselves, our way of using the goods of the earth and even our way of walking through the world, without arrogance, without domination, without scorn, without put-downs, without disdain, without self-centeredness. The more we know ourselves, the gentler we will be with others.
_Now, therefore, after ascending all these steps of humility, we will quickly arrive at the "perfect love" of God which "casts out fear" (1 John 4:18). Through this love, all that we once performed with dread, we will now begin to observe without effort, as though naturally, from habit, no longer out of fear of hell, but out of love for Christ, good habit, and delight in virtue. All this God will by the Holy Spirit graciously manifest in us now cleansed of vices and sins._
The chapter on humility is a strangely wonderful and intriguingly distressing treatise on the process of the spiritual life. It does not say, "Be perfect." It says, "Be honest about what you are and you will come to know God." It does not say, "Be flawless and you will earn God." It says, "If you recognize the presence of God in life, you will soon become more and more perfect." But this perfection is not in the twenty-first-century sense of impeccability. This perfection is in the biblical sense of having become matured, ripened, whole.
The entire chapter is such a nonmechanistic, totally human approach to God. If we reach out and meet God here where God is, if we accept God's will in life where our will does not prevail, if we are willing to learn from others, if we can see ourselves and accept ourselves for what we are and grow from that, if we can live simply, if we can respect others and reverence them, if we can be a trusting part of our world without having to strut around it controlling it, changing it, wrenching it to our own image and likeness, then we will have achieved "perfect love that casts out fear" (1 John 4:18). There will be nothing left to fear—not God's wrath, not the loss of human respect, not the absence of control, not the achievements of others greater than our own whose success we have had to smother with rejection or deride with scorn.
Humility, the lost virtue of our era, is crying to heaven for rediscovery. The development of nations, the preservation of the globe, the achievement of human community may well depend on it.
CHAPTER 8
THE DIVINE OFFICE AT NIGHT
_Feb. 10 – June 11 – Oct. 11_
_During the winter season, that is, from the first of November until Easter, it seems reasonable to arise at the eighth hour of the night. By sleeping until a little past the middle of the night, the community can arise with their food fully digested. In the time remaining after Vigils, those who need to learn some of the psalter or readings should study them._
_Between Easter and the first of November mentioned above, the time for Vigils should be adjusted so that a very short interval after Vigils will give the members opportunity to care for nature's needs. Then, at daybreak, Lauds should follow immediately._
Among the sayings of the Desert Monastics there is a story that may explain best Benedict's terse, clear instructions on prayer: "Once upon a time the disciples asked Abba Agathon, 'Among all good works, which is the virtue that requires the greatest effort?' Abba Agathon answered, 'I think there is no labor greater than that of prayer to God. For every time we want to pray, our enemies, the demons, want to prevent us, for they know that it is only by turning us from prayer that they can hinder our journey. Whatever good work a person undertakes, if they persevere in it, they will attain rest. But prayer is warfare to the last breath.'"
There are three dimensions of the treatment of prayer in the Rule of Benedict that deserve special attention. First, it is presented immediately after the chapter on humility. Second, it is not a treatise on private prayer. Third, it is scriptural rather than personal. Prayer is, then, the natural response of people who know their place in the universe. It is not designed to be a psychological comfort zone though surely comfort it must. And lastly, it is an act of community and an act of awareness.
Prayer, as Abba Agathon implies, is hard and taxing and demanding work. It breaks us open to the designs of God for life. It brings great insights and it demands great responses. It is based on the psalms, the very prayers that formed Jesus himself. And, most of all, it is unceasing. Day and night, Benedict says, day and night we must present ourselves before the face of God and beg for the insight and the courage it will take to go the next step.
There are volumes written on the structure and the history of the Divine Office: psalms, Scripture readings, and prayers that are identified as the official prayer of the church. What is most noteworthy here is not so much the ordering of the parts of the Office, which Benedict himself says in another place is not absolute, but the demonstration of humanity that undergirds the place of the Divine Office in the life of the monastic. The way Benedict deals with prayer says a great deal about the place of prayer in the life of us all even fifteen centuries later.
At first reading, the prayer life of Benedict's communities seems to be inhumanly rigorous and totally incompatible with modern life, either religious or lay. The monks are "to arise at the eighth hour of the night," the Rule says, and that is at least impossible for most people if not downright fanatical or destructive. It is important for a modern reader to realize, however, that the Roman night in a world without electric lights was computed from 6:00 P.M. to 6:00 A.M., from sundown to sunup. In this culture, in other words, the monks went naturally to bed at about 6:00 P.M. To wake at the eighth hour, then, was to wake at about 2:00 A.M., after eight full hours of sleep and the natural restoration of the body, to use the remaining hours before the beginning of the workday in prayer and study. The difference between us and the early monastic communities is that we extend our days at the end of them. We go to bed hours after sundown. They extended their days at the beginning of them; they got up hours before sunrise. The only question, given the fact that we both extend the workday hours, is what we do with the time. We stay up and watch television or go to parties or prolong our office hours. We fill our lives with the mundane. They got up to pray and to study the Scriptures. They filled their souls with the sacred.
CHAPTER 9
THE NUMBER OF PSALMS AT THE NIGHT OFFICE
_Feb. 11 – June 12 – Oct. 12_
_During the winter season, Vigils begin with the verse: "O God, open my lips and my mouth shall proclaim your praise" (Ps. 51:17). After this has been said three times, the following order is observed: Psalm 3 with doxology; Psalm 9 with a refrain, or at least chanted; an Ambrosian hymn; then six psalms with refrain._
_After the psalmody, a versicle is said and the prioress or abbot gives a blessing. When all are seated on the benches, the members in turn read three selections from the book on the lectern. After each reading a responsory is sung. The doxology is not sung after the first two responsories, but only after the third reading. As soon as the cantor begins to sing the doxology, let all rise from their seats in honor and reverence for the Holy Trinity. Besides the inspired books of the Old and New Testaments, the works read at Vigils should include explanations of Scripture by reputable and orthodox writers._
_When these three readings and their responsories have been finished, the remaining six psalms are sung with an "Alleluia" refrain. This ended, there follows a reading from the apostle recited by heart, a versicle and the litany, that is, "Christ, have mercy." And so Vigils are concluded._
In his instruction on the Night Office, Benedict supplies even the technologically advanced twenty-first century with valuable insights into prayer that may, at first sight, get lost in the strange details of the prayer format itself. Benedict asks for four elements of prayer, each of which gives a special dimension to the spiritual life: a specific versicle, the doxology, or Glory Be, responsories, and explanations of Scripture.
The versicle Benedict puts in our minds is a simple but important one: "O God, open my lips and my mouth shall proclaim your praise" (Ps. 51:17), he teaches us to pray. All life is in the hands of God. Even the desire to pray is the grace to pray. The movement to pray is the movement of God in our souls. Our ability to pray depends on the power and place of God in our life. We pray because God attracts us and we pray only because God is attracting us. We are not, in other words, even the author of our own prayer life. It is the goodness of God, not any virtue that we have developed on our own, that brings us to the heart of God. And it is with God's help we seek to go there.
The doxology, or Glory Be, gave solemnity to the Office but it gave witness as well to the divinity of all three persons of the Trinity, a concept disputed by the priest Arius, who held that Jesus was merely human, but one held to firmly by the church. To rise for the Glory Be, then, was to make a public witness to the divinity of Christ in an era when people were still divided on the subject, politically as well as theologically. It was a literal call to stand up for the faith, to claim the gospel publicly, a task that is demanded in every day and age including our own.
The direction to include responsories at prayer was a clear expectation that every member of the group would participate consciously in the act of prayer by reciting the responses that captured the spirit of what was being read to them in an age when manuscripts were rare, members were often illiterate, and prayer was more oral than written. Prayer is not something that is done to us or on us under any conditions. It is meant to engage us wholly—our minds, our bodies, and our souls—whatever its form. It is not a passive exercise. It is the work of God in us, and it demands our full attention.
Finally, Benedict introduces in chapter 9 what is central to Benedictine spirituality: immersion in the Scriptures. He wants us to do more than read them. He wants us to study them, to wrestle with them, to understand them, to make them part of us, to let them grow in us through the work of traditional and contemporary scholarship so that the faith can stay green in us.
Here, as a result of these concepts, is a prayer life grounded in faith, witness, attention, and serious study. Here is a prayer life that is serious, not superficial; concentrated not comfortable; full of witness, full of faith.
CHAPTER 10
THE ARRANGEMENT OF THE NIGHT OFFICE IN SUMMER
_Feb. 12 – June 13 – Oct. 13_
_From Easter until the first of November, the winter arrangement for the number of psalms is followed. But because summer nights are shorter, the readings from the book are omitted. In place of the three readings, one from the Old Testament is substituted. This is to be recited by heart, followed by a short responsory. In everything else, the winter arrangement for Vigils is kept. Thus, winter and summer, there are never fewer than twelve psalms at Vigils, not counting Psalms 3 and 95._
The very fact that when the seasons change and the nights grow shorter Benedict chooses to abbreviate the prayer time of the community rather than the sleeping time of the community gives short shrift to the notion of Benedictine spirituality as asceticism for its own sake or religion gone berserk or self-destruction masking as piety. No, the message of Benedictine spirituality is a consistent one: live life normally, live life thoughtfully, live life profoundly, live life well. Never neglect and never exaggerate. It is a lesson that a world full of cults and fads and workaholics and short courses in difficult subjects needs dearly to learn.
_Fill your bowl to the brim and it will spill._
_Keep sharpening your knife and it will blunt._
_Chase after money and security_
_and your heart will never unclench._
_Care about people's approval and you will be_
_their prisoner._
_Do your work, then step back._
_The only path to serenity._
CHAPTER 11
THE CELEBRATION OF VIGILS ON SUNDAY
_Feb. 13 – June 14 – Oct. 14_
_On Sunday the community should arise earlier for Vigils. In these Vigils, too, there must be moderation in quantity: first, as we have already indicated, six psalms are said, followed by a versicle. Then the members, seated on the benches and arranged in their proper order, listen to four readings from the book. After each reading a responsory is sung, but the doxology is added only to the fourth. When the cantor begins it, all immediately rise in reverence._
_**After these readings the same order is repeated: six more psalms with refrain as before, a versicle, then four more readings and their responsories, as above. Next, three canticles from the prophets, chosen by the prioress or abbot, are said with an "Alleluia" refrain. After a versicle and the blessing of the abbot or prioress, four New Testament readings follow with their responsories, as above. After the fourth responsory, the prioress or abbot begins the hymn "We Praise You, God." When that is finished, they read from the Gospels while all stand with respect and awe. At the conclusion of the Gospel reading, all reply "Amen," and immediately the prioress or abbot intones the hymn "To You Be Praise." After a final blessing, Lauds begin.**_
_This arrangement for Sunday Vigils should be followed at all times, summer and winter, unless—God forbid—the members happen to arise too late. In that case, the readings or responsories will have to be shortened. Let special care be taken that this not happen, but if it does, the one at fault is to make due satisfaction to God in the oratory._
By treating the recitation of the Sunday Office in a special way Benedict teaches all of us something fresh about prayer even today. The fact is that prayer is not to be a series of mindless mechanics in life. Prayer is the development of an attitude of mind that is concentrated and contemplative. For Benedict, therefore, the Sunday Office is a centerpiece that is fixed and solemn. The message is clear: Sunday, the weekly celebration of creation and resurrection, is always a reminder of new life, always special, always meant to take us back to the Beginning and the End, the Alpha and the Omega, the Center of life. It is a day full of tradition and rhythm and rememberings of the simple but important concepts of existence. It is a return to basic truths that are never to be sacrificed for variety and always reinforced through repetition.
The idea of Sabbaths that are fixed and solemn is for the most part gone in North American culture. Our Sundays are spent in hectic activity designed to make us relax by drowning out the pressures of the rest of the week with the inane uselessness of the weekends. In Benedictine spirituality, on the other hand, the Sabbath is the moment for returning to the surety and solemnity of life, for setting our sights above the daily, for restating the basics, for giving meaning to the rest of the week so that the mundane and the immediate do not become the level of our existence.
CHAPTER 12
THE CELEBRATION OF THE SOLEMNITY OF LAUDS
_Feb. 14 – June 15 – Oct. 15_
_Sunday Lauds begin with Psalm 66, said straight through without a refrain. Then Psalm 50 follows with an "Alleluia" refrain. Lauds continues with Psalms 117 and 62, the Canticle of the Three Young Men, Psalms 148 through 150, a reading from the Apocalypse recited by heart and followed by a responsory, an Ambrosian hymn, a versicle, the Gospel canticle, the litany, and the conclusion._
Every Sunday morning, just as day breaks, Benedict asks us to say five specific psalms: Psalm 67 asks for God's continuing blessings; Psalm 51 gives voice to our contrition; Psalm 118 recounts God's goodness in times past; Psalm 62 pours out a longing for God; and Psalms 148–150 bring the soul to a burst of praise. The structure itself, in other words, models the disposition of the soul before its God. At the beginning of the week, we ask for the energy of grace to go from this Sabbath to the next; we acknowledge the struggles of the week before us and the failings of the week that is past; we remember God's eternal fidelity in good times and bad; we recognize publicly that the great desire of our life is the desire for God, whatever else distracts us on the way; and, finally, we give our lives in thanksgiving to the One Who has brought us this far and Who is our final goal and our constant hope.
Sunday Lauds in the monastic liturgy is a soul-splitting commitment to go on. The point is that every life needs points along the way that enable us to rise above the petty daily problems, the overwhelming tragedies of our lives and begin again, whatever our circumstances, full of confidence, not because we know ourselves to be faithful, but because our God is.
CHAPTER 13
THE CELEBRATION OF LAUDS ON ORDINARY DAYS
_Feb. 15 – June 16 – Oct. 16_
_On ordinary weekdays, Lauds are celebrated as follows. First, Psalm 67 is said without a refrain and slightly protracted as on Sunday so that everyone can be present for Psalm 51, which has a refrain. Next, according to custom, two more psalms are said in the following order: on Monday, Psalms 5 and 36; on Tuesday, Psalms 43 and 57; on Wednesday, Psalms 64–65; on Thursday, Psalms 88 and 90; on Friday, Psalms 76 and 92; on Saturday, Psalm 143 and the Canticle from Deuteronomy, divided into two sections, with the doxology after each section. On other days, however, a Canticle from the prophets is said,according to the practice of the Roman Church. Next follow Psalms 148 through 150, a reading from the apostle recited by heart, a responsory, an Ambrosian hymn, a versicle, the Gospel canticle, the litany, and conclusion._
At the break of dawn, every day of the week, Benedict, through his organization of the morning psalms, reminds the monastic of two unfailing realities. The first is that life is not perfect, that struggle is to be expected, that the human being lives on the brink of danger and defeat at all times. As proof of that, the first Psalm of Lauds, every day of the week, is a cry for help (Psalm 5), a cry for vindication (Psalm 43), a cry for protection even from secret enemies (Psalm 64), a cry to be saved from depression, the death of the spirit, and on Friday, in Psalm 76, a review of the power of God in their lives.
In the second Psalm of Lauds, Benedict arranges a paean of praise, one after another, every day of the week in Psalms 36, 57, 65, 90, and 92 until, on Saturday, having lived through everything life had to give that week, the community bursts into unending praise for having survived it, learned faith in God from it again, and been saved one more time by a loving God.
Lauds becomes an unending lesson in reality and faith, in accepting what life brings, sure in the knowledge that the God who loves us is with us upholding us all the way.
_Feb. 16 – June 17 – Oct. 17_
_Assuredly, the celebration of Lauds and Vespers must never pass by without the prioress or abbot reciting the entire Prayer of Jesus at the end for all to hear, because thorns of contention are likely to spring up. Thus warned by the pledge they make to one another in the very words of this prayer: "Forgive us as we forgive" (Matt. 6:12), they may cleanse themselves of this kind of vice. At other celebrations, only the final part of this prayer is said aloud, that all may reply: "But deliver us from evil" (Matt. 6:13)._
"Each of us should have two pockets," the rabbis teach. "In one should be the message, 'I am dust and ashes,' and in the other we should have written, 'For me the universe was made.'"
These ideas are clearly Benedict's as well. Two things he does not want us to omit from our prayer lives are Psalm 67's plea for continued blessing and Psalm 51's need for continual forgiveness; a sense of God's goodness and our brokenness; a sense of God's greatness and our dependence; a sense of God's grandeur and our fragility. Prayer, for Benedict, is obviously not a routine activity. It is a journey into life, its struggles and its glories. It is sometimes difficult to remember, when days are dull and the schedule is full, that God has known the depth of my emptiness but healed this broken self regardless, which, of course, is exactly why Benedict structures prayer around Psalm 67 and Psalm 51. Day after day after day.
Then Benedict arranges the rest of the morning psalmody for the remainder of the week to remind us of the place God takes in human life. On Monday Benedict requires the saying of Psalms 5 and 36 to remind us at the beginning of every week that God is a God who "hears the voice" of those who "at daybreak lay their case" before the holy temple and who "maintains a faithful love." On Tuesday he prescribes Psalms 43 and 57 to remind us in the weight of the day that God is our hope, our joy, our defense. On Wednesday he prescribes Psalms 64 and 65 to recall to us when we are tempted to give in to our lesser selves, out of fatigue, out of stress, out of the ennui of the week, that God does punish evildoers, those who "shoot at the innocent from cover," and God does indeed "calm the turmoil of the seas." On Thursday, as the week wears on, Benedict's prayer structure assures us in Psalms 88 and 90 that distress is that part of life in which God is present in absence but that God "is our refuge" who each morning "fills us with faithful love" so that "we shall sing and be happy all our days." On Saturday, at the end of the week, with new lessons learned and new problems solved and new deaths survived, Benedict puts Psalm 143 and the Canticle of Deuteronomy in our hearts.
Moses reminds us by an excursion through history that God is "a trustworthy God who does no wrong." Whatever has happened to us in these days has been for our good, too, we are very subtly instructed, so that we can pray Psalm 143 in confidence of the week to come: "Show me the road I must travel for you to relieve my heart."
Monastic morning prayer is not an idle ordering of psalms. It is a treatise on the monastic mind-set that is to characterize those who claim to be giving their lives to God.
Finally, Benedict's prayer form requires a realistic appraisal of community life. "The celebration...must never pass by without reciting the entire Prayer of Jesus at the end for all to hear, because thorns of contention are likely to spring up." The Prayer of Jesus is designed to heal and cement and erase the pain and struggle of community life, of family life, of global life where we all live together at one another's expense.
Benedictine prayer is not an escape into a contrived or arcane life. It is prayer intended to impel us through the cold, hard, realities of life in the home, life in the community, life in the world, life with people whom we love enough to hate and whom we hate enough to dampen every other kind of love in us.
CHAPTER 14
THE CELEBRATION OF VIGILS ON THE ANNIVERSARIES OF SAINTS
_Feb. 17 – June 18 – Oct. 18_
_On the feasts of saints, and indeed on all solemn festivals, the Sunday order of celebrations is followed, although the psalms, refrains, and readings proper to the day itself are said. The procedure, however, remains the same as indicated above._
The meaning of this chapter is not so much in its content as in its existence. The fact that it is here at all in a document written when the identification of saints was largely a matter of public acclamation and their number far fewer than now says something about Benedict's ideas both about church and the meaning of prayer. Benedict's theology of prayer is as much attuned to the Communion of Saints, to our connectedness to those who have gone before us in the faith, to those who stand as sign to us that the Christian life is possible, as it is to the feasts that mark the Paschal Mystery of Christ.
We all need heroes. We all need someone in our lives who brings courage. We all need to get to know how the Christian life looks at its best, at its most difficult, at its most joyous.
The lesson is that we must keep the human dimensions of the faith very much in mind and find in models from the past proof that daily chaos can be ordered and the ordinary transfigured for us, too.
CHAPTER 15
THE TIMES FOR SAYING ALLELUIA
_Feb. 18 – June 19 – Oct. 19_
_From the holy feast of Easter until Pentecost, "Alleluia" is always said with both the psalms and the responsories. Every night from Pentecost until the beginning of Lent, it is said only with the last six psalms of Vigils. Vigils, Lauds, Prime, Terce,Sext, and None are said with "Alleluia" every Sunday except in Lent; at Vespers, however, a refrain is used. "Alleluia" is never said with responsories except from Easter to Pentecost._
The Navahos wrote, "We felt like talking to the ground, we loved it so." Ralph Waldo Emerson wrote, "The earth laughs in flowers." Benedict of Nursia wrote, say "Alleluia" always, no matter the time of day, no matter the season of life.
The use of the Alleluia dates back to the earliest of liturgical formularies, both Jewish and Christian, as an endless chant of joy. In the Christian community it was an expression of praise and a foretaste of eternal gladness. "We are an Easter people," Augustine wrote, "and Alleluia is our cry."
Benedict of Nursia did not originate the use of the Alleluia but one thing he did do was to extend its use to every day of the year except during Lent.
The prescription is a telling one. To the Benedictine mind, life in all its long nights and weary days is something to be praised, death is the rivet of joy, there is no end to the positive. Even life in hot fields and drab offices and small houses is somehow one long happy thought when God is its center, and blessings, however rare, however scant, are blessed.
CHAPTER 16
THE CELEBRATION OF THE DIVINE OFFICE DURING THE DAY
_Feb. 19 – June 20 – Oct. 20_
_The prophet says: "Seven times a day have I praised you" (Ps. 119:164). We will fulfill this sacred number of seven if we satisfy our obligations of service at Lauds, Prime, Terce, Sext, None, Vespers, and Compline, for it was of these hours during the day that it was said: "Seven times a day have I praised you" (Ps. 119:164). Concerning Vigils, the same prophet says: "At midnight I arose to give you praise" (Ps. 119:62). Therefore, we should "praise our Creator for just judgments" at these times: Lauds, Prime, Terce, Sext, None, Vespers, and Compline; and "Let us arise at night to give praise" (Ps. 119:164, 62)._
"Prayer is the service of the heart," the Talmud says. Benedict clearly thought the same. In forming his communities in prayer, Benedict had two realities with which to deal. The first was the biblical injunction "to pray always," around which the monastics of the desert had centered their lives. The second was the reality of community life itself: "We earn our bread by the toil of our hands," the Rule says.
The problem was that Benedict's monks were not hermits who scratched their daily fare out of a dry desert, living on locusts and honey. They were not gyrovagues, wandering monks, who, to demonstrate their dependence on God, begged their way through life. Benedict's monks were cenobites, community people with a family to support. They were each as responsible for their inexperienced young and worn-out elderly as they were for themselves. They were, in other words, just like us.
To sanctify both situations Benedict instructs his communities to rise early in the night, as his culture allowed, to study and to pray and then, during the day, to recite brief, simple, scriptural prayers at regular intervals, easy enough to be recited and prayed even in the workplace, to wrench their minds from the mundane to the mystical, away from concentration on life's petty particulars to attention on its transcendent meaning.
Benedict scheduled prayer times during the day to coincide with the times of the changing of the Roman imperial guard. When the world was revering its secular rulers Benedict taught us to give our homage to God, the divine ruler of heaven and earth. There was to be no stopping at the obvious, at the lesser, for a Benedictine.
The point is clear: there is to be no time, no thing, that absorbs us so much that we lose contact with the God of life; no stress so tension producing, no burden so complex, no work so exhausting that God is not our greatest agenda, our constant companion, our rest and our refuge. More, whatever other people worship, we are to keep our minds and hearts on God.
CHAPTER 17
THE NUMBER OF PSALMS TO BE SUNG AT THESE HOURS
_Feb. 20 – June 21 – Oct. 21_
_We have already established the order for psalmody at Vigils and Lauds. Now let us arrange the remaining hours._
_Three psalms are to be said at Prime, each followed by Glory Be. The hymn for this hour issung after the opening versicle, "God, come to my assistance" (Ps. 70:2), before the psalmody begins. One reading follows the three psalms, and the hour is concluded with a versicle, "Lord, have mercy," and the dismissal._
_Prayer is celebrated in the same way at Terce, Sext, and None: that is, the opening verse, the hymn appropriate to each hour, three psalms, a reading with a versicle, "Lord, have mercy," and the dismissal. If the community is rather large, refrains are used with the psalms; if it is smaller, the psalms are said without refrain._
_At Vespers the number of psalms should be limited to four, with refrain. After these psalms there follow: a reading and responsory, an Ambrosian hymn, a versicle, the Gospel Canticle, the litany, and, immediately before the dismissal, the Lord's Prayer._
_Compline is limited to three psalms without refrain. After the psalmody comes the hymn for this hour, followed by a reading, a versicle, "Lord, have mercy," a blessing, and the dismissal._
Perhaps the most important point to be made about the structure of prayer during the day hours, during the periods of distraction and the times of work, is simply this. Even then, prayer is to be prayer, not a glancing thought, not a shrug or a gesture or a mindless moment of empty daydreaming. It is to be brief, yes. It is not, however, to be superficial. Benedict wants us to pray the psalms. His own monks, many of them illiterate and all of them without manuscripts, memorized the psalms of the day hours so that they could be prayed in the fields as well as in the prayer place.
This chapter, consequently, of all the chapters in the Rule on prayer is a real challenge to a modern society. What psalm prayers can we say without reading? What prayers ring in our hearts? What do we think about when we're not thinking about anything special? Do we ever simply stop the work we are doing during the day, look straight ahead, and pray? What memorized material does run through our minds and why do we memorize what we do but not our prayers?
CHAPTER 18
THE ORDER OF THE PSALMODY
_Feb. 21 – June 22 – Oct. 22_
_Each of the day hours begins with the verse, "O God, come to my assistance; O God, make haste to help me" (Ps. 70:2), followed by the doxology and the appropriate hymn._
_Then, on Sunday at Prime, four sections of Psalm 119 are said. At the other hours, that is, at Terce, Sext, and None, three sections of this psalm are said. On Monday three psalms are said at Prime: Psalms 1, 2, and 6. At Prime each day thereafter until Sunday, three psalms are said in consecutive order as far as Psalm 20. Psalms 9 and 18 are each divided into two sections. In this way, Sunday Vigils can always begin with Psalm 21._
The psalms for Prime and the day hours of the psalmody—Terce, Sext and None—are relatively ordinary. They simply recite Psalms 1–20 in order. But they do it with two major emphases. The first is the opening of the Office with the verse, "O God, come to my assistance," the continuing reminder that even prayer is a gift from God.
The second is to form a kind of drumbeat for the highlight of the next week, the Vigil of Sunday that opens always with Psalm 21, which stands as both warning and promise. It details the underlying truth of life: the monastic is to remember, however powerless they may feel, that no ruler is as powerful as God; no ruler deserves our praise as does God; no ruler really rules anyone. However powerful particular rulers may seem, we know that in the end it is God who will prevail, it is God in whom we must put our trust.
_Feb. 22 – June 23 – Oct. 23_
_On Monday at Terce, Sext, and None, the remaining nine sections of Psalm 119 are said, three sections at each hour. Psalm 119 is thus completed in two days, Sunday and Monday. On Tuesday, three psalms are said at each of the hours of Terce, Sext, and None. These are the nine Psalms 120–128. The same psalms are repeated at these hours daily up to Sunday. Likewise, the arrangement of hymns, readings, and versicles for these days remains the same. In this way, Psalm 119 will always begin on Sunday._
The minor hours—Terce, Sext and None—are descants in the structure of Benedict's daily office. They repeat the same messages over and over. Over and over, every day of their lives the monastic hears the same message: God delivers us, God is our refuge, God will save us from those who seek to destroy us, God will bring us home. The words are haunting: "When I am in trouble, I call to Yahweh and God answers me..."; "Pity us, Yahweh, take pity on us...."; and finally, "What marvels indeed Yahweh did for us...for those who once sowing in tears now sing as they reap."
In the minor hours, the psalms carry us from hardship to joy, from inner captivity to liberation, from despair to trust. It is a message to us all that remembering to trust in God can be enough to carry us for a lifetime.
_Feb. 23 – June 24 – Oct. 24_
_Four psalms are sung each day at Vespers, starting with Psalm 110 and ending with Psalm 147, omitting the psalms in this series already assigned to other hours, namely, Psalms 118 through 128, Psalm 134 and Psalm 143. All the remaining psalms are said at Vespers. Since this leaves three psalms toofew, the longer ones in the series should be divided: that is, Psalms 139, 144, and 145. And because Psalm 117 is short, it can be joined to Psalm 116. This is the order of psalms for Vespers; the rest is as arranged above: the reading, responsory, hymn, versicle, and canticle._
In determining the order of the psalms for the prayer life of his community, Benedict grounds Prime, Terce, Sext, and None, the Little Hours of the Divine Office, in the wisdom psalm, 119. Wisdom psalms were not liturgical hymns of lament or praise. They were meant to instruct the assembly in divine truths and were often built on the alphabet in order to make memorization easier. Modern educators write children's books or songs in the same way and for the same reason. Psalm 119, therefore, has twenty-two sections, with each of the eight verses of each section beginning with one of the letters of the Hebrew alphabet.
It is this longest of all psalms, with its theme of the trustworthiness of God's law, the richness of God's will for us, the excellence of God's loving design for us, that Benedict wants us to learn and say daily and never forget.
_Feb. 24 (in leap year, otherwise add to preceding) – June 25 Oct. 25_
_The same psalms—4, 91, and 134—are said each day at Compline._
_The remaining psalms not accounted for in this arrangement for the day hours are distributed evenlyat Vigils over the seven nights of the week. Longer psalms are to be divided so that twelve psalms are said each night._
Compline, the night prayer of the community, was built around three psalms designed to do what we all need to do at night: recognize that what we did that day was not perfect, hope that the next day will be better, praise the God whose love and grace brought us through another day, and go to bed trusting that the God who sees our every action is more concerned with our motives than with our failures.
_Above all else we urge that if people find this distribution of the psalms unsatisfactory, they should arrange whatever they judge better, provided that the full complement of one hundred and fifty psalms is by all means carefully maintained every week, and that the series begins anew each Sunday at Vigils. For members who in a week's time say less than the full psalter with the customary canticles betray extreme indolence and lack of devotion in their service. We read, after all, that our holy ancestors, energetic as they were, did all this in a single day. Let us hope that we, lukewarm as we are, can achieve it in a whole week._
Finally, Benedict implies very clearly in this chapter on the order of the psalms that a full prayer life must be based on a total immersion in all the life experiences to which the psalms are a response. The order of the psalms is not nearly so important to Benedict as the fact that the entire 150 psalms are to be said each and every week. The Benedictine is not to pick and choose at random the psalms that will be said. The Benedictine is not to pick some psalms but not others. The Benedictine is to pray the entire psalter in an orderly way, regardless of mood, irrespective of impulses, despite personal preferences. Anything other than regular recitation and total immersion in the psalms is, to Benedict's way of thinking, spiritual sloth. Ours is to be a full spiritual palate. Readings may be shortened if situations warrant but the psalms, never. We are to tap into every human situation that the psalms describe and learn to respond to them with an open soul, an unfettered heart, and out of the mind of God.
CHAPTER 19
THE DISCIPLINE OF PSALMODY
_Feb. 24 (25) – June 26 – Oct. 26_
_We believe that the divine presence is everywhere and "that in every place the eyes of God are watching the good and the wicked" (Prov. 15:3). But beyond the least doubt we should believe this to be especially true when we celebrate the Divine Office._
_We must always remember, therefore, what the prophet says: "Serve the Holy One with reverence" (Ps. 2:11), and again, "Sing praise wisely" (Ps. 47:8); and, "in the presence of the angels I will sing to you" (Ps. 138:1). Let us consider, then, how we ought to sing the psalms in such a way that our minds are in harmony with our voices._
"The unexamined life is not worth living," the philosopher Socrates said. Benedict implies the same. If indeed we walk in the womb of God, then reflection on the meaning of every action and the end of every road is the constant to which we are called. There must be no such thing as the idle decision, the thoughtless act. Every part of our lives must be taken to prayer and the scrutiny of Scripture must be brought to every part of our lives because we believe "beyond the least doubt" that the God we seek is there seeking us.
Prayer in the Benedictine tradition, then, is not an exercise done for the sake of quantity or penance or the garnering of spiritual merit. Benedictine prayer is not an excursion into a prayer-wheel spirituality where more is better and recitation is more important than meaning. Prayer, in the spirit of these chapters, if we "sing praise wisely," or well, or truly, becomes a furnace in which every act of our lives is submitted to the heat and purifying process of the smelter's fire so that our minds and our hearts, our ideas and our lives, come to be in sync, so that we are what we say we are, so that the prayers that pass our lips change our lives, so that God's presence becomes palpable to us. Prayer brings us to burn off the dross of what clings to our souls like mildew and sets us free for deeper, richer, truer lives in which we become what we seek.
CHAPTER 20
REVERENCE IN PRAYER
_Feb. 25 (26) – June 27 – Oct. 27_
_Whenever we want to ask a favor of someone powerful, we do it humbly and respectfully, for fear of presumption. How much more important, then, to lay our petitions before the God of all with the utmost humility and sincere devotion. We must know that God regards our purity of heart and tears of compunction, not our many words. Prayer should therefore be short and pure, unless perhaps it is prolonged under the inspiration of divine grace. In community, however, prayer should always be brief; and when the prioress or abbot gives the signal, all should rise together._
The rabbis taught that "the first time a thing occurs in nature, it is called a miracle; later it becomes natural, and no attention is paid to it. Let your worship and your prayer be a fresh miracle every day to you. Only such worship, performed from the heart, with enthusiasm, is acceptable." The function of prayer is not to establish a routine; it is to establish a relationship with the God who is in relationship with us always. The function of times of prayer, then, is not to have us say prayers; it is to enable our lives to become a prayer outside of prayer, to become "pure of heart," one with God, centered in the truth that is Truth and the power that is Power and the love that is Love.
The function of prayer is to bring us into touch with ourselves, as well. To the ancients, "tears of compunction" were the sign of a soul that knew its limits, faced its sins, accepted its needs, and lived in hope. That's what Benedict wants for those who live the prayer life he describes: not long hours spent in chapel but a lifetime lived in the Spirit of God because the chapel time was swift and strong, quick and deep, brief but soul shaking. Prayer is "to be short and pure," he says, not long and tedious, not long and majestic, not long and grand. No, Benedictine prayer is to be short and substantial and real. The rest of life is to be impelled by it. To live in church, as far as Benedict is concerned, is not necessarily a sign of holiness. To live always under the influence of the Scriptures and to live in the breath of the Spirit is.
There are some who would look at the Rule of Benedict and be surprised that it does not contain a discourse on prayer instead of simply the description of a form of prayer. The fact is, of course, that Benedict does not theorize about the nature and purpose of prayer. All he does, with every choice he makes of the versicles and Alleluias and Jesus Prayers and psalms and length of it, is to demonstrate it and steep us in it until the theory becomes the thing.
CHAPTER 21
THE DEANS OF THE MONASTERY
_Feb. 26 (27) – June 28 – Oct. 28_
_If the community is rather large, some chosen for their good repute and holy life should be made deans. They will take care of their groups of ten, managing all affairs according to the commandments of God and the orders of their prioress or abbot. Anyone selected as a dean should be the kind of person with whom the prioress or abbot can confidently share the burdens of office. They are to be chosen for virtuous living and wise teaching, not for their rank._
In one simple paragraph Benedict does away with the notion of absolute hierarchy and the divine right to anything. The abbot and prioress are to be the last word in a Benedictine community, but they are not to be its only word. They are to "share the burdens of their office," not simply delegate them, with those members of the community who themselves are models of the monastic life. The age of a person or the number of years a person has been in the monastery has nothing to do with the decision to give one person rather than another a position of responsibility or authority in the group. What counts is the quality of their community life, the prayerfulness of their lives, their commitment to Benedictine values.
Whoever the leaders, the central thesis of the chapter remains: the community belongs to the community. Its sanctity and success do not rise and fall on the shoulders of one leader alone. They rise and fall on the shoulders of its members. What they are the community shall be.
It is an important concept in a culture that calls itself classless but that relies heavily on connections and prestige and money to define its centers of power and so overlooks the values and voices of many.
_If perhaps one of these deans is found to be puffed up with any pride, and so deserving of censure, they are to be reproved once, twice, and even a third time. Should they refuse to amend, they must be removed from office and replaced by another who is worthy. We prescribe the same course of action in regard to the subprioress or prior._
To share authority is not the same as to give it away. To share authority means that those who are responsible for the group must arrive at common decisions, share a common wisdom, come to a common commitment, and then teach it together in such a way that the community is united, not divided, by the people chosen to lead it. To give authority away is to abdicate it, to leave the group open to division, disunity, and destruction.
The government of a Benedictine community is to come out of a common vision, a common heart. There is one interpreter of the Rule in every Benedictine monastery, the abbot or prioress, who themselves are immersed in Scripture and who have listened to the experience of the community and bring those elements to bear on every present situation. The unity of the community depends on the centrality of that teaching. To divide a group into factions until the unity of the teaching pales, to tear at its center until its fabric frays and rends, to refuse to give focus to its focus, is to strike at the very heart of Benedictine spirituality. It is not possible to form a group when the group is being divided over the very items on which it should be being developed.
What Benedict is inveighing against, then, is the spirit of the coup d'état, that war that is waged against authority by the very people named by the authority to uphold it. The person with a Benedictine mind-set goes into the parish council or the union office or the hospital board to cooperate with the leadership, to carry the group, not to tug it to pieces over inconsequential matters for some gain of personal aggrandizement and ego satisfaction. A Benedictine family does not draw and quarter the children with two different sets of expectations. Benedictine spirituality uses authority to weld a group, not to fracture it.
CHAPTER 22
THE SLEEPING ARRANGEMENTS OF MONASTICS
Feb. 27 (28) – June 29 – Oct. 29
_Members are to sleep in separate beds. They receive bedding as provided by the prioress or abbot, suitable to monastic life._
At first glance, the paragraph seems pathetically mundane for anything so exalted as "the most influential monastic rule of all time." It is, on the contrary, exactly paragraphs like this that make the Rule so influential.
In a culture of peasants who came out of clans where whole families slept in one room—and still do in many poorer areas of the world—Benedict proclaims a policy of at least limited privacy and simplicity and adaptation. Benedict wants an atmosphere of self-sacrifice, true, but he also wants people to have opportunity for reflection. He wants no living situation to be so austere that both sleep and thinking become impossible in the cold of winter. In Benedictine spirituality people get what they need, both beds and bedding, both privacy and personal care.
The lesson is a good one when we are tempted to think that extremism is a virtue. As far as Benedictine spirituality is concerned, there is a very limited spiritual value in denying the body to the point where the soul is too agitated to concentrate on the things of the spirit.
_If possible, all are to sleep in one place, but should the size of the community preclude this, they will sleep in groups of ten or twenty under the watchful care of elders. A lamp must be kept burning in the room until morning._
The dormitory is of ancient origin in the monastic tradition. It carried the concept of community living from the chapel to the dining room to bedtime itself. The common life was indeed a common life for twenty-four hours of every day, with all the difficulty and all the virtue that implied. Nevertheless, the sleeping arrangements present in monastic communities of the sixth century were not very different from family circumstances of the same period. Nor were bedrooms in communities of manual laborers the study centers they were to become as monastics of later centuries became more engaged in intellectual labors.
What is important in the paragraph is not so much the sleeping arrangement itself as the underlying caution it presents to an era in which independence, individualism, and personal space have become values of such magnitude that they threaten the communal quality of the globe itself. The question becomes, what part of our lives do we really practice with others? Has our claim to the private and the personal evicted the world from our space, from our hearts?
_They sleep clothed, and girded with belts or cords; but they should remove their knives, lest they accidentally cut themselves in their sleep. Thus the members will always be ready to arise without delay when the signal is given; each will hasten to arrive at the Opus Dei before the others, yet with all dignity and decorum. The younger members should not have their beds next to each other, but interspersed among those of the elders. On arising for the Opus Dei, they will quietly encourage each other, for the sleepy like to make excuses._
In this instruction, monastics are formed to be modest—dressed even in bed, unlike a good proportion of the population of the time; and simple—willing to wear the same thing at night that they did during the day; and ready—quick to respond to the will of God at the first sound of the call. They are trained, too, to "quietly encourage each other" in the daily effort of rousing the soul when the body is in revolt.
Personal modesty, simplicity, readiness, and encouragement in life may well be the staples of community living, of family life, or decent society even today. What, after all, can shatter any group faster than the one person who is dedicated to being conspicuous, overdone, resistant, or self-centered?
CHAPTER 23
EXCOMMUNICATION FOR FAULTS
_Feb. 28 (29) – June 30 – Oct. 30_
_If monastics are found to be stubborn or disobedient or proud, if they grumble or in any way despise the holy rule and defy the orders of the elders, they should be warned twice privately by them in accord with Christ's injunction (Matt. 18:15, 16). If they do not amend, they must be rebuked publicly in the presence of everyone. But if even then they do not reform, let them be excommunicated, provided that they understand the nature of this punishment. If however they lack understanding, let them undergo corporal punishment._
One of the sages said, "I never met anyone in whom I failed to recognize something superior to myself: if the person was older, I said this one has done more good than I; if younger, I said this person has sinned less; if richer, I said this one has been more charitable; if poorer, I said this one has suffered more; if wiser, I honored their wisdom; and if not wiser, I judged their faults more lightly." Community is the place where we come to honor the world.
In one of the gentlest monastic rules ever written, Benedict devotes eight straight chapters to punishment and its techniques, none of them either very acceptable or very applicable today. His concept of punishment, if not his form of punishment, however, may well bear considerable reflection in our own time.
In the first place, Benedict does not punish severely for everything. He does not punish for incompetence or lack of spiritual intensity or ignorance or weaknesses of the flesh. No, Benedict punishes harshly only for the grumbling that undermines authority in a community and the rebellion that paralyzes it. Benedict punishes severely only for the destruction of the sense of community itself.
It is community that enables us both to live the Christian life and to learn from it. Human growth is gradual, Benedict knows—the grumblers and defiant are to be warned about their behavior twice privately—but grow we must. Otherwise those who do not honor the community, those who in fact sin against the development of community in the worst possible way, by consistent complaining, constant resistance, or outright rebellion, must be corrected for it.
In the second place, Benedict does not set out simply to reason with us about the disordered parts of our lives. Benedict intends to stop an action before it takes root in us. Physical punishment was common in a culture of the unlettered. Many monastic rules, in fact—the Penitential of St. Columbanus, the Rule of St. Fructuosus, the Rule of the Master—specify as many as a hundred lashes for offenses against the rules. At the same time, Benedict prefers another method more related to the nature of the sins. If we refuse to learn from the community and to cooperate with it, he implies, we have no right to its support and should be suspended from participation in it. Once we have separated ourselves from the community by withdrawing our hearts, then the community must withdraw from us in order to soften them.
There may be another point to be made, as well. Mild as it may have been according to the standards of the day, Benedict did mandate punishments and he did require atonement. The Rule would certainly expect the same attitudes from us even now. Things broken must be mended; things running away with us must be curbed; things awry in us must be set straight. What we may have to face in a culture in which self-control is too often seen as self-destructive is that none of that happens by accident. It requires discipline—conscious, honest, continuing discipline, not in the ways that discipline may have been prescribed in the sixth century, surely, but in some way that is honest and real.
CHAPTER 24
DEGREES OF EXCOMMUNICATION
_March 1 – July 1 – Oct. 31_
_There ought to be due proportion between the seriousness of a fault and the measure of excommunication or discipline. The prioress or abbot determines the gravity of faults._
_If monastics are found guilty of less serious faults, they will not be allowed to share the common table. Members excluded from the common table will conduct themselves as follows: in the oratory they will not lead a psalm or a refrain nor will they recite a reading until they have made satisfaction, and they will take meals alone, after the others have eaten. For instance, if the community eats at noon, they will eat in midafternoon; if the community eats in midafternoon, they will eat in the evening, until by proper satisfaction pardon is gained._
Chapter 24 makes two important points in the psychology of punishment and human association: first, the need to punish is no excuse for the arbitrary wielding of power and anger and vengeance; second, sins against community rupture the community and must be recognized as such.
Obedience is not a license to destroy another human being for the whims and fancies of an authority figure. To be a parent does not give anyone the right to beat a child. To be an official does not give anyone—the police, the president, the teacher—the right to vent either their force or their frustration on simple people for doing simple things. The nature of the punishment is always to be weighed against the nature of the offense.
The pursuit of holiness ought not to be a fearsome thing. Benedictine spirituality is a gentle manifestation of a loving and parenting God who wants us to be all that we can be.
What Benedict prescribes is one of two kinds of excommunication. In the first, for lighter offenses against the unity and peace of the community, a person is separated from the common table and denied the right to lead prayer. In the second, for more significant attacks on community well-being, the person is banished from community prayer, social life, and table sharing at the same time.
Benedict is teaching very clearly that to disturb the human community is serious. It makes us outcasts to our own kind. It eats away in the style of acid at the very things that a community needs to flourish and to be effective—love, trust, and cooperation. And, Benedict insinuates, once you have broken the bonds that make a community a community, a family a family, a team a team, there is no growth possible until we all face the fact.
CHAPTER 25
SERIOUS FAULTS
_March 2 – July 2 – Nov. 1_
_Those guilty of a serious fault are to be excluded from both the table and the oratory. No one in the community should associate or converse with them at all. They will work alone at the tasks assigned to them, living continually in sorrow and penance, pondering that fearful judgment of the apostle: "Such a person is handed over for the destruction of the flesh that the spirit may be saved on the day of Jesus Christ" (1 Cor. 5:5). Let them take their food alone in an amount and at a time the prioress or abbot considers appropriate. They should not be blessed by anyone passing by, nor should the food that is given them be blessed._
"There is no failure except in no longer trying," it is said. Benedict has no intention of letting anyone sink to the point where the intolerable is unnoticed and unremarked and institutionalized. Each of us is capable of betraying the best in us. We cut corners in the office, we stop cleaning the house, we let the study and the reading and the praying go. We sit around in life letting the juice turn black in us. We let the family down. We let the business slide. We let our minds and souls go to straw. We fight the call to growth and goodness with everything in us. We let the world carry us instead of carrying our part of the world. And, at that point, Benedict's Rule calls for the group whose life we affect to say "Enough," to quit bearing us up on the litter of community, to quit rewarding our selfish and surly behavior with security and affirmation and a patina of holiness. Excommunication, for all practical purposes, says, "You want to be a world unto yourself? Fine, be one."
The problem, of course, is that a human being needs help to be a human being. At our worst we seek the solace of another's hand. So, before expelling the rebellious, Benedict isolates them to give them time to decide if being out of the community is really what they want, really what they need, really what will bring them happiness. It is a time for making choices all over again.
It's not a bad idea to distance ourselves from what we say we do not want in order to discover whether the problem is actually in it or, perhaps, in us. Sabbaticals and long vacations and discernment retreats, even going away to college when we're young, all can help us see our parents and our family and our function in life in a completely different way. The point of the rule is simply that we have to take intervals to explore consciously what we ourselves are holding back from the group that depends on us.
CHAPTER 26
UNAUTHORIZED ASSOCIATION WITH THE EXCOMMUNICATED
_March 3 – July 3 – Nov. 2_
_If anyone, acting without an order from the prioress or abbot, presumes to associate in any way with an excommunicated member, to converse with them or to send them a message, they should receive a like punishment of excommunication._
Contemporary psychology talks a great deal about the need to be a support to people under stress. Popular psychology has not often made a distinction between positive and negative support. It is not supportive to take away a person's heart medicine simply because they do not like the taste of it. It is not supportive to fail to set a broken leg simply because the setting will be painful. It is not supportive to deny people the right and the environment to think a situation through, to recommit themselves, to gain perspective, to work things out without dividing the community over them. Sometimes pain itself cures. Benedict wants the cure to have the time to heal. Meddling, agitating, distracting a person from the great work of growth at such an important time in a person's life is a grave fault itself.
CHAPTER 27
THE CONCERN OF THE ABBOT AND PRIORESS FOR THE EXCOMMUNICATED
_March 4 – July 4 – Nov. 3_
_The abbot and prioress must exercise the utmost care and concern for the wayward because "it is not the healthy who need a physician, but the sick" (Matt. 9:12). Therefore they ought to use every skill of a wise physician and send in senpectae, that is, mature and wise members who, under the cloak of secrecy, may support the wavering sister or brother, urge them to be humble as a way of making satisfaction, and "console them lest they be overwhelmed by excessive sorrow" (2 Cor. 2:7). Rather, as the apostle also says: "Let love be reaffirmed" (2 Cor. 2:8), and let all pray for the one who is excommunicated._
The place of punishment in the Rule of Benedict is never to crush the person who is corrected. The purpose of excommunication is to enable a person to get life in perspective and to start over again with a new heart. So, although not just anyone with any agenda—personal dissatisfaction, a misguided sense of what support implies, community division—is encouraged to talk to the person who is enduring excommunication, someone must. The abbot and prioress themselves are expected to see that the confused or angry or depressed persons get the help they need to begin fresh again from discerning and mature people who are skilled in the ways of both the mind and the soul, who know life and its rough spots, who realize that humility is what saves us from the blows of failure.
Excommunication is no longer a monastic practice but help from the wise through periods of resistance and reluctance must be a constant or the spiritual life may never come to fullness. Community—family—is that place everywhere where we can fail without fear of being abandoned and with the ongoing certainty that we go on being loved nevertheless. Perfection is not an expectation in monastic life any more than it is an expectation in any healthy environment where experience is the basis both of wisdom and of growth.
A contemporary collection of monastic tales includes the story of the visitor who asks of the monk, "What do you do in the monastery?" And the monastic replies, "Well, we fall and we get up and we fall and we get up and we fall and we get up." Where continual falling and getting up is not honored, where the senpectae—the wise ones who have gone before us—are not present to help us through, life runs the terrible risk of drying up and blowing away before it is half lived.
_**It is the responsibility of the abbot or prioress to have great concern and to act with all speed, discernment, and diligence in order not to lose any of the sheep entrusted to them. They should realize that they have undertaken care of the sick, not tyranny over the healthy. Let them also fear the threat of the prophet in which God says: "What you saw to be fat you claimed for yourselves, and what was weak you cast aside" (Ezek. 34:3–4). They are to imitate the loving example of Christ, the Good Shepherd,who left the ninety-nine sheep in the mountains and went in search of the one sheep that had strayed. So great was Christ's compassion for its weakness that "he mercifully placed it on his sacred shoulders" and so carried it back to the flock (Luke 15:5).**_
The idea that the spiritual life is only for the strong, for those who don't need it anyway, is completely dispelled in the Rule of Benedict. Here spiritual athletes need not apply. Monasticism is for human beings only. The abbot and prioress are told quite clearly that they are to see themselves as physicians and shepherds tending the weak and carrying the lost, not as drill sergeants, not as impresarios. What we have in monasteries and parishes and all fine social movements and devoted rectories and most families are just people, simple people who never meet their own ideals and often, for want of confidence and the energy that continuing commitment takes, abandon them completely. Then, our role, the Rule of Benedict insists, is simply to try to soothe what hurts them, heal what weakens them, lift what burdens them and wait. The spiritual life is a process, not an event. It takes time and love and help and care. It takes our patient presence. Just like everything else.
CHAPTER 28
THOSE WHO REFUSE TO AMEND AFTER FREQUENT REPROOFS
_March 5 – July 5 – Nov. 4_
_If anyone has been reproved frequently for any fault, or even been excommunicated, yet does not amend, let that member receive a sharper punishment: that is, let that monastic feel the strokes of the rod. But if even then they do not reform, or perhaps become proud and would actually defend their conduct, which God forbid, the prioress or abbot should follow the procedure of a wise physician. After applying compresses, the ointment of encouragement, the medicine of divine Scripture, and finally the cauterizing iron of excommunication and strokes of the rod, if they then perceive that their earnest efforts are unavailing, let them apply an even better remedy: they and all the members should pray for them so that God, who can do all things, may bring about the health of the sick one. Yet if even this procedure does not heal them, then finally, the prioress or abbot must use the knife and amputate. For the apostle says: "Banish the evil one from your midst" (1 Cor. 5:13); and again, "If the unbeliever departs, let that one depart" (1 Cor. 7:15), "lest one diseased sheep infect the whole flock."_
The _Tao Te Ching_ reads, "If you realize that all things change, there is nothing you will try to hold on to. If you aren't afraid of dying, there is nothing you can't achieve." Benedict's call to growth is a pressing and intense one, even shocking to the modern mind. Physical punishment has long been suspect in contemporary society. Beating people with the rod is considered neither good pedagogy nor good parenting now, and the notion of whipping fullgrown adults is simply unthinkable. Times have changed; theories of behavior modification have changed; the very concept of adulthood has changed; this living of the Rule has changed. What has not changed, however, is the idea that human development demands that we grow through and grow beyond childish uncontrol to maturity and that we be willing to correct things in ourselves in order to do it, whatever the cost.
Benedict clearly believes that there are indeed things we must be willing to die to in life if we want to go beyond the fruitless patterns we're in right now. We aren't allowed to hang on to useless ideas or things or behaviors regardless of how good they might seem to us, regardless of their effect on others. We aren't allowed to live without dying to self. The Rule insists that people be called to growth. The entire community is in the process together and the process is not to be ignored, however painful the process may be.
The spiritual life in the Benedictine tradition is not a series of overnight stays where we do what we want without care for the impact of it on the lives of others, no matter how right we think we are. Human community is the universal obligation to live fully ourselves and to live well with others. So important is personal growth in community life for Benedict that when people refuse to grow in community virtues, to be a blessing to others as well as to be open to the blessings that are there for themselves, Benedict asks them to leave.
There can come a point, it seems, after every effort has been made to deal with a problem and every attempt has been made to correct a spiritual disease in life, when enough is enough and ought not to be tolerated any longer. The person may be a very good person but, the implication is, this just may not be the place for that person. The shoe simply does not fit and the foot should not be wrenched to it.
The lesson is a universal one. There are a number of good things that it would not be good for us to do. People who become priests because their parents wanted a priest in the family are often unhappy priests. Children who stay on the farm when they should have gone to art school run the risk of twisting their lives into gnarled deadwood—and the farm with it. People with the courage to put us out of something may be the best spiritual guides we ever get.
CHAPTER 29
READMISSION OF MEMBERS WHO LEAVE THE MONASTERY
_March 6 – July 6 – Nov. 5_
_If any community members, following their own evil ways, leave the monastery but then wish to return, they must first promise to make full amends for leaving. Let them be received back, but as a test of humility they should be given the last place. If they leave again, or even a third time, they should be readmitted under the same conditions. After this, however, they must understand that they will be denied all prospect of return._
Life is often a series of false starts while we find out who we are and determine where we really want to go. Benedict understands the struggle of uncertainty and indecision and makes room for it. After all, the giving of oneself to anything is no small thing and should be done with reflection and with peace of mind. So, Benedict allows candidates to the life to try again and again. What he does not permit them to do, however, is to ignore the fact that behavior has consequences or that sometime, somehow, they must finally commit to something if they are going to get on with the process of both psychological and spiritual growth. With those two concepts in mind, Benedict allows candidates to enter and leave the monastery no more than three times and then only provides that they realize that every new beginning begins at the beginning again.
There are in this chapter good insights for all of us: eventually we must all settle down and do something serious with our lives and every day we must make a fresh beginning of it.
CHAPTER 30
THE MANNER OF REPROVING THE YOUNG
_March 7 – July 7 – Nov. 6_
_Every age and level of understanding should receive appropriate treatment. Therefore, as often as the young, or those who cannot understand the seriousness of the penalty of excommunication, are guilty of misdeeds, they should be subjected to severe fasts or checked with sharp strokes so that they may be healed._
In the early centuries of monasticism, it was not uncommon for people to dedicate their children to religious life at a very early age or, much in the style of later boarding schools, to send them to an abbey for education where they lived very like the monastics themselves. The monastery, then, was a family made up of multiple generations. Benedict made provisions for every member of the community. Life in the Benedictine tradition was not a barracks or a prison or an exercise in deindividuation. On the contrary.
In the age of Benedict, however, the corporal punishment of children was a given. It was a given, in fact, in the homes and schools of our own time until, in the late twentieth century, social psychology detected the relationship between violence in society and violence against children. Only in our time has it finally become questionable for a teacher to whip a student or for a parent to spank a child. The question is, then, should this chapter now be discounted in the Rule? Children don't enter monastic communities anymore and children are not raised in them. The answer surely is no. The real lesson of the chapter is not that young people should be beaten. The continuing value of the chapter is that it reminds us quite graphically that no one approach is equally effective with everyone. No two people are exactly the same. In bringing people to spiritual adulthood we must use every tool we have: love, listening, counsel, confrontation, prayer that God may intervene where our own efforts are useless, and, finally, if all else fails, amputation from the group.
The real point of this and all seven preceding chapters of the penal code of the Rule is that Benedictine punishment is always meant to heal, never to destroy; to cure, not to crush.
CHAPTER 31
QUALIFICATIONS OF THE MONASTERY CELLARER
_March 8 – July 8 – Nov. 7_
_As cellarer of the monastery, there should be chosen from the community someone who is wise, mature in conduct, temperate, not an excessive eater, not proud, excitable, offensive, dilatory, or wasteful, but God-fearing, and like a parent to the whole community. The cellarer will take care of everything, but will do nothing without an order from the prioress or abbot. Let the cellarer keep to those orders._
Benedictine spirituality refuses to glorify a life of false frugality or fabricated irritations. The person who handles the supplies of the monastery, the cellarer, is to distribute the goods of the monastery calmly, kindly, without favoritism, and under the guidance of the abbot or prioress, not to put people under obligation to them or to wreak vengeance on those who rebuff them.
The cellarer does more than distribute goods. The cellarer becomes a model for the community, a person who is to be "temperate," not a person who is "an excessive eater," not someone in other words with rich tastes and a limitless appetite for material things. Benedict wants the cellarer to be someone who knows the difference between needs and desires, who will see that the community has what is necessary but does not begin the long, slippery road into excess and creature comforts and indolence and soft-souledness. In the house of Benedict, the principles of the life live in ways no words can convey, in the people who carry them out. The call to be what we say we believe becomes a measure of authenticity for teachers, parents, and administrators everywhere.
_The cellarer should not annoy the members. If anyone happens to make an unreasonable demand, the cellarer should not reject that person with disdain and cause distress, but reasonably and humbly deny the improper request. Let cellarers keep watch over their own souls, ever mindful of that saying of the apostle: "They who serve well secure a good standing for themselves" (1 Tim. 3:13). The cellarer must show every care and concern for the sick, young, guests, and the poor, knowing for certain that they will be held accountable for all of them on the day of judgment. The cellarer will regard all utensils and goods of the monastery as sacred vessels of thealtar, aware that nothing is to be neglected. Cellarers should not be prone to greed, not be wasteful and extravagant with the goods of the monastery, but should do everything with moderation and according to the order of the prioress or abbot._
If chapter 31 is anything at all, it is a treatment of human relationships. The one with power is not to annoy the powerless. The one with needs is not to demand. The chapter stands as stark warning to people in positions of authority and responsibility, whatever their station. They are to "keep watch of their own souls," guarding themselves against the pitfalls of any position: arrogance, disinterest, unkindness, aloofness from the very people the position is designed to serve. Then, to make the point clear, Benedict describes the people who are not to get overlooked for the sake of efficiency in the bureaucratic game of hurry up and wait. And they are everybody who cannot possibly be expected to want things when the office is open: the sick, the young, the guests, and the poor. The one who has power and resources, the Rule says, must know for certain that "they will be held accountable for all of them on the day of judgment." As will we all who find ourselves too busy, too insensitive, too uncaring to see that the goods of the earth are given to the poor ones who have as much claim on the Garden as we but no way to get the staples of life for themselves. As will we all who use our positions to diminish the people in behalf of whom we bear responsibility by wearing them down and wearing them out while we dally with their needs. The spouse who lets the door swell to sticking before fixing it or serves the meal an hour after its time; the employer who never buys the new file cabinet; the superior who never sees the staff personally—all fail in the Benedictine spirituality of service for the sake of the person that is taught in this chapter.
But the cellarer must do more than take care of people. A Benedictine cellarer has a responsibility to take care of things, too. Waste is not a Benedictine virtue. Planned obsolescence is not a Benedictine goal. Disposability is not a Benedictine quality. A Benedictine soul is a soul that takes care of things, that polishes wood and scrapes away rust and keeps a room clean and never puts feet on the furniture and mulches the garden and leaves trees standing and "treats all utensils and goods of the monastery like the sacred vessels of the altar." A Benedictine cares for the earth and all things well. The Benedictine heart practiced ecology before it was a word.
_March 9 – July 9 – Nov. 8_
_Above all, let the cellarer be humble. If goods are not available to meet a request, the cellarer will offer a kind word in reply, for it is written: "A kind word is better than the best gift" (Sir. 18:17). Cellarers should take care of all that the prioress or abbot entrusts to them, and not presume to do what they have forbidden. They will provide the members their allotted amount of food without any pride or delay, lest they be led astray. For cellarers must remember what the Scripture says that person deserves "who leads one of the little ones astray" (Matt. 18:6)._
_If the community is rather large, the cellarer should be given helpers, so that with assistance it becomes possible to perform the duties of the office calmly. Necessary items are to be requested and given at the proper times, so that no one may be disquieted or distressed in the house of God._
The cellarer gets a lesson from Benedict that we all need to learn sometime in life: we have a responsibility to serve others "without any pride or delay, lest they be led astray." It is not right, in other words, to tax other people's nervous systems, to try other people's virtues, to burden other people's already weary lives in order to satisfy our own need to be important. We don't have to lead them into anger and anxiety, frustration and despair. We don't need to keep them waiting; we don't need to argue their requests; we don't need to count out every weight to the ounce, every bag to the gram, every dollar to the penny. We can give freedom and joy with every gift we give or we can give guilt and frugality. The person with a Benedictine tenor learns here to err on the side of largesse of spirit.
CHAPTER 32
THE TOOLS AND GOODS OF THE MONASTERY
_March 10 – July 10 – Nov. 9_
_The goods of the monastery, that is, its tools, clothing, or anything else, should be entrusted to members whom the prioress or abbot appoints and in whose manner of life they have confidence. The abbot or prioress will, as they see fit, issue to them the various articles to be cared for and collected after use. The prioress and abbot will maintain a list of these, so that when the members succeed one another in their assigned tasks, they may be aware of what they hand out and what they receive back._
_Whoever fails to keep the things belonging to the monastery clean or treats them carelessly should be reproved. If they do not amend, let them be subjected to the discipline of the rule._
To those who think for a moment that the spiritual life is an excuse to ignore the things of the world, to go through time suspended above the mundane, to lurch from place to place with a balmy head and a saccharine smile on the face, let this chapter be fair warning. Benedictine spirituality is as much about good order, wise management, and housecleaning as it is about the meditative and the immaterial dimensions of life. Benedictine spirituality sees the care of the earth and the integration of prayer and work, body and soul, as essential parts of the journey to wholeness that answers the emptiness in each of us.
CHAPTER 33
MONASTICS AND PRIVATE OWNERSHIP
_March 11 – July 11 – Nov. 10_
_Above all, this evil practice [of private ownership] must be uprooted and removed from the monastery. We mean that without an order from the prioress or abbot, no members may presume to give, receive, or retain anything as their own, nothing at all—not a book, writing tablets, or stylus—in short not a single item, especially since monastics may not have the free disposal even of their own bodies and wills. For their needs, they are to look to the prioress or abbot of the monastery, and are not allowed anything which the prioress or abbot has not given or permitted. "All things should be the common possession of all, as it is written, so that no one presumes ownership of anything" (Acts 4:32)._
_But if any members are caught indulging in this most evil practice, they should be warned a first and a second time. If they do not amend, let them be subjected to punishment._
There are two concerns at issue in this chapter of the Rule: the development of personal freedom and the preservation of human community. Private ownership touches both of them.
The Hasidim tell the story of the visitor who went to see a very famous rabbi and was shocked at the sparsity, the bareness, the emptiness of his little one-room house. "Why don't you have any furniture?" the visitor asked. "Why don't you?" the rabbi said. "Well, because I'm only passing through," the visitor said. "Well, so am I," the rabbi answered.
On the journey to heaven, things tie us to the earth. We can't move to another city because we have a huge mortgage on the house in this one. We can't take care of a sick neighbor because we are too busy taking care of our own hedges. We go poor giving big parties in the hope for big promotions. We get beholden to the people who give big parties back. We take things and hoard things and give things to control our little worlds and the things wind up controlling us. They clutter our space; they crimp our hearts; they sour our souls. Benedict says that the answer is that we not allow ourselves to have anything beyond life's simple staples in the first place and that we not use things—not even the simplest things—to restrict the life of another by giving gifts that tie another person down. Benedictine simplicity, then, is not a deprivation. It frees us for all of life's surprises.
Simplicity is more than the key to personal freedom, however. Simplicity is also the basis of human community. Common ownership and personal dependence are the foundations of mutual respect. If I know that I literally cannot exist without you, without your work, without your support, without your efforts in our behalf, without your help, as is true in any community life, then I cannot bury myself away where you and your life are unimportant to me. I cannot fail to meet your needs, as you have met my needs, when the dearth in you appeals for the gifts in me. It is my ability to respond to your needs, in fact, that is my claim, my guarantee, of your presence in my own life. In community life, we genuinely need one another. We rely on one another. Community life is based on mutual giving.
The family, the relationship that attempts to reconcile the idea of community with the independent and the independently wealthy, the perfectly, the totally, the smugly self-sufficient, is no community, no family, no relationship at all. Why stay and work a problem out with people when you can simply leave them? And never notice that they're gone.
CHAPTER 34
DISTRIBUTION OF GOODS ACCORDING TO NEED
_March 12 – July 12 – Nov. 11_
_It is written: "Distribution was made as each had need" (Acts 4:35). By this we do not imply that there should be favoritism—God forbid—but rather consideration for weaknesses. Whoever needs less should thank God and not be distressed, but those who need more should feel humble because of their weakness, not self-important because of the kindness shown them. In this way all the members will be atpeace. First and foremost, there must be no word or sign of the evil of grumbling, no manifestation of it for any reason at all. If, however, anyone is caught grumbling, let them undergo more severe discipline._
Destitution and deprivation are not monastic virtues. Benedict immediately follows the chapter on the pitfalls of private ownership with a chapter insisting that people be given what they need to get through life. Benedictine spirituality is not based on a military model of conformity. Pianists need pianos, writers need computers, principals need to go to meetings, administrators need to get away from the group every once in a while, workers need places to work, the sick need special kinds of food, people with bad backs need the proper kinds of beds. Benedictine spirituality says get them and don't notice the differences; get them and don't count the cost; get them and don't complain about it. Just thank God that your own needs have yet to reach the level of such a burden.
It's an important chapter in a world where poverty is clearly an evil and not to be spiritualized while the children of the earth die with bloated stomachs. The person whose spirituality is fed by the Rule of Benedict would be acutely concerned about that, painfully disturbed about that, as was Benedict. The Benedictine spirit would not rest, in fact, until the imbalance was righted and the needs were met.
CHAPTER 35
KITCHEN SERVERS OF THE WEEK
_March 13 – July 13 – Nov. 12_
_The members should serve one another. Consequently, no members will be excused from kitchen service unless they are sick or engaged in some important business of the monastery, for such service increases reward and fosters love. Let those who are not strong have help so that they may serve without distress, and let everyone receive help as the size of the community or local conditions warrant. If the community is rather large, the cellarer should be excused from kitchen service, and, as we have said, those should also be excused who are engaged in important business. Let all the rest serve one another in love._
Benedict leaves very little to the imagination or fancy of the spiritually pretentious who know everything there is to know about spiritual theory and think that is enough. Benedict says that the spiritual life is not simply what we think about; it is what we do because of what we think. It is possible, in fact, to spend our whole lives thinking about the spiritual life and never develop one. We can study church history forever and never become holier for the doing. There are theology courses all over the world that have nothing whatsoever to do with the spiritual life. In the same way, we may think we are a community or assume we are a family but if we do not serve one another we are, at best, a collection of people who live alone together.
So Benedict chooses the family meal to demonstrate that point of life where the Eucharist becomes alive for us outside of chapel. It is in kitchen service that we prepare good things for the ones we love and sustain them and clean up after them. It was woman's work and Roman men were told to do it so that they, too, with their own hands and over their own hot fires, could know what it takes to spend their own lives to give life to the other.
_On Saturday the ones who are completing [the kitchen] work will do the washing. They are to wash the towels which the members use to wipe their hands and feet. Both the one who is ending service and the one who is about to begin are to wash the feet of everyone. The utensils required for the kitchen service are to be washed and returned intact to the cellarer, who in turn issues them to the one beginning the next week. In this way the cellarer will know what is handed out and what is received back._
Community love and accountability are focused, demonstrated, and modeled at the community meal. In every other thing we do, more private in scope, more personal in process, our private agendas so easily nibble away at the transcendent purpose of the work that there is often little left of the philosophical meaning of the task except our own translation of it. In the Middle Ages, the tale goes, a traveler asked three hard-at-work stone masons what they were doing. The first said, "I am sanding down this block of marble." The second said, "I am preparing a foundation." The third said, "I am building a cathedral." Remembering the greater cause of why we are doing what we do is one of life's more demanding difficulties. But that's not the case in a kitchen or in a dining room that is shaped around the icon of the Last Supper where the One who is first washes the feet of the ones who are to follow. "Do you know what I have just done?" the Scripture reads. "As I have done, so you must do."
In Benedict's dining room, where everyone serves and everyone washes feet and everyone returns the utensils clean and intact for the next person's use, love and accountability become the fulcrum of community life.
_March 14 – July 14 – Nov. 13_
_An hour before mealtime, the kitchen workers of the week should each receive a drink and some bread over and above the regular portion, so that at mealtime, they may serve one another without grumbling or hardship. On solemn days, however, they should wait until after the dismissal._
Work done in the Benedictine tradition is supposed to be regular, it is supposed to be productive, it is supposed to be worthwhile, but it is not supposed to be impossible. Give help where it is needed, the Rule says. Give whatever it takes to make it possible, the Rule says. Give people whatever they need to do it without grumbling. The servers are to serve, not starve. They are to eat before the others so that they don't wind up resenting the fact that others are eating and become bitter or reluctant in their service. It is a salutary and sobering thought in an age that exploits the poor and the illiterate with impunity for the sake of the comfort of the rich, paying workers too little to live on and working them too hard to live and then calling it "working your way up" or the "plight" of the unskilled laborer.
Benedictine spirituality does not set out to burden some for the sake of the others in the name of community. It sets out to make work possible for all so that the community can thrive in joy. Any group, any family, that makes life wonderful for some of its members at the expense of the others, no matter how good the work or how satisfied the group, is not operating in a Benedictine spirituality. It is, at best, simply dealing in some kind of holy exploitation, but it is exploitation nevertheless.
_On Sunday immediately after Lauds, those beginning as well as those completing their week of service should make a profound bow in the oratory before all and ask for their prayers. Let the server completing the week recite this verse: "Blessed are you, O God, who have helped me and comforted me" (Dan. 3:52; Ps. 86:17). After this verse has been said three times the server receives a blessing. Then the one beginning the service follows and says: "O God, come to my assistance; O God, make haste to help me" (Ps. 70:2). And all repeat this verse three times. When they have received a blessing, the servers begin their service._
In _The Sayings of the [Jewish] Fathers_ it is written, "It is wise to work as well as to study the Torah: between the two you will forget to sin." To make sure we do not forget that humble work is as sacred and sanctifying as prayer, Benedict blesses the kitchen servers of the week in the middle of the chapel. With that simple but powerful gesture all of life begins to look different for everyone. Suddenly it is not made up of "higher" and "lower" activities anymore. It is all—manual labor and mystical meditation—one straight beam of light on the road to fullness of humanity. One activity without the other, prayer without the creative and compassionate potential of work or work without the transcending quality of prayer, lists heavily to the empty side of life. The blessing prayer for the weekly servers in the midst of the community not only ordains the monastic to serve the community but it also brings together both dimensions of life, the transcendent and the transforming, in one clear arc: prayer is not for its own sake and the world of manual work is not a lesser world than chapel.
We are all meant both to pray and work, each of them influencing and fulfilling the other.
CHAPTER 36
THE SICK
_March 15 – July 15 – Nov. 14_
_Care of the sick must rank above and before all else so that they may truly be served as Christ who said: "I was sick and you visited me" (Matt. 25:36) and, "What you did for one of these least of my people you did for me" (Matt. 25:40). Let the sick on their part bear in mind that they are served out of honor for God, and let them not by their excessive demands distress anyone who serves them. Still, the sick must be patiently borne with, because serving them leads to a greater reward. Consequently, the prioress or abbot should be extremely careful that they suffer no neglect._
The rabbis say, "The purpose of maintaining the body in good health is to make it possible for you to acquire wisdom." Benedictine spirituality is about coming to a sense of the fullness of life. It is not about being self-destructive or living sour lives or dropping down pits of privacy so deep that no other ever dare intrude. Benedictine spirituality never gives up on life even though death is known to be the entry to its everlasting joy. Why? Because, the rabbi shows us, every day we have gives us another chance to become the real persons we are meant to be. Why? Because, the Scripture says, to serve the sick is to serve the Christ.
The point for us all, perhaps, is never to give up on life and never to doubt that every bit of kindness, every tender touch we lay upon another in life can heal what might otherwise have died, certainly in them, perhaps even in ourselves.
_Let a separate room be designated for the sick, and let them be served by an attendant who is God-fearing, attentive, and concerned. The sick may take baths whenever it is advisable, but the healthy, and especially the young, should receive permission less readily. Moreover, to regain their strength, the sick who are very weak may eat meat, but when their health improves, they should all abstain from meat as usual._
_The abbot and prioress must take the greatest care that cellarers and those who serve the sick do not neglect them, for the shortcomings of disciples are their responsibility._
Care for the sick, in the mind of Benedict, is not a simple warehousing process, though that in itself could have been a great contribution to a society without hospitals. Care for the sick, in Benedictine spirituality, is to be done with faith, with attention, and with a care beyond the technical. The infirmarian is to be "concerned." Baths, a very important part of Roman therapy and hygiene in a hot and sticky climate, and red meat, a treat used only rarely in early monastic houses both because of its scarcity and because of its purported relationship to sexual agitation, are both given generously and recklessly. Care of the sick, you see, is done in the name of God and to the person of the suffering Christ. Nothing was too much. Nothing was to be spared. Nothing that could do good was to be called forbidden.
We have to ask ourselves, in a society of technological health care, how much of it we do with faith and lavish attention and depth of soul and a love that drives out repulsion. We have to ask ourselves how willing we are to take a little of our own energy on behalf of those who are no longer the life of the party, the help on the job. How much of our own precious time do we spend on those with little time left?
CHAPTER 37
THE ELDERLY AND THE YOUNG
_March 16 – July 16 – Nov. 15_
_Although human nature itself is inclined to be compassionate toward the elderly and the young, the authority of the rule should also provide for them. Since their lack of strength must always be taken into account, they should certainly not be required to follow the strictness of the rule with regard to food, but should be treated with kindly consideration and allowed to eat before the regular hours._
There are two ages of life that lack the energy of the prime: youth and old age. Both, Benedict implies, have something to give us provided that we give them something as well. It is a vital lesson. People do not become useless simply because they do not have the strength or stamina of middle age. Life is a series of phases, each of them important, all of them worthwhile. Nothing must ever deter that, not even religious rigor or pious fervor. Fasting is good for the soul, but if it takes too much from the body of the old or the young, it ceases to be an expectation or a virtue. Prayer at the proper hours is good for the spiritual memory of life, but if it taxes the physical energy beyond the bearable, then those times are to be "anticipated," adjusted, changed for the person rather than destroy the person for the sake of the prayer. Exceptions are the way of life, and when they are not, something is wrong with life itself, Benedict reasons. Benedict builds compassion right into the Rule so that oppression in the name of God will not become a monastic sin. It is a sobering thought, this commitment to moderation and good sense, for people who set out to make the spiritual life central to their own.
CHAPTER 38
THE READER FOR THE WEEK
_March 17 – July 17 – Nov. 16_
_Reading will always accompany the meals. The reader should not be the one who just happens to pick up the book, but someone who will read for a whole week, beginning on Sunday. After Mass and Communion, let the incoming reader ask all to pray so that God may shield them from the spirit of vanity. Let the reader begin this verse in the oratory: "O God, open my lips, and my mouth shall proclaim your praise" (Ps. 51:17), and let all say it three times. When they have received a blessing, they will begin their week of reading._
Benedictine spirituality was rooted in prayer, study, and work. Every hour of the short days was filled with one or the other, and mealtime, too, was no exception. Monastics used food for energy, not for pleasure. Spiritual nourishment was the food that restored them and impelled them and made them strong, and mealtime was a good time to get it. They rested in body and in spirit there and, even at a moment of physical need, centered their hearts on higher things. They filled their hearts as well as their stomachs.
Benedict considers reading such an important part of the meal, in fact, that he insists that the person doing the reading be a good reader, someone who would inspire rather than irritate the souls of the listeners. The reading was to be an artistic event, an instructive experience, a moment of meditation, not a wrestling match with words. Nor was it to be a moment of personal display or lordship by those few educated who could read while the rest of the community could not.
This paragraph is just as important now as the day it was written. Maybe more so. People who give too much attention to the body give too little attention to anything else. They make themselves the idol before which they worship and run the risk of forgetting to raise their minds to higher things because they are more intent on the rich sauces and fine meats and thick desserts that fill their days than to the gaping emptiness in their minds and hearts and souls.
