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/ OUR WORK
/ WHO WE ARE
/ GET IN TOUCH
RODE was drawn to the opportunity to create a space that would anchor Savin Hill's local retail district, a mission that was shared by developers James Baker and John McDonough. "When we were kids... every corner had a market. It was really part of a fabric of the community, and we're looking to do that, replicate that, and make sure people feel this is a gathering place, a place to commune, quite frankly," offers Baker. The market is located across from the Savin Hill Redline station, offering a convenient stop for local commuters.
The building strengthens the streetscape of Savin Hill Ave, bringing new vitality to this neighborhood village. The folding forms tie into the street wall, activate the sidewalk, and reference the adjacent Flats on Savin and Cristo Rey High School, while offering outdoor spaces for the commercial tenants.
110 Savin Hill Avenue
535 Albany Street #405
© 2020 RODE Architects Inc All Rights Reserved.
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The following is the standings of the Iran Football's 1974–75 football season.
Final
Tractor and Bargh Shiraz promoted to Takht Jamshid Cup 1975–76.
References
See also
1974–75 Takht Jamshid Cup
League 2 (Iran) seasons
Iran
2
|
{
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| 1,318
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Finding History
Finding the Past | Living the Present
Around the U.S.
National Historic Sites
Informational Resources
SAAHATPA
Chief Juan Antonio and his band of Cahuilla Indians helped white settlers in the San Bernardino area defend their property and livestock against outlaws during the 1840s and 1850s. In late 1851, Juan Antonio, his warriors and their families, settled at nearby Saahatpa. During the winter of 1862-63, a smallpox epidemic swept through Southern California killing many Native Americans, including Juan Antonio. Cahuilla tradition asserts that the U.S. Government sent Army blankets that were contaminated with smallpox. After this disaster, Saahatpa was abandoned.
• State Historical Site (749)
Visited: 02/08/2016
Location: Map
(c) 2017 Living History || All Rights Reserved
|
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Q: Update one DataTable from another create table employee
(
id int,
name varchar(10),
dept_id int,
dept_name varchar(10)
);
insert into employee values(1, 'ABC1', 1, '');
insert into employee values(2, 'ABC2', 2, '');
insert into employee values(3, 'ABC3', 1, '');
insert into employee values(4, 'ABC4', 2, '');
insert into employee values(5, 'ABC5', 1, '');
create table dept
(
dept_id int,
dept_name varchar(10)
);
insert into dept values(1, 'XYZ1');
insert into dept values(2, 'XYZ2');
UPDATE e
SET e.dept_name = d.dept_name
FROM employee AS e
JOIN dept AS d
ON e.dept_id = d.dept_id
How do I convert above query (in bold letters) to a LINQ query?
A: I'm not sure you can. Linq to SQL is meant as a means to get data in/out of your database for use in your application code. What you want here is a lower-level data manipulation, which is best done in T-SQL as your example shows.
You could load the records from both tables using a join into a different object, then iterate the results.
// I'm doing this this way because I can't remember the syntax for
// LoadOptions and want to be sure to avoid the SELECT N+1 issue.
var query = from e in db.Employees
select new EmployeeDepartment
{
Employee = e,
Department = e.Department
};
foreach (var item in query)
{
item.Employee.DepartmentName = item.Department.DepartmentName;
}
db.SubmitChanges();
However
You probably already know this, but it might be better to normalise your database, so that the department name is only on the department table, and use your employee->department relationship to get the department name for a given employee.
A: using join:
var employeeDept = from e in db.employees
Join d in db.Depts on e.dept_id equals d.dept_id
select new
{Employee = e ,
Department = d};
foreach(var ed in employeeDept)
{
ed.Employee.DepartmentName = ed.Department.DepartmentName;
}
db.submitChanges();
A: One of the solution is as follows:
var enumEmp = employee.AsEnumerable();
var enumDept = dept.AsEnumerable();
var employeeDept = from empl in enumEmp
join d in enumDept on empl.Field<int>("dept_id") equals d.Field<int>("dept_id")
select new {enumEmp = empl , enumDept = d};
foreach(var ed in employeeDept)
{
ed.enumEmp.SetField<string>("dept_name",ed.enumDept.Field<string>("dept_name"));
}
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 1,347
|
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xmlns:android="http://schemas.android.com/apk/res/android"
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android:id="@+id/fullscreen_content"
android:layout_width="match_parent"
android:layout_height="match_parent"
tools:context="com.trackandtalk.pafos17.ui.explore.ImageViewerActivity"
android:fitsSystemWindows="true"
>
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android:id="@+id/landmark_viewpager"
android:layout_width="match_parent"
android:layout_height="wrap_content"
>
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| 4,656
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An American Blog
"…when the Son of Man comes, will He really find faith on the earth?" (Luke 18:8)
Totally destroy the Johnson Amendment
I can think of no better topic to start a new month—especially one in which we honor and express gratitude for our freedom and independence—than religious liberty.
One of the earliest very positive indicators me about then-candidate Trump was his particular emphasis on crushing the Johnson Amendment.
This is not some random pandering issue for him. If anything, if you listen to his nomination acceptance speech, one could have the impression he was advised against including this issue. From conversations I've had with a few connected people, I know that he talks about this in private, not just publicly.
This nation was founded on free speech. It's among the first freedoms in the First Amendment. For more than a hundred years, pastors were able and willing to speak out on issues, candidates, and whatever else they felt would be honoring to God based on Scripture.
Churches were also tax-exempt because "the power to tax was the power to destroy." (These days, the thinking is flipped, and the power to tax-exempt is the power to control, because a tax-exemption is considered a tax expenditure and tantamount to a subsidy.)
Fast-forward to 1954. Lyndon Johnson was a senator and wanted to shut-up a couple anti-war groups, so he inserts an amendment into a bill that became law without any debate. Covered under this anti-free speech provision was any non-profit organization, including churches.
Supposedly the First Amendment no longer applies to churches or anyone working in a non-profit, and if they violate this, they are threatened with the revoking of their tax-exempt status. In the financially fragile non-profit sector, that's usually more than enough motivation for self-policing on this matter.
President Trump has seen how this has weakened and neutered the effectiveness of the Church today, and he wanted to fix this. Even in the absence of a popular outcry, he was willing to lead on this issue. While he has not been able to have the statute repealed outright, he has taken executive action to mitigate the effects of this policy.
As far has his purposes are concerned, as long as he's President the Johnson Amendment is no longer a threat. He uses language as if it was fully repealed, and that's where he gives room for the Washington Post to claim he is "shifting" his claims about the policy. The bottom line for him is he wants it gone as much as possible. It's not right, and more people should be speaking up on his behalf on this issue.
There are some who criticize his action and policy stance claiming we should not have politics "pollute the Gospel" by bringing politics into the pulpit. For them, I have two questions: (1) Does the Gospel apply to all sin? (2) Is it possible to sin in politics and use of power? If yes to both, then there is no such thing as politics polluting the Gospel. The Gospel applies to all sin everywhere. A Gospel that has no relevance to politics, policy, lawmaking, and governing is an impotent Gospel.
In his address to the Faith and Freedom Coalition last week, President Trump cited his action on the Johnson Amendment. "We're allowed to talk without having to lose your tax exemption, your tax status, and being punished for speaking. And the people that we most want to hear, our great clergy, is now able to speak without fear of retribution."
President Trump went on to joke, "They can speak — unless they speak against me, in which case, we'll bring it back. (Laughter.) We'll bring back that Johnson Amendment so fast, Shaun." He went on to predict an out-of-context reaction in the media, "I'm only kidding. I'm only kidding. They're going to take it seriously. You know, they're going to go out — 'We have breaking news.' (Laughter.) They're going to say, 'See, I told you he wants to be a dictator. I told you that.' (Laughter.)"
Here's the thing: In this case, they may not because they like the Johnson Amendment. They like it that the people who have dedicated their lives and professions to reading, studying, and teaching God's Word are not allowed to speak publicly on public issues. To them, that's not being a dictator; they still think of that as the status quo where we are now, as if nothing has changed. What is truly more dictator-like is the status quo. It is evil to silence important wisdom that should be allowed under free speech, and this silencing is enabling evil to proliferate in this land.
It's as if those who fear repeal of this policy think that there would be far-right presidential endorsements everywhere, nothing but partisanship, and a complete breakdown of the entire non-profit sector that works to do good in this country. The fundamental mistake they're making is there is nothing about a repeal of a prohibition against something that turns it into a requirement for action. This isn't saying everyone has to get political. This is saying, if you think politics is relevant to your mission, that's your choice to make without fear of threats and intimidation.
I'm thankful for what President Trump has done so far, and for his commitment on this issue. There is still work Congress and the President can do to crush this unconstitutional encroachment on religious liberty. This policy is still in the statute and needs to be fully repealed so that it cannot be easily reinstated by an administration not friendly to religious liberty.
Possibilities for taking action is not limited to those in public office. The Alliance Defending Freedom has a standing offer to defend anyone willing to flout this unconstitutional policy and defend them all the way to the Supreme Court at no cost. Who will be the next brave man of God to take them up on this offer?
Tags: 1000, Freedom, Gospel
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Governing AI
A teacher with no limits?
The objectives of AI
Two categories of productivity
Rewriting the search of history
Congress Updates
The story of America
Ready for self-programming AI
2-year, 2-page budget deal agreement reached
The meaning not found in numbers
The key to getting things done
The reasons for the Church gathering
Using AI to encourage self-censorship of abusive c...
History is no judge
The Future of Value, Generalist Edition
A Civic Biology
Maximize your unique contribution
Making it home
The Billy Graham Rule
Evangelical Support for President Trump
Enhancing his earthly joys
How human nature is constructed
A career of usefulness
Only God has heart knowledge
Internet access = later nights
Not Afraid of Poverty
Understanding Appreciation
The last and final and most precious reward
Subscribe — Follow by Email
A Message of Hope
Un mensaje de esperanza
TimMcGhee.com
The message needed at a funeral
Not the best-sounding offer
Not the worst-sounding consequence
What's hard for the rich
The meaning of repentance
How to lead a soul to Christ
Evaluating salvation invitations
Evangelism vs. sin
Judgment Day: motivation for missions
The evangelistic nature of suffering
Understanding the nature of law
Life is not about commands
Life is about love
"Lord willing"
Eternal rewards
What is freedom?
What is a right?
First Amendment + the Gospel
Campaign finance laws vs. free speech
Evangelicals + Second Amendment
The 2 purposes of government
How to evaluate a law
How to evaluate a political argument
The right size for government
The purpose of taxes
Understanding compromise
Understanding religious liberty
Government is not "a force for good"
Government is not a business
Government is not a competitor
Communism threatens your soul
Cultural resiliency
"History" is not our Judge
The Increase of His Government
Evangelism, Gospel
Freedom, Government
Know the 3 main types of abortion
1,000 days of writing
Day 400 — Topic Categorizing
Day 700 — Information Tools
Now I beg you, brethren, through the Lord Jesus Christ, and through the love of the Spirit, that you strive together with me in prayers to God for me, that I may be delivered from those in Judea who do not believe, and that my service for Jerusalem may be acceptable to the saints, that I may come to you with joy by the will of God, and may be refreshed together with you. Now the God of peace be with you all. Amen.
Romans 15:30-33
Copyright Timothy T. C. McGhee. Simple theme. Powered by Blogger.
|
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| 9,034
|
{"url":"http:\/\/nrich.maths.org\/public\/leg.php?code=72&cl=3&cldcmpid=8665","text":"Search by Topic\n\nResources tagged with Generalising similar to Card Trick 1:\n\nFilter by: Content type:\nStage:\nChallenge level:\n\nCard Trick 2\n\nStage: 3 Challenge Level:\n\nCan you explain how this card trick works?\n\nReverse to Order\n\nStage: 3 Challenge Level:\n\nTake any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?\n\nHappy Numbers\n\nStage: 3 Challenge Level:\n\nTake any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.\n\nCunning Card Trick\n\nStage: 3 Challenge Level:\n\nDelight your friends with this cunning trick! Can you explain how it works?\n\nGOT IT Now\n\nStage: 2 and 3 Challenge Level:\n\nFor this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?\n\nLoopy\n\nStage: 4 Challenge Level:\n\nInvestigate sequences given by $a_n = \\frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?\n\nRepeaters\n\nStage: 3 Challenge Level:\n\nChoose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.\n\nElevenses\n\nStage: 3 Challenge Level:\n\nHow many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?\n\nMini-max\n\nStage: 3 Challenge Level:\n\nConsider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .\n\nConverging Means\n\nStage: 3 Challenge Level:\n\nTake any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .\n\nOdd Differences\n\nStage: 4 Challenge Level:\n\nThe diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n\u00b2 Use the diagram to show that any odd number is the difference of two squares.\n\nChocolate Maths\n\nStage: 3 Challenge Level:\n\nPick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .\n\nThree Times Seven\n\nStage: 3 Challenge Level:\n\nA three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?\n\nStage: 3 Challenge Level:\n\nThink of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .\n\nIntersecting Circles\n\nStage: 3 Challenge Level:\n\nThree circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?\n\nAP Rectangles\n\nStage: 3 Challenge Level:\n\nAn AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?\n\nMaths Trails\n\nStage: 2 and 3\n\nThe NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.\n\nStage: 3 Challenge Level:\n\nA little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .\n\nTake Three from Five\n\nStage: 3 and 4 Challenge Level:\n\nCaroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?\n\nStage: 3 Challenge Level:\n\nList any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?\n\nNim\n\nStage: 4 Challenge Level:\n\nStart with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.\n\nConsecutive Negative Numbers\n\nStage: 3 Challenge Level:\n\nDo you notice anything about the solutions when you add and\/or subtract consecutive negative numbers?\n\nGot It\n\nStage: 2 and 3 Challenge Level:\n\nA game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.\n\nNim-interactive\n\nStage: 3 and 4 Challenge Level:\n\nStart with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.\n\nNim-like Games\n\nStage: 2, 3 and 4 Challenge Level:\n\nA collection of games on the NIM theme\n\nPlus Minus\n\nStage: 4 Challenge Level:\n\nCan you explain the surprising results Jo found when she calculated the difference between square numbers?\n\nMake 37\n\nStage: 2 and 3 Challenge Level:\n\nFour bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.\n\nMagic Letters\n\nStage: 3 Challenge Level:\n\nCharlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?\n\nOne O Five\n\nStage: 3 Challenge Level:\n\nYou can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .\n\nMagic Squares\n\nStage: 4 and 5\n\nAn account of some magic squares and their properties and and how to construct them for yourself.\n\nSpecial Sums and Products\n\nStage: 3 Challenge Level:\n\nFind some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 \u00d7 12 = 48 and 16 is a factor of 48.\n\nWhat Numbers Can We Make Now?\n\nStage: 3 and 4 Challenge Level:\n\nImagine we have four bags containing numbers from a sequence. What numbers can we make now?\n\nSumming Consecutive Numbers\n\nStage: 3 Challenge Level:\n\nMany numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?\n\nPair Products\n\nStage: 4 Challenge Level:\n\nChoose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. 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| null | null |
How COVID-19 Is Harming State and City Budgets
The coronavirus pandemic is placing enormous budget pressure on state and local governments, threatening deep and potentially lasting cuts to education, infrastructure, and other important investments.
People in Arkansas wait in line to file for unemployment after the COVID-19 outbreak. Nick Oxford/Reuters
Anshu Siripurapu and Jonathan Masters
Last updated March 19, 2021 11:47 am (EST)
State and local governments are responsible for the bulk of U.S. education and infrastructure spending.
Many are facing severe budget shortfalls as a result of the coronavirus pandemic.
Subnational governments have few options for dealing with budget crises, and the federal government has stepped in to provide aid.
Many U.S. state and local governments, on the front lines of the response to the coronavirus pandemic, are facing severe budget shortfalls. A distressing combination of dwindling tax revenues, record unemployment, and rising health costs have pushed them to cut back on spending for infrastructure and education—of which states and cities are by far the primary funders. Some still bear the scars of the 2008 financial crisis, which forced painful spending cuts to public services.
Even before the pandemic, subnational governments were grappling with ballooning costs, including health care and pensions for public employees. Some had already sought bankruptcy protection.
State and Local Governments (U.S.)
In response to COVID-19, states and municipalities have made cuts, frozen spending and hiring, laid off workers, and drawn down rainy day funds. Some states have seen revenues rise due to the uneven nature of this recession, but by and large the pandemic has drained state and local coffers. The federal government has stepped in to provide substantial aid.
How do state and local governments budget?
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Most state and local governments keep two budgets. The operating budget, often simply referred to as "the budget," funds ongoing outlays such as employee salaries, payments for services, and interest on long-term debt. The capital budget—which is used to fund major infrastructure projects such as bridges, roads, or waterworks—issues long-term debt, such as bonds.
Unlike the federal government, states cannot run operating budget deficits. Every state in the union, with the exception of Vermont, has some type of balanced budget requirement—though many states have in the past used gimmicks, such as selling assets and then leasing them back, to circumvent the law. Under state laws, most municipalities must also keep balanced books. The cost of borrowing is also greater for subnational governments, as their bonds typically carry higher interest rates than U.S. Treasuries.
Some federal lawmakers have expressed concerns about mounting municipal debt, but aggregate levels as a percentage of gross domestic product (GDP) have remained within historical parameters. According to the U.S. Census Bureau, state and local governments had a little more than $3 trillion in total outstanding debt in 2017, of which nearly 99 percent was long term. Interest payments on this debt were roughly 3.3 percent of total expenditures—no more than in the 1970s. In contrast, the U.S. national debt has risen dramatically over the same time period, and is projected to soon exceed GDP.
Why are state and local budgets relevant?
The U.S. national economy is composed of a vast collection of local and regional economies, where state and local policies play a significant role. Collectively, state and local governments outspend Washington on direct goods and services, employ more workers than the domestic manufacturing sector, and are responsible for about 15 percent of national GDP.
States and cities supply nearly 80 percent of the $441 billion spent nationally on transportation and water infrastructure, according to the Congressional Budget Office's most recent data. These investments help fuel the economy directly by creating jobs in sectors such as construction, experts say, and are also essential for improving long-term economic efficiency and international competitiveness. (For example, less congestion lowers business costs.) The 2019 Global Competitiveness Report from the World Economic Forum (WEF) ranked the United States second overall in economic competitiveness, but thirteenth in quality of infrastructure. The American Society of Civil Engineers gave the United States a D+ in its 2017 Infrastructure Report Card—the same grade it gave in 2013.
State and local governments contribute more than 90 percent of the money spent nationally on K–12 education, as well as provide substantial financing for the public university system, which graduates the majority of U.S. college students. While education has typically been the largest budget category of total state spending, Medicaid's share has been growing over the past decade.
At a more basic level, states and cities are where life is lived and economic decisions are made. While budget austerity at the federal level can seem removed to many families and businesses, the effects of cutbacks in their home states and localities are tangible. Cuts to police and fire departments, for instance, mean some neighborhoods are less safe. Fewer teachers and more students per classroom erode the quality of education, making U.S. workers less competitive in the long run. Crumbling roads and bridges discourage businesses from investing.
What are the policy options for dealing with budget crises?
On their own, state and local governments have few options for dealing with budget shortfalls of great magnitude. Some states' laws prohibit borrowing to fund day-to-day operations, while others' require voter approval, which is generally difficult to obtain quickly. Most states have to pay back any debt within the fiscal year. States can, however, borrow from the federal government to pay out unemployment benefits.
That effectively leaves two options: raise revenue or cut spending. States have historically chosen the latter during recessions. Education and Medicaid, the two biggest budget items, cannot usually escape cuts; nor can state payrolls, another big expense. Mental health services, courts, law enforcement, and other public services can also face trims.
How did the Great Recession affect state and local budgets?
The deep economic recession of December 2007 to June 2009 and slow recovery put many subnational budgets in unusually dire straits. Depressed tax revenues, elevated spending on social welfare programs such as unemployment insurance and Medicaid, and, in many cases, rising personnel costs squeezed public purses. The situation was particularly bleak at the local level, where many balance sheets were battered by the collapse of home values and property tax revenues.
Despite federal aid, states were compelled to slash spending by $290 billion and hike taxes by $100 billion to try to close the budget gap, according to the Center on Budget and Policy Priorities (CBPP). States continued to lay off workers for years after the recession and cut back on infrastructure and education spending. State support for public higher education dropped by 13 percent, on average, in constant dollars between fiscal years 2006 and 2011. California, with the largest state budget in the country, cut its transportation spending by 31 percent from 2007 to 2009; Texas shrank its funding by 8 percent.
Spending recovered slowly. By 2017, some states' education funding was still more than 10 percent below prerecession levels. Total public infrastructure spending, meanwhile, fell in real terms by nearly $10 billion between 2007 and 2017, according to the Brookings Institution, and a larger share now goes toward maintenance than toward new projects.
By 2017, some states' education funding was still more than 10 percent below prerecession levels.
Other headwinds that hit state budgets include: the disproportionate growth of Medicaid spending; a decline in aid due to federal deficit reduction; shrinking tax bases and unstable revenues; and the fiscal plight of local governments. The recession also widened the hole in state pension funding. In 2017, states collectively could cover only 69 percent of their pension liabilities, down from 86 percent before the recession, according to a 2019 Pew report, though levels varied widely by state.
Unlike states, municipalities—of which there are more than eighty-seven thousand, including cities, towns, counties, school districts, and other public entities—can file for federal bankruptcy protection, known as Chapter 9 under the U.S. bankruptcy code. Though they are historically rare, there have been some high-profile cases, such as Detroit's in 2013.
Bankruptcies in the wake of the recession came about for various reasons. The experiences of Harrisburg, Pennsylvania, and Jefferson County, Alabama, both of which filed in late 2011, were largely the result of poor infrastructure investments and political dysfunction. The challenges faced by many cities in California, such as Stockton and San Bernardino, largely stemmed from a precipitous drop in home values. In other cities, such as Vallejo, California, and Central Falls, Rhode Island, the main problem was escalating personnel costs.
How will the COVID-19 pandemic shape state and local budgets?
The fiscal shock from the pandemic is expected to be comparable to that of the last recession. Before the pandemic's start, state budgets had mostly recovered from the 2007–2009 crisis, and many had built up rainy day funds to draw from during downturns. But the cushion is nowhere near sufficient to prevent the pandemic from wreaking considerable economic damage.
States ended the 2020 fiscal year in better shape than was initially forecast because of hundreds of billions of dollars in federal aid and the unusual nature of the pandemic recession. Low-wage earners in the service industries, who pay less in taxes, were devastated by job losses, while the stock market boomed and high-wage workers were relatively unscathed. But states were still projected to see huge shortfalls in the years ahead, even before the surge of COVID-19 cases in late 2020. The situation varies significantly across states, depending on the structure of their economies and their prepandemic fiscal health: for example, California has seen record revenues, fueled by the stock market and income gains among its wealthiest residents. But the majority of states have seen tax revenues shrink, according to an analysis by the Washington Post; five states, including Alaska and Florida, suffered double-digit percentage declines.
Tax revenues have fallen sharply as states faced record unemployment claims and rising public health costs. Though the economy is slowly adding jobs, nearly ten million of them have been lost since the pandemic began, with restaurants and hotels among the hardest-hit businesses. As people lose jobs, more qualify for Medicaid, the costs for which are shared by states and the federal government. Moreover, in the first quarter of 2020, state pension funding levels fell to their lowest point in three decades, as the pandemic caused the value of pension fund assets to plummet. Though the stock market later recovered, there are still concerns about its performance, as well as the effect of low interest rates from bonds on pension funds.
The pandemic has also starved state governments of sales taxes, one of their largest sources of revenue, as social-distancing measures curtailed activities such as shopping and dining out. Municipalities are being hit just as hard: a December 2020 survey by the National League of Cities found that revenues had declined by more than one-fifth.
To make ends meet, state and local governments have rolled out austerity measures. In the second quarter of 2020, state and local spending fell by 6 percent on an annual basis, the biggest drop since 1952, and in the third quarter it declined an additional 4 percent. By the end of the year, state and local governments had shed 1.3 million jobs, the overwhelming majority of those in education.
Many states have cut funding for K–12 and higher education. The pandemic has also contributed to a precipitous drop in enrollment by international students, who typically pay higher tuition than the average U.S. resident at public schools. A survey of U.S. colleges found that enrollment of new international students fell by 43 percent in fall 2020. Infrastructure spending is also on the chopping block. By August, state and local governments had delayed or canceled almost $10 billion in infrastructure projects due to the pandemic, according to the American Road & Transportation Builders Association.
The federal government has stepped in to help states manage the pandemic response. The Coronavirus Aid, Relief, and Economic Security (CARES) Act, passed at the start of the crisis, set aside roughly $150 billion for state, local, and tribal governments. The law also expanded unemployment benefits and created the Paycheck Protection Program (PPP), which helped lessen the economic blow. An earlier measure, the Families First Coronavirus Response Act, increased the federal government's contributions to Medicaid.
In December 2020, Congress approved an additional stimulus package. This law extended unemployment benefits and PPP funds, and allocated some money that state and local governments can use, including for education and health programs such as vaccine distribution, but it did not include direct aid.
A few months later, with Democrats in control, Congress approved another massive stimulus bill, which set aside $350 billion in aid for states, cities, counties, and tribal governments, and an additional $10 billion specifically for infrastructure projects. That money is on top of funds the law provides for schools, coronavirus testing, and vaccine distribution.
With the new influx of cash, state and local officials say they will be able to prevent additional layoffs; improve government services; and revive delayed or canceled projects, such as broadband internet expansion. But some experts, along with many Republican lawmakers, say the federal funds are excessive; in some cases, the money that state and local governments will receive exceeds their projected deficits.
Experts discuss the fiscal plight of state and local governments at this CFR Virtual Meeting in December 2020.
The New York Times examines the effect of the pandemic on public services and jobs.
The National Conference of State Legislatures tracks the measures states have taken to close budget gaps caused by the pandemic.
For media inquiries on this topic, please reach out to [email protected].
How can they manage budget crises?
What was the effect of the Great Recession?
How will the COVID-19 pandemic shape budgets?
The National Debt Dilemma
by James McBride and Anshu Siripurapu
Economic Recovery From the Coronavirus Pandemic
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Q: Why is it that when I run this code, I get "Hello from C++" on my android emulator? When I run the code down below, I get "Hello from C++" instead of the expected "Hello World!" on my USB connected android phone and emulator. Why is that?
I've tried running it on both my android phone and an android emulator (PIXEL XL API 28) and they both output the same thing, "Hello form c++"
<?xml version="1.0" encoding="utf-8"?>
<android.support.constraint.ConstraintLayout
xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:layout_height="match_parent"
tools:context=".MainActivity">
<TextView
android:id="@+id/sample_text"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:text="Hello World!"
android:textSize="35dp"
app:layout_constraintBottom_toBottomOf="parent"
app:layout_constraintLeft_toLeftOf="parent"
app:layout_constraintRight_toRightOf="parent"
app:layout_constraintTop_toTopOf="parent" />
</android.support.constraint.ConstraintLayout>
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"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{The Solution: An AI-Powered Agile Project Helper}
\section{An AI--Powered Agile Project Assistant}
The above challenges and the serious lack of effective tools presents an opportunity for AI to significantly improve the practice of agile project management. AI-based tools are able to process massive amounts of data generated from software projects, harvest useful insights, and train to perform complex tasks such as estimating effort, task refinement, resource management, and sprint planning. Figure \ref{fig:framework} shows our proposal for the architecture of an AI-powered agile project management assistant. The core of this AI system are an \emph{\textbf{analytics engine}}, a \emph{\textbf{planning engine}} and an \emph{\textbf{optimization engine}}. These machineries depends on the \emph{\textbf{learning representation engine}} to learn and generate representations of project data that are mathematically and computationally convenient to process. The \emph{\textbf{conversational dialog engine}} converses with users and brings the support provided by the other engines to the users.
\begin{figure}[ht]
\centering \includegraphics[width=\linewidth]{framework}
\caption{The architecture of an AI-powered agile project management assistant}
\label{fig:framework}
\end{figure}
\subsection{Representation learning engine}
Agile project artifacts contain both structured and unstructured data. For example, backlog items may have structured attributes such as type and priority (which are easily extracted to form a vector representation), whereas product visions, sprint goals, description of backlog items, and communication among team members (e.g. comments on backlog items) are written in natural text. Codebases contain documentations such as release notes and comments written in natural text, and source code written in programming languages. Hence, the representation learning engine is an important component of this AI system, responsible for learning meaningful vector representations for each project artifact. These representations can automatically be learned from unlabelled data, and are then used by the other machineries in the AI system.
The representation learning engine has a \texttt{NLP} component which performs automatic analysis on project textual artifacts and then generates good representations of those artifacts. Traditional NLP techniques (e.g. Bag of Words) produce very high dimensional and sparse vector representations. By contrast, latest advances in deep learning-based NLP techniques \cite{manning2016computational} such as word2vec, paragraph2vec, Long Short-Term Memory (used in Google Translate), or Convolutional Neural Networks (used in Facebook's DeepText engine) are able to generate dense vector representations that produce superior results on various NLP tasks. Source code is another important source of project data. The \texttt{Code Modeling} component is responsible for learning meaningful vector representations which reflect the semantic and syntactic structure of source code. State-of-the-art statistical language modeling techniques, including deep learning models, have demonstrated their effectiveness for source code and thus can be leveraged here \cite{allamanis2018survey,Tufano:2018:DLS}.
We have leveraged the powerful deep learning architecture, Long Short-Term Memory (LSTM) to automatically learn vector representations for both backlog items and source code\footnote{References omitted due to double-blind review requirement.}. LSTM enables us to learn the semantics and syntactic structures, particularly the long-term dependencies, existing in both natural text and source code. We will extend these models to learn representations for other textual artifacts such as product visions, sprint goals, and developer communications.
Any useful AI machinery must take into account the capability and dynamics of agile teams. Obtaining a representation for a team requires modeling of its members (e.g. developers). A developer can be represented through the project artifacts they have involved with, such as the backlog items they have completed or the code they have written. The \texttt{Feature Extraction and Aggregation} extracts all the vector representations of the artifacts related to a developer, and learn to aggregate them to form a vector representation of the developer. A number of feature aggregation techniques proposed in recent work \cite{Choetkiertikul-TSE-2018} which derive vector features of a sprints based on the features of the backlog items assigned to it. We will extend those aggregation methods to learn features for representing team members. This representation will be enriched with features representing work and social dependencies between team members, extracted from communication logs (e.g. comments or discussions on work items).
\subsection{Analytics engine}
The analytics engine aims to provide decision support in the following aspects:
\subsubsection{\textbf{Descriptive analytics}} Most existing agile project management tools support this basic level of analytics: data visualization via reports, dashboards, and scorecards. Common agile reports such as burndown charts, velocity charts, and sprint reports are created by summarizing what happened using historical project data and presented to the users in an intuitive and easily interpretable manner. Knowing what happened (e.g. team velocity from sprint to sprint) is useful, but \emph{diagnosing} why something happened (e.g. why the team's velocity dropped significantly in some sprints) is even more useful. AI equipped with machine learning can augment descriptive analytics by discovering patterns, identifying anomalies and detecting ``unusual'' events.
\subsubsection{\textbf{Predictive analytics}} Most existing agile tools are not yet capable of providing this advanced level of analytics. Two challenging areas are effort estimation and risk prediction that are specifically for agile contexts. Machine learning techniques are suited to build prediction models. For example, recent work \cite{Choetkiertikul-TSE-2019} used deep learning to estimate the size of user stories through learning a team's previous estimates. Estimation tools could be used as a decision support system and takes part in the existing estimation process (e.g. planning
poker) or in a completely automated manner. Forecasting future risks requires the capability of processing large amounts of historical project data, memorizing a long history of past experience, and inferring the current ``health'' state of the project. Recent work has moved forwards in this direction to predict delay risks \cite{Choetkiertikul:2017:PDI} or sprint delivery risks \cite{Choetkiertikul-TSE-2018}.
\subsubsection{\textbf{Prescriptive analytics}} This is the most advanced level in the project analytics stack. Using the results from descriptive analytics and predictive analytics, prescriptive analytics recommends the best course of actions for agile teams in a specific situation. We identify here three important areas in agile PM that prescriptive analytics would be useful:
\begin{itemize}
\item \emph{Backlog item identification:} Using the NLP component in the representation learning engine, prescriptive analytics will automatically process and extract new backlog items from different data sources such as a requirement specification, new feature requests from customers, bugs reported by end users, previous bug fixes, discussions among agile teams (e.g. technical debts, design changes or action items from retrospective meetings), end users' reviews of the product, and even experiences from previous projects. It will also able to recommend inter-dependencies between new item and existing ones using machine learning and representation learning.
\item \emph{Backlog item refinement:} prescriptive analytics will suggest how a user story is split into a smaller user stories or how a user story is decomposed into tasks. Learning decompositions is highly challenging since it requires a background knowledge. It is
still a new topic in AI and machine learning.
\item \emph{Risk mitigation:} Using results from predictive analytics, prescriptive analytics recommends a course of actions to take advantage of a future opportunity or mitigate a future risk and shows the implication of each decision option.
\end{itemize}
\subsection{Reasoning capability}
Reasoning is the capacity to infer new knowledge by algebraically manipulating existing knowledge base to respond to a query \cite{bottou2014machine}. Traditionally, it works on symbolic knowledge representation through the means of induction or deduction. This permits domain knowledge provided by agile teams (e.g. project rules). Recently, deep neural reasoning offers an alternative for producing answers from sub-symbolic (vector) representation \cite{jaeger2016artificial}, which are output of the representation engine. Inferred knowledge can be put back to enrich the knowledge base. The reasoning capability of our AI system is provided by two engines: planning and optimization.
\subsubsection{Planning engine}
Planning for a sprint can be formulated as an AI planning problem in which the initial state is the state of the project and the product prior a sprint, a goal state is specified in the sprint's goal, and selecting items from the product backlog can be viewed as the act of choosing plan operators to be executed from the initial state to a goal state. The planning engine needs to consider a range of input such as the existing product backlog items, the sprint's goal, the existing codebase, the team's capacity and previous performance in previous sprints, and the duration of the sprint. These data are often not formally expressed. The representation learning engine has convert them into vector representations but further formal encoding would be needed.
In addition, the plan needs to be executed in a manner that is \emph{robust} and \emph{resilient} to changes. The challenge is not only to be flexible enough to deal with immediate impediments to the sprint execution, but to also anticipate future states of affairs that might impede sprint execution or the achievement of the sprint goal. Impediments to the successful sprint execution can appear in many forms. For instance, a task might not be completed by the due date, preventing other dependent tasks from being started. Hence, the relationship between a sprint plan and its
operating environment can been seen as adversarial. Recent successful work in deep reinforcement learning (e.g. \cite{mnih2015human}) can be thus leveraged to build this part of the planning engine.
\subsubsection{Optimization engine}
The optimization engine helps the planning engine to compute the optimal set of actions given a certain situation. For example, it can be used to compute the optimal selection of backlog items for the upcoming sprint given multiple constraints and objectives. It can also be used for hyper-parameters tuning of machine learning models used in the analytics engine. Search-based software engineering techniques can be leveraged here to build the optimization engine.
\subsection{Conversational dialog engine}
The conversation dialog engine is envisioned to converse meaningfully with agile teams. It is a form of a software chatbot \cite{Lebeuf2018}, acting as an interface between the users and the remaining part of the AI system. The chatbot can be asked different types of questions, such as ``Show me your estimate of this user story'' or ``Can you help split this user story?''. Through conversations with the users, it receives input and requests, and passes them to relevant engines in the system. Future chatbots can be trained end-to-end \cite{serban2016building} and person-specific instead of task-specific \cite{kottur2017exploring}.
\section{Next Steps}
We are developing prototype tools to realize each component of the proposed AI-powered agile project management assistant. We plan to first evaluate it using our existing dataset of 150 open source projects. We will also collaborate with our existing industry partners to perform an evaluation on commercial software agile projects. We however believe that AI will assist, not substitute, human teams. Individuals, interactions, and collaboration are still the key elements of project success. AI can serve as a distinctive accelerator for agile teams and thus help increase project success rates.
\bibliographystyle{IEEEtran}
\section{Introduction}
Artificial Intelligence (AI) has started making a substantial impact to many parts of our society, and is predicted to disrupt how we produce, manufacture, and deliver. The rise of AI is empowered by the growth and availability of big data, breakthroughs in AI algorithms (e.g. deep learning), and significantly increased computational power. The pervasiveness of software products has resulted in a massive amount of data about software projects which AI techniques can leverage. We envision that AI will transform (software) project management practice in many aspects, from automating basic administration tasks to delivering analytics-driven risk predictions and estimation, facilitating project planning and making actionable recommendations. In this paper, we present a framework of how various AI technologies are adapted and integrated to support various areas of agile project management (\emph{agile PM}).
Agile methods (e.g. Scrum) have been widely used in industry to manage software projects \cite{hoda2018}. This relatively new approach to project management empowers software teams to focus on rapid delivery of business value to customers, thus significantly reducing the overall risk of project failures. Project management has thus witnessed a shift away from the traditional ``waterfall'' process and towards a more adaptive, agile model. The number of projects following agile has increased significantly in the recent years, not only in the software industry but also in other non-IT domains \cite{StateofAgile}.
An agile project are centered around a \emph{product backlog}, which is typically a collection of items to be completed in the project \cite{Cohn2005}. Items in a product backlog can be, for example, customer requirements for the product (user stories), requests for bug fixes, changes to existing features, and technical improvements. Product backlog is evolved through regular updates and refinement to ensure that it contains items that are relevant the project's scope and objectives, sufficiently detailed, and appropriately estimated. Important updates to the product backlog include adding or removing backlog items based on current needs, estimating the size of items, and refine large items into small fine-grained items.
\begin{figure}[ht]
\centering \includegraphics[width=\linewidth]{agile-process}
\caption{A typical agile process}
\label{fig:agile-process}
\end{figure}
An agile project consists of multiple iterations (or alternatively referred as \emph{sprints}). Each sprint is often a short period in which the team aims to complete a subset of items in the product backlog. Prior to a sprint, the team performs \emph{sprint planning} to identify the goal of that upcoming sprint, and select items from the product backlog which they will complete to meet the sprint's goal. During sprint planning, many agile teams decompose each product backlog items into a set of \emph{tasks}. These tasks and their corresponding product backlog items form the \emph{sprint backlog}. The team then executes the sprint to complete items in the sprint backlog to deliver a potentially shippable product increment.
\section{Challenges in Agile Project Management}
Many tools have been developed to support agile project management such as Atlassian's JIRA Software\footnote{\url{https://www.atlassian.com/software/jira}}, Axosoft\footnote{\url{https://www.axosoft.com}}, and Assembla\footnote{\url{https://www.assembla.com}}. Those tools allow agile teams to create and manage various agile artifacts such as user stories, product backlogs, sprints, and sprint backlogs. For example, they support teams in creating user stories and tasks, linking related tasks and user stories, assigning team members to tasks and issues, creating deadlines, setting priorities and estimates (e.g. story points). They also enable team members to see the amount of work required for individuals and teams during each sprint, track progress of the sprint and its associated user stories and tasks. They facilitate real-time information exchange and collaboration via centralised project information.
Although existing agile tools are useful, their support is limited to creating, managing, and tracking project artifacts, and visualising historical project data such as burndown charts and other agile reports. Current agile project management tools lack advanced analytical methods that are capable of harvesting valuable insights from project data for prediction, estimation, planning and action recommendation. Many decision-making tasks in agile projects are still performed by agile teams without machinery support. We identify a number of important areas in agile project management that remain challenging due to this lack of effective support.
\subsubsection{\textbf{Identifying backlog items}} Items in the product backlog can be derived from different sources such as a requirement specification, new feature requests from customers, bugs reported by end users, previous bug fixes, discussions among agile teams (e.g. technical debts, design changes or action items from retrospective meetings), end users' reviews of the product, and even experiences from previous projects. It is difficult and time consuming for agile teams, especially product owners, to process this large amount of heterogenous data in order to identify and create new items for the product backlog. In addition, for each newly created backlog item, it is necessary to consider \emph{inter-dependencies} between the new item and existing ones. This is challenging as a typical project has a large product backlog with more than 100 items.
\subsubsection{\textbf{Refining backlog items}} Some items (e.g. user stories) in the product backlog are initially large, thus do not fit within a single sprint. Agile teams are often required to refine these large items into small ones such that they not only facilitate implementation but are sufficiently large, allowing stakeholders to understand business value \cite{Cohn2005}. There are typically three levels of refinement: (1) decomposing an epic into a number of user stories; (2) splitting user stories into small stories; and (3) breaking a user story into a number of specific project tasks. Different rules and guidelines have been proposed to help teams refine backlog items, but rules often overlap with or even conflict with one another. Teams struggle to refine backlog items and rely on their own intuition and experience.
\subsubsection{\textbf{Sprint planning}} The key part of sprint planning is selecting a subset of items in the backlog which can realistically be accomplished by the team in the upcoming sprint to deliver a product increment. The customer expects the team to deliver what have been planned for a sprint, thus meeting this expectation is important in maintaining the customer's faith in the team's ability to deliver. Sprint planning is however highly challenging since many important factors must be considered, including items contributing toward the sprint goal, their priority and business value to customers, the dependencies among items, appropriate allocations to bug fixing and other technical work (e.g. resolving technical debts) and the availability of team members and the team's capacity. Risks impeding a sprint execution should also be forecasted and factored into a sprint plan. Sprint planning thus requires not only in-depth understanding of the current project and team but also experience learned from previous projects. Tool support is needed to manage complexities for large projects.
\subsubsection{\textbf{Pro-actively monitoring sprint progress and managing risks}} As the sprint unfolds, the team needs to track sprint progress and manage risks. Current practices in risk management mostly rely on high-level guidance and subjective judgements. Predicting future risks is highly challenging due to the inherent uncertainty, temporal dependencies, and especially the dynamic nature of software. There is currently a gap in providing agile teams with insightful and actionable information about the current existence of risks in a sprint, and recommending concrete measures to deal with those risks.
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Chen Yi-wen (; born 1966) is a Taiwanese filmmaker and actor.
Career
While Taiwanese directors are often associated with slower-paced, personal art films, Chen decided early on when he entered the film industry that he wanted to produce high quality, entertainment-oriented movies.
"The performing arts shouldn't be inhibited by theory." Chen has established a solid foundation with a career that includes screenwriting, directing, as well as theatre, instilling his films with a unique and distinguished style.
Chen's first short film, Scenes of Violence, cost NT$3,000 to produce and earned NT$600,000 in returns from television sales in Taiwan and Japan. This success gave him the confidence to devote himself to the film industry. Chen was even interviewed by Wealth Magazine for an in-depth report on the success of his short film for its high ROI (return on investment).
In 1998, a Japanese corporation invested in Chen's first feature film JAM. The film set a record of continuously running for over three months in theatres.
After the success of JAM, Chen completed a gangster film, A Chance to Die, once again getting financing from Japan. He asked Miki Mizuno, a well-known Japanese actress, and Takashi Kashiwabara, a famous Japanese idol who is also popular in Taiwan, to play the main characters. This was his second feature film.
For his third feature film, The Cabbie, Chen was able to get Rie Miyazawa, an accomplished Japanese actress, to play the leading role of the movie. The Cabbie was a fresh and inventive take on the Taiwanese comedy.
Chen's creativity and skills are on full display in the work of these three films. In being vivid and confident in the shooting, drawing on a strong foundation of the visual arts, the dramatics of storytelling, and with a focus on the shaping each characters' unique inner lives, Chen has always been able to effectively create entertaining and audience-friendly films while still maintaining his a strong vision.
Chen has continued to search for new and innovative storytelling methods. He returned from a short-term sabbatical in New York, with a renewed focus on producing high quality films.
In 2006, Chen finished a 35mm feature film, Tripping, also known as Time Tripper, which combined the road movie with a martial arts film. In 2013, he produced and directed As the Winds Blow.
After 2015, Chen worked as an actor in many feature films, Godspeed, The Great Buddha+, Xiao Mei, High Flash, A Sun, The Falls ... and so on, and he also acted on Netflix programs, Wake Up 2, On Children, and Monstrous Me.
In 2019, Chen got the Best Leading Actor Award by the feature film A Sun in the 56th Golden Horse Awards and he also got the nominee of the Best Leading Actors by a short film A Taxi Driver in the 54th Golden Bell Awards.
The Best Actor Award (Short Film) went to Yi-Wen by a short film Growing Pains in the 40th Hawaii International Film Festival in 2020.