_Let there be complete silence. No whispering, no speaking—only the reader's voice should be heard there. The members should by turn serve one another's needs as they eat and drink, so that no one need ask for anything. If, however, anything is required, it should be requested by an audible signal of some kind rather than by speech. No one should presume to ask a question about the reading or about anything else, "lest occasions be given to the devil" (Eph. 4:27; 1 Tim. 5:14). The abbot or prioress, however, may wish to say a few words of instruction._
In the course of the meal, the monastics are to concentrate on two things: the words of the reading and the needs of their neighbors. It is an astounding demonstration of the nature of the entire Christian life frozen in a single frame. We are to listen intently for the Word of God and be aware of those around us at the same time. Either one without the other is an incomplete Christianity. And never, at any time, are we to concentrate solely on ourselves in the name of religion.
_Because of Communion and because the fast may be too hard for them to bear, the one who is reader for the week is to receive some diluted wine before beginning to read. Afterward they will take their meal with the weekly kitchen servers and the attendants._
On Sundays and solemn feast days, when the community received Communion, the fast from the night before to the meal that followed the Eucharist was a long one. It would have been even longer for the reader who could eat only after the meal was ended. So Benedict, the one more full of compassion than of law, allowed the reader to take a little wine before starting in order to hold him over. The reader still fasts, in other words, but with help.
If anything, this chapter on a now defunct practice is a lesson in the way that gentleness softens rigor without destroying either the practice or the person. Legalists too often opt for practice, whatever the cost to the people who are trying to do it; liberals too often opt for people's convenience, whatever the loss of spiritual practice. Benedict opts for a way of life that cares for people physically while it goes on strengthening them spiritually.
The contemporary question with which the chapter confronts us is an extremely powerful one: When we eliminate a spiritual discipline from our lives, because it is out of date or impossible to do anymore or too taxing to be valuable, what do we put in its place to provide the same meaning? Or do we just pare away and pare away whatever demands spiritual centering makes of us until all that is left is a dried-up humanism, at best?
"Prayer without study is like a soul without a body," the rabbis say. Benedict clearly felt the same. The purpose of reading at table was to prepare the monastic for prayer. It is necessary to understand the Scriptures before it is possible to pray them. It is essential to be steeped in the Scriptures before it is possible to exude them. Table reading, in other words, was not a way to get away from people; it was a way to get closer to God. It was also one of the few times in the monastic day, outside of prayer times, that the spiritually thirsty but hardworking Benedictine could spend concentrated time on the things of God.
The point is that it isn't so much the practice of reading at table that is important in this chapter; it is the idea of groundedness in the spiritual life that should make us stop and think. We're all busy. We're all overscheduled. We're all trying to deal with people and projects that consume us. We're all spiritually thirsty. And, we're all responsible for filling the mind with rich ideas in order to leaven the soul. Prayer, contemplation, and spiritual adulthood don't happen by themselves. We have to work at them. If mealtime isn't a good time for study because the children or the family or the guest demand an attention then that no other time will provide, the question becomes, What periods do we set aside to become as comfortable with the ideas of God in life as we do the television schedule or the daily paper?
_Monastics will read and sing, not according to rank, but according to their ability to benefit their hearers._
The proclamation of the Word is the sowing of the soul. It is not to be done idly. It is not to be done without artistry. The proclamation of the Word of God must become part of the process of experiencing God. Prima donnas who do it more for their own sake than for the sake of the assembly, who come to perform rather than to blend in with the tone and theme of the liturgy, do not enrich a service. They distract from it. On the other hand, the ungifted or the unprepared interrupt the flow of the prayer and call equally disturbing attention to themselves. Lectors, homilists, and musicians, liturgy teams and pastors and teachers, all have something to learn here that is just as important for our own time as it was for this one. Goodwill is no excuse for a lack of artistry. Authority is no substitute for education. The spiritual nourishment of an entire people is in our hands. We do not have the right to treat liturgy lightly. We do not have the right to reduce the sacraments to such rote in the name of tradition that their dryness leaves the people dry. We do not have the right to make performance a substitute for the participation of the praying community.
CHAPTER 39
THE PROPER AMOUNT OF FOOD
_March 18 – July 18 – Nov. 17_
_For the daily meals, whether at noon or in midafternoon, it is enough, we believe, to provide all the tables with two kinds of cooked food because of individual weaknesses. In this way, the person who may not be able to eat one kind of food may partake of the other. Two kinds of cooked food, therefore, should suffice for all, and if fruit or fresh vegetables are available, a third dish may also be added. A generous pound of bread is enough for a day whether for only one meal or for both dinner and supper. In the latter case the cellarer will set aside one third of this pound and give it to the community at supper._
Chapter 39 is on generosity and trust that flies in the face of a tradition of stern and demanding asceticisms. Benedict of Nursia never takes food away from the community. On the contrary, he assures himself that the fare will always be ample and will always be simple but pleasing. These were working monastics who needed energy to toil and peace to pray. Benedict decides that food is not to be the penance of their lives.
Everybody needs something in life to make the rest of life doable and uplifting. The important thing in the spiritual life is that while we are creating penances for ourselves to build up our moral fiber we are also providing possibilities for ourselves to build up our spiritual joy.
_Should it happen that the work is heavier than usual, the abbot and prioress may decide—and they will have the authority—to grant something additional, provided that it is appropriate, and that above all overindulgence is avoided, lest anyone experience indigestion. For nothing is so inconsistent with the life of any Christian as overindulgence. Our God says: "Take care that your hearts are not weighted down with overindulgence" (Luke 21:34)._
Exceptions. Exceptions. Exceptions. The Rule of Benedict is full of rules that are never kept, always shifting, forever being stretched. Only two Benedictine principles are implied to be without exception: kindness and self-control. The abbot is to make exceptions always; the monastic is never to take advantage of them or to lose control, to slip into dissipation, to fail to keep trying to keep the mind in charge of the body. Soft living, slouch-heartedness, a dried-up soul are not what give life meaning. It is stretching ourselves that keeps us supple and keeps us trim. We believe it about the body. We are inclined to overlook it in the soul. Let them have what they need, the Rule says, but let them forego what they don't so that they can run through life with their bodies unburdened and their souls unsurfeited. It is good, clean living that Benedictine spirituality is about, living that keeps us young in heart and sharp of vision, living that has something for which to strive.
_The young should not receive the same amount as their elders, but less, since in all matters frugality is the rule. Let everyone, except the sick who are very weak, abstain entirely from eating the meat of four-footed animals._
The meat of four-footed animals was not part of the monastic diet because it was thought to heighten the animal facet of human nature. In a society whose philosophy was highly dualistic and whose world separated out neatly into things that were of the spirit and things that were of the flesh, the consideration was a serious one. Monastic life was about higher things and nothing was to be allowed to interfere with that.
The question for the modern world has seldom been what effect diet has on spirit—though interest in the field is certainly growing—but we have come to some conclusions about other things. We do know that colors, weather, light, environment all affect the spirit. Too much of anything, we have discovered, can weigh us down. Each of us needs to fast from something to bring ourselves to the summit of our spiritual powers. The question is, Have we lost a sense of the value of fasting or do we simply fill ourselves, glut ourselves, without limit, without end, with the useless and the disturbing?
CHAPTER 40
THE PROPER AMOUNT OF DRINK
_March 19 – July 19 – Nov. 18_
_"Everyone has personal gifts from God, one this and another that" (1 Cor. 7:7). It is, therefore, with some uneasiness that we specify the amount of food and drink for others. However, with due regard for the infirmities of the sick, we believe that a half bottle of wine a day is sufficient for each. But those to whom God gives the strength to abstain must know that they will earn their own reward._
The Rule of Benedict does not pretend to know the sacrifices that each of us needs to make in life. A tale from the Sufi may explain why, in the face of multiple spiritual disciplines, all of which specify many and sundry exercises as basic to the spiritual life, Benedict avoids this road of defined penances. "How shall we ever change," the disciples asked, "if we have no goals?" And the master said, "Change that is real is change that is not willed. Face reality and unwilled change will happen."
It is so easy to make cosmetic changes in the name of religion. It is so easy to make up rules and keep them so that we can feel good about doing something measurable in the spiritual life. We can fast and fast and fast from food or drink and nothing changes because fasting from food is not what we really need at that moment to turn our hearts of stone to hearts of flesh. We can kneel and kneel and kneel but nothing changes because kneeling is not what we need to soften our souls just then. We can fast and kneel and tithe and nothing changes because we do not really want anything to change.
Growth is not an accident. Growth is a process. We have to want to grow. We have to will to move away the stones that entomb us in ourselves. We have to work at uprooting the weeds that are smothering good growth in ourselves. Benedict doesn't tell us how much to eat. He simply provides the food and trusts us to make a choice to discipline ourselves somehow, some way, so that we do not sink into a mire of self-satisfaction so thick that there is no rescue for our sated souls.
_The abbot or prioress will determine when local conditions, work, or the summer heat indicates the need for a greater amount. They must, in any case, take great care lest excess or drunkenness creep in. We read that monastics should not drink wine at all, but since the monastics of our day cannot be convinced of this, let us at least agree to drink moderately, and not to the point of excess, for "wine makes even the wise go astray" (Sir. 19:2)._
The Rule of Benedict devotes itself more to the virtue of moderation than it does to the anesthetizing of the soul that can come with mortification. To forgo a thing completely is to prepare to forget it. If I never eat another piece of chocolate, I may forget all about chocolate, but I may also soon substitute something even more dangerous for its taste: drugs, consumerism, a hardened selfishness. To do something commonly but to do it in right proportion, on the other hand, is to win the struggle with it every day. To have one handful of salted peanuts, one piece of chocolate, one glass of wine in the midst of plenty, one car in a culture that counts its wealth in two-car garages, now that is mortification! Benedict knows that culture dictates the use of many things in life. What he cares about is that we control them rather than allowing them to control us.
_However, where local circumstances dictate an amount much less than what is stipulated above, oreven none at all, those who live there should bless God and not grumble. Above all else we admonish them to refrain from grumbling._
If Benedictine spirituality understands anything about life at all, it understands the corrosive effects of constant complaining. Complaining is the acid that shrivels our own souls and the soul of the community around us as well. Complaining is what shapes our mental set. Feelings, psychology tells us, do not affect thoughts. Thoughts affect feelings. What we allow ourselves to think is what we are really allowing ourselves to feel. When we learn how to correct our thought processes, then we learn not only how to stabilize our own emotions but how to change the environment around us at the same time. What we see as negative we make negative and feel negative about. What we are willing to think about in a positive way becomes positive.
Complaining, in other words, undermines the hope of a community and smothers possibility in a group. The whiner, the constant critic, the armchair complainer make an office, a family, a department, a community a polluted place to be. What we accept wholeheartedly that fails, we can always correct. What we condemn to failure before we have ever really tried to accept it, is not corrected; it is doomed to an untimely and, more than likely, an unnecessary death.
Benedictine spirituality tells us to open our hearts and our minds to let grace come in from unlikely places, without preplanning and prejudgments. "When there is no desire," the _Tao Te Ching_ instructs, "all things are at peace."
CHAPTER 41
THE TIMES FOR MEALS
_March 20 – July 20 – Nov. 19_
_From Easter to Pentecost, the monastics eat at noon and take supper in the evening. Beginning with Pentecost and continuing throughout the summer, the members fast until midafternoon on Wednesday and Friday, unless they are working in the fields or the summer heat is oppressive._
_On the other days they eat dinner at noon. Indeed, the abbot or prioress may decide that they should continue to eat dinner at noon every day if they have work in the fields or if the summer heat remains extreme. Similarly, they should so regulate and arrange all matters that souls may be saved and the members may go about their activities without justifiable grumbling._
The Rule of Benedict divides the year's meal schedules into four parts. From Easter to Pentecost there are no fast days, and the meals are taken at noon and before sundown. After Pentecost, Wednesdays and Fridays are fast days, as they were for all Christians of the period, and the meal, probably the only meal of the day, was to be delayed, the Rule mandates, until about three o'clock. But the law is no sooner made until Benedictine spirituality raises its fresh and liberating head again and softens the prescription with "unless." Unless it would be too hard to do. Unless they are too tired to wait. Unless the day is too hot to add one more difficulty to it. Then, the abbot or prioress and only the abbot or prioress may decide to mitigate the rule, to change the law, to allow the relaxation. And that is the issue. It is the abbot or prioress who decides what the change will be, not the individual monastic. Life, in other words, is not of our own choosing. The vagaries of life are not under our control. Circumstances change things and real spirituality demands that we be prepared at all times to accept them with faith and hope.
It isn't that Benedictine spirituality is meant to be lax; it is that it is meant to be sensible and it is meant to be serene. What is the use of making up difficulties when all we really have to do in life is to learn to bear well what must, under any circumstances, be borne?
_From the thirteenth of September to the beginning of Lent, they always take their meal in midafternoon. Finally, from the beginning of Lent to Easter, they eat toward evening. Let Vespers be celebrated early enough so that there is no need for a lamp while eating, and that everything can be finished by daylight. Indeed, at all times let supper or the hour of the fast day meal be so scheduled that everything can be done by daylight._
The third period of the year, from September 13 to Ash Wednesday, was the period known as "the monastic Lent." Here, Benedictine spirituality called for a measure above and beyond the norm. To do simply what was required was not enough. Benedictine spirituality called for extra effort in the development of the spiritual life. It is an interesting insertion in a rule that otherwise seems to be based on exceptions, mitigation, differences, basic Christian practice, and the law of averages.
Indeed, Benedictine spirituality is clearly rooted in living ordinary life with extraordinary awareness and commitment, a characteristic, in fact, that is common to monasticism both East and West. As the Zen masters teach, "One day a new disciple came up to the master Joshu. 'I have just entered the brotherhood,' the disciple said, 'and I am anxious to learn the first principle of Zen. Will you please teach it to me?' he asked. So Joshu said, 'Have you eaten your supper?' And the novice answered, 'Yes, I have eaten.' So Joshu said, 'Then now wash your bowl.'"
The first principle of Benedictinism, too, is to do what must be done with special care and special zeal so that doing it can change our consciousness and carve our souls into the kind of beauty that comes from simple things. It is so easy to go through life looking feverishly for special ways to find God when God is most of all to be found in doing common things with uncommon conscientiousness.
CHAPTER 42
SILENCE AFTER COMPLINE
_March 21 – July 21 – Nov. 20_
_Monastics should diligently cultivate silence at all times, but especially at night. Accordingly, this will always be the arrangement whether for fast days or for ordinary days. When there are two meals, all will sit together immediately after rising from supper. Someone should read from the Conferences or the Lives of the early church writers or at any rate something else that will benefit the hearers, but not the Heptateuch or the books of Kings, because it will not be good for those of weak understanding to hear these writings at that hour; they should be read at other times._
_On fast days there is to be a short interval between Vespers and the reading of the Conferences, as we have indicated. Then let four or five pages be read, or as many as time permits. This reading period will allow for all to come together, in case any were engaged in assigned tasks. When all haveassembled, they should pray Compline; and on leaving Compline, no one will be permitted to speak further. If monastics are found to transgress this rule of silence, they must be subjected to severe punishment, except on occasions when guests require attention or the prioress or abbot wishes to give someone a command, but even this is to be done with the utmost seriousness and proper restraint._
Silence has two functions. The first effect of exterior silence is to develop a sense of interior peace. The second value of silence is that it provides the stillness that enables the ear of the heart to hear the God who is "not in the whirlwind."
The constantly blaring iPods, the slammed door, the ceaseless, empty chatter in the hall, the constantly harsh voice all break the peace of the heart and agitate the soul. Day after day, month after month of them thickens the walls of the mind until it becomes impossible to hear the talk within us that shows us our pain and opens our mind to the truths of life and the presence of God.
Silence is not enough, however. Benedict wants night to rest our spirits as well as our bodies. He wants to send us to bed with instruction on the gentle Word of God, not on the scriptural history narratives with their blood and struggles, so that the stresses of the day can be softened by the thoughts of something beyond them.
We live with noise pollution now and find silence a great burden, a frightening possibility. Muzak fills our elevators and earbuds wire us to MP3 files and TVs blare from every room in the house from morning till night. We say we do not have the time to think, but what we actually lack is the quiet to think. Yet, until we are able to have at least a little silence every day, both outside and in, both inside and out, we have no hope of coming to know either God or ourselves very well.
CHAPTER 43
TARDINESS AT THE OPUS DEI OR AT TABLE
_March 22 – July 22 – Nov. 21_
_On hearing the signal for an hour of the Divine Office, monastics will immediately set aside what they have in hand and go with utmost speed, yet with gravity and without giving occasion for frivolity. Indeed, nothing is to be preferred to the Opus Dei._
_If at Vigils monastics come after the doxology of Psalm 95, which we wish, therefore, to be said quite deliberately and slowly, they are not to stand in their regular place in choir. They must take the last place of all, or one set apart by the prioress or abbot for such offenders, that they may be seen by them and by all, until they do penance by public satisfaction at the end of the Opus Dei. We have decided, therefore, that they ought to stand either in the last place or apart from the others so that the attention they attract will shame them into amending. Should they remain outside the oratory, there may be those who would return to bed and sleep, or, worse yet, settle down outside and engage in idle talk, thereby "giving occasion to the Evil One" (Eph. 4:27; 1 Tim. 5:14). They should come inside so that they will not lose everything and may amend in the future._
_At the day hours the same rule applies to those who come after the opening verse and the doxology ofthe first psalm following it: they are to stand in the last place. Until they have made satisfaction, they are not to presume to join the choir of those praying the psalms, unless perhaps the prioress or abbot pardons them and grants an exception. Even in this case, the one at fault is still bound to satisfaction._
Benedictine spirituality does not ask for great feats of physical asceticism, but it does require commitment to community and a sincere seeking of God through prayer. Tardiness is not to be tolerated. Indolence is not to be overlooked. Halfheartedness will not be condoned. Benedict does not want people sleeping in or dawdling along or "preferring anything to the Opus Dei," the work of God. Nothing in life qualifies as an exchange for the Word of God, not good work, not a job almost finished, not an interesting conversation, not the need for privacy.
Benedictine life centers around the chapel, and chapel must never be overlooked. What is being asked for in monastic spirituality is a life of fidelity to prayer and to the praying communities of which we are a part. Prayer is a community act in Benedictine life. It is at community prayer, in the midst of others, that we are most reminded that we are not a world unto ourselves.
Benedict will go so far as to have the community pray the opening psalm slowly to give the slow a chance to get there in an age without alarm clocks, but he will not allow such a lack of personal spiritual discipline to grow. Tardiness, the attempt to cut corners on everything in life, denies the soul the full experience of anything.
It is a lesson to be relearned in a modern age perhaps. There is nothing more important in our own list of important things to do in life than to stop at regular times, in regular ways to remember what life is really about, where it came from, why we have it, what we are to do with it, and for whom we are to live it. No matter how tired we are or how busy we are or how impossible we think it is to do it, Benedictine spirituality says, Stop. Now. A spiritual life without a regular prayer life and an integrated community consciousness is pure illusion.
_March 23 – July 23 – Nov. 22_
_But, if monastics do not come to table before the verse so that all may say the verse and pray and sit down at table together, and if this failure happens through their own negligence or fault, they should be reproved up to the second time. If they still do not amend, let them not be permitted to share the common table, but take their meals alone, separated from the company of all. Their portion of wine should betaken away until there is satisfaction and amendment. Anyone not present for the verse said after meals is to be treated in the same manner._
_No one is to presume to eat or drink before or after the time appointed. Moreover, if anyone is offered something by the prioress or abbot and refuses it, then, if the monastic later wants what was refused or anything else, that one should receive nothing at all until appropriate amends have been made._
In a world of fast food drive-in restaurants, multiple family schedules, and three-car garages, the family meal has taken a decided second place in the spiritual and social formation of the culture. In Benedictine spirituality, however, the sacramental value of a meal is that the human concern we promise daily at the altar is demonstrated in the dining room where we prepare and serve and clean up after one another. The Rule is at least as firm on presence at meals at it is about presence at prayer. No one is to be late. No one is to eat before or after meals, or on her own, or on the run because monastic spirituality doesn't revolve around food, either having it or not having it. Monastic spirituality revolves around becoming a contributing part of a people of faith, living with them, learning with them, bearing their burdens, sharing their lives. The meal becomes the sanctifying center that reminds us, day in and day out, that unless we go on building the community around us, participating in it and bearing its burdens, then the words family and humanity become a sham, no matter how good our work at the office, no matter how important our work in the world around us.
The Sufi tell a story. To a group of disciples whose hearts were set on a pilgrimage, the elder said, "Take this bitter gourd along. Make sure you dip it into all the holy rivers and bring it into all the holy shrines." When the disciples returned, the bitter gourd was cooked and served. "Strange," said the elder slyly after they had tasted it, "the holy water and the shrines have failed to sweeten it." All the prayer in the world, Benedict knows, is fruitless and futile if it does not translate into a life of human community made richer and sweeter by the efforts of us all. Both community and prayer, therefore, are essential elements of Benedictine spirituality, and we may not neglect either.
CHAPTER 44
SATISFACTION BY THE EXCOMMUNICATED
_March 24 – July 24 – Nov. 23_
_Those excommunicated for serious faults from the oratory and from the table are to prostrate themselves in silence at the oratory entrance at the end of the celebration of the Opus Dei. They should lie face down at the feet of all as they leave the oratory, and let them do this until the prioress or abbot judges they have made satisfaction. Next, at the bidding of the prioress or abbot, they are to prostrate themselves at the feet of the prioress or abbot, then at the feet of all that they may pray for them. Only then, if the prioress or abbot orders, should they be admitted to the choir in the rank the prioress or abbot assigns. Even so, they should not presume to lead a psalm or a reading or anything else in the oratory without further instructions from the prioress or abbot. In addition, at all the hours, as the Opus Dei is being completed, they must prostrate themselves in the place they occupy. They will continue this form of satisfaction until the prioress or abbot again bids them cease._
_Those excommunicated for less serious faults from the table only are to make satisfaction in the oratory for as long as the prioress or abbot orders.They do so until they give them blessing and say, "Enough."_
"A community is too heavy for any one to carry alone," the rabbis say. Benedict argues that the community enterprise is such an important one that those who violate their responsibilities to it must serve as warning to others of the consequences of failing to carry the human community. The point, of course, is not that the group has the power to exclude us. The point is that we must come to realize that we too often exclude ourselves from the relationships we promised to honor and to build by becoming the center of our own lives and ignoring our responsibilities to theirs.
The correction seems harsh and humiliating by modern standards, but the Rule is working with the willing if not with the ready who seek to grow rather than to accommodate. The ancients tell the story of the distressed person who came to the Holy One for help. "Do you really want a cure?" the Holy One asked. "If I did not, would I bother to come to you?" the disciple answered. "Oh, yes," the master said. "Most people do." And the disciple said, incredulously, "But what for then?" And the Holy One answered, "Well, not for a cure. That's painful. They come for relief."
This chapter forces us to ask, in an age without penances and in a culture totally given to individualism, what relationships we may be betraying by selfishness and what it would take to cure ourselves of the self-centeredness that requires the rest of the world to exist for our own convenience.
CHAPTER 45
MISTAKES IN THE ORATORY
_March 25 – July 25 – Nov. 24_
_Should monastics make a mistake in a psalm, responsory, refrain, or reading, they must make satisfaction there before all. If they do not use this occasion to humble themselves, they will be subjected to more severe punishment for failing to correct by humility the wrong committed through negligence. Youth, however, are to be whipped for such a fault._
"To know all of the Talmud is a great thing," the rabbis teach, "but to learn one virtue is greater." In Benedictine spirituality, two constants emerge clearly: first, community prayer is central to the life, and, second, whatever is done must be done well. To fail to prepare the prayer, then, to pray poorly, and sloppily, to read the Scripture to people who do not have books and to read it without care, without sense, without accuracy is to strike at the very core of the community life. It is a fault serious enough to undermine the spiritual life of the community. It is not to be endured.
"Those who pray without knowing what they pray," Maimon Ben Joseph wrote, "do not pray." If anything, this chapter requires us to ask even to this day how it is that we can hear the Scripture but never study it, pray prayers but never contemplate the universal implications of them, go through rituals but never immerse ourselves in their meaning. How is it that we too pray without thinking, pray carelessly, pray poorly, or pray without thought?
CHAPTER 46
FAULTS COMMITTED IN OTHER MATTERS
_March 26 – July 26 – Nov. 25_
_If monastics commit a fault while at any work—while working in the kitchen, in the storeroom, in serving, in the bakery, in the garden, in any craft or anywhere else—either by breaking or losing something or failing in any other way in any other place, they must at once come before the prioress or abbotand community and of their own accord admit their fault and make satisfaction. If it is made known through another, they are to be subjected to a more severe correction._
Accountability is the Benedictine value on which all community life is based. Benedict clearly never supposes perfection in a Benedictine community. People have bad days and recalcitrant spirits and limited education and difficult periods in life, all of which are acknowledged and even provided for in a rule that concerns itself with single-minded seeking of God. What Benedict does require, however, is a sense of responsibility. There is nothing in community life, he implies here, that is so unimportant that it can be ignored or overlooked. Nothing in life is so meaningless that we have the right to do it unthinkingly. What each of us does affects all the others and it is to everyone that we owe accounting and apology and reparation.
The notion that everything we do affects others and stands to be judged by them constitutes a concept of human community that is long lost. In this world, corporations gut the center out of forests and say not one word of sorrow to the children of the world who will inherit the dry and eroded mountainsides on which the trees once grew. Bankers take profits that close businesses and say nothing to the people made homeless by the deal. Politicians make policies that rape the third world and say not a thing to whole nations held hostage to greed. Individuals overheat, overconsume, and overbuy until the resources of the globe are wasted away to nothing and we think nothing of it.
Clearly, chapter 46 is not about punishment. Chapter 46 is about social consciousness.
_When the cause of the sin lies hidden in the conscience, the monastic is to reveal it only to the prioress or abbot or to one of the spiritual elders, who know how to heal their own wounds as well as those of others, without exposing them and making them public._
Everybody needs somebody to whom they can reveal themselves without fear of punishment or pain. Everybody, at some time in life, wrestles with an angel that threatens to overpower them. Contemporary society, with its bent for anonymity and pathological individualism and transience, has institutionalized the process in psychological consulting services and spiritual direction centers. Benedict would have approved. He wanted people to work skillfully with the souls of others. He would probably also have found some of it unnecessary. What we need, he says, are people in our lives who care enough about us to lead us through life's various stages gently. If we chose spiritual people for our friends and our leaders, if we respected our elders for their wisdom, if we wanted growth rather than comfort, if we ripped away the masks that hide us and were willing to have our bleeding selves cauterized by the light of spiritual leadership and the heat of holy friendship, we would, this chapter indicates, come to the humility that brings real peace.
Another facet of this chapter looms equally important. The challenge of community lies in whether we ourselves care enough about anyone else to be willing to be their light, to treat their wounds well, to protect their reputations when they try to talk to us.
The _Tao Te Ching_ reads, "Knowing others is intelligence; knowing yourself is true wisdom. Mastering others is strength; mastering yourself is true power." Benedictine spirituality asks for both.
CHAPTER 47
ANNOUNCING THE HOURS FOR THE OPUS DEI
_March 27 – July 27 – Nov. 26_
_It is the responsibility of the abbot and prioress to announce, day and night, the hour for the Opus Dei. They may do so personally or delegate the responsibility to a conscientious member, so that everything may be done at the proper time._
_Only those so authorized are to lead psalms and refrains, after the prioress or abbot according to their rank. No monastics should presume to read or sing unless they are able to benefit the hearers; let this be done with humility, seriousness, and reverence, and at the bidding of the prioress or abbot._
Prayer in a Benedictine community is to be both regular and artistic, and it is the role of leadership to see that this is so. In a culture without alarm clocks and in a community that prayed in the middle of the night, the responsibility was a major one. Even centuries later, however, when we all rouse ourselves to the sound of clock radios or a dozen other automatic devices and have no need for bellringers, the situation is just as serious. The message under the message is that unless the group becomes more and more immersed in prayer and the Scriptures, giving them priority no matter what the other pressures of the day, the group will cease to have any authenticity at all. It will cease to develop. It will dry up and cave in on itself and become more museum than monastery. This stress on our responsibility to call ourselves to prayer is an insight as fresh for the twenty-first century as it was for the sixth. For all of us, prayer must be regular, not haphazard, not erratic, not chance. At the same time, it cannot be routine or meaningless or without substance. Prayer has to bring beauty, substance, and structure to our otherwise chaotic and superficial lives or it is not long before life itself becomes chaotic and superficial. A life of spiritual substance is a life of quality. The _Tao Te Ching_ puts it this way:
She who is centered in the Tao
can go where she wishes, without danger.
She perceives the universal harmony,
even amid great pain,
because she has found peace in her heart.
CHAPTER 48
THE DAILY MANUAL LABOR
_March 28 – July 28 – Nov. 27_
_Idleness is the enemy of the soul. Therefore, the community members should have specified periods for manual labor as well as for prayerful reading._
There is little room for excursion into the quixotic in the Rule of Benedict. If any chapter proves that point best, it may well be the chapter on work. Benedict doesn't labor the point but he clearly makes it: Benedictine life is life immersed in the sanctity of the real and work is a fundamental part of it. The function of the spiritual life is not to escape into the next world; it is to live well in this one. The monastic engages in creative work as a way to be responsible for the upbuilding of the community. Work periods, in fact, are specified just as prayer periods are. Work and prayer are opposite sides of the great coin of a life that is both holy and useful, immersed in God and dedicated to the transcendent in the human. It is labor's transfiguration of the commonplace, the transformation of the ordinary that makes cocreators of us all.
_We believe that the times for both may be arranged as follows: From Easter to the first of October, they will spend their mornings after Prime till about the fourth hour at whatever work needs to be done. Fromthe fourth hour until the time of Sext, they will devote themselves to reading. But after Sext and their meal, they may rest on their beds in complete silence; should any members wish to read privately, let them do so, but without disturbing the others. They should say None a little early, about midway through the eighth hour, and then until Vespers they are to return to whatever work is necessary. They must not become distressed if local conditions or their poverty should force them to do the harvesting themselves. When they live by the labor of their hands, as our ancestors and the apostles did, then they are really monastics. Yet, all things are to be done with moderation on account of the fainthearted._
Benedictine spirituality exacts something so much harder for our century than rigor. Benedictine spirituality demands balance. Immediately after Benedict talks about the human need to work, to fill our lives with something useful and creative and worthy of our concentration, he talks about lectio, about holy reading and study. Then, in a world that depended on the rising and the setting of the sun to mark the days rather than on the artificial numbers on the face of a clock, Benedict shifts prayer, work, and reading periods from season to season to allow for some of each and not too much of either as the days stretch or diminish from period to period. He wants prayer to be brief, work to be daily, and study to be constant. With allowances for periodic changes, then, the community prayed and studied from about 2:00 A.M. to dawn and then worked for a couple of hours until the hour of Terce at about 10:00 A.M. Then, after Terce they read for a couple of hours until Sext before the midday meal. After dinner they rested or read until about 2:30 and then went back to work for three or four hours until Vespers and supper in the late afternoon. After saying a very brief Compline or evening prayer they retired after sundown for the night. It was a gentle, full, enriching, regular, calm, and balanced life. It was a prescription for life that ironically has become very hard to achieve in a world of light bulbs and telephones and cars, but it may be more necessary than ever if the modern soul is to regain any of the real rhythm of life and so its sanity as well.
_March 29 – July 29 – Nov. 28_
_From the first of October to the beginning of Lent, the members ought to devote themselves to reading until the end of the second hour. At this time Terce is said and they are to work at their assigned tasks until None. At the first signal for the hour of None, all put aside their work to be ready for thesecond signal. Then after their meal they will devote themselves to their reading or to the psalms._
_During the days of Lent, they should be free in the morning to read until the third hour, after which they will work at their assigned tasks until the end of the tenth hour. During this time of Lent each one is to receive a book from the library and is to read the whole of it straight through. These books are to be distributed at the beginning of Lent._
During Lent, the monks are to go on working but to increase their reading time. In this period, they are to be assigned a book to read "straight through." In Lent they are to put themselves on a regimen and study what they are told to study in a serious and ordered way. Nevertheless, the work continues. Benedictines were to "earn their bread by the labor of their hands," and no devotion was to take the place of the demands of life. These were working monastics who depended on God to provide the means of getting food but who did not, as the ancients said, depend on God to put it in the nest.
At the same time, work is not what defines the Benedictine. It is the single-minded search for God that defines Benedictine spirituality. That is what the monastic pursues behind every other pursuit. That is what gives the monastic life meaning. That is what frees the monastic heart. The monastic does not exist for work. Creative and productive work are simply meant to enhance the Garden and sustain us while we grow into God.
In today's culture in which people are identified more by what they do than what they are, this is a lesson of profound importance. Once the retirement dinner is over and the company watch is engraved, there has to be something left in life that makes us human and makes us happy or life may well have been in vain. That something, Benedictine spirituality indicates, is a mind and a heart full of a sense of meaning and an instinct for God.
_Above all, one or two elders must surely be deputed to make the rounds of the monastery while the members are reading. Their duty is to see that no one is so apathetic as to waste time or engage in idle talk to the neglect of their reading, and so not only harm themselves but also distract others. If such persons are found—God forbid—they should be reproved a first and a second time. If they do not amend, they must be subjected to the punishment of the rule as a warning to others. Further, membersought not to associate with one another at inappropriate times._
Study is hard work. It is so much easier to find something else to do in its place than to stay at the grind of it. We have excuses aplenty for avoiding the dull, hard, daily attempt to learn. There is always something so much more important to do than reading. There is always someone we have to talk to about something that can't wait until the reading time is over. There is always some overwhelming fatigue to be dealt with before we can really begin to concentrate. There is always some excuse for not stretching our souls with new ideas and insights now or yet or ever. But Benedictine spirituality says life is to be struggled through and worked at and concentrated on and cultivated. It is not a matter of simply going through it and hoping that enough of the rust of time is removed by accident to make us burnished spiritual adults.
_March 30 – July 30 – Nov. 29_
_On Sunday all are to be engaged in reading except those who have been assigned various duties. If any are so remiss and indolent that they are unwilling or unable to study or to read, they are to be given some work in order that they may not be idle._
_Those who are sick or weak should be given a type of work or craft that will keep them busy without overwhelming them or driving them away. The prioress or abbot must take their infirmities into account._
A midrash on Genesis reads, "Weeds spring up and thrive; but to get wheat how much toil we must endure." The Rule of Benedict treats work and lectio interchangeably. One focuses the skills of the body on the task of cocreation. The other focuses the gifts of the mind on the lessons of the heart. One without the other is not Benedictine spirituality. To get the wheat of life we need to work at planting as well as reaping, at reaping as well as planting. And everyone in the community is expected to do both. For those for whom study is an impossible burden, then physical labor is allowed to suffice for both, but never is the Benedictine mind to be left simply awash in idle emptiness. Even the sick and the weak are to be given simple tasks that upbuild the house of God because, Benedict knows, no matter how frail, no matter how old, no one is useless; every one of us is given a gift to give and a task to fulfill. At every stage of our lives, every one of us has a sign of hope and faith and love and commitment to share with the people around us. Sometimes, perhaps, it is precisely when we feel that we have least to give that our gifts are needed most. The sight of a grandmother in a garden or an uncle on a lawn mower, an old monastic tatting lace or a crippled young man lurching stiffly to the office may be just what the rest of us need to begin again down our healthy but tiresome paths.
CHAPTER 49
THE OBSERVANCE OF LENT
_March 31 – July 31 – Nov. 30_
_The life of a monastic ought to be a continuous Lent. Since few, however, have the strength for this, we urge the entire community during these days of Lent to keep its manner of life most pure and to wash away in this holy season the negligences of other times. This we can do in a fitting manner by refusing to indulge evil habits and by devoting ourselves to prayer with tears, to reading, to compunction of heart and self-denial. During these days, therefore, we will add to the usual measure of our service something by way of private prayer and abstinence from food or drink, so that each of us will have something above the assigned measure to offer God of our own will with the joy of the Holy Spirit (1 Thess. 1:6). In other words, let each one deny themselves some food, drink, sleep, needlesstalking, and idle jesting, and look forward to holy Easter with joy and spiritual longing._
"Once upon a time," an ancient story tells, "the master had a visitor who came to inquire about Zen. But instead of listening, the visitor kept talking about his own concerns and giving his own thoughts.
"After a while, the master served tea. He poured tea into his visitor's cup until it was full and then he kept on pouring.
"Finally the visitor could not bear it any longer. 'Don't you see that my cup is full?' he said. 'It's not possible to get anymore in.'
"'Just so,' the master said, stopping at last. 'And like this cup, you are filled with your own ideas. How can you expect me to give you Zen unless you first empty your cup?'"
A monastic Lent is the process of emptying our cups. Lent is the time for trimming the soul and scraping the sludge off a life turned slipshod. Lent is about taking stock of time, even religious time. Lent is about exercising the control that enables us to say no to ourselves so that when life turns hard of its own accord we have the spiritual stamina to say yes to its twists and turns with faith and with hope. Most interesting of all, perhaps, is the fact that Benedict wants us to do something beyond the normal requirement of our lives "of our own will." Not forced, not prescribed for us by someone else. Not required by the system, but taken upon ourselves because we want to be open to the God of darkness as well as to the God of light.
Benedict tells us that Lent is the time to make new efforts to be what we say we want to be. We applaud the concept in most things. We know, for instance, that even people who were married years ago have to keep working at that marriage consciously and intently every year thereafter, or the marriage will fail no matter how established it seems. We know that people who own businesses take inventories and evaluations every year or the business fails. We too often fail to realize, however, that people who say that they want to find God in life have to work every day too to bring that Presence into focus, or the Presence will elude them no matter how present it is in theory.
_All should, however, make known to the prioress or abbot what they intend to do, since it ought to be done with their prayer and approval. Whatever is undertaken without the permission of the prioress or abbot will be reckoned as presumption and vainglory, not deserving a reward. Therefore, everything must be done with their approval._
An ancient people tells us that when the moment of a great teacher's death was near, the disciples said, "What is it we will see when you are gone?" And the master said, "All I did was sit on the river bank handing out river water. After I'm gone I trust you will notice the water." Spiritual mentoring is a staple of the Benedictine tradition. The role of the abbot or prioress is to evaluate the directions the seeker intends to take. Like anything else, the spiritual life can become an elixir of novelties, a series of fads, an excursion into the whimsical. Benedict counsels the zealous to submit themselves to the scrutiny of wisdom so that the spiritual remedies they fancy have the merit of the tried and the true, the sensible and the measured. It is so easy to ply extremes and miss the river of tradition. This chapter reminds us that the purpose of personal restraint is to develop us, not to ravage our energies or confuse our perspective on life.
CHAPTER 50
MEMBERS WORKING AT A DISTANCE OR TRAVELING
_April 1 – Aug. 1 – Dec. 1_
_Members who work so far away that they cannot return to the oratory at the proper time—and the prioress or abbot determines that is the case—are to perform the Opus Dei where they are, and kneel out of reverence for God._
_So too, those who have been sent on a journey are not to omit the prescribed hours but to observe them as best they can, not neglecting their measure of service._
In Sanskrit it is written: necessity changes a course but never a goal. Benedictine spirituality—flexible, sensible, realistic at all times—sets loud, clear goals but models a number of ways to achieve them. Perhaps there is no surer proof of Benedict's awareness that spirituality is neither a formula nor a straightjacket than this chapter. Benedict values nothing more than community prayer, the Opus Dei. In other chapters he organizes it minutely and schedules it for seven times a day. "Nothing," he writes "is to be preferred to the Work of God." And yet, when the ideal is confronted by the real, Benedict opts for the sanctification of the real rather than the idealization of the holy. If there is work to be done at a great distance from the chapel, the monastic is to see that it's done. Holiness is not an excuse to avoid responsibility. Spirituality is not an escape from life. Spirituality leavens life. Spirituality is what stabilizes us in the middle of confusion and gives us energy to go on doing what must be done even when the rest of life taxes and fatigues and separates us from our own resources.
CHAPTER 51
MEMBERS ON A SHORT JOURNEY
_April 2 – Aug. 2 – Dec. 2_
_If members are sent on some errand and expect to return to the monastery that same day, they must not presume to eat outside, even if they receive a pressing invitation, unless perhaps the prioress or abbot has ordered it. Should they act otherwise, they will be excommunicated._
Benedictine spirituality, this chapter implies, is not a set of rules; it is a way of life. Being out of the monastery does not relieve the monastic of the obligation to be what we say we are—simple, centered in God, in search of higher things. What life demands from us is the single-minded search for God, not a series of vacations from our best selves. The point is a clear one: being a religious is full-time identity; being business people does not give us the right to do during the week what we tell ourselves on Sunday that we shun; being American does not give us the right to be less Christian in order to be more patriotic; being rich does not give us the right to forget the poor. No Christian ever has the right to be less than the Gospels demand of them wherever they are.
CHAPTER 52
THE ORATORY OF THE MONASTERY
_April 3 – Aug. 3 – Dec. 3_
_The oratory ought to be what it is called, and nothing else is to be done or stored there. After the Opus Dei, all should leave in complete silence and with reverence for God, so that anyone who may wish to pray alone will not be disturbed by the insensitivity of another. Moreover, if at other times some choose to pray privately, they may simply go in and pray, not in a loud voice, but with tears and heartfelt devotion. Accordingly, those who do not pray in this manner are not to remain in the oratory after the Opus Dei, as we have said; then they will not interfere with anyone else._
Richard Sullivan, a professor of creative writing at the University of Notre Dame in the 1960s and a writer himself, taught his classes that the two most important physical dimensions of the writing profession were time and space. "Write every single day at the same time and in the very same place," he said. "Whether you have anything to say or not, go there and sit and do nothing, if necessary, until the very act of sitting there at your writer's time in your writer's place releases the writing energy in you and begins to affect you automatically." Teachers of yoga, too, prescribe a set of basic postures and places to dispose the soul to the transcendent. Teachers of meditation prescribe times and places and mantras, a type of personal chant, to center the soul. In every tradition we are taught that it is not a matter of separating the sacred and the secular. It is a matter of staying conscious of the fact that the sacred is in the secular. There is, in other words, such a thing as a spiritual well where simply being in that place can tap open that special part of our souls and enable us to touch the sacred in the secular.
"Let the oratory be what it is called," Benedict said. Have a place where you can go in order to be about nothing but the business of being in the presence of God so that every other space in your life can become more conscious of that Presence as well. More than that, Benedict asks us to be there in a special way—with quiet and with awareness, not laughing or talking or lounging or distracting but alert and immersed and enshrouded in the arms of God. Americans, of course, have made of God a casual circumstance. We have prayer meetings with coffee cups in our hands and listen to psalmody with our legs crossed and our arms spread-eagled on the backs of our pews. We avoid churches and say that since God is everywhere, any place is good enough. All of which is true, at one level. But Benedictine spirituality says also that to know God in time and space we must regularly seek to find God in one time and space that enables us to recognize God more easily in every other one.
CHAPTER 53
THE RECEPTION OF GUESTS
_April 4 – Aug. 4 – Dec. 4_
_All guests who present themselves are to be welcomed as Christ, who said: "I was a stranger and you welcomed me" (Matt. 25:35). Proper honor must be shown "to all, especially to those who share our faith" (Gal. 6:10) and to pilgrims._
_Once guests have been announced, the prioress or abbot and the community are to meet them with all the courtesy of love. First of all, they are to pray together and thus be united in peace, but prayer must always precede the kiss of peace because of the delusions of the devil._
Stereotypes come hard in the Benedictine tradition. Is this a spirituality that centers on prayer or work? Does it recommend fleeing the world or embracing it? Does it set out to create a world unto itself or to leaven the wider one? The difficulty with understanding Benedictine spirituality comes in reading some sections of the Rule without reading the entire document. The fact is that Benedictine spirituality is not based in dualism, in the notion that things of the world are bad for us and things of the spirit are good. We are not to pray too long but we are to pray always. Self-discipline is a given, but wine and food and the creature comforts of a bed with bedding are also considered necessary. The Rule is for everyone, including the abbot or prioress, and yet everyone is a potential exception to it.
In this chapter on guests and hospitality, the wholism out of which it emerges is startlingly plain: this is a monastery and guests are to be received. As Christ. "Hospitality is one form of worship," the rabbis wrote. Benedictine spirituality takes this tendency seriously. The welcome at the door is not only loving—a telephone operator at a jail can do that. It is total, as well. Both the community and the abbot receive the guest. The message to the stranger is clear: come right in and disturb our perfect lives. You are the Christ for us today.
And to assure us all, guest and monastic alike, that this hospitality is an act of God that we are undertaking, the community and the guest pray together first and then extend the kiss of welcome so that it is understood that our welcome is not based on human measurements alone: we like you, we're impressed with you, you look like our kind, you're clean and scrubbed and minty-breathed and worthy of our attention.
Hospitality in a culture of violence and strangers and anonymity has become the art of making good connections at good cocktail parties. We don't talk in elevators, we don't know the security guard's name, we don't invite even the neighbors in to the sanctuary of our selves. Their children get sick and their parents die and all we do is watch the comings and goings from behind heavy blinds. Benedict wants us to let down the barriers of our hearts so that this generation does not miss accompanying the innocent to Calvary as the last one did. Benedict wants us to let down the barriers of our souls so that the God of the unexpected can come in.
_All humility should be shown in addressing a guest on arrival or departure. By a bow of the head or by a complete prostration of the body, Christ is to be adored and welcomed in them. After the guests have been received, they should be invited to pray; then the abbot or prioress or an appointed member will sit with them. The divine law is read to all guests for their instruction, and after that every kindness is shown to them. The prioress or abbot may break their fast for the sake of a guest, unless it is a day of special fast which cannot be broken. The members, however, observe the usual fast. The abbot or prioress shall pour water on the hands of the guest, and the abbot or prioress with the entire community shall wash their feet. After washing they will recite this verse: "God, we have received your mercy in the midst of your temple" (Ps. 48:10)._
"In India," Ram Dass writes, "when people meet and part they often say, 'Namaste,' which means: I honor the place in you where the entire universe resides; I honor the place in you of love, of light, of truth, of peace. I honor the place within you where if you are in that place in you and I am in that place in me, there is only one of us....'Namaste.'" In Benedictine spirituality, too, hospitality is clearly meant to be more than an open door. It is an acknowledgment of the gifts the stranger brings. "By a bow of the head or by a complete prostration...Christ is to be adored and welcomed in them."
But Benedictine hospitality is also a return of gifts. The stranger is shown both presence and service. After a trip through hard terrain and hot sun, the guest is given physical comfort and a good meal, spiritual instruction, and human support. Not even a fast day is counted as important as eating with a guest. Not even asceticism is counted as holy as care for the other. Obviously, from the point of view of the Rule of Benedict, it isn't so much what we do for those curious others in our lives, the strange, the needy, the unscrubbed, as it is the way we do it. We can give people charity or we can give them attention. We can give them the necessities of life or we can give them its joys. Benedictine hospitality is the gift of one human being to another. Benedictine hospitality is not simply bed and bath; it is home and family.
_Great care and concern are to be shown in receiving poor people and pilgrims, because in them more particularly Christ is received; our very awe of the rich guarantees them special respect._
"It's a barren prayer," St. Cyprian wrote, "that does not go hand in hand with alms." For the Benedictine heart the reception of the poor is an essential part of going to God. We cannot be too busy, too professional, too removed from the world of the poor to receive the poor and sustain the poor. Anything else, Benedict warns in a society that is by nature class structured, is not hospitality. It is at best more protocol than piety. Those who can buy their comforts or demand their rights are simply receiving what they can get, with us or without us. Those who have been thrown upon the mercy of the world are the gauge of our open hearts.
It is an important distinction in a culture in which strangers are ignored and self-sufficiency is considered a sign of virtue and poverty is a synonym for failure. Hospitality for us may as much involve a change of attitudes and perspectives as it does a handout. To practice hospitality in our world, it may be necessary to evaluate all the laws and all the promotions and all the invitation lists of corporate and political society from the point of view of the people who never make the lists. Then hospitality may demand that we work to change things.
_April 5 – Aug. 5 – Dec. 5_
_The kitchen for the abbot and prioress and guests ought to be separate, so that guests—and monasteries are never without them—need not disturb the community when they present themselves at unpredictable hours. Each year, two monastics who can do the work competently are to be assigned to this kitchen. Additional help should be available when needed, so that they can perform this service without grumbling. On the other hand, when the work slackens, they are to go wherever other duties are assigned them. This consideration is not for them alone, but applies to all duties in the monastery; members are to be given help when it is needed, and whenever they are free, they work wherever they are assigned._
_The guest quarters are to be entrusted to a God-fearing member. Adequate bedding should be available there. The house of God should be in the care of members who will manage it wisely._
_No monastics are to speak or associate with guests unless they are bidden; however, if the members meet or see guests, they are to greet them humbly, as we have said. They ask for a blessing and continue on their way, explaining that they are not allowed to speak._
With the fall of the Roman Empire, travel through Europe on unguarded and unkept roads through hostile territory and at the prey of marauding bands became both difficult and dangerous. Benedictine monasteries became the hospice system of Europe. There, anyone was received at any time. Rich and poor alike were accepted as equals and given the same service: food, bedding, immediate attention day or night. Yet, so that the monastery could remain a monastery in the midst of a steadily growing need for this monastic service, a special kitchen and special workers were assigned to provide the necessary care. It's an important addition to a chapter that could otherwise be read to mean that the monastic life itself was at the mercy of meandering peasants. The fact is that we all have to learn to provide for others while maintaining the values and structures, the balance and depth, of our own lives. The community that is to greet the guest is not to barter its own identity in the name of the guest. On the contrary, if we become less than we must be then we will be no gift for the guest at all. Parents must parent, and all the good work in the world will not substitute for that. Wives and husbands must be present to the other, and all the needs in the world will not forgive that. Balance and order and prayer in the life of those who practice Benedictine spirituality are keys to being a genuine support in the lives of others. Somehow, we must take on the needs of the world with a humble heart. As Edward Everett Hale said, "I cannot do everything but I can do something, and what I can do I will do, so help me God."
CHAPTER 54
LETTERS OR GIFTS
_April 6 – Aug. 6 – Dec. 6_
_In no circumstance are monastics allowed, unless the prioress or abbot says they may, to exchange letters, blessed tokens, or small gifts of any kind, with their parents or anyone else, or with another monastic. They must not presume to accept gifts sent them even by their parents without previously telling the prioress or abbot. If the prioress or abbot orders acceptance, they still have the power to give the gift to whomever; and the one for whom it was originally sent must not be distressed, "lest occasion be given to the devil" (Eph. 4:27; 1 Tim. 5:14). Whoever presumes to act otherwise will be subjected to the discipline of the rule._
Rabbi Mordecai said, "If a single coin is left over in my house at bedtime, I cannot fall asleep. But if totally penniless, I sleep soundly, knowing that when the moment comes to awaken, I must immediately look to the Lord for aid." And the rabbi of Porissover taught, "If a person is poor and meek, it is easy for that one to be joyful, inasmuch as there is nothing to guard against losing." In a community based on equality in the midst of a highly stratified society, Benedict had no desire to create a subset of the independently wealthy whose parents or friends could provide for them beyond the means of the monastery. The purpose of monastic life was to discover that the possession of God was far more satisfying than anything we could receive from family or friends, that it was freeing, that it was enriching far beyond what we could collect for ourselves.
We live in a culture that sees having things as the measure of our success. We strive for a life that sees eliminating things as the measure of internal wealth. Enoughness is a value long dead in Western society. Dependence on God is a value long lost. Yet, enoughness and dependence on God may be what is lacking in a society where consumerism and accumulation have become the root diseases of a world in which everything is not enough and nothing satisfies.
CHAPTER 55
CLOTHING AND FOOTWEAR
_April 7 – Aug. 7 – Dec. 7_
_The clothing distributed to the members should vary according to local conditions and climate, because more is needed in cold regions and less in warmer. This is left to the discretion of the prioress or abbot. We believe that for each monastic a cowl and tunic will suffice in temperate regions; in winter a woolen cowl is necessary, in summer a thinner or worn one;also a scapular for work, and footwear—both sandals and shoes._
_Monastics must not complain about the color or coarseness of all these articles, but use what is available in the vicinity at a reasonable cost. However, the prioress and abbot ought to be concerned about the measurements of these garments that they not be too short but fitted to the wearers._
Maimonides, one of the finest and best-educated minds in twelfth-century Jewish history, writes in the _Mishneh Torah,_ "The dress of the wise must be free of stains; they should not wear the apparel of princes, to attract attention, nor the raiment of paupers, which incurs disrespect." Clothing, in other words, was to clothe, neither to adorn nor to diminish the human person. Clothing was clothing.
Benedictines differ in their literal interpretation of the passage on clothing in the Rule. Some groups focus on the types of clothing described and devise a uniform from a sixth-century wardrobe—a long dress, a cowl to protect against weather that was cold and damp, a scapular. Other groups emphasize that the clothing worn should simply be local and approved by the local prioress or abbot. Whatever the present demonstration of the passage, both groups believe in simplicity, sufficiency, and a guard against excess. Slavery to style is not Benedictine. Excess is not Benedictine. Ostentation and pretension and fads are not Benedictine. Slovenliness and dirt are not Benedictine. The Benedictine is clean, simple, and proper to the time and place because the stewardship of the universe demands a commitment to order, harmony, and rightness if it is to survive. The Benedictine is one of the world's uncomplicated types who have what is necessary for every occasion and nothing more.
Dress is a mark of values and aspirations and ideals. It is as easy to call attention to ourselves by too little as too much; as easy to lose sight of what we really are about in life by too much as too little. If the chapter on clothing has anything to say to the modern world at all, it is certainly that we need to be who we are. We need to look inside ourselves for our value and not pretend to be what we are not. We need to stop putting on airs and separating ourselves out and pretending to be what we are not. Fraud is an easy thing. The honesty of humility, the humility of honesty are precious and rare.
_Whenever new clothing is received, the old should be returned at once and stored in a wardrobe for the poor. To provide for laundering and nightwear, every member will need two cowls and two tunics, but anything more must be taken away as superfluous. When new articles are received, the worn ones—sandals or anything old—must be returned._
_Those going on a journey should get underclothing from the wardrobe. On their return they are to wash it and give it back. Their cowls and tunics, too, ought to be somewhat better than those they ordinarily wear. Let them get these from the wardrobe before departing, and on returning put them back._
Taking care of the self has something to do with taking care of the universe. If we do not care about our presentation of self, it is unlikely that we will worry about littering the countryside or care about preservation of resources or stewardship of the earth. Being sloppy is not a monastic ideal. Just because a thing is not useful in the monastery anymore does not necessarily make it useless. It may, in fact, still be very useful to someone else and so should be given away. We owe what is useless to us to the poor. What is no longer important to us is to be made available to the other, in good condition, with quality and care. There is a Benedictine virtue in washing things and hanging them up and folding them nicely and keeping them neat and giving them to people who can use them, not because they are not worth anything but precisely because they are still worth something.
Benedictine spirituality recognizes that a thing may become valueless to us before it actually becomes valueless. In that case it is to be given to someone else in good condition. Benedictine spirituality does not understand a world that is full of gorgeous garbage while the poor lack the basics of life.
_April 8 – Aug. 8 – Dec. 8_
_For bedding monastics will need a mat, a woolen blanket, and a light covering as well as a pillow._
_The beds are to be inspected frequently by the prioress or abbot, lest private possessions be found there. Anyone discovered with anything not given by the prioress or abbot must be subjected to very severe punishment. In order that this vice of private ownership may be completely uprooted, the prioress or abbot is to provide all things necessary: that is, cowl, tunic, sandals, shoes, belt, knife, stylus, needle, handkerchief, and writing tablets. In this way every excuse of lacking some necessity will be taken away._
"The best way to know God," Vincent Van Gogh wrote, "is to love many things." Things do not destroy us. It is the way we approach things that entraps us. The Rule of Benedict provides for human needs without frugality, without abstemious control, without small-mindedness, and without indulgence. False asceticism is not a Benedictine virtue. Deprivation is not a Benedictine ideal. On the contrary, the point of Benedictine life is to live simply, joyfully, and fully. Benedict wants the monastic to have enough, to have it from the community, and to avoid hoarding, accumulating, consuming, and conniving. The Rule recognizes that people who lack the necessities of life often spend their time either consumed with thoughts of subsistence or struggling against bitterness and clawing for survival. On the other hand, people smothered by things run the risk of slipping into indolence or becoming blinded to the important things of life. In striking a balance between the two, Benedictine spirituality seeks to free the body so that the soul can soar. It is a gift long lost in a consumer society.
_The abbot and prioress, however, must always bear in mind what is said in the Acts of the Apostles: "Distribution was made as each had need" (Acts 4:35). In this way the prioress and abbot will take into account the weakness of the needy, not the evil will of the envious; yet in all their judgments they must bear in mind God's retribution._
Self-control is one value in the lexicon of monastic spirituality but compassion is another. Benedict may expect simplicity from the monastic, but he clearly expects great largesse from the abbot and the prioress. The function of authority, in other words, is to hold the Rule aloft in the community, to be clear about its standards and respectful of its values, without ever using the Rule as an excuse to frustrate people or irritate them or control them.
There is a great deal of pain administered in the interest of virtue. Righteousness allows no exceptions. As a result, laws meant to free the spirit so often enslave it to ideals far beneath its purpose. Benedictine spirituality, practiced in the little things of life like the distribution of clothing that calls for a minimum and then allows more, says that we must always grasp for what we cannot reach, knowing that the grasping itself is enough.
CHAPTER 56
THE PRIORESS'S OR ABBOT'S TABLE
_April 9 – Aug. 9 – Dec. 9_
_The table of the prioress or abbot must always be with guests and travelers. Whenever there are no guests, it is within their right to invite anyone of the community they wish. However, for the sake of maintaining discipline, one or two seniors must always be left with the others._
This tiny chapter introduces a major question into Benedictine history and interpretation: Did the abbot eat in a separate dining room away from the monastics or did the abbot and guests eat at a special table in the midst of the community? And, whatever the case, what was the implication of this separate table for the rest of community life? If the monastic meal was a central symbol of community life, then the presence or absence of an abbot or prioress is of serious import, to say nothing of the notion that the ideas of cloister may then have been flexible enough to make guests, too, part of the monastery meal. There have been cogent arguments brought to bear on both interpretations that are both interesting and historically important.
It seems, however, that the greater point of the chapter for us today is not the geography of the table but the fact that the leader of the community was expected to model the gift of self with strangers. It was the abbot and prioress themselves who showed the community the price and the process of availability and hospitality and presence to the other. Hospitality was not a warm meal and a safe haven. Hospitality in the Benedictine community was attention and presence to the needs of the other. Hospitality was a public ministry designed to nourish the other in body and in soul, in spirit and in psyche.
Welfare agencies give clothes; parishes collect food for the poor; flea markets provide rare goods at cheap prices. The problem is that too many of the handouts come with hardly a look and never a personal moment for the people they set out to serve. Benedictine spirituality sets a standard of comfort and care, conversation and respect—the things that make a human being human—as well as bed and board. And, as the presence of the abbot and prioress proves, none of us can afford to be too busy or too important to do the same.
CHAPTER 57
THE ARTISANS OF THE MONASTERY
_April 10 – Aug. 10 – Dec. 10_
_If there are artisans in the monastery, they are to practice their craft with all humility, but only with the permission of the prioress or the abbot. If one of them becomes puffed up by skillfulness in the craft, and feels that they are conferring something on the monastery, they are to be removed from practicing the craft and not allowed to resume it unless, after manifesting humility, they are so ordered by the prioress or abbot._
There are three major points made in the chapter on the artists of the monastery: first, that there may be artists in a monastery; second, that they must themselves be humble about it; and third, that an art is not to be practiced for the sake of money. All three points have a great deal to do with the way we look at religious dedication, personal development, and contemporary society in the development of spiritual life today.
The points made in the Rule are relatively plain: the development of the spiritual life does not depend on the suppression of beauty or the destruction of the self. The gifts we have been given are for the doing of them, not the denial of them. We do not smother great gifts in the name of great spirituality. The painter, the writer, the musician, the inventor, the scholar, all have to figure out how to put their gifts at the disposal of their spiritual life, not how to build a spiritual life at the expense of the gift.
The unusually gifted person or the person with the unusual gift, however, is also required to see that their giftedness does not get in the way of their striving for sanctity. No gift is given to tyrannize the community. On the contrary, we are expected to learn to take our gifts in stride, to practice them because they deserve to be practiced and because the community can profit from them. Aristotle wrote, "The aim of art is not to represent the outward appearance of things, but their inward significance." Any great gift is a revelation of the more in life, a natural expression of the spiritual, a necessary expression of the sacred. To stamp out the artist in the name of religious rigor is to stamp out the spiritual eye itself, and that kind of blindness plunges any group, any family, any person into darkness indeed. Without the artist to show us what we ourselves do not see of the beauty of the world around us, we lose sight of the beauty of God as well. Benedictine spirituality never substitutes conformity in discipline for fullness of expression in life. The function of the artist in the monastery—and in the life of us all—is to make the transcendent visible; to touch the soul in ways that match the soul; to enshrine beauty so that we may learn to see it; and to make where we live places of wonder.
A monastery without an artist could be a poor place spiritually indeed.
_Whenever products of these artisans are sold, those responsible for the sale must not dare to practice any fraud. Let them always remember Ananias and Sapphira, who incurred bodily death (Acts 5:1–11), lest they and all who perpetrate fraud in monastery affairs suffer spiritual death._
_The evil of avarice must have no part in establishing prices, which should, therefore, always be a little lower than people outside the monastery are able to set, "so that in all things God may be glorified" (1 Pet. 4:11)._
Of all the paragraphs in the Rule that are contrary to the cultural climate in which we live, this is one of the clearest. "Money often costs too much," Ralph Waldo Emerson wrote, and Benedictine spirituality would surely agree. Not just dishonesty but even the standards of the marketplace are un-Benedictine according to this chapter. Benedictine spirituality develops goods so that people can have them, not in order to make them available only to the highest bidder or to make excessive profits. Money gained in that fashion costs us compassion and community and our role as cocreators of the reign of God. It hollows out our souls and leaves us impoverished of character and deprived of the bounty of largesse. It is Benedictine to develop our gifts and distribute their fruits as widely and broadly as possible so that justice, but not profit, is the principle that impels us.
CHAPTER 58
THE PROCEDURE FOR RECEIVING MEMBERS
_April 11 – Aug. 11 – Dec. 11_
_Do not grant newcomers to the monastic life an easy entry, but, as the apostle says, "Test the spirits to see if they are from God" (1 John 4:1). Therefore, if someone comes and keeps knocking at the door, and if at the end of four or five days has shown patience in bearing harsh treatment and difficulty of entry, andhas persisted in the request, then that one should be allowed to enter and stay in the guest quarters for a few days. After that, the person should live in the novitiate, where the novices study, eat, and sleep._
The spiritual life is not a set of exercises appended to our ordinary routine. It is a complete reordering of our values and our priorities and our lives. Spirituality is not just a matter of joining the closest religious community or parish committee or faith-sharing group. Spirituality is that depth of soul that changes our lives and focuses our efforts and leads us to see the world differently than we ever did before. The Mezeritizer rabbi taught, "There are sparks of holiness in everything. They constitute our spirituality." Benedict, too, wants proof of this commitment to truth and perseverance in the search before a new member is even admitted to the community. "Test the spirits," the Rule says, and test he does, in more than one place. Even the newcomer is left sitting in the guesthouse until the community is sure that the applicant is sure. No one is to enter a Benedictine community on impulse and, once there, no one is to treat life as a series of hapless circumstances. In fact, life itself is a discipline. Life is something that we are to live with purpose and control right from the very beginning. Life is not easy and life is not to be lived as if it were, for fear that when we really need internal fortitude we will not have developed it.
It is an important insight for all of us. We must develop the rigor it takes to live through what life deals us. We can't set out to get holy in the hope that we will then automatically become faithful. We must require fidelity of ourselves even when we fail, in the hope that someday, as a result, we will finally become holy.
_A senior chosen for skill in winning souls should be appointed to look after the newcomer with careful attention. The concern must be whether the novice truly seeks God and shows eagerness for the Opus Dei, for obedience, and for trials. The novices should be clearly told all the hardships and difficulties that will lead to God._
There are two elements of this paragraph that may come as a surprise in the wake of early twentieth-century spirituality with its emphasis on particular examens and reparation for sin. The first is that it is not perfection that Benedict insists on in a newcomer to the spiritual life; it is direction. "The aim, if reached or not, makes great the life," Robert Browning wrote. The Rule of Benedict wants to know at what we're aiming: prayer, concern for the will of God, commitment—whatever the cost—or lesser things?
The second surprise in a document that was written in a century of harsh penances and rigorous pious disciplines is that the director is not asked to be harsh and demanding but "skilled in winning souls," someone who can make a hard way possible.
In the spiritual life we may fail often but we may never change course and we must always seek the help of those whose ways are wiser and more tried than ours.
_If they promise perseverance in stability, then after two months have elapsed let this rule be read straight through to them, and let them be told, "This is the law under which you are choosing to serve. If you can keep it, come in. If not, feel free to leave." If they still stand firm, they are to be taken back to thenovitiate, and again thoroughly tested in all patience. After six months have passed, the rule is to be read to them, so that they may know what they are entering. If once more they stand firm, let four months go by, and then read this rule to them again. If after due reflection they promise to observe everything and to obey every command given them, let them then be received into the community. But they must be well aware that, as the law of the rule establishes, from this day they are no longer free to leave the monastery, nor to shake from their neck the yoke of the rule which, in the course of so prolonged a period of reflection, they were free either to reject or to accept._
Benedict allows no one to take on the monastic life without knowing what it entails—in full and without gloss. At the same time, the Rule makes it quite clear that this is the process of a lifetime. It is not a year's experience; it is not a degree once gotten and then ignored. This is not a spiritual quick fix. It is a way of life and it takes a lifetime to absorb. Nothing important, nothing life altering, nothing that demands total commitment can be tried on lightly and easily discarded. It is the work of a lifetime that takes a lifetime to leaven us until, imperceptibly, we find ourselves changed into what we sought.
_April 12 – Aug. 12 – Dec. 12_
_When they are to be received, they come before the whole community in the oratory and promise stability, fidelity to the monastic life, and obedience. This is done in the presence of God and the saintsto impress on the novices that if they ever act otherwise, they will surely be condemned by the one they mock._
_They state their promise in a document drawn up in the name of the saints whose relics are there and of the prioress or abbot, who is present. Novices write out this document themselves, or if they are illiterate, then they ask someone else to write it for them, but put their mark to it and with their own hand lay it on the altar. After they have put it there, the novice begins the verse: "Receive me, O God, as you have promised, and I shall live; do not disappoint me in my hope" (Ps. 119:116). The whole community repeats the verse three times, and adds the doxology. Then the novices prostrate themselves at the feet of each member to ask prayers, and from that very day they are to be counted as one of the community._
Benedictine life is rooted in three dimensions: commitment to a community, fidelity to a monastic way of life, and obedience. It is a life that sees sanctification as a by-product of human society, the development of a new way of thinking and living, and a total openness to the constantly emerging challenges of the God-life within us. To pursue a Benedictine spirituality, we must carry our part of the human race and allow it to mold and polish and temper us. We are to be people who see the globe through eyes softened by the gospel. We are to see change and challenge in life as God's voice in our ears. Benedictine spirituality goes into the heart in order to embrace the world. It forms us differently than the world forms us but it does not attempt to shape us independent of the real world around us. The whole point of the profession ceremony itself is quite the opposite. We are, in fact, to make this commitment consciously and knowledgeably and publicly, in the presence of the community, the communion of saints that are represented by the relics of the church, and the leader of the community. This is a declaration that binds us to others and raises us beyond the changing feelings of the day to the obligations of a lifetime.
_If they have any possessions, they should either give them to the poor beforehand, or make a formal donation of them to the monastery, without keeping back a single thing for themselves, well aware that from that day they will not have even their own body at their disposal. Then and there in the oratory, they are to be stripped of everything of their own that they are wearing and clothed in what belongs to the monastery. The clothing taken from them is to be put away and kept safely in the wardrobe, so that, should they ever agree to the devil's suggestion and leave the monastery—which God forbid—they can be stripped of the clothing of the monastery before they are cast out. But that document of theirs which the prioress or abbot took from the altar should not be given back to them but kept in the monastery._
This passage of the Rule points out in a particularly graphic way that Benedictine spirituality demands a total change of the way we relate to life. In the first place, monastics are to depend entirely on the community for their support. They don't bring with them the family wealth, and they don't have any claim to personal property, not even their clothes. They give everything that they have gained up to the time of their entry into the community either to the poor or to the monastery itself. From then on, it is the support of the community and the providence of God upon which they are to depend, not on their savings, not on their business acumen, not on their relatives and connections. From then on they go through life as a people whose trust is in God and who are responsible for one another. The purpose, of course, is to free a person forcibly from the agenda of the world. "Those who have cattle have care," an African proverb teaches. We "can't serve God and mammon," the Scriptures say. The point of Benedictine spirituality is that we have to decide, once and for all, what we are about and then live in a way that makes that possible and makes that real.
CHAPTER 59
THE OFFERING OF CHILDREN BY NOBLES OR BY THE POOR
_April 13 – Aug. 13 – Dec. 13_
_If a member of the nobility offers a child to God in the monastery, and the child is too young, the parents draw up the document mentioned above; then, at the presentation of the gifts, they wrap the document itself and the child's hand in the altar cloth. That is how they make their offering._
_As to their property, they either make a sworn promise in this document that they will never personally, never through an intermediary, nor in any way at all, nor at any time, give the child anything or afford the child the opportunity to possess anything; or else, if they are unwilling to do this and still wish to win their reward for making an offering to the monastery, they make a formal donation of the property that they want to give to the monastery, keeping the revenue for themselves, should they so desire. This ought to leave no way open for the child to entertain any expectations that could deceive and lead to ruin. May God forbid this, but we have learned from experience that it can happen._
The dedication of children to God by their parents, the designation of their professions, or even the selection of their marriage partners was a common practice for centuries. The gifting of a child to a monastery, in particular, was believed to assure the salvation of the parents as well as the child. Not until the Council of Trent did the church itself define a legal profession age. In a period of history in which dedication of a child to God was a common pious practice, Benedict takes pains to see that the piety is not corrupted by the inexorable tension between the high ideals of the family and the test of time on the decision. The fact is that when the full realization of what we have promised begins to dawn on us, it is often more common to come to dubious terms with the demise of the commitment than it is to quit it. We marry in haste and then, as the years go by, we find ourselves starting to live life in two different parts of the house. We promise to spend more time with the children but read in the car while they play in the park. We take a job as a night security guard and go to sleep at the desk. Benedict wants to avoid that kind of silent erosion of zeal by binding both the child who is being given and the parents who do the giving to the promise to let the thing go on being what it set out to be. Benedict does not want the child torn between two identities, community member and family member, as it gets older. More than that, he does not want the parents themselves to begin to take back the spiritual covenant they have promised for the sake of their posterity or influence.
It is a chapter concerned about simplicity and community and equality, true, but it is also a chapter dedicated to the spirituality of the long haul. We must learn to complete in faith what we began in enthusiasm; we must learn to be true to ourselves; we must continue to become what we said we would be, even when accommodation to the immediate seems to be so much more sensible, so much more reasonable, so much easier.
_Poor people do the same, but those who have nothing at all simply write the document and, in the presence of witnesses, offer their child with the gifts._
The ability to eliminate distinctions between people is a hallmark of Benedictine simplicity and community. In the preceding paragraph it is obvious that Benedict is not accepting the children of the wealthy because their parents will endow the monastery. Whether they do or whether they don't makes no difference to him at all. What matters is that the children accepted as monastics out of the fervor of their parents' hearts be allowed to develop as monastics. Otherwise, he clearly fears, the community life and spirituality of the house will be corrupted by the independently wealthy who, as the years go by, grow more into the family fortune than into the monastic life. The poor have nothing whatsoever to give except their children, and Benedict accepts them on the same grounds, with the same ceremony, in the same spirit. Benedictine spirituality does not fear poverty; it fears the kind of self-sufficiency that frees people from the smelting effects of a communal spirituality.
CHAPTER 60
THE ADMISSION OF PRIESTS TO THE MONASTERY
_April 14 – Aug. 14 – Dec. 14_
_If any ordained priest asks to be received into a male monastery, do not agree too quickly. However, if he is fully persistent in his request, he must recognize that he will have to observe the full discipline of the rule without any mitigation, knowing that it is written: "Friend, what have you come for?" (Matt. 26:50). He should, however, be allowed to stand next to the abbot, to give blessings, and to celebrate the Eucharist, provided that the abbot bids him. Otherwise, he must recognize that he is subject to the discipline of the rule, and not make any exceptions for himself, but rather give everyone an example of humility. Whenever there is question of an appointment orof any other business in the monastery, he takes the place that corresponds to the date of his entry into the community, and not that granted him out of respect for his priesthood._
_Any clerics who similarly wish to join the community should be ranked somewhere in the middle, but only if they, too, promise to keep the rule and observe stability._
Benedictine life was monastic and lay, not diocesan and clerical. Its role was not to serve parishes or to develop dioceses but to create a way of life immersed in the Scriptures, devoted to the common life, and dedicated to the development of human community. It was simple, regular, and total, a way of living, not a way of serving; it was an attitude toward life, not a church ministry. Benedict, in other words, is not trying to create a clerical system. He is trying to create a human family. He is not out trying to collect priests, though he does recognize that a priest may well have a monastic vocation.
More interesting, then, than the fact that he does not see priesthood as essential to the achievement of his vision of life is the fact that he actually seems to discourage the idea. If they come and ask to be received, "do not agree too quickly," he cautions, and actually puts some restrictions on their membership: no elevated rank, no special attention, no official place. Why? And what can that possibly say to the rest of us now?
Benedict knew what most of us learn sooner or later: it is hard to let go of the past, and yet, until we do, there is no hope whatsoever that we can ever gain from the future. Priests, Benedict knew, came to the monastery having already been formed in another system. They were accustomed to living a highly independent and highly catered life. They had been a world unto themselves and leaders of others. In the monastery, they would have to be formed in a whole new way of life and spirituality. They would have to defer to the presence and needs of others. They, who had given so many orders, would have to take some. They would have to begin again. It could be done but it would not be easy. The _Tao Te Ching_ reads,
The Master leads
by emptying people's minds
and filling their cores,
by weakening their ambition
and toughening their resolve.
He helps people lose everything
they know, everything they desire,
and creates confusion
in those who think that they know.
The insights are important ones for all of us. Everyone has to put down some part of their past sometime. Everyone makes a major life change at some time or other. Everyone has to be open to being formed again. The only thing that can possibly deter the new formation is if we ourselves refuse to let go of what was. If we cling to the past, the future is closed to us.
CHAPTER 61
THE RECEPTION OF VISITING MONASTICS
_April 15 – Aug. 15 – Dec. 15_
_Visiting monastics from far away will perhaps present themselves and wish to stay as guests in the monastery. Provided that they are content with the life as they find it, and do not make excessive demands that upset the monastery, but are simply content with what they find, they should be received for as long a time as they wish. They may indeed with all humility and love make some reasonable criticisms or observations, which the prioress or abbot should prudently consider; it is possible that God guided them to the monastery for this very purpose._
Dietrich Bonhoeffer wrote once, "There is a meaning in every journey that is unknown to the traveler." The Benedictine Rule presumes the fundamental truth of that. In an era when monastics from small monasteries made regular pilgrimages to the shrines of Europe, Benedictine life not only welcomed them, fed them, kept them, and accepted them as one of its own, Benedictine life opened itself to learn from them. And we can learn from that kind of radical acceptance, too. Wisdom is welcome from any direction. Our task is to open ourselves to it, to see criticism as an occasion for growth, to see the value of continued evaluation, and never to close ourselves off from challenge, even when it comes from places we don't expect and people we don't know.
_If after a while they wish to remain and bind themselves to stability, they should not be refused this wish, especially as there was time enough, while they were a guest, to judge their character. But if during their stay they have been found excessive in their demands or full of faults, they should certainly not be admitted as a member of the community. Instead, they should be politely told to depart, lest their ways contaminate others._
Benedictine spirituality never requires perfection. It does, however, demand effort and openness. Complaining and complacency are the two evils that community life most abhors and can least afford. Any community, any group is poisoned by people who criticize constantly and exert themselves little. Benedict warns against them both here. "Don't keep them," he insists. Better to do with fewer and do the life well than to swell the numbers of a group with what will eventually corrode it. It is a hard lesson in a culture that measures its success in numbers.
_April 16 – Aug. 16 – Dec. 16_
_If, however, they have shown that they are not the kind of persons who deserve to be dismissed, let them, on their request, be received as a member of the community. They should even be urged to stay, so that others may learn from their example, because wherever we may be, we are in the service of the same God. Further, the prioress or abbot may set such a person in a somewhat higher place in the community, if they see that they deserve it. The prioress or abbot has the power to set any one of them above the place that corresponds to the date of their entry, if they see that their life warrants it._
_The prioress and abbot must, however, take care never to receive into the community anyone from another known monastery, unless the prioress or abbot of that community consents and sends a letter or recommendation, since it is written: "Never do to another what you do not want done to yourself" (Tob. 4:16)._
Elie Wiesel writes, "What God gave Adam was not forgiveness from sin; what God gave Adam was the chance to begin again." Life is made up of a series of opportunities to begin again. Benedictine spirituality builds that possibility and that obligation right into the Rule. Even monastics may move from monastery to monastery in their search for God. No one, in other words, has a call simply to a particular place, as good as it may be. The call of God is to the will of God. Consequently, though every institution mediates the call of God for us, every vocation transcends any particular institution. The question is always is this group, this place, calling out the best in me? Is this where I fit? Is this the place where I can most become what God created me to be? Is this the path on which I see the footsteps of God most clearly in front of me?
It is not a matter of one place being better than another. It is a matter of finding our way through life with an eye for turns in the road. It is a matter of always taking the right turn when settling for less would be so much easier. It is a matter of seeing change as a creative possibility in life.
CHAPTER 62
THE PRIESTS OF THE MONASTERY
_April 17 – Aug. 17 – Dec. 17_
_Any abbot of a male monastery who asks to have a priest or deacon ordained should choose from his monks one worthy to exercise the priesthood. The monk so ordained must be on guard against conceit or pride, must not presume to do anything except what the abbot commands him, and must recognize that now he will have to subject himself all the more to the discipline of the rule. Just because he is a priest, he may not therefore forget the obedience and discipline of the rule, but must make more and more progress toward God._
_He will always take the place that corresponds to the date of his entry into the monastery, except in his duties at the altar, or unless the whole community chooses and the abbot wishes to give him a higher place for the goodness of his life. Yet, he must know how to keep the rule established for deans and priors; should he presume to act otherwise, he must be regarded as a rebel, not as a priest. If after many warnings he does not improve, let the bishop too be brought in as a witness. Should he not amend even then, and his faults become notorious, he is to be dismissed from the monastery, but only if he is so arrogant that he will not submit to or obey the rule._
In chapter 60 the Rule makes it plain that monasticism, not clericalism, is the nature of the monastic life, that it demands an entirely different kind of formation and that simply coming to the monastery is not enough to claim conversion, even for priests. It is necessary as well to become a community person whose sanctification hinges on being open to being shaped by the Word of God in the human community around us. The question in chapter 60 is, "Can the cleric take monasticism?" and the answer is "Probably." The question in chapter 62 is, "Can the community take clericalism?" and the answer is "No."
In this chapter, Benedict reminds those priests who have been ordained from the ranks of the community itself that they, too, are under the discipline of the Rule and the abbot. Clearly, they are not, by virtue of their ordination, excused of their essential character as simple monk. In this chapter Benedict reminds us all to hold fast to our humanity, to make it our priority and never to let what we have become obscure what we are. It is so easy to take on a role in life with its trappings and privileges—doctor, judge, nun, monk, mother, teacher—and to lose, therefore, our own chance to be fully alive.
CHAPTER 63
COMMUNITY RANK
_April 18 – Aug. 18 – Dec. 18_
_Monastics keep their rank in the monastery according to the date of their entry, the virtue of their lives, and the decision of the prioress or abbot. The prioress or abbot is not to disturb the flock entrusted to them nor make any unjust arrangements, as though they had the power to do whatever they wished. They must constantly reflect that they will have to give God an account of all their decisions and actions. Therefore, when the members come for the kiss of peace and for Communion, when they lead psalms or stand in choir, they do so in the order already existing among them or decided by the abbot or prioress. Absolutely nowhere shall age automatically determine rank. Remember that Samuel and Daniel were still boys when they judged their elders (1 Samuel 3; Dan. 13:44–62). Therefore, apart from those mentioned above whom the abbot or prioress have for some overriding consideration promoted, or for a specific reason demoted, all the rest should keep to the order of their entry. For example, someone who came to the monastery at the second hour of the day must recognize that they are junior to someone who came at the first hour, regardless of age or distinction. Theyoung, however, are to be disciplined in everything by everyone._
A Benedictine community is obviously a motley place. It has locals and foreigners, old and young, cleric and lay, nobles and poor, educated and illiterate all going the same way, all intent on a life of the spirit, and all from vastly different backgrounds. All of them were conditioned to very defined expectations of privilege or oppression. Benedictine spirituality detoxifies the entire environment by putting the spotlight on the time of a person's entrance to the monastery, on the time at which they publicly began their total seeking of God, rather than on their previous status or position.
The purpose and effect of rank, then, was not the suppression of the person. It was designed to free people from their past castes or demands. The purpose of rank was to achieve equality, humility, and a new definition of self in groups rife with social hierarchies, systemic differences, and groundless exaltations. The date of entrance was the date before and after which all other events in life were marked and noted. The image of a world unskewed by material values and social definitions is the vision thrust before us in Benedictine spirituality. In a world where sex and race and money mark our spaces on the social ladder it is a picture of human liberation gone outrageously giddy with the freeing power of God as the sign of its sanctity.
_April 19 – Aug. 19 – Dec. 19_
_The younger monastics, then, must respect their elders, and the elders must love their juniors. When they address one another, no one should be allowed to do so simply by name: rather, the elders call the younger "sister" or "brother" and the younger members call their elders nonna or nonnus, which are translated as "venerable one." But the abbot and prioress, because we believe that they hold the place of Christ, are to be called "abbot" or "prioress" not for any claim of their own, but out of honor and love for Christ. They for their part, must reflect on this and in their behavior show themselves worthy of such honor._
_Wherever members meet, the junior asks the elder for a blessing. When older members come by, the younger ones rise and offer them a seat and do not presume to sit down unless the older bids them. In this way, they do what the words of Scripture say: "They should each try to be the first to show respect for the other" (Rom. 12:10)._
_In the oratory and at table, the young are kept in rank and under discipline. Outside or anywhere else, they should be supervised and controlled until they are old enough to be responsible._
This paragraph is clearly about the place of respect, experience, and wisdom in life. Obviously, the chapter on rank is not meant to grind the community down to its least common denominator. It is not meant to diminish in us the natural respect that differences should bring. Quite the opposite, in fact. This chapter is meant to freshen our eyes so that we can see all the gifts of the human community clearly: the gifts of old peasant farmers and the gifts of young artists, the gifts of young thinkers and the gifts of old keepers of the monastery door. Age, the Rule teaches, does not give us the right to dismiss the values of the young as if they were useless. Social class does not give us the right to overlook the insights of the poor. Education does not give us the right to snub the needs of the simple. We are to call one another by titles of love and respect. We are to care for the needs of the elderly, no matter our own needs or rank or station. We are to teach what we know so that the next generation grows in good air.
Once upon a time, the Zen masters teach, wealthy donors invited Master Ikkyu to a banquet. The master arrived there dressed in beggar's robes. His host, not recognizing him in this garb, hustled him away: "We cannot have you here at the doorstep. We are expecting the famous Master Ikkyu any moment." The master went home, changed into his ceremonial robe of purple brocade, and again presented himself at his host's doorstep where he was received with great respect and ushered into the banquet room. There, he took off his stiff robe, sat it upright at the dinner table and said, "I presume that it is my robe you have invited since when I first arrived without it a little while ago, you showed me away." In Benedictine spirituality reverence for the other based on the spark of the divine that is in us all is a gift to be given to a century alive with distinctions it will not admit and an insight into the sacred, scarred and bleeding, that it does not see.
CHAPTER 64
THE ELECTION OF A PRIORESS OR ABBOT
_April 20 – Aug. 20 – Dec. 20_
_In choosing an abbot or prioress, the guiding principle should always be that the one placed in office be the one selected either by the whole community acting unanimously out of reverence for God, or by some part of the community, no matter how small, which possesses sounder judgment. Goodness of life and wisdom in teaching must be the criteria for choosing the one to be made abbot or prioress even if they are the last in community rank._
The way an abbot or prioress is chosen is, like most other things in the Rule, left up to the changing needs of the group. Why an abbot or prioress is chosen is not. As far as the Rule is concerned, only "those who show goodness of life and wisdom in teaching" are fit for the position. Fundraisers and business people, efficiency experts and pious ascetics, administrators and philosophers are not ruled out; they are simply not defined in as categories that demand consideration. The implication is that if we choose those good of life and wise of heart then everything else will follow. We, of course, are always tempted to look for shortcuts to success: we look for the people who can trim our organizations or shape up our projects or stabilize our ministries. Benedictine spirituality cautions us always to follow only the good and the wise, only those who call us to our best selves, our fullest selves, knowing that if we live according to the Scriptures and choose according to the deepest and highest and greatest of human ideals, then life cannot fail for us, whatever its struggles, whatever its cost. "If I do not acquire ideals in my youth," Maimonides wrote, "when will I? Not in old age."
Benedictine spirituality tells us to choose for ideals at every turn, even at those times when management seems more important than vision.
_May God forbid that a whole community should conspire to elect a prioress or abbot who goes along with its own evil ways. But if it does, and if the bishop of the diocese or any Benedictine leaders or other Christians in the area come to know of these evil ways to any extent, they must block the success of this wicked conspiracy, and set a worthy person in charge of God's house. They may be sure that they will receive a generous reward for this, if they do it with pure motives and zeal for God's honor. Conversely, they may be equally sure that to neglect to do so is sinful._
There is no such thing as a private life in a globalized world. For a monastery, there never was. The monastery is that model of a place where the doors are always open, the environment is always gentle, the rhythm is always ordered, and God is always the center of life. A monastery is to be a light to remind all of us how beautiful the world would be if we shaped our own lives out of the same values. A Benedictine monastery is not of the church in the sense that a diocesan seminary or diocesan college is. It is not built by the church or operated by local diocesan officials. But it is definitely in the church and for the church. What happens in a Benedictine monastery should touch the spiritual life of an entire region. For that reason, whatever might erode monastic life—a breakdown of lifestyle, a contrived election, a loss of authenticity—is definitely everybody else's business. And the Rule takes care to guarantee and to maintain that sense of public acknowledgment and accountability. "The voice of the people is as the voice of God," a Jewish midrash writes. In this paragraph, Benedict requires the people to be the voice of God so that the house of God can be saved. If the monastery calls the public to commitment, there is no doubt that Benedict intends the public to call monasteries to authenticity as well so that Benedictine spirituality can continue to permeate the church. We are all guides for one another.
_April 21 – Aug. 21 – Dec. 21_
_Once in office, the abbot and prioress must keep constantly in mind the nature of the burden they have received, and remember to whom they will have "to give an account of their stewardship" (Luke 16:2). Let them recognize that the goal must be profit for the community members, not preeminence for themselves. They ought, therefore, to be learned in divine law, so that they have a treasury of knowledge from which they can "bring out what is new and what is old" (Matt. 13:52). The abbot and prioress must be chaste, temperate, and merciful, always letting "mercy triumph over judgment" (James 2:13) so that they too may win mercy. They must hate faults but love the members. When they must punish them, they should use prudence and avoid extremes; otherwise, by rubbing too hard to remove the rust, they may break the vessel. They are to distrust their own frailty and remember "not to crush the bruised reed" (Isa. 42:3). By this we do not mean that they should allow faults to flourish, but rather, as we have already said, they should prune them away with prudence and love as they see best for each individual. Let them strive to be loved rather than feared._
At the end of an entire series of injunctions and prescriptions, Benedict suddenly reintroduces a description of the kind of abbot or prioress whom he believes should guide a Benedictine community. He is, in other words, giving us a theology of authority or parenting or leadership. The Talmud reads, "Happy is the time where the great listen to the small, for in such a generation the small will listen to the great." In the Rule of Benedict the prioress and abbot are told to display the good like a blazing fire but always to "let mercy triumph over judgment" and to "strive to be loved rather than feared." Authority in Benedictine spirituality is not an end in itself nor is it an excuse to oppress the people for whom all law is made. Law is simply a candle on the path of life to lead us to the good we seek. Any authorities that make the law the end rather than the path are themselves worshiping at a lesser shrine.
_Excitable, anxious, extreme, obstinate, jealous, or overly suspicious the abbot or prioress must not be. Such a person is never at rest. Instead, they must show forethought and consideration in their orders and whether the task they assign concerns God or the world, they should be discerning and moderate, bearing in mind the discretion of holy Jacob, who said: "If I drive my flocks too hard, they will all die ina single day" (Gen. 33:13). Therefore, drawing on this and other examples of discretion, they must so arrange everything that the strong have something to yearn for and the weak nothing to run from._
In the midrash _Genesis Rabbah_ the rabbi says, "A farmer puts a yoke on his strong ox, not on his weak one." The function of Benedictine leadership is not to make life difficult; it is to make life possible for both the strong and the weak. If a leader gives way to moodiness or institutional paranoia, if a leader is not emotionally balanced and spiritually grounded, a whole climate is poisoned. This chapter on the abbot or prioress is an important signal for parents and teachers and superiors everywhere: what we cannot model, we cannot expect, not of children, not of the professionals who work for us, not even of the people who love us enough to marry us. The people around us can take our emotional battering only so long. Then they leave or rebel or batter back. Benedictine leadership models a guidance that is firm but loving; clear but understanding; just but merciful; itself authentically committed to its own principles for, indeed, the rabbis also teach, "A little sin is big when a big person commits it."
_They must, above all, keep this rule in every detail, so that when they have ministered well they will hear from God what that good servant heard who gave the other members of the household grain at the proper time: "I tell you solemnly, God will put this one in charge of greater things" (Matt. 24:47)._
In ancient civilizations, the law was the lawgiver's law. Subjects had no rights, only responsibilities. The lawgiver could change the law on a whim or a fancy. In the Roman empire, the paterfamilias, the Roman father, could do no wrong in his own home. No court of law would try him; no one would convict him. He himself according to the principles of Roman jurisprudence was judge and jury, king and lawgiver. In a climate and culture such as this, the chapter on the abbot or prioress, and this paragraph in particular, are extremely revolutionary. This section issues a clear warning: authority has limits; authority is not a law unto itself; authority is responsible to the persons under it for their welfare and their growth; authority itself is under the law. It is a theology such as this that makes people free and keeps people free because the knee we bow to government must really be bowed only to God.
CHAPTER 65
THE PRIOR AND SUBPRIORESS OF THE MONASTERY
_April 22 – Aug. 22 – Dec. 22_
_Too often in the past, the appointment of a subprioress or prior has been the source of serious contention in monasteries. Some, puffed up by the evil spirit of pride and thinking of themselves as a second prioress or abbot, usurp tyrannical power and foster contention and discord in their communities. This occurs especially in monasteries where the same bishop and the same prioress or abbot appoint both the abbot and prioress and the prior or subprioress. It is easy to see what an absurd arrangement this is, because from the very first moment of appointment they are given grounds for pride, as their thoughts suggest to them that they are exempt from the authority of the prioress or abbot. "After all, you were made subprioress or prior by the same members who made the prioress or abbot."_
_This is an open invitation to envy, quarrels, slander, rivalry, factions, and disorders of every kind, with the result that, while the prioress and subprioress or abbot and prior pursue conflicting policies, their own souls are inevitably endangered by this discord; and at the same time the monastics under them take sides and so go to their ruin. The responsibility forthis evil and dangerous situation rests on the heads of those who initiated such a state of confusion._
In any group—a political system, an athletic team, a social organization, even a monastery—authority is one thing, leadership is often another. Authority comes from being given or elected to a position. Leadership comes from vision and charism in concert. It is often the case that the two realities—authority and leadership—do not reside in the same person. Then the stage is set for tension.
If the legally deputed authority is insecure or bullying, uncertain or authoritarian, weak or controlling, the group is bound either to resist or to defect. Authority figures without the vision to identify their own weaknesses, who then appoint people to provide for those needs in the group, risk the loss of the only authority they have—which is clearly only a legal one.
On the other hand, charismatic figures in a group, people who deal well with people and have a clear vision of the future, who use those gifts to undermine the legal authority of the group, run the risk of dividing it and, eventually, of destroying it completely.
It is up to leadership figures to cooperate with authority, to uphold the unity of the group, to remember that there can be only one authority in a community at a time and no second-in-command, no department chair, not even any idea agent, is ever it.
Then the community, united in the tenuous search for the will of God together, can come to see that there are seldom instances in life when there is only one way to do anything. Then we learn that everything we do and every way we set out to do it together has something to teach us all.
_April 23 – Aug. 23 – Dec. 23_
_For the preservation of peace and love we have, therefore, judged it best for the abbot or prioress to make all decisions in the conduct of the monastery. If possible, as we have already established, the whole operation of the monastery should be managed through deans under the directions of the abbot or prioress. Then, so long as it is entrusted to more than one, no individual will yield to pride. But if local conditions call for it, or the community makes a reasonable and humble request, and the prioress or abbot judges it best, then let them, with the advice of members who reverence God, choose the one they want and themselves make that one the subprioress or prior. The subprioress and prior for their part areto carry out respectfully what the prioress or abbot assigns, and do nothing contrary to their wishes or arrangements, because the more they are set above the rest, the more they should be concerned to keep what the rule commands._
The problems dealt with in this chapter are the problems of loyalty, honesty, humility, and role and their effect on a group. The prior or subprioress in a Benedictine monastery is equivalent to the first assistant of any organization. They act as vicars, representatives, of the abbot or prioress but they do not have any specific role description or authority of their own. Most local constitutions of Benedictine communities to this day, in fact, say simply that the subprioress or prior is appointed by the prioress or abbot to "do whatever the abbot bids them to do." The point is that every community has one, single, ultimate authority, the abbot or prioress, and that any other arrangement or assumption is not only incorrect, it is dangerous to the unity and formation of the community.
Underlying the theological and organizational considerations, however, is the dark warning that the temptation to use a position, any position—vice principal, vice president, assistant, department director—to wrest authority away from the center or to promote our own careers by undermining the legitimate leader in order to make ourselves look good, is a sin against community. It uses a group for personal gain instead of for the good of the group. It is the story of a Rasputin or a Lucretia Borgia. It is a grasp at power for its own sake. It corrodes what we say we support. It eats like acid at anything in us that we say is real. It is cheap popularity and expensive advancement because, eventually, it will destroy what we say we value, the very community for which we are responsible.
_If these subprioresses or priors are found to have serious faults, or are led astray by conceit and grow proud, or show open contempt for the holy rule, they are to be warned verbally as many as four times. If they do not amend, they are to be punished as required by the discipline of the rule. Then, if they still do not reform they are to be deposed from the rank of subprioress or prior and replaced by someone worthy. If after all that, they are not peaceful and obedient members of the community, they should even be expelled from the monastery. Yet the abbot or prioress should reflect that they must give God an account of all their judgments, lest the flames of jealousy or rivalry sear their soul._
The _Tao Te Ching_ teaches, "Shape clay into a vessel; it is the space within that makes it useful." Every group has a distinct structure and history but without a single driving spirit, it may lack the heart to make a common impact. In Benedictine spirituality the abbot and prioress are the center of the community. They are the one voice, the one light, the one heart that the entire community can trust to act always in its true and total interest. In every group, in fact, it is that inspiriting space within that gives it energy. Destroy the axis, stop the heart, collapse the core of a world, and the world shrivels or shatters or disintegrates in space. That's what rivalry between the leaders of a group does to a community. That's what divergence between husband and wife does to the family. That's what tension between idols does to a world. Benedictine spirituality sees the community as something to mold us, not something to be used for the interests and vanity and power struggles of a few. It is a life dedicated to the spirit, not enmeshed in the agendas of the political. Where the authority of the abbot or prioress is constantly contested, routinely ignored, mockingly ridiculed, or sharply questioned, then the eye of the soul is taken off the Center of the life and shifted instead to the multiple minor agendas of its members. At that moment, the mystical dimension of the community turns into just one more arm-wrestling match among contenders. At that point, the Rule says, get rid of the people who lower the purpose of the group to the level of the mundane, making light of the great enterprise of life and diminishing its energy.
It is good advice in any human endeavor whose higher purpose is being fed to the appetites of the immature and the selfish to rid itself of those who have given over the lodestar of the group to a lesser direction.
CHAPTER 66
THE PORTER OF THE MONASTERY
_April 24 – Aug. 24 – Dec. 24_
_At the door of the monastery, place a sensible person who knows how to take a message and delivera reply, and whose wisdom keeps them from roaming about. This porter will need a room near the entrance so that visitors will always find someone there to answer them. As soon as anyone knocks or a poor person calls out, the porter will reply, "Thanks be to God" or "Your blessing, please," then, with all the gentleness that comes from reverence of God, provide a prompt answer with the warmth of love. Let the porter be given one of the younger members if help is needed._
Of all the questions to be asked about the over fifteen-hundred-year-old Rule of Benedict, and there are many in the twenty-first century, one of the most pointed must surely be why one of the great spiritual documents of the Western world would have in it a chapter on how to answer the door. And one of the answers might be that answering the door is one of the archactivities of Benedictine life. The way we answer doors is the way we deal with the world. Benedict wants the porter to be available, "not roaming around," so that the caller is not left waiting; responsible and "able to take a message," so that the community is properly informed; full of welcome; prompt in responding to people "with the warmth of love"; and actually grateful for the presence of the guest. When the person knocks—whenever the person knocks—the porter is to say, "Thanks be to God" or "Your blessing, please," to indicate the gift the guest is to the community. The porter is to be warm and welcome at all times, not just when it feels convenient. In the Rule of Benedict, there is no such thing as coming out of time to the monastery. Come in the middle of lunch; come in the middle of prayer; come and bother us with your blessings at any time. There is always someone waiting for you.
The chapter on the porter of the monastery is the chapter on how to receive the Christ in the other always. It is Benedict's theology of surprise.
_The monastery should, if possible, be so constructed that within it all necessities, such as water, mill, and garden are contained, and the various crafts are practiced. Then there will be no need for the members to roam outside, because this is not at all good for their souls._
_We wish this rule to be read often in the community, so that none of the members can offer the excuse of ignorance._
If there is any chapter in the Rule that demonstrates Benedictine openness to life and, at the same time, models a manner of living in the midst of society without being consumed by it, this is surely the one. Guests are welcomed enthusiastically in Benedictine spirituality but, at the same time, life is not to be frittered away on work, on social life, on the public bustle of the day. The community is to stay as self-contained as possible so that centered in the monastery they stay centered in their hearts. More, this balance between public and private, between openness and centeredness, between consciousness of the outside world and concentration on interior growth is to be remembered and rehearsed over and over again: "We wish this rule to be read often," the Rule says plaintively so that the monastic never forgets that the role of committed Christians is always to grow richer themselves so that they can give richly to others. Abba Cassian, a Desert Monastic, told the following story:
Once upon a time, we two monks visited an elder. Because he offered us hospitality we asked him, "Why do you not keep the rule of fasting when you receive visiting brothers?" And the old monastic answered, "Fasting is always at hand but you I cannot have with me always. Furthermore, fasting is certainly a useful and necessary thing, but it depends on our choice, while the law of God lays it upon us to do the works of charity. Thus, receiving Christ in you, I ought to serve you with all diligence, but when I have taken leave of you, I can resume the rule of fasting again."
The person with a monastic heart knows that the Christ and salvation are not found in religious gyrations alone. They are in the other, our response to whom is infinitely more important than our religious exercises.
CHAPTER 67
MEMBERS SENT ON A JOURNEY
_April 25 – Aug. 25 – Dec. 25_
_Members sent on a journey will ask the prioress or abbot and the community to pray for them. All absent members should always be remembered at the closing prayer of the Opus Dei. When they come back from a journey, they should, on the very day of their return, lie face down on the floor of the oratory at the conclusion of each of the customary hours of the Opus Dei. They ask the prayers of all for their faults, in case they may have been caught off guardon the way by seeing some evil thing or hearing some idle talk._
The Desert Monastic Samartus had written in a culture that called material things evil and only spiritual things good: "If we do not flee from everything, we make sin inevitable." This fear of things outside the monastery was clearly still alive in the time of Benedict and well beyond. Monastics who traveled outside, then—and they did, as we do, for reasons of business and personal need—were reminded in this paragraph to call themselves consciously into the presence of God and the purpose of their lives before leaving their monasteries. Two things in particular make the paragraph valuable today. First, however they saw the risks of the world in which they lived, they continued to confront them. They did not become less human in their search for the spiritual life. Second, however they counted their own commitment, they did not underestimate the lure of lesser things in life, even on them. They begged the prayers of the community while they were away, a practice continued to this day, and they kept as close as possible to the prayer schedule of the monastery while they were gone. Then, when the trip was over, they returned to their monasteries alert to the effects of the baubles and bangles of loose living. And they redoubled their efforts at monastic life. They started over again, prostrating themselves on the floor of the oratory as they had at the time of their profession, praying to be reconcentrated on the real meaning of life.
The value of the chapter is clear even today: no one lives in a tax-free world. Life costs. The values and kitsch and superficiality of it take their toll on all of us. No one walks through life unscathed. It calls to us for our hearts and our minds and our very souls. It calls to us to take life consciously, to put each trip, each turn of the motor, each trek to work in God's hands. Then, whatever happens there, we must remember to start over and start over and start over until, someday, we control life more than it controls us.
_No monastics should presume to relate to anyone else what they saw or heard outside the monastery, because that causes the greatest harm. If any do so presume, they shall be subjected to the punishment of the rule. So too shall anyone who presumes to leave the enclosure of the monastery, or go anywhere, or doanything at all, however small, without the order of the abbot or the prioress._
A Zen story tells of two monks walking down a muddy, rain-logged road on the way back to their monastery after a morning of begging. They saw a beautiful young girl standing beside a large deep puddle unable to get across without ruining her clothes. The first monk, seeing the situation, offered to carry the girl to the other side, though monks had nothing whatsoever to do with women. The second monk was astonished by the act but said nothing about it for hours. Finally, at the end of the day, he said to his companion, "I want to talk to you about that girl." And the first monk said, "Dear brother, are you still carrying that girl? I put her down hours ago."
The things we ruminate on, the things we insist on carrying in our minds and hearts, the things we refuse to put down, the Rule warns us, are really the things that poison us and erode our souls. We dull our senses with television and wonder why we cannot see the beauty that is around us. We hold on to things outside of us instead of concentrating on what is within that keeps us noisy and agitated. We run from experience to experience like children in a candy store and wonder how serenity has eluded us. It is walking through life with a relaxed grasp and a focused eye that gets us to where we're going. Dwelling on inessentials and, worse, filling the minds of others with them distract from the great theme of our lives. We must learn to distinguish between what is real and what is not.
CHAPTER 68
ASSIGNMENT OF IMPOSSIBLE TASKS
_April 26 – Aug. 26 – Dec. 26_
_Monastics may be assigned a burdensome task or something they cannot do. If so, they should, with complete gentleness and obedience, accept the order given them. Should they see, however, that the weight of the burden is altogether too much for their strength, then they should choose the appropriate moment andexplain patiently to the prioress or abbot the reasons why they cannot perform the task. This they ought to do without pride, obstinacy, or refusal. If after the explanation the abbot or prioress is still determined to hold to their original order, then the junior must recognize that this is best. Trusting in God's help, they must in love obey._
An old Jewish proverb teaches, "When you have no choice, don't be afraid." A modern saying argues, "There's no way out but through." The straight and simple truth is that there are some things in life that must be done, even when we don't want to do them, even when we believe we can't do them. Is the Rule cruel on this point? Not if there is any truth in experience at all. The reality is that we are often incapable of assessing our own limits, our real talents, our true strength, our necessary ordeals. If parents and teachers and employers and counselors and prioresses somewhere hadn't insisted, we would never have gone to college or stayed at the party or tried the work or met the person or begun the project that, eventually, changed our lives and made us more than we ever knew ourselves to be. Benedict understood clearly that the function of leadership is to call us beyond ourselves, to stretch us to our limits, to turn the clay into breathless beauty. But first, of course, we have to allow it to happen.
CHAPTER 69
THE PRESUMPTION OF DEFENDING ANOTHER IN A MONASTERY
_April 27 – Aug. 27 – Dec. 27_
_Every precaution must be taken that one member does not presume in any circumstance to defend another in the monastery or to be their champion, even if they are related by the closest ties of blood. In no way whatsoever shall monastics presume to do this, because it can be a most serious source and occasion of contention. Anyone who breaks this rule is to be sharply restrained._
"Stay away from your enemies but guard yourself against friends," Ben Sirach wrote in Ecclesiasticus. The Rule knows that false friendship is bad for the person and bad for the community as well. In a life dedicated to spiritual growth and direction, there is no room for multiple masters. Friends who protect us from our need to grow are not friends at all. People who allow a personal agenda, our need to be right or their need to shield, block the achievement of a broader vision in us and betray us. Supporters who risk dividing a group into factions over personal tensions rather than allowing individuals to work their way positively through the hard points of life barter the spirit and peace of the whole community. We are taught in the Rule not to take sides in issues of personal interpretation and spiritual challenge. We are to hold one another up during hard times, chapter 27 indicates, but we are not to turn personal difficulty into public warfare. The groups that would be better off if individuals had refused to turn differences of opinion into moral irreconcilables are legion. The Desert Monastics say that one of the disciples asked Abba Sisoes one day, "If I am sitting in the desert and a barbarian comes to kill me and if I am stronger than he, shall I kill him?" The old man said to him, 'No, leave him to God.' In fact, whatever the trial is that comes to a person, let them say, 'This has happened to me because of my sins,' and if something good comes say, 'This has happened to me because of the providence of God.'"
Life is not perfect; some of life just is. A great deal of mental, psychological, and spiritual health comes from learning to endure the average heat of the average day and to wear both its banes and its blessings with a tempered heart. No warfare. No armies mobilized on the plain. No identification of enemies. Just life.
CHAPTER 70
THE PRESUMPTION OF STRIKING ANOTHER MONASTIC AT WILL
_April 28 – Aug. 28 – Dec. 28_
_In the monastery every occasion for presumption is to be avoided, and so we decree that no one has the authority to excommunicate or strike any member of the community unless given this power by the prioress or abbot. "Those who sin should be reprimanded in the presence of all, that the rest may fear" (1 Tim. 5:20). The young up to the age of fifteen should, however, be carefully controlled and supervised by everyone, provided that this too is done with moderation and common sense._
_If any member, without the command of the abbot or prioress, assumes any power over those older or, even in regard to the young, flares up and treats them unreasonably, let that one be subjected to the discipline of the rule. After all, it is written: "Never do to another what you do not want done to yourself" (Tob. 4:16)._
This chapter of the Rule is not about fistfighting. It is about the arrogant usurpation of authority and the legitimization of violence. Even in a culture that routinely disciplined its young or unlettered with physical whippings, Benedict simply does not allow a culture of violence. Benedictine spirituality depends on personal commitment and community support, not on intimidation and brutality. Benedict makes it clear that the desire for good is no excuse for the exercise of evil in its behalf. This is an important chapter, then, for people whose high ideals lead them to the basest of means in the name of the achievement of good. To become what we hate—as mean as the killers, as obsessed as the haters—is neither the goal nor the greatness of the spiritual life.
CHAPTER 71
MUTUAL OBEDIENCE
_April 29 – Aug. 29 – Dec. 29_
_Obedience is a blessing to be shown by all, not only to the prioress and abbot but also to one another, since we know that it is by this way of obedience that we go to God. Therefore, although orders of the prioress and abbot or of the subprioress or prior appointed by them take precedence, and no unofficial order may supersede them, in every other instance younger members should obey their elders with all love and concern. Anyone found objecting to this should be reproved._
Into a democratic country and a highly individualistic culture, into a society where personalism approaches the pathological and independence is raised to high art, the Rule brings a chapter on listening and wisdom. The Rule says that we are not our own teachers, not our own guides, not our own standard setters, not a law unto ourselves. In addition to the "officials" in our lives—the employers, the supervisors, the lawgivers, and the police—we have to learn to learn from those around us who have gone the path before us and know the way. It is a chapter dedicated to making us see the elderly anew and our colleagues with awe and our companions with new respect. In a society that depends on reputation to such a degree that people build themselves up by tearing other people down, the chapter on mutual obedience turns the world awry. Monastic spirituality says that we are to honor one another. We are to listen to one another. We are to reach across boundaries and differences in this fragmented world and see in our differences distinctions of great merit that can mend a competitive, uncaring, and foolish world.
The _Tao Te Ching_ teaches,
If you want to become whole,
let yourself be partial.
If you want to become straight,
let yourself be crooked.
If you want to become full,
let yourself be empty.
_If a member is reproved in any way by the abbot or prioress or by one of the elders, even for some very small matter, or gets the impression that one of the elders is angry or disturbed with them, however slightly, that member must, then and there withoutdelay, fall down on the ground at the other's feet to make satisfaction, and lie there until the disturbance is calmed by a blessing. Anyone who refuses to do this should be subjected to corporal punishment or, if stubborn, should be expelled from the monastery._
What monastic spirituality wants among us is respect and love, not excuses, not justification, not protests of innocence or cries of misunderstandings. The Rule wants respect for the elder and love for the learner. The Rule wants a human response to the mystery of misunderstanding—not standoffs, not pouting, not rejection, not eternal alienation. The Rule wants relationships that have been ruptured to be repaired, not by long, legal defenses but by clear and quick gestures of human sorrow and forgiveness. The question in the Rule is not who is right and who is wrong. The question in the Rule is who is offended and who is sorry, who is to apologize and who is to forgive. Quickly. Immediately. Now.
The rabbi of Sassov, the Hasidic masters tell us, once gave away the last money he had in his pocket to a man of ill repute who quickly squandered it all. The rabbi's disciples threw it up to him. He answered them, "Shall I be more finicky than God, who gave it to me?" What monastic spirituality teaches in this paragraph of the Rule is that we must all relate to one another knowing our own sinfulness and depending on the love we learn from one another.
CHAPTER 72
THE GOOD ZEAL OF MONASTICS
_April 30 – Aug. 30 – Dec. 30_
_Just as there is a wicked zeal of bitterness which separates from God and leads to hell, so there is a good zeal which separates from evil and leads to God and everlasting life. This, then, is the good zeal which members must foster with fervent love: "They should each try to be the first to show respect to the other" (Rom. 12:10), supporting with the greatest patience one another's weaknesses of body or behavior, and earnestly competing in obedience to one another. No monastics are to pursue what they judge better for themselves, but instead, what they judge better for someone else. Among themselves they show the pure love of sisters and brothers; to God, reverent love; to their prioress or abbot, unfeigned and humble love. Let them prefer nothing whatever to Christ, and may Christ bring us all together to everlasting life._
Here is the crux of the Rule of Benedict. Benedictine spirituality is not about religiosity. Benedictine spirituality is much more demanding than that. Benedictine spirituality is about caring for the people you live with and loving the people you don't and loving God more than yourself. Benedictine spirituality depends on listening for the voice of God everywhere in life, especially in one another and here. An ancient tale from another tradition tells that a disciple asked the Holy One,
"Where shall I look for Enlightenment?"
"Here," the Holy One said.
"When will it happen?"
"It is happening right now," the Holy One said.
"Then why don't I experience it?"
"Because you do not look," the Holy One said.
"What should I look for?"
"Nothing," the Holy One said. "Just look."
"At what?"
"Anything your eyes alight upon," the Holy One said.
"Must I look in a special kind of way?"
"No," the Holy One said. "The ordinary way will do."
"But don't I always look the ordinary way?"
"No," the Holy One said. "You don't."
"Why ever not?" the disciple demanded.
"Because to look you must be here," the Holy One said. "You're mostly somewhere else."
Just as Benedict insisted in the Prologue to the Rule, he requires at its end: We must learn to listen to what God is saying in our simple, sometimes insane, and always uncertain daily lives. Bitter zeal is that kind of religious fanaticism that makes a god out of religious devotion itself. Bitter zeal walks over the poor on the way to the altar. Bitter zeal renders the useless invisible and makes devotion more sacred than community. Bitter zeal wraps us up in ourselves and makes us feel holy about it. Bitter zeal renders us blind to others, deaf to those around us, struck dumb in the face of the demands of dailiness. Good zeal, monastic zeal, commits us to the happiness of human community and immerses us in Christ and surrenders us to God, minute by minute, person by person, day after day after day. Good zeal provides the foundation for the spirituality of the long haul. It keeps us going when days are dull and holiness seems to be the stuff of more glamorous lives, of martyrdom and dramatic differences. But it is then, just then, when Benedict of Nursia reminds us from the dark of the sixth century that sanctity is the stuff of community in Christ and that any other zeal, no matter how dazzling it looks, is false. Completely false.
CHAPTER 73
THIS RULE ONLY A BEGINNING OF PERFECTION
_May 1 – Aug. 31 – Dec. 31_
_The reason we have written this rule is that, by observing it in monasteries, we can show that we have some degree of virtue and the beginnings of monastic life. But for anyone hastening on to the perfection of monastic life, there are the teachings of the early church writers, the observance of which will lead them to the very heights of perfection. What page, what passage of the inspired books of the Old and New Testaments is not the truest of guides for human life? What book of holy writers does not resoundingly summon us along the true way to reach the Creator? Then, besides the Conferences of the early church writers, their Institutes and their Lives, there is also the Rule of Basil. For observant and obedient monastics, all these are nothing less than tools for the cultivation of virtues; but as for us, they make us blush for shame at being so slothful, so unobservant, so negligent. Are you hastening toward your heavenly home? Then with Christ's help, keep this little rule that we have written for beginners. After that, you can set out for the loftier summits of the teaching and virtues we mentioned above, and under God's protection you will reach them. Amen._
This last chapter of the Rule leaves us with a reading list for future spiritual development: the Bible, the Mothers and Fathers of the Church and their commentaries on Scripture, and the classic contributions of other writers on the monastic life. But Benedict does not believe that the simple reading or study of spiritual literature is sufficient. He tells us to keep this Rule, its values, its concepts, its insights. It is not what we read, he implies; it is what we become that counts. Every major religious tradition, in fact, has called for a change of heart, a change of life rather than for simply an analysis of its literature. The Hasidim, for instance, tell the story of the disciple who said to the teacher, "Teacher, I have gone completely through the Torah. What must I do now?"
And the teacher said, "Oh, my friend, the question is not, Have you gone through the Torah? The question is, Has the Torah gone through you?"
Even at the end of his Rule, Benedict does not promise that we will be perfect for having lived it. What Benedict does promise is that we will be disposed to the will of God, attuned to the presence of God, committed to the search for God, and just beginning to understand the power of God in our lives. Why? Because Benedictine simplicity gentles us into the arms of God. Benedictine community supports us on the way to God. Benedictine balance makes a wholesome journey possible. Monastic prayer, rooted in Scripture, lights the way. It is a way of life, a spirituality that makes the humdrum holy and the daily the stuff of high happiness. It is a way of living that leads us to pursue life to its fullest. As this final chapter promises, the meaning of the human enterprise is for our taking if we will only follow this simple but profoundly life-altering way.
ACKNOWLEDGMENTS
It comes as no surprise to anyone that though books are written by one person, they are the product of many. My own writing is always an ongoing dialogue between myself and a bank of readers who argue and puzzle and struggle their way through a first draft so that, given their concerns, the second draft can be more usable to many others. The readers for this book were especially helpful. I am grateful for the time, the effort, and the insights of each of them. I took most of their suggestions but I did not take them all so what the manuscript lacks is no fault of theirs. For instance, though one reader recommended it strongly, I did not include examples from my own monastic life to demonstrate the specific concepts here because I felt that the text itself is too ancient and too venerable to be tied down to any one local practice or history of it. This book, I judged, is not about Joan Chittister and a specific community; it is about being able to read the Rule and imagine its potential for everyone everywhere, monastics and nonmonastics alike. Also, though I have included a book list of stories and proverbs that I used, I did not cite each of the tales separately since the same stories exist in multiple forms in all the great literary traditions of the world. If those were mistakes in judgment on my part I take full responsibility for them. I did, nevertheless, profit mightily from the marginalia and the questions each of the readers contributed and have tried to answer them one by one.
The readers who helped, then, to create this book include Gerald Trambley, Gene and Lisa Humenay, John and Karen Dwyer, Lawreace Antoun, S.S.J., Mary Lou Kownacki, O.S.B., Stephanie Campbell, O.S.B., Patrick Henry, Bro. Thomas Bezanson, Ann Marie Sweet, O.S.B., Kathy Stevens, and Diane Wilson.
Special acknowledgment is due a few: Marlene Bertke, O.S.B., brings precision, style, and consistency to all of my work and stayed with this one from beginning to end. Maureen Tobin, O.S.B., as personal assistant and appointments secretary, made the work logistically possible and my life quiet enough to do it. Mary Grace Hanes, O.S.B., brought the manuscript from the dark and tricky recesses of a computer to the light of day.
My editor, John Farina, who conceived the idea for this series, gave me encouragement, direction, and generous scope. Without him the world would be one attempt poorer to make the best of Western spirituality alive and accessible today.
Whatever the book is or is not able to bring to the lives of other people, it brought me hours of the most refreshing lectio of my life for all the months of its making. I am more grateful for that than for anything.
PHOTO CREDITS:
p. 1: St. Benedict delivering his Rule to St. Maurus and other monks of his order, (France, Monastery of St. Gilles, Nimes), public domain; p. 4: Pietro Perugino: Polittico di San Pietro (San Benedetto), public domain; p. 6: Croce e mura, © infoseat/PixMac; p. 8: baby hand 3, © Peter Galbraith/Fotolia; p. 10: Time, © Paul Morley/Fotolia; p. 18: Jean-François Millet, Angelus, public domain; p. 22: Climbers, © Oleg Kozlov/Fotolia; p. 24: Andrea Mantegna, Saint Lucas altarpiece, detail: Saint Benedict of Nursia, public domain; p. 28: Trabajar la tierra, © Jose Juan Castellano/Fotolia; p. 38: Old wagon wheel, © Olivier/Fotolia; p. 48: Chapel, © Mirek Hejnicki/Fotolia; p. 52: Tipi village, © Melissa Schalke/Fotolia; p. 55: Saint Scholastica, © zatletic/Fotolia; p. 59: Mahatma Gandhi on postage stamp from India, © PictureLake/Fotolia; p. 65: Sapling, © smartsun/Fotolia; p. 70: Night Prague fog, © Peter Zurek/Fotolia; p. 74: Kreuzgang, © Harald Schmid/Fotolia; p. 79: Calvaire, Bretagne, © Xavier29; p. 81: Waimea bay wave, © NorthShoreSurfPhotos/Fotolia; p. 87: Wooden bridge, © Marko Cerovac/Fotolia; p. 92: Trekking path, © Jaroslaw Grudzinski/Fotolia; p. 96: Mountain stream, © Carl Southerland/Fotolia; p. 99: Fra Angelico, Annunciation, 1433–1434, Museo Diocesano, Cortona, public domain; p. 102: Old bronze bell on a church, © Tjommy/Fotolia; p. 109: Dipinto, © LittleSteven65/Fotolia; p. 115: Christ Panocrator, Hagia Sophia, Istanbul, photo Eve Vaterlaus; p. 117: Mother Teresa on postage stamp from India, © PictureLake/Fotolia; p. 120: live-forever, © Kaj Gardemeister/Fotolia; p. 123: Fürbitte, © Alterfalter/Fotolia; p. 126: Footprints, © Paulus Rusyanto/Fotolia; p. 129: Bible, © Kwest/Fotolia; p. 131: Musik, © Berchtesgaden/Fotolia; p. 135: Indian fishermen, © Alexey Kuznetsov/Fotolia; p. 136: San Pietro, © M.M/Fotolia; p. 137: Relief from the abbey of Saint-Denis, 1250–1260. Épisode in the life of Saint Benedict Photo Marsyas, public domain (permission granted by photographer, Marsyas, on the Internet); p. 139: Stanza per la meditazione, © Ernesto Notarantonio/Fotolia; p. 151: Lüneburger Heide 080922 026, © crimson/Fotolia; p. 154: Old suitcase, © rimglow/Fotolia; p. 159: Frisches gemüse, © Dirk Houben/Fotolia; p. 161: Broom, © Liette Parent/Fotolia; p. 164: Plant in hands of people, © EdwardSV/Fotolia; p. 171: Elderly woman with vegetables, © Sandor Kacso/Fotolia; p. 176: Commitment, © marilyn barbone/Fotolia; p. 179: Hildegard of Bingen receives a divine inspiration, Miniature from the Rupertsberg Codex de Liber Scivias, public domain; p. 181: In church, © MassonOdessa/Fotolia; p. 185: Cut bread, © aragami/Fotolia; p. 188: Eiche #2, © RalfenStein/Fotolia; p. 193: Man washing tableware 2, © Valeriy Kirsanov/Fotolia; p. 196: Angel silence, © Jason Cosburn/Fotolia; p. 199: Kreuz mit See in den Alpen no. 1, © studali/Fotolia; p. 201: St. Benedict Eating with his Monks (Painting by Giovanni Sodoma) Abbey of Monteoliveto Maggiore, Siena, public domain; p. 205: Prayer old rabbi at the Western Wall in Jerusalem, © Mikhail Levit/Fotolia; p. 208: HuddleCLR, © monique delatour/Fotolia; p. 213: Équilibre en pleine nature, © minicel73/Fotolia; p. 215: Age of contemplation, © Kelly Ann/Fotolia; p. 217: Senior woman gardening, © Elenathewise/Fotolia; p. 219: Tea bowl, photo Eve Vaterlaus; p. 221: Water, photo Eve Vaterlaus; p. 223: Three Nuns worship outdoors, © Scott Hales/Fotolia; p. 226: Sonnenuhr, © Christa Eder/Fotolia; p. 228: Door and vines, © ADV/Fotolia; p. 231: Hands of the old woman, © Sergey Galushko/Fotolia; p. 237: Blessed Hildegard von Bingen, © zatletic/Fotolia; p. 239: Flea market, © reb/Fotolia; p. 242: Flowers for grandma, © Marzanna Syncerz/Fotolia; p. 245: Korbflecfhten, © fotografci/Fotolia; p. 252: Pflug mit wasserbüffel, © Digitalpress/Fotolia; p. 255: Scenes Depicting the Life of St. Hildegard by Fr. Paulus Krebs, in the Benedictine Abbey of Saint Hildegarde, permission Abbey of St. Hildegard in Ruedesheim, Germany; p. 260: Hiker on the trail, © Jens Ottoson/Fotolia; p. 262: Abbazia di montecassino, © kenzo/Fotolia; p. 264: Maske, © DarkRamses/Fotolia; p. 266: Seashells, © BVDC/Fotolia; p. 268: Freunde, © Michael Kempf/Fotolia; p. 272: Canada geese in flight, © Xiaodong Ye/Fotolia; p. 274: Potters team, © Ints Vikmanis/Fotolia; p. 275: Charrette a boeufs, © photlook/Fotolia; p. 279: The Twelve Apostles, © jvimages/Fotolia; p. 283: La portr cloutée, © Pierre Bonnel/Fotolia; p. 285: Still life with flowers, baking and lemon, © Alexey Khromushin/Fotolia; p. 288: Icon paintings in monastery interior, St. George, © Dessie/Fotolia; p. 289: Two boy monks walking on the street in Luang Prabang, Laos, © chrswbrwn/Fotolia; p. 291: Old bridge, © Hunta/Fotolia; p. 299: Africa, © africa/Fotolia; p. 303: Walking in alley, © .shock/Fotolia
Of related interest
**David Robinson**
**The Busy Family's Guide to Spirituality
Practical Lessons for Modern
Living from the Monastic Tradition
ISBN 978-0-8245-2524-8, paperback, 227 pages**
"The most innovative book I have seen in the area of family spirituality."
_—Spiritual Life Magazine_
Drawing on the timeless principles of monastic communal living, this spiritual guide for families offers effective tools to meet the countless challenges and distractions of contemporary family life.
Using the tested suggestions, exercises, and activities in this book, you'll learn about:
• Choosing a blueprint
• Family meetings
• Spiritual practices and rituals
• Rest and recreation
• The heart of prayer
• Coaching the family team
• Living simply
• Caring for others
• Fasting and feasting
• Family schedules
• Bedtime rituals
• Time together and time apart
• Hospitality
• Friends and mentors
• Transforming conflict
• Leaving home
• Benedictine spirituality
David Robinson (D.Min.) is a spiritual guide, teacher, and father of three grown sons. He is pastor at Community Presbyterian Church and a Benedictine Oblate of Mount Angel Abbey, Oregon. David and his wife live in Cannon Beach, Oregon.
From the introduction:
"Every year, just before we set out on our annual weeklong family hiking adventure, we gather together at the trailhead, hold hands in a circle, and pray for God's presence to be with us in our journey ahead. This simple act of devotion is the essence of family spirituality. Fifteen hundred years ago, Benedict gathered a family of monks around a common life of prayer and work. The insights he learned over several decades of life together with his family, he wrote down in a practical, spiritual guidebook, _The Rule of St. Benedict._ I believe Benedict has wisdom and guidance to offer the 21st century family. As your family begins to practice this way of spiritual life together, you will discover you've set out together upon a new path, a wonderful journey to a fulfilled family life. Like _The Rule of St. Benedict,_ this book is written for beginners (RB, 73), for busy parents who may be looking for support and encouragement in the adventure of raising children."
From Chapter 1:
**Create Your Home
Choosing a Proven Blueprint for Your Life**
Everyone who hears these words of mine and puts them into practice is like a wise man who built his house on the rock.
—Matthew 7:24
I vividly recall my first visit to a monastery in October 1986, and the deep sense of peace which came upon me as I sat quietly in the cloister garden. I had come home. Monasteries offer a beautiful design that serves the needs and purposes of the monastic community as well as those who come as guests. You'll almost always find an enclosed garden at the center of the monastery, surrounded by the sanctuary, dining hall, kitchen, and dormitory. Nearby, you'll find the guest house along with other rooms including workshops, an infirmary, and the library.
Like the design of a monastery, blueprints for family spirituality are designed to serve the needs and purposes of families seeking to live together in love. My wife and I shared many conversations regarding patterns of parenting when we became pregnant with our firstborn. Choosing a family blueprint is not easy, but it is essential for the crafting of a well-built family. Benedict's design for communal spiritual life has stood the test of time, and can offer busy families today wisdom and guidance for finding their way home. Benedict's family was a community of monks, living under the leadership of an abbot or "father," and guided by a common "rule of life." The word "rule," used throughout, comes from the Latin word _regula,_ meaning a measuring tool or guidebook. From the foundation of Benedict's _Rule,_ written in the sixth century, there developed thousands of long-enduring spiritual communities across the landscape of Europe, and later around the world, providing stability and wisdom across generations.
Every family lives according to some type of design whether consciously or not. But without a wise blueprint and a good foundation, a family may not endure the storms that lie ahead. Many parents I know have combined family blueprints from their upbringings. Most parents also seek guidance in the ongoing challenge of raising children. In this chapter we will survey a variety of blueprints for family spirituality, and then look at patterns for family life within those structures.
**Madeleine Delbrel
The Little Monk
Wisdom from a Little Friend of Big Faith**
"More than six billion people bear the burden of life. It helps to know that you're no exception," says the Little Monk, a fictional character drawn from all the faithful around the globe. Madeleine Delbrel, famous atheist, philosopher, convert and Carmelite nun, created the Little Monk from the depths of her faith and understanding of modern spirituality. The following is a selection from _The Little Monk_ (All images will be displayed in color).
| {
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Viewers turn to streaming services as they self-isolate amid coronavirus pandemic
Want to discuss? Please read our Commenting Policy first.
By LYNN ELBER The Associated Press
5:05 What to watch when you're stuck at home
WATCH: What to watch when you're stuck at home – Mar 18, 2020
Sports are on hold, theaters are closed and so are amusement parks, a disaster-movie scenario that has Hollywood reeling.
But Americans held captive at home by the coronavirus can turn to Netflix, Amazon, Hulu and other streaming services, outliers in an entertainment industry brought to an unprecedented standstill.
The recent launch of Disney and Apple services and the upcoming arrival of NBCUniversal's Peacock and WarnerMedia's HBO Max unleashed speculation about winners and losers in an increasingly crowded field. With self-imposed or required isolation the abrupt reality, emerging and niche streamers could draw new subscribers — gains that may even outlast the coronavirus crisis.
READ MORE: Canada-U.S. border closure amid coronavirus likely to start Friday night
The viral outbreak "has caused so much pain across industries globally," said Dan Ives, an analyst with Wedbush Securities. "Yet on the streaming side, the demand for those services is going to increase exponentially over the next three to six months" as consumers around the world remain stuck in place.
Up to a 20 percent increase is likely in the amount of time subscribers spend watching streamed fare, and millions of new customers will hop aboard worldwide, Ives predicted.
Pay TV channels could benefit as well as more people become shut-ins and reconsider cutting the cord, slowing an accelerating trend, said analyst Jeffrey Wlodarczak of Pivotal Research Group. Broadcast networks facing rating declines also could see a boost in viewership, he said.
1:24 Is it okay to still take a walk amid the COVID-19 pandemic?
Is it okay to still take a walk amid the COVID-19 pandemic? – Mar 19, 2020
Streaming companies are reacting to the moment in varied ways, but always carefully. Media companies want to be seen as good citizens who are serving up an antidote to anxiety over the virus and housebound boredom, not capitalizing on a disaster.
For most people, the coronavirus causes only mild or moderate symptoms, such as fever and cough. For some, especially older adults and people with existing health problems, it can cause more severe illness, including pneumonia. The vast majority of people recover.
Netflix, sitting comfortably in the front ranks of streamers, emailed customers with a gentle nudge to "Rewind. Replay. Rewatch," followed by suggestions of previously viewed titles such as "The Crown," "Schitt's Creek" and the 2012 movie "Frances Ha" starring Greta Gerwig.
(Disney via AP)
The Walt Disney Co. put the box-office hit "Frozen 2" on its Disney Plus streaming service three months earlier than planned, "surprising families with some fun and joy during this challenging period." The animated film became available in the past few days on the service in the U.S., Canada, Holland, Australia and New Zealand.
Hulu, controlled by majority owner Disney, is relying for now on such high-profile programs as the newly released Reese Witherspoon-Kerry Washington series "Little Fires Everywhere," based on Celeste Ng's bestselling novel.
READ MORE: Archived: Live updates on coronavirus in Canada
For Disney, streaming is the outlier in a corporate portfolio otherwise slammed by the coronavirus: The suspension of theatrical and TV productions and delays in new movie releases, including its long-anticipated "Black Widow," but also the closure of Disney resorts in the U.S., France and Asia and the sports lull's impact on its ESPN channels.
Disney and the other major streaming services didn't respond or declined requests for comment for this article.
0:28 Grandma celebrates 95th birthday while in self-isolation
Grandma celebrates 95th birthday while in self-isolation – Mar 19, 2020
As financial hardship builds for people in lockstep with the crisis response, the services have yet to cut fees for new or current subscribers. The current monthly tabs range from the modest (Apple TV Plus, with its fledgling roster of shows is US$5, Disney Plus is US$7 for a deep bench of old and new fare) to Netflix's top-tier plan (US$16, compared to US$9 for basic access).
As financial hardship builds for people in lockstep with the crisis response, the services have yet to cut fees for new or current subscribers. The current monthly tabs range from the modest (Apple TV Plus, with its fledgling roster of shows is $5, Disney Plus is $7 for a deep bench of old and new fare) to Netflix's top-tier plan ($16, compared to $9 for basic access).
AP graphic.
Niche services are taking a different tack. Acorn TV, which offers British and international programming, expanded its free trial offer for new subscribers from a week to 30 days, as have its AMC Networks siblings Sundance Now, Shudder and Urban Movie Channel (UMC).
"As the extent of the Covid-19 crisis became clear, we realized that a service like Acorn TV could make a meaningful difference in many people's lives, especially those who are facing prolonged isolation and potential economic hardship as a result of the virus," said Matthew Graham, the service's general manager.
READ MORE: Your coronavirus questions, answered: Medical experts respond to your COVID-19 concerns
Peacock's U.S. launch isn't until July, but with an upside for those with budget pressures: It will include a no-cost, ad-supported version, Peacock Free, with content including next-day access to new NBC shows and full seasons of older series. Two other tiers, including Peacock Premium for $5 monthly (free to Comcast and Cox subscribers), will offer more programming and live sports.
Peacock Free is going to be "a no-brainer for people," Wlodarczak said.
The PBS Video app, public TV's free streaming service, stepped up its offerings amid virus fallout. With Major League Baseball spring training games suspended and a delayed start to the season, the acclaimed Ken Burns' "Baseball" documentary was added to the service and online at PBS.org.
4:29 Learn guitar with Max Kerman of the Arkells
Learn guitar with Max Kerman of the Arkells – Mar 19, 2020
"As many of us hunker down in the days ahead, it's important that we find things that bring us together and show us our common humanity," Burns said in a video message. "That's why, in the absence of many of our favorite sports, I've asked PBS … to stream my film about America's pastime. Stay healthy, and let's look after each other. Play ball."
(Experts have said the U.S. internet won't get overloaded by spikes in traffic from the millions of at-home workers and students, although it's possible that households may find connections slowed because of their service level.)
READ MORE: Hitting 'paws' on life: Coronavirus self-isolating with pets has been a saving grace for many
The duration of the crisis could affect even streamers with well-stocked libraries, Wlodarczak said, given the widespread halt to filming. The fourth season of "Fargo" starring Chris Rock, which was to arrive on Hulu after its April debut on FX, has been postponed until production is back on track.
"The longer this lasts, you've got to find content to put on," the analyst said.
© 2020 The Canadian Press
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\section*{Appendix A - Example of Ethereum transaction in \ONT{}}
\begin{sloppypar}
Fig. \ref{fig:blockOntoexmple} shows an Ethereum transaction storing the smart contract for emitting NFTs of the Sicilian Wheat Bank (SWB) S.p.A.\footnote{https://www.bancadelgrano.it/en/} The individual \textit{block\_node\_10452395\_tran\_1} represents the Ethereum transaction and is related with the individual \textit{SWB\_SmartContractCreation} describing the smart contract by means of the object-property \textsf{describes}. The description of the agent identifying the SWB smart contract is introduced by the individual \textit{SWB\_smart\_contract\_agent} which is linked to \textit{SWB\_smart\_contract\_creation} by means of the object-property \textsf{describes}. Finally, the node (\textit{SparkPool}) that mines the block (\textit{block\_node\_10452395}) including the transaction is also associated to the Ethereum main-net (\textit{ethereum\_mainnet}) by means of the object-property \textsf{constitutes}.
\end{sloppypar}
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{img/blockchain-example.pdf}
\caption{Example of representing an Ethereum transaction in \ONT{}}
\label{fig:blockOntoexmple}
\end{figure}
\clearpage
\section*{Appendix B - ERC721 transferring function in \ONT{}}
\begin{sloppypar}
The behaviour for the ERC721 function for transferring tokens is depicted in Fig. \ref{fig:transferERC721template}. It provides three input parameters, one for the token to be transferred and one for each externally owned account involved in the transferring of the token, namely the source wallet and the destination wallet. The source and destination wallets are introduced by exploiting the object-property \textsf{refersAsNewTo} by means of two individuals instances of the class \textit{EOA-EthereumAccount}, namely, \textit{transfer-2\_ERC721\_EOA-account} (the source) and \textit{transfer-3\_ERC721\_EOA-account} (the destination). To ensure that the token is transferred from the wallet identified as source to the wallet identified as destination, the conditional illustrated in Fig. \ref{fig:transfercondERC721template} is provided.
\end{sloppypar}
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{img/erc721-tokentransfer.pdf}
\caption{\ONT{} behaviour template for the ERC721 transfer function}
\label{fig:transferERC721template}
\end{figure}
The conditional ensures the existence of a transfer activity for each token to be transferred. The conditional has as conditional object a fresh transfer activity and as operator the individual \textit{exist}. In its turn, the transfer activity indicates as transfer source the wallet used as first parameter in the token transferring, as transfer destination the wallet used as second parameter, and as transferred object the token passed as input parameter.
\begin{figure}[H]
\centering
\includegraphics[scale=0.73]{img/erc721-tokentransfer-conditional.pdf}
\caption{Conditional for the ERC721 transfer function}
\label{fig:transfercondERC721template}
\end{figure}
\clearpage
\section*{Appendix C - ERC721 token burning function in \ONT{}}
\begin{sloppypar}
The behaviour for the burning function of the ERC721 standard is depicted in Fig.~\ref{fig:burnERC721template}. In this case, the individual \textit{ethereum\_ERC721\_smart\_contract\_behavior\_template} (representing the ERC721 behaviour template) is also connected to the behaviour describing the burning function, whose structure is analogous to the minting function but with a different action and with an input parameter template instead of an output parameter template. The action introduced in the burning function is the individual \textit{burn}, whereas the input parameter template is connected with an individual representing the token to be burnt that is passed as input to the burning function.
\end{sloppypar}
\begin{figure}[H]
\centering
\includegraphics[scale=0.65]{img/erc721-tokenburn.pdf}
\caption{\ONT{} behaviour template for the ERC721 burning function}
\label{fig:burnERC721template}
\end{figure}
\clearpage
\section*{Appendix D - ERC721 token approve function in \ONT{}}
The standard ERC721 allows the owner of tokens to delegate wallets to manage tokens on his behalf. Authorizations may be carried out either on a single token or on any token stored in his wallet. The ERC721 function \textsf{approve} and \textsf{setApprovalForAll} are introduced for the former and the latter case, respectively.
In case that an externally owned account is authorized to manage a single token, the behaviour in Fig. \ref{fig:approveERC721template} is adopted. Specifically, the behaviour introduces as input parameter the account to be authorized and the granted token, as operator the instance \textit{delegate}, and as operator argument the instance \textit{ownership}. The conditional in Fig. \ref{fig:approvecondERC721template} ensures that only the operation of burning and transferring may be pursued when the wallet is authorized to operate on behalf of his owner. Such a condition is guaranteed by the existence of a delegation activity (instance of \textit{DelegationActivity}), having as delegation property the instances \textit{burn} and \textit{transfer}. The subject and the object of the delegation activity are the authorized wallet and the granted token, respectively.
\begin{figure}[H]
\centering
\includegraphics[scale=0.63]{img/erc721-tokenapprove.pdf}
\caption{\ONT{} behaviour template for the ERC721 approve function}
\label{fig:approveERC721template}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=0.65]{img/erc721-tokenapprove-conditional.pdf}
\caption{Conditional for the ERC721 approve function}
\label{fig:approvecondERC721template}
\end{figure}
\clearpage
\section*{Appendix E - ERC721 setApprovalForAll function in \ONT{}}
The Ethereum ERC721 standard allows wallet owners to authorize an account to manage all the tokens owned. The corresponding behaviour is illustrated in Fig.~\ref{fig:approveallERC721template}. The behaviour is close to the behaviour for granting a single token, with the only difference that there is no input parameter concerning the granted token. In such a case, the conditional in Fig. \ref{fig:approveallcondERC721template} ensures that authorization is extended to each token owned. Indeed, the delegation object expresses the object-property \textit{hasSpecificity} with value the individual \textit{any}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{img/erc721-tokenapproveAll.pdf}
\caption{\ONT{} behaviour template for the ERC721 \emph{setApprovalForAll} function}
\label{fig:approveallERC721template}
\end{figure}
\clearpage
\begin{figure}[H]
\centering
\includegraphics[scale=0.65]{img/erc721-tokenapproveAll-conditional.pdf}
\caption{Conditional for the ERC721 approve-all function}
\label{fig:approveallcondERC721template}
\end{figure}
\clearpage
\section*{Appendix F - ERC721 token ownership retrieval function in \ONT{}}
The Ethereum ERC721 standard allows one to retrieve the wallet owner of a token. The corresponding behaviour is illustrated in Fig. \ref{fig:ownerofERC721template}.
The behaviour is related with a task description providing as recipient and as input parameter the selected token, whose owner has to be retrieved. The conditional in Fig.~\ref{fig:ownerofcondERC721template} ensures that the wallet retrieved is the effective owner of the token considered.
\begin{figure}[H]
\centering
\includegraphics[scale=0.62]{img/erc721-tokenowner.pdf}
\caption{\ONT{} behaviour template for the ERC721 function for retrieving the token's owner}
\label{fig:ownerofERC721template}
\end{figure}
\clearpage
\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{img/erc721-tokenowner-conditional.pdf}
\caption{\ONT{} conditional for the ERC721 function for retrieving the token's owner}
\label{fig:ownerofcondERC721template}
\end{figure}
\clearpage
\section*{Appendix G - ERC721 token representation in \ONT{}}
Tokens are depicted in Fig. \ref{fig:token}. There are four main types of token:
\begin{itemize}
\item non-fungible tokens, represented by instances of the class \textit{EthereumSemiFungibleToken}, the latter containing the class \textit{EthereumTokenERC721} that represents non-fungible tokens compliant with the ERC721 standard protocol;
\item fungible tokens, represented by instances of the class \textit{EthereumFungibleToken}, the latter containing the class \textit{EthereumTokenERC20} that represents fungible tokens compliant with the ERC20 standard protocol;
\item semi-fungible tokens, represented by instance of the class \textit{EthereumSemiFungibleToken}, the latter containing the class \textit{EthereumTokenERC1155} that represents semi-fungible tokens compliant with the ERC1155 standard protocol;
\item custom user-defined tokens not compliant with the ERC standard protocols, represented by instances of the class \textit{EthereumCustomToken}.
\end{itemize}
\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{img/EthereumToken.pdf}
\caption{Ethereum token representation in \ONT{}}
\label{fig:token}
\end{figure}
The four mentioned classes are defined as subclass of the class \textit{EthereumToken}. Additionally, tokens that have been definitively destroyed are also instances of the class \textit{BurnedEthereumToken}.
Tokens carry two types of features~\cite{gangemi2002dolce}, a) perdurant features such as the token ID, that never change and are embedded with the entity representing the token and b) endurant features, that change during the life-span of the token and are associated with an instance of the class \textit{EthereumTokenEndurantFeatures} (subclass of \textit{EndurantFeature}), by means of the object-property \textit{hasEthereumTokenEndurantFeature}. The most notable subclass of \textit{EndurantFeature} is the class \textit{EthereumWalletOwnerEndurantFeature}, which describes the wallet of the token's owner (by means of the data-properties \textit{isInTheWalletOf}). When the endurant features of a token are modified by the smart contract managing it, they became deprecated and replaced by a new set of features by means of a modification activity. Those new features are introduced by means of a fresh instance of the class \textit{EndurantFeature}. Modification of tokens is allowed only if it involves endurant features and hence perdurant features cannot change. Endurant features may be replaced with other endurant features by introducing an instance of the class \textit{EthereumTokenFeatureModificationActivity} which is connected with:
\begin{itemize}
\item the changed endurant feature, which is also instance of the class \textit{DeprecatedEthereumTokenEndurantFeature}, by means of the object-property \textit{hasEthereumTokenFeatureModificationSource};
\item the new endurant feature, by means of the object-property \textit{hasEthereumTokenFeatureModificationResult}.
\end{itemize}
Moreover, the modified endurant feature is connected with the endurant feature that replaces it by means of the object-property \textit{isEthereumTokenFeature
ModifiedIn}, whereas the token embedding the features is connected with the new endurant feature by means of the object-property \textit{hasEthereumTokenEndurantFeature} as described above.
\section{Representing Ethereum through \ONT{} } \label{sec:onto}
\begin{sloppypar}
In this section, we describe how the Ethereum blockchain is modelled in \ONT{}.\footnote{The ontology is reachable at \\ https://www.dmi.unict.it/santamaria/projects/oasis/sources/ether-oasis.owl} \ONT{} provides a different representation of blockchains with respect to Ethon \cite{ethon} and BLONDiE \cite{Rojas2017AMP}, since the description of blockchain has to be aligned with the definitions of agent and agent behavior.
\end{sloppypar}
\begin{sloppypar}
Ethereum is represented in \ONT{} by following the schema illustrated in Fig.~\ref{fig:blockOnto}. Ethereum blocks embedding transactions are represented by instances of the class \textit{EthereumBlock} (subclass of \textit{BlockchainBlock}) and connected to the transactions contained in them by means of the object-property \textsf{embeds}. Each Ethereum transaction is identified by an instance of the class \textit{EthereumTransaction} (subclass of \textit{BlockchainTransaction}), encapsulating all the transaction information.
Block miners are identified by instances of the class \textit{EthereumNode} (subclass of \textit{BlockchainNode}) and linked to the mined block through the object-property \textsf{mines}: instances of \textit{BlockchainNode} are also instances of the class \textit{Agent}, representing agents and provided with a behavior as in the \ONT{} fashion. Moreover, instances of \textit{BlockchainNode} (resp., \textit{EthereumNode}) are connected to instances of the class \textit{System} (resp., \textit{EthereumSystem}) by means of the object-property \textsf{constitutes}. Such a characterization of nodes, blocks, and transactions is the main difference with analogous approaches such as Ethon and BLONDiE, since it allows one to describe activities carried out by both in-chain and out-of-chain agents, thus providing a higher-level model of the two ecosystems and a means to unify them. Specifically, Ethereum activities are classified into two main categories, namely, deployments of smart contracts, represented by instances of the class \textit{EthereumSmartContratCreation} (subclass of \textit{BlockchainSmartContractCreation}), and interactions with smart contracts, represented by instances of the class \textit{EthereumSmartContractInteraction} (subclass of the \textit{BlockchainSmartContractInteraction}).
\end{sloppypar}
In \ONT{}, smart contracts deployed into the Ethereum blockchain correspond to agents with well-defined behaviors: interactions with smart contracts are represented by \ONT{} \emph{plan executions} and linked to the behavior that induced the action. Specifically, a smart contract creation is represented by an instance of the class \textit{BlockchainSmartContractCreation}, which is related with the description of the agent describing its behavior by the object-property \textsf{describes}, the latter represented by an instance of the class \textit{BlockchainSmartContractAgent} (subclass of the class \textit{Agent}). Instances of \textit{BlockchainSmartContractCreation} are also associated by means of the object-property \textsf{associatedWith} with the related Ethereum accounts, represented by instances of the class \textit{EthereumSmartContractAccounts} (sublcass of the class \textit{BlockchainSmartContractAccount}, which, in turn, is a subclass of \textit{BlockchainAccount}). Users are instead associated with \emph{Ethereum externally owned accounts} (EOA) represented by instances of the class \textit{EOA-EthereumAccount} (sublcass of \textit{EOA-BlockchainAccount}, which is a subclass of \textit{BlockchainAccount}).
\begin{sloppypar}
\ONT{} identifies four main general categories of smart contract agents: a) smart contracts providing non-fungible token exchange mechanisms compatible with the Ethereum standard ERC721, which are represented by instances of the class \textit{EthereumERC721SmartContractAgent} (subclass of the class \textit{NonFungibleBlockchainSmartContractAgent}); b) smart contracts providing fungible token exchange mechanisms compatible with the Ethereum standard ERC20, which are represented by instances of the class \textit{EthereumERC20SmartContractAgent} (subclass of the class \textit{FungibleBlockchainSmartContractAgent}); c) agents responsible for exchanging Ether cryptocurrency, which are represented by instances of the class \textit{EtherExchangeSmartContractAgent} (subclass of the class \textit{CryptocurrencyExchangeBlockchainSmartContractAgent}); d) general purpose and user-defined smart contract agents that do not enjoy the characteristics of the aforementioned smart contracts, which are introduced by instances of the class \textit{GeneralPurposeBlockchainSmartContractAgent}.
\end{sloppypar}
\begin{sloppypar}
In \ONT{}, agents may perform actions autonomously or as response to requests of executing some operations submitted by a peer. Concerning the blockchain ecosystem, we mainly limit ourselves to take into account only requests that modify the state of the chain (both internal and external), and hence induce transactions, even through \emph{view functions}, namely functions that do not modify the state of the chain may be represented as well.
The transactions (represented by instances of the class \textit{EthereumTransaction}) induced by interaction requests submitted to smart contracts are related with instances of the class \textit{EthereumContractInteraction} (subclass of the class \textit{SmartContractInteraction}) by means of the object-property \textsf{describes}. Instances of \textit{EthereumContractInteraction} introduce plan descriptions as in the \ONT{} fashion by means of the object-property \textsf{describes}. Plan descriptions are ways of characterizing requests of actions and the related actions performed by agents. The most notable subclass of \textit{EthereumContractInteraction} is the class \textit{EtherExchangeSmartContractInteraction}, representing the transferring of Ether cryptocurrency from a wallet to another. The class \textit{EtherExchangeSmartContractInteraction} is also a subclass of the class \textit{CryptocurrencyExchangeBlockchainSmartContractInteraction}, which in turn is a subclass of the class \textit{BlockchainSmartContractInteraction}.
\end{sloppypar}
An example of representing Ethereum transactions in \ONT{} is illustrated in Appendix A.
\begin{figure}[H]
\centering
\includegraphics[scale=0.63]{img/blockchain.pdf}
\caption{Representing the Ethereum blockchain in \ONT{}}
\label{fig:blockOnto}
\end{figure}
\section{The ERC721 protocol in \ONT{}} \label{sec:erc}
\begin{sloppypar}
In this section we show how \ONT{} represents the main ERC721 standard functions for managing non-fungible tokens on the Ethereum blockchain. For space limitations,
we illustrate how the ERC721 standard for minting non-fungible tokens is modelled in \ONT{}, whereas the ERC721 functions for transferring tokens, burning tokens, granting ownership of single tokens, of all the tokens stored in the wallet, and to verify the ownership of tokens, as represented in \ONT{}, are described in Appendices B, C, D, E, and F, respectively. Finally, ERC721 tokens are described in Appendix G.
As seen in the previous section, the smart contracts defined for the ERC721 token management are introduced by means of instances of the class \textit{EthereumERC721SmartContractAgent} (subclass of the class \textit{Agent}). In \ONT{}, agents having similar behaviors inherit the representation of their behavior from a common template providing general descriptions that may be customized by single agents. For this purpose, \ONT{} provides a template for the ERC721 standard introduced by the individual \textit{ethereum\_ERC721\_smart\_contract\_behavior\_template}, which describes the behaviors of agents minting, burning, and transferring Ethereum NFTs according to the guidelines of the standard. Other functions admitted by the standard ERC721, namely the \emph{approve} function (delegating wallets for managing single tokens), the \emph{setApprovalForAll} function (delegating wallets for managing all the tokens owned), and \emph{ownerOf} (for retrieving the owner of a given token), are also represented in \ONT{}.
\end{sloppypar}
\begin{figure}[H]
\centering
\includegraphics[scale=0.65]{img/erc721-tokenmint.pdf}
\caption{\ONT{} behavior template for the ERC721 minting function}
\label{fig:mintERC721template}
\end{figure}
Fig.~\ref{fig:mintERC721template} illustrates the behavior template provided by \ONT{} describing the ERC721 function for generating new tokens. The behavior template consists of a single goal, which in turn consists of a single task. Tasks comprise four elements. The first element is the task operator, providing the description of the action to be performed, namely minting. The latter is introduced by means of the individual \textit{mint} (instance of the class \textit{Action}), through the the object-property \textsf{refersExactlyTo}. We recall that in \ONT{} the object-properties \textsf{refersExacltyTo} and \textsf{refersAsNewTo} are introduced to describe the way how constituting elements of agent behaviors must be matched when a verification of compatible behaviors occurs. The object-property \textsf{refersExactlyTo} introduces well-known entities whose IRIs must correspond to the IRIs of the matched entities or for which the OWL object-property \textsf{sameAs} has been expressed. On the contrary, the object-property \textsf{refersAsNewTo} introduces entities (instances of the class \textit{ReferenceTemplate}) that are used as general descriptions encapsulating the features that the matching entities must satisfy.
The second element of the ERC721 token minting task is the operator argument introducing the individual \textit{blockchain\_digital\_token} by means of the object-property \textsf{refersExactlyTo}. Operators and operator arguments identify unambiguously that the referred operation consists in the generation of (digital) tokens on the blockchain. The third and the fourth elements represent the recipient and the outcome of the operation, respectively. The recipient is introduced by means of a template of the task object, whereas the outcome is introduced by means of a template of the output parameter of the task. The object template and the output parameter are both connected through the object-property \textsf{refersAsNewTo} to the entity \textit{mint\_ERC721\_token}, which describes the features that the recipient of the action must have, i.e., being an instance of the class \textit{EthereumTokenERC721}.
\section{Conclusions}\label{sec:conclusion}
This paper leveraged the \ONT{} ontology towards the representation of the Ethereum blockchain and the smart contracts deployed on it. Specific focus was on those that comply with the ERC721 standard for NFTs management. \ONT{} is exploited as a means of semantically representing Ethereum with the aim of probing the blockchain for locating smart contracts and related NFTs by specifying the desired features.
In particular, the ontological representation of \ONT{} allows one to find smart contracts and related NFTs by inspecting their behavioural descriptions through purposely crafted SPARQL queries.
It was already clear that the \ONT{} approach to semantic representation had the power of generality but our findings demonstrate it at an applied level.
Future work is dense. The very next step is to extend \ONT{} so as to model the standard protocols ERC20 and ERC1155 for fungible and semi-fungible tokens, respectively, and to represent different blockchains such as Stellar Lumens.
Moreover, we intend to take up the design of a search engine exploiting \ONT{} to find desired smart contracts and tokens using a mechanism of auto-generating parametric \emph{ad-hoc} SPARQL queries that could be borrowed from sibling applications~\cite{woa2019}.
The present work supports the claim that the potential of the semantic representation of blockchains has much to be unveiled in the near future.
\section{Introduction}
\label{sect:intro}
The last decade reports vast interest on blockchain technology and related applications from various realms, including economic, social, business, and academic ones. Beyond the financial speculation concerning cryptocurrencies, the interest in blockchain technologies is mainly motivated by the fact that they realize decentralized and publicly shared ledgers, where third-party intermediaries demanding the client's total and unquestioned trust are no longer required. Blockchain technologies~\cite{christidis16} were precisely introduced to allow users to interact and run programs in a distributed way without the requirement of trusted entities, yet guaranteeing ownership, transparency, traceability, availability, continuity, and immutability of digital shared assets.