Filmography
Worked as an Actor in Feature Films
Worked as an Director
Feature Films Directed
1994 A Confucian Confusion(Assistant Director)
1998 Jam(Screenwriter, concurrently)
2000 A Chance to Die(Screenwriter & Costume Designer, concurrently)
2000 The Cabbie
2006 Tripping (aka: Time Tripper, aka: Gen Yu Den)(Screenwriter, concurrently)
2009 File No. 1689(Screenwriter & Producer, concurrently)
2013 As the Winds Blow(Screenwriter & Producer, concurrently)
Short Films Directed
1994 Scenes of Violence(Screenwriter, concurrently)
1995 Lessons(Screenwriter, concurrently)
Awards and Nominations for Actor or Director
References
External links
Chen Yi-wen's Facebook
1966 births
Living people
Taiwanese filmmakers
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Heat a large saucepan over a medium heat. Place butter, olive oil, onion and garlic in the saucepan and cook for 3-5 minutes or until onion is translucent.
Add meat and herbs to the pan, stir and add beef stock.
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Brush with whisked egg and bake for 20 minutes or until golden. Serve immediately.
Smokey paprika, sweet corn, spicy chilli and tangy lime combine to make this salad as filling as it is fabulous.
Combine prawns, garlic, chilli, sweet paprika and ½ the olive oil in a bowl. Toss to coat. If time permits, cover and refrigerate for 10-15 minutes to marinate.
Meanwhile, combine McCain Corn Kernels, tomatoes, red kidney beans, coriander, red onion and remaining oil in a large bowl. Place all ingredients for the dressing in a small bowl and whisk briskly to combine.
Preheat a barbecue or char grill to a medium-high heat, add remaining olive oil, place prawns on the grill and cook, turning, for 2 minutes on each side, or until cooked through.
Divide McCain Corn Kernel salad mixture among serving plates and top with prawns, drizzle with salad dressing, and serve with fresh lime cheeks.
Recently a friend came over for morning tea, having a new bub I find it next to impossible to get out of the house, let alone have the time to whip up a morning tea treat! After rummaging through the cupboards I decided I had almost enough ingredients to make banana bread. I was down the eggs, but, I'd make it work. And, work it did! This is one of the fastest, easiest and tastiest banana breads I have ever made. Cafe style, moist and moreish. YUM!
Preheat a fan forced oven to 180℃ and grease a loaf tin with a little olive oil.
Mix dry ingredients together in a large bowl, mix wet ingredients together in a medium bowl. Fold wet and dry ingredients together until combined just combined.
Pour mixture into a greased loaf tin, place decoration banana on top, and bake for 1 hour or until a skewer comes out clean.
Bruschetta is an exercise in getting the basics right; balancing the flavour, and texture of simple ingredients to create something greater than the sum of its parts.
The flavours of my Pea and Persian Feta Bruschetta are perfection. The creamy peas, are accentuated with fresh mint, then hit with the sharpness of garlic, parmesan and lemon, and finished with crunchy bread and smooth Persian Feta. So good.
Bring a pot of salted water to a boil, add McCain Peas and cook 3 minutes or until tender. Scoop peas out of the pot, into an ice bath to cool. Drain. The ice bath will help the peas retain their bright colour, juiciness and plumpness.
Add McCain Peas, mint leaves, olive oil, lemon juice, parmesan and garlic to a food processer and blitz for 1 minute or until roughly crushed. Season with salt and black pepper to taste.
Preheat a chargill pan over medium-high heat. Brush both sides of bread with oil and chargrill, in batches, for 3 minutes each side.
Spread pea mixture over grilled bread and serve scattered with Persian Feta, mint leaves and lemon cheeks.
Did you know that Abe's Bagels – Bagel Crisps contain 70% less fat than regular crisps? And, their scrumptious flavours contain no artificial colours, flavours or preservatives. Hells yeah! Meaning now you can have your crisps, and eat them too.
While chomping on these crisps, I thought I would try something different with my crisps, rather than cracking a packet and plonking on the couch or serving them with dip. Enter, Beef, Bean and Bagel Nachos; by replacing the standard nacho corn chips with bagel crisps not only will you be reducing your families fat intake, but the sturdy nature of the bagel crisps gives you superior scooping power as your crisp wont buckle under topping pressure or go soggy. Win.
Heat oil in a large pan over a medium-high heat. Add onion and garlic and cook stirring for 2-3 minutes until onion begins to soften.
Add beef mince, bean mix, tomato paste and herbs. Cook, using a fork to break up the mince, for 3-5 minutes or until beef mince is browned and herbs are fragrant.
Add water and allow to simmer uncovered for 5 minutes or until sauce has reduced and thickened.
Pile Abe's Bagels – Marlborough Sea Salt Bagel Crisps on an oven proof serving dish, top with beef and bean mix, top with cheese and grill under a medium grill to melt and brown.
Top with fresh coriander leaves and diced avocado, and serve immediately with sour cream and sweet chilli sauce.
Remember, if there is a specific ingredient you'd like to know how to use,m or dish you'd like to know how to make, just pop it in the comments section below.
This dish is a modern take on the classic Italian dish of Spaghetti with Pangrattato. "Pangrattato" is basically Italian for breadcrumbs – but so much more romantic! Coming from the Calabria district, like me! Traditionally this is seen as 'peasant food', given the cheap and readily available ingredients of bread and pasta; and was eaten during lent or at Christmas when Catholics avoid meat.
In this dish I have chosen to mix it up a bit and use Abe's Bagels – Bagel Crisps, for their crunch and in this case their roasted garlic flavour. This dish is easy and delicious, with the subtle hum of the anchovies, the kick of chilli and crunch of the garlic bagel crumbs.
Place Abe's Bagels – Roasted Garlic Bagel Crisps into the bowl of a food processor and process until roughly chopped.
Cook pasta in a large pot of salted boiling water until al dente. Drain, reserving 1/3 cup of cooking liquid.
Heat oil in a large, non-stick pan over medium heat. Add chilli and anchovies and stir cooking for 1-2 minutes or until fragrant. Add parsley and Abe's Bagels – Bagel Crisp crumbs and cook for 1-2 minutes or until bagel crumbs are golden and parsley is beginning to crisp. Add pasta and lemon juice to the pan and toss, adding enough of the reserved cooking liquid to moisten.
Divide between bowls and serve immediately.
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SFARI Launches SPARK Online Research Initiative
The initiative aims to recruit 50,000 individuals with autism to advance our understanding of the condition
The Simons Foundation Autism Research Initiative (SFARI) today announced the launch of SPARK, an online research initiative designed to become the largest autism study ever undertaken in the United States. SPARK, which stands for Simons Foundation Powering Autism Research for Knowledge, will collect information and DNA for genetic analysis from 50,000 individuals with autism — and their families — to advance our understanding of the condition's causes and accelerate the development of new treatments and supports.
Autism is already known to have a strong genetic component. To date, approximately 50 genes have been identified that almost certainly play a role in autism, and researchers estimate that an additional 300 or more are involved. By studying these genes, associated biological mechanisms and the interactions of environmental factors with genes, researchers can better understand the condition's causes and link them to the spectrum of symptoms, skills and challenges of those affected.
"Knowledge is power, and SPARK was created because we simply haven't learned enough about the genetics and other possible causes of autism," says Wendy Chung, SPARK's principal investigator and director of clinical research at SFARI. "SPARK will help researchers make new discoveries that will ultimately lead to the development of new supports and treatments to improve the lives of people living with challenges. Together, we can 'spark' a movement in autism research."
SPARK aims to speed up autism research by inviting participants from this large, diverse autism community, including individuals of both sexes and all ages, backgrounds, races, geographic locations and socioeconomic situations with a professional diagnosis of autism. The initiative catalyzes research by creating long-term access to a large number of study participants for whom detailed genomic, medical and behavioral information will be available. SPARK will connect participants to researchers, offering them the unique opportunity to impact the future of autism research by joining any of the multiple studies offered through SPARK. SPARK will also take feedback from individuals with autism and their parents to develop a robust research agenda that is meaningful for these families.
This new initiative is funded and centrally coordinated by SFARI. A total of 21 university-affiliated clinical sites and numerous national and local autism community organizations across the U.S. are partnering with SFARI to help recruit participants and spread the word about this landmark study. De-identified genetic and phenotypic data will be made available to any qualified researcher throughout the duration of the project, and researchers will have the opportunity to contact participants for potential enrollment in their research and clinical studies.
"A major goal of SPARK is to accelerate clinical research in autism by providing a large resource to the entire research community," says Pamela Feliciano, scientific director of SPARK and senior scientist at SFARI. "All qualified researchers will be able to access SPARK genomic, medical and behavioral data and recruit for their studies from SPARK as soon as possible."
Anyone interested in learning more about SPARK or in participating can visit www.SPARKforAutism.org.
A Many-Method Attack on the Many Electron Problem
Watch: Maarten de Hoop Turns the Earth Inside Out
On the Homepage: A Calabi-Yau Manifold
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Probably… Disqualification Warning
by Kristina Sperkova Posted on June 12, 2016 in
Alcohol Industry, Alcohol Norm, Alcohol's Harm To Others, Corporate Consumption Complex, Prevention, Social Justice
Let's call it what it is: UEFA, FIFA have to quit to put their profits over the well-being of children and families, fans and football itself. The events these days involving alcohol fueled violence should cause an urgent disqualification warning to Carlsberg as well. Like at the Fifa World Cup in Brazil, where violence erupted, after alcohol sales were re-introduced – we can clearly see that the alcohol norm needs to be disqualified…
I've been watching the EURO 2016 game between Croatia and Turkey today. I've good friends in Croatia and in the Balkan region that have Croatian roots – so I wanted to share the moment with them. Croatia played well and won the game but a different news story overshadowed everything else: England and Russia have been given disqualification warnings.
Just a few minutes ago, Swedish TV SVT1, the BBC and other EURO 2016 broadcasters reported that the UEFA Executive Committee has issued a warning directed at the Russian and English football associations, after violence in Marseille during the recent days. UEFA, European football's governing body, said in a press release:
The UEFA Executive Committee would like to express its disgust at the violent clashes which occurred in the city of Marseille. Such unacceptable behaviour by so-called supporters of the national teams of England and Russia has no place in football, a sport we must protect and defend. The UEFA Executive Committee has warned both football associations that – irrespective of any decisions taken by the independent disciplinary bodies relating to incidents inside the stadium – it will not hesitate to impose additional sanctions on the Football Association (FA) and the Russian Football Union (RFS), including the potential disqualification of their respective teams from the tournament, should such violence occur again."
This strongly-worded statement is a big deal. It's a big deal for what it says and it is a big deal for what it does not say.
UEFA holds the football associations, and quintessentially also the respective national teams, accountable for the behavior of their "fans". Reports say that Marseille, the city where the England-Russia football game took place, has been shaken by violent clashes between French youth, Russian, and English supporters and the French police and security forces. UEFA makes clear that it will not hesitate to take the most far-reaching measures in order to stop and eliminate the violence. I applaud them for what they have said now.
Calling it, what it is
But I also have to say that more could have and should have been done to prevent this violence. But what? Let's look at the elephant in the room and call UEFA actions what they are. The BBC writes:
In the steamy, humid environment of this port city, with alcohol flowing freely, all the ingredients were in place for events that unfolded in the Vieux-Port de Marseille."
In another article by BBC writer Ben Mundy reports:
There was a heavy police presence and at one point officers asked a bar to stop serving alcohol …"
Violence in the city days before the match. Violence inside the stadium right after the match. Reporting on the violence inside the stadium, the Independent's Mark Ogden asked:
Where were the police inside the stadium? Why was more not done to influence an alcohol ban ahead of such a potentially-volatile fixture? And why was the segregation in Stade Velodrome so obviously inadequate before the trouble erupted at the end of the game?"
And the Herald Scotland reported that Samia Ghali, a Marseille senator, said:
When you see drunken hooligans hurling bottles, I think you realise that it's necessary to impose restrictions on the sale of alcohol to make it easier to maintain order."
On Twitter, Martin Fricker, a Daily Mirror senior reporter, delivers news from Marseille and this picture struck me.
This pretty much sums up the day (top snap by Getty's @Carl_Court) … #EURO2016 #ENG #RUS pic.twitter.com/5EdV6YJUXS
— Martin Fricker (@martinfricker) June 11, 2016
In this picture, a lot is going on. But at center stage, too, is beer..
Lot's of alcohol.
Lot's of green bottles…
Carlsberg promotion after the game
Carlsberg promotion during the game
That's what Carlsberg and the UEFA Euro 2016 twitter actiicites look like. And this is an example of what it looks like in every single game, frequently…
Croatia-Turkey, Sunday 12 June 2016
Slovakia-Wales, Saturday 11 June, 2016
England-Russia, Saturday 11 June 2016
Pervasive alcohol marketing. Perverse alcohol norm
Alcohol is everywhere. When the biggest stars give interviews about their performance right after the game, Carlsberg is there. When the biggest stars celebrate their goals, Carlsberg is there. In the beginning, in the middle and in the end of games, Carlsberg is there. And Carlsberg profits from it. And so does UEFA. In the words of Mark Ogden:
It was Uefa and Marseille's job to be ready for the game, but both were patently unprepared…"
Football is not for hooligans and thugs. No question about it. And no question about this fact either: alcohol does NOT cause violence. But it is also high-time for UEFA and other football governing bodies and associations, and for our politicians and decision-makers to end the romance with alcohol companies.
The alcohol norm in football is perverse. Together with the pervasive alcohol marketing it fuels violence. Let's call it what it is: UEFA (and others) have to quit putting their profits over the well-being of children and families, fans and football itself. The events these days involving alcohol fueled violence should cause an urgent disqualification warning to Carlsberg as well.
Like at the Fifa World Cup in Brazil, where violence erupted, after alcohol sales were re-introduced, EURO 2016 is sadly another example where we can clearly see that the alcohol norm needs to be disqualified.
For more evidence
about how harmful alcohol sports sponsorship is to kids, click here: Big Alcohol Exposed
about how pervasive alcohol brands are in sports in particular and in the lives of children and youth in general, click here: Collection of unethical marketing cases.
to help end the harmful practices of the alcohol industry in football, click here: #OutbidChang
Tags: #OutbidChang Alcohol Harm alcohol marketing Alcohol Policy Alcohol Violence Big Alcohol Carlsberg Child Health EURO2016 Europe Evidence Fifa Football France Global Marseille Social Media UEFA Under-Age
Knowing Where The Goal Is: EURO 2016 And Alcohol Brands
Maik Dünnbier - Jun 10 '16
Calculated Loss
Professional Sports Glamorising Ill-Health
Big Alcohol's Attack On Women
Maik Dünnbier - Mar 14 '17
Get to know Kristina
Ahoj and warm welcome to my blog!
I am International President of this beautiful organization. I see alcohol as an obstacle to development – both for individuals and for communities and societies around the world.
I feel honoured and grateful to work with these questions and raise awareness about the beauty and benefits of an alcohol-free lifestyle – the lifestyle of the 21st century and the way of being for the majority of the world's population.
Other recent posts by Kristina Sperkova
6 Reasons Why New WHO Comment on Health and Cancer Risks From Low Dose Alcohol Use Is a Game Changer
Season's Greetings 2022 from the International Board
What Standing With Ukraine Is All About
Sober Vibes – A Super Power that Everyone Has
The Number One Action to Achieve WHO European Region Free From Alcohol Harm
What the 'Another Round' Movie Tells Us About Hollywood and Alcohol
Holiday Reflections: Looking Back and Looking Ahead
Kristina's most recent tweet
Great recent news #AlcPolPrio and fabulous work done by @actbr talking about alcohol policy as a catalyst of sustai… https://t.co/X4mHMw6gN8
11:04 AM Dec 12
Find more of Kristina's tweets »
2 comments on "Probably… Disqualification Warning"
Only Fair on June 18th, 2016 - 11:12pm
How about an update on this including the Croatian fans themselves and their behaviour?
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Dopo la sua formazione musicale a Francoforte Heinrich Anton Föppel l'amico d'infanzia aveva di Richard Wagner prime apparizioni al Teatro di Corte di Kassel sotto il direttore d'orchestra e compositore Louis Spohr.
Maria Theresia Löw sposò il tenore protagonista Karl-August Lehmann e divenne la madre di Elisabeth Maria "Lilli" Lehmann e Marie Lehmann, che dovrebbe sia tardi anche essere cantanti. Dopo la separazione dal marito nel 1853 ha preso cura della loro formazione vocale. Oltre alle sue figlie, era responsabile di altri artisti della sua epoca, tra cui il Teatro Nazionale tedesca in Praga, dove ha vissuto dal 1853.
Suo zio era il cavaliere del governo bavarese Johann Löw (1771-1833) da Spira, la cui figlia Amalie Löw (1811-1879) aveva sposato il principe Karl Theodor von Wrede.
Fonte e Letteratura
Lilli Lehmann La mia arte vocale. Berlino 1902
Lilli Lehmann My Life. Lipsia 1913, ristampa 1977
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\subsection{Training Setup}
We used a proprietary speech dataset containing 405 hours of speech data; 347,872 utterances including 45 speakers in 3 English accents (32 US English speakers, 8 British English, and 5 Australian English speakers).
The Parallel Tacotron models were trained with Nesterov\footnote{Adam optimizer with the typical Transformer schedule performed worse.} momentum optimization and $\alpha=0.99$. We used a linear warmup from 0.1--1.0 for the first 10K steps followed by the exponential decay from step 20K to 100K with a minimum value of 0.01.
All models were trained for 120K steps with global gradient norm clipping of 0.2 and a batch size of 2,048 using Google Cloud TPUs. Training took less than one day.
The fine-grained phoneme-level VAE model was trained differently. It used a KL-weight schedule where $\beta$ was increased linearly to 1.0 from step 6K to 50K. Then the fine-grained phoneme-level VAE model was trained with exponential decay between steps 40K and 120K and we chose the checkpoint at step 215K.
As a baseline we used the Tacotron~2 model \cite{tacotron2} with some small modifications: specifically it used Gaussian mixture model (GMM) attention \cite{graves2013generating,transfer} and a reduction factor of 2 \cite{tacotron}, both of which have been shown to improve the robustness \cite{shen-arxiv-2020}. Tacotron~2 was trained for 500K steps with batch size 1,024 using Adam with exponential decay from step 50K to 400K.
Both baseline and proposed models were combined with the same pretrained WaveRNN neural vocoder \cite{kalchbrenner2018efficient} to reconstruct audio signals from predicted mel-spectrograms.
\subsection{Evaluation Setup}
Subjective evaluations were conducted using 1,000 sentences.
These sentences were different from the training data and was used in the previous papers.
They were synthesized using 10 US English speakers (5 male \& 5 female) in a round-robin style (100 sentences per speaker).
The naturalness of the synthesized speech was evaluated through subjective listening tests, including 5-scale Mean Opinion Score (MOS) tests and side-by-side preference tests.
For MOS tests, a five-point Likert scale score (1: Bad, 2: Poor, 3: Fair, 4: Good, 5: Excellent) was adopted with rating increments of 0.5.
For side-by-side preference tests each rater listened to two samples then rated each with integral scores $[-3,3]$; where a positive score indicated that the first sample sounded better than the second one \cite{transfer, tacotron2}.
\subsection{Experimental Results}
\begin{table}[th]
\caption{Subjective evaluations of Parallel Tacotron with and without the iterative loss. Positive preference scores indicate that the corresponding Parallel Tacotron model was rated better than the reference Tacotron~2.}
\centering
\begin{tabular}{lrr}\\\toprule
\textbf{Model} & \multicolumn{1}{c}{\textbf{MOS}} & \multicolumn{1}{c}{\textbf{Preference}}
\\\midrule
Tacotron~2& $4.45 \pm 0.04$ & Reference \\
\midrule
\multicolumn{3}{l}{Parallel Tacotron (LConv, w/o VAE)} \\
\quad single loss & $4.32 \pm 0.04$& $\bf{-0.09 \pm 0.06}$\\
\quad iterative loss & $4.33 \pm 0.04$ & $\bf{-0.08 \pm 0.06}$ \\
\bottomrule
\end{tabular}
\label{evaluation_iter}
\vspace{1mm}
\caption{Subjective preference scores between Parallel Tacotron with and without the iterative loss.
Positive preference scores indicate that the corresponding model with the iterative loss was rated better than the one without the iterative loss.
}
\centering
\begin{tabular}{lr}\\\toprule
\textbf{Model} & \multicolumn{1}{c}{\textbf{Preference}}
\\\midrule
LConv w/o VAE & $\bf{0.09 \pm 0.05}$\\
LConv w/ Global VAE &$\bf{0.07 \pm 0.05}$\\
Transformer w/ Global VAE & $\bf{0.08 \pm 0.05}$\\
\bottomrule
\end{tabular}
\label{preference_iter}
\vspace{1mm}
\caption{Subjective evaluations of Parallel Tacotron with different self-attention (with Global VAE and iterative loss). Positive preference scores indicate that the corresponding Parallel Tacotron was rated better than Tacotron~2.}
\centering
\begin{tabular}{lrr}\\\toprule
\textbf{Model} & \multicolumn{1}{c}{\textbf{MOS}} & \multicolumn{1}{c}{\textbf{Preference}}
\\\midrule
\multicolumn{3}{l}{Parallel Tacotron (w/ iterative loss \& Global VAE)} \\
\quad Transformer & $4.36 \pm 0.04$ & $-0.03 \pm 0.06$ \\
\quad LConv & $4.40 \pm 0.04$ & $-0.01 \pm 0.05$ \\
\bottomrule
\end{tabular}
\label{evaluation_decoder}
\vspace{1mm}
\caption{Subjective preference score between Parallel Tacotron using LConv and Transformer-based self-attention.
Positive preference scores indicate that LConv was rated better than Transformer.
}
\centering
\begin{tabular}{lrr}\\\toprule
\textbf{Model} & \multicolumn{1}{c}{\textbf{Preference}}
\\\midrule
LConv \textit{vs} Transformer & $\bf{0.05 \pm 0.04}$\\
\bottomrule
\end{tabular}
\label{preference_decoder}
\end{table}
The first experiment evaluated the effect of the iterative loss.
Tables~\ref{evaluation_iter} and \ref{preference_iter} show the experimental result.
Although there was no significant difference in MOS and preference against Tacotron~2, the direct comparison between models with and without the iterative loss indicate that it can give small improvement.
The second experiment evaluated the impact of VAEs.
Tables~\ref{evaluation_vae} and \ref{preference_vae} show the experimental results.
Parallel Tacotron without VAE was significantly worse than the baseline Tacotron~2 in both MOS and preference.
The introduction of global VAE made it comparable to the baseline Tacotron~2 in both evaluations.