Applications of blockchain technologies range from the \emph{Internet of Things} (IoT) and robotics~\cite{xu2017taxonomy}, to commerce, healthcare, insurance, energy, laws, and communications.
One of the most popular applications of Turing-complete blockchains such as Ethereum~\cite{Szabo97} is the \emph{smart contract}. Smart contracts are self-executing and immutable programs, autonomously running and verified on a distributed and decentralized public network, which implement decentralized applications on blockchain systems called \emph{Dapps}. In 2020, Dapps have particularly grown as an exchange tool for non-fungible tokens (NFTs), namely digital certificates stored on the blockchain representing predetermined rights on certain unique assets. NFTs are mainly used as a proof of ownership of physical or digital products. Such tokens are routinely exchanged by users to witness that assets whose uniqueness is hard to demonstrate (for example, digital images) are owned in an exclusive way. At the end of 2020, the market capitalization of NFTs reached the amount of 338 millions of U.S. dollars.\footnote{https://www.statista.com/statistics/1221742/nft-market-capitalization-worldwide/ (last access: 08/07/2021).}
However, one of the main limitations of blockchains is the hard-coded nature of the transactions stored on them. As a consequence, it is hard to probe a blockchain, for example, to find smart contracts trading specific NFTs that satisfy certain requirements in terms of quality or quantity.
\begin{sloppypar}
Therefore, a formal semantic knowledge representation capturing the blockchain smart contracts as well as the activities carried out on it
facilitates the understanding of blockchain concepts, the interlinking with other out-of-chain information, and also formal reasoning. Moreover, a semantic conception of blockchains enables the automatic discovery of smart contracts, the interconnection of services running on different blockchains (i.e., cross-chain integration) and the integration between on-chain and off-chain services. These features turn out to be more interesting when smart contracts are implemented as mechanisms for generating and exchanging tokens. A desirable feature of token exchange systems is a precise and intelligent query mechanism capable
of determining what, when, and how certain assets have been generated, exchanged or destroyed. For example, intelligent systems may be aware of the activation of smart contracts for generating tokens with specific characteristics, e.g., of the type of exchanged asset, of the exchange of particular tokens at certain conditions, or of their destruction. More in general, intelligent systems may be aware of the activation of smart contracts and of all the related activities over the blockchain.
\end{sloppypar}
Beyond a semantic representation of transactions and information stored in blocks, a real, semantically represented blockchain is effectively achievable if smart contracts are interpreted as reactive agents operating on a common environment, with a fully specified semantics of available operations, committed actions, and stored data.
The representation of blockchain actors requires ontological capabilities for fully representing agents and their interactions in a detailed way. This paper adopts the \emph{Ontology for Agents, Systems, and Integration of Services} (``\ONT{}'') towards the full, semantic representation of the Ethereum blockchain and the smart contracts running on it, with a special focus on the smart contracts compliant with the ERC721 standard\footnote{https://ethereum.org/it/developers/docs/standards/tokens/erc-721/} for NFTs management.
The paper is organized as follows. Section \ref{sec:related} presents related work. Section \ref{sec:prelim} outlines \ONT{}, whereas Sections \ref{sec:onto} and \ref{sec:erc} depict the ontology implementing the \ONT{} representation of, respectively, the Ethereum blockchain and the ERC721 standard. Finally, Section \ref{sec:conclusion} draws some conclusions and delineates future research directions.
\section{Preliminaries} \label{sec:prelim}
The \emph{Ontology for Agents, Systems, and Integration of Services} (``\ONT{}'') \cite{woa2019} is an OWL 2 ontology for representing agents and their activities. On one hand, the ontology models (web) agents and, in particular, the way they interact and operate in a collaborative environment, regardless of the framework and language adopted for their implementation. Agents are mainly represented by means of the mentalistic notion of agent behavior inspired by \cite{tropos}, encompassing goals and tasks that are achievable (either publicly available or exposed on request) by the agent, together with actions, sensors, and actuators used to perform operations. On the other hand, \ONT{} is used to define actions that may be requested to other agents and their related information such as operation inputs and outputs. Such requests are submitted by exchanging suitable fragments of \ONT{}, whereas agents whose capabilities are compatible with the requested actions are discovered by means of SPARQL queries performed over their behaviors. \ONT{} was applied to build a TRL3 prototype of a home assistant that
activates and manages applications, devices, and users interacting with each other within the environment~\cite{woa2019}. \ONT{} was also used to define agent agreements and store these in the IPFS file system \cite{ipfs} in order to reduce the transaction data stored directly on blockchain~\cite{idc2021}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{img/oasis.pdf}
\caption{Agent representation in \ONT{}} \label{fig:oasis}
\end{figure}
OASIS models agents by publicly representing their behaviors. By exposing behaviors, agents report to the communication peers the set of operations that they are able to
perform and, eventually, the type of data required to execute these and the expected output. The representation of agents and their interactions in OASIS is carried out along three main steps. The first step consists in \textit{defining the template} for agent behaviour: templates are high-level descriptions of behaviors of abstract agents that can be implemented to define more specific and concrete behaviors of real agents. For example, a template may be designed for agents to sell and ship products to buyers, and it may be implemented by any phone seller that ships its products using the Fedex courier: templates are useful to guide developers to define the most suitable representation of their agents. The second step consists in \textit{implementing a template},
which requires a specification of the full operational details about the sought behavior.
As depicted in Fig.~\ref{fig:oasis}, agent behaviors are represented by the goals to achieve, which in turn are related with their constitutional elements, namely tasks.
Tasks represent atomic operations that agents execute, and are described by the actions to be performed. Actions are drawn from a shared and common vocabulary, and can be simple or composed, eventually associated with requested input parameters and expected outputs. Finally, agent behaviors in \ONT{} may be associated with conditionals \cite{idc2021}, adding constraints on the execution of actions and ensuring that certain conditions are verified before or after a task is executed. \ONT{} conditionals are OWL sentences that have the fashion of \emph{Semantic Web Rule Language} (SWRL) rules \cite{swrl}, describing operations that must be triggered when certain conditions hold.
\section{Related works} \label{sec:related}
Interest in symbiotically combining semantic web technologies and blockchains is quite recent \cite{English2015, CanoCimmino19}. One of the areas of
investigation concentrates in developing a characterization of blockchain concepts and technologies through ontologies and of blocks and transactions meta-data. An ontological albeit theoretical approach at blockchain representation exists~\cite{Kruijff2017}.
Ruta \emph{et al.} propose a blockchain framework for \emph{semantic web of things} (SWoT) contexts settled as a \emph{Service-Oriented Architecture} (SOA), where nodes can exploit smart contracts for registering, discovering, and selecting annotated services and resources~\cite{RSICPD18}.
Blockchain technologies are also exploited as a secure and public storage system for small data, including linked data, and to realize a more resilient architecture for the Semantic Web \cite{English2015}.
Other works aim at representing ontologies within a blockchain context. In \cite{KimL18}, ontologies are used as common data format for blockchain-based applications such as the proposed provenance traceability ontology, but are limited to implementation aspects of the blockchain.
Fill discusses blockchains applied for tracking the provenance of knowledge, for establishing delegation schemes, and for ensuring the existence of patterns in formal conceptualizations using zero-knowledge proofs~\cite{Fill2019ApplyingTC}.
A semantic interpretation of smart contracts as services bases on the \emph{Ethon} ontology \cite{ethon} exist~\cite{baqa2019}. The main limitation of that approach is the poor semantic description of smart contracts, thus hindering the discovery of unknown smart contracts and of the related operations fulfilled during their life-span.
\begin{sloppypar}
The \emph{Blockchain Ontology with Dynamic Extensibility} (BLONDiE) project \cite{Rojas2017AMP} provides a comprehensive vocabulary that covers the structure of different components (wallets, transactions blocks, accounts, etc.) of blockchain platforms (Bitcoin and Ethereum) and that can be easily extended to other alternative systems.
\end{sloppypar}
Finally, in \cite{idc2021} the authors illustrated how the ontology \ONT{} is applied for ontologically describing digital contracts (called \emph{ontological smart contracts}), intended as agreement among agents, and how they can be secured on Ethereum.
In this contribution, the definition of digital contracts in \cite{idc2021} is generalized to include blockchain smart contracts, intended as programs running on the blockchain and interpreted as digital agents in the \ONT{} fashion. | {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,152 |
\section{Introduction}
The idea that massive Galactic globular clusters (GCs) may be formed within
dwarf galaxies is not new (e.g., \citealt{bek03}). Recently, theoretical
considerations and new observations demonstrated that massive GCs
like $\omega$ Cen or M~54 probably had their origin in dwarf spheroidal
galaxies (dSphs), which are currently either lost or still surrounding the cluster
(\citealt{bok09,bel08,car10a}). This idea was extended by
\cite{car10b} to practically $all$ GCs, sketching a scenario which unifies the
view of GCs and dSphs. The ancestral progenitors of both kinds
of systems started as cosmological fragments, but the
evolution of dSphs proceeded undisturbed in near isolation from the
distant main Galaxy. On the contrary, strong interaction with the Galactic main
body triggered a chain of events in the so called $precursor$ of current GCs,
whose final products are the systems we are seeing (see \citealt{car10b} for a
detailed description). The same mechanism may also work within a single dSph,
as in the case of Fornax or Sagittarius.
This scenario, although not entirely new (e.g., \citealt{sea78}), may
explain several characteristics of GCs, including the multiple
stellar generations, found in $all$ objects investigated so far
(\citealt{car09a,car09b} and \citealt{gra04} for a review).
It is currently well assessed that the second-generation stars (presently
constituting the bulk of the cluster population) should have formed from
the ejecta of only a fraction of the first-generation (primordial) stars (e.g.,
\citealt{gra01,pra06}). To account for the present chemical composition, a precursor
baryonic mass about 20-50 times
larger than the current mass of the GCs is required \citep{car10b}.
However, the proposed scenario is still qualitative; for
instance, it is not obvious that only $one$ GC should be the final
output of an individual precursor. Examples of binary and multiple clusters
are frequently observed in the Large Magellanic Cloud (e.g.,
\citealt{die02}).
To clarify these issues, we added NGC~1851 to our ongoing FLAMES
survey studying the Na-O anticorrelation in GCs \citep{car06}.
NGC~1851 has a bimodal horizontal branch (HB) and several other
peculiarities. The color-magnitude diagram (CMD) shows a double subgiant
branch (SGB; \citealt{mil08}). Two distinct red
giant branch (RGB) sequences were discovered by \cite{lee09a} with Ca$uvby$
photometry, and confirmed by \cite{han09} using broad-band filters. In
\cite{car10c} we challenged the hypothesis advanced by \cite{lee09b} of Ca
variations in GCs, ascribed to a possible pollution by core-collapse SN~II.
However, the case of NGC~1851 was left open, since
we had not yet adequate data.
A photometric approach alone is probably not enough to understand the complex
nature of multiple populations in GCs.
The most striking photometric feature in NGC~1851, the split SGB, has been
explained in terms either of an age/metallicity/helium effect or a different
total CNO content between the two populations (e.g., \citealt{cas08,ven09}).
A mix of two or more factors cannot be excluded. Furthermore, the
spatial
distribution of these sub-populations is still controversial
(\citealt{zoc09,mil09}). Only a precise chemical tagging allows the accurate
separation of different stellar generations, and helps providing a first
relative ranking in age between them. The only existing abundance analysis from
high resolution spectroscopy in this GC is the one by \cite{yon08}, on eight RGB
stars. That analysis (complemented by \citealt{yon09}) showed a number of
additional interesting features: a possible small metallicity spread,
correlations between abundances of p-capture elements and elements produced in
$s-$processes, and possibly a variable value of the total C+N+O sum. However,
these data are only available for a frustrating small number of stars.
In this Letter we partially fill this gap, presenting the results on the
chemical composition of more than 120 red giants. We think that our results
support the hypothesis that NGC~1851 formed starting from two clusters (likely
born within a single dSph), an idea already circulating in the literature for
GCs with composite CMDs, in particular for NGC~1851 (e.g.
\citealt{van96,cat97}). These two clusters have slightly different metallicity,
and later underwent a merger, leaving however detectable traces of that past
event.
\section{Data and analysis}
The FLAMES spectra (GIRAFFE and UVES) of 124 RGB members of NGC~1851 were
obtained in April, August, and September 2009. The stellar parameters were
determined using the same techniques described, e.g., in \cite{car10a}.
A full description of the analysis will be presented elsewhere
(Carretta et al., in preparation); here we only show abundances
of some interesting species: Fe, Na, O, Ca, a few $s-$process elements, such
as Ba, La, Ce, and the $r-$process element Eu.
Details of our abundance analysis trace as closely as possible the homogeneous
procedures adopted for other GCs (\citealt{car09a,car09b} and references
therein)\footnote{For instance, the temperature is derived from a relation
between $T_{\rm eff}$ (from $V-K$ and the \citealt{alo99} calibration) and $K$
magnitudes, much more reliably measured than colors, leading to very small
internal errors in the atmospheric parameters, hence in the derived
abundances.}
\section{Metallicity spread in NGC~1851}
Our first result {\it is that there is a small but real spread in metallicity}
in NGC~1851. We find an average [Fe/H]{\sc i} $=-1.179 \pm 0.019$
($\sigma=0.067$ dex) from 13 stars with UVES spectra. The analysis of
the GIRAFFE spectra yields [Fe/H]{\sc i} $=-1.158 \pm 0.005$
($\sigma=0.051$ dex, 121 stars),
after correcting the equivalent widths to the
system defined by the higher resolution UVES spectra (\citealt{car07a}).
The observed dispersions in [Fe/H]{\sc i} for UVES spectra are statistically
significant when compared to internal errors in [Fe/H], which are estimated to
be 0.017 dex (internal errors for GIRAFFE spectra, 0.060 dex, are too large
to be used in this context)\footnote{For a detailed discussion of the larger
scatter found among brighter and cooler stars see \cite{car09c}. {\it The
observed spreads we found are only lower limits}, due to the way we derive the
final temperatures, using a $mean$\ relation between T$_{\rm eff}(V-K)$ and the
$K$ magnitude along the RGB, which tends to decrease real abundance spreads.
Finally, the reddening is very low \citep[$E(B-V)$=0.02,][]{har96} and not
differential.}.
There are additional ways to show that the spread in [Fe/H] is real. First,
\cite{yon08} also found some range in [Fe/H], although they
did not expand on this due to the small size of their sample. Second, in
Fig.~\ref{f:fig1} we compare the metallicity distribution function (MDF) in
NGC~1851 with the MDFs of M~4 and M~5, GCs of similar
metallicity analyzed through an identical procedure (\citealt{car09c}).
The metallicity spread in
NGC~1851 is clearly larger. The MDF of M~4 would reproduce the one of NGC~1851
if we split it into two equal components with a difference in metallicity of
0.06-0.08 dex; the same is obtained for M~5. Finally, we also find a spread in
[Ca/H], similar to the one in [Fe/H], confirming photometric results by
\cite{lee09a} and \cite{han09}.
For the sake of the following discussion we arbitrarily assume that NGC~1851 is
made of a metal-poor (MP) and a metal-rich (MR) component. Since there is no
obvious gap between these components, we simply assume that stars with
[Fe/H] larger than the average -1.16 dex belong to the MR component, the others
to the MP one. Fig.~\ref{f:fig2} shows the RGB accordingly divided. If we fit
the RGB with a line and compute the residuals in color between the individual
stars and the line, the MR stars have average $B-V$ color redder by
0.02$\pm$0.008 mag than the MP ones, in very good agreement with what is
expected (\citealt{gra10}) from the difference in metallicity.
One of our most striking results is shown in the right panel of Fig.1:
{\it these two metallicity components have a clearly different spatial
distribution}, with the MP one being more concentrated
than the MR component. All our stars are in the external regions, at
more than 2 half-mass radii (Fig.~\ref{f:fig1}), where the relaxation time is very
long so that mass segregation between the two components cannot be responsible,
since the mass difference implied by the different metallicity and age (see Sect.
6) is about 0.04 M$_\odot$. Our new result, statistically very robust, is fully
independent from any claimed radial distribution of SGB stars.
\section{The Na-O anticorrelation in NGC~1851}
We find the familiar Na-O anticorrelation also in NGC~1851. Interestingly,
this signature remains visible even considering the MR and MP
populations separately. In Fig.~\ref{f:fig3} we show the Na-O anticorrelation for both
components, together with the run of the [O/Na] ratios as a function of the
metallicity. We use [Na/H] and [O/H] to account for the
well known metallicity-dependence of Na as a function of the metal-abundance for
stars of the primordial (P) component (see the definition of primordial,
intermediate I, and extreme E populations in \citealt{car09a}).
The Na-O anticorrelation is slightly different for the MR and MP components,
but in both cases the extension is quite modest, more similar to that in
M~4 than in NGC~2808. This suggests only a modest spread in He in NGC~1851
because a high Y fraction seems to be associated only to very long tails of very O-poor
stars, not present in this GC. Our finding from a chemical approach
confirms the claims by \cite{sal08}, based on the absence of a tilt along the HB
and the lack of a splitting in the MS.
In addition, we observe a slight change of the mean value of O and Na abundances
at the level of the bump on the RGB. We already showed
\citep{car07b,bra10} that this variation is expected from theoretical
models, which predict a change in the bump luminosity with He
content \citep{sal06}, i.e. with
elements involved in p-capture reactions.
The presence of a mix of first and second generation stars results in a
concentration of Na-poor/He-poor stars just before the bump accompanied by an
accumulation of Na-rich/He-rich stars just
above the bump level. This accounts for the observed abundance changes at the
bump without resurrecting the internal mixing scenario for O and Na (e.g.
\citealt{lee10}) and then overcoming the unpalatable requirement of basic
stellar structure differences between field and GC stars. For NGC~1851, where
we hypothesize two distinct clusters (see below), we expect a further smearing
of the bump in the RGB luminosity function (Carretta et al., in prep).
\section{The location of stars on the RGB in NGC~1851}
In Fig.~\ref{f:fig4}, we use the Str\"omgren $u,u-b$ CMD to test where stars of
different components and populations
are located on the RGB.
This plane is optimally suited to separate first and second
generation stars, probably because of N (enhanced in O-depleted,
second generation stars) via the formation of NH, CN and their relevance on the $u-b$
(or the Johnson $U-B$), see \cite{yon008,mar08,car09a}.
Stars of the first generation (P, \citealt{car09a}) in NGC~1851 lie along a narrow
strip to the blue
of the RGB (Fig. \ref{f:fig4}, bottom panel), as
expected from their unprocessed chemical abundances. On the contrary, the
second generation stars are spread out to the red, as in NGC~6752
(\citealt{car09a}): the I stars are in the middle and the extreme E
component, with the lowest O abundances, is located at the reddest edge.
This segregation is followed also within each metallicity
component, and it is ``orthogonal" to the separation of MR and MP stars
(Fig.~\ref{f:fig4}, top panel), which are well intermingled across all the RGB in this
color. The same holds if we separate the RGB stars using the average value for Ca
([Ca/H]$=-0.83$): stars with low and high Ca are spread across the entire RGB
(Fig. ~\ref{f:fig4}, middle panel).
Therefore the spread of Ca does not track the abundances of
p-capture elements. A K-S test on the cumulative distributions of [Ca/H] for
stars of the first and second generations in NGC~1851 indicates that they
are indistinguishable. Instead, the Ca abundances
closely track those of Fe. The cumulative distribution of [Ca/H] values for
the MR and MP components on the RGB are definitively different. Moreover, also
the radial distributions of Ca-rich and Ca-poor stars confirm the close
correspondence with metallicity: the Ca-poor giants are more concentrated,
while Ca-rich stars show a tendency toward more
external regions.
\section{Is NGC~1851 a relic of a merger of two clusters?}
Is there a comprehensive scenario able to account for all the evidence found here
and in previous works in NGC~1851? In our view, the answer is affirmative if we
consider NGC~1851 as the result of a chain of events that started with two
distinct clusters. Several suggestions of duplicity come from the bimodal
distribution of HB stars, the double SGB, and hints of double sequences on the
RGB. Up to now, the main objection was the absence of a metallicity spread,
owing to the lack of precise abundances
for a statistically significant number of
stars. This limitation has finally been overcome by our study.
As a tentative working hypothesis we can think of two different clusters,
born in a much larger system, perhaps a dSph. Being distinct, each one might have
formed with a slightly different metallicity and with a different level of
$\alpha-$elements\footnote{We find that also Mg, Si, and Ti track iron in
the MR and MP components, as Ca does.}. Each object is rightfully a GC, since each
component show the Na-O anticorrelation, the classical
signature of the processes ending in a GC (\citealt{car10b}). After a while, the
two clusters underwent a merger, likely because both were dragged to the center of the
dSph by dynamical friction (see \citealt{bel08}) and the result is NGC~1851.
Finally, the dSph merged with the Milky Way. We think that this is the simplest
scenario, that with a minimum of hypothesis may account for many observational
constraints.
The two MR and MP components do not show any significant difference in
kinematics, the velocity dispersion being the same for both components. A
comprehensive dynamical model would be very welcome, although we do not know
when the merging occurred. We will rely on the observed chemistry.
The observables include: a double SGB, where the faint SGB (fSGB) includes
45\% of the stars and the bright SGB (bSGB) the remaining 55\% (\citealt{mil08}),
and with controversial evidence of different concentration; the MR and MP
components on the RGB, with a clear difference in radial concentration; a
bimodal distribution on the HB, with $\sim 40\%$ of the stars on the BHB
and $\sim 60\%$ on the RHB (\citealt{mil08}); the observed luminosity of HB stars
and the moderate extension of the Na-O anticorrelation in both the MR and MP RGB
components, which both suggests small He abundance variations.
We may explain these observables in different ways:
(i) A single GC with two populations having a different total CNO abundance
(but a similar He abundance). This may explain the SGB but fails to
reproduce the number ratios on the HB (if the same efficiency for
mass loss on the RGB is assumed for both sub-populations), because the minor fSGB component
should be associated with the major RHB one.
(ii) A single GC with two populations having a different He and total
CNO abundance. In this scenario there is much more He in CNO-rich than in
CNO normal stars. The CNO effect dominates on the SGB, and He effect on
the HB. However, in this scenario the BHB should also be brighter than
the RHB, which is not observed (e.g. \citealt{sal08}).
(iii) A merging of two GCs differing in age (although this does not exclude
{\it a priori} a difference in the total CNO). In this scenario, one of the GC
(including some 55-60\% of the stars) would be responsible for the bright, younger
(possibly less concentrated) SGB, of the MR (less concentrated) RGB, and of the
younger RHB. The other GC (including some 40-45\% of the stars) would be responsible
of the faint, older (possibly more concentrated) SGB, of the MP (more concentrated)
RGB, and of the older BHB. In this case, the two SGBs can be fitted by isochrones
differing in age by $\sim 1.5$ Gyr, if the same CNO content is assumed for
both GCs, and the difference in Fe is as derived from the MP and MR
RGB components.
Should the solution (iii) be the right one, the age difference that we need
($\sim 1-1.5$~Gyr) is not unlikely if the clusters were born in a
dSph (see e.g., the case of the Fornax dSph GCs, \citealt{buo98, buo99}).
As a consistency check, we explored Lee's diagram HB-type vs [Fe/H]
(see e.g., Fig. 9 in \citealt{rey01})\
using the average values of metallicity [Fe/H]$=-1.20$ ($\sigma=0.03$ dex) and
[Fe/H]$=-1.12$ ($\sigma=0.03$ dex) for the MP and MP components, respectively. If we
tentatively adopt for the two components the HB type of NGC~288 (0.98) and
NGC~362 (-0.87), the diagram shows that the MR
component may easily be about 1-1.5 Gyr younger then the MP one.
An additional constraint comes from the ratio between the abundances of $s-$
and $r-$process elements. Fig. ~\ref{f:fig5} shows the [Ba/Eu] ratio as a function of
metallicity for the 13 stars with UVES spectra. This ratio is close to that
expected from a pure-$r$ component for MP stars, while it indicates a larger
contribution by $-s$ process for the MR ones. The trend with [Fe/H] is even
cleaner when using the average from Ba, La, and Ce. It suggests a larger
contribution of polluters of smaller masses for the MR component, which fits
well with the proposed age difference.
To conclude, we note that \cite{carb10} reported the existence of a
distinct metal-poor main sequence around the GCs NGC~1851 and NGC~1904, which
they interpreted as a very low surface brightness stellar system. This is maybe
consistent with the structure identified by \cite{ols09} in the form of a halo
of main-sequence stars surrounding NGC~1851 up to a distance of 250 pc. Both
these observations suggest the existence of a residual structure that might be what
is left by the destruction of the ancestral dwarf where the progenitor of
NGC~1851 originated.
A really clear-cut test for the presence of two distinct GCs is to probe the
Na-O anticorrelation among HB stars. If the merger hypothesis is correct, each
one of the HB populations (coming from individual GCs) should
present the Na-O anticorrelation, as we find in the MR and MP components of the
RGB. On the contrary, for a single proto-cluster scenario, we should expect the
RHB stars be almost all O-rich, with the O-poor stars only confined to the BHB.
Specific proposals of observations aimed to perform this test have been already
submitted.
\acknowledgements
Partial funding come from the PRIN MIUR 2007 CRA 1.06.07.05, PRIN INAF 2007 CRA
1.06.10.04, the DFG cluster of excellence ''Origin and Structure of the
Universe''.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,273 |
\section{}
\section{Introduction}
The last decade has witnessed the discovery of striking phenomena
associated with geometric and topological aspects of the band structure,
brought about by the
presence of spin-orbit coupling and the breaking of spin rotation symmetry.\cite{Bruno:2007_B,Resta:2010_JPCM}
In the case of ferromagnets the Berry curvature of bands hosting spin-polarized
carriers was found to result in sizable contributions
to transport coefficients, such as the anomalous Hall conductance.\cite{Jungwirth:2002_PRL,Nagaosa:2010_RMP}
It has been suggested more recently that the Berry curvature also describes
the orbital part of carrier magnetization coming from delocalized circulation.\cite{Xiao:2005_PRL,Thonhauser:2005_PRL,Shi:2007_PRL,Thonhauser:2011_IJMPB,Zhu:2012_PRB,Schulz-Baldes:2013_CMP}
This recent development is particularly worthwhile, as it has delivered formulas for
carrier orbital magnetization in the form that can be directly implemented into {\em ab initio} methods,
allowing us to interpret theoretically experimental values of the orbital magnetic moment provided by, for instance, x-ray circular magnetic dichroism (XMCD).\cite{Ceresoli:2010_PRB,Lopez:2012_PRB} The verification of the modern theory in this way is especially
meaningful since, compared to transport coefficients, thermodynamic properties are less sensitive
to scattering and localization.
In this paper, we examine quantitatively magnetization of spin-polarized valence band holes in
dilute ferromagnetic semiconductors (DFSs).\cite{Dietl:2014_RMP} A particular versatile method
to model semiconductor properties and devices is the Kohn-Luttinger (KL) envelope function approximation,\cite{Winkler:2003_B}
whose six-band version has been exploited to describe various thermodynamic\cite{Dietl:2014_RMP}
and transport data in DFSs, including the anomalous Hall effect.\cite{Jungwirth:2002_PRL,Nagaosa:2010_RMP} Within this scheme, we
compare carrier magnetization obtained from the modern approach\cite{Xiao:2005_PRL,Thonhauser:2005_PRL,Shi:2007_PRL,Thonhauser:2011_IJMPB,Zhu:2012_PRB,Schulz-Baldes:2013_CMP}
and determined\cite{Dietl:2001_PRB,Jungwirth:2006_PRB,Sliwa:2006_PRB} employing the time-honored Landau theory.\cite{Landau:1930_ZP}
According to the combined KL and Landau's method, the spin-orbit interaction generates two contributions to orbital magnetization $M_{\text{orb}}$ in DFSs.\cite{Dietl:2001_PRB}
The first one, $M_{\text{L}}$, stems from Landau's quantization. The second contribution $M_{I}$ is proportional to the orbital angular momentum operator $\mathbf{\hat I}$.
As we demonstrate here, only $M_{\text{L}}$ is reproduced by the modern approach, but the second term emerges within the modern approach if the set of the basis of the Bloch wave functions $\{u_n\}$ is enlarged.
Furthermore, we show that both contributions have to be taken into account to describe quantitatively
experimental results
on XMCD,\cite{Freeman:2008_PRB,Wadley:2010_PRB} and on
the dependence of the chemical potential on the magnetic field in (Ga,Mn)As.\cite{Ciccarelli:2012_APL}
\section{Landau theory within the envelope function approach}
Within the KL method and neglecting the lack of inversion symmetry, the six-band Hamiltonian of holes in a magnetic field $\mathbf{B}$ and in the presence of Mn magnetization $\mathbf{M}$ consists of three terms in DFSs:\cite{Dietl:2001_PRB} (i) $\mathcal{{H}}_{\text{L}}$ that describes Landau's
quantization of the valence band in terms of the Luttinger band structure parameters $\gamma_1$, $\gamma_2$, and $\gamma_3$; (ii) the Zeeman-like contribution $H_{\text{Z}}$, and (iii) $\mathcal{H}_{pd}$ accounting for $p$-$d$ coupling between hole and Mn spins. In the basis employed previously,\cite{Winkler:2003_B,Sliwa:2006_PRB,Trebin:1979_PRB}
\begin{equation}
\mathcal{{H}}_{\text{Z}} = - (1 + 3\kappa)\mu_B \mathbf{\hat I}\cdot \mathbf{B}
+ g_0 \mu_B \mathbf{\hat s}\cdot \mathbf{B},
\label{eq:H_Z}
\end{equation}
where $\kappa$ is one more Luttinger parameter,\cite{Luttinger:1956_PR,Roth:1959_PR,Winkler:2003_B} the free electron Land\'e factor $g_0 \approx 2.002$, and the dimensionless angular-momentum tensor operators $\mathbf{\hat I}$ and~$\mathbf{\hat s}$ are given by,
\begin{equation}
\mathbf{\hat I} = \begin{pmatrix} \displaystyle \frac{2}{3} \mathbf{\hat J}&
\mathbf{\hat U} \\ \mathbf{\hat T}&
\displaystyle \frac{2}{3} \mathbf{\hat{\boldsymbol{\sigma}}}
\end{pmatrix}, \qquad
\mathbf{\hat s} = \begin{pmatrix} \displaystyle \frac{1}{3} \mathbf{\hat J}&
-\mathbf{\hat U} \\ -\mathbf{\hat T}&
\displaystyle -\frac{1}{6} \mathbf{\hat{\bm{\sigma}}}
\end{pmatrix}.
\end{equation}
Following Ref.~\onlinecite{Trebin:1979_PRB}, we denote by $\mathbf{\hat{\bm{\sigma}}}$
the Pauli matrices, by $\mathbf{\hat J}$ the set of spin-$3/2$ angular-momentum matrices,
and by $\mathbf{\hat U}$, $\mathbf{\hat T}$ the sets of matrices for the cross-space.
In Eq.\,(\ref{eq:H_Z}), besides the ordinary Pauli spin part $g_0 \mu_B \mathbf{\hat s}\cdot \mathbf{B}$, there is an orbital term
$\mathcal{{H}}_{I} = -(1 + 3\kappa)\mu_B\mathbf{\hat I}\cdot\mathbf{B}$. This contribution is brought about by coupling of the six valence subbands to remote bands in the presence of an external magnetic field.
That is,
$H_I$ accounts for an admixture of the orbital magnetic moment to the carrier effective Land\'e factor.\cite{Roth:1959_PR}
Finally, the $p$-$d$ coupling to the spin-polarized Mn ions is taken into account in the virtual-crystal and
molecular-field approximations, leading to additional giant spin splitting of Landau levels, described
by the Hamiltonian $\mathcal{H}_{pd} = (\Delta_{\text{v}}/ M )\mathbf{M} \cdot \mathbf{\hat s}$,
where $\Delta_{\text{v}}$ is the $p$-$d$ exchange splitting
of the valence band top.\cite{Jungwirth:2002_PRL,Dietl:2001_PRB,Sliwa:2006_PRB}
Within Landau's method\cite{Landau:1930_ZP} the carrier
magnetization $\mathbf{M}_{\text{c}}(T,\mathbf{B})$ is given by
the derivative of the grand thermodynamic potential,
\begin{eqnarray}
\label{eq:M_Landau}
\lefteqn{\Omega_{\text{c}} = -\mu_{\text{B}} B k_{\text{B}}T
\sum_j\int\limits_{-\infty}^{\infty} \frac{m_0\, dk_3}{2(\pi\hbar)^2}
} \\ & & \qquad \qquad
\ln\left\{1+\exp \left( -[E_j(k_3) - \mu]/k_{\text{B}}T \right)\right\},
\nonumber
\end{eqnarray}
with respect to the magnetic field,
$\mathbf{M}_{\text{c}} = -\partial \Omega_{\text{c}}/\partial \mathbf{B}$, where $m_0$ in Eq.\,(\ref{eq:M_Landau}) is the free-electron mass.
Here,
$E_j(k_3)$ is the $j$-th eigenenergy of $\mathcal{H}_{\text{L}}+\mathcal{H}_{\text{Z}} + \mathcal{H}_{pd}$
for a carrier
with the $\mathbf{k}$ component along the direction of the
magnetic field denoted as $k_3$, and $\mu$ is the chemical potential. The values of $\mathbf{M}_{\text{c}}$
computed in this way for (Ga,Mn)As were reported previously.\cite{Sliwa:2006_PRB}
This approach allows us to evaluate orbital parts of carrier magnetization, $M_{\text{L}}$ and $M_{I}$, associated with $\mathcal{H}_{\text{L}}$ and $\mathcal{H}_{\text{I}}$, respectively, at a given $p$-$d$ exchange splitting of bands described by $\mathcal{H}_{pd}$. The key question we address in this paper is how these two contributions are related to orbital magnetization $M_{\text{mod}}$ obtained from the modern theory. A formulation of the modern theory within the KL method is discussed in the subsequent section.
\section{Modern theory of orbital magnetization}
Within the modern approach the orbital part of $\mathbf{M}_{\text{c}}$ at $B = 0$ for $N$ bands is given by,
\begin{eqnarray}
\label{eq: modern}
\lefteqn{\mathbf{M}_{\text{mod}} = \mu_{\text{B}}
\int \frac{d^3\mathbf{k}}{(2\pi)^3} \sum_{n, n' = 1}^N \mathcal{M}_T(E_{n'\mathbf{k}}, E_{n\mathbf{k}}) } \\
& & \qquad \qquad
\mathop{\mathrm{Im} } \left[
m_0
\left\langle u_{n\mathbf{k}} \middle| \mathbf{\hat{v}} \middle| u_{n'\mathbf{k}} \right\rangle
\times
\left\langle u_{n'\mathbf{k}} \middle| \mathbf{\hat{v}} \middle| u_{n\mathbf{k}} \right\rangle \right],
\nonumber
\end{eqnarray}
where $u_{n\mathbf{k}}$ is
the Bloch function corresponding to the eigenenergy $E_{n\mathbf{k}}$
of the KL Hamiltonian $\mathcal{H}_{\mathbf{k}}$ at $B =0$ including $\mathcal{{H}}_{pd}$;
$\hbar \mathbf{v} = \partial \mathcal{{H}}_{\mathbf{k}}/\partial \mathbf{k}$,
and
\begin{eqnarray}
\lefteqn{\mathcal{M}_T(E_{n'\mathbf{k}}, E_{n\mathbf{k}}) = {}} \nonumber \\ & & \qquad
\mathcal{M}\bigl[ (E_{n'\mathbf{k}}-\mu)/k_{\text{B}}T,
(E_{n\mathbf{k}}-\mu)/k_{\text{B}}T \bigr] / k_{\text{B}}T,
\end{eqnarray}
where the dimensionless function~$\mathcal{M}$ reads
\begin{eqnarray}
\label{eq:m}
\lefteqn{\mathcal{M}(x_{n'}, x_n) = \frac{1}{x_{n'} - x_n} \biggl[
\frac{f(x_{n'}) + f(x_n)}{2} + {}} \\ & & \qquad {} + \frac{
\ln[1+\exp(-x_{n'})] - \ln[1+\exp(-x_n)]}{x_{n'} - x_n} \biggr], \nonumber
\end{eqnarray}
with the Fermi-Dirac distribution function $f(x) =[1 + \exp(x)]^{-1}$
(notice that the cross product of velocity matrix elements is purely imaginary).
The contribution coming from the first term in Eq.\,(\ref{eq:m}) corresponds to magnetization of the carriers' wave packets,\cite{Chang:2008_JPCM} whereas the second is proportional to the Berry curvature.
The definition of $x$ implies that $x =0$ for states at the Fermi level, whereas $x > 0$ and $x < 0$ correspond to the empty and occupied states, respectively.
Since the formula for magnetization involves a symmetric summation over a pair of indices running over the same set of bands, and the cross product is antisymmetric, we have adopted ${\mathcal{M}}(x_1, x_2)$ in an antisymmetrized form that allows us to tackle better with a possible divergence at $x_1 = x_2$. As seen in Fig.~\ref{fig: m}, $\mathcal{M}(x_2, x_1) = - \mathcal{M}(x_1, x_2)$,
and $\mathcal{M}$ vanishes rather than diverges at the band crossings, $\mathcal{M}(x, x) =0$, as required for degenerate bands.
The function $\mathcal{M}(x_1, x_2)$
also obeys $\mathcal{M}(-x_1, -x_2) = \mathcal{M}(x_1, x_2)$ (electron-hole symmetry) and $\mathcal{M}(x, -x) = 0$.
Furthermore, according to Fig.~\ref{fig: m}, $\mathcal{M}(x_1, x_2)$ decays exponentially to zero with inverse temperature in the first and third quadrant, i.e., when $x_1$ and $x_2$ have the same sign (either
positive or negative, corresponding to pairs of empty or pairs of occupied states, respectively). This formulation substantiates
a picture in which orbital magnetization is described by a sum
over \emph{pairs} of subbands, with significant contributions only from empty--occupied states.
\begin{figure}[tb]
\includegraphics[width=3in]{fig_1_fun_Mn}
\caption{Plot of the function $\mathcal{M}(x_1,x_2)$ that appears in the formula [Eq.\,(\ref{eq:m})] for orbital magnetization.}
\label{fig: m}
\end{figure}
\section{Comparison of the two approaches}
\begin{figure*}
\leavevmode
\hbox to 0.2\hsize{\hspace{1.25em}(a)\hfill}\hfill
\hbox to 0.2\hsize{\hspace{1.25em}(b)\hfill}\hfill
\hbox to 0.2\hsize{\hspace{1.25em}(c)\hfill}\hfill
\hbox to 0pt{}\\[0.2em]
\hbox to \hsize{%
\hfill\includegraphics[scale=0.7]{fig_2a_GaAs_orb}\hfill
\hfill\includegraphics[scale=0.7]{fig_2b_GaAs_tot}\hfill
\hfill\includegraphics[scale=0.7]{fig_2c_InAs}\hfill
}
\caption{(Color online) Contributions to hole magnetization at 10\,K computed within the six-band Kohn-Luttinger model of the valence band for parameters of GaAs ($\gamma_1 = 6.85$, $\gamma_2 = 2.1$, $\gamma_3 = 2.9$, $\kappa = 1.2$, $\Delta_{\text{SO}} = 0.341 \, \mathrm{eV}$) and InAs ($\gamma_1 = 20.0$, $\gamma_2 = 8.5$, $\gamma_3 = 9.2$, $\Delta_{SO} = 0.39 \, \mathrm{eV}$, $\kappa = 7.60$), and $\mathbf{M}\parallel \langle 100\rangle$ and the parameter of valence band exchange splitting $\Delta_{\text{v}} = -180 \, \mathrm{meV}$, corresponding to the magnitude of saturation magnetization in $\mathrm{Ga}_{0.95}\mathrm{Mn}_{0.05}\mathrm{As}$. Orbital magnetization $M_{\text{orb}}$ in (Ga,Mn)As from Landau's method (solid line) is decomposed into $M_{\text{L}}= M_{\text{mod}}$ provided by the modern method and the remaining (missing) part $M_{I}$ (dashed and dotted lines, respectively) (a). Total hole magnetization $M_{\text{c}}$ (solid line) in (Ga,Mn)As (b) and (In,Mn)As (c) decomposed into $M_{\text{orb}}$ and the spin part $M_{\text{spin}}$ (dashed and dotted lines, respectively). Inset in (c) shows $M_{\text{orb}}$ for (In,Mn)As and (Ga,Mn)As in an expanded scale.}
\label{fig:six_bands}
\end{figure*}
We first compare hole orbital magnetization determined within the KL method from the modern approach, $M_{\text{mod}}$ [Eq.\,(\ref{eq: modern})] to $M_{\text{L}}$ determined from the grand thermodynamic potential [Eq.\,(\ref{eq:M_Landau})] in the limit $B \to 0$. In order to evaluate $M_{\text{L}}$, i.e., orbital magnetization resulting from Landau quantization of the hole spectrum, we assume $\mathcal{{H}}_{Z} =0$, i.e., neglect the contribution $M_I$ to orbital magnetization. For $\mathrm{Ga}_{1-x}\mathrm{Mn}_{x}\mathrm{As}$, in the explored parameter space ($T = 10 \, \mathrm{K}$, $5\times10^{19} \leq p \leq 10^{21} \, \mathrm{cm}^{-3}$, and $\Delta_{\text{v}} =-180\,\mathrm{meV}$, i.e., $x \simeq 0.05$), the relative difference between the data obtained by these two methods is within our numerical uncertainty of $10^{-5}$. This finding highlights a major progress provided by the modern approach that allows one to circumvent the computational load associated with the determination of Landau level energies for complex band structures.
However, quantitative agreement between the Landau and modern approach to orbital magnetization is obtained neglecting $M_{I}$. This indicates that the term arising from the coupling to remote bands, $-(1 + 3\kappa)\mu_B\mathbf{\hat I}$, is not taken into account within the modern approach. The magnitude of the missing magnetization $M_{I}$ can be evaluated from the grand thermodynamic potential [Eq.\,(\ref{eq:M_Landau})] with eigenenergies $E_{n\mathbf{k}}$ of the Hamiltonian ${\mathcal{H}}_{\mathbf{k}} + {\mathcal{H}}_{I}$. According to results presented in Fig.\,\ref{fig:six_bands}(a), $M_{I}$ is quite sizable and, in fact, compensates largely $M_{\text{L}} = M_{\text{mod}}$ provided by the modern approach. For comparison, we also show the total hole magnetization $M_{\text{c}}$ that is seen to be dominated by the spin part $M_{\text{spin}}$, obtained from ${\mathcal{H}}_{\mathbf{k}} + g_0\mu_B\mathbf{\hat s}\cdot{\mathbf{B}}$, in both (Ga,Mn)As [Fig.\,\ref{fig:six_bands}(b)] and (In,Mn)As [Fig.\,\ref{fig:six_bands}(c)].
\section{Discussion}
The results presented in the previous section point to disagreement between the two theories of orbital magnetization. A question then arises whether $M_{I}$ is an artifact of the Landau approach or rather it is the modern theory that disregards the quantitatively important contribution $M_{I}$.
In order to address this issue we note that the modern approach requires information on both eigenenergies and eigenfunctions. In contrast, the Landau method is developed in terms of eigenenergies only. Within the KL method, the second order perturbation theory serves to determine the contribution to carrier eigenenergies of bands beyond the valence band states. However, no effect of the remote bands on the eigenfunctions $u_{n\mathbf{k}}$ is considered within such an approach. This suggests that by taking into account a contribution of remote bands to $u_{n\mathbf{k}}$, either perturbatively or by enlarging the basis $\{u_n\}$ of the KL scheme, the accuracy of the modern approach can be improved. To verify this hypothesis we have computed the magnitude of orbital magnetization within the eight-band model that incorporates the conduction band states to $\{u_n\}$.\cite{Winkler:2003_B}
Within the six-band model the grand thermodynamic potential has been derived in the hole picture. Since in the eight-band model the energies are bound neither from below nor from above, it is necessary to use the electron picture in order to describe the states residing above a fixed energy in the band gap. That is, we exploit the identity $-\ln(1+e^{-x}) = -\ln(1+e^{x}) + x$ to split the grand thermodynamic potential into a sum of the hole contribution and a hole-concentration independent shift. The shift describes the magnitude of orbital magnetization for the fully occupied valence band brought about by transitions to the conduction band, and it vanishes in the absence of band spin splittings.
Within the modern approach, an equivalent approach is to decompose $\mathcal{M}(x_{n'}, x_n)$ as follows:
\begin{eqnarray}
\lefteqn{\mathcal{M}(x_{n'}, x_n) = \frac{1}{x_{n'} - x_n} \biggl[
\frac{f(x_{n'}) - f(-x_n)}{2} + {}} \\ & & \qquad {} + \frac{
\ln(1+e^{-x_{n'}}) - \ln(1+e^{x_n})}{x_{n'} - x_n}
\biggr] + \frac{1}{2} \frac{x_{n'} + x_n}{(x_{n'} - x_n)^2} . \nonumber
\end{eqnarray}
As can be shown by inspection, also here the second term leads to a shift independent of the hole concentration but dependent on band spin splittings; it assumes a nonzero value if spin splittings of the valence and conduction bands differ, $\Delta_{\text{v}} \ne \Delta_{\text{c}}$. Since it provides just an additional contribution to the magnitude of orbital magnetization coming from fully occupied bands, a comparison between the two approaches is still meaningful even if we disregard the shift.
\begin{figure}[b]
\includegraphics[scale=1.0]{fig_3_8b}
\caption{(Color online) Orbital magnetization $M_{\text{orb}}$ of (Ga,Mn)As computed
by the Landau method within the eight- and six-band models (solid and dashed lines,
respectively). These results differ substantially from the outcome
of the eight-band modern model (the dotted line).
However, the modern model (except for a concentration-independent shift describing magnetization of the
fully occupied band) agrees with the truncated eight-band Landau model
(no coupling to remote bands; dash-dotted line).
The computations have been carried out for splitting of the valence and conduction
bands, $\Delta_{\text{v}} = -180$\,meV and $\Delta_{\text{c}} = 30$\,meV,
respectively.}
\label{fig:eight_bands}
\end{figure}
As shown in Fig.\,\ref{fig:eight_bands}, within the Landau theory there is a minor change in the magnitudes of $M_{\text{orb}}$ on going from the six- to the eight-band model, as eigenenergies are fairly accurately provided by either of these two KL schemes. In contrast, there is a considerable difference between magnetization values for these two KL implementations within the modern approach, as seen comparing the data in Figs.\,\ref{fig:six_bands} and \ref{fig:eight_bands}. This demonstrates that the enlargement of the set $\{u_n\}$ has a substantial influence on the magnitude of $M_{\text{mod}}$. However, according to the data in Fig.\,\ref{fig:eight_bands}, $M_{\text{mod}}$ obtained in this way still disagrees with $M_{\text{orb}}$ from the Landau method. Actually, according to the results in Fig.\,\ref{fig:eight_bands}, the modern method is in accord with a truncated variant of the eight-band Landau method, in which the coupling to bands beyond the eight-band manifold is disregarded (i.e. $\kappa\prime = -1/3$ in the notation of Ref.\,\onlinecite{Winkler:2003_B}). This indicates that for the modern approach the eight-band basis is still too small for obtaining accurate values of orbital magnetization.
Altogether these findings imply that it is possible to determine orbital magnetization
without referring to carrier spectrum in the magnetic field but to achieve the
same quantitative accuracy the set of basis wave functions $\{u_{n}\}$
must be much larger in the modern approach than needed within the Landau theory.
However, there exists an efficient method
to compute $M_{\text{orb}}$ at $B = 0$ exploiting advantages of
these two theoretical schemes. The hybrid method we propose consists of evaluating
orbital magnetization as
$M_{\text{orb}} = M_{\text{mod}} + M_I$,
where both $M_{\text{mod}}$ and $M_I$ are to be computed within the minimal KL scheme
for the problem at hand (typically either six- or eight-band model).
Thus, the hybrid method requires only a small set of basis wave functions $\{u_{n}\}$ and supplies accurate values of $M_{\text{orb}}$ without
computing Landau level energies. Below, we compare experimental data for (Ga,Mn)As to our theoretical results obtained by the hybrid procedure within the eight-band KL scheme.
\section{Comparison to available experimental data}
\begin{figure}[b]
\includegraphics[scale=0.8]{fig_4_Wadley_8b_MSat}
\caption{(Color online) Orbital magnetization of As $4p$ states
determined experimentally at $\sim 10$\,K and in 2\,T by Wadley {\em et
al.}\cite{Wadley:2010_PRB} as a function of saturation magnetization
$M_{\text{Sat}}$ for (Ga,Mn)As (open circles) and (In,Ga,Mn)As (open squares)
compared to theoretical values of orbital magnetization $M_{\text{orb}}$
obtained from the hybrid method within the eight-band KL model for (Ga,Mn)As directly (empty
diamonds) and including a possible contribution $\Delta m_{\text{orb}}
= a \Delta_{\text{v}} + b$, where $a$ and $b$ are fitting parameters
(full squares).}
\label{fig:comparison}
\end{figure}
Figure \ref{fig:comparison} presents the orbital moment of As $4p$ states
determined by XMCD for (Ga,Mn)As and (In,Ga,Mn)As films with different
saturation magnetizations $M_{\text{Sat}}$ and Curie temperatures
$T_{\text{C}}$.\cite{Wadley:2010_PRB} Since the orbital moment of cations
appears to be much smaller,\cite{Wadley:2010_PRB,Freeman:2008_PRB} we
compare these data to our theory, evaluating $\Delta_{\text{v}}$ and hole
concentrations $p$ from $M_{\text{Sat}}$ and $T_{\text{C}}$ within the
eight-band $sp$-$d$ Zener model.\cite{Hankiewicz:2004_PRB,Winkler:2003_B} As seen, our theory explains
both the sign and the small magnitude of $m_{\text{orb}} =
M_{\text{orb}}/N_0$ observed experimentally, where $N_0$ is the anion
concentration. Since contributions to $m_{\text{orb}}$ coming from the
cations and fully occupied bands have been neglected, we may
expect an additional term proportional to $\Delta_{\text{v}}$.
Furthermore, experimental data were taken in 2\,T. This may lead to a
diamagnetic shift of $M_{\text{orb}}$, which should weakly depend on
$\Delta_{\text{v}}$. Accordingly, we supplement the theoretical values
of $m_{\text{orb}}$ with $\Delta m_{\text{orb}} = a\Delta_{\text{v}} + b$.
The fitting procedure implies $a = -14.8\times 10^{-3}$\,$\mu_{\text{B}}$/eV and $b
= -1.59 \times 10^{-3}$\,$\mu_{\text{B}}$. Although the quality
of the fit is excellent, such a large value of the offset~$b$
calls for further attention.
Another relevant experiment concerns variations of the chemical potential
$\mu$ with the magnetic field $B$, as provided by studies of
an Al single electron transistor (SET) with a
(Ga,Mn)As gate.\cite{Ciccarelli:2012_APL}
Figure \ref{fig:SET} shows $\mu(B)$ determined from the field-induced shift of Coulomb blockade
peaks for a SET with the $\mathrm{Ga}_{0.97}\mathrm{Mn}_{0.03}\mathrm{As}$ gate
in respect to the shift in a control SET with an Au gate.\cite{Ciccarelli:2012_APL}
We are interested in the region $B \gtrsim 7$\,T, in which the Mn spins become saturated but
nevertheless $\mu$ varies with the magnetic field.
In order to explain these data we make use of relations $\mu = \partial \Omega_{\text{c}}/\partial p$
and $M_c = - \partial \Omega_{\text{c}}/\partial B$, which lead to the thermodynamic identity,
\begin{equation}
\left.\frac{\partial \mu}{\partial B}\right|_p =
\frac{\partial^2 \Omega_\text{c}}{\partial B \, \partial p} =
-\left.\frac{\partial M_{\text{c}}}{\partial p}\right|_B,
\label{first}
\end{equation}
implying $\partial M_{\text{c}} / \partial p = -\partial \mu / \partial B$,
which relates the derivative of the carrier magnetization (with respect to carrier
concentration) to changes of the electron's chemical potential in an external
magnetic field.
\begin{figure}[tb]
\includegraphics[width=7cm]{fig_5_Ciccarelli}
\caption{(Color online) Chemical potential determined experimentally for $\mathrm{Ga}_{0.97}\mathrm{Mn}_{0.03}\mathrm{As}$ at 0.3 K (squares) by Ciccarelli {\em et al}.\cite{Ciccarelli:2012_APL} Slopes of dashed and solid lines are computed for interstitial concentrations
$x_{\text{I}} = 0$ and 0.5\%, respectively.} \label{fig:SET}
\end{figure}
Because of virtual cancellations between $M_{\text{mod}}$
and $M_{I}$, the total hole magnetization $M_{\text{c}}$ is dominated
by the spin part [see, Fig.~\ref{fig:six_bands}(b)] that is isotropic.
This explains why $\mu(B)$ was independent of the field direction in
respect to crystallographic axes.\cite{Ciccarelli:2012_APL} In order
to evaluate $\partial M_{\text{c}} / \partial p$ information on
saturation magnetization and hole concentration are needed, which at given $x$
depend on density of Mn interstitials $x_{\text{I}}$.\cite{Dietl:2014_RMP}
As shown in Fig.\,\ref{fig:SET}, theoretical results obtained for
$x_{\text{I}} = 0$ and 0.5\% are consistent with the experimental data.
\section{Conclusions}
In summary, we have proposed a numerically efficient method that
combines advantages of the modern and Landau approach to carrier orbital magnetization. The computed hole magnetization within
the formalism developed here explains the magnitude of orbital and spin magnetizations implied
by experimental studies of XMCD and the Coulomb blockade in (Ga,Mn)As.
A timely question arises about implications of our findings to the theory
of anomalous and spin Hall effects in semiconductors.
\section*{Acknowledgments}
We thank B.\,L. Gallagher, K.\,Edmonds, and P.\,Wadley for instructive discussions on XMCD results.
This work was supported by the European Research Council through the FunDMS Advanced
Grant (No. 227690) within the ``Ideas'' 7th Framework Programme of the EC.
| {
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package de.hftstuttgart.projectindoorweb.persistence.entities;
public class PositionResult {
private double x;
private double y;
private double z;
private boolean wgs84;
protected PositionResult() {
}
public PositionResult(double x, double y, double z, boolean wgs84) {
this.x = x;
this.y = y;
this.z = z;
this.wgs84 = wgs84;
}
public double getX() {
return x;
}
public void setX(double x) {
this.x = x;
}
public double getY() {
return y;
}
public void setY(double y) {
this.y = y;
}
public double getZ() {
return z;
}
public void setZ(double z) {
this.z = z;
}
public boolean isWgs84() {
return wgs84;
}
public void setWgs84(boolean wgs84) {
this.wgs84 = wgs84;
}
}
| {
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} | 5,357 |
sap.ui.define([
'sap/ui/core/mvc/Controller',
'sap/ui/model/json/JSONModel'
], function(Controller, JSONModel) {
"use strict";
var ListController = Controller.extend("sap.m.sample.ListGrowing.List", {
onInit : function (evt) {
// set explored app's demo model on this sample
var oModel = new JSONModel(sap.ui.require.toUrl("sap/ui/demo/mock/products.json"));
this.getView().setModel(oModel);
}
});
return ListController;
}); | {
"redpajama_set_name": "RedPajamaGithub"
} | 457 |
Zoe Sugg, aka Zoella, is a 25-year-old vlogger from Brighton, UK. Her beauty, fashion, and lifestyle vlogs have gained her millions of YouTube subscribers, and her views often exceed 12 million a month. She won the 2011 Cosmopolitan Blog Award for 'Best Established Beauty Blog' and went on to win the 'Best Beauty Vlogger' award the following year. Zoe has twice received the 'Best British Vlogger award', at the 2013 & 2014 Radio 1 Teen Awards, and the 2014 & 2015 Nickelodeon Kids' Choice award for 'UK's Favourite Vlogger'. She was also named Web Star for Fashion and Beauty at the 2014 Teen Choice Awards. | {
"redpajama_set_name": "RedPajamaC4"
} | 50 |
\section{Introduction}
The Gas Electron Multiplier (GEM) was introduced by Fabio Sauli, who was working at European Center for Nuclear Research (CERN) in 1997 \cite{Sauli, Sauli2, Sauli3, Hoch} and since then it has been used for various practical applications. Currently, many high energy and nuclear physics experiments are using or proposing to use GEM technology, which is creating a big demand for GEM foils. Therefore it is difficult for CERN, which is the main provider of GEM foils, to keep up with such an increasing demand. There is a need for commercially available GEM foils to help fulfill the surge in demand. The Micropack Pvt. Ltd., an India based company, acquired a license from CERN under Transfer of Technology (ToT) to produce GEM foils. The University of Delhi (DU) then began collaborating with Micropack to help them establish a consistent manufacturing procedure for GEM foils utilizing both double-mask and single-mask techniques \cite{MerlinThesis, DblMask}. The Micropack $\rm 100~mm \times 100~mm$ double-mask GEM foils that have been optically analyzed for geometric properties and electrically tested via leakage current at DU have been discussed elsewhere~\cite{Foils}. The basic quality controls on the detector built using these foils are described in~\cite{PerfDU}.
In this paper, we will present studies performed on a detector built using commercially manufacture GEM foils. These studies are helpful to understand the characteristic of the insulating material (Apical \footnote[1]{Apical is a polyimide film similar to Kapton and is available from Kaneka Texas Corporation, Pasadena, TX.}) and how they affect the stability of GEM detector. The gain stability \cite{Pol_Volt, Stab_GEM_TPC} over time is a very important parameter of gaseous detectors because any unwanted variation in gain can cause a loss of efficiency. The rate capability \cite{EveRC} of a gaseous detector shows the variation of detector gain with respect to the incident flux and for Micro Pattern Gaseous Detector (MPGD) the gain stays stable up to very high incoming rates but it also depends upon space charge effects \cite{Space_charge}. Several systematic studies on such effects have been performed in the past to determine the principle causes of the gain variation observed in the detector due to the properties of foils. We will report on some of these studies here.\\
In the following section, we describe the experimental set-up used for performing the studies mentioned in this paper. Section 3 is dedicated to understanding the gain stability of the detector whereas section 4 provides a quantitative idea of the behaviour of the triple GEM detector under very high incoming particle flux which is $O(MHz/mm^2)$. Finally, we conclude with discussions on the results obtained.
\section{Experimental set-up for measurements}
The active area of the GEM foils used for this study is $\rm 100~mm \times 100~mm$; they have been manufactured with the double-mask etching technique to have 70 $\rm\mu m$ (50 $\rm \mu m$) outer (inner) hole diameter and pitch of 140 \rm $\mu m$. The triple GEM detector was built with 3mm/1mm/2mm/1mm for Drift/Transfer1/Transfer2/Induction gap configuration and all the foils were powered up using a High Voltage (HV) resistive divider whose schematics is shown in fig.~\ref{fig:Grid}(a). The active area of the GEM detector was divided into 5 $\times$ 5 equal sectors each covering an area of $\rm 20~mm \times 20~mm$ and their nomenclature was fixed as shown in fig. \ref{fig:Grid}(b). A 1 mm collimator was placed in the front of the X-Ray and it was placed perpendicular to the detector touching the drift volume. Therefore, while exposing the particular sector with X-Rays, our setup ensured that no other neighboring sectors got exposed to the beam.
\begin{figure}[h]
\hspace{1 cm}
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/HV_Schematic.png}
\caption{}
\end{subfigure}
\hspace{2.5 cm}
\begin{subfigure}[b]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Grid_RO.png}
\caption{}
\end{subfigure}
\caption{Schematic of (a) high voltage resistive divider used for triple GEM detector prototype, and (b) Nomenclature used for 5 $\times$ 5.}
\label{fig:Grid}
\end{figure}
The gas mixture used in all the measurements was a mixture of noble gas and a quencher \cite{GasOperation}. For the studies reported here, a mixture of Argon ($\rm Ar$) and Carbon-Dioxide ($\rm CO_2$) in the ratio of 70\% and 30\% respectively have been used. The main advantage of those mixtures is that these gases are non-flammable, eco-friendly and that they are easily available and relatively cheap \cite{GasProp,Gases}. A gas mixing unit is used for supplying the fixed mixture and the flow rate was controlled by the mass flow controllers which has been set to 5 $\ell$/h for all the measurements.
The GEM detector has been powered up using a CAEN N1470 multichannel power supply. A high voltage filter is introduced between the power supply and the detector to reduce the electronics noise. The output is read by the readout board having an active area of $\rm 100~mm \times 100~mm$ and consisting of 128 1-D strips. Signal amplification was performed in two stages; first it was amplified by ORTEC 142 charge sensitive pre-amplifier and then using the ORTEC 474 timing filter amplifier. For counting measurements, the signal passed through ORTEC 974 discriminator with a threshold of 140 $\rm mV$ to cut the noise. The rate has been measured using CAEN N1145 Scaler \& Counter over the desired period of time. The current was measured with the help of Keithley 6517B pico-ammeter \cite{PicoAmp} and recorded with a Labview program via a GPIB interface.
\section{Gain stability}
The effective gain of a detector is a unique parameter used to relate the general properties which include the electrical and geometrical properties together with the gas composition. This is defined as the ratio of the charge arriving at the anode and the charge created in the drift volume.
\begin{center} $G = \frac{I_{with~Source} - I_{Without~Source}}{No.~of~Primaries*rate~on~Anode*charge~of~electron}$ \end{center}
Figure~\ref{fig:IV_UP_DOWN}(a) shows the block diagram of how the current and rate was measured from the detector. The gain stability is very important for gaseous detectors because any unwanted variation in gain causes a change in efficiency. As mentioned high voltage divider is used to power up the GEM foils and using it a current-voltage (IV) curve is obtained when the voltage is first ramped up and then ramped down across the divider. The measured current shows slope for ramping up and ramping down and hence no hysteresis in IV has been observed \cite{Hysteresis_IV}. Figure~\ref{fig:IV_UP_DOWN}(b) shows an ohmic behaviour at a given range of divider current across the detector with a total resistance of the circuit as 5 $\rm M\Omega$ (which includes the resistance of HV filter as 0.3 $\rm M\Omega$ and HV divider as 4.7 $\rm M\Omega$).
\begin{figure}[h]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Effective_Gain.png}\qquad
\caption{}
\end{subfigure}
\hspace{0.5cm}
\begin{subfigure}[b]{0.42\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/IV_UP_DOWN.png}
\caption{}
\end{subfigure}
\caption{(a) Effective gain measurement block diagram.(b) I-V Characteristics of the detector showing ohmic behaviour while ramping up (black circles) and ramping down (red diamonds) at different operating voltages(right).}
\label{fig:IV_UP_DOWN}
\end{figure}
In order to understand the hysteresis as a function of other parameters of the GEM detector, a measurement of effective gain has been performed. The measurement has been done while ramping up and then ramping down the divider current across the detector. The outcome of this exercise is shown in Figure~\ref{fig:70_30_UP_DOWN}. The measurements were taken by varying the current in steps of 10 $\rm \mu A$ from 620 $\rm \mu A$ to 690 $\rm \mu A$ by setting the corresponding voltage and then in the reverse direction with waiting time of 2 minutes between each measurement. As expected, the gain varies exponentially with respect to the current. However, looking at the variation in the log scale, the slope of the gain has been found to be different while ramping up and ramping down. This is a manifestation of hysteresis effect in gain \cite{Hysteresis_Gain} of the detector which is due to the charging up or polarisation effects in the detector/foils.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{figures/70_30_UP_DOWN.png}
\caption{Effective gain of the detector as a function of current across the detector while ramping up (black circles) and ramping down (red diamonds) with Ar/$\rm CO_{2}$ = 70/30.}
\label{fig:70_30_UP_DOWN}
\end{figure}
The gain stability of a triple GEM detector is affected mainly by two effects: the charging up and the polarisation effects. The charging up effect occurs due to the multiplication of the charge near the surface of the foil which results in the trapping of the charge in the hole~\cite{IEEE} . The polarisation effect, on the other hand, is due to the movement of the charges inside the polyimide layer after applying the voltage across the foil. It is independent of the charge deposited by the particle but depends upon the geometrical and electrical properties of the foil. The polarisation effect can be explained with the help of a trapping model, as described in~\cite{Pol_Trap} for Kapton-H film. Due to the absorption of the photon by polyimide during the irradiation of X-rays, electron excitation in the polyimide occurs. The presence of an electric field across the foil makes these electrons move in its direction unless they get captured by some trapping center in the foil. This process results in an increase of the anode current which in turn increases the gain. However, when the dynamic equilibrium is reached the gain starts to saturate.\\
To estimate the gain stability in time and variations in the gain due to polarisation effect, a series of measurements of gain of the detector was carried out for several hours. The anode current was recorded while irradiating a sector of the detector with Amptek Mini-X X-ray \cite{Amptek} source for 30 seconds. This process was repeated at an interval of 5 minutes for the same sector until a stable gain value had been achieved. These measurements were performed at various values of gain of the detector to estimate the dependency of divider current (or gain) on polarisation effect. During the measurement ambient temperature was continuously monitored using an ARDUINO based system.\\
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{figures/Pol_6k.png}
\caption{Variation of effective gain (blue squares) of the detector as a function of time having initial gain of 6,800 for triple GEM detector in Ar/$\rm CO_{2}$ = 70/30. Variation of temperature was also recorded every second and plotted (red continuous line).}
\label{fig:Pol_15k}
\end{figure}
Figure~\ref{fig:Pol_15k} shows the value of gain with respect to time. The measurement started with a gain of 6.8k and the plateau was observed after 6 hours showing a difference of $\sim$29\% in gain. During the measurement, the ambient temperature was monitored continuously. From the measurement it is visible that the percentage difference of change in gain and the time taken to reach the stable gain depends upon the initial value of the gain of the detector. For an initial gain of 11k and 15k, the time to reach the gain plateau is 5 hours and 2.5 hours respectively as shown in Figure~\ref{fig:Pol_10k_6k}. From this measurement, we conclude that the gain variation due to the polarisation can be mitigated by switching ON the chamber for several hours prior to the start of precise measurements of timing and efficiency of the detector.
\begin{figure}[h]
\begin{subfigure}[b]{0.455\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Pol_10k.png}\qquad
\caption{}
\end{subfigure}
\hspace{0.5cm}
\begin{subfigure}[b]{0.42\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Pol_15k.png}
\caption{}
\end{subfigure}
\caption{Variation of effective gain (blue squares) of the detector as a function of time having initial gain of (a) 10,000, and (b) 15,000 for triple GEM detector in Ar/$\rm CO_{2}$ = 70/30. Variation of temperature was also recorded every second and plotted (red continuous line).}
\label{fig:Pol_10k_6k}
\end{figure}
Since the polarisation effect is a global phenomena, a gain scan was performed before and after the test to disentangle the local and global fluctuations. A comparison of gain at different positions before and after the polarisation effect is shown in fig. \ref{fig:Global}. Initially, the X-ray source was placed at sector 2C according to the nomenclature in section 2 for $\sim$6 hours. The gain at the position (3D, 4B,and 5B) which were not irradiated is increased by a few tens percent with respect to the initial gain value. And this increase in gain is due to the polarisation which is a global phenomena.
\begin{figure}[h]
\centering
\includegraphics[width=0.42\textwidth]{figures/Global_Effect_Pol.png}
\caption{Comparison between the initial and final gain scans for the polarisation measurement of the triple GEM detector. The detector was irradiated at position 2C by X-ray source with an initial gain of 6,751.}
\label{fig:Global}
\end{figure}
\section{Rate capability}
The MPGD technologies were mainly introduced in response to the limited rate capability of the Multi-wire Proportional Counters (MWPC) to handle fluxes higher than several $\rm kHz/mm^2$. Reducing the amplifying structure to the microscopic scale helps quickly mitigate the effect of space charge which results in higher gains even for the high incoming particle flux \cite{RC1}.
GEM detectors are known for their stable operation even at very high particle flux. In the particular case of the triple-GEM technology, we can distinguish three different regions depending on the incoming flux of particles. The detector shows stable gain when the incoming flux is of the order of a few tens of $\rm kHz/mm^2$ (horizontal region), at a particular value of divider current (or gain) across the detector. As the flux increases to few $\rm MHz/mm^2$ (upward region) the gain increases as well. Further increase in the value of flux results in the decrease of the gain (downward region).
To check the dependency of the effective gain with the X-ray flux, known as rate capability, a collimated beam of 22.1 KeV X-rays of about 1 mm beam diameter from a silver X-ray generator has been used to produce the primary ionisation in the conversion volume.
\begin{figure}[!ht]
\begin{subfigure}[b]{0.46\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/RC_3D_Flux.png}\qquad
\caption{}
\end{subfigure}
\hspace{0.5cm}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/RC_1.png}
\caption{}
\end{subfigure}
\caption{(a) Flux provided by the X-ray source using different layers of Copper attenuators vs. the X-ray source supply current, and (b) Rate capability for triple GEM detector operated at a nominal effective gain of approximately 6.5$\rm k$ in Ar/$\rm CO_{2}$ = 70/30.}
\label{fig:RC}
\end{figure}
To estimate the effective gain as a function of the particle flux, the X-ray was placed on a particular sector to measure the amplified detector current. The X-ray flux was adjusted by changing the X-ray tube power or by attenuators. The flux delivered to the detector was calculated by taking into account the X-ray rate measured by the discriminator and the known diameter of the collimator. Since, for the higher rate, discriminator starts to saturate due to pile up, the interaction rate on the detector was measured using the copper attenuators. Once the attenuation factor is known, the interaction rate can be extrapolated to obtain the rate without attenuator. Figure~\ref{fig:RC}(a) shows the estimated flux for different power of the X-ray source. Figure~\ref{fig:RC}(b) shows the measurement of rate capability at the initial detector gain of 6800. The detector gain is stable from lowest flux used up to about 50 $\rm kHz/mm^2$. For higher fluxes, up to approximately 0.4 $\rm MHz/mm^2$, the effective gain increases as a function of flux. Further increasing the flux results in a decrease of the effective gain.
\begin{figure}[!ht]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/DiffColl.png}\qquad
\caption{}
\end{subfigure}
\hspace{0.5cm}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/SameColl.png}
\caption{}
\end{subfigure}
\caption{Dependence of effective gain as a function of flux (a) having same initial gain but different collimator, and (b) different initial gain but same collimator for triple GEM detector in Ar/$\rm CO_{2}$ = 70/30.}
\label{fig:Comb_RC}
\end{figure}
The rate capability studies were further extended by using two different collimators in front of the X-ray source, and the result is shown in Figure~\ref{fig:Comb_RC}(a) with an initial detector gain of 6,500 for both collimators. By changing the collimator from 1 $\rm mm$ to 2 $\rm mm$, the area of the sector under irradiation increases 4 times which causes a steeper increase in the effective gain. The rate capability measurement was performed for initial gains of 6500 and 9800 with 2 mm collimator, as shown in Figure \ref{fig:Comb_RC}(b). The observed change for different initial gain is related to the charge density in the detector. And higher nominal gain leads to the appearance of the observed effects at lower particle fluxes with the increase being steeper while a decreased gain will shift the effect towards higher fluxes. This is due to the fact that the field distortion depends upon the number of ions generated and accumulated in the detector. For a higher nominal effective gain of 9800, the number of ions in the detector volume is larger than compared to a detector operated at a gain of 6500.
\section{Summary}
A study has been carried out to investigate the gain stability and rate capability of the triple GEM detector assembled using commercially manufactured GEM foils. During the high voltage scan of the detector, no hysteresis was observed while a significant effect was observed in the effective gain measurement during ramping up and down the divider current. The short term stability of the GEM foil was measured and a direct correlation has been established with the initial gain of the triple GEM detector. For an initial gain of 6.8K, the plateau observed in 6 hr was found to have a percentage difference of $\sim$29\% with respect to the initial gain. However, increasing the divider current sees an increase in the initial gain as the plateau effect has been observed much earlier. Also, the measurement of rate capability of the triple GEM detector used shows stable gain up to flux of about 50 $\rm kHz/mm^2$ with 1 mm collimator. A correlation between the rate capability with respect to the different initial gain as well as different collimators placed in front of X-ray has also been performed and results compared. Using a collimator of relatively bigger diameter causes a stepper increase in the gain and varying the initial gain results in the occurrence of the effect earlier due to charge density.
\section{Conclusions}
Micropack has been successful in manufacturing the small-area Double mask GEM foils. However, before these foils can be utilised for various applications, it is important to characterize these foils and the detectors assembled from them for various properties. In this study, we tested these foils for their short term stability and the rate capability.
The polarization and rate capability of detectors obtained using commercially manufactured foils in India is similar to the detectors built with CERN made foils \cite{RC_Comp}. Therefore, these foils can be used for various purposes and the technology can be exploited in interdisciplinary applications such as medical, cultural heritage studies, muography, etc. Currently, Micropack Pvt. Ltd. is one of the candidates to build the foils for the smallest modules of GE2/1 size detectors as well as for the ME0 \cite{TDR} CMS muon chamber upgrade. The studies reported in this manuscript are a step in the direction of utilising these foils in the CMS experiment and elsewhere.
\section{Acknowledgements}
We would like to acknowledge the funding agency, Department of Science and Technology (DST), New Delhi (grant nos. SR/MF/PS-02/2014-DUA(G) and SB/FTP/PS-165/2013) for providing financial support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,808 |
\section{Our Approach}
\label{ourapp}
Our framework for question generation consists of a generator and an evaluator. From the reinforcement learning (RL) point of view, the generator is the \textit{agent} and the generation of the next word is an \textit{action}. The probability of decoding a word $P_\theta(word)$ gives a stochastic \textit{policy}. On every token that is output, an evaluator assigns a reward for the output sequence predicted so far using the current policy of the generator. Based on the reward assigned by the evaluator, the generator updates and improves its current policy. Let us denote the reward (\textit{return}) at time step $t$ by $r_t$. The cumulative reward, computed at the end of the generated sequence is represented by $R = \sum_{t=0}^T r_t$.
The goal of our framework is to determine a generator (policy) that maximizes the expected return:
\begin{equation}
\label{rlloss}
Loss_{RL}(\theta) = - \mathrm{E}_{P_\theta(Y_{0:T}|\textbf{X})} \sum\limits_{t=0}^{T}r_t(Y_t;\textbf{X},Y_{0:t-1})
\end{equation}
where $X$ is the current input and $Y_{0:t-1}$ is the predicted sequence until time $t-1$ and $\theta$ is the trainable model parameter. This supervised learning framework allows us to directly optimize task-specific evaluation metrics ($r_t$) such as BLEU.
The generator is a sequence-to-sequence model, augmented with (i) an encoding for the potentially best pivotal answer, (ii) the copy mechanism~\cite{gu2016incorporating} to help generate contextually important words, and (iii) the coverage mechanism~\cite{DBLP:conf/acl/TuLLLL16} to discourage word repetitions. The evaluator provides rewards to fine-tune the generator. The reward function can be chosen to be a combination of one or more metrics.
The high-level architecture of our question generation framework is presented in Figure~\ref{arch}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{RL-arch}
\caption{Our generator-evaluator framework for question generation. $p_{cg}$ is the probability which determines whether to copy a word from source text or sample it from vocabulary distribution.}
\label{arch}
\end{figure}
\subsection{Generator}
\label{gen}
Similar to AutoQG~\cite{kumarpakdd2018}, we employ attention and boundary pointer network to identify pivotal answer spans (most important answer spans in the text to ask question about) in the input sentence. The generator then takes as input the sequence of words in the sentence, each augmented with encoding of most probable pivotal answer, along with a set of linguistic features such as POS tag, NER tag, {\it etc.} At each step, the generator outputs a word with the highest probability, to eventually produce a word sequence. Additionally, as we will see, the generator employs copy and coverage mechanisms.
\paragraph{Sentence Encoder:} Each word in the input text is fed sequentially into the encoder along with its linguistic features as well as with the encoded pivotal answer (identified by the boundary pointer network).
Our encoder is a two-layer bidirectional LSTM network, consisting of
$\overrightarrow{h_t} = \overrightarrow{LSTM_2}(x_t,\overrightarrow{h_{t-1}})$ and
$\overleftarrow{h_t} = \overleftarrow{LSTM_2}(x_t,\overleftarrow{h_{t-1}})$,
which generates a sequence of hidden states.
Here $x_t$ is the given input word at time step $t$, and $\overrightarrow{h_t}$ and $\overleftarrow{h_t}$ are the hidden states at time step $t$ for the forward and backward passes respectively.
\paragraph{Question Decoder:} Our question decoder is a single-layer LSTM network, initialized with the state $s= [\overrightarrow{h_t}; \overleftarrow{h_t}]$, which is concatenation of hidden state from forward and backward passes.
We also model the attention~\cite{bahdanau2014neural} distribution over words in the source text.
We calculate the attention ($a^t_i$) over the $i^{th}$ source word as $a^t_i = softmax(e^t_i)$, where
\begin{align}
\label{att}
e^t_i &= v^ttanh(W_{eh}h_i+W_{sh}s_t+b_{att})
\end{align}
Here $v^t$, $W_{eh}$, $W_{sh}$ and $b_{att}$ are model parameters to be learned, and $h_i$ is the concatenation of forward and backward hidden states of the encoder.
We use this attention $a_i^t$ to generate the \emph{context vector} $c_t^*$ as a weighted sum of encoder hidden states: $c_t^* = \sum_i a_i^th_i$. We further use the $c_t^*$ vector to obtain a probability distribution over the words in the vocabulary as: $P = sofmax(W_v[s_t,c_t^*]+b_v)$,
where $W_v$ and $b_v$ are model parameters. Thus during decoding, the probability of a word is $P(qword)$.
During the training process for each timestamp, the loss is calculated as $L_t = -\log P(qword_t)$. The loss associated with the generated question is:
\begin{equation}
Loss = \frac{1}{T}\sum_{t=0}^{T}L_t = -\frac{1}{T}\sum_{t=0}^{T} \log P(qword_t)
\end{equation}
\subsubsection{The Copy and Coverage Mechanisms:}
\label{cnc}
The copy mechanism facilitates the copying of important entities and words from the source sentence to the question.
We calculate $p_{cg} \in [0,1]$ as the decision of a binary classifier that determines whether to generate (sample) a word from the vocabulary or to copy the word directly from the input text, based on attention distribution $a_i^t$:
\begin{equation}
p_{cg} = sigmoid(W_{eh}^Tc_t^*+W_{sh}^Ts_t+W_{x}x_t+b_{cg})
\end{equation}
Here $W_{eh}$, $W_{sh}$, $W_{x}$ and $b_{cg}$ are trainable model parameters.
The final probability of decoding a word is specified by the mixture model:
\begin{equation}
\label{pd}
p^*(qword) = p_{cg}\sum_{i:w_{i}=qword} a_i^t + (1-p_{cg})p(qword)
\end{equation}
Where $p^*(qword)$ is the final distribution over the union of the vocabulary and the input sentence.
As discussed earlier, Equation (\ref{pd}) addresses the rare words issue, since a word not in vocabulary will have probability $p(qword)=0$. Therefore, in such cases, our model will replace the {$<$unk$>$} token for out-of-vocabulary words with a word in the input sentence having the highest attention obtained using attention distribution $a_i^t$.
To discourage meaningless multiple repetitions of words in the question (as illustrated in row 3 of Table~\ref{tab:sample}), we maintain a word coverage vector ($wcv$) for the words already predicted as the sum of all the attention distributions ranging over timesteps $0$ until $t-1$. Specifically, at time step $t$, $wcv = \sum^{t-1}_{t'=0} a^{t'}$.
No word is generated before timestep $0$, and hence \textit{wcv} will be a zero vector then.
After storing the word coverage vector until $t-1$, while attending to the next word, we will need to inform our attention mechanism about words covered until then. Hence, equation (\ref{att}) is now modified to be:
\begin{equation}
e^t_i = v^ttanh(W_{wcv}wcv_i^t+W_{eh}h_i+W_{sh}s_t+b_{att})
\end{equation}
Here $W_{wcv}$ are trainable parameters that inform the attention mechanism about words that have been previously covered while choosing to attend over the next word.
Following the incorporation of the copy and coverage mechanism in our generator, the generator's final loss function will be:
\begin{equation}
\label{finalloss}
Loss_{copy+cov}= \frac{1}{T}\sum_{t=0}^{T}\log P^*(w_t)-\lambda_cL_{cov}
\end{equation}
where $\lambda_c$ is the coverage hyperparameter and the coverage loss $L_{cov}$ is defined as: $L_{cov}=\sum_i min(a_i^t,wcv_i^t)$ \\
We note that this cross-entropy based loss function still does not include task-specific metrics such as BLEU that were motivated earlier.
We employ an evaluator to refine the model pre-trained on this loss function to directly optimize the task specific reward. We also empirically show that the refinement of maximum likelihood models using task-specific rewards such as BLEU improves results considerably. In the next subsection we describe our evaluator.
\subsection{Evaluator}
\label{eval}
The evaluator fine-tunes the parameters of the generator network by optimizing task-specific reward functions through policy gradient. It takes as input the predicted sequence and the gold sequence, evaluates a policy, and returns a reward (a score between $0$ and $1$) that reflects the quality of the question generated. For question generation, the choice of reward functions include task-specific metrics BLEU, GLEU and ROUGE-L~\cite{du2017learning,kumarpakdd2018}, as well as the decomposable attention~\cite{parikh2016decomposable} described below. More importantly, we present two new reward functions that are specifically designed for question generation, QSS and ANSS, for the conformity of questions and answers respectively.
Combining Equation (\ref{finalloss}) with a reward function $R$ (BLEU, GLEU, ROUGE, DAS, QSS and ANSS), we obtain the overall loss function using the expected reward objective as follows:
\begin{align}
L_{overall} = & \alpha Loss_{copy+cov} + \beta \sum\limits_{t=0}^T Loss_{RL}(\theta)
\label{overall_loss}
\end{align}
where $Loss_{RL}(\theta)$ is reinforcement loss using expected reward (refer to equation \ref{rlloss}), $\mathcal{Y}$ is a set of sequences sampled from the final distribution, and $\alpha$ and $\beta$ are tunable hyperparamters.
\subsubsection{Decomposable attention based evaluator}
\label{das}
The use of a lexical similarity based reward function such as BLEU or ROUGE does not provide the flexibility to handle multiple possible versions of the ground truth. For example, the questions \textit{``who is the widow of ray croc?''} and \textit{``ray croc was married to whom?''} have almost the same meaning, but due to word order mismatch with the gold question, at most one of them can be rewarded using the BLEU score at the cost of the other(s). Empirically, we find this restriction leading to models that often synthesize questions with poor quality.We therefore, design a novel reward function, a decomposable attention~\cite{parikh2016decomposable} based similarity scorer (DAS).
Denoting by $\hat{q}$ a generated question and by $q$ the ground-truth question, we compute a cross attention based similarity using the following steps:
\paragraph{Cross Attention:} The generated question $\hat{q}$ and the ground-truth question $q$ are inter-attended as:
\begin{equation}
\begin{split}
\hat{q}^*_i &= \sum\limits_{j=0}^{L_{q}} a_{ji}e(q_j),\ a_{ji} = \frac{\exp (e(\hat{q}_i)^Te(q_j))}{\sum_{k=0}^{L_{\hat{q}}}\exp(e(\hat{q}_i)^Te(q_k))}, \\
q^*_j &= \sum\limits_{i=0}^{L_{\hat{q}}} b_{ji}e(\hat{q}_i),\ b_{ji} = \frac{\exp(e(\hat{q}_i)^Te(q_j))}{\sum_{k=0}^{L_{q}}\exp(e(\hat{q}_k)^Te(q_j))}
\end{split}
\end{equation}
where $e(.)$ is the word embedding of dimension size $d$, $\hat{q}^*$ is the cross attention vector for a generated question $\hat{q}$, and $q^*$ is the cross attention vector for a question $q$ in the ground truth.
\paragraph{Comparison:} Each n-gram $\hat{q}_i$ in the generated question (through its embedding $e(\hat{q}_i)$) is compared with its associated cross-attention vector $\hat{q}^*$ using a feed forward neural network $N_1$. Similarly, each n-gram $q_j$ in the ground-truth question (through its embedding $e(q_j)$) is compared with its associated attention vector $q^*$ using another network $N_2$ having the same architecture as $N_1$. The motivation for this comparison is that we would like to determine the soft alignment between n-grams in the generated question and the gold question. As an illustration, while comparing the gold question ``\emph{why do rockets look white?}'' with a generated question ``\emph{why are rockets and boosters painted white?}'', we find that an n-gram ``\emph{rockets and boosters}'' is softly aligned to ``\emph{rockets}'' while ``\emph{look}'' is softly aligned to ``\emph{painted}''.
\begin{equation}
\bm{\hat{q}_{1,i}} = N_1([e(\hat{q}_i),\hat{q}^*]),\ \bm{q_{2,j}} = N_2([e(q_j),q^*])\label{fm:das}
\end{equation}
where $\bm{\hat{q}_{1,i}}$ and $ \bm{q_{2,j}}$ are vectors containing comparison scores of aligned phrases in generated question and gold question respectively and $N_1$ and $N_2$ are the feed forward neural nets.
\paragraph{Matching Score:}
The vectors \bm{$\hat{q}_{1,i}$} and \bm{$q_{2,j}$} are aggregated over each word or phrase in the predicted question and gold question respectively before feeding them to a linear function ($L$):\\
$DAS = L(\sum_{i=1}^{L_q}\bm{\hat{q}_{1,i}},\sum_{j=1}^{L_{\hat{q}}}\bm{q_{2,j}})$\\
This matching score between the predicted question and the gold question is the reward returned by the decomposable attention based evaluator.
\subsubsection{QG quality specific reward functions}
\label{qgquality}
We introduce two new reward functions that specifically designed to evaluate the conformity of the generated question (QSS) and answer (ANSS) against the ground truth.
\textbf{Question sentence overlap score (QSS): } This reward function is specific to QG. We compute the sentence overlap score as the number of common n-grams between predicted question and the source sentence. This reward ensures that generated question is relevant to the given sentence.
Thus, if $precision_n(s,q)$ computes the $n-$gram precision match between sentence and question,\\
$QSS=(\prod_{i=1}^n precision_i(sentence, question))^{\frac{1}{n}}$
\textbf{Predicted and encoded answer overlap score (ANSS):} In order to ensure that the generated question is about the pivotal answer/ground truth answer we calculate answer overlap score. Answer overlap score is the number of common n-grams between the encoded answer and the answer predicted (ans$_{qa}$) for the generated question using the best performing question answering model over SQuAD\footnote{\url{https://github.com/huggingface/pytorch-pretrained-BERT}}\\
$ANSS=(\prod_{i=1}^n precision_i(\text{ans$_{qa}$}, pivotal\_answer))^{\frac{1}{n}}$
\section{Conclusion}
\label{conc}
We presented a novel, holistic treatment of question generation (QG) using a generator-evaluator framework. Our generator provisions for explicitly factoring in question syntax and semantics, identifies pivotal answers, recognizes contextually important words and avoids meaningless repetitions. Our evaluator allows us to directly optimize for conformity towards the structure of ground-truth question(s). We propose two novel reward functions account for conformity with respect to ground-truth questions and predicted answers respectively. In conjunction, the evaluator makes use of task-specific scores, including BLEU, GLEU, ROUGE-L, and decomposable attention (DAS) that are naturally suited to QG and other seq2seq problems. Experimental results on automatic evaluation and human evaluation on the standard benchmark dataset show that our framework, especially with the incorporation of the new reward functions, considerably outperforms state-of-the-art systems.
\section{Experimental Setup}
\label{exp}
In this section, we present our evaluation framework on the publicly available SQuAD~\cite{rajpurkar-EtAl:2016:EMNLP2016} dataset. We first explain various reward functions employed in our experiments. We then describe our baseline and the evaluation methods.
\paragraph{Reward Functions:}
We experimented with the five reward functions discussed in Section \ref{eval}: (1) BLEU, (2) GLEU, (3) ROUGE-L, (4) DAS, and (5) the QG-specific reward QSS+ANSS.
In our experiments we considered BLEU for up to 4-grams. For the GLEU score, we recorded all sub-sequences of up to $4$-grams.
\paragraph{Baselines and Evaluation Methods:}
We reimplemented two state-of-the-art question generation models as baselines for comparison: L2A~\cite{du2017learning} and AutoQG~\cite{kumarpakdd2018}. A direct (and fair) comparison with another recent technique, NQG$_{LC}$~\cite{song2018leveraging}, is not feasible, as unlike us, NQG$_{LC}$ requires ground-truth answers, whereas both AutoQG and our model predict pivotal answers. L2A does not consider answers. Moreover, their context (input is sometimes more than one sentence) is different also the train/test split is different from ours. Hence, we only report the original numbers reported in their paper. We also did not perform human evaluation on NQG$_{LC}$ as their source code has not been made available.
We also use an existing implementation of a recent RL-based abstractive summarization technique \cite{paulus2018} to train baseline models SUM$_{\textnormal{BLEU}}$\ (with BLEU as reward function) and SUM$_{\textnormal{ROUGE}}$\ (with ROUGE as reward function). This comparison studies the effectiveness of state-of-the-art abstractive summarization techniques applied to question generation as-is, as the two are conceptually similar tasks.
We report automatic and human evaluation results on eight variants of our model, each of which is equipped with the copy and coverage mechanism, the pointer network, as well as one of the four reward functions: BLEU, GLEU, ROUGE-L, DAS or one of the four rewards in combination with QG quality specific rewards (QSS+ANSS). Hence, our models are named GE$_{\textnormal{BLEU}}$, etc.
For automatic evaluation, we employ BLEU, ROUGE-L and METEOR, which are standard evaluation measures used to evaluate sequence prediction tasks. We use the evaluation scripts released by \cite{chen2015microsoft} that was originally used to evaluate the image captioning task.
We also performed human evaluation to further analyze the quality of questions generated for their syntactic correctness, semantic correctness and relevance. Syntactic correctness measures the grammatical correctness of a generated question, semantic correctness measures meaningfulness and naturalness of the question, and relevance measures how relevant the question is to the text. We perform human evaluation for each model on a randomly selected subset of 100 sentences. Each of the three judges is presented the 100 sentence-question pairs for each model and asked for a binary response on each quality parameter. The responses from all the judges for each parameter is then averaged for each model.
\begin{table*}[htb]
\begin{center}
\scalebox{0.8}{
\begin{tabular}{| l | c | c |c | c | c | c | }
\hline
Model& BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 &METEOR &ROUGE-L \\ \hline
L2A~\cite{du2017learning} &43.21 (43.09) &24.77 (25.96) &15.93 (17.50) &10.60 (12.28) &16.39 (16.62) &38.98 (39.75)\\
AutoQG~\cite{kumarpakdd2018} &44.68 (46.32) &26.96 (28.81) &18.18 (19.67) &12.68 (13.85) &17.86 (18.51) &40.59 (41.75) \\
NQG$_{LC}$~\cite{song2018leveraging} & - & - & - & - (13.98) & - (18.77) & - (42.72)\\
SUM$_{\textnormal{BLEU}}$~\cite{paulus2018} &11.20- &3.50- &1.21- &0.45- &6.68- & 15.25-\\
SUM$_{\textnormal{ROUGE}}$~\cite{paulus2018} &11.94- &3.95- &1.65- &0.082- &6.61- &16.17- \\
\hline
GE$_{\textnormal{BLEU}}$ &46.84 &29.38 &20.33 &14.47 &19.08 &41.07\\
\rowcolor{Gray}
GE$_{\textnormal{BLEU+QSS+ANSS}}$ &46.59 &29.68 &20.79 &15.04 &19.32 &41.73\\
GE$_{\textnormal{DAS}}$ &44.64 &28.25 &19.63 &14.07 &18.12 &42.07\\
\rowcolor{Gray}
GE$_{\textnormal{DAS+QSS+ANSS}}$ &46.07 & 29.78 & 21.43 & 16.22 & 19.44& 42.84\\
GE$_{\textnormal{GLEU}}$ & 45.20&29.22&20.79 & 15.26&18.98 &43.47\\
\rowcolor{Gray}
GE$_{\textnormal{GLEU+QSS+ANSS}}$ &47.04 &30.03 &21.15 & 15.92 & 19.05 & 43.55 \\
GE$_{\textnormal{ROUGE}}$ &47.01 &30.67 &21.95 &16.17 &19.85 &43.90\\
\rowcolor{Gray}
GE$_{\textnormal{ROUGE+QSS+ANSS}}$ & \textbf{48.13} & \textbf{31.15} & \textbf{22.01} & \textbf{16.48} & \textbf{20.21}& \textbf{44.11}\\
\hline
\end{tabular}}
\end{center}
\caption{Experimental results on the test set on automatic evaluation metrics. Best results for each metric (column) are \textbf{bolded}. The numbers in parentheses for L2A, AutoQG and NQG$_{LC}$ are those from the best models reported in their respective original papers. The slight difference of up to 1.7\% from our reproduced numbers can be attributed to reimplementation and different versions of various libraries used. Models with new QG-specific reward functions (QSS+ANSS) are highlighted in gray for easy comparison.}
\label{results}
\end{table*}
\section{Introduction}
\label{Intro}
Recent years have seen rapid development in conversational systems, as represented by widely-used personal assistants such as Siri, Cortana, and Alexa, as well as a myriad of online chatbots.
Asking intelligent and relevant questions is a very important yet challenging tasks for such systems.
Question generation (QG) is the task of generating syntactically correct, semantically sound and relevant questions from various input formats such as text, a structured database or a knowledge base \cite{mannem2010question}.
Recent neural network based techniques~\cite{du2017learning,song2018leveraging,kumarpakdd2018,zhao2018paragraph} have achieved remarkable success on QG.
These methods typically falls under the sequence-to-sequence (Seq2Seq) setup, employing an RNN-based architecture and additional features such as answering encoding, copy and coverage mechanisms.
These state-of-the-art models are not without their disadvantages. They usually are trained to minimise the cross-entropy loss, which ignores the important sequence information. Moreover, as the training set and the test set may not have the same word distribution, the use of cross-entropy loss may make the training process brittle.
In this paper, we first present a framework in which a {\em generator} mechanism that is employed for generating a question-answer pair invokes or pulls the {\em evaluator} mechanism that is employed for evaluating the generated pair. Our clearly delineated {\em generator}-{\em evaluator} framework lets us (a) easily incorporate several best practices from the above referred previous models in the {\em generator} while (b) also letting us employ in the {\em evaluator}, other complex non-decomposable rewards that are consistent with performance measures (such as BLEU and ROUGE) on test data. We also propose some novel reward functions that {\em evaluate} the syntax of the question and semantics of the question-answer pair in its entirety.
More specifically, since the generated question is in anticipation of some specific answer, we find it most natural to incorporate candidate answer generation (using Pointer Networks) alongside QG right in our generator module, so that the evaluator can optionally take into cognizance the conformity of the generated answer to the ground-truth answer, along with text conformity. Likewise, we also incorporate copy and coverage mechanisms for QG into the generator module so that they can be specifically trained by leveraging a suite of holistically designed and structure-sensitive reward functions in the evaluator module.
\begin{table*}[htb]
\footnotesize
\centering
\scalebox{0.8}{
\begin{tabular}{|l|l|l|}
\hline
\multicolumn{3}{|c|}{\parbox[t][20pt][t]{.96\linewidth}{\textbf{Text:}
``new york city traces its roots to its 1624 founding as a trading post by colonists of the dutch republic and was named new amsterdam in 1626 .''}}\\
\hline
\textbf{Row} & \textbf{Model} & \textbf{Question generated} \\
\hline
1 & Seq2Seq model optimized on vanilla (cross entropy) loss without answer prediction & in what 1624 did new york city traces its roots ?\\
\hline
2 & Seq2Seq model optimized on vanilla (cross entropy) loss with answer prediction & what year was new york named ?\\
\hline
3 & Copy aware Seq2Seq model & what year was new new amsterdam named ? \\ \hline
4 & Coverage and copy aware Seq2Seq model & in what year was new amsterdam named ? \\ \hline
5 & Seq2Seq model optimized on BLEU (using RL) & what year was new york founded ? \\ \hline
\end{tabular}}
\caption{Sample text and questions generated using variants of our model.}\label{tab:sample}
\end{table*}
\begin{table}[htb]
\footnotesize
\centering
\scalebox{0.8}{
\begin{tabular}{|l|l|p{6.1cm}|}
\hline
\multicolumn{3}{|l|}{\parbox[t][48pt][t]{1.2\linewidth}{\textbf{Text:}
``even with the five largest cities in sichuan suffering only minor damage from the quake , some estimates of the economic loss run higher than us \$ 75 billion , making the earthquake one of the costliest natural disasters in chinese history .''}}\vspace{-12pt}\\
\multicolumn{3}{|l|}{\textbf{Expected answer:} five}\\
\hline
\textbf{Row} & \textbf{Model} & \textbf{Question generated} \\
\hline
1 & GE$_{\textnormal{BLEU}}$ & how much did it making for the earthquake of the economic ? \\ \hline
2 & GE$_{\textnormal{BLEU+QSS+ANSS}}$ & how many largest cities in sichuan experience only minor damage from the quake ? \\ \hline
3 & GE$_{\textnormal{DAS}}$ & how many cities were in sichuan ? \\ \hline
4 & GE$_{\textnormal{DAS+QSS+ANSS}}$ & how many largest cities in sichuan suffering only minor damage from the quake ? \\
\hline
4 & GE$_{\textnormal{ROUGE}}$ & how much did the economic loss run in sichuan ? \\ \hline
5 & GE$_{\textnormal{ROUGE+QSS+ANSS}}$ & what is the largest cities in sichuan ? \\
\hline
\end{tabular}}
\caption{Sample text and questions generated using different reward functions, with and without our new QG-specific rewards QSS+ANSS.}\label{tab:samplerl}
\end{table}
\subsubsection*{The Generator}
In Table~\ref{tab:sample}, in rows 1 through 4, we illustrate through examples, the incremental benefits of introducing answer prediction and the copy and coverage mechanisms \cite{see2017get} in the generator. The evaluator associated with the corresponding three generator models employs the conventional and simplistic cross-entropy loss. The motivation for answer prediction in the generator module is obvious and will be further discussed in Section 2.1. In row 3 we illustrate the influence of our copy mechanism, where a rare phrase `new amsterdam' has been rightly picked up in association with the name of the city.
We however note that in row 3, the word `new' has been erroneously repeated twice, since an encoder-decoder based model could generate questions with meaningless repetitions
We introduce a mechanism for discouraging such repetitions in our generator by quantitatively emphasizing the \emph{coverage} of sentence words while decoding. Row 4 shows the improved and relevant question generated by our model trained by incorporating both the copy and coverage mechanisms.
\subsubsection*{Evaluator}
In row 5 of Table~\ref{tab:sample}, we observe the high-quality question that is generated when the simplistic cross-entropy loss in the evaluator is replaced with the more complex and non-decomposable (across words) BLEU reward that accounts for proximity of `founded' to `new york'.
In Table~\ref{tab:samplerl}, we further illustrate the effect of employing other reward functions (described in Section 2.2) in the evaluator. As can be seen, the model that incorporates QG-specific reward functions (QSS and ANSS) generates a significantly better question when compared to the question generated without these rewards.
\noindent \textbf{Limitations of simple decomposable losses:}
A Seq2Seq model trained using a vanilla cross-entropy loss function (decomposable over words in the question) generates the question \textit{``what year was new york named ?''} (row 1 in Table~\ref{tab:sample}), which is not addressed in the sentence. The passage talks only about the founding of the city and its naming two years later. The inaccuracy of the question is possibly caused by the use of a loss that is agnostic to sequence information. In other words, given its decomposable nature, the cross-entropy loss on the ground-truth question or any of its (syntactically invalid) anagrams will be the same.
Moreover, use of the cross-entropy loss in the sequence prediction model could make the process brittle, since the model trained on a specific distribution over words is used on a test dataset with a possibly different distribution to predict the next word given the current predicted word. This creates exposure bias~\cite{ranzato2015sequence} during training, since the model is only exposed to the data distribution and not the model distribution. Thus, performance suffers due to inadequately evaluating the \emph{structure} of the generated question against the ground-truth question.
The standard metrics for evaluating the performance of question generation models such as BLEU~\cite{papineni2002bleu}, GLEU, and ROUGE-L~\cite{lin2004rouge} are based on degree of n-gram overlaps between a generated question and the ground-truth question. It would be desirable to be able to directly optimize these \emph{task-specific metrics}.
However, these n-gram based metrics do not decompose over individual words and are therefore hard to optimize.
We explicitly employ an evaluator that rewards each generated question based on its conformance to one (or more than one using decomposable attention) questions in the ground-truth set using these possibly non-decomposable reward functions. We find such learning to be a natural instance of reinforcement learning (RL)~\cite{sutton1998introduction} that allows us to use policy gradient to directly optimize task-specific rewards (such as BLEU, GLEU and ROUGE-L), which are otherwise non-differentiable and hard to optimize. In Table \ref{tab:samplerl} we illustrate questions generated using different reward functions. It can be observed that questions generated using combination of standard reward functions with reward functions specific to QG quality (QSS+ANSS) exhibit higher quality.
\paragraph{Contributions}
We summarize our main contributions as follows:
\begin{itemize}
\item A comprehensive, end-to-end \textbf{generator-evaluator framework} naturally suited for automated question generation. Whereas earlier approaches employ some mechanism for generating the question, intertwined with an evaluation mechanism, we show that these approaches can benefit from a much clearer separation of the generator of the question from its evaluator.
\item A {\em generator} founded on the \textbf{semantics} and \textbf{structure} of the question by (a) identifying target/pivotal answers (Pointer Network), (b) recognizing contextually important keywords in the answer (copy mechanism), and (c)
avoiding redundancy (repeated words) in the question (coverage mechanism).
\item An {\em evaluator} that (a) directly optimizes for conformity to the \textbf{structure} of ground-truth sequences (BLEU, GLEU, etc.), and (b) matches against appropriate questions from a set of ground-truth questions (Decomposable Attention).
\item Novel reward functions that ensure that the generated question is relevant to the text and conforms to the encoded answer.
\end{itemize}
When evaluated on the benchmark SQuAD dataset~\cite{rajpurkar-EtAl:2016:EMNLP2016}, our system considerably outperforms state-of-the-art question generation models~\cite{du2017learning,kumarpakdd2018,song2018leveraging} in automatic and human evaluation.
\section{Related Work}
\label{relwork}
Neural network-based methods represent the state-of-the-art in automatic question generation (QG) from text. Motivated by neural machine translation, Du et al \shortcite{du2017learning} proposed a sequence-to-sequence (Seq2Seq) architecture for QG. Kumar et al \shortcite{kumarpakdd2018} proposed to augment each word with linguistic features and encode the most relevant \emph{pivotal answer} to the text while generating questions. Similarly, Song et al \shortcite{song2018leveraging} encode ground-truth answers (given in the training data), use the copy mechanism and additionally employ context matching to capture interactions between the answer and its context within the passage. They encode ground truth answer for generating questions which might not be available for test set in contrast we train a Pointer Network based model to predict the pivotal answer to generate question about.
\par Very recently deep reinforcement learning has been successfully applied to natural language generation tasks such as abstractive summarization~\cite{paulus2018,N18-1150} and dialogue generation~\cite{D16-1127}. In summarization, one generates and paraphrases sentences that capture salient points of the text. On the other hand, generating questions additionally involves determining question type such as what, when, etc., being selective on which keywords to copy from the input into the question, leaving remaining keywords for the answer. This also requires the development of a specific probabilistic generative model. \cite{yao2018teaching} proposed generative adversarial network (GAN) framework with modified discriminator to predict question type. Recently Fan et al \shortcite{fan2018reinforcement} proposed a bi-discriminator framework for visual question generation. They formulate the task of visual question generation as a language generation task with some linguistic and content specific attributes.
\section{Results and Discussion}
\label{rnd}
We show and compare results on automatic evaluation in Table~\ref{results}. Note the numbers in parentheses for L2A~\cite{du2017learning}, AutoQG~\cite{kumarpakdd2018}, and NQG$_{LC}$~\cite{song2018leveraging} are those reported in their original papers. The slight difference of up to 1.7\% in the original and reproduced numbers can be attributed to reimplementation and different versions of various libraries used. As can be seen, all our eight models outperform L2A and AutoQG on all evaluation metrics. Two of our models, GE$_{\textnormal{GLEU}}$\ and GE$_{\textnormal{ROUGE}}$, also outperform NQG$_{LC}$. Hence, using evaluation metrics as the reward function during reinforcement based learning improves performance for all metrics. We also observe that GE$_{\textnormal{ROUGE+QSS+ANSS}}$, the model reinforced with ROUGE-L (that measures the longest common sequence between the ground-truth question and the generated question) as the reward function in combination with QG quality specific rewards(QSS+ANSS), is the best performing model on all metrics, outperforming existing baselines considerably. For example, it improves over AutoQG on BLEU-4 by 29.98\%, on METEOR by 13.15\%, and on ROUGE-L by 8.67\%.
In Table \ref{heresults} we present human evaluation results for the models evaluated on three quality parameters (a) syntactic correctness, (b) semantic correctness, and (c) relevance.
Consistent with automatic evaluation results shown in Table~\ref{results}, seven of our eight models outperform the two baselines, with GE$_{\textnormal{DAS+QSS+ANSS}}${} being the best model on syntactic correctness and semantic correctness quality metrics, outperforming all the other models by a large margin. However, model GE$_{\textnormal{BLEU+QSS+ANSS}}${} generates highly relevant questions and is the best model on relevance metrics.
It is noteworthy that for each of our models (e.g.\ GE$_{\textnormal{BLEU}}$), adding QG-specific rewards (e.g.\ GE$_{\textnormal{BLEU+QSS+ANSS}}$) significantly improves question quality in human evaluation, even though there is less noticeable improvements in automatic evaluation. This clearly demonstrates the effectivess of our new QG-specific reward functions.
We measure inter-rater agreement using Randolph's free-marginal multirater kappa~\cite{randolph2005free}. This helps in analyzing level of consistency among observational responses provided by multiple judges. It can be observed that our quality metrics for all our models are rated as \emph{moderate agreement}~\cite{viera2005understanding}.
\subsection{Analyzing Choice of Reward Function}
BLEU\cite{papineni2002bleu} measures precision and ROUGE\cite{lin2004rouge} measures recall, we believe that cross-entropy loss was already accounting for precision to some extent and using it in conjunction with ROUGE (which improves recall) therefore gives best performance.
\begin{table}[htb]
\begin{center}
\scalebox{0.8}{
\begin{tabular}{| l | c | c | c | c |c | c | }
\hline
\multirow{2}{*}{Model}& \multicolumn{2}{c|}{Syntax} & \multicolumn{2}{c|}{Semantics} & \multicolumn{2}{c|}{Relevance} \\ \cline{2-7}
& Score & Kappa & Score & Kappa & Score & Kappa \\
\hline
L2A &39.2 &0.49 & 39 &0.49 &29 & 0.40 \\
AutoQG &51.5 &0.49 &48 &0.78 &48 & 0.50 \\
\hline
GE$_{\textnormal{BLEU}}$ &47.5 &0.52 &49 &0.45 &41.5 &0.44 \\
\rowcolor{Gray}
GE$_{\textnormal{BLEU+QSS+ANSS}}$ & 82 &0.63 &75.3 &0.68 & \textbf{78.33}& 0.46\\
GE$_{\textnormal{DAS}}$ &68 &0.40 &63 & 0.33 &41 & 0.40 \\
\rowcolor{Gray}
GE$_{\textnormal{DAS+QSS+ANSS}}$ & \textbf{84} &0.57 & \textbf{81.3} &0.60 &74 & 0.47 \\
GE$_{\textnormal{GLEU}}$ & 60.5 &0.50 &62 & 0.52 &44 & 0.41 \\
\rowcolor{Gray}
GE$_{\textnormal{GLEU+QSS+ANSS}}$ & 78.3 &0.68 & 74.6 &0.71 & 72& 0.40 \\
GE$_{\textnormal{ROUGE}}$ & 69.5& 0.56 & 68 & 0.58 & 53 & 0.43 \\
\rowcolor{Gray}
GE$_{\textnormal{ROUGE+QSS+ANSS}}$ & 79.3 &0.52 & 72 &0.41 & 67 & 0.41 \\
\hline
\end{tabular}}
\end{center}
\caption{Human evaluation results (column ``Score'') as well as inter-rater agreement (column ``Kappa'') for each model on the test set. The scores are between 0-100, 0 being the worst and 100 being the best. Best results for each metric (column) are \textbf{bolded}. The three evaluation criteria are: (1) syntactically correct ({Syntax}), (2) semantically correct ({Semantics}), and (3) relevant to the text ({Relevance}). Models with new QG-specific reward functions (QSS+ANSS) are highlighted in gray for easy comparison.}
\label{heresults}
\end{table}
DAS calculates semantic similarity between generated question and the gound-truth question.
As discussed in section \ref{das} DAS will give high reward even though the generated question has low BLEU score. Thus, the performance of the model on automatic evaluation metrics does not improve with DAS as the reward function, though the quality of questions certainly improves. Further, ROUGE in conjunction with the cross entropy loss improves on recall as well as precision whereas every other combination overly focuses only on precision.
Error analysis of our best model reveals that most errors can be attributed to intra-sentence dependencies such as co-references, concept dependencies {\em etc.} In a camera ready version of the paper, we will share link to a detailed report containing extensive experiments that include ablation tests. Also link to the source code will be provided then.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,208 |
12:00 AM, November 08, 2016 / LAST MODIFIED: 03:09 AM, November 08, 2016
Faraaz named for Mother Teresa Award
IANS, Mumbai
The Harmony Foundation yesterday named the braveheart of the Dhaka terror attack, Faraaz Ayaaz Hossain, for this year's Mother Teresa Memorial International Award for Social Justice.
The award will be conferred posthumously upon Faraaz, who was killed during the terror strike at a restaurant in Gulshan on July 1, said Harmony Foundation President Abraham Mathai.
"On July 1, Faraaz courageously stood up against the terrorists, refused to let his Hindu friends bear the brunt of the terrorists' ire by standing up for them, earning him the title of a 'hero'," Mathai told IANS.
Faraaz's parents will land in Mumbai to receive the award on his behalf on November 20. This is the first time the award would be conferred posthumously upon anyone in its 12th edition, he added.
The award, instituted in 2005 in the memory of Mother Teresa, is the only one in the world recognised by the institution she founded, the Missioneries of Charity, Kolkata, and the first time after she was canonised to Sainthood on September 4 by Pope Francis at the Vatican.
Some of the past recipients have been Medecines Sans Froneiters, the Dalai Lama, Malala Yousafzai, Dr Mahathir Mohammad, Baroness Caroline Cox and other global personalities or organisations, Mathai said.
Accepting the Harmony Foundation's decision, Faraaz's family said: "On behalf of my son, it is an absolute honour and a privilege to have the opportunity to accept the award. Though in deep grief, we cling onto the immense pride with which Faraaz has filled our hearts. He stood up for what was right by paying the highest cost any human can pay and we hope every day that his sacrifice does not go in vain."
Mathai said the 20-year-old Faraaz was "the obvious choice" and the world needs heroes like him for his act of selflessness and choosing to lay down his life for his friends.
"He displayed courage and bravado when faced with life and death. We consider it our most humble privilege to bestow this honour on Faraaz," Mathai said.
As part of the award ceremony, the Harmony Foundation has organised an International Conference on Combating Terrorism, which is likely to be addressed by former Afghan president Hamid Karzai, Saarc Secretary-General Arjun Thapa, former union minister Farooq Abdullah and others.
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DSCC will be tourist destination | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,309 |
package org.apache.activemq.artemis.core.security.jaas;
import javax.security.auth.Subject;
import org.apache.activemq.artemis.core.security.CheckType;
import org.apache.activemq.artemis.core.security.Role;
import org.apache.activemq.artemis.core.security.impl.SecurityStoreImpl;
import org.apache.activemq.artemis.spi.core.security.ActiveMQJAASSecurityManager;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import java.lang.invoke.MethodHandles;
import org.junit.Rule;
import org.junit.Test;
import org.junit.rules.TemporaryFolder;
import org.junit.runner.RunWith;
import org.junit.runners.Parameterized;
import java.io.UnsupportedEncodingException;
import java.net.URL;
import java.net.URLClassLoader;
import java.net.URLDecoder;
import java.nio.charset.StandardCharsets;
import java.util.Arrays;
import java.util.Collection;
import java.util.HashSet;
import java.util.Set;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertNotNull;
import static org.junit.Assert.assertTrue;
@RunWith(Parameterized.class)
public class JAASSecurityManagerTest {
private static final Logger logger = LoggerFactory.getLogger(MethodHandles.lookup().lookupClass());
@Parameterized.Parameters(name = "newLoader=({0})")
public static Collection<Object[]> data() {
return Arrays.asList(new Object[][] {{true}, {false}});
}
static {
String path = System.getProperty("java.security.auth.login.config");
if (path == null) {
URL resource = PropertiesLoginModuleTest.class.getClassLoader().getResource("login.config");
if (resource != null) {
try {
path = URLDecoder.decode(resource.getFile(), StandardCharsets.UTF_8.name());
System.setProperty("java.security.auth.login.config", path);
} catch (UnsupportedEncodingException e) {
throw new RuntimeException(e);
}
}
}
}
@Parameterized.Parameter
public boolean usingNewLoader;
@Rule
public TemporaryFolder tmpDir = new TemporaryFolder();
@Test
public void testLoginClassloading() throws Exception {
ClassLoader existingLoader = Thread.currentThread().getContextClassLoader();
logger.debug("loader: {}", existingLoader);
try {
if (usingNewLoader) {
URLClassLoader simulatedLoader = new URLClassLoader(new URL[]{tmpDir.getRoot().toURI().toURL()}, null);
Thread.currentThread().setContextClassLoader(simulatedLoader);
}
ActiveMQJAASSecurityManager securityManager = new ActiveMQJAASSecurityManager("PropertiesLogin");
Subject result = securityManager.authenticate("first", "secret", null, null);
assertNotNull(result);
assertEquals("first", SecurityStoreImpl.getUserFromSubject(result));
Role role = new Role("programmers", true, true, true, true, true, true, true, true, true, true);
Set<Role> roles = new HashSet<>();
roles.add(role);
boolean authorizationResult = securityManager.authorize(result, roles, CheckType.SEND, "someaddress");
assertTrue(authorizationResult);
} finally {
Thread.currentThread().setContextClassLoader(existingLoader);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,149 |
{"url":"http:\/\/openstudy.com\/updates\/50b23707e4b0e906b4a640ec","text":"anonymous 3 years ago Evaluate the integral using the substitution rule. (Use C for the constant of integration.) Integration of 2y\/((2+3y^2)^3) dy\n\n1. anonymous\n\n$\\int\\limits \\frac{2y}{(2+3y^2)^3} dy$ $u=2+3y^2$ $\\frac{du}{dy}=6y \\Rightarrow dy=\\frac{du}{6y}$ $\\Rightarrow \\int\\limits \\frac{2y}{(u)^3} \\frac{du}{6y}$You can do the rest\n\n2. anonymous\n\narent' the variables supposed to be the same?\n\n3. anonymous\n\ni'm not getting the answer. i thought it would be 6y\/(3(2+3y^2)^3) +C\n\n4. anonymous\n\nIt cancels to$\\frac{1}{3}\\int\\limits\\frac{du}{u^3}=\\frac{1}{3}\\frac{1}{-4u^4}$\n\n5. anonymous\n\noh i forgot to integrate. i just got so confused on the substitution. oops! :) thanks a bunch!\n\n6. anonymous\n\nyeah, it was wrong. remember that 1\/u^3 is the same as u^-3. thus the integration of u^-3 is u^-2 \/ -2 or 1\/ (-2u^2)\n\n7. anonymous\n\nSorry about that- I half misread it as $u^3$, not $\\frac{1}{u^3}$. You're right","date":"2016-06-29 20:11:55","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\": 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\": 0.9776597023010254, \"perplexity\": 4042.567127084827}, \"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-2016-26\/segments\/1466783397797.77\/warc\/CC-MAIN-20160624154957-00191-ip-10-164-35-72.ec2.internal.warc.gz\"}"} | null | null |
Q: Showing that a certain stochastic process does not have normal distributed increments Edit: Question Resolved. See below.
As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz.
In this paper he defines a stochastic process.
Let $G_1, G_2, W_t$ be independent. Here we have $G_1$, $G_2$ $\sim \mathcal{N}(0,1)$
and $W_t$ Brownian motion.
For $a\geq 0$ and $t\geq e^{-a}$ define the filtration $\mathcal{F}_t^{(a)}=\sigma(G_1,G_2,(W_s)_{0\leq s\leq a+\ln t}),$
and the process
$$X_t^{(a)} = \sqrt{t}\left(G_1\cos{W_{a+\ln t}}+ G_2\sin{W_{a+\ln t}}\right).$$
Now Oleskiewicz claims,
(1) $X_t^{(a)}$ is a continuous martingale,
(2) $X_t^{(a)}\sim \mathcal{N}(0,t)$,
(3) $X_e^{(a)}-X_1^{(a)}$ is not gaussian,
without actually proofing any of these statements.
While I managed to proof the first two statements, the proofs are rather long and involved, so I wanted to ask if I am missing something obvious.
Furthermore, I am absolutely clueless regarding the third statement.
I hope you can help me.
A: This is the answer he send me regarding these increments.
"As for the increments, $X^{(a)}_e-X^{(a)}_1=
(\sqrt{e}\cos W_{a+1}-\cos W_a)G_1+(\sqrt{e}\sin W_{a+1}-\sin W_a)G_2.$
Since the pair $(W_a, W_{a+1})$ is independent of the pair $(G_1, G_2)$, this
increment has the same distribution as
$[(\sqrt{e}\cos W_{a+1}-\cos W_a)^2+
(\sqrt{e}\sin W_{a+1}-\sin W_a)^{2}]^{1/2} G$,
where $G$ is $\mathcal{N}(0,1)$ independent of the pair $(W_a, W_{a+1})$.
Now, if $Z$ and $G$ are independent, $G$ is $\mathcal{N}(0,1)$-distributed and $ZG$ is
Gaussian then $|Z|$ must be constant a.s. -- indeed, $\mathbb{E}ZG=\mathbb{E}Z \mathbb{E}G=\mathbb{E}Z \cdot 0=0$,
so that $ZG$ is a centered Gaussian random variable; now it suffices to
notice that $\mathbb{E}G^4=(\mathbb{E}G^2)^2, \mathbb{E}(ZG)^4=3(E(ZG)^2)^2$, so that $EZ^4=(EZ^2)^2$ and
hence $Var(Z^2)=0$, i.e., $Z^2$ is constant a.s.
However, the expression standing inside $[...]^{1/2}$ obviously is not
constant a.s., as it is equal to $e+1-2\sqrt{e}\cos(W_{a+1}-W_a)$, and
$W_{a+1}-W_{a}$ is a $\mathcal{N}(0,1)$ Gaussian random variable."
It's a quite impressive and interesting approach.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,207 |
## CONTENTS
1. INTRODUCTION
2. CHAPTER ONE PRESSURE CANNING BASICS
3. CHAPTER TWO STORAGE, TROUBLESHOOTING, AND OTHER CONSIDERATIONS
4. CHAPTER THREE CLASSIC VEGETABLE RECIPES
5. CHAPTER FOUR GOING FURTHER WITH VEGETABLES
6. CHAPTER FIVE PRESSURE CANNING FRUITS
7. CHAPTER SIX PRESSURE CANNING MEAT
8. CHAPTER SEVEN SALSA, SAUCE, AND BROTH RECIPES
9. RESOURCES
10. INDEX
11. ABOUT THE AUTHOR
## Guide
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## INTRODUCTION
The glass clinks gently in the hot, soapy water as I wash my jars for this year's canning. Sitting on the counter are colanders full of tomatoes and peppers that need to be saved for the winter. Although canning involves plenty of work, it's something I look forward to every year. Just like cooking dinner, the work is rewarding because of the taste: there's nothing better than opening a jar full of flavor in the middle of winter.
The jump from canning to pressure canning comes at a different point for different people. I decided to take the plunge into pressure canning as my skills grew in the garden. My family was living on a mountain in Montana and most of my entertainment came from growing vegetables and fruit. Although I started canning jams and jellies, it became obvious I needed to be able to preserve much more—and that required pressure canning.
But that's me! Why should _you_ learn to pressure can? In short, pressure canning puts the power of food preservation at your fingertips. Whether your inspiration is the farmers' market or a treasured family recipe, you can preserve it. Pressure canning lets you preserve the foods you and your family like to eat so you can enjoy them at their peak, year-round. It's great for the spontaneous cook, who likes to have a variety of ingredients to work with. It's also wonderful for those who want convenience without sacrificing health benefits or taste. Unlike store-bought canned food, there are never any preservatives or questionable ingredients in home-canned food. And you simply can't beat the taste and texture attainable when home canning.
I enjoy teaching people the ins and outs of pressure canning. It's so much fun to walk to the garden with new canners and load up a basket with fruits and veggies. After a couple of hours prepping and canning, we all go home with jars of beautiful preserved food. It's a lesson those new to canning won't soon forget. And it's exciting to share not only a hobby, but a way of life: home cooking from your own pantry. Welcome to _Modern Pressure Canning_.
### WHAT IS THE DIFFERENCE BETWEEN PRESSURE CANNING AND WATER BATH CANNING?
Pressure canning and water bath canning are similar in that they are both ways to preserve food in your own kitchen. The general process of putting food in jars, attaching lids, heating to seal, and storing are the same. However, there are some major differences you should know.
All foods have natural acidity levels. Many fruits and tomatoes are highly acidic. These foods can be water bath canned. Or, take pickles: cucumbers (a low-acid food) in an acidic pickling solution. These can also be water bath canned.
_Water bath canning_ , or hot water canning as it's sometimes called, is the method of submerging filled jars into hot water and boiling for a specific period of time. This means the internal temperature of the food in the jars is heated to 212°F. It's a great way to start canning, and the most common way to make popular pantry items such as jams, jellies, and pickles. However, to get a wider range of foods in jars, you need to learn the art of pressure canning, which is what this book covers.
_Pressure canning_ is the preserving method by which filled jars are placed into a large pot with just a few inches of water in it. A locking lid is placed on the pot and steam develops inside the pot. The jars' contents reach an internal temperature of 240°F under a specific pressure (in pounds, see here), and the recipe states a specific period of time to hold that pressure. Pressure canning can be used for a wider variety of foods. You can process vegetables and fruits with low or high acidity, meats, poultry, fish, sauces, and even whole recipes—such as soups or stews. Pressure canning generally means less mess, and my jars always seem to seal when I pressure can. (I also admit some bias—I love canning meat. On the farm, we raise much of our own meat, and being able to process it myself means I can fill my pantry with healthy food that has been raised ethically.)
Pressure canning, as you might expect, is done in a _pressure canner_. A pressure canner is a large pot designed specifically for the canning process; do not confuse a pressure _canner_ with a pressure _cooker_. A pressure cooker is very useful, but it's a different piece of equipment meant for general cooking. Although there is some crossover, and electric pressure cookers can be used to can a couple of jars at a time, they are two distinctly different kitchen items with similar names.
We'll soon dive into more detail on all of this and there's much more to learn in the chapters that follow—and of course, there are plenty of recipes! From everyday staples like tomato sauce and green beans to creative canned creations like pineapple-flavored zucchini and Mexican-style chicken soup, I hope this book not only gives you the confidence to can, but gets you excited about it as well!
Water bath canning and pressure canning are the only two approved methods for canning that exist today. You may have heard of some old-fashioned ways, like canning using the oven or dishwasher—or using aspirin. However, these alternative methods have passed out of favor with time for a reason: they are not reliably safe!
## CHAPTER ONE
## PRESSURE CANNING BASICS
HAVE YOU EVER WONDERED WHY PRESSURE CANNING WORKS? It's actually a simple concept, and, if you are a bit of a science geek like me, quite fascinating!
The goal of pressure canning is to expose food to a high temperature under a specific pressure for a specific period of time to destroy microorganisms that are harmful if eaten. Pressure canning allows food to be heated to 240°F—the temperature necessary to destroy these microorganisms, including the botulism bacteria.
Botulism is rightfully one of the biggest fears for those new to canning, or those who don't can. It is flavorless and odorless. Even worse, it cannot be destroyed by boiling or hot water bath canning, which only reaches 212°F regardless of how long you boil the food. Botulism thrives in low-acid foods, in moist environments, and in an anaerobic (no oxygen) environment. All these conditions are present inside an _improperly_ processed jar of food. Luckily, proper pressure canning creates a higher-temperature environment and can eliminate the risk of botulism in hours.
The other big fear when it comes to pressure canning is the safety of the pressure canner itself. Through the rest of this chapter, we'll discuss pressure canners, common terms, essential equipment, and using the pressure canner for the first time. By the end of this chapter, your fear should be a thing of the past, replaced by excitement to try your first recipe!
### COMMONSENSE PRESSURE CANNING
Keeping your workspace clean is essential when canning.
Canning is a method of food preservation that requires common sense. You can play around with ingredients, invent new recipes, and even make artistic labels that wow your friends, but _cleanliness_ and _organization_ are the keys to germ-free jars of food.
The most important rule for pressure canning is **keep the process clean and simple**. This means keeping the work area and equipment as clean as you possibly can. You should also avoid nonstandard jars and any clutter—such as jar decorations—until after the food is ready for the pantry. You want as tidy a workspace as possible, especially as you're just getting started. Following are other important things to consider when planning for canning:
* Start in the garden: grow what you like.
* Choose your ingredients wisely—freshest and ripest.
* Know the proper steps to canning.
* Prepare and pack the right way.
* Use reliable, tested recipes.
* Update your canning knowledge.
Let's look at each more closely.
#### START IN THE GARDEN
After cleanliness, my most important piece of advice as you begin your canning journey is to grow what you and your family love to eat. You may have a really tasty-sounding recipe for pickled beets, but if you're the only one in your family who enjoys beets, they probably shouldn't take up much space in the garden! It's helpful to make a list of ingredients you find yourself using most frequently in your cooking, or the recipes you always rely on. Break down the recipes into ingredients and see what you might be able to grow and can yourself.
Take my family: We love tomatoes—fresh from the garden or in sauce, it matters not. My family wants to eat a tomato-based recipe every few days, so it only makes sense that I stock up on tomatoes in every form. Quart jars fit about one pound of processed tomatoes, so it's easy for me to plan the quantity I need to grow or buy from a local farmer.
This past year, I asked my favorite farmer to grow 100 pounds of tomatoes for me to can. Sound like a lot? Not when you do the math for my crew. We use two quarts of canned tomatoes in one meal. My plan was to have enough tomatoes for one tomato dish per week for a year. (We eat more than that, but this makes sure we don't waste tomatoes either.) You can do the same with meat, side-dish vegetables, soups, and more. By storing what you grow or buying during the season, when winter comes, your family will enjoy those summer flavors, without the sticker shock of winter shopping in the produce section.
Once you have a list of ingredients you want to can, make a general plan. Don't let that scare you. Planning for your pantry simply means thinking about the kinds of foods you will find useful to "shop" from in your own cupboards. I know my family likes soups for lunches, so I make plenty of soup bases. They also eat as much canned chicken as I can put up. Why buy tuna when you can preserve your own fish?
Canning may be simple enough, but it is work. Careful consideration about what you will preserve will keep you inspired and satisfied all year.
#### CHOOSE YOUR INGREDIENTS WISELY
Using fresh, local produce that your family loves is the best place to start with canning.
Always start with the freshest and ripest ingredients you can. This means you should be prepared to can before you shop. Even if you don't have the specific recipe ready to go, you can have all the equipment—jars, lids, rings, jar lifter, funnel, etc.—ready before you start. At a minimum, take inventory so if you're out of something, like jars or lids, you can pick them up when you go shopping.
It might be tempting to can less-than-perfect produce. Sure, canning "ugly" vegetables is okay. Sometimes you'll find tomatoes, carrots, or other vegetables that taste great but have an odd shape or other imperfection. These are fine for canning! However, _avoid canning wilted, soft, or bruised foods_. Starting with bruised or damaged food can give decay and bad bacteria that come with it a head start. Softer vegetables will also be even softer after canning, which most often works against your recipe. Use only firm, ripe foods at their peak of freshness.
#### KNOW THE STEPS TO CANNING
Successful canning means getting food quickly from the market or garden into jars. We'll cover the basic steps shown here and you should be familiar with them and your recipe _before_ you start your work. The middle of a recipe isn't the time to look something up. Even now, at the beginning of a season, I do a dry run through the process, pretending to can. It may look silly, but many times I realize I don't have a funnel or enough towels. My canning kitchen looks quite different from my regular kitchen; many items on my counters are put away and I have clear access around the room.
#### PREPARE AND PACK THE RIGHT WAY
Cutting uniformly sized pieces is key to ensuring the food in the jars cooks evenly.
Even the most pristine fruits and vegetables require preparation for canning. For one, you'll want to wash off any dirt—or scrub, in the case of root vegetables. You'll also remove stems, leaves, and any part of the plant you don't want to eat (such as a tough end of a rhubarb stalk).
Beyond cleaning and cutting, you'll also be following a recipe to cold pack or hot pack your produce. Cold packing, also known as raw packing, is exactly what it sounds like: packing raw vegetables or fruit in the hot jars and pouring hot liquid over top before canning. Hot packing, on the other hand, will have you cooking your vegetables or fruit before adding them—and often the cooking liquid—to the jars. If this seems overwhelming, don't worry; your recipe will always be your guide on whether to cold or hot pack the jars.
#### USE RELIABLE, TESTED RECIPES
This can be a sensitive topic. Many cooks have emotional ties to recipes—and their historical significance—especially when it comes to something as classic as canning. The conflict comes from techniques or recipes that have since been proven unsafe. This is one area where caution rules. If you have an heirloom recipe, you can certainly find an updated version of it from a reliable source. You may also be able to troubleshoot the unsafe issue—say, by increasing the time of cooking to match modern standards. When all else fails, can the jars based on the most delicate ingredient. So, if you were to can your Great-Auntie M's famous chicken soup, use the most modern techniques for canning chicken soup you can find, and can it in a pressure cooker under the correct pressure for that ingredient.
As much as I am a scratch cook who doesn't always stick to a recipe, when it comes to canning foods, following the rules is the only way. Changing things such as seasonings and how sweet you make the sugar syrup (see here) is fine, but be certain to follow canning recipes carefully and **do not experiment with time and temperature.**
#### UPDATE YOUR CANNING KNOWLEDGE
Each year I review the USDA's online guidelines for safe canning techniques. While my grandmother certainly knew what she was doing, I do things a little differently, and the next generation of canners will, too. We continue to learn ways to improve techniques, safety, and recipes as the years go by. It's always a good idea to keep abreast of best practices. Recipes and techniques are often handed down through families, and it can be hard to change or give something up, but remember safety is key.
In this book, we'll be referencing the most current USDA guidelines at the time of writing in 2017. While there may be further updates or minor adjustments if a new style of pressure cooker is introduced, these guidelines will serve you well overall no matter the year.
### EQUIPMENT
A lid with a gasket will twist or clamp to fasten, removing the need for wing nuts.
When I started canning, I was a purist. I wanted to pretend my grandmother was a homesteader and she was teaching me her ways in the kitchen. I wanted to use only equipment she would have used—and everything needed to have a history. Soon, though, the reality was I needed to can large quantities to feed my growing family. There just weren't enough hours in the day to hand-shred 50 pounds of zucchini!
As I started using my food processor, a professional-quality chef's knife, and recipes with precise measurements and clear instructions, my canning became more efficient—and productive. I still have some of my old-fashioned equipment, but it has found a new home—lovingly displayed on the top of my cupboards!
That said, the equipment needed for canning is not extensive and it is affordable. If you're an avid home cook, you may already own much of what you need except, perhaps, the canner itself. Other than the canner, jars, and lids, most items on the following pages are recommended for your convenience.
#### PRESSURE CANNER
A pressure canner is, essentially, a heavy kettle designed to withstand higher pressure than a normal pot. While models vary in their features, all pressure canners offer a few common elements. For a quick reference, see the photo shown here. For a more detailed explanation, read on!
Before we get into the nuts and bolts (for some models, I mean that literally!), know this: a pressure cooker is not the same as a pressure canner. Yes, both devices use pressure and they look similar. However, pressure cookers are not designed for canning. Most models aren't large enough for cans and they don't have pressure gauges.
Pressure canners come in a wide range of models and sizes. When choosing the right model for you, consider how much you'll be canning at one time; for most people, a pressure canner in the 16- to 22-quart range will do the trick. The other main differences will be the gauge type (dial versus weighted) and cover type (metal-to-metal or lock-on with gasket). Let's get to it.
COMMON CANNING TERMS
**These terms are used throughout this book and the canning world. Once you understand the terminology, canning becomes a breeze.**
**Altitude:** The elevation above sea level.
**Blanch:** Submerging food under boiling water for a few minutes, to loosen the skin or to stop enzymatic action in it. After blanching, the food is often placed in ice-cold water to stop the blanching process.
**Boil:** To heat liquid to 212°F, when bubbles break the surface.
**Botulism:** Food poisoning caused by ingesting the spores of _Clostridium botulinum_. The spores need an anaerobic environment (no oxygen) to survive. Botulism can be fatal.
**Brine:** A salt and water mixture used when pickling foods.
**Canning salt:** Salt that contains no iodine or noncaking agents. It is the salt used in most canning recipes.
**Chutney:** A thick recipe of sweet fruits and savory vegetables cooked with spices and vinegars.
**Clear Jel:** A modified food starch that will not break down when heated to a high temperature. It is a common thickener for canning foods.
**Cold pack (or raw pack):** Raw food placed in jars to be canned.
**Headspace:** The distance between the top of the food and the top of the mouth of the canning jar. Each canning recipe has a specified headspace.
**High acid:** High acid refers to a food's pH of 4.6 or lower. High-acid foods include most fruits and pickled foods. Most of these foods do _not_ need to be canned in a pressure canner.
**Hot pack:** Hot pack is the method of heating food in hot liquid before placing it into jars.
**Low acid:** Foods with a pH higher than 4.6. Low-acid foods _must_ be pressure canned.
**Mason jar:** A glass jar designed for heating foods and liquids. Mason jars can withstand the high pressure of a canner and should not be substituted with other food jars.
**Oxidation:** The reaction that occurs when cut fruits and vegetables are exposed to the air, such as apples turning brown.
**Processing:** The length of time necessary for canned food to remain in the pressure cooker to be completely heated throughout.
**Purée:** Food blended to a smooth, even consistency.
**Saucepan:** A heavy pot with a broad flat bottom and deep sides. Saucepans come in many sizes.
**Screw band:** A metal band with threads, used to attach the canning lid to the jar.
**Simmer:** To heat a liquid until numerous tiny bubbles rise to the surface.
**Sterilize:** To kill all microorganisms.
**Locking cover:** All pressure canners have a locking cover that can be closed in only one correct position. There are canners with covers that lock on with a rubber gasket between the cover and kettle, and other canners that have individual locking wing nuts around the edge. I have both styles and still use them both every year. Neither one is better, objectively, but I find I reach for the model with the gasket and lock-on cover more often.
I do this because the locking process for the pots is different and I prefer the locking procedure on the cover with the gasket. Unlike a metal-to-metal cover, which requires manual tightening, lock-on covers with gaskets have a specific closing procedure with matching markings for the lid and pot and/or clamps. This means there's either a "locked" or "not locked" situation, and no guessing as to the tightness of wing nuts. (Is this tight enough? Did I overtighten?)
No matter which lid type you have, always follow the manufacturer's instructions for locking the lid. Also, note that if your lid has a gasket, you'll need to clean it according to your manufacturer's instructions and use care. For example, some models may tell you to lubricate gaskets with cooking oil while others do not. Any damage to a gasket can affect the machine's ability to hold pressure correctly. Regularly inspect your gasket and replace at the first sign of wear.
**Pressure gauge:** Another feature that all pressure canners share is a gauge that displays the pressure inside the canner. There are two types of gauges and both work equally well. The first type of gauge is a weighted gauge. This gauge has no dial face, but instead has numbers engraved around its edges. This gauge will rattle when the correct pressure is reached; then it will rattle faster and allow some steam to escape if the pressure gets higher than necessary. Note that it's not better to have slow or fast jiggling with this type of pressure gauge. The fact that it is moving means it is maintaining the recommended pressure. In my opinion, this gauge is more foolproof as it requires less heat adjustment than a dial gauge.
The second type, a dial gauge, is an easy-to-read, clock face–style gauge. It has a hand that moves as the pressure increases. You adjust the heat while you watch the gauge. Start the timer once the canner gauge reaches the correct pressure and adjust the heat level with the goal of keeping the pressure gauge at the correct number for the period of time specified in the recipe.
As mentioned elsewhere, it's important to check your gauge for accuracy at least annually. The start of a new canning season is a great time to do this. Your county's cooperative extension office should be able to help and some manufacturers offer this service as well.
**Release valve and overpressure plug:** On the cover of your pressure canner, there is a release valve, which can also be referred to as a vent tube, petcock, or pipe vent in the manufacturer's manual. This release valve needs to be checked at the start of each canning session. Simply hold the cover up and make sure you can see light through the vent tube. If not, clean it with a pipe cleaner or however the manufacturer recommends. Note that this valve is where steam will come out and there are models where the weighted gauge will be on the valve.
Your pressure canner will also have an overpressure plug or safety fuse. This is a simple release that functions as a backup to the primary release valve/vent tube. The overpressure plug opens if the pressure gets too high due to the release valve being blocked. It is a safety feature not present on old-style pressure canners and you should not use a pot that's so old it does not have this important feature. This backup release mechanism is there for your safety. You should also look for the Underwriter's Laboratory (UL) seal to ensure the device's safety.
The manufacturer of your pot may have recommendations for keeping the release valve and overpressure plug clean and operational. Always follow the best practice directions for your pot.
**Canning rack:** Finally, your pressure canner will come with a rack. The rack will look like a circular dish with holes in it and it's designed to sit on the bottom of the canner. Jars will sit on top of the rack; its design allows for the circulation of steam around the jars. It also helps stabilize your jars so they don't knock together or against the sides of the canner (possibly breaking them). If your canner did not come with a rack or your rack breaks, you need to buy a new one. The best place to start is the canner's manufacturer. _Don't can without a rack_.
Weighted gauge models do not self-correct for altitude adjustments. This means at altitudes above sea level (higher than 1,000 feet), the pressure you set should be adjusted upward. See the guidelines shown here.
Ball or Kerr jars ( _right_ ) with 2-piece lids are the best option for beginners.
#### OTHER ESSENTIAL EQUIPMENT
**Jars:** These come in a variety of sizes and your recipe will indicate the correct size for the food you are canning. Jars are typically sold by the dozen for the most common jar sizes, though higher-end, fancier jars may be sold individually. For more information on jars, see here.
**Jar lids:** The lid is critical to a well-sealed jar, not just for the covering you can see it provides but also for the seal it creates underneath. The most common type of canning jars and lids are the variety made by large manufacturers such as Ball. The lids have two pieces: the flat lid itself and a screw band that goes around the outside of the lid (see the next item in this list). If you look at the underside of one of these canning jar lids, you will see a reddish-brown ring around the edge. This ring softens during processing and forms an airtight seal as it cools. Jar lids with this sealing compound are _not_ reusable; however, you can reuse the jars and screw bands. Just buy a new pack of lids before the next canning session.
**Screw bands:** These bands are designed to hold the lid in place during processing. Since they can be reused, carefully inspect them for nicks, rust, and other signs the band has been weakened. As you'll read repeatedly in the recipes, the bands are screwed onto the jars until hand-tight and no tighter. They need to be loose enough to allow the jar lid to release air during processing. Screw bands can be removed after the jars cool completely to room temperature. Removing the band does not affect the lid seal and with no band to hide the seal it is easier to watch for any signs of leakage while the jar is in storage.
**Canning funnel:** Use a canning funnel to fill your jars. You might think this would be an optional piece of equipment; however, keeping the rim of the jar clean is very important. Because you will be working with very hot foods and liquids, using a funnel will help keep your hands from getting burned as well. A canning funnel has a wide mouth and is wider than a normal funnel at the bottom to accommodate big pieces of food. They are inexpensive, and I recommend buying a few. I have both metal and plastic funnels in my canning supplies and both work equally well.
**Electronic scale:** I recommend a digital scale that is reliable, and that includes a tare function. (Tare means you can place your container on the scale and then set it back to 0 before weighing your ingredients.) I also prefer battery-powered scales so you don't have to deal with a cord.
**Jar lifter:** This specialized item makes your canning experience much easier. You'll use the jar lifter to safely lift hot jars from the canner and protect your hands from hot water and steam. This tool has handles that stay cool, and the business end is rubber-dipped and curved to match the curvature of a jar. I have an old one and a newer model; the only difference is the newer one has plastic-covered handles and my older one has wooden handles. They both work equally well.
**Wooden chopsticks (or a similar tool):** When canning, there will be times when you need a long, straight object like a chopstick or the long handle of a wooden spoon. Really, you can use almost any (nonmetal) object that can safely be poked into a jar to release trapped air bubbles and move food around as needed so it fits better. I like using chopsticks because they are cheap and effective. You can also use skewers, as long as they're wooden—you don't want to damage the glass jars. One other advantage or using chopsticks or skewers is you can use them in a pinch for other tasks, such as lifting bands out of hot water.
**Lid lifter:** Not to be confused with a jar lifter, this little tool is a must have. Once again, it comes into play when you're handling hot items—the best way to do anything at this stage is to do it safely. Your lids will be sitting in steaming-hot water and keeping them hot while they are being placed on jars is important. A lid lifter has a small magnet on the end that lifts the lid easily, allowing for an easy transfer to the jar. Before I owned one, I had to use tongs and it was often difficult to get the lids out of the water without burning myself at least a little. Save yourself from the same experience and buy a lid lifter.
**Lid wrench:** While the recipes in this book recommend hand-tightening lids before pressure canning, there are times when lids are tough to remove after canning. If you sometimes struggle with the lids on jars, this tool will make your life easier.
**Dishtowels:** I can't imagine canning without a clean stack of dishtowels for a variety of tasks. I use linen towels so there is no lint, but you can also use tightly woven cotton blends that have no nap. Just avoid fluffy towels that leave lint behind! Use your towels as a landing place for jars when removing them from the canner, for wiping jar rims before adding the lid, and for cleaning up spills as soon as they happen. I also use my dishtowels at various other times, to protect my hands from steam, for wiping off spoon handles if needed, etc. I start my canning session with at least 6 on the counter and usually end up using every one. I recommend white towels that don't look like your everyday towels, so you keep them just for canning and bleach them as needed.
**Knives:** I was taught a sharp knife is safer than a dull one and I have yet to find any evidence to the contrary. Canning involves a lot of food prep, and that means a lot of cutting. Sharp knives reduce hand fatigue and help you cleanly cut uniform pieces of food. You will not need a large number of knives; a paring knife and an 8- or 10-inch chef's knife will suffice. Keep them supersharp and learn how to cut properly with each if you don't already have strong knife skills. A little time spent learning will immediately pay you back in saved prep time.
**Measuring cups:** Plastic, metal, or glass, it doesn't matter which you choose as long as the markings are clear. What is important is having more than 1 set of measuring cups available. Having a second (or third) set at the ready will save you from stopping to clean in the middle of canning.
**Measuring spoons:** Just as with measuring cups, it's essential to have a spare set of spoons if you don't want to be caught unprepared while canning. Also, as much as I love the novelty of newer measuring spoons that slide or adjust to the measurement you need, I don't think they're as reliable for exact measurements—especially when measuring liquids. I stick to the easy-to-read, standard measuring spoons that have served cooks well for generations.
**Rubber spatulas:** While not necessary for every recipe, they are perfect for scraping out and moving sticky items such as jams and jellies. Rubber spatulas are easy to clean and they don't absorb flavors. They are also safer to use inside glass jars. I buy the type with a rubber end that can be removed from the handle for better washing and sanitizing.
**Tongs:** You probably have a pair of tongs in your kitchen already. They're perfect for handling hot food while keeping your hands far away—and that's true whether you're grilling or canning. I prefer the longer style that most people use for the grill, but shorter tongs will work as well.
**Pot holders:** Remember that even the thickest pot holder will allow steam through the fabric, so use them with caution. Having said that, pot holders are the tool of choice for moving hot, heavy pots around your stove. They can also insulate hot jars if you need to set one down on a cool countertop. Use a pot holder in a situation like this to avoid the possibility of cracking the glass jar or damaging your counter.
**Ladles:** A ladle should have a long handle and, although I love metal for many utensils, for ladles, plastic rules! Why? A ladle will spend quite a bit of time in boiling-hot foods when canning and a metal handle can conduct heat. The last thing you want is a ladle that's too hot to handle! I most often use a plastic ladle rated for high temperatures that measures 1/2 cup of liquid, and can double as a measuring device in a pinch.
**Wooden spoons:** Wooden spoons will not become soft or excessively bendable when submerged in hot liquid, unlike some cheaper plastic spoons. I use wooden spoons when cooking, and canning is no different. Actually, it just gives you the opportunity to use a wooden spoon even more—for example, you can use the handle for releasing trapped air bubbles.
#### NONESSENTIAL (BUT HELPFUL) EQUIPMENT
These items are not necessary for canning most recipes. However, some will make your life easier for every recipe and others are essential for a smaller group of recipes—such as jams and jellies.
**Food processor:** This is definitely not a "need it" item, but once you have one it is hard to go back. I use my food processor most often for shredding; it produces even results, and it greatly speeds up the job. I also use my food processor's slicing blade to cut thin slices for pickles. If you're on the fence about buying a food processor, borrow a friend's for a weekend. You will quickly see whether the speed and convenience are worth the cost (and cleanup).
**Food mill:** A food mill is designed to purée fruits and vegetables. It removes the seeds and skins as well. It's the surefire way to get silky smooth applesauce and the smooth consistency you crave in some recipes. Food mills were once hard to find, but now are sold in most big box stores. You also may get lucky and find a used model in good shape. While I don't recommend buying an old pressure canner, a used food mill is a different story. If cleaned well after every use and used carefully, this useful tool will last for many years.
**Jelly bag:** A jelly bag might sound like a very specialized tool, but it can be used for much more than just jelly. It is used to strain juice from fruits and vegetables after cooking. You transfer the produce to the bag and hang it over a pot or bowl so the juice can drip out and be collected. When making Tomato Ketchup, cooked tomatoes and other vegetables are cooked down and placed in a jelly bag. The excess tomato water drains away, leaving the thick pulp behind. The concentrated tomato pulp is the perfect base for thick, tasty ketchup. The tomato water can be discarded, or you could find a creative use—perhaps a new soup recipe!
**Timer:** Yes, you can use the timer on your stove for canning. However, having a dedicated timer you can slip in your pocket is handy. Find a timer you can hear easily, and one that can get splashed without damage. (This probably means buying one with a midrange price point.) I set my timer as soon as my pressure canner reaches the correct pressure, and slip it into my apron. If I get distracted (as much as I tell you not to, life does happen), I have a backup reminder that my jars are ready.
**Labels:** Labels are one item that may seem frivolous to beginners. After all, you can simply write the contents and date you preserved them on the metal lid. Yet labels have their place. Good labels adhere to the glass jars, making it simple enough to identify the contents and date without handling the jars to peek at the tops. Labels are also more attractive and, thus, are a nice touch if you're making the recipes as gifts—or if your cupboard is in view from your kitchen or dining room.
I use labels on my jars and try to change them each year. This means at a glance I know the quantity that is left from a given year—helpful especially on a shelf that has many of the same things, like tomatoes. With unique year-based labels, I know quickly that there are four more quarts to be used up before we start eating this year's supply. So labels are not critical to the canning process, but the more you can, the more you'll want to invest in at least simple labels.
#### LET'S TALK ABOUT JARS
Jars are about to become your new best friend. There are as many styles, sizes, shapes, colors, and uses for these sturdy glass containers as there are foods to put in them. Yet despite the fact that most jars have the same purpose (food preservation), not all jars are created equal.
The canning jars I recommend are the modern-style canning jars made by Ball and similar manufacturers. They are distinguished by their sturdy glass, clearly marked volumes, and 2-piece lids.
The main alternative to this style of jar is experiencing a resurgence, though more as a dry storage jar than for canning. This alternate style of jar, called a bail-type jar, made famous by Weck has two glass pieces—the jar and the lid—that are sealed together using a rubber gasket. Metal clips or a wire were used in conjunction to hold the lid on while canning, but the rubber seal acts like the modern seal under the lid to create the airtight environment you need to can.
_While this type of jar may be used for canning, I tend to recommend it for **refrigerated or dry storage** instead. Weck jars with brand-new rubber seals may be used if you follow the manufacturer's instructions, but they tend to fail the beginning canner more often. Even worse, similar-looking vintage jars may not be intended for pressure canning at all. If you collect glass jars to reuse for food storage like I do, keep them in rotation only for dry goods_.
So, what other options do you have within the recommended type of jars? You have a variety of sizes to choose from. The recipes in this book all come with a recommended jar size. Other considerations include the size of your canner, which may not accommodate the largest jars.
The other important consideration is the size of the jar's mouth opening. Canning jars commonly come with two options:
**1. Regular-mouth jars** have a small opening, about 23/8 inches, but are good for liquids such as stocks or juices, and brothy soups. While many canners use only regular-mouth jars and have no trouble, keep in mind that regular-mouth jars are a little harder to clean if there is residue inside.
Keep the opening size of the jar in mind when shopping for new jars.
**2. Wide-mouth jars** are my personal favorite for a few reasons. You can easily get a utensil inside to scrape them clean. I also like wide-mouth jars because, with their opening at about 3 inches, the jars are easier to fill for recipes with larger pieces. If your recipe will retain a firm texture, it's easier to remove the vegetables when serving as well.
There are many canning jar sizes, from half pints all the way up to gallons (these large jars are not usually used by home canners). With even grocery stores selling canning jars these days, you can easily find the right jar for the job. The recipes in this book, and elsewhere, recommend a specific size. For safety reasons, use the recommended size and don't switch up sizes unless there is a corresponding recommendation for canning. Let's look at some of the jars you'll use!
* **Jelly jar:** Not just for jellies and jams, these jars come in a range of sizes from 4 ounces to 12 ounces. They are the perfect size for gifting, and for the little condiments you want on the table, without having to eat from the container for an extended period of time. Can you imagine a quart of mustard, for example? Use jelly jars for mustards, ketchups, barbecue sauce, and small-batch recipes.
Jelly jars can also highlight a particular food. Jelly jars have wide openings so you can get out every last bite, and they are easy to clean. Due to their size, they require less processing time, which can be an advantage come canning day.
* **Half pint:** The half-pint jar gets quite a bit of mileage in my kitchen. When it comes to volume, these jars hold 8 ounces of food—and they also have the benefit of a wide mouth. I think it's the perfect size jar for relishes, pickles, and chunky chutneys. I use them exclusively for foods I want to use up quickly.
* **Pint:** Pint jars hold 16 ounces and can be used for many items. I use pints for soups I want to package as single lunches, salsas, and specialty recipes like cocktail onions. Pints are a manageable size for most recipes unless you are canning for a large family.
* **Quart:** Speaking of canning for the whole family, quart jars are my biggest workhorse. Quarts hold 2 pints, so you can package double the soup, sauce, etc. I always use quarts to can tomatoes, family-size recipes of sides like baked beans, and soups and broths meant to serve more at once. If you are canning sweet fruits, you may want to consider quart jars. It all depends on how fast your family goes through the food—in my house, the sweets go fast!
Jars also come in larger half-gallon or gallon sizes, but they are beyond the scope of the home canner. I use these supersize jars for dry food storage only.
### CARING FOR YOUR CANNING GEAR
Your pressure canner is a sturdy piece of equipment with simple parts. It does require some care, however. Before the canning season, inspect your canner and all the parts for any nicks or cracks. Check the cover and make sure the overpressure plug moves freely and the vent tube is unblocked. If your pressure canner has a rubber gasket, make sure it is pliable and clean.
Every year, the dial gauge on your pressure canner should be checked for accuracy. This is usually a free or inexpensive service that can be done at your local county extension office or hardware store. Do this early, as many services require that you drop off your equipment. If you wait until the last minute, you may have to wait for the return and miss a week or two of prime produce.
Between canning sessions, a quick rinse and dry of the pot is all that is needed before storage as long as you keep it clean as a matter of course. I keep my canner in the original box once it is clean and dry. That helps keep it from gathering dust and ensures it doesn't get dinged up. It also protects the gauge from damage.
Every canning season should start with a visual inspection. This includes removing the lid and checking the gasket for some models.
When it comes to your canning jars, they are sturdy but, at the end of the day, they are glass. Run your hand gently over the rim of each jar the day before you want to use it to find any small nicks. Also, hold each jar up to the light and look at the glass. You will be able to see small cracks if there are any.
Chips, nicks, and cracks are all signs the glass is no longer fit for canning. It is for this reason I never recommend letting your family use canning jars as drinking glasses. Something that seems as harmless as the teaspoon you stir your sweet tea with can weaken the glass. If you love the look of canning jars, buy a set just for drinking and keep the ones you use for canning stored separately.
The screw bands are the backbone of a good seal. When you purchase a box of canning jars, they come with the lid and band on each jar. The lids are not reusable, but the bands are. Once your jars cool, carefully remove the screw band and store the sealed jar. Screw bands can be reused until they become bent or corroded. Wash and dry your bands before storing them and plan to replace a few each year. I recommend keeping a box on hand for this purpose.
Likewise, keep measuring cups and spoons in good shape by keeping them clean. I like to keep all my canning supplies in the same drawer and I like to sanitize my supplies before each canning session (see sidebar). This ensures my surfaces and items are as clean as possible before I start. Using plastic items means they can be sanitized without the risk of corrosion.
CLEAN IS BEST
**We all have clutter in our kitchens. As much as _things_ make our house our home, they also collect dust and dirt.**
When you are canning, clear your kitchen counters completely of anything you aren't using for the canning process. I remove all my small appliances, jars of utensils, and my extra measuring cups and storage containers. If I am not going to touch the item for canning, it gets removed.
This may seem like extra work, but it ensures the work area can be cleaned thoroughly. It also means you won't accidentally knock into something and ruin some of your hard work. If you're wondering where to put everything, do what I do: Make your kitchen table a temporary countertop! Sure, it clutters up the kitchen table, but that's okay. You won't be using it while you can.
**CLEANING**
Okay, is your counter free of all clutter? Great! Now it's time to wipe down all the surfaces with hot, soapy water to remove any grease and dirt. This is the cleaning step; it will remove any small particles of food or other impurities from your countertop. Only once your counter and work surfaces are clean are you ready to move on to sanitizing.