Furthermore, the introduction of fine-grained phoneme-level VAE further boosted the naturalness.
\begin{table}[th]
\caption{Subjective evaluations of Parallel Tacotron with different VAEs. Positive preference scores indicate that the corresponding Parallel Tacotron was rated better than Tacotron~2.}
\centering
\begin{tabular}{lrr}\\\toprule
\textbf{Model} & \multicolumn{1}{c}{\textbf{MOS}} & \multicolumn{1}{c}{\textbf{Preference}}
\\\midrule
\multicolumn{3}{l}{Parallel Tacotron (w/ iterative loss \& LConv)} \\
\quad No VAE & $4.33 \pm 0.04$ & $\bf{-0.08 \pm 0.06}$ \\
\quad Global VAE & $4.40 \pm 0.04$ & $-0.01 \pm 0.05$ \\
\quad Fine VAE & $4.42 \pm 0.04$ & $\bf{0.07 \pm 0.07}$ \\
\bottomrule
\end{tabular}
\label{evaluation_vae}
\vspace{1mm}
\caption{Subjective preference scores between Parallel Tacotron using the global and fine-grained VAEs. %
Positive preference scores indicate that left models were rated better than the right ones.
}
\centering
\begin{tabular}{lr}\\\toprule
\textbf{Model} & \multicolumn{1}{c}{\textbf{Preference}}
\\\midrule
Global VAE \textit{vs} No VAE & $\bf{0.13 \pm 0.05}$\\
Fine VAE \textit{vs} Global VAE & $\bf{0.06 \pm 0.06}$\\
\bottomrule
\end{tabular}
\label{preference_vae}
\vspace{1mm}
\caption{Subjective preference scores between synthetic and natural speech. The preference scores become positive when synthetic speech was rated better than natural one.}
\centering
\begin{tabular}{lr}\\\toprule
\textbf{Model} & \multicolumn{1}{c}{\textbf{Preference}}
\\\midrule
Tacotron~2 &$\bf{-0.09 \pm 0.07}$\\ \midrule
\multicolumn{2}{l}{Parallel Tacotron w/ LConv \& iterative loss} \\
\quad Global VAE & $-0.07 \pm 0.07$\\
\quad Fine VAE & $-0.01 \pm 0.06$\\
\bottomrule
\end{tabular}
\label{preference_human}
\vspace{1mm}
\caption{Inference speed to predict mel-spectrograms for $\sim$20-second long utterance on a TPU (aggregated over ten trials). }
\label{performance}
\centering
\begin{tabular}{lrr}\\\toprule
\textbf{Model} & Mean \textbf{($1 / \textsc{RTF}$)} & Stddev
\\\midrule
Tacotron~2& 79 & 1\\ \midrule
\multicolumn{2}{l}{Parallel Tacotron} & \\
\quad Transformer w/ Global VAE & 804 & 5\\
\quad LConv w/ Global VAE & 1051 & 13\\
\quad LConv w/ Fine VAE & 1013 & 10\\
\bottomrule
\end{tabular}
\end{table}
The third experiment compared Transformer and LConv for self-attention.
Tables~\ref{evaluation_decoder} and \ref{preference_decoder} show the experimental results.
Parallel Tacotron with both Transformer and LConv-based self-attention matched the baseline Tacotron~2 both in MOS and preference.
When they were directly compared, Parallel Tacotron with LConv was more preferred.
The last evaluation compared was done over 1,000 utterances from a held-out test set.
This allowed us to make direct comparisons with natural speech.
The MOS of natural speech was $4.54 \pm 0.04$ on this test set.
Experimental results are shown in Table~\ref{preference_human}.
It can be seen from the table that all neural TTS models are doing well compared to human speech, however there is still a room for further improvement.
\subsection{Inference Efficiency}
Table~\ref{performance} shows the inference time to predict mel-spectrograms for a 20 second-long utterance on a tensor processing unit (TPU).
It can be seen from the table that Parallel Tacotron was about $13$ times faster than Tacotron 2.
It also shows that the use of lightweight convolutions is faster than of Transformer.
It should be noted that little work has been done to trade-off model size and speed against the naturalness thus further speedup is highly possible.
\section{Introduction}
\label{sec:intro}
\input{introduction.tex}
\section{Parallel Tacotron}
\label{sec:architecture}
Figure~\ref{fig:architecture} illustrates the architecture of the Parallel Tacotron model.
It consists of an input encoder, a residual encoder, a duration decoder, an upsampling block, and a spectrogram decoder stack.
The model heavily relies on self-attention blocks with either Transformer or lightweight convolutions (LConv).
The LConv is a depth-wise convolution which shares certain output channels whose weights are normalized across time \cite{wu2019pay}.
It has a few orders of magnitude less parameters than a standard non-separable convolution.
Unlike Transformer-based self-attention, lightweight convolutions have a fixed context window and reuse the same weights for context elements, regardless of the current time-step.
This can be more useful in TTS as relevant context elements can be more local than machine translation or other language tasks.
The LConv block is also illustrated in Fig.~\ref{fig:architecture}.
It consists of a gated linear unit (GLU), a lightweight convolution, and a feedforward (FF) layer with residual connections.
As in \cite{wu2019pay} we use dropout 0.1, and perform FF mixing in the FF layer using the structure $\text{ReLU}(\bm{W}_1\bm{X} +\bm{b_1})\bm{W}_2 +\bm{b}_2$ where $\bm{W}_1$ increases the dimension by a factor 4.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{figures/PTacoBlock.pdf}
\caption{Block diagram of the Parallel Tacotron model. The residual encoder (purple blocks) in this figure correspond to the global VAE in Section~\ref{sec:global_vae}.}
\label{fig:architecture}
\end{figure*}
\subsection{Input Encoder}
As in Tacotron 2 \cite{tacotron2}, the input encoder consists of a phoneme embedding lookup followed by three convolutional blocks.
Thereafter, sinusoidal positional embeddings are added, then six Transformer blocks are applied.\footnote{An LConv stack works equally well.} The output of the encoder can be viewed as a contextual phoneme representation.
The encoder outputs are concatenated with a speaker embedding of dimension 64 and latent representation from the residual encoder.
\subsection{Variational Residual Encoder}
Two different VAE models are investigated; a global VAE per speaker similar to \cite{hsu2018hierarchical} and a fine-grained phoneme-level VAE akin to \cite{sun2020fullyhierarchical}.
\subsubsection{Global VAE per Speaker}
\label{sec:global_vae}
Like multi-speaker modeling in \cite{hsu2018hierarchical}, the model is augmented by a global VAE per speaker, each with a speaker specific learned prior.
The latent representation from the VAE can be viewed as the residual information which cannot be represented by input phoneme sequences and associated speaker IDs, such as prosodic variations \cite{hsu2018hierarchical}.
The residual encoder, also known as the posterior network, takes target mel-spectrograms as its input and processes it using a stack of LConv blocks. The stack consists of; three $17 \times 1$ LConv blocks, followed by five $17 \times 1$ LConv blocks interleaved with strided $3 \times 1$ convolutions, where all LConv blocks have eight heads.
This lets the model successively downsample the representation before applying global average pooling to get the final global latent representation.
Although the dimension of the latent representation is 8, it is projected up to 32.
During inference, a speaker specific prior mean is used.
In addition to modeling the residual information, this latent representation works to implicitly capture the dependencies among output mel-spectrogram frames given input tokens \cite{gu2017,lee-arxiv-2018,shu-aaai-2020}.
\subsubsection{Phoneme-Level Fine-Grained VAE}
A fine-grained phoneme-level VAE \cite{sun2020fullyhierarchical} has the posterior network which aligns the groundtruth spectrogram with the encoder outputs using attention.
To achieve this, the positional embeddings from Section~\ref{sec:positional} and the speaker embedding are concatenated to the groundtruth spectrogram frames, subsequently five 8-headed $17 \times 1$ LConv blocks are applied. The attention is then computed with the layer normalized \cite{ba2016layer} encoder outputs. We use an 8-dimensional latent representation, concatenate the speaker embedding and encoder outputs, and then project it to dimension 32.
Furthermore, similar to \cite{sun2020fullyhierarchical}, we train a separate autoregressive LSTM to predict a sequence of latent representation during inference.
Similarly with the posterior network, this LSTM takes both speaker embedding and encoder outputs as its input.
The LSTM is trained to predict the posterior mean using the $L_2$ loss with teacher forcing. This loss is computed independently of the posterior network; no error gradient is back-propagated to the posterior network.
\subsection{Duration Decoder}
The encoder outputs augmented by the latent representation and speaker embeddings are fed to a duration decoder which predicts phoneme duration.
We assume that groundtruth phoneme durations are provided by an external aligner, such as hidden Markov model (HMM)-based one \cite{talkin1994aligner}.
Here the input sequence consists of regular phonemes, silences at all word boundaries, and punctuation marks.
Whereas regular phonemes usually have non-zero duration, zero-length duration are often associated with silences and punctuation marks (\textit{e.g.}, word boundaries without pauses).
To model such sequences, the duration decoder predicts two types of outputs: binary zero/non-zero duration and continuous phoneme duration in seconds.
The duration decoder contains four LConv blocks with 3 $\times$ 1 lightweight convolutions followed by two independent projections.
One is followed by a sigmoid activation predicting the probability of non-zero duration $p_z$.
Another is followed by a softplus activation predicting the phoneme duration in seconds.
If $p_z<0.99$ then the predicted duration is zeroed out.
Here the cross-entropy (CE) and $L_1$ loss functions are used for the former and latter, respectively.
\subsection{Upsampling and Positional Embeddings}
\label{sec:positional}
Upsampling takes the activation of the last LConv block in the duration decoder and upsamples them to the length of target spectrogram frame sequence using the phoneme duration.
After upsampling, three different types of positional embeddings are added to make the spectrogram decoder aware of the frame positions within phonemes:
(1) Transformer-style sinusoidal embedding \cite{transformer} of a frame position within a phoneme, (2) Transformer-style sinusoidal embedding \cite{transformer} of phoneme duration, (3) Fractional progression of a frame in a phoneme (1D CoordConv \cite{liu-neurips-2018}).
Since several positional embeddings are added, per channel a learned weighted sum is used to combine them with the output of the duration decoder.
Specifically, the weights are softmax normalized per channel; which lets the network select for each channel which embedding that is preferred by the spectrogram decoder.
\subsection{Spectrogram Decoder with Iterative Loss}
\label{iterative_loss}
The output of the upsampling stack is fed to the spectrogram decoder.
It has six 8-headed self-attention blocks with 17 $\times$ 1 LConv with dropout 0.1 as in \cite{wu2019pay}.
The output of the last FF layer in each block is then projected to 128-bin mel-spectrogram.
We investigate the use of an iterative loss and its impact on the naturalness of synthesized speech;
the outputs of the last FF layers in the self-attention blocks are independently projected to the mel-spectrogram then $L_1$ losses between the predicted and target mel-spectrogram are summed to form the final loss.
This type of iterative loss has previously been used in \cite{lee-arxiv-2018,dejavu}.
\subsection{Training Objective}
The overall loss functions for the Parallel Tacotron models with global and fine-grained VAEs become
\begin{align}
\mathcal{L}_\text{global} &= \frac{1}{KT} \sum_i{\mathcal{L}_{\text{spec}_i}} + \frac{1}{N} \lambda_{\text{dur}} \mathcal{L}_{\text{dur}} - \beta D_\mathrm{KL}, \\
\mathcal{L}_\text{phone} &= \frac{1}{KT} \sum_i{\mathcal{L}_{\text{spec}_i}} + \frac{1}{N} \lambda_{\text{dur}} \mathcal{L}_{\text{dur}} - \beta D_\mathrm{KL} + \frac{1}{N} \mathcal{L}_{\text{prior}},
\end{align}
where $\mathcal{L}_{\text{spec}_i}$ is the $L_1$ spectrogram loss for the $i$-th LConv block in the spectrogram decoder, $\mathcal{L}_{\text{dur}}$ is the two-level duration loss, $D_\mathrm{KL}$ is the KL divergence between prior and posterior from the residual encoder, $\mathcal{L}_{\text{prior}}$ is the learned prior loss in the phoneme-level VAE, $T$ is the total number of frames, $K$ is the size of spectrogram, and $N$ is the number of tokens.
\section{Experiments}
\label{sec:experiments}
\input{experiments.tex}
\section{Conclusions}
\label{sec:conclusion}
A non-autoregressive neural TTS model called Parallel Tacotron was proposed.
It matched the baseline Tacotron~2 in naturalness and offered significantly faster inference than Tacotron~2.
We also showed that both variational residual encoders and an iterative loss improved the naturalness, and the use of lightweight convolutions as self-attention improved both naturalness and efficiency.
Future work includes the direct comparison against duration-based models with the autoregressive decoder and those using additional representation such as $F_0$ and energy.
Investigating better fine-grained variational models is also necessary.
\section{Acknowledgements}
The authors would like to thank Mike Chrzanowski, Tejas Iyer, Vincent Wan, and Norman Casagrande for their help.
\clearpage
\footnotesize
\bibliographystyle{IEEEbib}
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\section{Introduction}
Let $X=(V,E)$ be a simple, undirected graph. An automorphism of $X$ is a permutation of the vertex set that preserves adjacency. The automorphism group of $X$, denoted by $\Aut(X)$, is the set of all automorphisms of the graph $X$, that is, $\Aut(X) := \{g \in \Sym(V): E^g = E\}$. A graph $X$ is said to be vertex-transitive if for any two vertices $u, v \in V(X)$, there exists an automorphism $g \in \Aut(X)$ that takes $u$ to $v$. A graph $X$ is said to be edge-transitive if for any two edges $\{u,v\}, \{x,y\} \in E(X)$, there exists an automorphism $g \in \Aut(X)$ such that $\{u^g, v^g\} = \{x,y\}$. In other words, $X$ is edge-transitive iff the action of $\Aut(X)$ on the edge set $E(X)$ has a single orbit.
Given a group $H$ and a subset $S$ of $H$ such that $1 \notin S$ and $S = S^{-1}$, the Cayley graph of $H$ with respect to $S$, denoted by $\Cay(H,S)$, is the graph with vertex set $H$ and edge set $\{ \{h, sh\}: h \in H, s \in S \}$. The automorphism group of a Cayley graph $\Cay(H,S)$ contains the right regular representation $R(H)$ as a subgroup, whence all Cayley graphs are vertex-transitive (cf. \cite{Biggs:1993}). Let $S$ be a set of transpositions in the symmetric group $S_n$. The transposition graph of $S$, denoted by $T(S)$, is the graph with vertex set $[n] = \{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in S$. A set $S$ of transpositions in $S_n$ generate $S_n$ if and only if the transposition graph $T(S)$ is connected (cf. \cite{Godsil:Royle:2001}).
If $S$ is a set of transpositions in $S_n$, then the Cayley graph $\Cay(S_n,S)$ is called a Cayley graph generated by transpositions. The family of Cayley graphs generated by transpositions has been well-studied because it is a suitable topology for consideration in interconnection networks (cf. \cite{Heydemann:1997}, \cite{Lakshmivarahan:etal:1993} for surveys). This family of graphs has better degree-diameter properties than the hypercube \cite{Akers:Krishnamurthy:1989}. The automorphism group of Cayley graphs generated by transpositions has also been determined in some cases (cf. \cite{Feng:2006}, \cite{Ganesan:DM:2013}, \cite{Ganesan:JACO}, \cite{Zhang:Huang:2005}). In the present note, we further study the symmetry properties of $\Cay(S_n,S)$, especially with regards to how symmetry properties of $\Cay(S_n,S)$ depend on the properties of the generating set $S$.
The main result of this note is the following:
\begin{Theorem} \label{thm:main:statement} Let $n \ge 5$.
(a) Let $S, S'$ be sets of transpositions generating $S_n$. Then, the Cayley graphs $\Cay(S_n,S)$ and $\Cay(S_n,S')$ are isomorphic if and only if the transposition graphs $T(S)$ and $T(S')$ are isomorphic.
(b) Let $S$ be a set of transpositions generating $S_n$. Then, the Cayley graph $\Cay(S_n,S)$ is edge-transitive if and only if the transposition graph $T(S)$ is edge-transitive.
\end{Theorem}
\begin{Remark} Three comments and corollaries of Theorem~\ref{thm:main:statement}:
1. The reverse implication of Theorem~\ref{thm:main:statement}(a) is proved in \cite[Theorem 4.5]{Lakshmivarahan:etal:1993}. Parts of Theorem~\ref{thm:main:statement} are stated in Heydemann et al \cite{Heydemann:etal:1999} and Heydemann \cite{Heydemann:1997} without a proof; they attribute the result to unpublished reports. We could not find a proof of Theorem~\ref{thm:main:statement} in the literature.
2. If the transposition graph $T(S)$ is the path graph on $n$ vertices, then the Cayley graph $\Cay(S_n,S)$ is called the bubble-sort graph of dimension $n$. Some of the literature (cf. \cite{Konstantinova:2012} \cite{Latifi:Srimani:1995} \cite{Latifi:Srimani:1996} ) incorrectly assumes the bubble-sort graph is edge-transitive. Since the path graph is not edge-transitive, Theorem~\ref{thm:main:statement}(b) implies that the bubble-sort graph is not edge-transitive.
On the other hand, if $T(S)$ is the complete graph $K_n$, the cycle $C_n$ or the star $K_{1,n-1}$, then the corresponding Cayley graphs $\Cay(S_n,S)$, which are referred to as the complete transposition graph, the modified bubble-sort graph and the star graph, respectively, are edge-transitive because $K_n$, $C_n$ and $K_{1,n-1}$ are edge-transitive.
3. The vertex-connectivity of a connected graph $X$, denoted by $\kappa(X)$, is the minimal number of vertices whose removal disconnects the graph (cf. \cite{Bollobas:1998}). Clearly, $\kappa(X)$ is at most the minimum degree $\delta(X)$. By Menger's theorem \cite{Menger:1927}, graphs with high connectivity have a large number of parallel paths between any two nodes, making communication in such interconnection networks efficient and fault-tolerant. Latifi and Srimani \cite{Latifi:Srimani:1995} \cite{Latifi:Srimani:1996} proved that the complete transposition graphs have vertex-connectivity equal to the minimum degree.
Watkins \cite{Watkins:1970} proved that the vertex-connectivity of a connected edge-transitive graph is maximum possible. Thus, Theorem~\ref{thm:main:statement}(b) (in conjunction with the theorem of Watkins \cite{Watkins:1970}) gives another proof that many families of graphs, including the complete transposition graphs, modified bubble-sort graphs and the star graphs, have vertex-connectivity that is maximum possible.
Incidentally, Mader \cite{Mader:1970} showed that if $X$ is a connected vertex-transitive graph that does not contain a $K_4$, then $X$ has vertex-connectivity equal to its minimum degree. Since all Cayley graphs generated by transpositions are bipartite, they do not contain a $K_4$, and so all connected Cayley graphs generated by transpositions have vertex-connectivity maximum possible. This gives an independent proof of the optimal vertex-connectivity of connected Cayley graphs generated by transpositions.
\qed
\end{Remark}
\section{Preliminaries}
Let $X = (V,E)$ be a graph. The line graph of $X$, denoted by $L(X)$, is the graph with vertex set $E$, and $e, f \in E(X)$ are adjacent vertices in $L(X)$ iff $e,f$ are incident edges in $X$. If two graphs are isomorphic, then clearly their line graphs are isomorphic. A natural question is the following: if $X$ and $Y$ are connected graphs with isomorphic line graphs, are $X$ and $Y$ also isomorphic? Whitney \cite{Whitney:1932} showed that the answer is in the affirmative, unless one of $X$ or $Y$ is $K_3$ and the other is $K_{1,3}$. Every automorphism of a graph induces an automorphism of the line graph. Whitney \cite{Whitney:1932} showed that we can go in the reverse direction: if $T$ is a connected graph on 5 or more vertices, then every automorphism of the line graph $L(T)$ is induced by a unique automorphism of $T$.
\begin{Theorem} \label{thm:Whitney:graph:linegraph:sameautgroup} (Whitney \cite{Whitney:1932}, Sabidussi \cite{Sabidussi:1961})
Let $T$ be a connected graph on 5 or more vertices. Then, every automorphism of the line graph $L(T)$ is induced by a unique automorphism of $T$, and the automorphism group of $T$ and of $L(T)$ are isomorphic.
\end{Theorem}
In the sequel, we shall refer to the following result due to Feng \cite{Feng:2006} and its proof (the proof sketch is given below).
\begin{Theorem} \label{thm:Feng:Aut:Sn:S:equals:AutTS}(Feng \cite{Feng:2006})
Let $S$ be a set of transpositions in $S_n$ ($n \ge 3$). Then, the group of automorphisms of $S_n$ that fixes $S$ setwise is isomorphic to the automorphism group of the transposition graph of $S$, i.e., $\Aut(S_n,S) \cong \Aut(T(S))$.
\end{Theorem}
\noindent \emph{Proof sketch:}
In the proof of this result, the bijective correspondence between $\Aut(S_n,S)$ and $\Aut(T(S))$ is as follows. If $g \in S_n$ is an automorphism of the transposition graph $T(S)$, then conjugation by $g$, denoted by $c_g$, is the corresponding element in $\Aut(S_n,S)$. In the other direction, every element in $\Aut(S_n,S)$ is conjugation $c_g$ by some element $g \in S_n$, and it can be shown that if $c_g \in \Aut(S_n,S)$, then $g \in \Aut(T(S))$.
\qed
We shall also refer to the following result.
\begin{Proposition} \label{prop:Ge:restrictedtoS:is:in:AutLT} (Ganesan \cite{Ganesan:JACO})
Let $S$ be a set of transpositions generating $S_n$ ($n \ge 5$) and let $G$ be the automorphism group of $X = \Cay(S_n,S)$. Let $g \in G_e$. Then, the restriction map $g|_S$ is an automorphism of the line graph of the transposition graph of $S$.
\end{Proposition}
The proof can be found in \cite{Ganesan:JACO}.
\section{Proof of Theorem~\ref{thm:main:statement}}
In this section, we prove both parts of Theorem~\ref{thm:main:statement}.
\begin{Theorem}
Let $S, S'$ be sets of transpositions generating $S_n$ ($n \ge 5$). Then the Cayley graphs $\Cay(S_n,S)$ and $\Cay(S_n,S')$ are isomorphic if and only if the transposition graphs $T(S)$ and $T(S')$ are isomorphic.