**SANITIZING**
To sanitize, you can use a commercial product, such as Star San. However, most people prefer to use what they have on hand: bleach. Create a sanitizing mixture by mixing 1 tablespoon food-safe bleach with 1 gallon water. This concentration of bleach in water, 200 ppm, doesn't need to be rinsed off but it does need to air-dry before the surface is considered sanitized. Do what I do and pour the mixture in a spray bottle. Once you've cleaned your counter, mist it with the sanitizing solution, and wipe so you have a thin layer across the surface. Let it air-dry and you're ready to move on to canning!
### THE PRESSURE CANNING PROCESS
No matter what food you are preserving, it requires the same steps to pressure can.
* If you're using an electric pressure canner, you will need to rely on additional instructions from the manufacturer to achieve the desired pressure for the appropriate length of time—you will not adjust it with a stovetop burner like the other models.
* Likewise, if you're using a weighted-gauge model, consult the manufacturer's instructions on setting the canning pressure and for information on how to evaluate whether the canner is maintaining pressure.
The USDA specifically calls out a serious error that can cause unsafe canning:
As mentioned previously, the **internal canning temperature drops at high altitudes**. To correct for this, adjust the pressure of your canner upward if you live above sea level. USDA guidelines recommend increases as follows for **dial gauge canners**. The recipes in this book specify dial gauge pressure for canners below 2,000 feet.
For recipes at 11 pounds:
2,001 to 3,000 feet ............................... 111/2 pounds
3,001 to 4,000 feet ............................... 12 pounds
4,001 to 5,000 feet ............................... 121/2 pounds
5,001 to 6,000 feet ............................... 13 pounds
6,001 to 7,000 feet ............................... 131/2 pounds
7,001 to 8,000 feet ............................... 14 pounds
8,001 to 9,000 feet ............................... 141/2 pounds
9,001 to 10,000 feet ............................. 15 pounds
For recipes at 6 pounds:
2,001 to 4,000 feet ............................... 7 pounds
4,001 to 6,000 feet ............................... 8 pounds
6,001 to 8,000 feet ............................... 9 pounds
8,001 to 10,000 feet ............................ 10 pounds
For **weighted gauge canners** , as soon as you get over 1,000 feet in elevation, you should switch from the 5-pound to the 10-pound or the 10-pound to the 15-pound setting for safety, depending on the recipe. The recipes in this book specify pressure for weighted gauge canners below 1,000 feet.
If **air is trapped in your canner** , it lowers the temperature obtained at any pressure. Take care to avoid this.
Here is a quick review of the steps we'll take:
1. Prepare the canning area.
2. Prepare the canning jars and utensils.
3. Prepare the canner.
4. Prepare the food.
5. Fill the jars to the correct level.
6. Fill the canner.
7. Follow the recipe instructions.
8. Cool the jars.
9. Check the seals; label and store the jars.
#### PREPARE THE CANNING AREA
If you skipped the last few pages (go back and read them!), preparing the area means removing whatever you don't need to pressure can from the counter: extra utensils, small appliances, daily dishes, and other items should be placed elsewhere while you focus on canning. Clean and sanitize your countertops and anything else you think will come in contact with the canning jars and lids before they go into the canner (see here). **A**
#### PREPARE THE CANNING JARS AND UTENSILS
The jars and utensils you will use should be examined beforehand to make sure nothing is chipped or broken. The jars and utensils you will use should be washed in hot, soapy water before use. The clean jars should then be placed in hot—not boiling—water. Single-use lids need to be clean. It is no longer necessary to heat Ball or Kerr brand lids, but if you choose to heat them, do not boil them.
One tip for being ready to can is to place your jars in the dishwasher and run it through a sanitizing cycle. You can leave your jars in the dishwasher until you actually need to use them. I often do this the night before so they are waiting for me first thing in the morning. Before I had a dishwasher, I would wash them the night before and place them on linen hand towels on the sideboard. In the morning, all I had to do was place them in hot water to heat while I prepped my food.
#### PREPARE THE CANNER
Follow the manufacturer's guidelines for filling your specific canner. You'll likely need to insert the canning rack, add 2 to 3 inches of water to the bottom of the canner, and get that water preheated.
#### PREPARE THE FOOD
Your recipe may require a variety of preparation techniques before you can it. Fruits or vegetables must be washed, and perhaps peeled, and cut according to the recipe guidelines. Some foods are also browned or otherwise precooked. When making broth or soup, you may cook the entire recipe before you can. In any case, your recipe will clearly provide these directions.
#### FILL THE JARS TO THE CORRECT LEVEL
Once your food is ready to can, the sanitized jars must be filled to the recipe's specifications. **B** Nearly all recipes in this book provide cooking liquid or heated water to top up your jars to the proper fill level. Note that all recipes require some amount of headspace, the room between the top of the food or liquid and the rim of the jar. (Proper headspace allows the food to expand and air to be forced out during the canning process.) Once the jars are full, use a nonmetallic tool to remove any air bubbles. **C** Wipe the rims with a clean damp cloth, **D** carefully place the lids on, **E** and hand-tighten the screw bands around the lids. **F**
#### FILL THE CANNER
Gently place your jars in the canner on top of the canning rack, making sure they do not touch (using a jar lifter, grab the jars below the screw band). **G** When the canner is full, double check that all jars are upright and not tilted. Place the lid on the canner and lock it according to the manufacturer's instructions.
If you accidentally prepared more food than will fit in your pressure canner, do not pack jars for the next batch and let them sit. Instead, I recommend repeating the steps here in the same order: prepare the food, prepare the canner, pack the jars, and then fill the canner once again.
Follow your manufacturer's guidelines to close and lock the pressure canner. If you're using a model with more manual controls, you typically need to secure the lid, open the petcock (vent pipe) or leave the weight off the vent port, and place the pot over high heat until steam flows continuously for about 10 minutes. **H** At this point, it is time to close the petcock or vent port and let the canner pressurize—it should take less than 5 minutes. **I** Again, always follow your machine's instructions as you must close and lock the lid properly to maintain pressure!
#### FOLLOW THE RECIPE
With your lid locked in place, use your canner's settings or a stovetop burner to get a steady stream of steam from your pressure canner for 10 minutes or according to the manufacturer's guidelines. _Only once you do this should you close the vent and start bringing the canner to the proper pressure._
For an electric pressure canner or a weighted-gauge model, you will most likely be instructed to set the pressure now. If you're using the stovetop to supply the heat, keep the pot at a temperature that maintains the pressure at or above the recommended level for the duration of the recipe's recommended processing time.
No matter which type of pressure canner you use, _start timing your recipe once the pressure on the dial gauge is correct or when the weighted gauge moves_ as described by the manufacturer. In other words, if your recipe says the pressure should be held at 11 pounds for 30 minutes, start the timer only once the pot reaches 11 pounds, not when you first apply the heat.
**_Pressure Dip?_**
Did your pressure dip below the recipe's recommended pressure? _The only safe way to remedy this is to bring the pot back to the recommended pressure and restart the timer_. **Never** let the timer continue once a pot falls below the recommended pressure amount as you can no longer guarantee the safety of the canned food.
Use care not to create large drops or increases in pressure, even above the recommended range, as this could prevent some of the jars from sealing properly. (Again, if you're using a weighted gauge, the manufacturer will provide instructions on how to monitor the pressure.)
#### COOL THE JARS
Once your recipe is finished cooking, turn off the heat. After some of the cooling time has passed, you can safely remove the lid. The amount of time it takes before you can safely release pressure and remove the lid varies by machine. It's often about 10 minutes after the machine reaches 0 pounds pressure.
Do not attempt to speed the cooling by putting your canner in the fridge or freezer, or by placing it in a sink full of cold water. The canner needs to cool without your assistance. Trying to cool the canner faster can result in unsafe canned foods, as well as imperfect seals. Once the lid is off, do not remove the jars immediately; let them continue to cool inside the canner for about 20 minutes. This is a good time to set up a place for the jars to rest once you remove them. I recommend setting a clean towel in an out-of-the way place in the kitchen, away from people who may be tempted to touch the cooling canned food and away from any drafts. You may need a spot for the cans to rest for up to 24 hours as they cool, so choose wisely!
Once you're ready to remove the jars, use a jar lifter to remove them and place them on the clean dishtowel, about 1 inch apart. **J** Do not disturb the jars while they cool and do not try to remove the bands yet. You will hear the jars "ping" as each lid finishes sealing during the cooling process.
#### CHECK THE SEALS AND LABEL AND STORE THE JARS
Once the jars cool to room temperature, remove the screw bands and check that each jar is sealed properly: Press gently on the lid. It should be concave (dip inward) if you have a good seal. It should feel firm to the touch and it should not move or dip when you push on it. Do not tip or shake the jars, but do wipe off any residue that may have escaped during the time in the canner. Label your jars and place the sealed jars in a cool, dry location. If you have any jars that did not seal properly, refrigerate them and use them within a week or two (depending on the recipe).
BE PRESENT, SAFE, AND EFFICIENT
**No matter what you are canning, the basic procedure is the same. Beginners will undoubtedly make a mistake here or there though, and even experienced canners can forget something. That's why when people ask me for my "secrets" for canning, I usually offer these three—my golden rules of canning.**
**BE PRESENT**
Never have a canning session when you are tired or distracted. Don't can if you are enjoying a glass of wine or having friends over for good conversation. Canning is not difficult, but it does require precision and careful monitoring. Skipping a step or missing a drop in pressure can let bacteria slip by or keep a batch of jars from being sealed properly. Your minor distraction can cause major illness. At the very least, it will probably result in wasted food.
**BE SAFE**
Canning involves sharp knives, hot water, scalding-hot steam, and hot jars. Always cut carefully, move hot water and other burning-hot items around your work area safely, and plan for children and pets to play elsewhere while you can. Few other hobbies require high heat as essential to success—even the food you can will be heated to above boiling temperatures. Always make sure you have a place for the hot jars on the counter before you need to transfer them and make double sure that place is far away from a place children could be tempted to touch.
**BE EFFICIENT**
This third rule helps you follow the first two rules: be efficient when canning. Keep only the items you need out in the open. Clear away all the tchotchkes and other homey decor. Clean work surfaces thoroughly and prepare enough food for one canner load at a time. Your jars and lids should be ready to go before you start so you can methodically go through the recipe steps and get the food into the jars in record time. It is important, no matter what type of cook you are the rest of the time, that you are a disciplined and efficient canner.
## CHAPTER TWO
## STORAGE, TROUBLESHOOTING, AND OTHER CONSIDERATIONS
NOW THAT YOU'VE READ THE FIRST CHAPTER, congratulations! You now know the basics of canning. Hopefully that also means you've purchased the equipment you need to can—maybe you even have a recipe or two under your belt.
But knowing how to can is just the beginning. Chances are you'll have ups and downs along the way. Maybe a jar or two didn't seal. Or maybe you have a cloudy jar and are wondering if it's safe to eat. (Alert: don't eat it!) This chapter covers some of the most common problems when canning, as well as how to spot unsafe food and jars.
This chapter also addresses a few other important issues, such as proper storage. After all, if you don't put up your canned goods with as much care as you can them, they won't last as long or taste as fresh when you open them.
### AFTER YOU CAN
Let's start this chapter where we ended the last one—at the end of a canning session. As you already know, the last step is removing the bands from your canning jars and checking the seals by examining the tops (which should be concave). Then it's on to wiping down the jars and labeling them for storage.
While you might be tired after an afternoon of canning, do not skip these important steps. It's easy to think "I'll do it later" when it comes to labeling. But I've found it's best to label the foods with the recipe name and date immediately after you finish canning when the information is fresh in my mind, and before I forget to do it.
As you get started in canning, it's also important to keep a notebook. Jot down your notes while the day is fresh in your mind. Did one jar fail to seal? Did you have any issues maintaining pressure? How long did it take for the pot to reach pressure? These notes are the beginning of your foundation of canning knowledge. Without detailed notes, it will take you much longer to develop your skills and master your own canning setup.
So, what's next? Clean jars should be stored upright, in neat rows, in a cool, dark place. You don't want the jars exposed to light, nor do you want them exposed to temperatures that fluctuate. In other words, while an outdoor shed or garage might be cool, dark, and the temperature never goes above 60°F in the winter, it might fluctuate by 20°F to 30°F some days or throughout a season. This is not an ideal environment for canned goods. Store them indoors in a pantry. Neat rows are essential so you can easily see that each jar looks the same as it did the day it was placed on the shelf. It's easier to spot a jar that looks different from the rest when they're aligned in a neat row.
If you find a white residue on your jars as you go to store them, don't panic. If it disappears when you wipe the jar but comes back when it dries, this is just a sign that your water has a high mineral content. Wiping it with a cloth dampened in a vinegar-water mixture will remove it.
As far as what to avoid when storing your jars, here are a few common mistakes to be aware of:
1. Do not tip and shake your jars from time to time to keep them evenly mixed. Let them settle and _only stir them after opening_.
2. Do not place jars with no labels on the shelf. Add the name of the food and date it was canned to each jar. If you're in a hurry, at least write that information on the lid with a marker!
3. Keep jars out of direct sunlight. If your pantry's light changes from season to season, double check to make sure there isn't a jar or two getting light for a couple hours a day.
4. Do not stack jars. This can keep you from seeing if a lid has loosened or become unsealed. If you need to fit more in the pantry, adjust your shelf height or install more shelves instead.
5. If you find an unsealed jar, do not try to save the food by reheating it. Your health is not worth a jar of food.
#### USE WHAT YOU CAN
If you enjoy cooking (and you probably do since you are reading this book), you will enjoy having shelves of canned goods based on your tastes and needs. When your basic foods are customized to your taste and ready at your fingertips, you can conveniently "shop" in your own store at home—with better quality and flavor than the grocery store. If you start planning meals to include at least one ingredient of your own, you will come to rely on your handiwork and see how that work is well worth the effort.
We have all seen beautiful photos of pantries bursting with rainbows of jars from floor to ceiling. I only wish my pantry was that colorful and full! The truth is, I can what my family will actually eat, and I run out of room before I run out of ideas.
I once wrote down the recipes my family uses the most. It was easy to see we eat mostly from scratch, but generally only used about two dozen or so recipes year-round with any regularity. Doing the same for your household will help you get a handle on what canned foods you'd use the most.
Make a simple spreadsheet listing your favorite recipes, with the ingredients you need, and identify the ingredients you could can. Of course, canning also depends on your access to certain fruits and vegetables. If you grow your own, the success of your garden may drive how much you can (unfortunately, I know this issue well). However, if you have access to a farmers' market, you can usually find whatever quantity you need to round out your own harvest.
You should also consider common serving sizes at your house. If you are a canning for just one or two people, it makes more sense to preserve your foods in pints and half pints. A simple way to think about this is by looking at a can of vegetables from the grocery shelf. Do you use one can of beans when preparing a meal? If so, you can safely can a pint or half pint of beans. For my family, I found we use a pint of beans, relish, or corn, but need a larger amount—a quart—for applesauce or pickles. (Yes, my family eats a pint of zucchini relish at a meal and would eat even more if I would give it to them!)
Consider what your family likes to eat when selecting your canning recipes.
## 3 REASONS TO EAT FROM YOUR PANTRY
Always leave the headspace recommended in the recipe so the food and liquid have room to expand under pressure.
**1. Can for inspiration.** I often go to the pantry, find a canned vegetable that catches my eye, and use that ingredient as a starting point for dinner. Alternatively, try the same exercise as forced inspiration: If you have way too many canned carrots beckoning, look for new recipes that will use up those carrots!
**2. Can for every day and for rainy days.** When I began canning, it was sometimes difficult to break out a jar I had worked so hard creating. You may find you hoard some of your more precious jars or favorite recipes. A great way around this is to compromise with yourself. If you find you have a favorite you're reluctant to use, save yourself just a jar or two for a special treat. This will let you work through the rest and ensure you don't end up with expired food, while also guaranteeing you have a jar or two of something special come February.
**3. Can to save money.** Utilize your pantry foods to save money. Were you about to run to the store because you have "nothing" on hand for dinner? What about all those canned tomatoes? Be resourceful about dinner, using mostly canned goods and grains you have on hand.
#### SPOTTING PROBLEMS IN THE PANTRY
No matter how hard you try, occasionally a jar will go off. When this happens, it is critical that you do _not_ try to salvage the food. Knowing the signs of spoilage and how to care for your canned goods is an essential part of pressure canning. Let's take a look at some things you might find in the pantry.
**Sealed jars that don't stay sealed.** First of all, your jars have to seal or they don't make it onto the shelves. Period. That said, a jar can unseal over time. Check your jars regularly and, _if one becomes unsealed, discard it immediately_. If more than one seal fails from the same batch, go back to your notes from that canning day to try to figure what went wrong. You'll then know if that same thing happens again, you'll need to keep a very close eye on your jars.
**Food fades in color.** Over time, food can fade yet still be safe to eat. However, with that fading comes loss of nutrients. Fading food can occur due to exposure to light though. With exposure to sunlight also comes additional heat, which will decrease the shelf life of your food. Make sure you store food in a cool, dark place and use it within one year.
**Foods that darken in color.** If food floats to the top of the liquid in the jar, it may darken where it is exposed to air. This is not dangerous as the canning killed any harmful bacteria, but it is visually unappealing. It is important to cover the food with liquid before canning to try to avoid this; if you have food that breaks the surface of the liquid, you run the risk of darkened food.
**Changes in food other than fading or darkening.** Your food should not bubble, swell, shrivel, or become cloudy. If any of those things happens, discard the food. Bacteria can cause a variety of visible changes in your foods and it is never safe to try even a tiny taste if you see changes other than fading or darkening.
**Mold in jars.** _Mold is never safe_ in your jars. You might see it grow on the contents of the jars, on the jar walls, or even on the underside of the lid. Do not try to remove the mold or think that heating it will help. Mold in jars is dangerous as it contains spores that become airborne when disturbed. When there is mold found in any can you should discard everything, including the jar and its contents.
### TROUBLESHOOTING
Even the best canners occasionally have issues. However, there are some common issues and questions I get from beginning canners. From the mysterious disappearance of liquid to cracked jars, let's run through a few of the greatest hits.
#### WHERE DID ALL THE LIQUID GO?
When you remove your jars from the canner, sometimes there is not enough liquid left in the jars to cover the food. There are many potential reasons for this, but it's usually one of the most common causes to blame. Here are the top culprits for losing liquid during the canning process and tips on how to avoid them.
* **Jars that are too full or too tightly packed.** When food and liquid are boiled, they expand. If there is not enough headspace in the jar, liquid is forced out. Always follow the recipe guidelines for recommended headspace. More is not always better when it comes to canning!
* **Air bubbles are trapped in the canned food.** Trapped air bubbles can indicate the food was not canned properly. Always make sure to release air bubbles with a straight wooden or plastic utensil before topping up the jar to the specified level and sealing.
* **Pressure fluctuations.** If pressure fluctuates during processing, liquid may be forced out of the jars. Control the pressure carefully and avoid frequent adjustments. If you experience a quick increase or decrease in pressure—even above the recommended recipe guideline for minimum pressure—make sure to check your jars carefully once they cool.
* **Pressure is released too rapidly.** A rapid release of pressure can also cause liquid to be forced out of the jars. This is one reason you should always allow your canner to cool to room temperature on its own. Never try to help it along by placing it in the fridge, freezer, or a bath of cool water.
#### WHY DID MY JARS BREAK?
It has happened to everyone who cans: The excitement of opening the canner quickly turns to shock as you are greeted with a mess of broken jars and wasted food. Even just one jar breaking can put a damper on a fun day of canning. Here are some of the most common reasons glass breaks in the canner and tips to prevent this from happening to you.
* **Reusing store product jars** instead of home canning jars is a surefire way to get a broken jar—or three—on canning day. Even if canning lids and rings fit the commercial jars, the glass may not be made to be reheated under pressure. This is why I recommend only using actual canning jars for your home canning.
* If a jar has been **damaged prior to canning** , it's probably going to crack under pressure. Inspecting each jar before you use it, making sure it has no chips or cracks, is crucial. Hold the jar up to the light and take your time during the inspection. The smallest crack is all it takes for the jar to break in the canner.
* Related to this last point, jars can be **damaged by metal objects**. Using metal spoons and other metal utensils to rearrange foods or release air bubbles can nick and weaken the jar. Even a small amount of damage to a canning jar can cause breakage once it's under pressure. It's important you don't damage the jars you've already inspected on canning day.
* If you place **jars in the canner without a rack** , they will not be stable during the canning process and they can break. Always use a rack when canning.
* **Hot jars** can crack if they are placed on a cold surface. In addition to saving your countertops, placing jars that come out of the canner onto a clean dishtowel, away from drafts, will keep you from losing part of your batch due to breakage.
* When **hot jars are filled with cold water, or cold jars are filled with hot food** , the glass can crack. Avoid this by keeping jars hot and only filling hot jars with hot food.
#### WHAT ARE THE MOST COMMON CANNING MISTAKES (AND HOW DO I AVOID THEM)?
All cooks make mistakes, but canners need to be precise and careful with their canning technique. There are common mistakes that many canners make, which could be, at the least, wasteful and, at the most, dangerous. Let's take a look at what to avoid and why.
* **Using outdated canning methods.** This is, perhaps, one of the most important things to avoid when canning. Just because your grandparents did something and survived doesn't mean you should do it as well. There's a reason today's safety standards differ from the standards decades ago. We have learned a great deal about safe canning and there's no reason not to follow the current canning guidelines.
* **Overpacking jars.** Perhaps the most common mistake beginning canners make is not leaving adequate headspace in the canning jars. Every recipe has proper headspace requirements and this requirement is essential for safe canning. Filling the jars beyond the recommended level means the food won't have room to expand while under pressure. The pressure on overfilled jars forces liquid out between the lid and rim, and can keep the lid from sealing properly as the pressure returns to normal. Always fill jars leaving the correct amount of headspace.
* **Not adhering to proper pressure requirements and times.** While not following a recipe's instructions is not a common mistake canners make, some canners are not even aware when they are "breaking the rules." The length of time needed for the recipe is essential for heating the food to the center of the jar. If the pressure drops below the recommended level, it is not safe simply to resume the timer once you're back at pressures—yes, even if you were nearly done with a recipe! If the pressure drops below the recommended level, you must restart your timer to ensure safe canned food.
* **Interchanging ingredients in a recipe.** Recipes for pressure canning are designed with a length of cook time and pressure designed around the ingredients, both to properly heat them through and to take into account ingredient variables such as acidity. Do not change the recipe or you will risk your food being canned improperly, and spoilage can develop. The only changes you can make safely are to spices, such as oregano, thyme, etc.
* **Leaving screw bands on jars.** Leaving bands on jars can lead to stuck lids and can mask other issues—namely, it is hard to tell if the lid sealed properly if the edge of the seal is hidden by a band.
* **Starting with overripe foods.** Food that is pressure canned past its prime will result in recipes that have an unappealing color, texture, or flavor. If a food has started to decay, simply cutting out the visible "bad" spots may not be enough to remove all the decay. Your food must be as perfectly ripe as possible to get the best, and safest, results.
* **Overcooking recipes before canning.** Since foods are cooked at such high heat during the pressure canning process, if recipes are cooked too long before placing the food in the jars, the resulting food can become mushy. While there's no safety concern here, your family will have to eat their way through your mistake!
#### FINDING AND PREPARING FOOD FOR CANNING
The best canning results comes from the best ingredients. Logically, the first step toward making that perfect recipe is starting with perfectly ripe ingredients. In addition to ripeness, check your produce for excess bruises or other damage. A small spot is acceptable if it's surface damage, but you don't want to can anything that shows signs or deep damage or rot. Always err on the side of caution. If you've already bought the produce, save those less-than-perfect pieces for your day-to-day meals; reserve only the best for your jars.
I am also a believer in using locally grown produce. If you garden, growing your own fruits and vegetables for canning makes it certain to know there are no dangerous pesticides or other chemicals used to treat what you can. Local farmers' markets are the next best option. Get to know your local farmers. Even farms that don't pursue organic certification may abide by the organic rules—or even stricter standards. Talk to the farmers and see if you can visit their farms. Once you know who grows the best produce, make a point of checking in with them throughout the growing season. Getting a sense of what's available when—and at what price—will help you plan recipes for next year.
There are other reasons a good relationship with your farmer or farmers' market vendor pays off. If you tell them you're looking to can, they can bring you enough to make a batch of whatever you want. I often do this when I want to can large numbers of tomatoes. I set the date with my vendor and arrive at the farmers' market before it opens for business. They tend to appreciate this as well; otherwise, you may buy all their tomatoes for the day and that risks angering other customers.
A visit to the local farmers market often leads to a canning session!
So, you've bought the best produce. Now what? Most foods require some basic preparation. You should thoroughly wash your produce, especially if you're not removing the skin. Your recipe will likely have you process the fruit or vegetables into pieces of a certain size; nearly every recipe requires uniform pieces. Similar-size pieces cook at the same rate, while pieces that vary too much in size may result in some being overcooked or some being undercooked. Some recipes also call for precooking the produce in a specific liquid; others simply use water with a bit of salt.
When it comes to meat and seafood, both require careful handling before you can them. While it should go without saying, you only want the best-quality meat and fish for canning—never can expired meat or fish! Generally, you keep meat and fish cold until the day of canning. Most recipes will instruct you to bring them to room temperature. While you can freeze meat and fish to extend their shelf life, do not do so if you are canning. Freezing results in an inferior texture for many recipes. It's not unsafe, as long as you follow the recipe, but, if you're doing the work to can, why not ensure the end result is as tasty as possible.
Finally, though it's been said before, never overpack a jar with anything you are canning. This overlaps with careful preparation as you need to use care for certain recipes to cut pieces to the right size. For example, if you cut carrot sticks too long, the tops may stick out of your canning liquid if you leave the proper headspace.
While you shouldn't use any past-prime or damaged produce, there is nothing wrong with canning "ugly" produce. Often farmers will have oddly shaped carrots, peppers, squash—even tomatoes—available at a discount due to their appearance. As long as they are ripe and show no signs of damage, they are a great find for canning!
## CHAPTER THREE
## CLASSIC VEGETABLE RECIPES
CANNED VEGETABLES CAN BE SO MUCH MORE than the shriveled peas served in the school cafeteria. Home-canned vegetables are bright, plump, and full of natural flavor—without all the added salt and preservatives you find in factory-processed varieties.
Canning pairs perfectly with another of my favorite hobbies—gardening. I know that no matter how much I grow, I can preserve it all for a later date. This justifies my seed and plant collection! Not a gardener? No problem. If you are lucky enough to have a well-stocked farmers' market in your area, get to know the farmers who sell there. Tell them you're planning to can some of what you buy and ask what they like to put up. If you become a regular, some farmers will begin setting aside new vegetables for you to try canning—you're sure to learn a new recipe or two.
Buying local is the best strategy for procuring ingredients whenever possible. These farmers pick their produce at its peak ripeness and bring it directly to market. They do not have to pick underripe fruits and vegetables and ship them thousands of miles, like traditional grocery store suppliers. You will have the best-quality food available in your area and you'll be supporting your local economy. As an added bonus, your local farmers' market is usually less expensive than the grocery store.
Because there are so many options for canning vegetables, I selected recipes for this chapter that are classics in my pantry. They are great introductions to the joy of canning.
### ◁ STEWED TOMATOES
On a cold night on the farm, one of our favorite comfort foods is a simple meal of hot, stewed tomatoes and buttered noodles. We pan fry some extra onions and peppers to go with the onion and pepper in these tomatoes, but, otherwise, the jarred goodness carries the meal. I've found my family doesn't much notice the skin on the tomatoes once they've been canned, but if you want to remove the skins before canning it's easy enough. Simply dip the tomatoes in boiling water and immediately transfer them to an ice-water bath. The skins will peel off easily.
**PROCESSING TIME: 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
12 to 14 large tomatoes, quartered
1/2 cup roughly chopped onion
1/3 cup roughly chopped green bell pepper
2 teaspoons sugar
1 teaspoon canning salt
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over medium-high heat, combine the tomatoes, onion, and green bell pepper.
3. Add the sugar and salt, stirring to dissolve. Bring the mixture to a boil and reduce the heat to a simmer. Cook for 15 minutes, stirring often to prevent burning.
4. Carefully fill the jars with the hot tomato mixture, leaving 1/2 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more tomatoes or liquid as needed to maintain the proper 1/2-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 15 minutes. (For complete canning instructions, see here.)
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the tomatoes within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ THE SWEETEST CANNED CORN
If summer to you means biting into a fresh ear of corn, you should be canning corn for a taste of summer come fall or winter. Just a bite of this canned sweet corn will make your taste buds very happy. Warning: it may make you want to fire up the grill in December! If you use good-quality corn, it remains plump and slightly crisp through the canning process. The secret to great canned corn is to can some of the sweetest corn you can find rather than adding sugar.
**PROCESSING TIME: 55 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 3 PINTS**
7 pounds fresh ears of corn, husks and silks removed
11/2 teaspoons canning salt
1. Prepare 3 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Using a sharp knife, cut the corn kernels from the cobs. Cut about 3/4 of the kernel to avoid cutting into the cob.
3. In a medium-size pot, bring at least 2 quarts of water to a boil.
4. In a large pot over medium heat, combine the corn kernels with 1 cup of boiling water for each 2 cups of corn. Bring the mixture to a simmer and cook for 5 minutes.
5. Using a slotted ladle or large slotted spoon, carefully ladle the corn evenly into the hot jars, leaving a bit more than 1 inch of headspace. Pour the hot cooking liquid over top, leaving 1 inch of headspace.
6. Add 1/2 teaspoon of salt to each pint jar.
7. Remove any air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1-inch headspace.
8. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
9. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 55 minutes.
10. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
11. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the corn within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ CREAMED CORN
You'd be forgiven for thinking there was actual cream in this creamed corn, yet this is the perfect side dish for vegans or even the base for a creamy vegan soup. It's rich yet lighter than dairy-filled creamed corn, thanks to the goodness of the natural corn "milk." Corn milk? Yes, and getting what I call corn milk is easy: After you blanch the corn and remove the kernels, run the back of your knife down the cob. You can even catch all the milk in the same bowl as the kernels as you go. It might just become your new cooking secret!
**PROCESSING TIME: 1 HOUR, 35 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 3 PINTS**
7 pounds fresh ears of corn, husks and silks removed
11/2 teaspoons canning salt
1. Prepare 3 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot of boiling water (or more pots as needed) over high heat, blanch the corn on the cob for 4 minutes. Remove the corn and let the cobs cool enough to handle safely.
3. With a sharp knife, cut the corn kernels from the cobs. Cut about 3/4 of the kernel to avoid cutting into the cob.
4. Run the back of your knife blade over the cob a second time to remove the corn "milk" and combine "milk" with kernels.
5. In a large pot over high heat, combine the corn kernels with 1 cup of water for every 2 cups corn–corn milk mixture. Bring the mixture to a boil.
6. In a separate pot over high heat, bring 2 quarts water to a boil in case you need more hot liquid to top up your jars.
7. Using a slotted ladle or large slotted spoon, carefully ladle the corn evenly into the hot jars, leaving a bit more than 1 inch of headspace. Pour the boiling water over the top, as needed, leaving 1 inch of headspace.
8. Add 1/2 teaspoon of salt to each pint jar.
9. Remove any air bubbles with a plastic or wooden utensil, adding more boiling water as needed to maintain the proper 1-inch headspace.
10. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
11. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 35 minutes.
12. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
13. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the corn within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ CORN RELISH
Corn relish is a versatile condiment. It can be used like creamed corn—for a pop of summery flavor in the deep of winter. However, it's also a summer classic, found at backyard barbecues across the country. At my house, we eat it on our burgers, mixed into chicken salad, and some even take an extra spoonful or two for a bite on the side!
**PROCESSING TIME: 20 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 6 PINTS**
16 to 20 fresh ears of corn, husks and silks removed
2 cups chopped onion
1 cup chopped green bell pepper
3/4 cup chopped red bell pepper
11/2 cups sugar
1 quart white or apple cider vinegar, 5%
1 tablespoon dry mustard
1 tablespoon mustard seeds
2 tablespoons canning salt
1. Prepare 6 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. With a sharp knife, cut the corn kernels from the ears until you have about 2 quarts of kernels.
3. In large pot over high heat, combine the corn, onion, green and red bell peppers, sugar, vinegar, dry mustard, mustard seeds, and canning salt. Bring to a boil, stirring occasionally.
4. Once the mixture reaches a boil, stir it again, cover the pot, and simmer for 20 minutes. Continue stirring occasionally to prevent scorching.
5. Carefully ladle the corn mixture into the hot jars, leaving 1 inch of headspace. You should have enough relish to fill your jars and there is no recommended top-up liquid for this recipe.
6. Remove any air bubbles with a plastic or wooden utensil, adding more relish as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 20 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the relish within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ PERFECTLY CANNED PEAS
Fresh peas are fun to pick and prepare. Stir-fry them, steam them, or snip the ends and eat 'em raw! However, almost nobody thinks of canning peas as the ideal way to treat these flavor-packed little morsels. I encourage you to get past the undeserved reputation of canned peas and try canning them at home. You'll soon find they're not mushy at all, and, in fact, pack nearly as much flavor as fresh peas. So, spread the word! Can those peas and share them with your friends.
**PROCESSING TIME: 40 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 3 PINTS**
7 pounds fresh peas in their pods, washed, and peas removed from the pods
11/2 teaspoons canning salt
1. Prepare 3 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Place the peas in a large pot, cover them with water, and bring to a boil over high heat. Boil for 4 minutes, or until they are bright green but not completely cooked.
3. Drain the peas over a large pot or bowl to reserve the cooking liquid. Pack the cooked peas loosely into the prepared jars.
4. Pour the hot cooking liquid over the peas, leaving 1 inch of headspace.
5. Add 1/2 teaspoon of salt to each pint jar.
6. Remove any air bubbles with a plastic or wooden utensil, adding more hot cooking liquid as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 40 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the peas within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ SPICED PICKLED BEETS
If you're looking for a recipe that will make you famous with friends and family, look no further! While beets are healthy, they can sometimes be a tough sell to picky eaters, or younger eaters who complain they "taste like dirt." This recipe is sure to change minds as the beets receive a treatment of sugar, spice, and everything nice. For those who already love beets, it will be hard to eat just one. Serve them as part of an appetizer spread and watch them disappear.
**PROCESSING TIME: 30 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
4 pounds firm unblemished beets, washed, trimmed, leaving the root intact as well as about 2 inches of the stems
3 cups thinly sliced white onion
2 cups sugar
21/2 cups distilled white vinegar
11/2 cups water
1 teaspoon canning salt
2 tablespoons whole-spice pickling mix, or a mix of 1 tablespoon mustard seeds, 1 teaspoon whole allspice berries, 1 teaspoon whole cloves, and 3 cinnamon sticks
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot, combine the beets with enough water to cover. Bring to a boil over high heat. Reduce the heat to low and simmer for about 20 minutes until fork-tender.
3. Drain the beets and run cold water over them to stop the cooking process. When cool enough to handle, use a sharp knife to remove the skins. The skins should easily come away from the flesh. Remove the stem and root at this point as well.
4. Slice the beets in 1/4-inch-thick slices and place them in a large bowl. Set the beets aside.
5. Rinse the pot you cooked the beets in, place it over high heat, and combine the onion, sugar, vinegar, water, salt, and pickling mix in it. Bring the mixture to a boil. Reduce the heat to low and simmer for 5 minutes.
6. Add the beet slices to the hot mixture and simmer for 3 minutes more. Turn off the heat and remove the cinnamon sticks.
7. Carefully ladle the hot beet and onion mixture into the hot jars. Ladle the hot cooking liquid over the beets, leaving 1/2 inch of headspace.
8. Remove any air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1/2-inch headspace.
9. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
10. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 30 minutes.
11. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
12. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the beets within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ GREEN BEANS
In my garden I try to pick beans just as fast as they ripen. Fresh beans are the key to canning them and keeping a slight bite. If your beans have any bend, they will surely not be crisp after canning. While I previously cautioned about switching ingredients in and out of recipes, for this one you have my permission to use any color string-type bean you wish. They all taste great! Use these beans as a last-minute addition to your favorite stir-fry or soup. They will add a burst of fresh flavor to any dish.
**PROCESSING TIME: 20 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
1 pound green beans, ends trimmed and cut into 2-inch pieces
2 teaspoons canning salt
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a medium-size pot or kettle, bring 1 quart water to a boil.
3. While the water boils, pack the green beans into the hot jars as tightly as possible.
4. Pour the boiling water over the beans, leaving 1 inch of headspace.
5. Add 1/2 teaspoon of salt to each pint jar.
6. Remove any air bubbles with a plastic or wooden utensil, adding more boiling water as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 20 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the vegetables within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ SPICY DILLY BEANS
Pickled green beans, also known as dilly beans in some circles, are a must-have for your pantry. They make a tangy component of bean salads, but they also work great as a condiment and as a unique topping for appetizers. Some even swear by them as a garnish for a Bloody Mary, used in place of the more traditional celery stalk. Mixing green and yellow beans adds a pop of color to the jars, but no matter the visual interest, these beans are tasty. You may not be able to make enough of them to last all year!
**PROCESSING TIME: 20 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
21/2 cups distilled white vinegar
21/2 cups water
1/4 cup canning salt
21/2 pounds green beans, ends trimmed; measured and cut so they leave 1/4-inch headspace in the jars
4 garlic cloves, peeled
4 stems fresh dill, cut in half if tall
4 dried red chile peppers
1 teaspoon cayenne pepper
1 teaspoon dill seed
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over high heat, combine the vinegar, water, and salt. Bring the mixture to a boil and cook for 1 minute. Stir well while the mixture boils to make sure the salt dissolves. Reduce the heat to maintain a simmer and keep the liquid very hot while you continue to work.
3. Pack the beans upright in the jars, leaving 1/4 inch of headspace. Cut the beans again, as needed, to leave the 1/4 inch of headspace.
4. In each jar, place 1 clove garlic, 1 dill sprig, 1 dried red chile pepper, 1/4 teaspoon of cayenne, and 1/4 teaspoon of dill seed.
5. Carefully ladle the hot liquid over the beans, leaving 1/4 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1/4-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 20 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the beans within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ PICNIC-FRIENDLY BAKED BEANS
The problem with store-bought baked beans isn't the taste or the lack of variety. These days you can find maple baked beans, baked beans with thick-cut bacon, vegetarian baked beans—the list goes on! However, so many recipes contain questionable preservatives or flavor enhancers. Even the best recipes contain a large amount of sugar or other sweetener. You'll find this recipe makes beans with just the right amount of sweetness. The sauce is rich and the beans themselves still have some texture (something rarely said for store-bought baked beans). Serve alongside sandwiches and lemonade at your next picnic, or plan an indoor "picnic" night in the middle of winter. Hot dogs, hamburgers, corn, coleslaw, and these baked beans are always a treat.
Note that this recipe requires soaking the beans for 12 hours before cooking and canning. You'll also need to bake the bean recipe for about 3 hours, so it's a good idea to soak the beans the night before you want to cook and can them.
**PROCESSING TIME: 1 HOUR, 20 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 6 PINTS**
2 pounds dried navy beans
6 quarts water
1/2 pound bacon, cut into small pieces
3 large onions, sliced
2/3 cup packed brown sugar
4 teaspoons salt
2 teaspoons dry mustard
1/3 cup molasses
1. The night before you want to can, place the beans into a large pot. Add 3 quarts of water to cover the beans. Soak the beans, covered, for 12 hours.
2. Drain the beans and return them to the pot. Add the remaining 3 quarts of water and bring the mixture to a boil over high heat. Reduce the heat to low and simmer the beans until soft and the skins begin to split, about 11/2 to 2 hours. Drain the beans and reserve the cooking liquid.
3. Preheat the oven to 350°F.
4. Transfer the drained beans to a large baking dish and add the bacon and onions.
5. In large bowl, stir together the brown sugar, salt, dry mustard, and molasses. Add 4 cups of the reserved cooking liquid (add more water if needed to make 4 cups). Stir this sauce until it is well combined. Pour the sauce over the beans—but do not stir. Cover the dish with aluminum foil and bake for 31/2 hours.
6. Meanwhile, prepare 6 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
7. After 31/2 hours, carefully ladle the bean into the hot jars, leaving 1 inch of headspace.
8. Remove any air bubbles with a plastic or wooden utensil, adding more beans as needed to maintain the proper 1-inch headspace.
9. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
10. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 20 minutes.
11. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
12. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the beans within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ HEIRLOOM BEANS
Dried beans are practically magical. They are inexpensive and filling, and can give a hearty boost to many recipes. You can even combine them with rice for a complete protein. If there's one downside they have, it's the time. Chances are, you reach much more for canned beans than dried on a daily basis—so why not can your own? While you can certainly use any dried beans you have on hand for this recipe, I recommend seeking out an heirloom bean to elevate this pantry staple. There are so many bean varieties you can order these days, and most are not available as canned beans in the grocery store! It is not necessary to add the salt pork or bacon, but it's a great way to infuse additional flavor into your beans and separate them from their store-bought counterparts. If you omit the salt pork or bacon, add a little salt, or not, depending on how you like to use your beans in recipes.
Just like the Picnic-Friendly Baked Beans, these beans require an overnight (12-hour) soak before you're ready to cook and can.
**PROCESSING TIME: 1 HOUR, 35 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 3 PINTS**
8 ounces dried beans of choice
1/2 cup cubed salt pork, or 1/4 pound good bacon, cut into bite-size pieces (optional; see headnote)
1. The night before you want to can, place the beans in large pot and cover them with warm water. Soak the beans, covered, for 12 hours.
2. Drain the beans, return them to the pot, and cover them with fresh water. Cook over medium-high heat until they begin to boil. Stir and continue cooking the beans according to the package directions, but stop just short of cooking them all the way through.
3. Meanwhile, prepare 3 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
4. Meanwhile, bring 1 quart of water to a boil in case you need more hot liquid to top up your jars. (Some beans absorb more water than others while cooking.)
5. Divide the salt pork, or bacon, evenly among the pint jars (if using).
6. Carefully ladle the hot beans into the hot jars. Pour the hot cooking liquid over the top, leaving 1 inch of headspace. If the cooking liquid does not go far enough, add the hot water to top up to the proper 1-inch headspace.
7. Remove any air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1-inch headspace.
8. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
9. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 35 minutes.
10. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
11. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the beans within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ LENTILS
Lentils are a popular base for healthy soup recipes. There are also many varieties of lentil salad, from Middle Eastern preparations to American classics. Unlike most beans, lentils do not need a long soak before cooking and that means there's no need to soak them before canning either. While lentils are one of the quicker legumes to cook from scratch, there's nothing as quick as cracking open a jar of your homemade canned lentils!
**PROCESSING TIME: 1 HOUR, 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 3 PINTS**
2 cups dried lentils
4 cups Vegetable Broth or store-bought broth
1 large onion, minced
1. Prepare 3 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Inspect the lentils carefully and remove any foreign objects, such as small stones. Place the lentils in a large pot and add the broth and onion. Bring the pot to a boil over high heat, reduce the heat, and simmer for about 5 minutes until the lentils are partially cooked.
3. Carefully ladle the lentils into the hot jars, filling each jar halfway. Ladle the hot cooking liquid over the lentils, leaving 1 inch of headspace in the jars.
4. Remove any air bubbles with a plastic or wooden utensil, adding more hot cooking liquid as needed to maintain the proper headspace.
5. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
6. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 15 minutes.
7. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
8. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the lentils within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### KIDNEY BEAN SOUP
Canned bean soups are a lot like canned beans—they can get the job done, but it's hard to get excited about them. Worse even, canned soups often pack lots of sodium—perhaps making up for their shortcomings in texture or flavor! This soup is a clear upgrade and, even better, kid approved. One of my family's favorite meals is a combination of this soup with fresh-from-the-oven biscuits. I've also done some testing and this recipe holds up just fine if you want to omit the ham and use vegetable broth instead of a meat-based broth. As far as the salt goes, I recommend canning this recipe as is and allowing your family (or guests) to season with salt and pepper when serving. The salt in the broth may well be enough for some.
**PROCESSING TIME: 1 HOUR
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
2 quarts Chicken or Turkey Broth or store-bought broth
1 cup diced peeled potatoes
1 cup fresh corn kernels
1/2 cup sliced celery
1/2 large yellow onion, diced
1 garlic clove, thinly sliced
1 cup cubed cooked ham
1 cup canned and drained kidney beans
1. In a large stockpot over medium-high heat, combine the broth, potatoes, corn, celery, onion, and garlic. Cover the pot and heat until the mixture comes to a simmer.
2. Add the ham and kidney beans. Bring the mixture to a boil. Cook for 10 minutes.
3. Meanwhile, prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
4. Carefully ladle the hot soup into the hot jars, leaving 1 inch of headspace.
5. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
6. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour.
7. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
8. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the soup within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### MIXED VEGETABLE MEDLEY
When I was a little girl, whenever my family had mixed vegetables I picked out every piece of mushy carrot. Now that I make them myself, I make sure every vegetable is ripe so the whole mixture is delicious. I'm happy to say my kids don't leave any vegetable behind! One quick note: make sure to wash and dice your vegetables to a uniform size before combining them to cook, or they may not cook at the same speed. Beyond making for an easy out-of-the-jar side dish, mixed vegetables are great to have on hand for soups or as a topping for healthy grain bowls.
**PROCESSING TIME: 55 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
1 cup chopped tomatoes
7 cups total of a combination of the following, all cut to uniform sizes (see headnote):
Diced carrots (1-inch dice)
Sweet corn
Green beans, cut into 1-inch pieces
Diced zucchini (1-inch dice)
2 teaspoons canning salt
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot, combine the tomatoes with the 7 cups of vegetables you selected and add enough water to cover. Bring the mixture to a boil over high heat and boil for 5 minutes. Stir frequently to prevent any burning or sticking on the bottom of the pot.
3. Carefully ladle the vegetables evenly into the hot jars. Top with the hot cooking liquid, leaving 1 inch of headspace.
4. Add 1/2 teaspoon of salt to each jar.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner Process the jars at the pressure listed above for 55 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the vegetables within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ CANNED POTATOES
I know fall has arrived when the potato man comes down from Aroostook County to sell his potatoes by the 50-pound bag—50 pounds! I buy two bags, one for my shelves and one for dry storage. Canning potatoes is one concept that can take some getting used to, as they're not a popular product in most grocery stores. However, I think once you start canning them you'll be converted. You can use the potatoes in nearly all the recipes you'd normally cook—except baked potatoes, of course! Having them precooked just means less time cooking dinner. Add them to soups and chowders or throw them on a sheet tray for roasting along with some quick-cooking vegetables. Since you just have to crisp the potatoes and not cook them from scratch, dinner is ready in no time.
**PROCESSING TIME: 35 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 6 PINTS**
3 pounds fresh potatoes, washed, peeled, and cut into 1/2-inch cubes
3 teaspoons canning salt
1. Prepare 6 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over high heat, bring 1 gallon of water to a boil.
3. Place the potatoes into the boiling water. Return the water to a boil and cook for 2 minutes.
4. Using a large slotted spoon, pack the potatoes into the hot jars. Reserve the cooking liquid.
5. Add 1/2 teaspoon of salt to each jar.
6. Ladle the hot cooking liquid over the potatoes, leaving 1 inch of headspace.
7. Remove air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1-inch headspace.
8. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
9. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 35 minutes.
10. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
11. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the potatoes within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ TASTE OF SUMMER CARROT STICKS
You might think canned carrot sticks would take away the fun of snapping into a fresh carrot, as you cook out some of the crunch. However, carrots take on a new flavor profile when canned. Cooked just enough to enhance the natural sweetness, canned carrot sticks are a healthy finger food whenever you need to mix up snack time.
**PROCESSING TIME: 25 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 3 PINTS**
6 pounds whole carrots, washed and peeled
11/2 teaspoons canning salt
1. Prepare 3 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Cut the carrots into sticks very close to the same size, long enough to fit into the jars leaving 1 inch of headspace.
3. Place the carrot sticks in a large saucepan and cover them with water. Bring to a boil over high heat, reduce the heat to a simmer, and cook for 5 minutes.
4. In another saucepan, bring a few cups of water to a boil as a reserve, if needed, to fill the jars to the proper headspace.
5. Tightly pack the carrots into the hot jars, leaving 1 inch of headspace.
6. Add 1/2 teaspoon of salt to each pint jar.
7. Ladle the hot cooking liquid over the carrots, leaving 1 inch of headspace.
8. Remove any air bubbles with a plastic or wooden utensil, adding the reserved hot water as needed to maintain the proper 1-inch headspace.
9. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
10. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 25 minutes.
11. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
12. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the carrots within 2 weeks. Label the remaining jars with the recipe name and date before storing.
## CHAPTER FOUR
## GOING FURTHER WITH VEGETABLES
I'M A BELIEVER IN KEEPING IT SIMPLE, especially when you're just starting out. Instead of trying to find fifty recipes with fifty different vegetables, it's more interesting and rewarding to focus on a dozen vegetables that grow well in your area and different ways to prepare them. As you return to the same season year after year, you'll naturally build on what you learned the year before.
Yet simple doesn't mean boring! From the first asparagus of the season to the last fall pumpkin, pressure canning unexpected vegetables is a wonderful way to preserve your garden's bounty—beyond peas, corn, and carrots. In fact, I've found that some of these more unusual vegetables—fiddlehead ferns, for example—make great food gifts with a real wow factor.
In this chapter, you'll also find new ideas for more common vegetables—Sweet and Spicy Ginger Carrot Coins with an unexpected hit of ginger, onions are re-imagined as The Perfect Cocktail Onions, and zucchini takes a trip to Hawaii where it's married with the flavor of pineapple (shown here). (It may sound crazy, but it really works!) These recipes will take your excitement for canning to a new level.
### ◁ SWEET AND SPICY GINGER CARROT COINS
If you want a carrot recipe with some kick, you've come to the right place. In this recipe, the sweetness from brown sugar and orange juice marries with spicy ginger, making these carrots your new favorite finger food. I've also made this recipe with baby carrots instead of carrot coins with great results. They are fantastic for a party!
**PROCESSING TIME: 30 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
21/2 pounds carrots, washed and peeled
2 cups packed brown sugar
1 cup freshly squeezed orange juice (from 2 to 4 oranges)
2 cups water
2 pieces crystallized ginger (about 1 inch each), minced
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Cut the carrots into 1/3-inch-thick coins or 1-inch chunks, unless you're using baby carrots, which you can process whole.
3. In a large saucepan over medium heat, combine the brown sugar, orange juice, water, and crystallized ginger. Cook until the sugar dissolves. Keep the syrup hot, stirring occasionally, as you continue to work. Reduce the heat if the mixture comes close to a simmer.
4. Pack the carrots tightly into the hot jars, leaving 1 inch of headspace.
5. Carefully ladle the hot syrup over the carrots. Leave 1 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, adding more hot syrup as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 30 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the carrots within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ FIRST-OF-THE-SEASON ASPARAGUS
Asparagus is one of the first signs of spring on our farm. Once you find these pencil-thin spears in the garden or at the farmers' market, it's officially time to get out the canner and jars. Canning asparagus is easy to do, though it takes practice to pack the jars. You need to fit enough to fill the jars, but you don't want to overpack them—and of course the spears can be too tall for the jars! If you're planning to gift some of the jars, make them especially beautiful by placing the best-looking spears around the outside of the jar where they will show. Then the not-as-pretty, but still delicious, spears will be in the center. If you don't care about the appearance of the asparagus in the jars, cut it up and fill the jars with bite-size pieces instead. It tastes every bit as good and is much easier to pack.
**PROCESSING TIME: 30 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
8 pounds fresh asparagus, washed and trimmed to fit the height of the jars (they shrink in the canning process)
2 teaspoons canning salt
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over high heat, bring 2 quarts water to a boil.
3. Pack the raw asparagus spears into the hot jars as tightly as possible.
4. Add 1/2 teaspoon of salt to each pint jar.
5. Using a funnel or ladle, carefully add boiling water to each jar leaving 1 inch of headspace.
6. Remove air bubbles with a plastic or wooden utensil, adding more hot water as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 30 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the asparagus within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ THE PERFECT COCKTAIL ONIONS
If you love cocktail hour, it's time to bring some of your favorite canning recipes to the bar. Many recipes in this book can do double duty as a happy hour snack, but a few recipes work as a cocktail garnish as well. Most often called pearl onions, these beautiful minis are especially delicious when pickled to perfection. My subtle variation on this classic recipe is to add a garlic clove to each jar. I think that extra complexity creates a depth of flavor, but feel free to leave out the garlic if desired.
**PROCESSING TIME: 11 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
51/2 cups distilled white vinegar
1 cup water
2 cups sugar
2 teaspoons canning salt
8 cups white pearl onions, peeled and washed
4 garlic cloves, peeled (optional)
8 teaspoons mustard seeds
4 teaspoons celery seed
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over medium-high heat, combine the vinegar, water, sugar, and salt. Bring to a boil, and boil for 5 minutes.
3. Add the onions, reduce the heat to a simmer, and cook for 5 minutes more.
4. Place the following into each hot jar: 1 garlic clove (if using), 2 teaspoons mustard seeds, and 1 teaspoon celery seed.
5. Using a slotted spoon, remove the onions from their cooking liquid (reserve the liquid for filling the jars) and firmly pack them into the prepared jars.
6. Carefully pour the hot cooking liquid over the onions, leaving 1 inch of headspace.
7. Release any air bubbles with a plastic or wooden utensil, adding more hot liquid as necessary to maintain the proper 1-inch headspace.
8. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
9. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 11 minutes.
10. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
11. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the onions within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ YEAR-ROUND PUMPKIN BITES
In my opinion pumpkin isn't as popular as it should be. Sure, there's the annual pumpkin spice madness come fall, where you can find "pumpkin" in everything from lattes to candles. However, the enthusiasm seems to be more about the spice than the pumpkin. By preserving cubes of this vitamin A–rich vegetable without typical pie spices, you taste the natural flavor of the pumpkin all the better. Though this recipe uses light sugar syrup to bring out the sweetness of the pumpkin, the jars can be used for more than just desserts. Try pumpkin anywhere you'd use sweet squash, such as in soups, grain bowls, or as a side dish garnished with fresh herbs.
**PROCESSING TIME: 30 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 6 PINTS**
1 recipe light sugar syrup (see here)
10 pounds pie (sugar) pumpkins, halved, peeled, seeds and pulp removed, cut into 1-inch cubes
1. Prepare 6 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a medium-size saucepan over high heat, bring the light sugar syrup to a boil.
3. Pack the raw pumpkin into the hot jars.
4. Carefully pour or ladle the hot syrup over the pumpkin, leaving 1 inch of headspace.
5. Remove air bubbles with a plastic or wooden utensil, adding more hot syrup as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 30 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the pumpkin within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ SPICY HOT PEPPER MIX
It seems as though every family has at least one eater who wants—no, needs—to turn the heat up. Yet what constitutes spicy varies not only from person to person but also family to family. My family's "hot" might seem mild to you! The nice thing about this recipe is that it can be dialed up or down when it comes to heat. Make a batch of habaneros for the capsaicin hound in your house or seed jalapeños and serranos from the garden for a milder mix. If your pepper mix turns out too hot, there's a way to soften the edge after processing. Chop some Taste of Summer Carrot Sticks and mix them in a bowl with your peppers for a homemade riff on a classic taqueria condiment.
You may want to start this recipe the night before, as you need 12 hours to soak the peppers before canning.
**PROCESSING TIME: 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 3 PINTS**
3 quarts assorted hot peppers
3/4 cup canning salt
5 cups distilled white vinegar
1 cup water
1/4 cup sugar
2 or 3 garlic cloves, sliced
1. Wearing rubber gloves, wash the peppers and cut 2 slits lengthwise into each pepper.
2. In a very large bowl or large pot, combine the salt with 1 gallon water. Stir to dissolve the salt completely. Add the peppers and let sit for 12 hours.
3. Prepare 3 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
4. Drain and rinse the peppers, and pack them into the hot jars.
5. In a medium-size saucepan over medium heat, combine the vinegar, water, sugar, and garlic. Bring to a simmer and cook for 15 minutes. Remove and discard the garlic. Carefully pour the hot solution over the peppers, leaving 1/2 inch of headspace.
6. Remove air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1/2-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 15 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the peppers within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ PICKLED CAULIFLOWER
Cauliflower is one versatile vegetable. It can be steamed and puréed for those who prefer it creamy. It can be riced or finely chopped, cooked for only a moment, and used in place of rice. It can be roasted, grilled, or tempura-battered and fried. Yet, unless you can, you probably haven't experienced the joy of tart, pickled cauliflower. While this recipe is so delicious it should be eaten straight from the jar, or as part of a cheese board, pickled cauliflower can also be a secret weapon. Imagine adding the crunch and tang of a homemade pickle, but without the cucumber flavor. I've found finely chopped pickled cauliflower is a welcome addition to potato salad, pasta salad, and even chicken salad.
**PROCESSING TIME: 20 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
4 cups water
4 cups distilled white vinegar
1/2 cup canning salt
4 pounds cauliflower, washed and cut into bite-size pieces, core discarded
1 yellow onion, sliced into 1/4-inch-thick slices, breaking up any rings
1 teaspoon red pepper flakes
4 sprigs dill
4 garlic cloves, peeled
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over high heat, combine the water, vinegar, and salt. Bring the liquid to a boil and boil for 1 minute. Stir well to dissolve the salt completely. Reduce the heat to a simmer and keep the liquid very hot.
3. Fill each jar with cauliflower pieces and evenly divide the onion among the jars.
4. To each jar, add 1/4 teaspoon red pepper flakes, 1 sprig dill, and 1 garlic clove.
5. Carefully ladle the hot liquid over the vegetables, leaving 1/4 inch of headspace.
6. Remove air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1/4-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 20 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the cauliflower within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ TOMATILLOS
When we lived in Nebraska, I grew both green and purple tomatillos for the farmers' market. Some customers had a little trepidation—after all, tomatillos are much less common than tomatoes in that part of the country. However, once I shared a few recipes for salsa verde and Mexican-inspired soups, people were hooked. Soon I couldn't grow enough to keep up with demand! While it's true you can make and pressure can salsa (see here), by canning the tomatillos as simply as possible, you'll have all options available when you crack open the jar.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
4 pounds fresh tomatillos, papery husks removed, washed and quartered
1/4 cup freshly squeezed lemon juice (from about 2 lemons)
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In large pot, combine the tomatillos with enough water to cover. Bring to boil over high heat and boil for 5 to 10 minutes until tender.
3. With a large spoon, carefully fill the jars with the tomatillos and some cooking water, leaving 1/2 inch of headspace.
4. Add 1 tablespoon of lemon juice to each jar.
5. Remove air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1/2-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the tomatillos within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### PINEAPPLE-FLAVORED ZUCCHINI
Zucchini is one of those vegetables that can quickly overwhelm you with a bumper crop. Some years my garden produces so many zucchini I don't know what I'd do if I couldn't can them. With that large quantity in mind, I'm always looking for new flavors to break the zucchini monotony. This one might sound wacky, but try it before you write it off. Fresh zucchini has a mild flavor that I've found takes on the flavor of its canning liquid more than other vegetables. Using pineapple juice with a hint of lemon creates a tropical twist the whole family will love. Shredding the zucchini increases the surface area to soak up as much pineapple goodness as possible.
**PROCESSING TIME: 25 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 6 PINTS**
9 cups shredded zucchini
3 cups sugar
53/4 cups (46 ounces) canned pineapple juice
1/2 cup freshly squeezed lemon juice (from about 4 lemons)
1. Prepare 6 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over medium-high heat, combine the zucchini, sugar, pineapple juice, and lemon juice. Bring to a simmer and cook and for 20 minutes.
3. With a large spoon, carefully pack the zucchini into the hot jars, dividing it and the juice evenly among the jars, leaving 1/2 inch of headspace.
4. Remove air bubbles with a plastic or wooden utensil, adding hot juice as needed to maintain the proper 1/2-inch headspace.
5. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
6. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 25 minutes.
7. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
8. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the zucchini within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### SWEET AND SPICY PICKLED BRUSSELS SPROUTS
From kimchi to sauerkraut, cabbage is one of the most popular pickling vegetables the world over. Brussels sprouts are a fellow member of the brassica family, and, while their flavor is unique, there's no denying they share a certain something with cabbage. So, for this recipe I let them shine in a pickle. Simply halving them creates perfect bite-size morsels; of course, you can break them down further by dicing them after pickling. I recommend letting these sit at least 2 weeks in the jar after pickling to develop the proper amount of flavor.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 5 PINTS**
2 pounds brussels sprouts, washed, trimmed, brown leaves discarded, and halved
10 tablespoons canning salt, divided
5 garlic cloves, peeled
5 cups distilled white vinegar
5 cups water
11/2 teaspoons red pepper flakes
1. Prepare 5 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Soak the brussels sprouts in a large bowl of lightly salted water (3 tablespoons of salt) for 20 minutes. Drain well.
3. Evenly divide the sprouts among the jars, leaving 3/4 inch of headspace. Add 1 garlic clove to each jar.
4. In a large pot over high heat, combine the vinegar, water, remaining canning salt, and red pepper flakes. Bring to a boil and stir until the salt dissolves completely.
5. Carefully ladle the hot pickling liquid into each jar, leaving 1/2 inch of headspace.
6. Remove air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1/2-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the brussels sprouts within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ RAINBOW PEPPERS
We eat first with our eyes, and this recipe is one of the best for adding a pop of color to your mid-winter recipes. That said, my family also loves these in summer. We'll often grill sausages and top them with sautéed onions and these peppers. It's like the best of the county fair brought home—only tastier. As an added bonus, you'll feel healthier the next day as you won't be eating those funnel cakes for dessert.
**PROCESSING TIME: 35 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 5 PINTS**
2 quarts water
5 pounds assorted bell peppers, any color, washed, quartered, stemmed, and seeded
5 tablespoons freshly squeezed lemon juice (from about 3 lemons)
21/2 teaspoons canning salt
1. Prepare 5 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a medium-size pot over high heat, bring the water to a boil.
3. Pack the peppers into the hot jars as firmly as possible.
4. Carefully pour or ladle the boiling water over the peppers, leaving 1 inch of headspace in the jars.
5. Add 1 tablespoon of lemon juice and 1/2 teaspoon of canning salt to each jar.
6. Remove air bubbles with a plastic or wooden utensil, adding more boiling water as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 35 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the peppers within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ VEGETABLE SOUP
It's a mystery to me why many children who avoid vegetables come dinnertime will eat a full bowl of vegetable soup. Who am I to argue? I'm always happy to find a way to make sure my kids eat the vegetables I put in front of them. This soup is a personal favorite. It's what I reach for to pack a thermos for lunch when I'm headed out for the day. It's hearty and filling on its own, but you can make it more of a meal with a thick slice of buttered bread. One important note: make sure you use chickpeas or another bean that has been precooked—don't use dried beans in this recipe!
**PROCESSING TIME: 1 HOUR
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 3 PINTS**
1 pint canned tomatoes
2 cups water
1 cup canned corn
1 cup canned and drained chickpeas, or other cooked bean
3 carrots, sliced
11/2 pounds potatoes, peeled and cubed
1 medium-size onion, chopped
1/2 cup chopped celery
2 garlic cloves, minced
Salt and pepper to taste
1 teaspoon dried Italian seasoning
1. Prepare 3 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large heavy stockpot over high heat, combine the tomatoes, water, corn, chickpeas, carrots, potatoes, onion, celery, and garlic. Season with salt and pepper. Stir to combine and bring to a boil. Reduce the heat to a simmer and stir in the Italian seasoning. Simmer the soup for 10 minutes.
3. Increase the heat and bring the soup to a boil again.
4. Carefully ladle the hot soup into the hot jars, leaving 1 inch of headspace.
5. Remove air bubbles with a plastic or wooden utensil, adding more hot soup as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the soup within 2 weeks. Label the remaining jars with the recipe name and date before storing.
## CHAPTER FIVE
## PRESSURE CANNING FRUITS
FRUITS ARE ONE OF THE MOST REWARDING FOODS TO CAN if you love sharing food with family and friends. Everyone loves a sweet treat, whether it's meant to be spread on warm bread, spooned over ice cream, or nibbled as part of an appetizer spread. Unlike canned vegetables, store-bought jams, jellies, and even canned fruit don't have a bad reputation. So why can? Because you can take good to great! And you get the same benefits as canning your vegetables: You know how your food was grown and every ingredient that goes into the jar.
I also enjoy canning fruits because they are precious. Our seasons in northern Maine are very short, so preserving as much as possible means my family can eat a lot more peaches and blueberry pie than we could if limited to the few weeks when the produce is locally available. Canned fruits are also beautiful. It's rewarding to look at a shelf overflowing with peaches, and who doesn't smile when given a jar of freshly canned berries?
As you get going, you'll notice a few differences between home-canned fruit and store-bought. For one, home-canned food doesn't look like factory-canned food—and that's okay. No matter how particular you are at home, perfection is not always picture perfect. Even when your fruits and vegetables are ripe, tasty, and nutritious, at times they are not the same size, shape, and color that is the industry standard. Yet, this doesn't stop canned fruit from being beautiful. The colors are simply wonderful and a cupboard full of colorful glass jars beats metal cans any day.
## SUGAR SYRUP
There are five options for sugar syrup, named according to the amount of sugar in the water. Even when a recipe calls for one type of syrup, you can use a different syrup if you know your family likes sweeter (or less sweet) results. Swapping out sugar syrups is one of the few dramatic changes you can make to these recipes without affecting food safety. Most recipes will tell you to add hot syrup to the jars, so making a syrup ahead of time and refrigerating it will not speed up anything the day you can.
To make any of these syrups, dissolve the sugar in the water as part of the canning process.
**Very light sugar syrup:** 4 cups water to 1/2 cup sugar
**Light sugar syrup:** 4 cups water to 1 cup sugar
**Medium sugar syrup:** 4 cups water to 2 cups sugar
**Heavy sugar syrup:** 4 cups water to 3 cups sugar
**Very heavy sugar syrup:** 4 cups water to 4 cups sugar
Home canning will bring a new appreciation for the actual taste of fruit as well. If you have ever looked on the back of a jar of canned fruit, you can see that there is a laundry list of extra ingredients you certainly never put into your own homemade fruit—thickeners, high fructose corn syrup, salt, the list goes on. In fact, there is no reason even to add heavy sugar syrup to your canned fruits. There are, in fact, many levels of sugar syrups (see above), and the end product is still delicious. (Even if you do prefer a heavy syrup, when you make it yourself, you know it's made from pure cane sugar and water, not high fructose corn syrup.)
### APPLE PIE FILLING
You might think that apple pie filling would simply be Canned Apples with the addition of apple pie spices. While it's true that apples meant for pie should be spiced when canned, there are a couple of other key differences. First, we'll use sugar and Clear Jel instead of a sugar syrup. This helps you get the consistency you want for a pie filling. Second, a hint of lemon juice is added as well. Don't worry, this won't make for a tart apple pie. It helps balance the sweetness and creates a better-quality pie filling. In addition to baking these apples into a classic apple pie, try them in hand pies, cobbler, and filled sweet bread, to name a few.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
6 pounds apples, washed, cored, peeled, and sliced or cubed into even bite-size pieces
2 cups sugar
1/2 cup Clear Jel
2 tablespoons freshly squeezed lemon juice (from about 1 lemon)
2 teaspoons ground cinnamon
1/2 teaspoon ground nutmeg
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over medium heat, combine the apples, sugar, Clear Jel, lemon juice, cinnamon, and nutmeg. Cook until just thickened.
3. Pack the hot jars with the apple pieces and pour the thickened syrup over them, leaving 1/2 inch of headspace.
4. Remove any air bubbles with a plastic or wooden utensil, adding more hot syrup as needed to maintain the proper 1/2-inch headspace.
5. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
6. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
7. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
8. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. One the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the filling within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ CANNED APPLES
We have apple trees in our yard, so I always begin my canned apple recipe by tasting our fresh apples. From there, I know if I want to adjust the taste by mixing them with another type of apple, and also which sugar syrup I want to use. Serve your canned apples warmed, over ice cream, or crush some graham crackers for an impromptu "crust" and top with freshly whipped cream. A (canned) apple a day keeps the doctor away!
**PROCESSING TIME: 10 MINUTES
PRESSURE: 5 POUNDS WEIGHTED GAUGE, 6 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
6 pounds apples, washed, cored, peeled, and sliced into uniform, bite-size pieces
1 recipe sugar syrup of choice (shown here), at a boil
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Pack your hot jars with apple pieces.
3. Carefully pour or ladle the hot syrup over the apples, leaving 1/2 inch of headspace.
4. Remove air bubbles with a plastic or wooden utensil, adding more hot syrup as needed to maintain the proper 1/2-inch headspace.
5. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
6. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
7. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
8. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the apples within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ APPLESAUCE
Apples are positively a delight for canners. They can be canned as sliced fruit, for pie filling, or applesauce—and I've included recipes for all three in this chapter. Part of the secret to the best apple recipes is figuring out what apples you and your family like best. It's generally more interesting to use a mix of apples—some sweeter and some more tart—when you make applesauce. The other thing to consider is the season. While summer apples aren't typically good for canning as sliced fruit, they actually make applesauce that's quite good. They are sweet and cook down quickly, and I've found they tend to produce a silky smooth sauce.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 5 POUNDS WEIGHTED GAUGE, 6 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
6 pounds apples, washed, cored, peeled, and halved
Cinnamon to taste
1 recipe sugar syrup of choice (shown here), at a boil
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large, heavy pot over medium heat, combine the apples and cinnamon, as desired. Cook the apples until very soft, stirring occasionally.
3. Carefully pack the jars with the hot apples.
4. Carefully pour or ladle the hot syrup over the apples, leaving 1/2 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot syrup as needed to maintain the proper 1/2-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the applesauce within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ BLUEBERRY FILLING
Here in Maine a freezer full of wild blueberries is considered normal. Our growing seasons may be short, but they can be bountiful! In my family, we celebrate these tasty berries by eating them in every conceivable way—including straight from the freezer. Of course, I also can some so we'll have luscious pie filling on hand when the season is over. Use this filling to make pies, cobblers, buckles, and tarts, and spoon it over ice cream and yogurt. While this recipe isn't meant to be as thick as a store-bought can of blueberry filling, it will be much more flavorful.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 2 PINTS**
3/4 cup sugar
1 cup water
1/4 cup Clear Jel
31/2 teaspoons freshly squeezed lemon juice (from about 1 lemon)
31/2 cups fresh or frozen and thawed blueberries
1. Prepare 2 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a medium-size saucepan over medium-high heat, combine the sugar, water, and Clear Jel. Cook until the mixture thickens.
3. Stir in the lemon juice and boil for 1 minute, stirring constantly so the mixture doesn't burn.
4. Carefully fold in the blueberries and immediately pack the hot mixture into the hot jars, leaving 1 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more blueberries and hot liquid as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the filling within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ RHUBARB FILLING
Rhubarb used to be an underrated fruit—farmers practically gave it away! However, that has changed in the past decade as people were drawn to its showstopping looks and unique flavor. Since rhubarb contains a decent amount of water, I like to unlock the fruit's moisture to can instead of adding water to the recipe. While adding water is certainly the faster way to can, it results in canned rhubarb with a fraction of the flavor. Instead, just allow the ingredients to sit for 4 hours before you can them. Once you try a bite you'll agree the wait is worth it.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 6 PINTS**
6 pounds rhubarb, leaves removed, sliced into 1/2-inch pieces
3 cups sugar
1. In a large pot, combine the rhubarb pieces and sugar and stir well to combine. Let stand for 4 hours, so the rhubarb can create its own juice.
2. Meanwhile, prepare 6 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
3. After the 4 hours is up, bring the rhubarb and sugar mixture to a boil over medium-high heat. Stir well to prevent any sticking.
4. Carefully pack the rhubarb and hot syrup into the hot jars, leaving 1 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot syrup as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the filling within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ RHUBARB SAUCE
If your family can't get enough rhubarb, or if rhubarb is particularly abundant in your neck of the woods, try this unique preparation. Note that rhubarb sauce is not a topping for ice cream or other desserts (though I suppose you could use it that way). It's actually a spin on applesauce! Sure, in its raw form, rhubarb doesn't taste a lot like apples, but once you cook it, it shares that addictive sweet/tart balance of a good Granny Smith. Take a step off the beaten path and try this rhubarb sauce.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
4 pounds rhubarb, leaves and white end removed and discarded, red and green stem parts chopped into 1/4-inch pieces
4 cups sugar
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over medium-high heat, combine the rhubarb pieces and sugar. Cook until it becomes a thick sauce, stirring often to prevent burning, about 30 minutes.
3. Carefully ladle the boiling-hot sauce into the hot jars, leaving 1 inch of headspace.
4. Remove any air bubbles with a plastic or wooden utensil, adding more hot sauce as needed to maintain the proper 1-inch headspace.
5. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
6. Carefully transfer the filled jars to the rack inside the pressure canner Process the jars at the pressure listed above for 10 minutes.
7. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
8. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the sauce within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ SPICED PEARS
When my family lived in Nebraska, there was a huge mature pear tree in our yard. With more pears than we could eat, I learned to make spiced pears and have included them in my summer canning repertoire ever since. These are lightly seasoned, so the true pear flavor shines through. My biggest piece of advice, besides using delicious ripe pears, is to use fresh spices as well. An aromatic cinn-amon stick and vanilla bean are what you want here—not one languishing for months (or years) in the pantry.
**PROCESSING TIME: 11 MINUTES
PRESSURE: 5 POUNDS WEIGHTED GAUGE, 6 POUNDS DIAL GAUGE
YIELD: 3 PINTS**
6 cups water, divided
2 tablespoons Fruit-Fresh
3 pounds pears, peeled, cored, and quartered (see step 2)
1 cup sugar
1 cinnamon stick
1/2 vanilla bean, split widthwise
1. Prepare 3 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large bowl, mix together 2 cups of water and the Fruit-Fresh. Immediately place the pears as they're cut in the bowl. This helps keep them from browning.
3. In a large pot over medium-high heat, combine the remaining 4 cups of water, the sugar, cinnamon stick, and vanilla bean quarters. Bring the mixture to a boil.
4. Drain the pears and add them to the pot. Return the mixture to a boil and cook for 5 minutes. Remove and discard the vanilla bean.
5. Carefully ladle the boiling-hot pears and liquid into the hot jars, leaving 1/2 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1/2-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 11 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the pears within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ CANNED CHERRIES
When I was a child growing up in upstate New York, our family went cherry picking most every year. Picking the flats of cherries was great fun, but pitting all those cherries was a nightmare. My mother made the work worth it by canning plenty of fresh cherries, which we enjoyed throughout the year. Cherries have such a short season in most parts of the country; you just need to can or freeze them if you love this fruit. These days, even with the invention of the cherry pitter, I tend to stick to one flat of cherries. Their appearance on the table from time to time is a special treat.
**PROCESSING TIME: 8 MINUTES
PRESSURE: 5 POUNDS WEIGHTED GAUGE, 6 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
1 cup sugar
7 cups water
6 pounds pitted cherries
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over medium-high heat, combine the sugar and water. Heat, stirring, until the sugar dissolves completely.
3. Add the pitted cherries to the hot sugar syrup. Return the mixture to a boil and cook for 5 minutes.
4. Carefully ladle the hot cherries and syrup into the hot jars, leaving 1/2 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot syrup as needed to maintain the proper 1/2-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 8 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the cherries within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ CANNED PEACHES
Of all the canned fruits, peaches are one of the most drastically different between homemade and store-bought. (While you might find tasty applesauce at the grocery store, good luck finding canned peaches you truly enjoy!) While most people think of them as a summery fruit, add the right spices to dishes like spiced peach cobbler and there's no arguing it's meant for eating fireside.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 5 POUNDS WEIGHTED GAUGE, 6 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
3 pounds ripe peaches
3 cups water
3/4 cup sugar
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Bring a medium-size pot of water to a boil and prepare a separate ice-water bath in a large bowl. This is to help remove the skins from the peaches—the 3 cups of water listed above is for the recipe.
3. Remove the peach skins: One at a time, place the peaches into the boiling water for 30 seconds. With tongs or a slotted spoon, remove the peach and place it into the bowl of ice water. The peels should come off easily. Repeat with the remaining peaches. Cut the peeled peaches into uniform slices.
4. In a large pot over medium-high heat, combine the 3 cups of water and sugar. Bring to a boil, stirring until the sugar dissolves completely.
5. Add the peaches and return the mixture to a boil.
6. Carefully ladle the hot peaches and syrup liquid into hot jars, leaving 1/2 inch of headspace.
7. Remove any air bubbles with a plastic or wooden utensil, adding more hot syrup as needed to maintain the proper 1/2-inch headspace.
8. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
9. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
10. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
11. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the peaches within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ WHOLE CRANBERRIES
Cranberries aren't a local fruit for most of the country but they sure are here in Maine. We know the owners of the nearby cranberry bogs, and perhaps that makes them taste even better. Cranberries travel well and are easy to find nationwide around Thanksgiving. You can freeze them until you have time to use them, too. I've found there isn't much difference between canning fresh or frozen cranberries. This recipe is designed to be sweet, but not so sweet it completely hides the tartness of the cranberries. Serve it next Thanksgiving topped with fresh orange zest or spread it on a turkey sandwich any time of year.