\end{Theorem}
\noindent \emph{Proof}:
Let $X = \Cay(S_n,S)$ and $X' = \Cay(S_n,S')$. Suppose $f$ is an isomorphism from the transposition graph $T(S)$ to the transposition graph $T(S')$. We show that the Cayley graphs $X$ and $X'$ are isomorphic. Suppose $f$ takes $i$ to $i'$, for $i \in [n]$. Since $f$ preserves adjacency and nonadjacency, the transposition $(i,j) \in S$ iff $(i',j') \in S'$. Let $\sigma$ be the map from $S_n$ to itself obtained by conjugation by $f$. Denote the image of $g \in S_n$ under the action of $\sigma$ by $g'$. Since $f$ is an isomorphism, it takes the edge set of $T(S)$ to the edge set of $T(S')$. Hence, $S^\sigma = S$.
We show that $\sigma: V(X) \rightarrow V(X')$ is an isomorphism from $X$ to $X'$. Suppose vertices $g,h$ are adjacent in $X$. Then there exists an $s \in S$ such that $sg=h$. Applying $\sigma$ to both sides, we get that $(sg)^\sigma = h^\sigma$, whence $s'g' = h'$. Note that $s' \in S'$. Hence, vertices $g'$ and $h'$ are adjacent in $X'$. By applying $\sigma^{-1}$ to both sides, we get the converse that if $g',h'$ are adjacent vertices in $X'$, then $g,h$ are adjacent vertices in $X$. We have shown that $X$ and $X'$ are isomorphic.
Now suppose the Cayley graphs $X$ and $X'$ are isomorphic, and let $f: V(X) \rightarrow V(X')$ be an isomorphism. Since $X'$ admits the right regular representation $R(S_n)$ as a subgroup of automorphisms, if $f$ takes the identity vertex $e \in V(X)$ to $h' \in V(X')$, then $f$ composed with $r_{h'}^{-1} \in R(S_n)$ takes $e$ to $e$. Therefore, we may assume without loss of generality that the isomorphism $f$ maps the identity vertex of $X$ to the identity vertex of $X'$. The neighbors of $e$ in the Cayley graphs $X$ and $X'$ are $S$ and $S'$, respectively. Hence, $f$ takes $S$ to $S'$. Consider the restriction map $f|_S$. By the proof of Proposition~\ref{prop:Ge:restrictedtoS:is:in:AutLT}, the restriction map is an isomorphism from the line graph of $T(S)$ to the line graph of $T(S')$. Denote these two transposition graphs $T(S), T(S')$ by $T, T'$, respectively, and their line graphs by $L(T), L(T')$, respectively. We have just shown that the line graphs $L(T)$ and $L(T')$ are isomorphic.
Since $S, S'$ generate $S_n$, their transposition graphs $T, T'$, respectively, are connected. Because $X$ and $X'$ are isomorphic, $|E(T)| = |E(T')|$ and $|V(T)| = |V(T')|$. Therefore, it is not possible that one of $T, T'$ is $K_3$ and the other $K_{1,3}$. Since the line graphs $L(T)$ and $L(T')$ are isomorphic, by Whitney's Theorem~\ref{thm:Whitney:graph:linegraph:sameautgroup}, the transposition graphs $T$ and $T'$ are isomorphic.
\qed
\begin{Proposition} \label{prop:Ge:Le:TS}
Let $S$ be a set of transpositions generating $S_n$ ($n \ge 5$). Let $G$ be the automorphism group of $X = \Cay(S_n,S)$ and let $L_e$ denote the set of element in $G_e$ that fixes the vertex $e$ and each of its neighbors. Then, $G_e = L_e \rtimes \Aut(S_n,S)$.
\end{Proposition}
\noindent \emph{Proof}:
Let $g \in G_e$. Then $g |_S$ is an automorphism of the line graph of $T(S)$ (cf. Proposition~\ref{prop:Ge:restrictedtoS:is:in:AutLT}). By Whitney's Theorem~\ref{thm:Whitney:graph:linegraph:sameautgroup}, the automorphism $g|_S$ of the line graph of $T(S)$ is induced by an automorphism $h$ of $T(S)$. Conjugation by $h$, denoted by $c_h$, which is an element of $\Aut(S_n,S)$, has the same action on $S$ as $g$, i.e., $g|_S = c_h|_S$. This implies that $g c_h^{-1} \in L_e$, whence $g \in L_e c_h$. It follows that $G_e$ is contained in $L_e \Aut(S_n,S)$. Clearly $L_e \Aut(S_n,S)$ is contained in $G_e$. Hence, $G_e = L_e \Aut(S_n,S)$.
Since $L_e$ is a normal subgroup of $G_e$ (cf. \cite{Biggs:1993}), it remains to show that $L_e \cap \Aut(S_n,S) = 1$. Each element in $L_e$ fixes $X_1(e)$ pointwise. The only element in $\Aut(S_n,S)$ which fixes $X_1(e)$ pointwise is the trivial permutation of $S_n$ because if $g \in \Aut(S_n,S)$ fixes $X_1(e)$ pointwise, then the restriction map $g|_S$ is a trivial automorphism of the line graph of $T(S)$, and hence is induced by the trivial automorphism $h$ of $T(S)$. Since $g$ is conjugation by $h$ (cf. proof of Theorem~\ref{thm:Feng:Aut:Sn:S:equals:AutTS}), $g=1$. We have shown that the only element in $\Aut(S_n,S)$ which fixes $S$ pointwise is the trivial permutation of $S_n$. It follows that $L_e \cap \Aut(S_n,S)=1$.
\qed
A graph $X=(V,E)$ is said to be arc-transitive if for any two ordered pairs $(u,v), (x,y)$ of adjacent vertices, there is an automorphism $g \in \Aut(X)$ such that $u^g=x$ and $v^g=y$.
\begin{Theorem} \label{thm:edge:trans}
Let $S$ be a set of transpositions generating $S_n$ ($n \ge 5$). Then, the Cayley graph $\Cay(S_n,S)$ is edge-transitive if and only if the transposition graph $T(S)$ is edge-transitive.
\end{Theorem}
\noindent \emph{Proof}:
Suppose the transposition graph $T(S)$ is edge-transitive. Let $G$ be the automorphism group of $X = \Cay(S_n,S)$. To prove $X$ is edge-transitive, it suffices to show that $G_e$ acts transitively on $X_1(e)$. Let $t, k \in X_1(e) = S$. Note that $t,k$ are edges of $T(S)$. By hypothesis, there exists an an automorphism $g \in S_n$ of $T(S)$ that takes edge $t$ to edge $k$. Conjugation by $g$, denoted by $c_g$, is an automorphism of $S_n$ that takes permutation $t \in S_n$ to $k$. Also, $c_g \in \Aut(S_n,S)$ (cf. proof of Theorem~\ref{thm:Feng:Aut:Sn:S:equals:AutTS}).
Since $\Aut(S_n,S) \le G_e$, $G_e$ contains an element $c_g$ which takes $t$ to $k$. It follows that $G_e$ acts transitively on $X_1(e)$.
For the converse, suppose the Cayley graph $\Cay(S_n,S)$ is edge-transitive. Fix $t \in S$. Let $r_t$ be the map from $S_n$ to itself that takes $x$ to $xt$. Observe that $r_t$ takes the arc $(e,t)$ to the arc $(t,e)$ since $t^2=e$. Hence, the Cayley graph $\Cay(S_n,S)$ is arc-transitive. This implies that $G_e$ acts transitively on $X_1(e) = S$. Since $G_e$ acts transitively on $X_1(e)$ and $L_e$ fixes $X_1(e)$ pointwise, the formula $G_e = L_e \Aut(S_n,S)$ (cf. Proposition~\ref{prop:Ge:Le:TS}) implies that $\Aut(S_n,S)$ acts transitively on $X_1(e)$.
Let $t, k$ be two edges of the transposition graph $T(S)$, so that $t, k \in X_1(e)$. By the argument in the previous paragraph, there exists an element $g \in \Aut(S_n,S)$ that takes vertex $t$ of $X$ to vertex $k$ of $X$. By the bijective correspondence between $\Aut(S_n,S)$ and $\Aut(T(S))$ (cf. proof of Theorem~\ref{thm:Feng:Aut:Sn:S:equals:AutTS}), there exists an automorphism $h$ of $T(S)$ such that $g=c_h$, where $c_h$ denotes conjugation by $h$, and such that $h$ takes edge $t$ of the transposition graph to edge $k$. Thus, the set of permutations $\{h \in S_n: c_h \in \Aut(S_n,S)\}$ is contained in $\Aut(T(S))$ and acts transitively on the edges of $T(S)$.
\qed
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Viva Topo! ist ein mehrfach ausgezeichnetes Brettspiel für Kinder ab 4 Jahren. Es wurde von Manfred Ludwig entwickelt und erschien 2003 bei Selecta.
Spielablauf und Material
Vier Mäusefamilien verlassen ihre sicheres Zuhause, um eine Reise zum "Käse-Schlaraffenland" anzutreten. Die Wanderung ist allerdings gefährlich, denn eine Katze folgt den Mäusen und fängt jene, die nicht rechtzeitig das Schlaraffenland oder eine der vier Höhlen entlang des Weges erreichen können.
Das Spielmaterial umfasst 20 Mäuse-Spielfiguren in vier Farben, eine Katzenfigur, 20 Käsestücke in fünf unterschiedlichen Größen, einen speziellen Spielwürfel sowie den Spielplan. Spielfiguren und Käsestücke sind aus Holz gefertigt und in verschiedenen Farben lackiert. Die Figuren haben zudem Ohren aus Filz sowie eine Kordel als Schwanz.
Abhängig von der Spielerzahl erhält jeder Spieler zu Beginn vier oder fünf Mausfiguren einer Farbe, die alle in einem gemeinsamen Unterschlupf starten. Reihum würfeln die Spieler nun mit einem speziellen sechsseitigen Würfel und bewegen eine ihrer Mäuse entsprechend der gewürfelten Augenzahl weiter. Der Weg zum Schlaraffenland führt dabei einmal um den Spielplan herum, die vier Höhlen liegen entlang des Weges jeweils in den Ecken des Spielplans. Die Katze startet mit einigem Rückstand zu den Mäusen und wird immer bewegt, wenn eines der beiden Katzensymbole gewürfelt wurde. Auf ihrem Weg fängt sie alle Mäuse, diese werden aus dem Spiel genommen. Im Verlauf des Spiels bewegt sich die Katze zunehmend schneller, so dass die Reise für die Mäuse immer riskanter wird.
Ziel ist es, die Mäuse entweder ins Schlaraffenland oder in eine der Höhlen zu bringen und dabei möglichst viel Käse zu sammeln: In jedem der Ziele liegen verschieden große Käsestücke, die umso mehr Punkte bringen, je weiter entfernt sich das Ziel vom Unterschlupf befindet. Das Spiel endet, wenn alle Mäuse eines der Ziele erreicht haben oder von der Katze gefangen wurden. Sieger ist, wer bei Spielende den meisten Käse sammeln konnte.
Kritiken und Auszeichnungen
2003 wählte die Jury Viva Topo! zum Kinderspiel des Jahres. Die Redaktion von Spiel des Jahres schrieb im Artikel Kinderspiel des Jahres 2003 – Viva Topo!:
Ebenfalls 2003 wurde Viva Topo! vom österreichischen Spielpreis Spiel der Spiele als "Spielehit für Kinder" sowie vom Japan Boardgame Prize als "The Best Childgame" ("Bestes Kinderspiel") ausgezeichnet.
Auch die Kritiken zu Viva Topo! fielen sehr positiv aus. Rezensenten lobten insbesondere das hochwertige Spielmaterial sowie die einfache Verständlichkeit des Spielprinzips bei gleichzeitig hohem Spaß- und Spannungsfaktor.
Weblinks
Einzelnachweise
Spiel 2003
Kinderspiel
Brettspiel
Kinderspiel des Jahres
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New set of officers of the Council of Filipino Associations in Austria (CFAA) were inducted into office. In a gathering, the new officers of CFAA were sworn in at the Philippine Embassy Hall, Ares Tower, 1220 Vienna.
The Council is headed by Ms. Hilaria Garcia, President of the Igorot Austria.
The Chairman thanked the member organizations of CFAA for the confidence accorded to her.
Philippine Ambassador to Austria, Hon. Maria Cleofe Nativitidad presided over the swearing in of the officers. She expressed hope that the Council will continue to support the programs of the Embassy and the Philippine government for Filipinos in Austria.
The Ambassador likewise recognized CFAA's distinguished service to the Filipino community in Austria.
Outgoing Chairman of the Council of Filipino Associations in Austria, Mr. Elmer Blanco also called for unity with all the organizations and its members and said that everyone can be good leaders, especially if they have become good followers.
The CFAA was formed in the year 2000 as council of organizations in the Filipino community. Its fundamental objective is to unify the different Filipino organizations here in Austria.
Since its birth, the CFAA, in cooperation with the Philippine Embassy, has taken charge of significant and considerable activities of the Filipino community, such as the yearly Philippine Independence Day Celebration and other activities, like the traditional Barrio Fiesta, the Paskong Pinoy or "Filipino Christmas".
Other activities like Philippine cultural shows and painting exhibits have been organized and supported by the CFAA . It has also conducted several symposia, providing information on topics regarding retiring here or in the Philippines, on health insurance, on acquiring properties in the Philippines, on inheritance and taxes, to name a few. Financial support has been endowed to various projects in the Philippines through its member organizations. It also actively participates in the activities of the other organizations in the other parts of Austria.
The Philippine community has always been considered one of the most integrated communities in the mainstream of Austrian society. The CFAA aims to continue its active participation in promoting the objectives of the integration office of the Republic of Austria. The vision of the CFAA is a unified, progressive and peaceful Filipino community in Austria.
|
{
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how can i able to set the read preference as prmiary ?
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1. Title Page
2. Acknowledgments
3. Introduction
4. Understanding Polyamorous Relationships
1. What Is Polyamory?
2. Who Does Polyamory, and Why?
3. Polyamorous Communities in the United States
4. Issues Facing Poly Relationships
5. Polyamorous Families with Children
1. Children in Poly Families
2. Adults in Poly Families
3. Benefits of Polyamorous Family Life
4. Difficulties in Polyamorous Families
5. Overcoming Obstacles
6. Conclusion
7. Appendix A
8. Appendix B
9. Notes
10. Bibliography
11. Index
The Polyamorists
Next Door
The Polyamorists
Next Door
Inside Multiple-Partner
Relationships and Families
Elisabeth Sheff
ROWMAN & LITTLEFIELD PUBLISHERS, INC.
Lanham • Boulder • New York • Toronto • Plymouth, UK
Rowman & Littlefield
4501 Forbes Boulevard, Suite 200, Lanham, Maryland 20706
www.rowman.com
10 Thornbury Road, Plymouth PL6 7PP, United Kingdom
Copyright © 2014 by Rowman & Littlefield
_All rights reserved._ No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without written permission from the publisher, except by a reviewer who may quote passages in a review.
British Library Cataloguing in Publication Information Available
**Library of Congress Cataloging-in-Publication Data**
Sheff, Elisabeth, 1969–
The polyamorists next door : inside multiple-partner relationships and families / Elisabeth Sheff.
pages cm
Includes bibliographical references and index.
ISBN 978-1-4422-2295-3 (cloth : alk. paper)—ISBN 978-1-4422-2296-0 (electronic)
1. Non-monogamous relationships. 2. Open marriage. 3. Families. 4. Sexual ethics. I. Title.
HQ980.A527 2010
306.84—dc23
2013024916
TM The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences Permanence of Paper for Printed Library Materials, ANSI/NISO Z39.48-1992.
Printed in the United States of America
# Introduction
"Ahhh, that was great! I was starved," Dani Warren said, pushing back from the table. In her mid-forties, white, highly educated, middle class, and liberal, Dani looked every inch the poly hippie mom. Next to her sat Lex, one of her husbands, and next to him sat Mike, their mutual husband and third member in the Warren triad. Lex had whipped up a Mexi-Cali feast that the four adults and four children seated around the table had just devoured, and we sat chatting before clearing the table. Mike commented, "The rice was excellent, just the right amount of spice. Thanks for making dinner, Lex." Lex responded "Eli helped. She was my sous Sheff. Get it? Sous chef?" Chuckling over the pun they had made with my last name, the two men smiled at each other and touched hands. "Well thanks to you, too, then, Eli," Mike responded.
* * * * *
In this book you will meet families like the Warrens, who are polyamorous. They are your bankers, information technology specialists, teachers, and dentists. Like your other neighbors, they love their children, still owe on their student loans, forget to floss, and could probably stand to lose a few pounds. The thing that sets them apart from your other neighbors is that they have (or are open to having) multiple romantic partners at the same time and with each other's consent.
Polyamory is not for everyone. Complex, time-consuming, and potentially fraught with emotional booby traps, polyamory is tremendously rewarding for some people and a complete disaster for others. While I explain it in far greater detail later in the book, here I briefly define _polyamory_ as consensual and emotionally intimate nonmonogamous relationships in which both women and men can negotiate to have multiple partners.
This book reports the results of my fifteen-year ethnographic study of polyamorous families with children.[1] I quote these poly folks throughout the book, using pseudonyms for everyone. People with first and last names are members of families I know well, usually because I interviewed them several times over the years and often interviewed many of their family members. People who only have first names are people I know less about because I only interviewed them once or chatted with them at a social event or online.[2] Because I quote many people, and it can be a little confusing, I have included a list in Appendix A of the families I frequently refer to for clarity. There is also Appendix B with more information on my research methods.
Initially, I approached polyamory as a "civilian" rather than a researcher. I was madly in love with a man who wanted to be nonmonogamous, and as an intellectual I try to understand things that frighten me. I was terrified of nonmonogamy, or what I learned in 1995 was called _polyamory_ when I heard a National Public Radio interview with Ryam Nearing, then publisher of the polyamorous magazine _Loving More_.[3] In an effort to master my fear, I sought out the local poly community and began asking members how they managed their multiple-partner relationships. Deep into the graduate school process by then, it eventually became clear to me that the social implications of such an unconventional relationship style would make an ideal dissertation, so I formalized my initial self-serving questions into an official study with the university's Institutional Research Board (IRB) approval in 1996. Sixteen years later, I am almost fully recovered from my near brush with polyamory that drove me to sell my house and move to a different state to run away from disaster, and which expanded my mind, broke my heart, and ended my fifteen-year romantic relationship.
While I do not identify as polyamorous myself, I see it as a legitimate relationship style that can be tremendously rewarding for adults and provide excellent nurturing for children. Most of the evidence I use in this book comes from the many wonderful people who volunteered their time and energy to participate in interviews, though I also include some of my own experiences in chapter 4 because they are emblematic of what can happen when poly relationships go awry.
Polyamorous families are increasingly common, though fairly little is known about them outside of their own social circles. This book provides the information for people who wish to understand these complex and unusual relationships that are springing up across the United States, Canada, Europe, and Australia. As these families spread, professionals from counselors and therapists or educators and clergy to medical staff and lawyers will need factual information based in sound research to help them serve this growing client base.
## Contemporary Families
Popular opinion among social conservatives in the United States harkens back to an idyllic 1950s family as the ideal familial form, portraying current society as floundering in a state of decay and lamenting a perceived loss or dilution of "the family"—a heterosexual, monogamous, legally married, two-parent, procreational unit that provides children with stable home environments run by a wage-earning father and supported by a mother who is a full-time parent.[4] In truth, families have always been in transition, and shifts toward single-parent and remarried families both cause and are affected by changes in labor markets and other social institutions.[5] The current cultural fascination in the United States with an idyllic vision of "traditional marriage" reinforces a romanticized, patriarchal family that never existed as we pretend it did. Pretending families used to be static institutions that never evolved and only began to change with the sexual revolution of the 1960s creates the false impression that families today are caught in an unprecedented state of chaos.
While "the" family has never been a static institution, changes in family life in the United States accelerated dramatically during the second half of the twentieth century. Most significantly, middle-class women entered the paid workforce en masse, precipitating dramatic shifts in gender norms and marital relationships.[6] Two especially important trends have been the rise in divorce and the subsequent creation of "blended families"[7] and serial monogamy, and the increase in single parenthood through divorce[8] and nonmarital childbirth.[9] For some, this move toward disengaging marriage from traditional gender roles and childbearing restrictions has opened fresh family possibilities, creating new options for people in same-sex relationships,[10] women who become pregnant through donor insemination,[11] and nonmonogamists.
### Nonmonogamies
During the 1970s, academic researchers studied nonmonogamous relationships such as swinging,[12] mate swapping,[13] and open marriage,[14] focusing almost exclusively on open relationships among heterosexual white people. Research on sexually nonexclusive relationships dwindled in the 1980s, as the sexual revolution collided with the spread of the AIDS epidemic and a backlash of political conservatism.[15] It was during this period of social and political turmoil that polyamory emerged as an identity and a familial form.
While polyamorists have written about their relationships and familial experiences,[16] outside of my own research that explains polyamorous parenting strategies[17] and examines the "slippery slope" between same-sex marriage and polyamory,[18] few scholars have studied polyamorous families. Rubin briefly mentions polyamory in his review of family studies in which he documents a decline in the study of nonmonogamous relationships.[19] Bettinger uses a family systems approach to introduce factors that impact a "stable and high functioning gay male polyamorous family" of seven people—five adults and their two teen-aged sons.[20] Using examples from lesbian, gay, and poly families, Riggs explores various possibilities for kinship structures that value children's definitions of and contributions to their families, rather than relying solely on the adults' views of the relationships.[21] Pallotta-Chiarolli examines polyamorous relationships among women and their actively bisexual husbands,[22] and "polyfamilies'" interactions with school systems, detailing the costs of invisibility[23] and the strategies these families use to manage their interactions with school personnel and bureaucracies.[24] In her most recent book, Pallotta-Chiarolli investigates the state of "border families" composed of bisexual members, those in "mixed-orientation" marriages (gay/straight, poly/mono), and polyamorous families with children, concluding that educational programs designed for gay relationships do not sufficiently address issues specific to bisexual or polyamorous students and families.[25]
### Serial Monogamy
The contemporary social reality is that families are changing—very few people in the general population expect to be monogamous in the classical sense of marrying as a virgin and having one sexual partner for their entire lifetimes. Rather, most people establish a monogamous relationship for a period of time with one person, break up,[26] and establish another monogamous relationship with someone else—a cycle called _serial monogamy_. As a consequence, a growing number of people in the United States today have children with one person and then create another family later with another person, thus involving multiple adults in the lives of children. These social shifts make it increasingly important to understand the fluidity of sexuality and families as well, and this research can be instrumental in translating research findings from polyamorous families to educational materials and policies useful to remarried and blended families with multiple adults in monogamous relationships responsible for children.