**PROCESSING TIME: 30 MINUTES
PRESSURE: 5 POUNDS WEIGHTED GAUGE, 6 POUNDS DIAL GAUGE
YIELD: 2 PINTS**
2 cups water
2 cups sugar
4 cups fresh whole cranberries
1. Prepare 2 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over medium-high heat, combine the water and sugar. Bring to a boil, stirring, until the sugar dissolves completely.
3. Fill the hot jars with the raw cranberries, leaving 1 inch of headspace.
4. Once the boiling-hot syrup is ready, carefully ladle it over the cranberries in the jars, leaving 1/2 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot syrup as needed to maintain the proper 1/2-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 30 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the cranberries within 1 week. Label the remaining jars with the recipe name and date before storing.
### CRANBERRY JELLY
Every family is different, but, given the choice, my family is a cranberry jelly crowd through and through. You can keep the chunky sauce—pass us the jelly! In fact, I've often wondered why cranberry jelly isn't available year-round. It makes the perfect condiment to many a dish, and my kids will eat it like a treat. Sure, there's sugar involved, but at least there's some fruit—unlike that chocolate pudding or peanut butter ice cream.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 5 POUNDS WEIGHTED GAUGE, 6 POUNDS DIAL GAUGE
YIELD: 2 PINTS**
4 cups fresh whole cranberries
2 cups sugar
Finely chopped peel of 1 orange
Juice of 1 orange plus enough water to equal 1 cup liquid
1 cinnamon stick
1. Prepare 2 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large pot over medium-high heat (the mixture will foam, so use a larger pot than you think you need), combine the cranberries, sugar, orange peel, orange juice plus water, and cinnamon stick. Cook the mixture until it cooks down just a bit, about 10 minutes.
3. Strain the cranberry mixture through a fine-mesh sieve into a smaller pot, pressing on the berries to remove as much cranberry liquid from the stems, seeds, and peelings as possible. Discard the solids.
4. Bring the strained liquid back to a boil over medium-high heat.
5. Carefully ladle the hot cranberry liquid into the hot jars, leaving 1/2 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, topping off the jars as needed to maintain the proper 1/2-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the jelly within 1 week. Label the remaining jars with the recipe name and date before storing.
### HOMEMADE GRAPE DRINK
I think this is one of the best recipes in the whole book, and it is fun for the entire family. Have you ever tried wild grapes? On our farm, they grow taller than our house and line the entire back of the yard. All summer long, we check and recheck the fruits until after the first frost, when the grapes suddenly go from super tart to sweet, juicy treats. Canning juice is the best way to keep that lovely flavor. Try this recipe with any sweet, juicy grapes you can get your hands on. While this recipe does have a good amount of sugar, you are making a concentrate that tastes better and is, at least, slightly more wholesome than those powdered drink mixes. To use this concentrate, pour the canned juice and grapes from one jar into a mesh strainer, catching the juice in a 1-quart container. Discard the grapes. Add enough cold water to make a full quart. Serve over ice and enjoy!
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 2 PINTS**
3 cups grapes, rinsed
11/4 cups sugar
1. Prepare 2 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Evenly divide the grapes between the 2 hot jars.
3. In a 2-cup (or larger) glass measuring cup, add the sugar and top it off with hot water until it reaches the 2-cup mark. Microwave on high power for a minute or two depending on your microwave's power, and stir until the sugar dissolves. Pour the sugar water over the grapes, leaving 1 inch of headspace.
4. Remove any air bubbles with a plastic or wooden utensil, adding more hot sugar water as needed to maintain the proper 1-inch headspace.
5. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
6. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
7. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
8. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the juice within 2 weeks.
## CHAPTER SIX
## PRESSURE CANNING MEAT
We raise our own meat here on the farm, so we know just how precious and valuable meat truly is. These days, I endeavor to make meat more of a supporting character in recipes than the main star. Yet reducing the amount of meat you eat makes it even more important that the meat you use is of the best quality and flavor.
The same goes when buying meat. More of us than ever are paying attention to where our meat comes from, asking important questions of our producers. How were the animals raised? What were they fed? The answers have a very real impact on the quality of the meat you buy.
Since meat tends to be the most expensive item on any grocery list, avoiding waste is always important. Buying the proper amount for a canning recipe will help you stretch your dollars, as will having a canning recipe at the ready to take advantage of a sale at the grocery store.
_**Note:** While most grocery stores are very good about reducing the price of meat well before a "best by" date, use caution when shopping sales. Even though the meat will be brought to a high temperature when pressure canning, there is no magical "undo" button for meat that is past its prime. Don't take the risk if you find meat with a bad date or that shows other signs it is past its prime. If you wouldn't grill it, don't can it._
Those new to canning may wonder why you should bother canning meat at all. Inferior canned meats, including heavily processed meat and fish, have given canned products a bad name. However, we need only look at gourmet foods and the foods of other countries to realize it doesn't have to be this way. Take, for example, some of the newer "canned" tunas on the market that come in pouches instead of cans. These packages often contain better quality tuna that has received better treatment during processing. Canned tuna can be delicious without drowning it in mayonnaise for tuna salad! Another example is the canned meat and fish in European countries, such as Spain and Portugal. It is not uncommon in a tapas bar to have something locally canned.
When canning meat at home, you're sure to change minds as well. Canning meat locks in all the flavor. While it doesn't necessarily look appetizing or sound appetizing (a jar of juicy cherries will always conjure up a nicer mental image than a jar of juicy meat), the reality is that canned meat enhances recipes. And don't forget about canned recipes that contain meat, too. That's right, in this chapter we'll also make soups and stocks.
While these recipes will guide you in a similar manner as those in the other chapters in this book, this seems like a good place to run through a few general rules and best practices for canning meat and fish.
### THE BASICS
When canning meat, your first task is to find the freshest meat. If you're used to shopping at the grocery store, you may need to talk to a local butcher about the signs of how to spot fresh meat from meat that's a few days old. Common signs of fresh meat include flesh that is firm but not too dry—it bounces back quickly when pressed with a finger. Meat should also not have any strong odors. That goes for fish as well—fishy smelling fish should never be canned. You will also need to confirm the meat hasn't been previously frozen. Frozen meat or fish will have an inferior texture when canned, even if thawed before canning.
Time will be your enemy from start to finish when canning meat. You want to move your meat through the canning cycle as soon as possible and that means getting it from butcher to jars quickly. Bacteria grow fast on meats and moving efficiently through the process is the only way to be safe. Only prepare enough meat for one canner's worth of jars at a time. Do not keep meat sitting at room temperature for too long at any time—including while you make the recipe.
If you do find you have too much meat to can all at once, keep it refrigerated at 32°F to 38°F. Can it no later than the next day or use it for another recipe that doesn't require canning.
There are numerous ways to cut the meat, but, before you decide on that, you'll need to trim the meat well. Remove any bruising, fat, and silverskin you see. Fat can keep your jars from sealing well, and it can also develop a strong flavor with some types of meats. Fatty meats can also spoil more easily than lean meats. Don't worry about removing every speck, but do a thorough job trimming and you will appreciate the result.
Before you cut the meat, review your recipe. It will probably have a suggestion for how to process the meat; deviating greatly from that could make the recommended canning time unsafe. The other consideration is the meat itself. You may want to cut it with or against the grain.
The most common ways to cut meat for canning are into cubes or strips, or grinding it. With cubed meat, the goal is to create those perfect bite-size pieces. Whether the meat will go in a soup recipe or be canned on its own, cutting the pieces as uniform as possible is key. Just like with vegetables or fruit, if you have pieces that are much smaller than the rest, they will overcook—and the big danger is letting a few large cubes slip through. If they don't hit the proper temperature at the center during canning, they can harbor harmful bacteria.
Cutting meat into strips is a common treatment for steaks or roasts. If you're cooking a recipe that calls for strips, I recommend cutting _against the grain_ of the meat so the strips will fit in the jar lengthwise. Cutting against the grain will result in more tender pieces. Just as with cubed meat, cutting uniformly thick strips is key. Do not can any strips that are much thicker than what the recipe specifies.
Ground meat probably looks the least appetizing once canned, yet it's also the most useful type of meat in my pantry. Ground meat is family friendly and used in many comforting recipes, from red sauce to stroganoff. Having canned ground meat on hand also makes for an easy addition to casseroles and soups. Ground meat doesn't have to be beef either. Your butcher will happily grind fresh pork, chicken, or turkey. I recommend seasoning ground meat lightly before canning. As you may use it in any number of recipes once you open the jar, if you have perfectly seasoned beef you may end up oversalting a sauce or stew.
You'll notice many recipes call for browning meat before canning. This step is key, as it's what makes your canned meat taste so much better than the "cooked in the can" variety. The browned layer goes through what is called the _Maillard reaction_ , a complex interaction of sugars and amino acids that is only possible under high heat. Since you'll be fully cooking the meat when canning, the goal is usually just to create that browned layer, not cook the meat all the way through.
To make things easier when packing and unpacking jars, use wide-mouth jars when canning meat. Always leave the proper headspace as stated in the recipe, and do not fill the jars too tightly. Also, while it might be tempting to thicken the canning liquid, resist the urge. You can always thicken it after you open the jars—it's a quick process to turn the thin canned sauce into a rich, thick gravy.
#### THINGS TO AVOID
Following are a few things to avoid when canning meat:
* **A water bath canner.** Canning meat and fish _must be done in a pressure canner_. All meats and seafood are low-acid foods and unsafe to can using a hot water bath canner. You may have heard otherwise, but doing so is highly dangerous. It not only ruins expensive food, but can make you or your loved ones incredibly sick.
* **Thick sauces or gravies.** When canning meats, you cannot include a very thick gravy in the jar. Your meat will release its own juices and you can certainly thicken that juice when reheated, so _any gravy added to canned meats should be thin_. A good rule of thumb is to make any gravy-type liquid no thicker than tomato sauce in the jar. This allows good penetration of heat throughout the jar and into the center of each piece of meat.
* **Anything experimental.** When you can meat, it is important to follow proven recipes from reliable sources. It is no time to try new recipes or tweak the one you have. If needed, adjust the seasoning when it comes out of the jar.
### ◁ CUBED PORK
We have a standing order with our butcher that I pick up each fall. He knows to call me the minute the meat is ready, as I want to get it home quickly to freeze or can. With pork, you have a good amount of leeway in the cuts you choose. As long as the meat is lean, you can cube it and can it according to the following recipe. I tend to can a variety of cuts in different base liquids to use throughout the year. Can pork in tomato juice for the beginning of a stew, in broth, or simply in salted water. The plainer the pork going in, the more versatile it is coming out—but you may find you like it best with the added flavor of broth. Whatever you choose, don't skip the browning! The flavor the pork picks up while browning can't be replicated by searing it after you can it.
**PROCESSING TIME: 1 HOUR, 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
3 pounds lean pork, trimmed of fat and cubed
1 quart water, tomato juice, or broth
1 teaspoon canning salt or 4 cubes of bouillon (optional)
1. Prepare 4 wide-mouth pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large skillet over medium-high heat, lightly brown the pork, stirring, until it's between halfway and fully cooked. Remove the pork from the pan and set aside.
3. In a medium-size saucepan over medium-high heat, heat your liquid of choice until it reaches a boil.
4. Divide the pork evenly among the hot jars. Pour the hot liquid over the top, leaving 1 inch of headspace. If you'd like, add 1/4 teaspoon salt or a bouillon cube to each jar.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 15 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the pork within 1 week. Label the remaining jars with the recipe name and date before storing.
### BACON JAM
How do you improve on jam? The same way you improve on almost any recipe—add bacon! Joking aside, this recipe packs big flavor with its salty-sweet goodness. Garlic, onion, and bacon meet brown sugar and maple—with a hit of tang from cider vinegar. Serve it on fancy crostini or bring it to the breakfast table. It adds punch to a bagel and cream cheese and even goes well with scrambled eggs. One other thing: This recipe is packed in tiny half-pint jars for a reason. It requires a large quantity of good-quality bacon and the last thing you want is to crack open a jar only to have it languish and expire in the fridge. On the upside, the tiny jars make great gifts. Finally, don't be in a rush when making this recipe. It requires time and care to keep the bacon from burning—trust me when I say it's worth every minute.
**PROCESSING TIME: 1 HOUR, 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 12 HALF PINTS**
5 pounds good-quality bacon, cut into 1-inch pieces
5 large yellow onions, sliced into 1/4-inch-thick slices
10 garlic cloves, minced
11/2 cups very strong coffee
1 cup apple cider vinegar
1 cup packed brown sugar
1/2 cup pure maple syrup
Pepper to taste
1. In a large skillet over medium heat cook the bacon, in batches so you don't crowd the pan, until it starts to crisp, being careful not to burn it. Drain the excess fat (you may need to do this more than once), reserving 2 tablespoons. Set the bacon aside.
2. In a large pot over medium heat, combine the reserved bacon fat, onions, and garlic. Cook, stirring frequently, until the onions are translucent and the garlic is soft.
3. Stir in the coffee, vinegar, brown sugar, maple syrup, and pepper and bring the mixture to a simmer.
4. Add the cooked bacon to the onion mixture. Reduce the heat until the mixture is at a low simmer and cook for 1 hour, stirring often. If the bacon is not tender after 1 hour, continue to cook for a few minutes longer until the bacon is tender.
5. While the bacon cooks, prepare 12 wide-mouth half-pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
6. Remove the bacon-onion mixture from the heat and let sit for 15 minutes to cool slightly.
7. Once the mixture cools slightly, use an immersion blender (or transfer the jam to a food processor and pulse) to break up the mixture just a bit. You want small pieces of bacon and onion, but not a smooth purée.
8. Carefully ladle the hot bacon-onion mixture into the hot jars, leaving 1 inch of headspace.
9. Remove any air bubbles with a plastic or wooden utensil, adding more hot jam as needed to maintain the proper 1-inch headspace.
10. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
11. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 15 minutes.
12. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
13. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the bacon jam within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ GROUND MEAT
Folks, this is it: the most useful (and some say the tastiest) protein in my pantry. It's always a family-wide disappointment when we run out of ground meat as I use homemade beef, pork, chicken, and turkey in a wide variety of recipes. You'll see every bit of flavor is kept in the jars during the canning process. With all that flavor, there's no need to oversalt the meat, or your dish. In fact, adding too much salt is a classic mistake—salty meat added to a perfectly seasoned soup or sauce can ruin it. As with all meats, I recommend adjusting the seasonings only after cooking and canning safely.
**PROCESSING TIME: 1 HOUR, 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
4 pounds ground meat of choice
1 quart water, tomato juice, or broth
1 teaspoon canning salt or 4 cubes of bouillon (optional)
1. Prepare 4 wide-mouth pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large skillet over medium-high heat, and working in batches if necessary, fry the meat until it is lightly browned and about halfway cooked. Don't overload your pan trying to cook all the meat at once—the browning is crucial to this recipe.
3. In a medium-size saucepan over medium-high heat, heat your liquid of choice until it reaches a boil.
4. Divide the ground meat among the jars, packing it loosely. Pour the hot liquid into the hot jars over the meat, leaving 1 inch of headspace. If you'd like, add 1/4 teaspoon salt or a bouillon cube to each jar.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot liquid as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 15 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the meat within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ CANNED CHICKEN
Commercially made canned chicken must be one of the more dreadful canned meats. Chicken doesn't have the strong flavor of foods like tuna to cover the flavor imparted by the can. As if the producers wanted to make it worse, the quality of meat used for canned chicken is often not as high as chicken sold fresh. It may well be a blend of cuts, labeled as "with rib meat," or similar. Obviously, homemade canned chicken doesn't suffer from the same problems. You choose the cuts and quality and you process it in jars instead of cans. This chicken is ready straight from the can, and not just for chicken salad. Try it in any recipe where you'd use a store-bought rotisserie chicken. My family is partial to wrapping it as chicken verde burritos.
**PROCESSING TIME: 1 HOUR, 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
4 pounds boneless, skinless chicken, trimmed of fat and cut into 1-inch cubes
1 teaspoon canning salt (optional)
1. Prepare 4 wide-mouth pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Fill the jars loosely with the raw chicken, leaving 11/4 inches of headspace (see Note). Add 1/4 tsp salt to each jar, if desired.
3. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
4. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 15 minutes.
5. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
6. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the chicken within 1 week. Label the remaining jars with the recipe name and date before storing.
#### NOTE
_Yes, the headspace for this recipe is more than 1 inch, so there's a bit of extra room in the jars. The chicken will release its natural juices during cooking, so you will not need to add water before canning._
### ◁ CANNED HADDOCK
Living on the coast, canning seafood and fish just makes sense. It's a "local" food, after all. Although I freeze much of it, having it ready to eat on the shelf provides another option. It's become a yearly tradition for my family to order haddock from a local fisherman and can it right after the catch. I use this mild-flavored fish like I would tuna: for fish burgers, in fish soup, and as the star of savory chowders. By keeping the flavoring plain when canning, it's easy to spice it up however I need to for a meal. I can dozens of jars, but there is no need to save so much of it if your family doesn't eat as much fish. If you can buy fresh fish in any quantity, try to save some for your shelf. It's wonderful!
**PROCESSING TIME: 1 HOUR, 40 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE |
YIELD: 6 PINTS**
12 pounds haddock, or similar freshwater fish fillets
6 teaspoons canning salt
1. Prepare 6 wide-mouth pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Make sure your fillets don't have any stray entrails, scales, or pieces of the head, tail, or fins. Cut the fish into pieces short enough to fit in the jars, keeping in mind the 1-inch required headspace. Cut the pieces as close as possible to uniform thickness. Fill the jars with the fish, placing the pieces around the outside of the jar for a neater look, and filling the center with the less than perfect pieces, leaving 1 inch of headspace.
3. Add 1 teaspoon of salt to each jar. Like canned chicken, you do not need to add any water to fill the jars.
4. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
5. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 40 minutes.
6. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
7. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the fish within 1 week. Label the remaining jars with the recipe name and date before storing.
## BE PREPARED WHEN CANNING FISH
Canning fish is not something I recommend for the beginning canner. It's best to start canning easier items, like vegetables and fruits, move on to meat and soups, and finally try your hand at fish.
Being a delicate meat, fish needs to be prepared as quickly and carefully as possible. Don't wait until the day you are going to buy or catch your fish for canning; be sure you have all your supplies beforehand.
Here on the coast we contact our local haddock fisherman and make plans to pick up 50 pounds of haddock on a specific time and date. I prepare my canning equipment and check that I have enough lids. (Because for some reason I never do!) When my husband brings home a heavy box of haddock fillets, I am ready to get to work.
### THANKSGIVING TURKEY SOUP
This soup tastes like Thanksgiving in a jar. I often use the turkey bones from the big day to make the broth and throw in a little of the leftover turkey from our meal. Since I brine my turkey the night before, the meat starts out moist. I experimented by adding the cranberries one year and now I do it every time. Try it and you'll see—they seem to pull the whole recipe together.
**PROCESSING TIME: 1 HOUR
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
2 quarts Chicken or Turkey Broth, or store-bought broth
1/2 cup sliced celery
1/2 cup sliced carrot
1/2 large yellow onion, chopped
1 cup fresh or canned and drained corn
1/2 cup chopped fresh cranberries
1 cup shredded cooked turkey
1 cup Great Northern beans, cooked and drained
1 fresh sage leaf, or 1/2 teaspoon dried sage
Salt and pepper to taste
1. Prepare 4 wide-mouth pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large stockpot over medium-high heat, combine the broth, celery, carrot, onion, corn, and cranberries. Cover the pot and bring to a simmer.
3. Add the turkey, beans, and sage. Season with salt and pepper. Continue cooking until the soup comes to a boil.
4. Carefully ladle the hot soup into the hot jars, leaving 1 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot soup as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the soup within 1 week. Label the remaining jars with the recipe name and date before storing.
## PRESSURE CANNING SOUPS
There is nothing like a comforting bowl of soup after a long day at work, as the perfect lunch, or when you don't feel well. Like most foods, home-canned soups taste better than store-bought cans—and you can leave out the questionable preservatives.
Making soups for canning is almost the same as making them to serve immediately for dinner. There are just a few simple rules to follow to be sure your soups are canned safely.
1. Do not add starches such as noodles and rice to your canned soups. These foods become mushy and fall apart during the canning process. Even if the soups look good after the jars come out of the canner, reheating the soup will result in a breakdown of the starches and an undesirable mushy texture. Add the rice or noodles to the soup after you crack the jar to serve it.
2. When canning soups, add more broth than you would when making it on the stovetop. I fill my jars one-fourth of the way with vegetables and meat and the rest of the way with broth.
3. Cooled soup may thicken slightly in the jar when using homemade meat broths. If this happens, it's completely normal. Once you reheat the soup, it will quickly thin again.
4. Some ingredients do not belong in canned soups as they become so strong in flavor they overwhelm the recipe—broccoli, cabbage, and brussels sprouts all taste too intense after canning, in my opinion. If you want to experiment with these ingredients, use less than you normally would so they don't overpower your recipe.
5. When adding dried beans to your soups, cook them before adding to the recipe. Dry beans absorb quite a bit of moisture and will absorb too much broth.
### MEXICAN CHICKEN SOUP
Slightly spicy, this soup is just right with a piece of fresh cornbread. I love serving it with a dollop of sour cream as well. Make it as spicy as you like. My family falls into the "medium heat" category, but the base recipe here is as mild as the taco seasoning you use. To spice it up, leave in some or all of the jalapeño seeds and add a few dashes of hot sauce when serving.
**PROCESSING TIME: 1 HOUR
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
2 quarts Chicken or Turkey Broth, or store-bought broth
1/2 cup sliced celery
1/2 large yellow onion, chopped
2 garlic cloves, sliced
1 cup diced tomatoes
1 cup fresh or canned and drained corn
1 jalapeño pepper, seeded and minced
1 cup shredded cooked chicken
1 cup canned and drained black beans
2 teaspoons taco seasoning
Salt to taste
1. Prepare 4 wide-mouth pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large stockpot over medium-high heat, combine the broth, celery, onion, garlic, tomatoes, corn, and jalapeño. Cover the pot and bring to a simmer.
3. Stir in the chicken, black beans, and taco seasoning. Taste and add salt, if desired. Continue to heat the broth to a boil.
4. Carefully ladle the hot soup into the hot jars, leaving 1 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot soup as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the soup within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ CHICKEN SOUP
I first started canning my own soup when I realized how much better the homemade versions of my childhood soups tasted. It's also a convenient meal: a pint of chicken soup and a couple of grilled sandwiches make for a filling meal in minutes. It saves me time and money for months if I can a few dozen jars for the winter. You may substitute your favorite vegetables in this recipe, but do _not_ add any starch, such as rice or noodles. Starches break down during the canning process.
**PROCESSING TIME: 1 HOUR
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 6 PINTS**
11/2 quarts Chicken or Turkey Broth, or store-bought broth
2 cups canned and drained corn kernels
1 cup chopped onion
1 cup sliced carrot
1 cup sliced celery
1 cup fresh or frozen peas
1 cup canned and drained chickpeas
1 whole bay leaf,
2 fresh sage leaves, crumbled
Salt and pepper to taste
2 cups chopped cooked chicken
1. Prepare 6 wide-mouth pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large stockpot over medium-high heat, combine the broth, corn, onion, carrot, celery, peas, chickpeas, bay leaf, and sage. Season with salt and pepper. Bring the soup to a simmer. Taste and adjust the seasonings, as you like.
3. Using a funnel to keep the rims clean, divide the chicken evenly among the jars, filling them about one-fourth full.
4. Carefully pour the hot soup over the chicken and fill the jars, leaving 1 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding liquid as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the soup within 1 week. Label the remaining jars with the recipe name and date before storing.
### BEEF SOUP WITH LENTILS
Beef soup with lentils is a go-to dish when I want to get my kids to eat some beans. This is the heartiest soup I make, which makes it perfect after a long day of sledding or climbing around the beach during low tide in winter. It's filling on its own but I recommend serving it with a thick slice of homemade bread on the side.
**PROCESSING TIME: 1 HOUR
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
2 quarts broth of your choice
1 cup peeled and cubed potato
1/2 cup sliced celery
1/2 large yellow onion, chopped
1/2 cup sliced carrot
1/2 cup fresh or canned and drained corn
1 cup shredded cooked beef
1 cup lentils, cooked and drained
1 bay leaf
Salt and pepper to taste
1. Prepare 4 wide-mouth pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a large stockpot over medium-high heat, combine the broth, potato, celery, onion, carrot, and corn. Cover the pot and bring to a simmer.
3. Add the beef, lentils, and bay leaf. Season with salt and pepper. Continue cooking until the soup comes to a boil.
4. Carefully ladle the hot soup into the hot jars, leaving 1 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot soup as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the soup within 1 week. Label the remaining jars with the recipe name and date before storing.
### MEATBALLS IN TOMATO JUICE
Nobody will turn down homemade meat sauce. Yet I think 9 out of 10 people prefer red sauce with meatballs. While I normally serve these simply, with buttered noodles, it also works as the base of a tomato and vegetable soup. Since you'll be forming meatballs instead of canning ground meat, you will need to pay careful attention both to how you pack the meatballs and how consistent in size they are. Use a large spoon or small ice cream scoop that gives you perfect 1-inch meatballs. That way, if you get a bit distracted toward the end, you will still maintain that all-important consistency.
**PROCESSING TIME: 1 HOUR, 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
3 pounds ground beef
1/2 cup minced onion
3 cups soft (fresh) bread crumbs
3 eggs
Salt and pepper to taste
1 quart tomato juice
1. Preheat the oven to 450°F.
2. Meanwhile, prepare 4 wide-mouth pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
3. In a large bowl, combine the ground meat, onion, bread crumbs, and eggs. Season with salt and pepper. Mix by hand until well combined. Using a small ice cream scoop, or by hand, form the meat mixture into 1-inch meatballs and place them on a jelly-roll pan or rimmed baking sheet. Bake for 15 minutes. Remove from the oven and set aside.
4. In a medium-size saucepan over medium heat, heat the tomato juice until it boils.
5. Pack the meatballs into the hot jars. Carefully pour the hot juice over the meatballs and fill the jars, leaving 1 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, adding more hot juice as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 1 hour, 15 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the meatballs within 1 week. Label the remaining jars with the recipe name and date before storing.
## CHAPTER SEVEN
## SALSA, SAUCE, AND BROTH RECIPES
NO MATTER HOW MANY OF THE RECIPES in this chapter I can and save, by the end of the winter they're always gone. The recipe for classic Tomato Sauce is a great example. As is, it has just enough seasoning to go a number of ways when you're ready to eat it. You can add seasoned meat to it or a mix of fresh stir-fried vegetables.
Canning salsa is a much-anticipated activity on our farm. I make a big deal out of how hot it might end up tasting each year, as we grow many types of hot peppers. My recipe is super simple, too. It showcases the fresh peppers, onions, and ripe tomatoes.
Wait, didn't I say these recipes make meals in the winter? How does salsa get a meal going? Well, in addition to spicing up taco night, it makes a flavorful addition to soups, can be the start of a simmer sauce for meats, and it's the perfect condiment for healthy veggie bowls.
### ◁ TOMATO SAUCE
This basic sauce is the beginning of a long list of meals around here. We tend to have pasta once a week, and I also use this red sauce to start many a chili, soup, and casserole. A well-seasoned sauce is one of the most useful foods on the pantry shelf.
**PROCESSING TIME: 25 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 5 PINTS**
2 medium-size onions, chopped
2 garlic cloves, minced
2 carrots, shredded
2 celery stalks, finely chopped
8 ounces mushrooms, sliced
2 quarts stewed tomatoes (canned is fine), including the liquid
2 (6-ounce) cans tomato paste
1 cup dry red wine
1 cup chopped fresh parsley leaves
1 tablespoon chopped fresh basil leaves
1 teaspoon salt (optional)
Freshly ground pepper
1. In a large, heavy-bottomed pot over medium heat, combine the onions, garlic, carrots, celery, and mushrooms. Cook, stirring frequently, until the vegetables start to sweat and soften.
2. Stir in the tomatoes, tomato paste, wine, parsley, basil, and salt (if using). Season with pepper. Stir again to combine. Cook the sauce until it comes to a boil, stirring frequently. Once the sauce reaches a boil, immediately reduce the heat to low and simmer for 2 hours, uncovered, stirring often.
3. Meanwhile, prepare 5 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
4. Carefully ladle hot sauce into the hot jars, leaving 1 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot sauce as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 25 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the sauce within 2 weeks. Label the remaining jars with the recipe name and date before storing.
### ◁ TACO SAUCE
Our version of Mexican night is pretty tame by most standards, but we love the slightly spicy taste of this recipe. Use it with the ground meat of your choice or get creative: I've found I can get as many vegetables as I want into my kids when I flavor them with this sauce. That said, it's not authentic. If you are a purist, be forewarned!
**PROCESSING TIME: 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
8 cups peeled, chopped, and drained tomatoes,
1 medium-size onion, chopped
1 cup chopped scallion
1/2 cup chopped seeded jalapeño peppers, or other green chile
4 garlic cloves, chopped
1 teaspoon salt
1 teaspoon chili powder
1/2 teaspoon ground cumin
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. In a saucepan over medium heat, combine the tomatoes, onion, scallion, jalapeños, garlic, salt, chili powder, and cumin. Stir to combine and bring to a simmer. Simmer for 30 minutes, stirring frequently.
3. Strain the sauce through a fine-mesh strainer and return it to the pan. Bring the sauce to a boil and cook for 5 minutes.
4. Carefully ladle the hot sauce into the hot jars, leaving 1 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot sauce as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 15 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. One the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the sauce within 2 weeks.
### ◁ TOMATO SALSA
Bright with the flavor of perfectly ripe summer tomatoes, this is the most versatile salsa I have ever made. I use it for everything from chips and salsa to seasoning my pot roast in the slow cooker. Make it as hot or mild as you like. I try to put up some with different spice levels to satisfy everyone.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
10 ripe paste-type tomatoes, such as Roma
2 onions, chopped
2 green bell peppers, seeded and chopped
2 chile peppers, seeded and chopped (or include the seeds for more heat)
1 jalapeño pepper, seeded and chopped (or include the seeds for more heat)
2 garlic cloves, minced
1/4 cup freshly squeezed lemon juice (from about 2 lemons)
1 tablespoon salt (or to taste)
1 teaspoon freshly ground black pepper (or to taste)
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Prepare an ice-water bath and bring a medium-size saucepan of water to a boil.
3. Remove the tomato skins: Working 1 tomato at a time for best results, place the tomato into the boiling water for 1 minute. Transfer to the ice-water bath. The skins will split and peel off easily. Chop the peeled tomatoes.
4. Empty the pan and return it to medium-high heat. In it, combine the tomatoes, onions, green bell peppers, chile and jalapeño peppers, and garlic. Stir in the lemon juice. Bring the salsa to a boil. Once the salsa is warm, taste and season with salt and pepper as needed. Reduce the heat to low and simmer the salsa for 15 minutes, stirring frequently.
5. Carefully ladle the hot salsa into the hot jars, leaving 3/4 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, adding more hot salsa as needed to maintain the proper 3/4-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the salsa within 2 weeks.
### ◁ PEACH TOMATO SALSA
While I normally advocate only the freshest fruits and vegetables when canning, this salsa is an exception. It is a wonderful way to use peaches that aren't quite ripe. As with most fruit salsas, this is a great condiment not only for tacos, but for fish and chicken as well.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 6 PINTS**
2 pounds paste-type tomatoes, such as Roma
3 pounds peaches, chopped
21/2 cups chopped onion
2 cups chopped green bell pepper
2 cups chopped peeled apple
1 tablespoon salt
1 teaspoon red pepper flakes
33/4 cups packed light brown sugar
21/4 cups apple cider vinegar
1/4 cup pickling spice, tied in a muslin bag
1. Prepare 6 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Prepare an ice-water bath and bring a large saucepan of water to a boil.
3. Remove the tomato skins: Working 1 tomato at a time for best results, place the tomato into the boiling water for 1 minute. Transfer to the ice-water bath. The skins will split and peel off easily. Chop the peeled tomatoes.
4. Empty the pan and return it to medium-high heat. In it, combine the tomatoes, peaches, onion, green bell pepper, apple, salt, red pepper flakes, brown sugar, vinegar, and the muslin bag containing the pickling spices.
5. As the mixture starts to warm, stir to incorporate everything. Bring the mixture to a boil, stirring frequently to keep it from burning. Once the mixture boils, reduce the heat to low, and simmer for 30 minutes.
6. Remove and discard the muslin bag and turn off the heat.
7. Carefully ladle the salsa into the hot jars, leaving 1/2 inch of headspace.
8. Remove any air bubbles with a plastic or wooden utensil, adding more hot salsa as needed to maintain the proper 1/2-inch headspace.
9. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
10. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
11. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
12. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the salsa within 1 week.
### ◁ TOMATILLO SALSA
For a change of pace, I like to make and serve this green, not-so-spicy salsa for everyday eating. If you're unfamiliar with tomatillo salsa, it is quite a departure from a classic red tomato salsa. The tomatillos have a unique flavor to start—tart, fruity, and slightly herbal. Throw in a good amount of vinegar and cilantro and you end up with a bright, fresh flavor. Use it as you would any salsa, as a dip or a topping for tacos. Or try a recipe that features the salsa, such as enchiladas verde.
**PROCESSING TIME: 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 2 PINTS**
2 pounds tomatillos, cleaned, cored, and diced into small pieces
1 large onion, roughly chopped
4 large Anaheim chiles, seeded
4 garlic cloves, peeled
1 cup distilled white vinegar
1/4 cup freshly squeezed lime juice (from about 2 limes)
2 tablespoons minced fresh cilantro leaves
2 teaspoons ground cumin
1/2 teaspoon canning salt
1/2 teaspoon red pepper flakes
1. Prepare 2 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Place the tomatillos in a large pot.
3. In a food processor, combine the onion, chiles, and garlic. Pulse a few times until the vegetables are broken into very small pieces. Add them to the tomatillos. Bring the mixture to a boil over high heat.
4. As the mixture starts to warm, add the vinegar, lime juice, cilantro, cumin, canning salt, and red pepper flakes. Stir well to incorporate the ingredients. Once the salsa boils, reduce the heat to low, and simmer for 10 minutes.
5. Carefully ladle the hot salsa into the hot jars, leaving 1 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, adding more hot salsa as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 15 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the salsa within 2 weeks.
### ◁ GREEN TOMATO CHUTNEY
No matter how early I plant my tomatoes, I always end up with plenty that don't have a hint of red on them by first frost. I used to be upset by these stragglers until I found recipes like this that put green tomatoes to good use. Now, I'm actually quite happy if there are plenty of green tomatoes remaining at the end of gardening season.
**PROCESSING TIME: 10 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
10 paste-type tomatoes, such as Roma
2 onions, chopped
2 green bell peppers, seeded and chopped
2 garlic cloves, minced
2 chile peppers, such as Thai chiles, seeded and chopped (or include the seeds for more heat)
1 jalapeño pepper, seeded and chopped (or include the seeds for more heat)
1/4 cup freshly squeezed lemon juice (from about 2 lemons)
1 tablespoon salt
1 teaspoon freshly ground black pepper
2 tablespoons chopped fresh cilantro leaves
1. Prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Prepare an ice-water bath and bring a large pot of water to a boil.
3. Remove the tomato skins: Working 1 tomato at a time for best results, place the tomato into the boiling water for 1 minute. Transfer to the ice-water bath. The skins will split and peel off easily. Chop the peeled tomatoes.
4. Empty the pot and return it to medium-high heat. In it, combine the tomatoes, onions, green bell pepper, garlic, chile and jalapeño peppers, lemon juice, salt, pepper, and cilantro. Stir to combine. Bring the mixture to a boil. Reduce the heat to low and simmer for 15 minutes, stirring frequently.
5. Carefully ladle the hot chutney into the hot jars, leaving 3/4 inch of headspace.
6. Remove air bubbles with a plastic or wooden utensil, adding more hot chutney as needed to maintain the proper 3/4-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 10 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the chutney within 2 weeks.
### BARBECUE SAUCE
Every spring, we break out the grill long before the last snow. It must be funny to see us gathered around a smoking grill in our winter coats! With such a strong urge to grill, it's no surprise we use more barbecue sauce than the average family. I have to put up quite a bit so we don't run out.
Barbecue sauces vary by region and by personal preference, so I suppose I should tell you what our house sauce is like—it's thick but not too sweet, with just enough smoky flavor to enhance a good cut of meat. We usually wait until the food is cooked and add the sauce during the last minute or two. That way it flavors well, but the sugars in the sauce don't have a chance to burn.
**PROCESSING TIME: 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 2 PINTS**
2 quarts Roma tomatoes
1 cup chopped onion
1 green bell pepper, seeded and chopped
1 hot pepper, such as jalapeño or serrano, seeded and minced
1 garlic clove, minced
1/2 cup packed dark brown sugar
11/2 teaspoons smoked paprika
11/2 teaspoons salt
1/2 cup plus 2 table-spoons white or apple cider vinegar, 5%
1. Prepare 2 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
2. Prepare an ice-water bath and bring a large saucepan of water to a boil.
3. Remove the tomato skins: Working 1 tomato at a time for best results, place the tomato into the boiling water for 1 minute. Transfer to the ice-water bath. The skins will split and peel off easily. Chop the peeled tomatoes.
4. In a large stockpot over medium heat, combine the chopped tomatoes, onion, green bell pepper, hot pepper, garlic, brown sugar, paprika, salt, and vinegar. Cook for about 30 minutes, until the vegetables are very soft.
5. Remove the pot from the heat and use an immersion blender or food mill to purée the sauce as much as possible. Strain the sauce through a fine-mesh to remove any large pieces. At this point you should have a nice smooth sauce. Return the sauce to the heat and simmer until it has reduced by half, about 45 minutes.
6. Carefully ladle the hot barbecue sauce into the hot jars, leaving 1/2 inch of headspace.
7. Remove any air bubbles with a plastic or wooden utensil, adding more hot barbecue sauce as needed to maintain the proper 1/2-inch headspace.
8. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
9. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 15 minutes.
10. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
11. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the barbecue sauce within 2 weeks.
### TOMATO KETCHUP
Richer in tomato flavor and much less sweet, this is a reason to make french fries! Perhaps the most-used condiment on the farm, homemade ketchup isn't in the same category as the store-bought variety. If you have a picky eater or two who must have their favorite brand, you may not convert them. However, I've found that most who have tried homemade ketchup never look back.
**PROCESSING TIME: 15 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
13 pounds tomatoes, chopped
3 onions, chopped
4 cups sugar
2 cups white or apple cider vinegar, 5%
3 teaspoons salt
1/2 teaspoon ground cloves
1/2 teaspoon ground cinnamon
1/2 teaspoon dry mustard
1/2 teaspoon red pepper flakes
1. In a large stockpot over medium heat, combine the tomatoes and onions. Cook, stirring frequently, until the onions are soft.
2. Transfer the mixture to a cloth juice bag and hang above the pot or a large bowl. Let drain for 2 hours.
3. Meanwhile, prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
4. Run the drained pulp through a food mill or blend with an immersion blender to smooth out any seeds and skins. Return the smooth pulp mixture to the pot and place it over medium-high heat. Stir in the sugar, vinegar, salt, cloves, cinnamon, mustard, and red pepper flakes. Bring to a boil and boil for 10 minutes, stirring frequently.
5. Carefully ladle the hot ketchup into the hot jars, leaving 3/4 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, adding more hot ketchup as needed to maintain the proper 3/4-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 15 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the ketchup within 2 weeks.
### ◁ CHICKEN OR TURKEY BROTH
Both chicken and turkey soup are frequent menu items on our farm. Yet I make this broth quite often because it's so useful beyond just the expected soups. A cup of broth with a pinch of cayenne pepper soothes even the worst cold. And my college-age children always ask for jars of it take back to their dorm rooms. I'm sure it ends up the base of many quick and easy dinners! Make your turkey broth after the holidays and get the most out of the bird—truly using every bit of it. For this recipe, make the broth the day before you can it so you can chill it and skim off the fat. You can use a 10- to 15-pound carcass for this or you can use cuts with meat. Just remove the meat and use it for something else after cooking instead of canning it.
**PROCESSING TIME: 20 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 8 PINTS**
3 pounds chicken or turkey pieces
1 gallon water
2 celery stalks
2 onions, halved, skin on
15 peppercorns
3 whole bay leaves
Salt to taste
1. In a large stockpot over high heat, combine the poultry pieces and water. Bring to a boil.
2. Add the celery, onions, peppercorns, bay leaves, and salt. Reduce the heat to low and simmer, covered, for 2 hours, stirring occasionally.
3. Strain the hot broth through a cheesecloth-lined colander or fine-mesh sieve. Chill overnight.
4. The next day, remove and discard the fat layer on top before proceeding with the recipe. Reheat the broth to boiling.
5. Meanwhile, prepare 8 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
6. Carefully ladle the hot broth into the hot jars, leaving 1 inch of headspace.
7. Remove any air bubbles with a plastic or wooden utensil, adding more hot broth as needed to maintain the proper 1-inch headspace.
8. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
9. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 20 minutes.
10. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
11. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the broth within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ BEEF BONE BROTH
Beef broth is perfect for those cold winter meals. It's a delicious treat on its own in a mug (just salt to taste) and is the ideal base for your favorite stew. Rather than shop for this recipe, I try to save all my beef bones in the freezer, until I have enough for a batch of stock. The good news is, no bones are better than others for this recipe—save all of them! Be sure to remove any visible fat before freezing and store them in an airtight freezer bag. For this recipe, make the broth the day before you want to can it so you can chill it and skim off the fat. If you have a couple of days and a slow cooker, this recipe can be made that way as well. The flavor develops beautifully if cooked overnight.
**PROCESSING TIME: 25 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 8 PINTS**
3 pounds assorted beef bones
1 gallon cold water
1 large yellow onion, halved, skin on
2 garlic cloves
Salt and pepper to taste (optional)
1. In a large stockpot over high heat, combine the bones, cold water, onion, and garlic. Bring to a boil. Reduce the heat to low, cover the pot, and simmer for 4 hours. Alternatively, place all ingredients in a slow cooker set on low heat, cover the cooker, and cook for at least 8 hours, or overnight.
2. Strain the broth through a cheesecloth-lined colander or fine-mesh sieve. Taste and season with salt or pepper, if desired. Chill overnight.
3. The next day, remove and discard the fat layer on top before proceeding with the recipe. Reheat the broth to boiling.
4. Meanwhile, prepare 8 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
5. Carefully ladle the hot broth into the hot jars, leaving 1 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, adding liquid as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 25 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the broth within 1 week. Label the remaining jars with the recipe name and date before storing.
### ◁ GARLIC BROTH
Broth is perhaps not the most glamorous of foods to can, but it is one of the nicest things to have on the shelf. Garlic broth is my little secret for restorative dishes. My favorite thing to do for a sick family member is to serve a mug of this with a few red pepper flakes floating in it. However, you can also use it in many recipes in place of Chicken or Turkey Broth. It's a great way to make your vegetable soups vegan without sacrificing the deep flavors that meat-based broths bring to a recipe.
**PROCESSING TIME: 20 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 8 PINTS**
20 garlic cloves, peeled
1 gallon water
2 onions, halved, skin on
Salt and pepper to taste
1. In a large stockpot over high heat, Combine the garlic and water. Bring to a boil.
2. Add the onions and reduce the heat to low. Simmer for 2 hours, stirring occasionally.
3. Meanwhile, prepare 8 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
4. Carefully strain the hot broth through a cheesecloth-lined colander. Taste, and season with salt and pepper. Return the broth to the pot and heat over high heat until it boils. Turn off the heat.
5. Carefully ladle the hot broth into the hot jars, leaving 1 inch of headspace.
6. Remove any air bubbles with a plastic or wooden utensil, adding more hot broth as needed to maintain the proper 1-inch headspace.
7. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
8. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 20 minutes.
9. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
10. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the broth within 1 week.
### ◁ VEGETABLE BROTH
The following vegetable broth recipe is my base—but it's a bit different each time I make it when it comes to the pound and a half of vegetables. That's because, just as with the Beef Bone Broth, the base is built on extras that I stash away. I keep a freezer bag in my freezer to add to throughout the month. I throw in leftover vegetables, like half an onion, the tops of carrots, and occasionally, extras from the farmers' market. When that bag is full, I empty it in a stockpot and cover everything with water. This mix of vegetables makes a wonderful broth for many recipes. Naturally, this recipe should be the foundation of your next vegetable stew, but don't stop there. Use this broth for bean soups or try it in a recipe in place of chicken broth.
**PROCESSING TIME: 20 MINUTES
PRESSURE: 10 POUNDS WEIGHTED GAUGE, 11 POUNDS DIAL GAUGE
YIELD: 4 PINTS**
2 quarts water
11/2 pounds assorted vegetables
2 large carrots, peeled and halved
1 celery stalk
1 onion, halved, skin on
2 garlic cloves
3 whole bay leaves
Salt and pepper to taste
1. In a large stockpot over high heat, combine all the ingredients and season with salt and pepper. Bring to a boil. Reduce the heat to low and simmer for 2 hours, stirring occasionally.
2. Meanwhile, prepare 4 pint jars and the canner: Clean the jars and prepare the 2-piece lids according to the manufacturer's guidelines. Keep the jars in hot but not boiling water until you're ready to use them. Prepare the canner by filling it with 2 to 3 inches of water and bringing it to a simmer, or according to your manufacturer's directions.
3. Carefully strain the hot broth through a cheesecloth-lined colander. Return the broth to the pot and heat over high heat until it boils. Turn off the heat.
4. Carefully ladle the hot broth into the hot jars, leaving 1 inch of headspace.
5. Remove any air bubbles with a plastic or wooden utensil, adding more hot broth as needed to maintain the proper 1-inch headspace.
6. Wipe the rims and seal the jars hand-tight with the 2-piece lids.
7. Carefully transfer the filled jars to the rack inside the pressure canner. Process the jars at the pressure listed above for 20 minutes.
8. Let the canner return to 0 pounds pressure. Wait 10 minutes more, then carefully open the canner lid according to the manufacturer's instructions.
9. With a jar lifter, remove the jars and place them on a clean dishtowel away from any drafts. Once the jars cool to room temperature, check the seals. If any jars have not sealed, refrigerate them and use the broth within 1 week.
## RESOURCES
### CANNING SUPPLIES
The internet makes it simple to find and buy anything you need for proper canning. Here are some of my favorite places to buy products that I actually can use and that will last.
**Canning Supply:** www.canningsupply.com
Pressure canners to jelly bags, you can find all of it, and things you never thought of.
**Fresh Preserving:** www.freshpreserving.com
For everything canning, this website from Ball is the place to go.
**Kitchen Krafts:** www.kitchenkrafts.com/category/home-canning-tools-and-supplies This site has a huge inventory of canning supplies. I love it because when all other canning supply companies are backordered, Kitchen Krafts seems to come through every time.
**Pressure Cooker Outlet:** www.pressurecooker-outlet.com/Canning-Supplies.htm Pressure canners that you will dream about are found at this site.
**Safe Canning Practices:**
nchfp.uga.edu/how/can_home.html
The National Center for Home Food Preservation is your source for current research-based recommendations for most methods of home food preservation.
nchfp.uga.edu/publications/publications_usda.html
USDA's _Complete Guide to Home Canning_. Download free. Adobe Reader 10 (or higher) is needed for proper viewing and printing of the USDA canning guide files.
**Stainless Steel Screw Bands:** www.Ecojarz.com
Stainless steel screw bands will last for years with careful cleaning. They will not rust and do not bend nearly as easily as the tin variety.
## INDEX
A
acidity levels, ,
air, trapped,
air bubbles,
altitude, ,
apples
Apple Pie Filling,
Applesauce,
Canned Apples,
Asparagus, First-of-the-Season,
B
Bacon Jam,
bail-type jars,
Baked Beans, Picnic-Friendly,
Barbecue Sauce,
beans, dried
Beef Soup with Lentils,
Heirloom Beans,
Kidney Bean Soup,
Lentils,
Picnic-Friendly Baked Beans,
Vegetable Soup,
beans, green
Green Beans,
Mixed Vegetable Medley,
Spicy Dilly Beans,
beef
Beef Bone Broth,
Beef Soup with Lentils,
Beets, Spiced Pickled,
being present,
blanch,
bleach,
Blueberry Filling,
boil,
botulism, ,
breakage,
brine,
browning meat,
Brussels sprouts, Sweet and Spicy Pickled,
C
Canned Apples,
Canned Cherries,
Canned Chicken,
Canned Haddock,
Canned Peaches,
Canned Potatoes,
canning area,
canning funnel,
canning rack,
canning salt,
canning supplies, resources for,
carrots
Mixed Vegetable Medley,
Sweet and Spicy Ginger Carrot Coins,
Taste of Summer Carrot Sticks,
Vegetable Broth,
Vegetable Soup,
Cauliflower, Pickled,
Cherries, Canned,
chicken
Canned Chicken,
Chicken or Turkey Broth,
Chicken Soup,
Mexican Chicken Soup,
chutney,
cleanliness, ,
clear jel,
cold packing, ,
color
darkening,
fading,
corn
Corn Relish,
Creamed Corn,
Mixed Vegetable Medley,
Sweetest Canned Corn,
Vegetable Soup,
cranberries
Cranberry Jelly,
Whole Cranberries,
Creamed Corn,
Cubed Pork,
D
dial gauge, ,
dishtowels,
E
efficiency,
F
fading colors,
fat on meats,
fillings
Apple Pie Filling,
Blueberry Filling,
Rhubarb Filling,
First-of-the-Season Asparagus,
fish
about,
basics of,
Canned Haddock,
preparing for,
food mill,
food processor,
freshness,
fruit, about, . _see also individual fruits_
funnel,
G
Garlic Broth,
Grape Drink, Homemade,
Green Beans,
Green Tomato Chutney,
ground meat
about,
recipe for,
H
Haddock, Canned,
half pint jars,
headspace, , ,
Heirloom Beans,
high acid,
Homemade Grape Drink,
hot packing, ,
I
ingredients
finding and preparing, , , ,
interchanging,
overripe,
J
jar lifter,
jars
about,
broken,
cooling,
filling,
overpacking,
preparing,
types of,
jelly bag,
jelly jars,
K
Kidney Bean Soup,
knives,
L
labels, , ,
ladles,
Lentils,
Beef Soup with,
lid lifter,
lid wrench,
lids,
liquid, troubleshooting,
locking cover,
low acid,
M
Maillard reaction,
mason jar,
measuring cups, ,
measuring spoons, ,
meat
about,
basics of,
browning,
cutting,
ground, ,
things to avoid with,
trimming,
_see also_ beef; chicken; pork; turkey
Meatballs in Tomato Juice,
Mexican Chicken Soup,
mineral content in water,
mistakes, common, ,
Mixed Vegetable Medley,
mold,
N
navy beans, Picnic-Friendly Baked Beans,
O
onions
Perfect Cocktail Onions,
Vegetable Soup,
organization,
outdated methods,
overcooking,
overpacking jars,
overpressure plug,
oxidation,
P
packing, , , ,
pantry
eating from, ,
spotting problems in,
peaches
Canned Peaches,
Peach Tomato Salsa,
Pears, Spiced,
Peas, Perfectly Canned,
peppers
Barbecue Sauce,
Rainbow Peppers,
Spicy Hot Pepper Mix,
Tomato Salsa,
Perfect Cocktail Onions,
Perfectly Canned Peas,
Pickled Cauliflowers,
Picnic-Friendly Baked Beans,
Pineapple-Flavored Zucchini,
pint jars,
planning,
pork
Bacon Jam,
Cubed Pork,
pot holders,
potatoes
Canned Potatoes,
Vegetable Soup,
preparation for canning,
pressure canner
caring for,
description of, , ,
filling,
preparing,
pressure canning
basics of,
caring for gear for,
commonsense aspects of,
description of,
equipment for,
knowing steps for,
overview of,
process for,
for soups,
pressure cooker, ,
pressure dip,
pressure fluctuations,
pressure gauge, , ,
pressure release, rapid,
pressure requirements,
processing,
Pumpkin Bites, Year-Round,
purée,
Q
quart jars,
R
Rainbow Peppers,
raw pack,
recipes
following,
overcooking,
using reliable,
regular-mouth jars,
release valve,
resources,
rhubarb
Rhubarb Filling,
Rhubarb Sauce,
rubber spatulas,
S
safety,
sanitizing,
saucepan,
sauces
Barbecue Sauce,
Peach Tomato Salsa,
Taco Sauce,
Tomatillo Salsa,
Tomato Salsa,
Tomato Sauce,
scale, electronic,
screw band, , , ,
seals
checking,
failed,
serving sizes,
simmer,
soups
Beef Bone Broth,
Beef Soup with Lentils,
Chicken or Turkey Broth,
Chicken Soup,
Garlic Broth,
Kidney Bean Soup,
Meatballs in Tomato Juice,
Mexican Chicken Soup,
pressure canning,
Thanksgiving Turkey Soup,
Vegetable Broth,
spatulas,
Spiced Pears,
Spiced Pickled Beets,
Spicy Dilly Beans,
Spicy Hot Pepper Mix,
spoons, wooden,
sterilize,
Stewed Tomatoes,
storage,
sugar syrup,
sunlight, ,
Sweet and Spicy Ginger Carrot Coins,
Sweet and Spicy Pickled Brussels Sprouts,
Sweetest Canned Corn,
T
Taco Sauce,
Taste of Summer Carrot Sticks,
terms,
Thanksgiving Turkey Soup,
timer,
timing,
tomatillos
Tomatillo Salsa,
Tomatillos,
tomatoes
Barbecue Sauce,
Green Tomato Chutney,
Mixed Vegetable Medley,
Peach Tomato Salsa,
Stewed Tomatoes,
Tomato Ketchup,
Tomato Salsa,
Tomato Sauce,
Vegetable Soup,
tongs,
trimming meat,
troubleshooting,
turkey
Chicken or Turkey Broth,
Thanksgiving Turkey Soup,
U
USDA guidelines, ,
utensils,
V
vegetables
about classic recipes,
about going further with,
Mixed Vegetable Medley,
Vegetable Broth,
Vegetable Soup,
_see also individual_ _vegetables_
W
water bath canning, differences from,
weighted gauge, , ,
white residue,
Whole Cranberries,
wide-mouth jars,
wooden chopsticks,
wooden spoons,
Y
Year-Round Pumpkin Bites,
Z
zucchini
Mixed Vegetable Medley,
Pineapple-Flavored Zucchini,
##
ABOUT THE AUTHOR
Amelia "Amy" Jeanroy has been canning and preserving foods for twenty-five years. She is coauthor of _Canning and Preserving for Dummies_ and author of _Fermenting for Dummies_ , as well as hundreds of magazine articles about food preservation, homesteading, cooking, and rural living. Amy is a master gardener and has owned a greenhouse business since 2010, growing plants, produce, and seeds for farmers' markets. She has been a guest on the _Martha Stewart Living_ radio show, the _Take Care_ show on NPR, PBS's _Earth Eats_ , and _The Wisconsin Vegetable Gardener_. Find her online as "The Farming Wife" on Facebook and YouTube.
© 2018 Quarto Publishing Group USA Inc. Text © 2018 Amelia Jeanroy
Photography © Kerry Michaels, except the following from Shutterstock: pages (bottom), , , , , , , , , , , , , 134, , , , , , , ,
First published in 2018 by Voyageur Press, an imprint of The Quarto Group, 401 Second Avenue North, Suite 310, Minneapolis, MN 55401 USA. T (612) 344-8100 F (612) 344-8692 www.QuartoKnows.com
All rights reserved. No part of this book may be reproduced in any form without written permission of the copyright owners. All images in this book have been reproduced with the knowledge and prior consent of the artists concerned, and no responsibility is accepted by producer, publisher, or printer for any infringement of copyright or otherwise, arising from the contents of this publication. Every effort has been made to ensure that credits accurately comply with information supplied. We apologize for any inaccuracies that may have occurred and will resolve inaccurate or missing information in a subsequent reprinting of the book.
Voyageur Press titles are also available at discount for retail, wholesale, promotional, and bulk purchase. For details, contact the Special Sales Manager by email at specialsales@quarto.com or by mail at The Quarto Group, Attn: Special Sales Manager, 401 Second Avenue North, Suite 310, Minneapolis, MN 55401 USA.
Digital edition: 978-0-76036-441-3
Softcover edition: 978-0-76035-210-6
Library of Congress Cataloging-in-Publication Data
Names: Jeanroy, Amelia, author.
Title: Modern pressure canning : recipes and techniques for today's home canner / Amelia Jeanroy ; photographs by Kerry Michaels.
Description: Minneapolis, MN : Voyageur Press, 2018. | Includes index.
Identifiers: LCCN 2017051171 | ISBN 9780760352106 (paperback)
Subjects: LCSH: Canning and preserving. | Pressure cooking. | BISAC: COOKING / Methods / Canning & Preserving. | COOKING / Methods / Special Appliances. | COOKING / Methods / General.
Classification: LCC TX603 .J39 2018 | DDC 641.4/2—dc23
LC record available at <https://lccn.loc.gov/2017051171>
ACQUIRING EDITOR: Todd Berger
PROJECT MANAGER: Alyssa Lochner
ART DIRECTOR: Cindy Samargia Laun
COVER DESIGN: Faceout Studio
PAGE DESIGN AND LAYOUT: Laura Shaw Design
| {
"redpajama_set_name": "RedPajamaBook"
} | 47 |
are the numbers for the Lines of Code that are shown on the index page for each type for the logical or physical lines of code?
The Lines of Code that are shown in the Index page of the TYPES on the HTML pages, are for the logical lines of code. Logical lines of code are the number of lines within the COBOL with copybooks included. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,142 |
package org.onosproject.kubevirtnode.cli;
import io.fabric8.kubernetes.api.model.Node;
import io.fabric8.kubernetes.client.KubernetesClient;
import org.apache.karaf.shell.api.action.Command;
import org.apache.karaf.shell.api.action.lifecycle.Service;
import org.onosproject.cli.AbstractShellCommand;
import org.onosproject.kubevirtnode.api.KubevirtApiConfig;
import org.onosproject.kubevirtnode.api.KubevirtApiConfigService;
import org.onosproject.kubevirtnode.api.KubevirtNode;
import org.onosproject.kubevirtnode.api.KubevirtNodeAdminService;
import java.util.Set;
import static org.onosproject.kubevirtnode.api.KubevirtNode.Type.WORKER;
import static org.onosproject.kubevirtnode.api.KubevirtNodeState.INIT;
import static org.onosproject.kubevirtnode.util.KubevirtNodeUtil.buildKubevirtNode;
import static org.onosproject.kubevirtnode.util.KubevirtNodeUtil.k8sClient;
/**
* Synchronizes kubevirt node states.
*/
@Service
@Command(scope = "onos", name = "kubevirt-sync-state",
description = "Synchronizes kubevirt node states.")
public class KubevirtSyncStateCommand extends AbstractShellCommand {
@Override
protected void doExecute() throws Exception {
KubevirtApiConfigService apiConfigService = get(KubevirtApiConfigService.class);
print("Re-synchronizing Kubevirt node states..");
KubevirtApiConfig config = apiConfigService.apiConfig();
bootstrapKubevirtNodes(config);
print("Done.");
}
private void bootstrapKubevirtNodes(KubevirtApiConfig config) {
KubevirtNodeAdminService nodeAdminService = get(KubevirtNodeAdminService.class);
Set<KubevirtNode> completeNodeSet = nodeAdminService.completeNodes();
KubernetesClient k8sClient = k8sClient(config);
if (k8sClient == null) {
log.warn("Failed to connect to kubernetes API server");
return;
}
for (Node node : k8sClient.nodes().list().getItems()) {
KubevirtNode kubevirtNode = buildKubevirtNode(node);
// we always provision VMs to worker nodes, so only need to install
// flow rules in worker nodes
if (kubevirtNode.type() == WORKER) {
if (completeNodeSet.stream().map(KubevirtNode::hostname)
.filter(name -> name.equals(kubevirtNode.hostname()))
.findAny().isPresent()) {
print("Initializing %s because the node was COMPLETE state.",
kubevirtNode.hostname());
KubevirtNode updated = kubevirtNode.updateState(INIT);
nodeAdminService.updateNode(updated);
} else {
nodeAdminService.updateNode(kubevirtNode);
}
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 576 |
{"url":"http:\/\/aas.org\/archives\/BAAS\/v25n4\/aas183\/abs\/S3711.html","text":"Pulse and Polarization Morphology for $\\gamma$-Ray Pulsars\nSession 37 -- Pulsars\nDisplay presentation, Thursday, January 13, 9:30-6:45, Salons I\/II Room (Crystal Gateway)\n\n## [37.11] Pulse and Polarization Morphology for $\\gamma$-Ray Pulsars\n\nRoger W. Romani \\& I.-A. Yadigaroglu (Stanford University)\n\nThe location and efficiency of pulsar $\\gamma$-ray emission is a subject of appreciable debate. The variety of profiles found in recent EGRET detections motivates improved modeling. We present a calculation of polarization and profile properties of high energy pulsar emission in an outer gap picture. Our sums indicate that comparison with the polarization data and with radio pulse profiles provides significant constraints on the emission zone geometry. We find that high energy emission is generally detected from the pole opposite that associated with the surface radio pulse; the result is robust to uncertainties in the outer magnetosphere currents and details of the field geometry. The Crab pulsar provides an important test case and we match the model to optical polarization measurements tracing the outer gap region, as well as the high energy light curves. With spin axis inclinations and viewing geometries provided by the model fit, we derive an effective beaming factor that can be used to estimate the total flux from the detected value. We also show that this picture provides good correspondence to the $\\gamma$-ray light curves and radio pulse offset and polarization properties for the other $\\gamma$-ray pulsars ($e.g.$ PSR 1706-44, Geminga) and provides useful constraints on the emission altitude; these provide a clue to the origin of the spectral index variations in the $\\gamma$-ray data.","date":"2015-07-04 19:07:53","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\": 0.5373547673225403, \"perplexity\": 2557.015847649694}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-27\/segments\/1435375096870.66\/warc\/CC-MAIN-20150627031816-00266-ip-10-179-60-89.ec2.internal.warc.gz\"}"} | null | null |
Louisiana-Exhibition. Installation Views & Videostills. ©1997.
A letter-correspondence with Robert Morris about his felt works, and my fascination and efforts to know about them and him.
• See close-up of felt-sculptures. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,148 |
July 17, 2014 Slideshows » News, Blogs
10 Things Going on in Cleveland this Weekend (July 18 - 20)
Irish blood runs in the veins of many Clevelanders; celebrate that heritage by paying a visit to the Cleveland Irish Cultural Festival that kicks off today at the Cuyahoga County Fairgrounds. From traditional Irish dancers to bands hailing from Dublin and Sligo, entertainment abounds at this festival dedicated to preserving Cleveland's rich Irish history. A new feature this year is the Temple Bar & Museum, where festival-goers can relax, grab a seat near the stage, or stroll through a variety of cultural exhibits and displays. Get green and join in on the fun beginning tonight at 5 p.m; general admission is $12, but parking is free. The festival continues through Sunday. (Cortni Dietz)
This year's Party in the Park, which takes place in downtown Painesville's Veteran's Park today through Sunday, features more than 30 acts from all over Ohio and across the U.S. performing continuously on three stages. The eclectic line-up includes New Orleans singer-songwriter Eric Lindell, Charlotte-based reggae rockers Simplified, Nashville roots rock act the Delta Saints, and local hero Carlos Jones and his PLUS band. Local favorites such as Alex Bevan, Cats on Holiday and Tropidelic are also on the bill. Last year's event drew some 55,000 people. "We try to cover as many musical genres as possible to attract a wide spectrum of music lovers and we book bands that are playing major music festivals that people pay a lot of money to see," says entertainment chief Gabe Cicconetti in a press release. The Painesville Community Improvement Corporation, the non-profit organization that hosts the event, has booked some 40 food vendors and a beer garden featuring Great Lakes Brewing Co., Oskar Blues, Miller, Guiness, Seven Brothers Spirits, and wine. The event also includes a community stage, a "Kidzone" with inflatables and a classic and custom car show. Admission is free. (Niesel) parkpartyinfo.com for more info.
The Jamboree has gone huge this year. With a line-up that boasts Keith Urban, Little Big Town, Tim McGraw and Brantley Gilbert, this is Country Central for the month of July. (Check out the Blossom Country Megaticket for all other related needs.) The Jamboree brings music lovers and campers to bucolic Belmont, Ohio, just a quick two-and-a-half hours down I-77/70. This event has been going on for decades and its 100,000+ annual attendees can all vouch for its cred. If you've got a cowboy hat that needs a few days in its indigenous habitat, then you've no choice but to pick up a nice four-day pass to the Jamboree (fest runs Thurs - Sun).
Photo via Cleveland Scene Archives
For the sixth year in a row, Scene will descend on Lincoln Park in Tremont to dazzle thousands with the best day-drinking of the summer. Dozens of vendors pour every craft beer you can name while restaurants serve the food you need to keep going all four hours. Music, entertainment, sun, trees, and, ya know, beer. It's all here. Tickets are $30 in advance. Buy tickets now.
The Hudson Wine Festival, a benefit for animal welfare in Summit County, will take place Friday, July 18 from 4 – 10 p.m. and Saturday, July 19 from 2 – 10 p.m. along Village Way in the First & Main shopping area. Showcasing over 150 wines from national & international producers, guests will enjoy musical entertainment, food, art & retail exhibits and raffles. Tickets are $22 in advance; $27 at the door & include 10 tastings & a souvenir wine glass. For more information visit www.hudsonwinefestival.com. Tickets are $22 in advance, $27 at the door
Photo via Browns Bunch, Facebook
Football season won't start up for a few more weeks, but it's never too early to get into the proper mind frame. Today from 5 to 9 p.m., Scoundrels in Berea hosts its seventh annual Dawgs for the Dogs charity tailgate. For $25, you get all the pizza, pasta, wings, wine and well drinks you can eat and drink. And kids under 12 get in for free. Proceeds go toward the animal shelter Secondhand Mutts. (Niesel) dawgs4dogs.com.
The Blossom Music Festival is in full swing this weekend, bringing you two more amazing concert performances by the Cleveland Orchestra. Tonight at 8, Bramwell Tovey conducts the orchestra in its performance of a suite from Georges Bizet's Carmen, an unabridged version of Manuel de Falla's The Three-Cornered Hat and Camille Saint-Saëns' Violin Concerto No. 3 with guest soloist Karen Gomyo. Tomorrow night at 7, Stanislaw Skrowaczewski will lead the orchestra in the Overture to Der Freischütz by Carl Maria von Weber and Symphony No. 5 by Dmitri Shostakovich. Pianist Francesco Piemontesi will join the festivities for Mozart's Piano Concerto No. 27. Tickets to each concert range from $23 to $85. (Gonzalez)
Local eateries hawk their wares today on Professor Avenue from noon to 8 p.m., during the annual Taste of Tremont. Satiate your appetite by sampling dishes from restaurants headed by Dante Boccuzzi and Michael Symon, among others. Listen to live music and enjoy the culture that Tremont has to offer as you make your way from food vendor to food vendor, and cool off with a drink while you're at it; beer and wine are $5 and $4, respectively. (Cortni Dietz)
The Avant-Garde Art & Craft show comes to Cleveland Heights for the first time this year. Featuring works from nearly 100 artists, the show offers unique and oddball pieces that take the idea of "arts and crafts" to a whole new level, like wine bottle lights and beachglass jewelry — stuff you can actually use at home and looks nice. The show runs from 10 a.m. to 5 p.m., and is held in Kangesser Ballroom at Park Synagogue Main. Admission is $3 at the door and kids under 12 get in free. (Trenholme) avantgardeshows.com.
The 5th Annual Big-Little Comedy Fest boasts some of the best improv, sketch, & stand-up performers from throughout the US for 4 nights of non-stop laughs (Fest runs Thurs - Sun). See playhousesquare.com for details and nightly performance lineup.
12 Things to Do in Cleveland When You're Bored and It's Cold and Snowy
12 Eerie Photos of the Randall Park Mall
31 Vintage Photos of Kids Growing Up in Cleveland
16 Photos of the Tragically Hip Performing at House of Blues
Joe Kleon16 images
Additional Slideshows
News, Blogs
By Joe Kleon | January 17, 2015
31 Vintage Photos of Kids Growing Up in Cleveland 2
All Photos are courtesy of the Cleveland Memory Project and Cleveland Press.
12 Eerie Photos of the Randall Park Mall 2
Photos by Johnny Joo
10 Things Going on this Weekend in Cleveland (January 9 -11)
10 Cleveland Lunch Joints Where You Can Eat for Under $10 2
By Jason Beudert | January 8, 2015
Monsters vs. Rochester Americans @ Rocket Mortgage Fieldhouse
Cavs vs. Oklahoma City Thunder @ Rocket Mortgage Fieldhouse
Sat., Jan. 22
NBA Rising Stars Game @ Rocket Mortgage FieldHouse
Thu., Feb. 17 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,127 |
We are excited to invite Yulia Hanansen, an internationally acclaimed mosaic artist to teach at MSoP. She is a second-generation mosaic artist and has been professionally involved in mosaic making since 2001. The mission of her Mosaic Sphere Studio is to build public awareness about the quality of mosaics, and to exhibit the superb qualities of mosaic techniques.
"Flower Exploration in Mosaic Art: Contemporary Renderings"
When: 2 days, October 22-23.
Description: This workshop offers an opportunity to learn about contemporary renderings of flowers in a mosaic technique. The workshop will start with a presentation that discusses in detail ancient and modern representations of flowers and floral elements. A demo will be given on techniques, stylizations, and layouts. As a hands-on part students will create their own mosaic interpretation of a flower based on a photograph or a sketch using a traditional layout or Yulia's signature brush-stroke technique. During the workshop students will also explore a color palette, flower details, composition, layout, and technical aspects of working with the stained glass modules. Images will be provided or bring your own upon approval of the instructor. All skill levels are welcome. Non-members are welcome to sign up!
Students will need to bring their own glass. Due to the crisis in the glass industry I am not able to supply any glass for the workshop. Please let the student know that they will have to bring their own glass. I hope they'll understand. The workshop fee will be reduced to $20 as I'll bring the rest of the supplies and handouts.
Costs : $350. (MSoP members), $390.(nonmembers) plus $45. materials. Workshop will be held in Willow Grove , Pa. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,009 |
\section{Introduction}
In this paper we extend the convergence result from
\cite{Sandberg-DI1}. The following differential inclusion is
considered:
\begin{equation}\label{eq:DI}
\begin{split}
x'(t)&\in F\big(x(t)\big),\\
x(0)&=x_0,
\end{split}
\end{equation}
where $x_0\in\Re\sp d$, and $F$ is a function from $\Re\sp d$ to the
compact subsets of $\Re\sp d$.
In \cite{Sandberg-DI1}, it is shown
that if $F$ is uniformly bounded in the sense that
\begin{equation}\label{eq:Fbound}
\abs{y}\leq K,\quad \text{for all }y\in\bigcup_{x\in\Re\sp d} F(x),
\end{equation}
and Lipschitz continuous with respect to Hausdorff distance,
\begin{equation}\label{eq:FLip}
\mathcal{H}\big(F(x),F(y)\big) \leq L\abs{x-y},
\end{equation}
then the Forward Euler method converges with rate one. For the
definition of the Hausdorff
distance we need the following notation (as in
\cite{Sandberg-DI1}). We denote by $B$ the closed unit ball in $\Re\sp
d$. The Minkowski sum of two non-empty sets $C,D \subset\Re\sp d$ is
defined by
\begin{equation*}
C+D = \big\{c+d\ |\ c\in C \text{ and } d\in D\big\},
\end{equation*}
the multiplication by a scalar, $\lambda>0$, by
\begin{equation*}
\lambda C = \big\{\lambda c\ |\ c\in C\big\},
\end{equation*}
and the sum of an element $c\in\Re\sp d$ and a set $C$ by
\begin{equation*}
c+C=\big\{c\big\}+C.
\end{equation*}
The Hausdorff distance is given by
\begin{equation*}
\mathcal{H}(C,D) = \inf \big\{\lambda \geq 0\ |\ C \subset D+\lambda B
\text{ and } D \subset C + \lambda B \big\}.
\end{equation*}
We will denote by $\abs{\cdot}$ the Euclidean norm, when applied to a
vector, and the Euclidean operator norm, when applied to a matrix.
We consider solutions $x:[0,T]\rightarrow\Re\sp d$ to the differential inclusion \eqref{eq:DI} in
the finite time interval $[0,T]$. A solution is an absolutely
continuous function which satisfies \eqref{eq:DI} a.e. For the Forward
Euler method we split the interval $[0,T]$ into $N$ parts of equal
length $\Delta t=T/N$. The Forward Euler scheme is given by
\begin{equation}\label{eq:FE}
\begin{split}
\xi_{n+1} &\in \xi_n + \Delta t F(\xi_n), \quad n=0,1, \ldots ,N-1, \\
\xi_0&=x_0.
\end{split}
\end{equation}
The convergence result in \cite{Sandberg-DI1} concerns the reachable sets
\begin{equation*
\begin{split}
C_n &= \big\{x(n\Delta t)\ |\ x:[0,T]\to \Re^d \text{ solution to
\eqref{eq:DI}}\big\}, \\
D_n &= \big\{\xi_n\ |\ \{\xi_i\}_{i=0}^N \text{ solution to
\eqref{eq:FE}}\big\}.
\end{split}
\end{equation*}
It was shown there that under the assumptions in \eqref{eq:Fbound} and
\eqref{eq:FLip} the following bound holds:
\begin{equation}\label{eq:ConvReachSets
\max_{0 \leq n \leq N} \mathcal{H}(C_n,D_n) \leq
\Big(Ke^{LT}\big(Kd(d+1)+LT\big) +2Kd \Big)\Delta t.
\end{equation}
This was an extension of the previous first order convergence result
in \cite{Dontchev-Farkhi1989} in the sense that the set-valued
function $F$ did not need to be convex. In \cite{Grammel}, the
non-convex case was presented, although in a different form (see
\cite{Sandberg-DI1}), but there only half-order convergence was
proved. Although the convergence of the reachable sets in
\eqref{eq:ConvReachSets} is what is needed in many situations, e.g.\
in optimal control (see \cite{Sandberg-DI1}), it is weaker than the
convergence of solution paths, the type of convergence used in e.g.\
\cite{Dontchev-Farkhi1989} and \cite{Grammel}. In section \ref{sec:results} we
show that the first-order convergence result for non-convex
differential inclusions can be extended so that it gives convergence
of solution paths. The proof is actually only a minor change of the
proof in \cite{Sandberg-DI1}.
Another weakness with the convergence result in
\eqref{eq:ConvReachSets} is that the constant depends quadratically
on the dimension. In \cite{Sandberg-DI1} it was shown that this
constant can not be expected to be smaller than of order $\sqrt d$ in
general. In section \ref{sec:results} a partial improvement is given for the case
where the differential inclusion is a control problem with few control
parameters.
In section \ref{sec:FD} two results which are needed in the proof of
the theorem involving few control parameters are presented.
\section{The Results}\label{sec:results}
We introduce the same set-valued maps that was used in
\cite{Sandberg-DI1}.
Let $\varphi$ and $\psi$ be
functions from $\Re^d$ into the non-empty compact subsets of $\Re^d$, defined by
\begin{equation*}
\begin{split}
\varphi(x) &= x + \Delta t F(x), \\
\psi(x) &= x + \Delta t\, \text{co}\big(F(x)\big),
\end{split}
\end{equation*}
where co denotes the convex hull.
If $A$ is a subset of $\Re^d$ we define
\begin{equation*}
\varphi(A)= \bigcup_{x\in A}\varphi(x),
\end{equation*}
and similarly for the set-valued maps $\psi$ and $F$
We will use the following result for convex differential
inclusions. It is taken from \cite{Dontchev-Farkhi1989}, where
it is formulated in a slightly more general setting than the one
presented here.
\begin{thm}\label{thm:ConvexPaths}
Let $F$ be a function from $\Re^d$ into the non-empty compact convex subsets of
$\Re^d$, which satisfies \eqref{eq:FLip} and \eqref{eq:FLip}.
For any solution
$x : [0,T]\to \Re^d$ to \eqref{eq:DI} there exists a
solution $\{\eta_n\}_{n=0}^N$ to \eqref{eq:FE} such that
\begin{equation}\label{eq:pathapprox}
\max_{0 \leq n \leq N} |x(n\Delta t)-\eta_n| \leq KLTe^{LT}\Delta t.
\end{equation}
Moreover, for any solution $\{\eta_n\}_{n=0}^N$ to \eqref{eq:FE} there
exists a solution $x:[0,T]\to\Re^d$ to \eqref{eq:DI} such that \eqref{eq:pathapprox} holds.
\end{thm}
The convergence of solution paths to non-convex differential
inclusions is given next.
\begin{thm}\label{thm:convpaths1}
Assume that $x:[0,T]\rightarrow\Re\sp d$ solves (1.1).
Let $F$ be a function from $\Re^d$ into the non-empty compact subsets of
$\Re^d$, which satisfies \eqref{eq:FLip} and \eqref{eq:FLip}.
Then there
exists a solution $\{\xi_n\}_{n=0}\sp N$ to \eqref{eq:FE}, such that
\begin{equation}\label{eq:pathconv}
\max_{0\leq n\leq N}\abs{x(n\Delta t)-\xi_n} \leq K(e\sp{LT}d(d+1) +
2d+ LTe\sp{LT})\Delta t.