### Family Resilience
Studies of family resilience emphasize a strengths-based perspective, examining the ways in which families deal with crises and develop adaptive behaviors to navigate the effects of adverse life events.[27] People who study resilient families seek to identify risks and protective mechanisms that help people through adversity, as well as tracking the strategies these families use as they attempt to balance risks with capabilities.[28] Resilient families are in a constant process of creation and recreation as they adapt to changing circumstances, and researchers have identified a number of protective processes that shield families in crises.[29] Two important protective processes are family cohesiveness, or the "balance between family separateness and connectedness," and the degree of flexibility, or the "balance between change and stability."[30] With their extensive communication and habit of tailoring their relationships to suit their needs, we shall see that the polyamorous families in this book generally have high levels of connectedness, flexibility, and resilience.
## Evolution of the Term _Polyamory_
The word _polyamory_ has a rich background. People involved in multiple-partner relationships in the 1960s, 1970s, and early 1980s sought words to express their ideas, found Standard English lacking, and began to create their own words. While the term _polyamory_ was certainly coined by a member of the polyamorous community, exactly which person created the term is a matter of contention. One version claims the word _polyamory_ is an outgrowth of the term _polyfidelity_ , which Judson "Bro" Jud of the Kerista group had coined to mean "faithful to many."[31] Kerista, a polyamorous commune based in San Francisco that existed from 1971 to 1991, was an important element in founding the polyamorous community in the Bay Area and then nationwide. Jud, cofounder of Kerista with Even Eve, intended the term _polyfidelity_ to mean, as a longtime Keristan told me, "closed and committed family units of up to a dozen bonded lovers, sexually faithful (exclusive) with each other."[32] Enacting this ideal for Keristans included creating an "equitable" sleeping schedule in which partners rotated nightly. Officially, Keristans were not to engage in same-sex lovemaking, though this rule was not always observed in practice.
"Janea," a woman who lived at Kerista for a number of years, credits Geo Barnes of Kerista with coining the term _polyfidelity_ during a group discussion. "They were looking for something positive to say rather than use the frequently used 'non-monogamous' term." Janea remembers that the initial term _polyfidelity_ branched to the more inclusive _polyamory_ when Morning Glory Zell-Ravenheart,[33] the "senior wife" of the foundational Ravenheart clan with Oberon Zell-Ravenheart[34]
> came up with the term in the early nineties . . . in reaction to the fact that Kerista coined polyfidelity and it included sexual fidelity to your group—many who were interested in being poly did not want fidelity as part of the form, so they used polyfidelity to describe themselves even if they weren't. This created discord in the community and infighting about what is fidelity, etc. So Morning Glory was part of the group searching for another umbrella term that would include those who wanted to be poly and love their partners, but could include those with or without any agreement to fidelity within a closed circle of lovers.
The Ravenheart website cites the first appearance of the term _polyamory_ in Morning Glory's foundational "A Bouquet of Lovers" (also referred to as "Rules of the Road"), which appeared in an article in _Green Egg_ , a Ravenheart Church of All Worlds publication. Morning Glory was searching for "a simple term to express the idea of having multiple simultaneous sexual/loving relationships without necessarily marrying everyone" and coined the term _polyamory_ to be both an expression of the lifestyle and a more positive way to express what practitioners had previously labeled _responsible nonmonogamy_ , a term that had contentiously evolved into _polyfidelity_. The Ravenheart clan also contributed the term _monamory_ or "love of one" to the polyamorous lexicon to provide an alternative to the cultural conception that monogamy fit all occasions, when in current usage it customarily refers to steady dating rather than simply marriage to one other person.
## Overview of the Book
The book is divided in to three parts. Part I, Understanding Polyamorous Relationships, provides an overview of the relationship style and communities. Chapter 1 defines polyamory and explains the different types of poly relationships, levels of emotional intimacy, and key terms. Chapter 2 explores who does polyamory and why, including demographic characteristics of sample and community members, common rules that structure poly relationships, and the motivations people report for establishing poly relationships. Polyamorous communities are the focus of chapter 3, including a brief history of nonmonogamy, the overlap with other communities, and some characteristics of poly communities. Chapter 4 explains issues facing poly relationships, such as jealousy, sexually transmitted infections, and dealing with stigma.
Part II focuses on polyamorous families with children. Chapter 5 explains how children in polyamorous families fare, and chapter 6 focuses on adults in poly families. Using data from adults and children, chapter 7 explains the benefits to poly family life, such as added resources, emotional intimacy, expanding family support, and being open-minded. Chapter 8 explores the down side of poly families, examining issues such as partners leaving, stigma, and family friction. It is important to note that the disadvantages poly families deal with are the same ones facing serial monogamous families, and that these difficulties are not distinctive to polyamorous families. In chapter 9, poly people reveal the strategies parents and kids use to deal with the disadvantages, such as emotional protection, stigma management, and creating chosen family.
Part III, Conclusions, takes the information from the study and examines how it can be useful to the many people who do not identify as polyamorous. Chapter 10 details the ideas and strategies that poly people use to navigate their complex relationships and the ways in which these techniques can be useful for people in monogamous relationships and families, as well as the policy implications this research indicates.
1.
Please see Appendix B for a more complete discussion of the research methods I used for this study.
2.
I would always ask these people if I could quote them, either in person if we were chatting or by private email if I was quoting them from an online discussion.
3.
_Loving More_ , http://www.lovemore.com/magazine/.
4.
(Popenoe 1996; Waite and Gallagher 2000; Wilson 2002)
5.
(Coontz 1988, 1992, 1998, 2005; Skolnik 1991; Stacey 1996)
6.
(Coontz 1992, 1998, 2005; Skolnik 1991; Stacey 1996)
7.
(Baxter et al. 2005)
8.
(Amato 2001; Hetherington & Stanley-Hagan 1999)
9.
(Bumpass & Raley 1995; Edin & Kefalas 2005)
10.
(Carrington 1999; Stacey 1996, 2003)
11.
(Sullivan 2004)
12.
(Bartell 1970, 1971; Fang 1976; Henshel 1973)
13.
(Denfeld and Gordon 1970; Spanier and Cole 1975)
14.
(Constantine and Constantine 1973; Smith and Smith 1974)
15.
(Rubin 2001)
16.
(Anapol 1997; Anderlini-D'Onofrio 2005; Block 2008; Easton & Liszt 1997; Nearing 1992)
17.
(Sheff 2011)
18.
(Sheff 2011)
19.
(Rubin 2001)
20.
(Bettinger 2005: 106)
21.
(Riggs 2010)
22.
(Pallotta-Chiarolli and Lubowitz 2003)
23.
(Pallotta-Chiarolli 2006)
24.
(Pallotta-Chiarolli 2010a)
25.
(Pallotta-Chiarolli 2010b)
26.
While ideally people in serial monogamous relationships break up with one person before beginning a new relationship with someone else, in practice many people actually establish another relationship prior to breaking up with their current partner.
27.
(Olsson et al. 2003)
28.
(McCubbin & McCubbin 1988)
29.
(Olsson et al. 2003; Patterson 2002)
30.
(Patterson 2002: 240)
31.
See http://www.kerista.com/ for more information on Kerista.
32.
http://www.kerista.com/
33.
http://en.wikipedia.org/wiki/Morning_Glory_Zell-Ravenheart
34.
http://en.wikipedia.org/wiki/Oberon_Zell-Ravenheart
# Acknowledgments
I would like to thank the following people and organizations for their support across the many years I have conducted the Polyamorous Families study and written this book. This book would not have been possible without all of you.
Patricia and Peter Adler; Serena Anderlini-D'Onofrio; the American Association of Sexuality Educators, Counselors, and Therapists (AASECT); Meg Barker; Jonathan, Jordan, Mike, and Tess Berger; Francesca Coin; the Community Academic Alliance for Research on Alternative Sexualities (CARAS); Stephanie Coontz; Dawn Davidson; Kris DeWelde; Denise Donnelly; Shari Dworkin; Maureen Ewell; Alice Fothergill; Terry Gross; Rhett Gayle; Corie Hammers; Ken Haslam; David LaPorta; Ryam Nearing; Maria Pallotta-Chiarolli; Erika Pluhar; Polyfamilies; PolyResearchers; Adina Nack; Donald Reitzes; Erin Ruel; Christine, Colby, Elaine, and Jonathan Sheff; Wendy Simonds; Suzanne Staszak-Silva; Sociologists for Women in Society (SWS); the Society for the Study of Social Problems (SSSP); Cascade and Zhahai Spring; Tristan Taormino; Geraldine Thompson; Robyn Trask; Chandra Ward; Mary Wolf; and the many participants who allowed me to ask them probing questions about their families—thanks for your time and your candor!
Part One
# Understanding Polyamorous Relationships
Chapter 1
# What Is Polyamory?
_Polyamory_ is consensual, openly conducted, multiple-partner relationships in which both men and women have negotiated access to additional partners outside of the traditional committed couple. It is not _polygamy_ (marriage of many) because polyamorists are not always married. Even more importantly, polygamy is almost always practiced as _polygyny_ , or one man married to multiple women. Usually in those relationships, the women are not allowed to have additional male partners and are prohibited from having sex with each other. Polyamory is also not _cheating_ because (ideally) everyone is aware of the other partners—the relationships have been negotiated with rules to structure scheduling and safer-sex agreements. It is also not _swinging_ , which tends to be more focused on sexual variety and less accepting of emotional intimacy. Some swingers in fact negotiate arrangements that prohibit emotional connection or even repeated interaction with the same lover. Polyamory can also overlap with the versions of swinging that allow emotional intimacy, and the intersection between polyamory and swinging is common enough that Ken Haslam (a well-known polyamory activist) coined the term _swolly_ to describe the juncture between the two relationship styles. Chapter 3 discusses the intersection of polyamory and swinging in greater detail.
The gender equality that exists (at least ideally) between women and men distinguishes polyamory from many other forms of nonmonogamies,[1] and this has important implications for where polyamory occurs in the world. Most popular in Australia, Canada, the United States, and Western Europe, polyamory only flourishes where women can be the social equal of the men around them. For many of my respondents, this translated to women actively pursuing the education and professional skills that allowed them to be financially self-sufficient, with or without male partners.
## Quick Facts about Polyamorous People
People who have polyamorous relationships are called _polyamorists_ , and they use the term _poly_ as a noun (a person who is poly engages in polyamorous relationships), an adjective (to describe something that has polyamorous qualities), and an umbrella term that includes polyfidelity, or relationships based in sexual and emotional fidelity among a group larger than a dyad. Some poly people are legally married, and others span a wide range of types and levels of commitment. Some live together, usually in groups of two to five, and others live alone or with roommates. Many have children, some of them from previous monogamous relationships, and others are born into poly households.
Many people in poly communities have a suspicion of institutions and a rebellious streak.[2] This can translate into a rejection of conventional relationships with institutions and a willingness to exist outside of institutions that most take for granted, such as homeschooling and homebirth. In line with this suspicion of institutions, most of my respondents did not practice any religion, but significant minorities were Pagan or Unitarian Universalist, with a smattering of Jews, Buddhists, and Christians. Like other sexual minorities, poly people tend to live in cities or suburban areas where it is easier to meet potential partners and have more privacy than it is in many small towns or rural areas. The polyamorists who have participated in research (mine and other's[3]) tend to be white, well-educated, liberal, and middle to upper-middle class.
Unfortunately, there are no reliable statistics about the number of polyamorous people. Like other sexual minorities, people with poly relationships tend to be closeted because being openly poly, gay, kinky, or otherwise sexually unconventional can have serious consequences, such as the loss of jobs, friends, family, housing, and custody of children. Further complicating the question of the number of poly people is that it is difficult to decide whom to include in the count. Should it include everyone who is in an openly conducted, nonmonogamous relationship, regardless of the boundaries that structure their relationships? Only the people who identify as polyamorous and interact with poly communities? Even with the difficulties inherent in identifying the population, folks on poly Internet sites estimate that between 1.2 and 9.8 million people in the United States are polyamorous and/or nonmonogamous.
Polyamorists hope to create a variety of relationships—both long and short term—that are primarily focused on emotional intimacy, with or without sexual intimacy. Poly relationships have a few distinctive characteristics, including the degree of sexual exclusivity, number of people involved, and level of emotional intimacy and commitment.
## Levels of Sexual Exclusivity
Sexual exclusivity, probably the single most important and distinguishing factor of monogamous relationships, is not expected in polyamorous relationships. Levels of sexual exclusivity, however, are a popular topic of conversation among polyamorous people, and it is the subject of intense negotiation.
Those in polyamorous relationships generally attempt to maintain sexually, and (ideally) emotionally, intimate relationships with no promise of sexual exclusivity. For ease of conversation, people in mainstream poly communities in the United States tend to use _polyamory_ as an umbrella term to encompass the practices of polyamory, polyfidelity, and polysexuality. In this book I will use _polyamory_ and _poly_ interchangeably as the umbrella terms, and I will specify if I mean something else like _polyfidelity_.
### Polyfidelity
_Polyfidelity_ most closely resembles a closed group marriage because, while the people in it might not be married, they do expect the others in the relationship to be sexually exclusive with people inside the relationship group. It differs from polyamory in that _polyfideles_ (the term for someone who is a polyfidelitist) generally expect the people in their group to be sexually exclusive, and polyamorists generally do not. The majority of polyfidelitous groups require that people who want to join their group get tested for sexually transmitted infections (STIs) prior to having sex of any kind with any group member, much less unprotected sex (which requires fluid bonding, see the definition later in this chapter). Members of polyfidelitous groups often see each other as family members, regardless of the degree (or lack) of sexual contact within their relationships. The larger the group is, the more likely it is to have members who do not have sex with each other.
Polyfidelitous groups sometimes experience cheating, when a member sneaks outside of the approved group to have sex with someone else who either has not been tested or approved or who might have been actively disapproved by other group members. While most polyamorists talk about avoiding making rules about how people should feel about each other, some polyfideles express a strong preference that all group members share equal feelings of affection or love for each other member of the group. Such equality seems much easier for smaller groups (especially triads) to maintain, and bigger groups inevitably develop some relationships that are more intense than others. The essential difference between polyamory and polyfidelity is that the polyfideles expect sexual exclusivity within their specific group and the polyamorists do not. Some polyamorists characterized those in polyfidelitous relationships as practicing "monogamy plus" and harboring a "closed-minded and grasping" approach to relationships. Some polyfideles, on the other hand, scorned polyamorists as "swinger wanna-bes" or "just screwing around." Each camp claimed to define the "real" form of polyamory and judged the other's practice as defective.
### Polysexuality
_Polysexuality_ is the practice of having sex with multiple people, either simultaneously (as a form of group sex) or in concurrent, dyadic (two-person) relationships. Depending on whom you talk to, polysexuality can cover a wide range, from dating many people casually or having lots of sex to using public sex environments or attending sex parties and orgies. Ryan, a white man in his late thirties, identified differences between his feelings of polysexuality and polyamory. With polysexuality, "there doesn't have to be any love to have good sex, it can be a carnal connection." He viewed the Judeo-Christian condemnation of multiple-partner relationships as a fundamental rejection of polysexuality. "The Bible is all about polyamory, what else do they mean by 'love your neighbors'? But they get real freaked out by polysexuality, [laughing] that's for sure."
Those who emphasized the distinction between polyamory and polysexuality often asserted that people could not participate in both, simultaneously. Someone who was polyamorous but not polysexual might develop a sexual relationship with one person and polyaffective relationships with others. In this scenario, polysexual persons have sex with many people, but they most likely love only one or no one at all. Others asserted that polyamory and polysexuality could coexist in the same person, and they have different expressions depending on the relationship.
### Polyaffectivity
Many respondents described emotionally intimate, sexually platonic relationships with their partners' partners. Inspired by poly community tradition, I coined the term _polyaffective_ to describe nonsexual relationships among people in polyamorous relationships. Adult polyaffective relationships with other adults appear as cospouses or quasi siblings, and with children as coparents, aunts/uncles, or quasi older siblings. Children's relationships with each appear as quasi sibling, cousin, friend, and rival.
### Poly Geometry
_Poly geometry_ refers to the number of people involved in the relationship and how they are related to each other. The number of people in the relationship can vary from one to many, and it includes _poly singles_ , _open couples_ , _vees_ , _triads_ , _quads_ , _moresomes_ , and _intimate networks_. As the number of people in the relationship gets larger, the relationships become less stable and less common.
### Poly Singles
_Poly singles_ are people with wide-ranging attitudes toward conducting relationships. _Free agents_ prefer to abstain from, or have not yet found, primary partnerships. _Seekers_ are actively dating and searching for the ideal polyamorous relationship. The people who are _poly-for-now_ are usually experimenting with polyamorous relationship styles while they are young, unattached, or recently split up from a serious relationship. Poly-for-nows usually plan to "settle down" in a monogamous relationship eventually but are not yet ready or have not yet found the right person for a serious monogamous relationship. Second in number only to open couples—the most common form of relationship among mainstream poly communities—poly singles are also quite common because newcomers, and those between relationships, are often single.
### Open Couples
The most common form of polyamorous relationship is the _open couple_. Most open couples in mainstream poly communities are composed of a woman and a man in a committed, long-term relationship, who often live together (some married, others unmarried) and who date other people in addition to their primary partner. The defining characteristic of an open couple is that they have (or are willing to have) sexual relationships with people outside the couple. Open couples take many forms, from those who date independently to those who only date as a pair, _poly/mono_ couples, and _fly-throughs_.
Most open couples, such as Summer and Zack, date others individually. Summer and Zack, both white, college-educated IT professionals, were an open couple who had been together for over thirty years. During their three decades together their relationship had been through many transitions, and some other partners had come and gone, but Summer and Zack stayed together. While they did occasionally cohabitate, most significantly with Jarvis, one of Summer's partners, they usually lived alone together and dated other people. Initially Summer and Zack established a few ground rules that helped their relationship thrive. For instance, if one of them had a lover come over to spend the night, the live-in and visiting lovers would share the guestroom and the other primary would sleep in the main bedroom. Later they came to know each other so well that rules were no longer necessary because they had internalized the ways they figured out to care for the relationship.
Some open couples date exclusively as a couple. Jane, a white woman in her late forties, and her husband Sam, a man in his early sixties, both bisexual, were a polyamorous open couple. They always dated others together, and either one of them could use _veto power_ when choosing lovers. If one person vetoed another lover, then neither one of them could have sex with that other person. Sam has never invoked the veto, but Jane "uses it liberally" when she does not like who Sam has chosen or to "conserve his energy." Jane recognized the imbalance in their use of the veto, but she felt it balanced the relationship in other ways, saying, "I am the brakes and he is the accelerator." Sexuality was central to life for Jane and Sam—both Tantra[4] practitioners and teachers: they had sex with each other at least twice daily as part of their spiritual practice and both wanted daily group sex as well. Their ideal was mutually dating two couples at the same time.
Because open couples are so prevalent in polyamorous communities, there are many discussions about how to best maintain a relationship in light of the pressures, emotional rigor, and relational complexity associated with polyamorous relationships. Numerous support group meetings focus on topics such as integrating another partner, soothing bruised feelings, and suggesting avenues to personal growth that will enable people to deal with jealousy. Polyamorous literature also includes extensive advice and suggestions about the best way to proceed as an open couple.
The _poly/mono_ is a variety of the open couple that includes one polyamorous person and one monogamous person in a serious, long-term relationship, often a marriage. Usually both members of the relationship could have additional partners, but the monogamist usually does not want to (or is physically/medically unable to) have more partners but allows his or her partner/spouse to have multiple relationships.
One such poly/mono couple was Ian and Meredith, a white couple in their early forties, married for twenty-two years. Ian wanted multiple-partner relationships and, while Meredith was reluctant to take outside lovers herself, she said that she was "fine" with Ian seeking additional lovers. Ian said that Meredith "has had a few external liaisons, she would rather use her time and energy to sculpt. That is her real passion, but unfortunately not her day job." Ian "loves her [Meredith] dearly" and spent an average of four nights a week at home with her. He also spent two nights a week with Shawna, his girlfriend with whom he shared many common interests. In explaining his relationship, Ian said that both Meredith and Shawna's wife, Nancy, "love to see us together 'cause it makes us so happy."
Not all poly/mono relationships go so smoothly, however, as evidenced by Sheryl's predicament. Sheryl, a white woman in her late forties, met Jack, a white man in his mid-fifties, while they were contra dancing, and she "instantly fell in love." When Jack explained to Sheryl that he was polyamorous, she initially decided not to get involved with him because "I didn't think I could handle it." Sheryl resolved to see Jack only as a dance partner, but eventually she fell even more deeply in love with him and "just gave in" to an open-couple relationship. "There are really parts of it I enjoy: the honesty, self-knowledge, he treats me great!" Even so, Sheryl remained uncomfortable with Jack's polyamorous sexuality. "When I really think about it, I want him all to myself and it hurts me that he wants to be with other women." Jack was sympathetic and willing to discuss her pain at length, but he was unwilling to be monogamous with anyone. He told me, "She knew what she was getting into that first night. This is who I am."
Some poly/monos deal with the difficulty of blending two relationship styles by using a "don't ask, don't tell" (DADT) strategy. These are romantic relationships in which the monogamous partner allows the polyamorous partner additional relationships but does not want to hear any of the details or meet the other partners. Some people in DADT relationships encourage their partners to have sexual liaisons only while traveling, others allow any type of sexual relationship as long as safer-sex protocols are observed, and still others sanction only online relationships, as long as participants never meet in person.
The Campo family had an important poly/mono relationship that has been ongoing for many years. Campo family members include: Lexi, a mother of one with two family partners; Samuel, Lexi's husband; Blake, Lexi's cohabitational partner; Zina, Samuel and Lexi's adolescent daughter; and Dia and Brian, Lexi's parents. Samuel, Dia, and Brian lived with Zina on a farm about an hour outside a liberal town in the Pacific Northwestern portion of the United States where Lexi shared a home with Blake. Blake and Lexi routinely visited the farm, and Zina frequently came in to town to stay with her mother and Blake, though Brian, Dia, and Samuel did not come to town as often. All white and with varying degrees of education, the Campos were struggling to make ends meet and sustain two households, although the income Lexi was able to earn at the job she kept by living in town subsidized the farm significantly. Samuel was the swim coach and manager of a local pool, and Brian and Dia cared for foster children on their ranch, along with many animals.
The Campo family had evolved slowly into its current configuration. Samuel and Lexi had married when they were fairly young, and shortly afterward Samuel was seriously injured. Lexi explained how that changed their lives and impacted their practice of polyamory:
> At one point my husband, Samuel, for three or fourish years, I was monogamous with my husband or celibate when he got in a bad car accident and had a traumatic brain injury that changed his personality dramatically. I kind of consider myself to be in a cosmically arranged marriage. I was initially married to the person I chose as a spouse for about six months, but after his accident he had a marked change in personality and we have become emotionally close again, mutually supportive . . . while he did not request monogamy I felt like the chaos after the accident required that I pull my energy into the relationship . . . I was not seeing anyone else for a while and once I was interested in seeing other people he was capable of requesting me to not do that for a while. Samuel has never asked me not to do this or not to be this way. He has said that will hurt me or that will make me sad, but he has never made a proactive request for me to be something I wasn't. But he was so unhappy at the idea and he was so needy that it just kept being put off for a while. Every four or six months I would stay up all night crying and he said he would support me in my choices but then forget about it until the next time he saw me crying and that went on for about three years. At first he was very much an egocentric, self-centered child for that time after the accident. I came to a decision to delay getting those needs met right after the accident, but let him know that I would have to return to being who I really am at some point.
Years of communicating and reestablishing their relationship allowed Samuel and Lexi to negotiate a poly/mono relationship in which both of their needs were met. Lexi's willingness to delay her desire for poly relationships for over three years gave Samuel the time he needed to heal and learn to trust Lexi as he got to know her again. Their patience and communication has produced positive results, in that the whole family appears to be quite comfortable with their polyamorous arrangement.
When I asked him how he felt about being monogamous while his wife was polyamorous, Samuel at first gave me monosyllabic answers like "fine, great." Later in the interview I returned to that question and Samuel responded:
> Samuel: You are just not going to be happy until I have something bad to say about that, huh? (laughing) It really is fine, I am sincerely OK with this. If I wanted other partners I could have them, but I am just not interested. Sex is not that important to me, and if I want to have sex Lexi is always happy to be with me, so that need is met. The ranch is a lot of work, and I am the only full-time employee of the Parks and Rec in town, so that takes up a lot of time and attention. In the summer I hire people to work at the pool, but the rest of the year it's just me. And I really love coaching [the local swim team]. Zina takes a lot of my time, too, in a good way. It's all good, in fact.
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> Elisabeth: Does it ever get to be a drag, living out here with Lexi's parents while she's in town with Blake?
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> Samuel: No, I get along great with Dia and Brian. Brian and I take care of the ranch, and Dia [who had lost her legs to diabetes and used a wheelchair], and Zina. It all goes fairly smoothly, and I really don't have any complaints. I like living in the country and working with the kids and the animals, this suits me just fine.
Samuel stated firmly that he was comfortable in his poly/mono relationship, and in her own interview Zina echoed his sentiment.
> Elisabeth: How do you feel about your dad being monogamous while your mom is polyamorous?
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> Zina: I don't think about it that much. I don't really see anything wrong with it, I dunno. It's just that, he could have other partners if he wanted to, but he just doesn't or whatever, so, I don't really know why. We don't usually talk about it, occasionally, but not usually. Sometimes when I'm picking on him I will say about some girl or something "you have a crush on her don't you?" And he will be like "I do not," so it usually gets brushed off or whatever. I can tell it doesn't hurt his feelings, it's not a sore spot the way he reacts. It's no big deal. We joke about it. We could talk about it seriously if we wanted to, we just haven't had a need for it.
Lexi and Samuel's poly/mono relationship went remarkably smoothly, more so than most poly/mono relationships of which I was aware. Regarding poly/mono relationships, Blake commented:
> I was part of a poly/mono relationship for eighteen years . . . and will never, ever do it again. There are several folks in the local poly scene who are also in poly/mono relationships, as is one of my sweeties. There seem to be two distinct and radically different kinds of poly/mono relationships. The first one in which a poly person is involved with a mono person, and that monogamous person really, really wishes the polyamorous person were monogamous too. This is the type I was in. My ex-wife was monogamous and wanted a monogamous relationship; she agreed to polyamory only reluctantly, and it was a constant source of stress and tension between us. Sadly, the people who got the worst of that tension were those who were luckless enough to be my partners. I've never seen this kind of relationship work out well. I've especially never seen it work out well for any third parties who happen along. There are a zillion ways for a person who's resentful about polyamory to make life miserable for a newcomer without ever quite being direct about it. The second kind of poly/mono relationship is one in which a polyamorous person is involved with a person who is "monogamous" in the sense that he doesn't want any additional partners himself, but is totally fine with the polyamorous person being poly. This is the relationship my partner is in right now with her husband. This kind of relationship can work very well.
In the Campo family's case, everyone involved agreed that the poly/mono configuration was working well for the family. This outcome was possible only because of the care and patience with which Samuel and Lexi had negotiated their agreement.
Another form of open couple is the _fly-through_ —an established couple who become interested in trying polyamory, but their first experience with new partners ends with such disastrous consequences that they do not want to do polyamory any more. Most fly-throughs leave the polyamorous community upon ending their poly relationships. Chapter 4 includes discussion of issues related to open couples, such as couple privilege, veto power, and "unicorn hunting."
### Vees
_Vees_ are relationships among three people, with one member who is intimately connected with each of the two others. The relationship between the other two nonlovers can range from virtual strangers, who are aware of and cordial with each other, through good friends, to enemies. The association between the two nonlovers is not as close as it generally is in a triad, a more intimate polyamorous grouping with three members.
### Triads
_Polyamorous triads_ are generally made up of three adults who are all sexually involved, commonly understood as a _ménage-a-trio._ While occasionally a triad will begin as a threesome, more often triads form when a single person joins an open couple or a larger group loses a member(s). Tina, a thirty-six-year-old white urban planner, has been involved in two separate triads with married couples across a period of several years. The first triad attended social events together and spent time at home with the couple's two children. "It was like I was a member of the family with them, hanging out and folding the laundry and stuff. The sex was not all that important, and didn't happen a lot." The second couple was more focused on Tina as a "sexual accessory. Really, the only reason they wanted me around was for group sexual activities. That did not last for very long." Triads, like any relationship, vary tremendously depending on who is involved, where they live, what kind of resources they have, and how they organize their relationships. Tina's second couple were characteristic of _unicorn hunters_ —a heterosexual man and bisexual or heteroflexible woman looking for a bisexual woman who will (the cliché implies) fit in to the couple's life at their convenience, bringing no additional partners of her own, disappear or pass as a friend when being openly poly might embarrass or inconvenience the couple, and hopefully wants to take care of the children and do the laundry.
In other instances, three people form a cooperative and loving family with sexual relationships between some members and platonic relationships between others. These _polyaffective triads_ are more emotionally intimate than vees, but they are less sexually interactive than polyamorous triads. The Tree polyaffective triad was composed of Bjorn, Gene, Leah, and a son, Will. Leah had a sexual relationship with both men, and the men (each heterosexual) were platonic cohusbands. Polyaffective triads with two men and one woman were some of the most lasting poly relationships I found, and when they broke up the men often helped each other remain in contact with the children (more so than in other families when the men had been lovers).
### Quads
_Quads_ , as the name suggests, are groups of four adults most commonly formed when two couples join, although they can also develop when a triad adds a fourth or a moresome loses a member(s). Notoriously unstable, conventional poly wisdom that "2 + 2 = 3" or "a quad makes a great triad" implies that most quads will lose someone to poly-style divorce.
The formerly monogamous couples of Monique and Edward and Alicia and Ben, all white and in their late thirties or early forties, formed the Mayfield quad when Monique and Edward's two daughters, Josie and Kate, were three and five years old. Monique worked as an administrative assistant, Edward as a computer network designer, and Ben as a music producer. Alicia had previously been injured and was disabled enough that paid work was difficult, but she was able enough to care for Kate, Josie, and many of the household chores. Edward recalled that they had been advised that quads were unstable:
> The whole quad, right, had gotten advice for, well-respected people that were in quads, they generally break down into triads or pairs or whatever, they break down into and of course we were in love so that was just so much gobbledygook. Now, one thing that's interesting about the quad and which where we went to these meetings at least I was so proud to talk about this. You know, you've got to be careful because there are six relationships amongst four people with three people you only have three maybe four depending on how you pattern it out but you have to be careful and make sure everybody's communicating. Well, that's all well and good but we weren't. It is very easy to say, oh yeah, you need to communicate, but communicating and _saying you need to communicate_ [his emphasis] are two entirely different things.
The Mayfields eventually broke up, with Monique and Ben forming a lasting relationship, pained distance between Monique and Alicia, and divorces for both legally married pairs.
In some quads all of the members have sexual relationships with each other either in groups or pairs, but more commonly people have sexual relationships with some and others are platonic. Morgan Majek (a white mother of two and office manager) remembered a quad she had been in when she was in her late twenties with her husband, Carl (a white father of two and real estate developer/city planner), then in his early thirties, and another married couple, Josh and Jessica, both white and in their late thirties who had been married for twelve years. Both heterosexual, Carl and Josh each had sexual relationships with Morgan and Jessica but not each other. The women, sometimes singly and sometimes as a pair, had sexual relationships with each man. They did not, however, tend to have independent sex alone together as a couple.
After two years of good times and bad, they could no longer maintain the emotional stress, and that dissolved the quad. Morgan, however, remained involved with all members for "as long as I could take it, I just hated to let it die!" Eventually, even Morgan gave up on her attempts to reunite the quad and stopped seeing both Josh and Jessica. Her poly journey did not end there, though, and we will hear more about all of these families later in the book.
### Moresomes
_Moresomes_ , groups with five or more adult members, are larger, more fragile, and more complicated than the quads. Jana Founder's moresome, which started with an open couple and progressed from a polyaffective triad and into a moresome, exemplified the tendency of large polyamorous relationships to change over time. Jana, a forty-seven-year-old[5] white editor and mother of one, and Mike, a fifty-two-year-old white writer, married when they were quite young. After several years of monogamy, they opened their relationship and met George, with whom each established a deep connection. The three lived as a polyaffective triad for thirteen years, with sexual relationships between Jana and both men and a platonic relationship between George and Mike. Jana and Mike divorced but continued to cohabitate, and they remained emotionally and sexually intimate. Jana reported, "We didn't want any of us more connected than the other, and with a marriage between me and Mike it seemed like our relationship was somehow more important than the one we had with George, and it wasn't really. So we got divorced, but nothing really changed."
Eventually Jana met Sam, a fifty-three-year-old white computer consultant. Sam and Jana maintained a long-distance relationship for several years until they decided they were serious enough to attempt cohabitation. In the meantime, Mike met Michelle, a white forty-six-year-old writer, and they established a similarly serious relationship. The five of them founded a family, with spousal relationships between Jana and Sam, George, and Mike, and also between Michelle and Mike. Michelle and Jana had a polyaffective relationship, as did Mike, George, and Sam. Jana and Sam had a child, and the moresome remained together through Zachariah's birth.
After a year of motherhood, Jana felt raising Zachariah while maintaining such a complex web of relationships was taking a toll on her, so she, Sam, and Zachariah moved out of the home they had shared with the entire moresome. Michelle and Mike moved a few blocks away and maintained close contact with the rest of the family. George lived with Sam and Jana for six months of every year, spending the other six months with a lover in Hawaii. Zachariah saw Michelle, George, and Mike regularly, and he continued to think of them as his family. Although the moresome changed over time, the core family connection remained.
### Intimate Network
One step larger than a moresome, an _intimate network_ is a group of closely connected people who do not generally cohabitate as a unit (though some segments might cohabitate) and are sexually intimate with various group members. Some intimate networks consider themselves families, though most do not. Intimate networks often have entrance procedures that include disclosure of, and testing for, sexually transmitted infections (STIs). These procedures include a discussion of appropriate precautions against transmission, as well as the specific norms and boundaries that structure member's expanded relationships.
Thaddeus, a thirty-five-year-old white musician, remembered how an intimate network of twenty men[6] introduced him to the polyamorous community:
> About twenty men in a relationship and I was _astounded_ and it was a good thing that this was a closed relationship because if it'd been open I would have tried to dive in and, at seventeen [years old], would have ended up being something of a bull in a china shop. I'm sure. It was beautiful to see. They were all over thirty-five and I was seventeen and obviously it just would not have functioned probably to their expectations. I think it would have been beyond mine. They had that number because a lot of them traveled and didn't want to play around on the side because it's [the AIDS epidemic] scary. And the reason why I think it would have been a bull in a china shop situation is that I wasn't stable. I had been coming from a rather difficult childhood. I would have invested an awful lot of [pause] an awful lot of energy in just trying to be whatever they would have needed me to be. And with twenty of them, all of them are highly attractive men I just feel like I would have torn a swath and caused problems and that's been something I've worked most of my life not to do.
While Thaddeus admired this network of men, there was an undertone of caution in his tale. That a visiting adolescent could "tear a swath" underlines his belief that the expanded relationship between the men are fragile, a delicately balanced unit that could inadvertently be destroyed.
Most large networks maintain brief (usually several months to several years) stability, and then membership changes. I interacted briefly with a member of another intimate network of twenty gay men in the San Francisco Bay area that owned a large house together. Two members had recently moved out to cohabit monogamously. The remaining men began searching for two new members. The member in attendance at the party quipped, "If you think it is hard to date others as a couple, just think about how hard it is for us to date a couple as eighteen!" He laughed, but his underlying message was clear: maintaining such a large and complex intimate network was a challenging and intricate task.
## Level of Intimacy and Commitment
In addition to the level of sexual exclusivity and numbers of people involved, polyamorists frequently categorized their relationships by their levels of intimacy, commitment, and duration.
### Primary, Secondary, and Tertiary
Community members often use the terms _primary_ , _secondary_ , and (less often) _tertiary_ to describe their varied levels of connection _._ Primary partners—sometimes corresponding to the larger cultural conception of a spouse—usually have long-term relationships, joint finances, cohabit, mutually make major life decisions, and some have children. Secondary partners share an emotional connection but tend to keep their lives more separate than primary partners. Secondaries often discuss major life decisions, but they do not usually make those decisions jointly. They typically have separate finances and residences, and some have less intense emotional connections than do primary partners. Tertiary relationships are usually allotted less time and energy than primary and secondary relationships. Although possibly the first phase of a deeper relationship or an enduring long-distance relationship, more often relationships with tertiary partners are less emotionally intimate and can resemble swinging.
These definitions, like so much in the polyamorous subculture, are subject to varied interpretations by polyamorists. Tina described a group discussion in which her partner, Edward Mayfield, changed his mind about her classification:
> And he was saying that he feels that since he's married that Monica [his wife] should be primary and I should be secondary. I was like, okay, whatever. As much as I was like, I don't know how that's going to work out so well. But it has been working out just fine. And secondary is secondary. You feel like, well, how is this secondary? In what ways is it secondary? So we were having this discussion at a group over at a potluck sometime about how do you define your primaries and your secondaries. And one person was saying I define it by, well, I don't have any secondaries, everybody is a primary. And she was saying—and somebody else was saying well I define it by the amount of time I spend with that person or however many primaries they have. I become a secondary. So after that conversation, Edward turned to me and said well, I guess we must be primaries because we spend a lot of time together.
Similarly, Morgan described the fluidity that defined what she and the other members of her quad thought of as primary or secondary:
> Carl is primary in that we're living together, he supports us, and I feel like, yeah, it's still primary. If we were to all live together, it would be equal I'd have two primaries. Josh feels, I think, I don't know how you can have another primary and not be living together. Because Josh considers me equal with Jessica, but they're still primary. It's because they're living together . . . It's different when you're actually living with someone. You don't, you answer to them differently. Just more, you know, it's more of a primary relationship.
Polys often disagree about specific definitions of primary, secondary, and tertiary; some refuse the distinctions altogether, preferring what they cast as "less hierarchical" and "more compassionate" terminology.
### Nesting/Non-Nesting
Polyamorous people regularly debate the categorization of relationships. Some view the primary/secondary/tertiary terminology as hierarchical and contrary to their desire for more compassionate forms of relationship. These people prefer _nesting_ and _non-nesting_ to differentiate between partners who live together and others who maintain intimate emotional lives but keep their residences, finances, and decisions separate. Thinking back on her poly triad, Melody Lupine, a thirty-six-year-old white magazine editor and mother of three, asserted that she did not feel that any of her lovers were primary to her, not only because she felt uncomfortable placing some beneath the others, but also:
> My number one relationship is with myself. And one thing I've learned is polyamory helps me, especially as a woman, to keep my autonomy so that I don't lose myself, whether it be within a relationship, like a man or a woman or my children. It helps me to define what I want and set my boundaries and take relationships at what I need.
Joya Starr, a mother of one and a costume designer, feels more comfortable focusing on a certain group synergy than on the type of relationship:
> Very rarely do I "primary partner." It's not my natural bent. I like ones, threes, and fives the best. I like myself a lot. I have been in some long-term triads where the energy for that really flows very well and I think has a stability that I haven't found in other numbers. But I would say my favorite is five because I really love boys together a lot and I don't get that in my triads.
Joya values the quality of interaction between people, and she feels that categories such as primary or non-nesting did not reflect her experience. She largely rejects primary partnerships:
> I haven't found in myself that ability to care for one partner more than my other partners. I've had places where I feel like I'm more expressed with one than the other, but I mean, there were times where it would be like, OK, is it the one that I've been involved with for a decade, is it the one that I had a child with, is it the one that makes my body sing, is it the one who I can talk to and explore the place when you are in conversation about what you don't already know, when you get into that kind of magic kind of talking. And I couldn't for the life of me say that one of those people were more primary to me than the other, like I needed all of that and more, and I felt whole not with one more than the other.
Some polys who reject hierarchical distinctions between lovers discard the notion of primaries relating altogether. For them, relating to lovers around a specific quality makes the differences among primary/secondary and nesting/non-nesting insignificant. These polys often point out that both categorizations relied upon the same distinction: the degree of practical interdependence was the truly important quality, rather than emotional depth.
## Useful Terms
Because polyamory is a relatively recent and unusual relationship innovation, conventional English provides few (if any) words to describe openly conducted, nonmonogamous relationships. Polyamorists have thus been forced to create their own words to reflect and describe their own experiences, several of which appear below.
* _Compersion_ (termed _frubbly_ in the United Kingdom) is the joy at seeing one's partner(s) happily in love with others. It is not precisely the opposite of jealousy, but close.
* _Fluid bonding_ is when people decide that they are willing to exchange bodily fluids during sexual encounters. Poly community norms dictate that, unless otherwise explicitly negotiated, everyone is assumed to be having safer sex in which no fluids will be transferred.
* A _metamour_ or an OSO (Other's Significant Other) is the partner of a partner (a girlfriend's boyfriend), people who do not share a sexual connection with each other but do have a partner in common. Metamours and OSOs are aware of each other and are usually friends or acquaintances, but they occasionally become enemies or rivals.
* _New Relationship Energy_ or _NRE_ is a term coined by Zhahai Stewart[7] to describe the overwhelming rush of love, characteristic of the beginnings of relationships when everything is exciting, new, and exhilarating.
* _Polyaffectivity_ is the term I have coined for emotionally intimate poly relationships that are nonsexual. People in poly relationships who see each other as family members but are not sexually connected (for instance, spice [see below] who share a lover in common but are not lovers themselves) have polyaffective relationships.
* The _polyamorous possibility_ is what I call the mind-set that acknowledges the potential to love multiple people at the same time, or the awareness of polyamory as a relationship option. Once it has occurred to someone that openly conducted, multiple-partner relationships are possible and can be managed in an ethical manner, they can never unthink that idea. They have become aware of the polyamorous possibility and, regardless of whether they consider polyamory themselves or simply reject it out of hand, they can never again be unaware of consensual nonmonogamy as an option.
* _Spice_ are multiple spouses.
* _Swolly_ is a term coined by Ken Haslam[8] to describe the intersection between swinging and polyamory, where the two different styles of nonmonogamy overlap and become difficult to distinguish.
* The _unicorn_ is an unattached bisexual woman who wants to date (or simply have a quick ménage a trios) an existing female/male couple. She is so rare as to be virtually mythical. In her most exaggerated form, she is a young, single woman, eager to move to the couple's dilapidated farm in rural North Dakota to care for their children, work on their farm, clean their house, be their sex toy, and disappear whenever it would be inconvenient to explain her presence to the couples' family or friends.
## Common Guidelines that Structure Polyamorous Relationships
1. _There are no rules_ that everyone has to follow. Each relationship makes their own guidelines, which tend to share a few common themes. Other than that, polyamorists have what one poly person called "designer relationships" in that each group can make their individual relationship whatever they wish it to be.
2. _Tell the truth_. It is impossible to feel safe without trust, and trust flourishes with honest communication.
3. _Communicate, communicate, communicate_. This helps to clarify expectations, manage complexity, and develop intimacy.
4. _If he gets more lovers then so does she_ (and vice versa). While there are poly/mono couples in which one partner is polyamorous and the other is monogamous, most commonly the monogamous people have the option to have other lovers and simply do not avail themselves of it.
5. _Make and follow safer sex agreements_ , or negotiated contracts among lovers stipulating what kinds of sex they can have with other people and how exactly they will protect against sexually transmitted infections. Popular elements include requiring extensive use of prophylactics (condoms, gloves, and dental dams), testing for sexually transmitted infections, and immediate disclosure of any infections.
6. _Take responsibility for self-growth_ , even if it is uncomfortable. Jealousy and insecurity are serious issues for many people in poly relationships, and the desire to exert veto power the moment these feelings become unmanageable can be overwhelming.
7. _Allow for change._ Not only do poly relationships often work out differently than people had anticipated that they might, but they also tend to change over time. If the people in them are not willing to change with them, things fall apart fairly rapidly.
8. And most important of all: _Treat people kindly and live an ethical life._ Or as the popular poly saying goes, "Don't be a dick." This guideline extends to everyone, not just lovers. In general, be willing to give others the benefit of the doubt, assume they are trying their hardest, treat them gently, and let ethical considerations guide behaviors.
1.
For more discussion on the importance of gender in polyamory see Ritchie and Barker 2006; Sheff 2005a, 2005b, and 2006).
2.
(Aviram 2007)
3.
See Sheff and Hammers, "The Privilege of Perversities," in _Psychology & Sexuality _2, no. 3 (2011) for a more comprehensive discussion of the racial and ethnic composition of polys in the United States, Europe, and Australia.
4.
Tantra is a form of sacred sexuality originally associated with a form of medieval Buddhism in India, introduced to the West in the 1960s and popularized in a much-Westernized form in Australia, Canada, Europe, and the United States.
5.
Ages were current at the time of the interview, not the time when the people met.
6.
Although my data includes information regarding two separate moresomes of twenty gay men in each, gay men are rare in the communities I studied, and moresomes this large are even rarer, especially ones that own large houses together. The appearance of two moresomes of gay men who own homes together is anachronistic in the polyamorous communities I studied. My sample is not representative of men who have sex with men. For more detailed information on that population, please see Yip (1997), Connell (2005), and Weeks, Heaphy, and Donovan (2001).