\end{equation}
\end{thm}
\begin{proof}
Let $\{\eta_n\}_{n=0}\sp{N}$ be a solution to the scheme
\begin{equation*}
\eta_{n+1}\in\psi(\eta_n),\quad \text{for }0\leq n\leq N-1,
\end{equation*}
which satisfies \eqref{eq:pathapprox}. By lemma 2.1 in
\cite{Sandberg-DI1} it follows that the set-valued function
$\text{co}\big(F(x)\big)$ is Lipschitz continuous in the Hausdorff
distance with the same constant as $F(x)$. Therefore Theorem
\ref{thm:ConvexPaths} guarantees the existence of such a solution
$\{\eta_n\}$.
Let $\varepsilon$ be any positive number, $n$ an integer such that
$d\leq n\leq N$, and $\xi_{n-d}$ a point in $\Re\sp d$ such
that
\begin{equation*}
\eta_n\in \psi\sp d(\xi_{n-d}) + \varepsilon B.
\end{equation*}
Similarly as in the proof of Theorem 3.4 in \cite{Sandberg-DI1} we have the
following inclusion:
\begin{equation*}
\psi\big(\psi\sp d(\xi_{n-d})+\varepsilon B\big) \subset \psi\sp
d\big(\varphi(\xi_{n-d})\big) +\big(KLd(d+1)\Delta t\sp 2
+\varepsilon(1+L\Delta t)\big)B
\end{equation*}
Therefore, there must exist an $\xi_{n-d+1}\in\varphi(\xi_{n-d})$, such
that
\begin{equation*}
\eta_{n+1}\in \psi\sp
d(\xi_{n-d+1}) +\big(KLd(d+1)\Delta t\sp 2
+\varepsilon(1+L\Delta t)\big)B.
\end{equation*}
It follows that there exists a solution $\{\xi_n\}_{n=0}\sp{N-d}$ to
\begin{equation}\label{eq:phievol}
\xi_{n+1}\in\varphi(\xi_n)
\end{equation}
for $0\leq n\leq N-d-1$,
such that
\begin{equation*}
\eta_{n+d}\in\psi\sp d(\xi_n) +\varepsilon_n B,
\end{equation*}
where
\begin{equation*}
\begin{split}
\varepsilon_{n+1}&=(1+L\Delta t)\varepsilon_n + KLd(d+1)\Delta t\sp
2,\\
\varepsilon_0&=0.
\end{split}
\end{equation*}
By the proof of Theorem 3.5 in \cite{Sandberg-DI1} it holds that
\begin{equation*}
\varepsilon_n\leq Ke\sp{LTn/N}d(d+1)\Delta t \leq
Ke\sp{LT}d(d+1)\Delta t.
\end{equation*}
Let us extend the solution $\{\xi_n\}$ up to $n=N$, by letting
$\{\xi_n\}_{n=N-d+1}\sp N$ be any solution to \eqref{eq:phievol} for
$N-d\leq n\leq N-1$. For $d\leq n\leq N$ we have
\begin{equation}\label{eq:xixbound}
\abs{\xi_n-\eta_n} \leq \abs{\xi_n-\xi_{n-d}}+\abs{\eta_n-\xi_{n_d}} \leq
Kd\Delta t +Kd\Delta t + Ke\sp{LT}d(d+1)\Delta t.
\end{equation}
For $0\leq n\leq d$ we have
\begin{equation*}
\abs{\xi_n-\eta_n}\leq \abs{\xi_n-x_0}+\abs{\eta_n-x_0}\leq 2Kd\Delta t.
\end{equation*}
Hence \eqref{eq:xixbound} holds for all $0\leq n\leq N$.
This together with \eqref{eq:pathapprox} gives \eqref{eq:pathconv}
\end{proof}
Let us now consider the situation where the set-valued function $F$ is
given by
\begin{equation}\label{eq:disccontrol1}
F(x)=\{f_i(x)\}_{i=1}\sp M,
\end{equation}
and where we have smoothness, in the sense that there exists a
constant $S>0$, such that
\begin{equation}\label{eq:disccontrol2}
\abs{f_i(x)-f_i(z)-f'_i(z)(x-z)}\leq S\abs{x-z}\sp 2,
\end{equation}
for all $1\leq i\leq M$ and $x,z\in\Re\sp d$. By \eqref{eq:Fbound} we
have
\begin{equation}\label{eq:disccontrol3}
\abs{f_i(x)}\leq K, \quad\text{for all $1\leq i\leq M$ and $x\in\Re\sp
d$.}
\end{equation}
Let us assume that we also have the following bound on the
Jacobians:
\begin{equation}\label{eq:disccontrol4}
\abs{f'_i(x)}\leq L,\quad\text{for every }x\in\Re\sp d.
\end{equation}
Under these assumptions we have
\begin{thm}\label{thm:fullydiscrete}
Assume that \eqref{eq:disccontrol1}, \eqref{eq:disccontrol2},
\eqref{eq:disccontrol3}, and \eqref{eq:disccontrol4} hold.
Assume that $M\geq d+1$, and that $x:[0,T]\rightarrow\Re\sp d$ solves \eqref{eq:DI}. Then
there exists a solution $\{\xi_n\}_{n=0}\sp{N}$ to \eqref{eq:FE}, such
that
\begin{multline}\label{eq:pathM}
\max_{0\leq n\leq N}\abs{x(n\Delta t)-\xi_n} \leq
\big(e\sp{LT}(KLT+K(8M-10))+2K(M-1)\big)\Delta t\\
+e\sp{LT}\big(KL(M-1)(M-2) + 2KL\frac{(M-1)\sp3-(M-1)}{3}(1+L\Delta
t)\sp{M-3}\\
+ 2SK\sp2\frac{M(M-1)(2M-1)}{3L}\big)\Delta t\sp2.
\end{multline}
\end{thm}
\begin{proof}
This proof follows the same basic lines as the proof of Theorem
\ref{thm:convpaths1}. Hence we assume that we have a solution
$\{\eta_n\}_{n=0}\sp N$ to the scheme
\begin{equation*}
\eta_{n+1}\in\psi(\eta_n),\quad\text{for }0\leq n\leq N-1,
\end{equation*}
which satisfies \eqref{eq:pathapprox}. Let $\varepsilon$ be any
positive number, $n$ an integer such that $M-1\leq n\leq N$, and
$\xi_{n-M+1}$ a point in $\Re\sp d$ such that
\begin{equation*}
\eta_n \in \text{co}\big(\varphi\sp{M-1}(\xi_{n-M+1})\big)+\varepsilon B.
\end{equation*}
Similarly as in the proof of Theorem 3.4 in \cite{Sandberg-DI1} we
have the following inclusion:
\begin{equation*}
\psi\big(\text{co}(\varphi\sp{M-1}(\xi_{n-M+1}))+\varepsilon B\big)
\subset
\psi\big(\text{co}(\varphi\sp{M-1}(\xi_{n-M+1}))\big)+\varepsilon(1+L\Delta t)B.
\end{equation*}
By Theorems \ref{thm:psiconvexhull} and \ref{thm:coco} we have
\begin{multline*}
\psi\big(\text{co}(\varphi\sp{M-1}(\xi_{n-M+1}))\big)+\varepsilon(1+L\Delta
t)B \subset
\bigcup_{x\in\varphi(\xi_{n-M+1})} \text{co}\big(\varphi\sp{M-1}(x)\big)\\
+\big(\varepsilon(1+L\Delta t) +(8M-10)KL\Delta t\sp 2\\
+(2KL\sp2\frac{(M-1)\sp3-(M-1)}{3}(1+L\Delta t)\sp{M-3}\\
+2SK\sp 2
\frac{M(M-1)(2M-1)}{3})\Delta t\sp3\big)B=:\big(\varepsilon(1+L\Delta
t) + C_1\Delta t\sp2 + C_2\Delta t\sp 3\big)B.
\end{multline*}
Therefore, there must exist a $\xi_{n-M+2}\in\varphi(\xi_{n-M+1})$, such
that
\begin{equation*}
\eta_{n+1}\in \text{co}\big(\varphi\sp{M-1}(\xi_{n-M+2})\big) +
\big(\varepsilon(1+L\Delta t) +C_1\Delta t\sp 2 + C_2\Delta t\sp 3\big)B.
\end{equation*}
Hence there exists a solution $\{\xi_n\}_{n=0}\sp{N-M+1}$ to
\begin{equation*}
\begin{split}
\xi_{n+1}&\in\varphi(\xi_n),\\
\xi_0=x_0,
\end{split}
\end{equation*}
for $0\leq n\leq N-M$, such that
\begin{equation*}
\eta_{n+M-1}\in \text{co}\big(\varphi\sp{M-1}(\xi_n)\big) +
\varepsilon_n B,
\end{equation*}
where
\begin{equation*}
\begin{split}
\varepsilon_{n+1}&=(1+L\Delta t)\varepsilon_n+ C_1\Delta t\sp
2+C_2\Delta t\sp 3,\\
\varepsilon_0&=KL(M-1)(M-2)\Delta t\sp2.
\end{split}
\end{equation*}
From this we have that
\begin{equation*}
\epsilon_n = (1+L\Delta t)\sp{n+1}\varepsilon_0 + (C_1\Delta
t\sp2+C_2\Delta t\sp3)\big(1+(1+L\Delta t)+\cdots+(1+L\Delta t)\sp n\big).
\end{equation*}
For $0\leq n\leq N-M$ we have $(1+L\Delta t)\sp n\leq e\sp{LT}$ and
\begin{equation*}
1+(1+L\Delta t)+\cdots+(1+L\Delta t)\sp n \leq
\frac{e\sp{LT}-1}{L\Delta t}\leq\frac{e\sp{LT}}{L\Delta t}.
\end{equation*}
Hence
\begin{equation*}
\varepsilon_n \leq KLe\sp{LT}(M-1)(M-2)\Delta
t\sp2+\frac{e\sp{LT}}{L}(C_1\Delta t+C_2\Delta t\sp2).
\end{equation*}
As in the proof of Theorem \ref{thm:convpaths1} we can extend
$\{\xi_n\}$ up to $N$ and have
\begin{equation*}
\abs{\xi_n-\eta_n}\leq KLe\sp{LT}(M-1)(M-2)\Delta
t\sp2+\frac{e\sp{LT}}{L}(C_1\Delta t+C_2\Delta t\sp2) + 2K(M-1)\Delta t.
\end{equation*}
This together with \eqref{eq:pathapprox} gives us \eqref{eq:pathM}.
\end{proof}
\section{The fully discrete case}\label{sec:FD}
We present here two results, Theorems \ref{thm:coco} and
\ref{thm:psiconvexhull}, that are useful for the proof of Theorem
\ref{thm:fullydiscrete}. We will use the following well-known result,
the Carath\'eodory Theorem:
\begin{thm}\label{thm:Caratheodory}
The convex hull of an arbitrary subset $A$ of $\Re^d$ is given by
\begin{equation*}
\text{co}(A)=\Big\{\sum_{i=1}^{d+1}\lambda_i a_i\ |\ a_i\in A,
\lambda_i \geq 0, \sum_{i=1}^{d+1} \lambda_i=1\Big\}.
\end{equation*}
\end{thm}
For a proof, see \cite{Berger}.
\begin{thm}\label{thm:coco}
Assume that \eqref{eq:disccontrol1}, \eqref{eq:disccontrol2},
\eqref{eq:disccontrol3}, and \eqref{eq:disccontrol4} hold, and that
$M \geq d+1$. Then
\begin{equation}\label{eq:convincl}
\text{co}\big(\varphi\sp{M}(x_0)\big)\subset\cup_{x\in\varphi(x_0)}\text{co}\big(\varphi\sp{M-1}(x)\big)+RB,
\end{equation}
where
\begin{multline*}
R=(8M-10)KL\Delta t\sp2 +
\big(2KL\sp2\frac{(M-1)\sp3-(M-1)}{3}(1+L\Delta t)\sp{M-3}\\
+SK\sp2\frac{M(M-1)(2M-1)}{3}\big)\Delta t\sp3.
\end{multline*}
\end{thm}
\begin{proof}
We start by introducing the notation
\begin{equation*}
b_i=f_i(x_0),\ A_i=f'_i(x_0),\quad \text{for }0\leq i\leq M.
\end{equation*}
To begin with, we will make the assumption that the functions $f_i$
are given by
\begin{equation}\label{eq:linearized}
f_i(x)=b_i+A_i(x-x_0).
\end{equation}
Afterwards, we will consider the general case.
For simplicity, we will prove \eqref{eq:convincl} for the case where
$M=d+1$. The general result follows directly from this. We will also
assume that $x_0=0$, to simplify the presentation.
\emph{Step 1.}
Every point $x$ in $\varphi\sp{d+1}(x_0)$ is given by
\begin{equation}\label{eq:pointinphidplusone}
x=x_0+\Delta t f_{i_1}(x_0)+\Delta t f_{i_2}(x_1)+\cdots+\Delta t
f_{i_{d+1}}(x_d),
\end{equation}
where $i_j\in\{1,\ldots,d+1\}$, for all $1\leq j\leq d+1$, and $x_1$,
$x_2$,..., are defined recursively by
\begin{equation*}
x_{n+1}=x_n+\Delta t f_{i_n}(x_n), \quad \text{for } n=0,\ldots,d.
\end{equation*}
Since we now assume that the functions $f_i$ are given by \eqref{eq:linearized},
we have
\begin{equation}\label{eq:discevolonestep}
x_{n+1}=x_n+\Delta t b_{i_n}+\Delta t A_{i_n}x_n.
\end{equation}
When we sum the terms in \eqref{eq:pointinphidplusone} under the
consideration of \eqref{eq:discevolonestep}, we see that
\begin{equation*}
x=x_0+\sum_{r=1}\sp{d+1} \sum_{1\leq
k_1<k_2<\ldots<k_r\leq d+1} A_{i_{k_r}}A_{i_{k_{r-1}}}\cdots
A_{i_{k_2}}b_{i_{k_1}}\Delta t\sp r.
\end{equation*}
Let $\tilde x$ be the approximation of $x$, where all terms of power
three or larger in $\Delta t$ have been dropped, i.e.
\begin{equation*}
\tilde x=x_0+\sum_{r=1}\sp{2} \sum_{1\leq
k_1<k_2<\ldots<k_r\leq d+1} A_{i_{k_r}}A_{i_{k_{r-1}}}\cdots
A_{i_{k_2}}b_{i_{k_1}}\Delta t\sp r.
\end{equation*}
With the bounds on $A_i$ and $b_i$ from \eqref{eq:disccontrol3}and
\eqref{eq:disccontrol4}, we have that
\begin{multline*}
\abs{\tilde x-x}\leq \sum_{r=3}\sp{d+1} {d+1 \choose r}
L\sp{r-1}K\Delta t\sp r\\
=\frac{K}{L}\sum_{r=0}\sp{d+1}{d+1 \choose r}
(L \Delta t)\sp r - \frac{K}{L}\big(1+(d+1)L\Delta t+ \frac{d(d+1)}{2}L\sp 2 \Delta
t\sp 2\big)\\
=\frac{K}{L}(1+L\Delta t)\sp{d+1}- \frac{K}{L}\big(1+(d+1)L\Delta t+ \frac{d(d+1)}{2}L\sp 2 \Delta
t\sp 2\big).
\end{multline*}
With a Taylor expansion of the function $f(x)=(1+x)\sp{d+1}$ around
$x=0$, we establish that
\begin{equation*}
(1+x)\sp{d+1}\leq 1+(d+1)x +
\frac{d(d+1)}{2}x\sp2+\frac{d\sp3-d}{6}(1+x)\sp{d-2}x\sp 3,
\end{equation*}
and hence
\begin{equation}\label{eq:higherorderbound}
\abs{\tilde x-x}\leq KL\sp2\frac{d\sp3-d}{6}(1+L\Delta t)\sp{d-2}\Delta
t\sp 3.
\end{equation}
\emph{Step 2.} We now consider the convex combination
\begin{equation}\label{eq:twoconvcomb}
(1-\frac{1}{N})x\sp1+\frac{1}{N}x\sp2,
\end{equation}
where $x\sp1$ and $x\sp2$ are two elements in $\varphi\sp{d+1}(x_0)$,
such that in the expression in \eqref{eq:pointinphidplusone} for
$x\sp1$, none of the indices $i_n$ equals one, while for $x\sp2$, $N$
of the indices equal one. We will see how well the convex combination
in \eqref{eq:twoconvcomb} can be represented by another convex
combination,
\begin{equation}\label{eq:twoconvcomb2}
(1-\frac{1}{N})\tilde x\sp1+\frac{1}{N}\tilde x\sp2,
\end{equation}
where $\tilde x\sp1$ and $\tilde x\sp2$ both have precisely one index
$i_n$ equal to one in the expression in
\eqref{eq:pointinphidplusone}.
Pick any $n\in\{1,2,\ldots,d+1\}$. Assume that we define $\tilde
x\sp1$ by changing the index $i_n$ (denote $i_n=k$) in the expression
\eqref{eq:pointinphidplusone} for $x\sp1$ to one. Then
\begin{multline}\label{eq:x1diff}
\tilde x\sp1-x\sp1 = \Delta t(b_1-b_{i_n}) + \Delta
t\sp2\big((A_1-A_{i_n})(b_{i_1}+b_{i_2}+\cdots+b_{i_{n-1}})\\
+(A_{i_{n+1}}+\cdots +A_{i_{d+1}})(b_1-b_{i_n}) \big) + \ \text{higher
order terms.}
\end{multline}
Simiarly, we define $\tilde x\sp2$ by exchanging all the indices
for which $i_n=1$ to $i_n=k$. Then the first order term in \eqref{eq:twoconvcomb2}
is the same as in \eqref{eq:twoconvcomb}. The second order term in
the difference $\tilde x\sp 1-x\sp1$ in \eqref{eq:x1diff} is bounded
in magnitude by $2KLd\Delta t\sp 2$.
Since this bound holds independently of which of the indices was
changed, it follows that the second order term in the difference
$\tilde x\sp 2-x\sp2$ is bounded in magnitude by $2NKLd\Delta
t\sp2$. Hence the difference in the second order term between the
convex combinations in \eqref{eq:twoconvcomb} and in
\eqref{eq:twoconvcomb2} is bounded by
\begin{equation*}
(1-\frac{1}{N})2KLd\Delta t\sp2+\frac{1}{N}2NKLd\Delta
t\sp2=(4-\frac{2}{N})KLd\Delta t\sp2 \leq 4KLd\Delta t\sp2.
\end{equation*}
In step 1, we established that the sum of all terms of order higher
than or equal to three in $\Delta t$ for every element in
$\varphi\sp{d+1}(x_0)$ is bounded as in
\eqref{eq:higherorderbound}. We thereby have
\begin{equation}\label{eq:convcomberror}
\big|(1-\frac{1}{N})(\tilde x\sp1-x\sp1)+\frac{1}{N}(\tilde x\sp
2-x\sp2)\big| \leq 4KLd\Delta t\sp 2 + KL\sp2\frac{d\sp3-d}{3}(1+L\Delta
t)\sp{d-2}\Delta t\sp3.
\end{equation}
\emph{Step 3.} Let $z$ be any element in
$\text{co}\big(\varphi\sp{d+1}(x_0)\big)$. By the Carath\'eodory
Theorem (Theorem \ref{thm:Caratheodory}), we have that there exists a
$G\leq d+1$ and points and constants $x\sp i\in\varphi\sp{d+1}(x_0)$ and
$\alpha_i> 0$, for $1\leq i\leq G$, such that
\begin{equation}\label{eq:zconvcomb}
z=\sum_{i=1}\sp{G}\alpha_i x\sp i,
\end{equation}
and
$\sum_{i=1}\sp{G} \alpha_i=1$.
We can then write
\begin{equation*}
z=x_0+\Delta t\sum_{i=1}\sp{d+1}\gamma_ib_i+\ \text{higher
order terms,}
\end{equation*}
where $\gamma_i\geq 0$ for $1\leq i\leq d+1$ and
$\sum_{i=1}\sp{d+1}\gamma_i=d+1$. It must hold that at least one of
the coefficients $\gamma_i\geq 1$. For simplicity, let us assume that
$\gamma_1\geq 1$.
We will now present an algorithm
which gives us an approximation of $z$ in the form of a convex
combination of points in $\varphi\sp{d+1}(x_0)$ which all have one
index $i_n=1$ in the formula \eqref{eq:pointinphidplusone}. We also
give an error bound of this approximation.
Let $I\subset\{1,2,\ldots,G\}$ be the index set of all the points
$x\sp i$ in \eqref{eq:zconvcomb} for which none of the indices $i_1$
in the formula for the points in $\varphi\sp{d+1}(x_0)$ equals
one. Let $J$ be a set which consists of the weights $\alpha_i$
corresponding to elements in $I$, i.e.\ $i\in I$ if and only if
$\alpha_i\in J$.
If $I=\emptyset$, we already have what we are aiming for. Let us
therefore assume that $I$ is nonempty, and for simplicity that $i=1$
is one of the elements therein. Take one element in
$\{1,2,\ldots,d+1\}\setminus I$, such that the corresponding element
$x\sp i$ in the convex combination \eqref{eq:zconvcomb} is of the form
\begin{equation*}
x\sp i = x_0+\Delta t(Nb_1+\cdots)+\ \text{higher order terms,}
\end{equation*}
with $N\geq 2$. Such an element must exist, since $\gamma_1\geq
1$. For simplicity, let us assume that $x\sp2$ is one such
element.
We may write
\begin{equation*}
\frac{1}{\alpha_1+\alpha_2}(\alpha_1 x\sp1 + \alpha_2 x\sp 2) = x_0 +
\Delta t(kb_1 +\cdots) +\ \text{higher order terms.}
\end{equation*}
One of the two following cases must hold:
\begin{enumerate}
\item $k > 1$. When this is the case we rewrite as follows:
\begin{multline}\label{eq:convcombrewritten}
\alpha_1 x\sp1+\alpha_2 x\sp2=\alpha_1 x\sp 1 +
\frac{\alpha_1}{N-1}x\sp2 +
\big(\alpha_2-\frac{\alpha_1}{N-1}\big)x\sp2\\
=\frac{N}{N-1}\alpha_1\big(\big(1-\frac{1}{N}\big)x\sp1+\frac{1}{N}x\sp
2\big)+\big(\alpha_2-\frac{\alpha_1}{N-1}\big)x\sp2.
\end{multline}
Since $k>1$ we have that $\alpha_2-\alpha_1/(N-1)$ is positive.
By the result from step 2, we have the approximation result in
\eqref{eq:convcomberror}, with some points $\tilde x\sp1$ and $\tilde
x\sp2$, both being of the form
\begin{equation*}
x_0+\Delta t(b_1+\cdots)+\ \text{higher order terms}.
\end{equation*}
Together with \eqref{eq:convcombrewritten}, this implies that
\begin{equation}\label{eq:kappacomb}
\sum_{i=1}\sp G \alpha_i x\sp i = \alpha_1\tilde x\sp1 +
\frac{\alpha_1}{N-1}\tilde x\sp2 +
\big(\alpha_2-\frac{\alpha_1}{N-1}\big)x\sp2 + \sum_{i=3}\sp G
\alpha_i x\sp i + \kappa,
\end{equation}
where
\begin{equation}\label{eq:kappabound}
\abs{\kappa}\leq \frac{N}{N-1}\alpha_1\big(4KL d\Delta t\sp 2 + KL\sp2\frac{d\sp3-d}{3}(1+L\Delta
t)\sp{d-2}\Delta t\sp3\big).
\end{equation}
\item $k\leq 1$. In this case we rewrite as follows:
\begin{multline*}
\alpha_1 x\sp 1+\alpha_2 x\sp 2 = (N-1)\alpha_2 x\sp 1 + \alpha_2
x\sp2 + \big(\alpha_1-(N-1)\alpha_2\big)x\sp1\\
=N\alpha_2\big(\big(1-\frac{1}{N}\big)x\sp1+\frac{1}{N}x\sp
2\big) + \big(\alpha_1-(N-1)\alpha_2\big)x\sp1.
\end{multline*}
Since $k\leq 1$ we have that $\alpha_1-(N-1)\alpha_2$ is
nonnegative. Similarly as in case (1), we have
\begin{equation}\label{eq:kappacomb2}
\sum_{i=1}\sp{G}\alpha_i x\sp i = (N-1)\alpha_2\tilde x\sp1 +
\alpha_2\tilde x\sp 2 + \big(\alpha_1-(N-1)\alpha_2\big)x\sp1 +
\sum_{i=3}\sp G \alpha_ix\sp i + \kappa,
\end{equation}
where
\begin{equation}\label{eq:kappabound2}
\abs\kappa\leq N\alpha_2\big(4KLd\Delta t\sp 2 + KL\sp2 \frac{d\sp3-d}{3}(1+L\Delta
t)\sp{d-2}\Delta t\sp3\big).
\end{equation}
\end{enumerate}
If case (1) holds we let
\begin{equation*}
\hat z = \alpha_1\tilde x\sp 1 + \frac{\alpha_1}{N-1}\tilde x\sp 2 +
\big(\alpha_2-\frac{\alpha_1}{N-1}\big)x\sp 2 + \sum_{i=3}\sp G
\alpha_i x\sp i,
\end{equation*}
and remove $i=1$ from $I$ and $\alpha_1$ from $J$. If case (2) holds
we let
\begin{equation*}
\hat z=(N-1)\alpha_2\tilde x\sp1 +
\alpha_2\tilde x\sp 2 + \big(\alpha_1-(N-1)\alpha_2\big)x\sp1 +
\sum_{i=3}\sp G \alpha_ix\sp i,
\end{equation*}
and replace $\alpha_1$ with $(N-1)\alpha_2$ in $J$.
We then iterate the process above with $z$ replaced by $\hat z$, and
the new sets $I$ and $J$. We continue this process until the sets $I$
and $J$ are empty, and we have an approximation $\tilde z$ of $z$ of
the form
\begin{equation}\label{eq:tildez}
\tilde z = \sum_{i}\tilde\alpha_i \tilde x\sp i,
\end{equation}
where every $\tilde x\sp i$ is of the form
\begin{equation*}
\tilde x\sp i = x_0 + \Delta t(mb_1 + \cdots) +\ \text{higher order terms,}
\end{equation*}
with $m$ an integer greater than or equal to one.
We note that the factor $N\alpha_2$, appearing in the error estimate
in \eqref{eq:kappabound2} is the same as the weights of the new points
$\tilde x\sp1$ and $\tilde x\sp2$ in \eqref{eq:kappacomb2}. The same
holds also for case (1), with \eqref{eq:kappabound} and
\eqref{eq:kappacomb}. Since the total weight of the points that have
been changed can not be larger than one, we have that
\begin{equation}\label{eq:zerr1}
\abs{\tilde z - z} \leq 4KLd\Delta t\sp2+KL\sp2 \frac{d\sp3-d}{3}(1+L\Delta
t)\sp{d-2}\Delta t\sp3.
\end{equation}
\emph{Step 4.} We now approximate the point $\tilde z$ in \eqref{eq:tildez}
by a point in $\varphi\sp d(x_0+\Delta t f_1(x_0))$. Consider any
point $\tilde x\sp i$ in the convex combination in
\eqref{eq:tildez}. When $\tilde x\sp i$ is computed by equation
\eqref{eq:pointinphidplusone} we know that at least one of the indices must
equal one. Let us assume that $i_n=1$. We denote by $\bar x\sp i$ the
element in $\varphi\sp d(x_0+\Delta t f_1(x_0))$ we obtain by
switching the indices $i_1$ and $i_n=1$ in the expression for $\tilde
x\sp i$ in \eqref{eq:pointinphidplusone}. We then have the difference
\begin{multline*}
\bar x\sp i -\tilde x\sp i= \Delta
t\sp2\big((A_{i_2}+\cdots+A_{i_{n-1}})(b_{i_n}-b_{i_1})
+A_{i_1}(b_{i_2}+\cdots+b_{i_n})\\
-A_{i_n}(b_{i_1}+\cdots+b_{i_{n-1}})\big)+\ \text{higher order terms.}
\end{multline*}
We may get a bound for the magnitude of the difference $\bar x\sp
i-\tilde x\sp i$ by using the bounds on $\abs{A_i}$ and $\abs{b_i}$ from
\eqref{eq:disccontrol3} and \eqref{eq:disccontrol4} and the bound on the higher order terms of $\bar
x\sp i$ and $\tilde x\sp i$ in \eqref{eq:higherorderbound}. We get the
largest possible difference if $n=d+1$:
\begin{equation*}
\abs{\bar x\sp i-\tilde x\sp i} \leq (4d-2)KL\Delta t\sp 2 + KL\sp
2\frac{d\sp 3-d}{3}(1+L\Delta t)\sp{d-2}\Delta t\sp 3.
\end{equation*}
Now let
\begin{equation*}
\bar z = \sum_i\tilde\alpha_i\bar x\sp i.
\end{equation*}
Since $\sum_i\tilde\alpha_i=1$, we therefore have that
\begin{equation}\label{eq:zerr2}
\abs{\bar z-\tilde z} \leq (4d-2)KL\Delta t\sp 2 + KL\sp
2\frac{d\sp 3-d}{3}(1+L\Delta t)\sp{d-2}\Delta t\sp 3.
\end{equation}
\emph{Step 5.} We now consider the contribution to the error from the
fact that we may not have \eqref{eq:linearized}, but instead the
functions $f_i$ satisfy \eqref{eq:disccontrol2}. Denote by $P$ the
set $\text{co}\big(\varphi\sp{d+1}(x_0)\big)$ when it is computed
using \eqref{eq:linearized}. The set
$\text{co}\big(\varphi\sp{d+1}(x_0)\big)$ in the general case
satisfies the inclusion
\begin{equation*}
\text{co}\big(\varphi\sp{d+1}(x_0)\big) \subset P+rB,
\end{equation*}
where
\begin{multline*}
r=SK\sp 2\Delta t\sp 3 + 4 SK\sp 2 \Delta t\sp 3 +\cdots + d\sp2
SK\sp2\Delta t\sp3\\
= SK\sp2 \frac{d(d+1)(2d+1)}{6}\Delta t\sp3.
\end{multline*}
An error of size $r$ is made also when
\begin{equation*}
\cup_{x\in\varphi(x_0)}\text{co}\big(\varphi\sp d(x)\big)
\end{equation*}
is approximated using \eqref{eq:linearized}.
This, together with \eqref{eq:zerr1}, \eqref{eq:zerr2}, and $M=d+1$
gives \eqref{eq:convincl}.
\end{proof}
\begin{thm}\label{thm:psiconvexhull}
Assume that \eqref{eq:disccontrol1}, \eqref{eq:disccontrol2},
\eqref{eq:disccontrol3}, and \eqref{eq:disccontrol4} hold. Then
\begin{equation}\label{eq:thirdorderincl}
\psi\big(\text{co}(\varphi\sp{M-1}(z))\big) \subset
\text{co}\big(\varphi\sp M(z)\big) +
SK\sp2\frac{M(M-1)(2M-1)}{3}\Delta t\sp3B.
\end{equation}
\end{thm}
\begin{proof}
To begin with, let us assume that
\begin{equation}\label{eq:simplefuncs}
f_i(x)=b_i+A_i(x-z),\quad \text{for }1\leq i\leq M.
\end{equation}
We will show that this implies that
\begin{equation}\label{eq:psico}
\psi\big(\text{co}(\varphi\sp{M-1}(z))\big) \subset
\text{co}\big(\varphi\sp M(z)\big).
\end{equation}
Let $v$ be any unit vector in $\Re\sp d$ and consider the function
\begin{equation*
(x,\alpha_1,\ldots,\alpha_M)\mapsto v\cdot(x+\sum_{i=1}\sp M \alpha_i f_i(x)),
\end{equation*}
over the set
\begin{equation*}
\big\{x,\alpha_1,\ldots,\alpha_M : x\in \text{co}(\varphi\sp{M-1}(z)),
\alpha_i\in\Re, \alpha_i\geq 0, \sum_{i=1}\sp M \alpha_i=1\big\}.
\end{equation*}
The function is continuous, and the set is compact, and hence there is
a maximizer
\begin{equation*}
x\sp*,\alpha_1\sp*,\ldots,\alpha_M\sp*.
\end{equation*}
We note two things:
\begin{enumerate}
\item Since the function
\begin{equation*}
x\mapsto v\cdot(x+\Delta t\sum_{i=1}\sp M \alpha_i\sp* f_i(x))
\end{equation*}
is linear,
its maximum over $\text{co}\big(\varphi\sp{M-1}(z)\big)$ is attained at a point in
$\varphi\sp{M-1}(z)$.
\item Since the function
\begin{equation*}
(\alpha_1,\ldots,\alpha_M)\mapsto v\cdot(x\sp*+\Delta t\sum_{i=1}\sp M \alpha_i f_i(x\sp*))
\end{equation*}
is linear, its maximum over
\begin{equation*}
\big\{\alpha_1,\ldots,\alpha_M:\alpha_i\in\Re,\alpha_i\geq0, \sum_{i=1}\sp{M}\alpha_i=1\big\}
\end{equation*}
is attained at a point where one of the $\alpha_i$:s are one.
\end{enumerate}
Since $v$ can be any element in $\Re\sp d$, these facts imply that
\begin{equation}\label{eq:inclusionequality}
\text{co}\big(\psi\big(\text{co}(\varphi\sp{M-1}(z))\big)\big) =
\text{co}\big(\varphi\sp N(z)\big),
\end{equation}
which implies \eqref{eq:psico}.
Now let $b_i=f_i(z)$ and $A_i=f'_i(z)$. Let us denote by $P$ the set
in \eqref{eq:inclusionequality} when the functions in
\eqref{eq:simplefuncs} are used.
By \eqref{eq:disccontrol2} and \eqref{eq:disccontrol3} we have that
\begin{equation}\label{eq:incl1}
\psi\big(\text{co}(\varphi\sp{M-1}(z))\big) \subset P + rB,
\end{equation}
where
\begin{multline*}
r=SK\sp 2\Delta t\sp 3 + 4 SK\sp 2 \Delta t\sp 3 +\cdots + (M-1)\sp2
SK\sp2\Delta t\sp3\\
= SK\sp2 \frac{M(M-1)(2M-1)}{6}\Delta t\sp3.
\end{multline*}
Similarly,
\begin{equation}\label{eq:incl2}
P\subset \text{co}\big(\varphi\sp{M}(z)\big)+rB.
\end{equation}
The inclusion \eqref{eq:thirdorderincl} follows by \eqref{eq:incl1} and \eqref{eq:incl2}.
\end{proof}
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,259 |
Q: Variable cannot be resolved error in struts2 My requirement is am trying to populate pid(projectid) from project table and name from userdetails table as an drop down for a form.Am new to struts framework. Could someone please throw us some light on this issue please.
Here is the code:
sprintform.jsp:
<%@ page language="java" contentType="text/html; charset=ISO-8859-1"
pageEncoding="ISO-8859-1"%>
<%@ taglib prefix="s" uri="/struts-tags"%>
<?xml version="1.0" encoding="UTF-8" ?>
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=ISO-8859-1">
<title>Insert title here</title>
<link href="css/jquery.ui.datepicker.css" rel="stylesheet"
type="text/css" />
<script src="js/jquery-1.7.1.min.js" type="text/javascript"></script>
<script src="js/jquery-ui-1.8.17.custom.min.js" type="text/javascript"></script>
<script type="text/javascript">
$(function(){$('.dateTxt').datepicker({
dateFormat : 'yy-mm-dd'
}); });
</script>
</head>
<body>
<h1 style="color: green">Sprint</h1>
<s:form action="sprintInsert" namespace="/" method="post"
name="sprintForm" theme="xhtml">
<s:textfield name="title:" size="40" maxlength="40" required="true"
label="Title" />
<p>
Begin Date: <input id="one" class="dateTxt" type="text"
name="begindate" />
</p>
<p>
End Date: <input id="two" class="dateTxt" type="text" name="enddate" />
</p>
<s:select label="ProjectId" headerKey="-1"
headerValue="Select Project Id" list="projectidList" name="pid" />
<%-- <s:select label="Owner" headerKey="-1"
headerValue="Select Sprint Owner" list="sprintownerList"
name="sprintowner" /> --%>
<tr>
<td>State:</td>
<td><select name="state">
<option value="">Choose a state..</option>
<option value="A">Active</option>
<option value="F">Future</option>
<option value="C">Close</option>
</select></td>
</tr>
<s:textfield name="targetestimatedpoints" size="40" maxlength="40"
required="true" label="Target Estimate pts:" />
<s:textfield name="totalestimatedpoints" size="40" maxlength="40"
required="true" label="Total Estimate pts:" />
<s:textfield name="totaldefaultestimatedhours" size="40"
maxlength="40" required="true" label="Total Detail Estimate Hrs: " />
<s:textfield name="todohours:" size="40" maxlength="40"
required="true" label="Total To Do Hrs:" />
<s:textfield name="description: :" size="40" maxlength="40"
required="true" label="Description: " />
<tr align="right">
<td><div align="center">
<input type="submit" value="save">
</div>
<td align="center"><input type="reset" value="Reset"></td>
</tr>
</s:form>
<s:if test="hasActionErrors()">
<div id="fieldErrors">
<s:actionerror />
</div>
</s:if>
</body>
</html>
SprintAction.java:
package com.bits.sprintanalyzer.action;
import java.util.List;
import org.apache.log4j.Logger;
import com.bits.sprintanalyzer.ResourceException;
import com.bits.sprintanalyzer.dao.SprintDAO;
import com.opensymphony.xwork2.ActionSupport;
public class SprintAction extends ActionSupport {
private static final Logger LOG = Logger.getLogger(SprintAction.class);
/**
*
*/
private static final long serialVersionUID = -6257623073537028210L;
private String title;
private String begindate;
private String enddate;
private String pid;
private String sprintowner;
private String state;
private int targetestimatedpoints;
private int totalestimatedpoints;
private int totaldefaultestimatedhours;
private int todohours;
private String description;
public String getPid() {
return pid;
}
public void setPid(String pid) {
this.pid = pid;
}
public String getSprintowner() {
return sprintowner;
}
public void setSprintowner(String sprintowner) {
this.sprintowner = sprintowner;
}
public String getState() {
return state;
}
public void setState(String state) {
this.state = state;
}
public int getTargetestimatedpoints() {
return targetestimatedpoints;
}
public void setTargetestimatedpoints(int targetestimatedpoints) {
this.targetestimatedpoints = targetestimatedpoints;
}
public int getTotalestimatedpoints() {
return totalestimatedpoints;
}
public void setTotalestimatedpoints(int totalestimatedpoints) {
this.totalestimatedpoints = totalestimatedpoints;
}
public int getTotaldefaultestimatedhours() {
return totaldefaultestimatedhours;
}
public void setTotaldefaultestimatedhours(int totaldefaultestimatedhours) {
this.totaldefaultestimatedhours = totaldefaultestimatedhours;
}
public int getTodohours() {
return todohours;
}
public void setTodohours(int todohours) {
this.todohours = todohours;
}
public String getDescription() {
return description;
}
public void setDescription(String description) {
this.description = description;
}
public String display() throws Exception {
return INPUT;
}
public String getBegindate() {
return begindate;
}
public void setBegindate(String begindate) {
this.begindate = begindate;
}
public String getEnddate() {
return enddate;
}
public void setEnddate(String enddate) {
this.enddate = enddate;
}
public String getTitle() {
return title;
}
public void setTitle(String title) {
this.title = title;
}
public List<String> getpidList() throws ResourceException {
return SprintDAO.getpidList();
}
public List<String> getOwnerList() throws ResourceException {
return SprintDAO.getOwnerList();
}
@Override
public void validate() {
}
@Override
public String execute() throws Exception {
LOG.info("title" + title);
LOG.info("begindate" + begindate);
LOG.info("enddate" + enddate);
LOG.info("pid" + pid);
LOG.info("sprintowner" + sprintowner);
LOG.info("state" + state);
LOG.info("targetestimatedpoints" + targetestimatedpoints);
LOG.info("totalestimatedpoints" + totalestimatedpoints);
LOG.info("totaldefaultestimatedhours" + totaldefaultestimatedhours);
LOG.info("todohours" + todohours);
LOG.info("description" + description);
// ProjectDAO.insert(projectname,description,scrummaster,productowner,begindate,enddate);
int i = SprintDAO.save(this);
if (i > 0) {
return "success";
}
return "error";
}
}
SprintDAO:
package com.bits.sprintanalyzer.dao;
import java.sql.Connection;
import java.sql.PreparedStatement;
import java.sql.ResultSet;
import java.sql.SQLException;
import java.util.ArrayList;
import java.util.List;
import com.bits.sprintanalyzer.ResourceException;
import com.bits.sprintanalyzer.action.SprintAction;
import com.bits.sprintanalyzer.util.ConnectionUtil;
public class SprintDAO {
private static final String PROJECTQUERY = "select pid from project";
private static final String USERQUERY = "select name from userdetail";
public static List<String> getpidList() throws ResourceException{
List<String> projectidList = new ArrayList<String>();
// this should be populated from DB
try (Connection con = ConnectionUtil.INSTANCE.getConnection();
PreparedStatement st = con.prepareStatement(PROJECTQUERY)){
ResultSet rs =st.executeQuery();
while(rs.next()){
projectidList.add(rs.getString(1));
}
return projectidList;
}
catch (SQLException | ResourceException e) {
throw new ResourceException("Failed to validate project id", e);
}
}
public static List<String> getOwnerList() throws ResourceException{
List<String> sprintownerList = new ArrayList<String>();
// this should be populated from DB
try (Connection con = ConnectionUtil.INSTANCE.getConnection();
PreparedStatement st = con.prepareStatement(USERQUERY)){
ResultSet rs =st.executeQuery();
while(rs.next()){
sprintownerList.add(rs.getString(1));
}
return sprintownerList;
}
catch (SQLException | ResourceException e) {
throw new ResourceException("Failed to validate productowner", e);
}
}
//insert into database
public static int save(SprintAction SA) throws Exception{
int status=0;
try{
Connection con = ConnectionUtil.INSTANCE.getConnection();
PreparedStatement ps = con.prepareStatement("insert into sprint(pid,title,begindate,enddate,owner,state,targetestimatedpoints,totalestimatedpoints,totaldefaultestimatedhours,todohours,description) values(?,?,?,?,?,?,?,?,?,?,?)");
ps.setString(1, SA.getPid());
ps.setString(2, SA.getTitle());
ps.setString(3, SA.getBegindate());
ps.setString(4, SA.getEnddate());
ps.setString(5, SA.getSprintowner());
ps.setString(6, SA.getState());
ps.setInt(7, SA.getTargetestimatedpoints());
ps.setInt(8, SA.getTotalestimatedpoints());
ps.setInt(9, SA.getTotaldefaultestimatedhours());
ps.setInt(10, SA.getTodohours());
ps.setString(11, SA.getDescription());
status=ps.executeUpdate();
}catch(Exception e){
e.printStackTrace();}
return status;
}
}
struts.xml:
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE struts PUBLIC
"-//Apache Software Foundation//DTD Struts Configuration 2.1.7//EN"
"/WEB-INF/classes/struts-2.1.7.dtd">
<struts>
<!--
You could also set the constants in the struts.properties file
placed in the same directory as struts.xml
-->
<constant name="struts.devMode" value="true" />
<package name="sprintanalyzer" extends="struts-default" namespace="/">
<!--
If no class attribute is specified the framework will assume success and
render the result index.jsp
If no name value for the result node is specified the success value is the default
-->
<action name="">
<result>/jsp/login.jsp</result>
</action>
<!--
If the URL is hello.action then call the execute method of class HelloWorldAction.
If the result returned by the execute method is success render the HelloWorld.jsp
-->
<action name="login" class="com.bits.sprintanalyzer.action.LoginAction"
method="execute">
<result name="success">/jsp/sprintanalyzer.jsp</result>
<result name="input">/jsp/login.jsp</result>
</action>
<action name="projectform" class="com.bits.sprintanalyzer.action.ProjectAction"
method="display">
<result name="input">/jsp/projectform.jsp</result>
</action>
<action name="projectInsert" class="com.bits.sprintanalyzer.action.ProjectAction"
method="execute">
<result name="success">/jsp/sprintanalyzer.jsp</result>
</action>
<action name="sprintform" class="com.bits.sprintanalyzer.action.SprintAction"
method="display">
<result name="input">/jsp/sprintform.jsp</result>
</action>
<action name="sprintInsert" class="com.bits.sprintanalyzer.action.SprintAction"
method="execute">
<result name="success">/jsp/sprintanalyzer.jsp</result>
</action>
</package>
</struts>
Issue is coming from both projectidlist and sprintownerList. Please advise accordingly.
Please find below stacktrace:
2016-10-01T18:23:38.916+0530|Info: 2016-10-01 18:23:38,916 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The first TextProvider in the ValueStack (com.opensymphony.xwork2.ActionSupport) could not locate the message resource with key 'Login'
2016-10-01T18:23:38.917+0530|Info: 2016-10-01 18:23:38,917 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The default value expression 'Login' was evaluated and did not match a property. The literal value 'Login' will be used.
2016-10-01T18:23:38.925+0530|Info: 2016-10-01 18:23:38,924 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The first TextProvider in the ValueStack (com.opensymphony.xwork2.ActionSupport) could not locate the message resource with key 'Login'
2016-10-01T18:23:38.925+0530|Info: 2016-10-01 18:23:38,925 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The default value expression 'Login' was evaluated and did not match a property. The literal value 'Login' will be used.
2016-10-01T18:23:57.121+0530|Info: 2016-10-01 18:23:57,120 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The first TextProvider in the ValueStack (com.opensymphony.xwork2.ActionSupport) could not locate the message resource with key 'Login'
2016-10-01T18:23:57.121+0530|Info: 2016-10-01 18:23:57,121 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The default value expression 'Login' was evaluated and did not match a property. The literal value 'Login' will be used.
2016-10-01T18:23:57.128+0530|Info: 2016-10-01 18:23:57,128 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The first TextProvider in the ValueStack (com.opensymphony.xwork2.ActionSupport) could not locate the message resource with key 'Login'
2016-10-01T18:23:57.129+0530|Info: 2016-10-01 18:23:57,128 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The default value expression 'Login' was evaluated and did not match a property. The literal value 'Login' will be used.
2016-10-01T18:24:03.841+0530|Severe: Sat Oct 01 18:24:03 IST 2016 WARN: Establishing SSL connection without server's identity verification is not recommended. According to MySQL 5.5.45+, 5.6.26+ and 5.7.6+ requirements SSL connection must be established by default if explicit option isn't set. For compliance with existing applications not using SSL the verifyServerCertificate property is set to 'false'. You need either to explicitly disable SSL by setting useSSL=false, or set useSSL=true and provide truststore for server certificate verification.
2016-10-01T18:24:04.022+0530|Info: 2016-10-01 18:24:04,021 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The first TextProvider in the ValueStack (com.bits.sprintanalyzer.action.LoginAction) could not locate the message resource with key 'welcome to Sprint Analyzer Tool'
2016-10-01T18:24:04.022+0530|Info: 2016-10-01 18:24:04,022 WARN org.apache.struts2.util.TextProviderHelper.warn:45 - The default value expression 'welcome to Sprint Analyzer Tool' was evaluated and did not match a property. The literal value 'welcome to Sprint Analyzer Tool' will be used.
2016-10-01T18:24:06.835+0530|Warning: Servlet.service() for servlet jsp threw exception
tag 'select', field 'list', name 'pid': The requested list key 'projectidList' could not be resolved as a collection/array/map/enumeration/iterator type. Example: people or people.{name} - [unknown location]
at org.apache.struts2.components.Component.fieldError(Component.java:237)
at org.apache.struts2.components.Component.findValue(Component.java:358)
at org.apache.struts2.components.ListUIBean.evaluateExtraParams(ListUIBean.java:80)
at org.apache.struts2.components.Select.evaluateExtraParams(Select.java:105)
at org.apache.struts2.components.UIBean.evaluateParams(UIBean.java:856)
at org.apache.struts2.components.UIBean.end(UIBean.java:510)
at org.apache.struts2.views.jsp.ComponentTagSupport.doEndTag(ComponentTagSupport.java:42)
at org.apache.jsp.jsp.sprintform_jsp._jspx_meth_s_select_0(sprintform_jsp.java:236)
at org.apache.jsp.jsp.sprintform_jsp._jspx_meth_s_form_0(sprintform_jsp.java:144)
at org.apache.jsp.jsp.sprintform_jsp._jspService(sprintform_jsp.java:88)
at org.apache.jasper.runtime.HttpJspBase.service(HttpJspBase.java:111)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:790)
at org.apache.jasper.servlet.JspServletWrapper.service(JspServletWrapper.java:411)
at org.apache.jasper.servlet.JspServlet.serviceJspFile(JspServlet.java:473)
at org.apache.jasper.servlet.JspServlet.service(JspServlet.java:377)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:790)
at org.apache.catalina.core.StandardWrapper.service(StandardWrapper.java:1682)
at org.apache.catalina.core.ApplicationDispatcher.doInvoke(ApplicationDispatcher.java:875)
at org.apache.catalina.core.ApplicationDispatcher.invoke(ApplicationDispatcher.java:739)
at org.apache.catalina.core.ApplicationDispatcher.processRequest(ApplicationDispatcher.java:575)
at org.apache.catalina.core.ApplicationDispatcher.doDispatch(ApplicationDispatcher.java:546)
at org.apache.catalina.core.ApplicationDispatcher.dispatch(ApplicationDispatcher.java:428)
at org.apache.catalina.core.ApplicationDispatcher.forward(ApplicationDispatcher.java:378)
at org.apache.struts2.dispatcher.ServletDispatcherResult.doExecute(ServletDispatcherResult.java:154)
at org.apache.struts2.dispatcher.StrutsResultSupport.execute(StrutsResultSupport.java:186)
at com.opensymphony.xwork2.DefaultActionInvocation.executeResult(DefaultActionInvocation.java:362)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:266)
at com.opensymphony.xwork2.interceptor.DefaultWorkflowInterceptor.doIntercept(DefaultWorkflowInterceptor.java:165)
at com.opensymphony.xwork2.interceptor.MethodFilterInterceptor.intercept(MethodFilterInterceptor.java:87)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.validator.ValidationInterceptor.doIntercept(ValidationInterceptor.java:252)
at org.apache.struts2.interceptor.validation.AnnotationValidationInterceptor.doIntercept(AnnotationValidationInterceptor.java:68)
at com.opensymphony.xwork2.interceptor.MethodFilterInterceptor.intercept(MethodFilterInterceptor.java:87)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.interceptor.ConversionErrorInterceptor.intercept(ConversionErrorInterceptor.java:122)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.interceptor.ParametersInterceptor.doIntercept(ParametersInterceptor.java:195)
at com.opensymphony.xwork2.interceptor.MethodFilterInterceptor.intercept(MethodFilterInterceptor.java:87)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.interceptor.ParametersInterceptor.doIntercept(ParametersInterceptor.java:195)
at com.opensymphony.xwork2.interceptor.MethodFilterInterceptor.intercept(MethodFilterInterceptor.java:87)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.interceptor.StaticParametersInterceptor.intercept(StaticParametersInterceptor.java:179)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at org.apache.struts2.interceptor.MultiselectInterceptor.intercept(MultiselectInterceptor.java:75)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at org.apache.struts2.interceptor.CheckboxInterceptor.intercept(CheckboxInterceptor.java:94)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at org.apache.struts2.interceptor.FileUploadInterceptor.intercept(FileUploadInterceptorcom.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.interceptor.PrepareInterceptor.doIntercept(PrepareInterceptor.java:138)
at com.opensymphony.xwork2.interceptor.MethodFilterInterceptor.intercept(MethodFilterInterceptor.java:87)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.interceptor.I18nInterceptor.intercept(I18nInterceptor.java:165)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at org.apache.struts2.interceptor.ServletConfigInterceptor.intercept(ServletConfigInterceptor.java:164)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.interceptor.AliasInterceptor.intercept(AliasInterceptor.java:179)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.interceptor.ExceptionMappingInterceptor.intercept(ExceptionMappingInterceptor.java:176)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at org.apache.struts2.impl.StrutsActionProxy.execute(StrutsActionProxy.java:52)
at org.apache.struts2.dispatcher.Dispatcher.serviceAction(Dispatcher.java:488)
at org.apache.struts2.dispatcher.ng.ExecuteOperations.executeAction(ExecuteOperations.java:77)
at org.apache.struts2.dispatcher.ng.filter.StrutsPrepareAndExecuteFilter.doFilter(StrutsPrepareAndExecuteFilter.java:91)
at org.apache.catalina.core.ApplicationFilterChain.internalDoFilter(ApplicationFilterChain.java:256)
at org.apache.catalina.core.ApplicationFilterChain.doFilter(ApplicationFilterChain.java:214)
at org.apache.catalina.core.StandardWrapperValve.invoke(StandardWrapperValve.java:316)
at org.apache.catalina.core.StandardContextValve.invoke(StandardContextValve.java:160)
at org.apache.catalina.core.StandardPipeline.doInvoke(StandardPipeline.java:734)
at org.apache.catalina.core.StandardPipeline.invoke(StandardPipeline.java:673)
at com.sun.enterprise.web.WebPipeline.invoke(WebPipeline.java:99)
at org.apache.catalina.core.StandardHostValve.invoke(StandardHostValve.java:174)
at org.apache.catalina.connector.CoyoteAdapter.doService(CoyoteAdapter.java:416)
at org.apache.catalina.connector.CoyoteAdapter.service(CoyoteAdapter.java:283)
at com.sun.enterprise.v3.services.impl.ContainerMapper$HttpHandlerCallable.call(ContainerMapper.java:459)
at com.sun.enterprise.v3.services.impl.ContainerMapper.service(ContainerMapper.java:167)
at org.glassfish.grizzly.http.server.HttpHandler.runService(HttpHandler.java:206)
at org.glassfish.grizzly.http.server.HttpHandler.doHandle(HttpHandler.java:180)
at org.glassfish.grizzly.http.server.HttpServerFilter.handleRead(HttpServerFilter.java:235)
at org.glassfish.grizzly.filterchain.ExecutorResolver$9.execute(ExecutorResolver.java:119)
at org.glassfish.grizzly.filterchain.DefaultFilterChain.executeFilter(DefaultFilterChain.java:283)
at org.glassfish.grizzly.filterchain.DefaultFilterChain.executeChainPart(DefaultFilterChain.java:200)
at org.glassfish.grizzly.filterchain.DefaultFilterChain.execute(DefaultFilterChain.java:132)
at org.glassfish.grizzly.filterchain.DefaultFilterChain.process(DefaultFilterChain.java:111)
at org.glassfish.grizzly.ProcessorExecutor.execute(ProcessorExecutor.java:77)
at org.glassfish.grizzly.nio.transport.TCPNIOTransport.fireIOEvent(TCPNIOTransport.java:536)
at org.glassfish.grizzly.strategies.AbstractIOStrategy.fireIOEvent(AbstractIOStrategy.java:112)
at org.glassfish.grizzly.strategies.WorkerThreadIOStrategy.run0(WorkerThreadIOStrategy.java:117)
at org.glassfish.grizzly.strategies.WorkerThreadIOStrategy.access$100(WorkerThreadIOStrategy.java:56)
at org.glassfish.grizzly.strategies.WorkerThreadIOStrategy$WorkerThreadRunnable.run(WorkerThreadIOStrategy.java:137)
at org.glassfish.grizzly.threadpool.AbstractThreadPool$Worker.doWork(AbstractThreadPool.java:591)
at org.glassfish.grizzly.threadpool.AbstractThreadPool$Worker.run(AbstractThreadPool.java:571)
at java.lang.Thread.run(Thread.java:745)
2016-10-01T18:24:06.844+0530|Warning: Servlet.service() for servlet jsp threw exception
tag 'select', field 'list', name 'pid': The requested list key 'projectidList' could not be resolved as a collection/array/map/enumeration/iterator type. Example: people or people.{name} - [unknown location]
at org.apache.struts2.components.Component.fieldError(Component.java:237)
at org.apache.struts2.components.Component.findValue(Component.java:358)
at org.apache.struts2.components.ListUIBean.evaluateExtraParams(ListUIBean.java:80)
at org.apache.struts2.components.Select.evaluateExtraParams(Select.java:105)
org.apache.catalina.core.ApplicationDispatcher.invoke(ApplicationDispatcher.java:739)
at org.apache.catalina.core.ApplicationDispatcher.processRequest(ApplicationDispatcher.java:575)
at org.apache.catalina.core.ApplicationDispatcher.doDispatch(ApplicationDispatcher.java:546)
at org.apache.catalina.core.ApplicationDispatcher.dispatch(ApplicationDispatcher.java:428)
at org.apache.catalina.core.ApplicationDispatcher.forward(ApplicationDispatcher.java:378)
at org.apache.struts2.dispatcher.ServletDispatcherResult.doExecute(ServletDispatcherResult.java:154)
at com.opensymphony.xwork2.interceptor.MethodFilterInterceptor.intercept(MethodFilterInterceptor.java:87)
at com.opensymphony.xwork2.DefaultActionInvocation.invoke(DefaultActionInvocation.java:237)
at com.opensymphony.xwork2.interceptor.ConversionErrorInterceptor.intercept(ConversionErrorInterceptor.java:122)
at
Here is the login code:
package com.bits.sprintanalyzer.action;
import org.apache.log4j.Logger;
import com.bits.sprintanalyzer.ResourceException;
import com.bits.sprintanalyzer.dao.LoginDAO;
import com.opensymphony.xwork2.ActionSupport;
public class LoginAction extends ActionSupport{
private static final Logger LOG = Logger.getLogger(LoginAction.class);
/**
*
*/
private static final long serialVersionUID = 6877145894906143530L;
private String username;
private String password;
public String getUsername() {
return username;
}
public void setUsername(String username) {
this.username = username;
}
public String getPassword() {
return password;
}
public void setPassword(String password) {
this.password = password;
}
@Override
public void validate(){
if (username==null || username.length()==0 || password ==null || password.length() ==0 )
addActionError(getText("User name or Password cannot be null"));
}
@Override
public String execute() throws Exception {
try{
if( LoginDAO.isValidUser(username, password) ){
return SUCCESS;
}
else
{
addActionError(getText("Invalid Username or Password"));
}
}catch(ResourceException e){
LOG.error("Failed to valid User", e);
addActionError(getText("Something Went wrong with DBConnection"));
}
return INPUT;
}
}
Latest stacktrace after changing connection string:
2016-10-01T18:55:04.396+0530|Warning: Servlet.service() for servlet jsp threw exception
tag 'select', field 'list', name 'pid': The requested list key 'projectidList' could not be resolved as a collection/array/map/enumeration/iterator type. Example: people or people.{name} - [unknown location]
at org.apache.struts2.components.Component.fieldError(Component.java:237)
at org.apache.struts2.components.Component.findValue(Component.java:358)
at org.apache.struts2.components.ListUIBean.evaluateExtraParams(ListUIBean.java:80)
at org.apache.struts2.components.Select.evaluateExtraParams(Select.java:105)
at org.apache.struts2.components.UIBean.evaluateParams(UIBean.java:856)
at org.apache.struts2.components.UIBean.end(UIBean.java:510)
at org.apache.struts2.views.jsp.ComponentTagSupport.doEndTag(ComponentTagSupport.java:42)
at va:111)
at javax.servlet.http.HttpServlet.service(HttpServlet.java:790)
A: 1) You have MySQL connection exception as below:
Establishing SSL connection without server's identity verification is not recommended.
According to MySQL 5.5.45+, 5.6.26+ and 5.7.6+ requirements SSL connection must be established by default if explicit option isn't set.
To resolve this use the connection string like below:(change the db name to your existing db name)
jdbc:mysql://localhost:3306/dbname?autoReconnect=true&useSSL=false
This needs to be sorted first, that's why the values of the variables are not been rendered
2) In the Latest trace the requested list key 'projectidList' could not be resolved as a collection/array/map/enumeration/iterator type.
This error occurs when you try to access a list/collection which haven't been created.
Try to initialize the collection objects List<String> projectidList at class level.
when particular action is triggered at that time, JSP page is not knowing the the type of field, in this case, when you write List<String> projectidList = new ArrayList<String>(); instead of this, change it to ArrayList projectidList = new ArrayList();.
Make sure that you access list after the action class is instantiated i.e. corresponding action is called.
If you want to directly access it before calling action make it static and access it inside jsp.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,173 |
\section{Introduction}
Sensitivity to the new physics phenomena at the TeV scales is one of the main
arguments for considering CLIC as the next energy-frontier machine.
While precision measurements will provide many indirect constraints on the Beyond Standard Model (BSM) physics already at the first CLIC stage, the main goals of CLIC will be to search for new physics directly, taking advantage of the lepton collider clear environment and availability of the simple signatures. This talk presents the latest studies of the CLIC sensitivity to the BSM physics.
\section{Dark matter searches in the mono-photon channel}
Mono-photon signature is the most general approach to the dark matter (DM) searches
at the e$^{+}$e$^{-}$\xspace colliders. In this channel, pair-produced DM particles leave detector unobserved and the event can be tagged by an initial-state photon emission, which is precisely described by the Standard Model and only indirectly depends on the DM production.
The CLIC potential for DM detection has been studied for the 3\,TeV running scenario, considering large WIMP and mediator masses~\cite{Blaising:2021vhh}.
CLIC sensitivity to the DM production is enhanced by the use
of $\pm80$\,\% electron beam polarisation.
For the vector, axial-vector and scalar mediators considered in the Simplified Dark Matter Models framework, the best limits were obtained using the cross section ratio for the left-handed and the right-handed electron beam polarisation.
The 95\% C.L. exclusion limits in the plane of mediator and DM particle masses,
($m_Y$, $m_\chi$), are shown in figure~\ref{fig:monophoton} (left) for three types of the mediator coupling and the coupling value $g_{eY}=1$.
The shape of the measured cross section ratio allows for the discrimination between vector and axial-vector model hypotheses and the WIMP mass can be determined with an accuracy of up to 1\%.
Scenarios with low mediator masses are not excluded, if the mediator coupling to SM particles is small. Sensitivity to different low mass scenarios with mono-photon signature at 3\,TeV CLIC was considered in a recent study~\cite{Kalinowski:2020lhp,Kalinowski:2021tyr}.
A novel ``experimental-like`` approach was proposed, where the expected exclusion limits are defined in terms of the DM pair-production cross section as a function of the mediator mass and width. In this approach, expected limits hardly depend on the assumed DM type and coupling structure. Expected 95\% C.L. limits on the mediator coupling to electrons are shown in figure~\ref{fig:monophoton} (right) as a function of the mediator mass and different mediator coupling scenarios. Results from combined analysis of data taken with different electron beam polarisations take into account systematic uncertainties from luminosity measurement, background normalisation, beam polarisation and beam energy spectra shape. For high mediator masses, limits on the mediator couplings were used to estimate the corresponding limits on the mass scale of new interactions in the EFT approach. Limits range from 6.1\,TeV for scalar and 6.6\,TeV for vector mediator scenarios, to 10.1\,TeV for mediator with V+A coupling structure.
\begin{figure}[bt]
\centering
\begin{subfigure}{0.56\textwidth}
\centering
\includegraphics[width=\linewidth]{./figures/exclusion_gDM1_glep1.pdf}
\end{subfigure}%
\begin{subfigure}{0.44\textwidth}
\centering
\includegraphics[width=\linewidth]{./figures/clic2_sys_all_coupling_0.03_model.pdf}
\end{subfigure}
\caption{DM searches in the mono-photon channel at 3\,TeV CLIC. Left: 95\% C.L. exclusion limits on the mediator masses $m_Y$ and DM masses $m_X$, for $g_{eY}=1$ and three mediator types~\cite{Blaising:2021vhh}. Right: 95\% C.L. limits on the mediator coupling to electrons (denoted in \cite{Kalinowski:2021tyr} as $g_{eeY}$) as a function of the mediator mass (denoted in \cite{Kalinowski:2021tyr} as $M_Y$), for different mediator coupling structures, the relative mediator width $\Gamma/M=0.03$ and fermionic DM with mass of 50\,GeV~\cite{Kalinowski:2021tyr}. }
\label{fig:monophoton}
\end{figure}
\section{Extended scalar sector}
Within the Standard Model, only about 0.1\% of the Higgs boson decays is ``invisible``, resulting from the ZZ$^*$ decay channel, where both Z boson decay into pair of neutrinos. However, increased branching fraction for the invisible channel is expected in many extensions of the SM, when Higgs decays into new, weakly-interacting particles that escape from the detector unobserved. Prospects for direct detection of invisible Higgs boson decays at CLIC have been studied for CLIC running at 380\,GeV~\cite{Mekala:2020zys}.
The analysis was based on Whizard 2.7.0~\cite{Kilian:2007gr} simulation of Higgs boson production and background processes and \textsc{Delphes} framework~\cite{deFavereau:2013fsa} for fast simulation of detector response. Expected CLIC beam spectra as well as backgrounds resulting from $\gamma\gamma$ and $\gamma$e$^\pm$ hard interactions were taken into account.
Assuming that the measured event distributions are consistent with the predictions of the Standard Model, the 95\% C.L. limit on the invisible Higgs branching ratio expected at 380\,GeV CLIC is 1\% (0.5\%) for integrated luminosity of 1\,ab$^{-1}$ (4\,ab$^{-1}$).
Limits on the invisible scalar decays can also be considered in the framework of the Higgs portal models, where another scalar, H', mixing with the SM Higgs boson, could be produced in e$^{+}$e$^{-}$\xspace collisions in a process similar to the SM Higgs-strahlung channel: e$^{+}$e$^{-}$\xspace$\to$ ZH'. Expected limits on the production cross section for this process, relative to the SM cross section for the given Higgs boson mass, are shown in figure~\ref{fig:scalars} (left) as a function of the new scalar mass, $m_{H'}$. The results are shown for CLIC running at 380\,GeV, with two luminosity scenarios, and for the 1.5\,TeV CLIC stage, compared with other e$^{+}$e$^{-}$\xspace colliders. These limits, when interpreted in terms of the Vector Fermion Dark Matter Model, correspond to limits on the scalar mixing angle much stronger than the current constraints from ATLAS, CMS and indirect measurements.
Production of DM particles at CLIC was also studied in the framework of the Inert Doublet Model (IDM), one of the simplest extensions of the SM, where just one additional doublet is introduced. It contains four new scalar fields, H$^\pm$, A and H, from which the latter is stable, what makes it a natural DM candidate. The CLIC potential for observing IDM scalar production in the leptonic channels is limited by small branching ratios~\cite{Kalinowski:2018kdn}.
Much higher sensitivity is expected using the semi-leptonic final state~\cite{Klamka:2728552}, which offers higher statistics. A set of 23 benchmark points \cite{Kalinowski:2018ylg} in the model five-dimensional parameter space was considered. For the five selected scenarios, full simulation of detector response was used and, to extend the scope of the study, \textsc{Delphes} toolkit provided fast simulation for all of the considered benchmarks.
For the IDM scenarios with small scalar mass splitting, reconstruction of the low-energetic final state is highly influenced by the background coming from beam-induced $\gamma\gamma$ interactions producing soft hadrons, the so called \textit{overlay events}. The standard mitigation of this background at CLIC is achieved by applying timing cuts on the reconstructed particles. As these cannot be implemented in the CLICdet model for \textsc{Delphes}, the overlay events were included in fast simulation using approximate cuts applied on the generator level to the $\gamma\gamma\to$ hadrons samples. With this procedure, good agreement between the fast and full simulation results is obtained~\cite{Klamka:2021vqp}. In figure~\ref{fig:scalars} (right) the expected statistical significance of IDM scalar production observation is presented as a function of the charged inert scalar mass, $m_{H^\pm}$. Shown are results obtained using \textsc{Delphes} simulation for all benchmark scenarios considered in the study, both at 1.5\,TeV and 3\,TeV CLIC running stages. CLIC sensitivity to charged IDM scalar production is significantly increased with respect to the previous study \cite{Kalinowski:2018kdn} and indicate that the inert scalars could be discovered at CLIC in almost all of the considered model scenarios, with significance reaching up to~50\,$\sigma$.
\begin{figure}[bt]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=\linewidth]{./figures/lep_ilc_clic.pdf}
\end{subfigure}%
\begin{subfigure}{0.51\textwidth}
\centering
\includegraphics[width=\linewidth]{./figures/signifVsMass_allDelphesWithOverlay_label.pdf}
\end{subfigure}
\caption{Discovery reach for the scalar sector extensions. Left: expected limits on the production of scalar H' with respect to the SM background, for 380\,GeV (1\,ab$^{-1}$ and 4\,ab$^{-1}$ option) and 1.5\,TeV CLIC running, compared to other e$^{+}$e$^{-}$\xspace colliders. Right: statistical significance of deviations from the SM background, expected at 1.5\,TeV and 3\,TeV CLIC for the selected IDM benchmark scenarios. }
\label{fig:scalars}
\end{figure}
\section{Heavy neutrinos}
Many problems of the SM, including the nature of dark matter or CP violation, can be related to the neutrino sector, which is not well constrained. We do not understand the very nature of neutrinos or the origin of their masses. All these problems can be solved by introducing the additional neutrinos (or ``neutral leptons``), that in principle could be heavy and might be produced in particle colliders.
A possibility of detecting heavy neutrino production at CLIC was studied for a model with additional right-handed massive neutrinos~\cite{Pascoli:2018heg}. Considered was the process of light-heavy neutrino pair production with the semi-leptonic final state allowing for full reconstruction of the new particle. Both Dirac and Majorana scenarios were studied, with masses in range 200-3200\,GeV~\cite{Mekala:PoS}. Signal events, as well as backgrounds from e$^{+}$e$^{-}$\xspace, $\gamma\gamma$ and $\gamma$e$^\pm$ hard interactions (both from the beamstrahlung and the Equivalent Photon Approximation), were simulated in Whizard 2.8.5. The analysis was based on fast detector response simulation with \textsc{Delphes} and the use of Boosted Decision Trees for event classification. Expected limits on the heavy neutrino production cross section were evaluated using CLs method and translated into constraints on the coupling parameter $V_{lN}^2$ describing heavy neutrino coupling to SM leptons (assumed to be independent of the lepton flavour). Results are presented in figure~\ref{fig:neutrinos} as a function of the neutrino mass, $m_N$, and compared with the corresponding predictions for the ILC and different hadron collider options (all concern Dirac neutrinos, except for current limits from the LHC). Presented results confirm that the sensitivity of CLIC to heavy neutrino production extends up to the kinematic limit, is comparable to the ILC (with higher mass reach) and much better than expected in the current or future hadron colliders.
\begin{figure}[bt]
\centering
\includegraphics[width=0.6\textwidth]{./figures/new_results.pdf}
\caption{Expected 3\,TeV CLIC sensitivity to coupling $V^{2}_{\mathrm{lN}}$ of heavy neutrino to SM leptons, compared with the expected limits from ILC and hadron colliders. }
\label{fig:neutrinos}
\end{figure}
\section{Conclusions}
With the clean environment of e$^{+}$e$^{-}$\xspace collisions and its high energy reach, CLIC offers a great possibility of direct searches for BSM physics. Presented results indicate that in many cases it surpasses not only the expected capabilities of the HL-LHC, but can also be competitive to future hadron colliders such as FCC-hh or to the ILC project. Apart from the presented direct searches, precision measurements at CLIC probe new physics scenarios extending up to ${\cal{O}}(100)$\,TeV scales. Example limits on the new physics effects extracted in the EFT framework are presented in figure~\ref{fig:eft} \cite{deBlas:2018mhx}. All direct and indirect search opportunities make CLIC a fantastic future collider option.
\begin{figure}[bt]
\centering
\includegraphics[width=0.8\textwidth]{./figures/1812.02093_Fig33.pdf}
\caption{Results of a global fit of universal EFT operators to precision
Higgs, top and EW observables at CLIC \cite{deBlas:2018mhx}. }
\label{fig:eft}
\end{figure}
\bibliographystyle{JHEP}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,351 |
using System.Data.Common.Utils;
using System.Collections.Generic;
using System.Text;
namespace System.Data.Mapping.ViewGeneration.Validation
{
/// <summary>
/// A class representing a set of constraints. It uses generic parameters
/// so that we can get strong typing and avoid downcasts
/// </summary>
internal class SchemaConstraints<TKeyConstraint> : InternalBase
where TKeyConstraint : InternalBase
{
#region Constructor
// effects: Creates an empty set of constraints
internal SchemaConstraints()
{
m_keyConstraints = new List<TKeyConstraint>();
}
#endregion
#region Fields
// Use different lists so we can enumerate the right kind of constraints
private List<TKeyConstraint> m_keyConstraints;
#endregion
#region Properties
internal IEnumerable<TKeyConstraint> KeyConstraints
{
get { return m_keyConstraints; }
}
#endregion
#region Methods
// effects: Adds a key constraint to this
internal void Add(TKeyConstraint constraint)
{
EntityUtil.CheckArgumentNull(constraint, "constraint");
m_keyConstraints.Add(constraint);
}
// effects: Converts constraints to human-readable strings and adds them to builder
private static void ConstraintsToBuilder<Constraint>(IEnumerable<Constraint> constraints, StringBuilder builder)
where Constraint : InternalBase
{
foreach (Constraint constraint in constraints)
{
constraint.ToCompactString(builder);
builder.Append(Environment.NewLine);
}
}
internal override void ToCompactString(StringBuilder builder)
{
ConstraintsToBuilder(m_keyConstraints, builder);
}
#endregion
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 958 |
{"url":"https:\/\/casper.astro.berkeley.edu\/astrobaki\/index.php?title=Main_Page&diff=prev&oldid=8","text":"# Difference between revisions of \"Main Page\"\n\nMediaWiki has been successfully installed.\n\nConsult the User's Guide for information on using the wiki software.\n\n## Measurement Equation\n\n${\\displaystyle V_{ij}=a_{i}a_{j}\\sum _{n}{I_{n}(\\nu )e^{{\\frac {2\\pi i}{\\lambda }}{\\vec {b}}{\\dot {s}}_{n}}}\\,\\!}$\n\n## Something Completely Different\n\n${\\displaystyle \\int _{0}^{\\infty }{\\sin x}\\,\\!}$","date":"2022-08-07 21:22:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 2, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.37703990936279297, \"perplexity\": 9489.344952940428}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882570730.59\/warc\/CC-MAIN-20220807211157-20220808001157-00075.warc.gz\"}"} | null | null |
Q: How to save two different jpegs in Python? I am trying to save to different scatter plots as two different jpegs.
fig,ax = plt.subplots(1,2)
ax[0].scatter(df_oz['o'], df_oz['t'], color='g', alpha=0.5, s=150)
ax[1].scatter(df_oz['o'], df_oz['r'], color='b', alpha=0.5, s=150)
fig.suptitle('T&R', fontsize = 20)
fig.set_size_inches(14,7)
fig.savefig('Hi and Lo.jpg')#this at least gets both graphs saved as one jpeg
Now I have tried:
fig.savefig[0]('Hi.jpg')
fig.savefig[1]('Lo.jpg')
But that doesn't work.
Any advice?
A: The figure.savefig() is a function and can't indexing like what you do. If you want 2 different jpg files, you should separate it into 2 figures then save them separately. Try this:
# plot 1st figure and save
plt.scatter(df_oz['o'], df_oz['t'], color='g', alpha=0.5, s=150)
plt.savefig("Hi.jpg")
# clear the current plot
plt.clf()
# plot 2nd figure and save
plt.scatter(df_oz['o'], df_oz['r'], color='b', alpha=0.5, s=150)
plt.savefig("Lo.jpg")
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,115 |
Microbiology Time – May 2022
Microbiology time
Here we are with the May edition of Copan's Microbiology Time. This month, the top three papers are:
A study published on Microchimica Acta investigating a probing system based on dual-site ligation-assisted loop-mediated isothermal amplification (dLig-LAMP) for the selective colorimetric detection of SARS-CoV-2. With this method – and by designing a proprietary probe for sensing pyrophosphate – the group developed a colorimetric signaling system for point-of-care detection of SARS-CoV-2 which offers higher selectivity than RT-LAMP. This new method could be an option for improving the selectivity of point-of-care detection of SARS-CoV-2.
An Israeli paper aiming at the isolation of bacteria secreting growth-inhibiting compounds. Guedj Dana and colleagues implanted cultures of Francisella tularensis in environmental samples and monitored for inhibition zones. As a result, the researchers isolated two bacterial strains secreting antibiotic-like compounds. The genomic sequence of these strains was deciphered, and taxonomic profiling analysis classified them as belonging to the Pantoea genus.
In the last study, the Chilean researchers collected tracheal, cloacal swabs, and sera samples from domestic backyard poultry and fresh feces from wild birds during two sampling seasons in 2019 on Easter Island, to perform surveillance of the avian influenza virus. They detected antibodies against avian influenza in almost half of the samples collected and identified a novel low pathogenic H6N1 virus closely related to the one circulating in South America. The results of this research are the first evidence of avian influenza circulation in domestic birds on a Polynesian island.
Discover the complete studies below, and see you next June!
Scientific study
Microchimica Acta
Dual-site ligation-assisted loop-mediated isothermal amplification (dLig-LAMP) for colorimetric and point-of-care determination of real SARS-CoV-2
Moon Hyeok Choi et al.
BMC Genomic Data
Whole genome sequencing and taxonomic profiling of two Pantoea sp. isolated from environmental samples in Israel
Yehoudit Guedj-Dana, et al.
Novel Low Pathogenic Avian Influenza H6N1 in Backyard Chicken in Easter Island (Rapa Nui), Chilean Polynesia
Francisca di Pillo et al. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,645 |
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