7.
http://aphroweb.net/articles/nre.htm
8.
Ken Haslam is a well-known polyamorous activist who speaks publicly and sponsored the polyamory collection at the Kinsey Library at the University of Indiana, http://www.kinseyinstitute.org/library/haslam.html.
Chapter 2
# Who Does Polyamory, and Why?
Most people in mainstream polyamorous communities in the United States and those who volunteer to participate in research on poly relationships—my own research and others'[1] —are white, middle or upper-middle class, highly educated, and employed in professional fields such as information technology, counseling, or education. The people in my study shared some personality traits that were also common in poly communities, such as being liberal, intellectual, open-minded, geeky, and devoted to social justice. They are aging hippies, young professionals, science fiction enthusiasts obsessed with steampunk, and families with children. Like the general population, the vast majority of polys are _cis-gendered_ , meaning that they identify with the body and gender in which they were born. Most are not _transgendered_ (people whose external sex or gender does not match their internal experiences of themselves) or _intersexed_ (people with a blend of chromosomes or ambiguous genitalia that means they are a mix of both male and female, used to be called hermaphrodite).
## Who Does Polyamory? Who Is Polyamorous?
While these two questions appear to be synonymous, they are actually quite different for polyamorists. Much like some monogamous people grit their teeth and force themselves to ignore or repress attractions for others (doing monogamy) and other monogamous people are simply and profoundly uninterested in other possible romantic partners when they are in a relationship (monogamy by orientation), polyamorists can approach the relationship style in a variety of ways, or combine several approaches at once. Some people _do_ polyamory, meaning they see it as an option, a lifestyle, or even a form of sacred sexuality practice they may choose depending on the circumstances in their lives and relationships. This group envisioned polyamory as a spiritual path based on practices of honesty, self-knowledge, and sacred sex, or a practice that augmented other forms of spirituality such as Paganism, Tantra, Taoism, or Quodoshka.[2] Rosie, the leader of a sacred sexuality group popular among a small group of polyamorists, pointed out:
> The vast majority of polys seek a deeper connection on all levels. They are seekers of knowledge, and sacred sex takes sex beyond, "OK, let's jump in the sack and get laid"—it takes sex to a much deeper connection. . . . Poly people are explorers, they are more open-minded and they seek alternatives to enhance their lives and have more real relationships. Sacred sex does that for them.
These polys craved and created in practice an integration between the Divine and the physical body that acted as a form of spirituality. People who practiced a form of sacred sexuality often linked a personal connection with the Divine to their increasingly intimate emotional and sexual connection with others.
Others saw polyamory as a lifestyle (or a "lovestyle" in poly lingo), a choice that gave polys much greater relationship flexibility than monogamists usually allowed themselves (at least openly). Polyamorists with this belief tended to emphasize freedom as central to their relationships and identities. Emmanuella Ruiz—a university professor and mother of three in her mid-forties—stated:
> It's about what I've been saying, a broader base of community can be built and shared and that if I choose to share my body with people within that constellation, that's my choice. And if I don't want to, that's my choice too.
Members of this group often avoided activism and other activities that might draw public attention (and attendant discrimination) to polyamory. Many thought it best to "pass" and had chosen to "blend in" and use their misattributed access to "monogamous privilege" to their own advantage.
For others, polyamory is a _belief or worldview_ based on abundance, multiplicity, and freedom. In some cases, people who are new to the idea of polyamory practice it as a belief before finding a partner with whom they can practice it in action (poly in theory). In other cases, some community leaders view polyamory as a movement intent upon securing equal rights and educating monogamous people regarding polyamorists and the issues they confronted. Those who viewed themselves as activists on behalf of the movement quipped that they were "polyactive." They made a variety of public appearances on radio and television broadcasts, granted newspaper interviews, spoke to public groups, wrote books, hosted conferences and workshops, and edited magazines.
A New York group called Polyamorous NYC was perhaps one of the most publicly visible organizations, having hosted an annual Poly Pride Day at Great Hill in Central Park annually since 2001. Polyamorous NYC hoped to promote awareness and acceptance of polyamory with the pride day, as well as bring community members together in a festival atmosphere. Small by public march standards, there were about one hundred attendees at the first gathering in 2001 and 150 the next year. Some attendees traveled from as far away as Seattle and Kentucky.[3] Finally, some people _are_ polyamorous, meaning it functions as an essential or innate sexual or relationship orientation in their lives.
Members of this latter group who experience polyamory as a sexual or relational orientation often report knowing they wanted multiple partners even as children and feeling profoundly uncomfortable in the monogamous relationships they did attempt. Joya Starr, a costume designer and mother of one, reported that, when she had previously been in monogamous relationships, she
> felt suffocated, as if my skin were crawling and I couldn't breathe. And I always ended up cheating and felt terrible about it. I never meant to hurt anyone but could not wrap my head around how to be exclusive with just one person when there were so many people in the world. By the end of high school I was not making monogamous agreements with anyone anymore, and haven't for the last 20 years now.
Lexi Campo, a white customer service professional and mother of one, also felt that she was polyamorous by orientation: "I have always been poly. When I was four years old I told my caretaker that I would have a wife and we would share three or four husbands. I have always had a multiple-adult ideal in my head." When I asked if she had ever been monogamous, Lexi replied:
> Not really. I was both celibate and monogamous in behavior for sections of our marriage, after my husband, Samuel, had a bad accident that severely injured his brain. For about three or four years I was physically monogamous, though I maintained cyber-relationships with people who were emotionally supportive of me, and that included some cybersex. Samuel knew all about it and was OK with it . . . I have a very high libido and am very motivated by touch and cuddling. At a very basic level I want more than one partner in my life and if I don't have that I don't do as well. More importantly, I like to be surrounded by people and creating a schedule and having dinner as a group, laughing and noisy. On a fundamental level I like being surrounded by that kind of collective energy . . . communal living is very important to me.
Lexi wondered if there was a biological component to her polyamorous orientation, stating that:
> Gayness and bisexuality are common on both sides of my natal families, as is multiple partners relating with mistresses and wives or multiple partners for many important relatives all the way back to 1604 when two brothers both lived with a woman for 40 years, and while it is not clear who slept with whom, they lived as a three-person unit.
Lexi's parents Brian and Dia reported a long-term triadic relationship in their own past, supporting Lexi's assertion that "poly might run in my genes." Whether or not there was a genetic component to desire for multiple partners, the fact that Lexi believed that there was reinforced her image of herself as innately polyamorous.
## Community Characteristics
### Age
Most of the people in the study were in their early thirties to mid-sixties, though there are certainly older and younger segments of the polyamorous population. This age distribution makes sense for at least three reasons. First, my focus has been on families with children, and most middle-class families with kids in the United States today have parents that are in their thirties and forties. Second, the poly pioneers of the 1960s and 1970s (what I call the second wave of polyamory) are still around, and some of them are eager to talk to researchers because they are pleased to see polyamory finally getting serious attention. Third, because poly relationships are outside the norm and have to be consciously negotiated, it often takes people a while to try them. Many people follow social conventions early in life out of ingrained training, lack of power to make other choices, or sheer habit. Discomfort with social norms or desire for alternatives may take years to germinate and grow in to taking action toward stepping outside of accepted norms and values. Because polyamory challenges one of the most cherished contemporary social norms—monogamy—it can be intimidating to broach and is often not the first step in a journey beyond conventional social bounds.
The _polygeezers_ , organized by Ken Haslam,[4] who coined the term, are a growing group of older polys, some of whom have been having multiple-partner relationships for many years and others of whom embark upon poly relations once their adult children are distracted with their own lives or after losing a spouse to death or divorce. As the baby boomers age and polyamory continues to grow in popularity, I predict that this segment of the poly population will expand dramatically. Retirement communities and nursing homes have already begun dealing with issues generated by residents in poly (as well as same-sex) relationships.[5]
People in their twenties are certainly practicing polyamory, and some consciously identify it as such, but they have not been as interested in participating in my research and tend not to have children, so they were not my focus. As I will explain later with sexual orientation, if a group already has a built-in mechanism to deal with nonmonogamy, then they are less likely to emerge as a significant subpopulation in poly communities. The college hook-up culture is so common among white, middle-class youth that it can make a polyamorous identity, with its tinge of families with kids and aging hippies, less compelling for many people in their twenties.
### Gender
One of the most distinguishing characteristics of polyamory is that it allows women multiple partners. Across time, most multiple-partner relationship styles have allowed men multiple wives (polygyny) but only rarely do women "get" multiple husbands (polyandry). When a woman is married to multiple men, they tend to be brothers or some other form of a preestablished social group, and the woman is often required to perform wifely tasks (cooking, cleaning, sex, childrearing) for them all.[6] Far more frequently, polygamy is organized as polygyny, which is also more common than monogamy across history and various cultures.
This emphasis on gender equality has significant implications for poly relationships. Although there are certainly important poly men in communities and as researchers, women have historically dominated leadership positions in both poly communities and in academic circles that study these groups. While equality is a complex and elusive ideal that can be difficult to achieve or sustain, in practice poly women often do have equal—and in some instances greater—power than either monogamous women or poly men. Much of this is class based: poly women tend to be highly educated and frequently able to support themselves financially, which gives them the autonomy to contemplate the end of the relationship without the dread of possibly ending up living in their cars with their children. As is also true for economically self-sufficient women in monogamous relationships, the ability to leave means that poly women are much less likely to tolerate objectionable relationship conditions. Being able to set firm boundaries and make relationship requirements is both a source and an expression of power that is denied women in most traditional or patriarchal marriages, and even more so in most contemporary polygynous marriages that are often based in religious systems that require women to submit to male dominance.
In addition to social class, poly women have the poly-specific advantage of being more highly sought-after partners. Whether a result of the enduring sexual double standard that allows men far greater sexual latitude than women or a biological propensity that compels them to spread their seed, men seem more willing to have multiple partners. While women outnumber men in a few poly communities, in general there are fewer women available to spark new relationships. This comparative rarity can provide poly women with a numerical and social advantage because they have more options than their male counterparts.
Bisexuality plays a significant role in poly women's social advantage, as well. The ability to be attracted to both genders not only allows the women to consider a far wider range of potential partners but also gives bisexual women the added social cache of being the "hot bi babe" (or HBB)[7] that so many female-male couples want to add to their existing relationships. This social dynamic of a couple—usually a heterosexual man and a woman who is bisexual, bicurious, or heteroflexible—looking for a bisexual woman is so common in poly circles that it is cliché. These eligible bisexual women are so rare that polys (and swingers) call them "unicorns," and the couples who want to hook-up and/or establish relationships with them are labeled "unicorn hunters." The cliché holds that (most often at the male's behest) unicorn hunters will approach a poly community either in person or online with a list of what they seek in a female partner. Usually this includes someone who is bisexual, unattached (meaning not bringing in her own partners as well), wants to help raise the children, clean the house, have sex with the couple, and disappear when it would be inconvenient to explain her presence. Poly community responses to these gaffes vary from encouraging the unicorn hunters to think about why the women would want them and what they have to offer in addition to what they want from her, to "flaming" in which the unicorn hunters are shamed and badgered until they leave (this happens primarily online). Chapter 4 addresses these issues in greater detail.
The men who are drawn to poly relationships tend to be interested in social justice and egalitarian relationships. Because polyamory contrasts sharply with other multiple-partner relationship styles in which men are allowed multiple women but are not required to share their female partners with other men, the men who select polyamory as an alternative to monogamy or polygyny must be willing to step outside of an ownership model. Not that all poly men are paragons of equality or that the ideal always lives in reality, but to be willing to consider sharing a woman with another man requires a flexible personality and openness to equality that goes against conventional masculinity that demands that "real" men have unquestioned, exclusive access to "their" women.[8] The poly men I interviewed often did not insist on sexual ownership of women, tended to be liberal, open-minded, and invested in social and gender justice (something I call _polyhegemonic masculinity_ ).[9]
### Sexual Orientation
While there is considerable variation, in my research and in mainstream poly communities there are mostly heterosexual men and bisexual women, with a significant minority of heterosexual women and a smaller minority of bisexual men.[10] The dominance of heterosexuality among men mirrors larger society in which most people identify as heterosexual. Bisexual men's comparatively lower status also mirrors both monogamous and swinging cultures in which women's bisexuality is highly valued as entertaining to men who, as Marlie, a thirty-seven-year-old white woman, put it, "get to watch while women warm each other up and then come in and finish them off with the big penis-inator." In sharp contrast, male bisexuality is cast as threatening to heterosexual masculinity and unappealing to women.
Another reason for the emphasis on heterosexuality and bisexuality is that both of these sexual orientations have traditionally been less developed as identities and thus do not already have an existing social niche. As the social norm, heterosexuality is so dominant that it remains unquestioned and is rarely taken as a primary identity. Much like whiteness masquerades as a nonrace, heterosexuality usually blends in to the social background unless something specifically emphasizes it. Bisexuality is often invisible, mistaken as homo or heterosexuality, and historically marginalized from or absorbed by gay and lesbian communities. Heterosexual and bisexual people who seek community based on shared sexual or relationship proclivity are required to build it for themselves, and they do so following a similar route blazed by gays and lesbians (who in turn patterned their political movement on the civil rights movement).
There are few exclusively same-sex relationships among polyamorous men, primarily because, as a friend once told me, "Gay men invented open relationships, we don't need another label to do what we are already doing." Lesbians also have nonmonogamous norms and groups already within their own communities, and they may be reluctant to join poly gatherings because of the potential to encounter unwanted male attention. I discuss reasons for the relative lack of people in exclusively same-sex relationships in greater detail in chapter 3.
### Race and Class
Having unconventional relationships can be dangerous: The stigma associated with nonmonogamy means that being exposed as a polyamorist can result not only in strained relationships with families of origin and friends but also job loss, eviction from housing, and losing custody of children. For white, middle-class people, race and class privileges shield them from some of the potential impacts of nonconformity and provide resources to deal with disadvantages or discrimination. People already laboring under the disadvantages of poverty and racism—externally fixed social realities that are difficult or impossible to change—are less likely to be willing or able to take on additional stigma voluntarily. Similarly, those who receive public assistance are under far more surveillance than are those who do not. Living in public housing or receiving food stamps means disclosing roommates, partners, and relationships to authorities, and living with multiple partners (even secretly) can result in the denial of housing or monetary benefits.
People with enough money to own their homes, attain the kind of education that makes them indispensible at work or able to be self-employed, and sufficient funds to hire good lawyers in case their mother-in-law seeks custody of their children have the latitude to take the risks associated with voluntary nonconformity. White people can do whatever they want without worrying about being seen as examples of their entire race, whereas, stereotypically, people of color are already labeled as dangerously hypersexual and run the risk of being used as proof of how "those people" have loose morals or bad values.[11]
I began this research in Colorado, a state that is predominantly white, and my initial sample was composed largely of white people with a few people of color. In an attempt to get a more diverse sample, I traveled to the California Bay Area to collect data in one of the most diverse areas of the nation with one of the largest known poly communities. In defiance of my best efforts, even in the Bay Area I found mostly white, middle-class poly people, and it has been difficult for me to find respondents of color to interview. While this seems as if it is beginning to shift now, there just weren't that many people of color in mainstream polyamorous communities when I started this research, and those people of color who are involved might be less willing to participate in research than their white counterparts who volunteered in droves.
The people of color who did participate in interviews gave several reasons why there might be so few people of color in mainstream poly communities. In addition to the dangers of surveillance and risks of job or housing loss, people of color risk rejection from their ethnic or racial communities and the possibility that white people in poly communities will stereotype or objectify them. Yansa, a twenty-nine-year-old kink- and poly-identified African American[12] health care provider, said she was extremely uncomfortable when she attended a house party in the San Francisco Bay Area in which most attendees were playing in a backyard swimming pool.
> I was not sure if they wanted me there. Like I felt like maybe I had walked in on somebody else's thing and I wasn't invited. . . . [there were] seventy-five, eighty naked people in this huge pool and I walked in and everybody just turned and looked . . . and I realized I am the only black person here. I was the only person in a swimming suit so that could have been another issue, too, like maybe she's lost her way, what is she doing here?
Not only the sole black person at the party, as the only person in a swimsuit Yansa felt as if she stood out even more. Though organizers called the event "clothing optional," meaning that people could wear clothes or not depending on their personal tastes, the community norm was that people were naked while in the swimming pool and wore whatever they wanted (nudity to full dress) on the pool deck.
While Yansa became progressively more comfortable socializing in her local poly community, she became increasingly nervous about the possible backlash from other people if they found out she was polyamorous. At one point she was the only black employee in her section of a large financial institution, and she was certain that if they found out she was poly they would fire her, because her employers were
> executives who went to Wharton and Harvard and were Republicans and assholes . . . very, very closed minded. And I got the impression that they were already not comfortable with me being a person of color. To throw in the other stuff that I did may confirm their stereotypes about black people or they may have just thought she's the weirdest shit on the planet, I don't trust her . . . We don't want her on this job anymore, someone may find her out.
Because she was unique in her environment and already faced the racism that permeates contemporary society in the United States, Yansa felt that she had to carefully guard her secret poly identity lest its disclosure confirm stereotypes her employers most likely held. In this instance, racism acted as a significant deterrent to Yansa's ability to be comfortably out as a poly person.
Not only did Yansa experience forces in conventional society repelling her from identifying as a polyamorist but also she reported discussions with other African Americans who were judgmental of or could not understand her involvement in poly relationships.
> I've heard from black folks that they think it's a nasty white person thing to do. And they throw out the whole scenario of slavery you know they raped us and they took our women and impregnated them and . . . that any respectable, educated, cultured black person in their right mind wouldn't even think about doing something so disgusting.
Not only are poly relationships a "nasty white person thing" and "disgusting" for these "black folks," they are also rife with the potential for sexual stereotype and exploitation.
> I've had black people in the community tell me that they don't want to feel like the token black . . . the novelty like the fat girl or the Asian girl. I don't want to feel like people are attracted to me and wanting to play with me or date me because they're trying to figure out something. Like I'm some anthropological experiment or something.
Yansa reported that the black people she knew rejected poly subcultures as white, foreign, and potentially corrupt social environments in which the white majority might see the black people in attendance as merely sex toys rather than multifaceted potential partners.
Like Yansa, Victor, a poly-identified thirty-six-year-old African American therapist, artist, and college instructor, thought the black people he knew would not approve of polyamory. "I can imagine being in a room of black people and them going that sounds like crazy white folks, that's some crazy shit." Victor felt more comfortable in his local poly community than Yansa felt in hers, and even though it was "monochromatic," Victor was not sure if that resulted from "issues of either privilege or even cultural interest." The fact that nearly all of his poly friends were white did not particularly bother him, partially because he grew up around white people and felt he was "acclimated" to them. Victor also thought that his poly friends seemed less racist than other people he knew in conventional society. "People who are interested in really relating with people and good whole truth telling are going to tend to be less racist . . . I've actually felt a lot of acceptance."
Not only did the apparent lower levels of racism make Victor feel at home in his local poly community, he also said that his education and work background allowed him to experiment with relationships in a way that other people with less privilege might not find accessible.
> It's sort of privilege related . . . if you're not worrying about certain things, then you have the privilege or the space to explore alternatives. . . . the freedom to explore polyamory sort of comes from a freedom either financially or just psychologically not having to [struggle to] survive in other ways.
Victor saw polyamorous relationships as less accessible to people who were forced by racism and poverty to struggle to survive, and he acknowledged that his class and education privileges allowed him to live more comfortably in his poly community.
This is not to say that African Americans are not having multiple-partner relationships: Black people in the United States do have nonmonogamous relationships, but they might be less likely to label them as polyamorous than their white counterparts. Victor thought it was unlikely that mainstream African Americans would embrace an organized poly identity, even though there were "communities of color where there are multipartner relationships going on. I don't know whether they would call it poly or not. Probably not. . . . I think that populations tend to self select."
Mikayla, a twenty-eight-year-old African American woman who worked as an educational consultant and performing artist and identified as bisexual, engaged in what she (in retrospect) saw as poly relationships, though they had not defined them so at the time.
> Deep down I think a lot of African American men are poly and just not open about it. They have the characteristics—do things that poly people do without saying it or admitting it so people call it cheating. I meet so many young men here in Georgia with multiple kids with multiple women, and that is a freakin' poly relationship to me. They can say cheater, when they see many women at the same time, but it sounds like to me deep down you are poly. I guess when you move from cheating to being open is when it can actually be defined as poly.
Mikayla reported that the black people she knew not only routinely engaged in de facto poly relationships but also that they were inflected with an entrenched sexual double standard that allowed men far more latitude than women.
> The act of poly versus the philosophy of poly—no, philosophically they are cheating. But the act—being in multiple homes, living with multiple women or in on and off relationships that sounds to me like poly . . . There is a silence in the African American community so certain things are not communicated but it is probably obvious . . . This is taboo, even though I am doing it. It is fine to do it as long as you don't talk about it . . . There remains a double standard there. I think [my boyfriend] Marlon knows that it happens—the women he is involved with having relationships with other men—but I know there is a double standard. He used to speak frequently about not trusting hoes (women in general and women he was involved with).
There is no question that some people—not only African Americans but also every other race or ethnicity—engage in nonmonogamous relationships but do not identify as polyamorous.
Similarly, there are undoubtedly people who identify as polyamorous but do not attend meetings or join groups. Again, it could be that those who feel marginalized or different from the more "visible" members of poly communities will remain outside the very organizations that purportedly represent their ilk. It is also possible that people of color involved in unconventional sexual practices are just as active but more clandestine and maintain their own, more exclusive, list-serves, events, and private sexual venues. Precisely how these more underground sexual networks might differ from the more visible sexual subcultures requires additional research.[13]
### Who Is Missing?
For the reasons I explore above, the most obvious group missing from mainstream poly communities is people of color, although recent shifts in crowd composition at poly events seem to indicate a trend toward greater racial and ethnic diversity among poly communities in the California Bay Area and the Southeastern United States. If poly communities continue to follow in the wake of LGBTQ communities, they will become increasingly racially and ethnically diverse. Still in short supply, working-class people and those in exclusively same-sex relationships remain scarce in mainstream poly communities. Working class or poor people may not have the time or the finances to pursue multiple-partner relationships because maintaining multiple relationships is so time-consuming that it can interfere with holding multiple jobs, something that many working-class and poor people must do to cope with rising prices and stagnant low wages. Much of the poly community organizing and communicating happens online, and to access the poly sites it is almost crucial to have private high-speed Internet. Libraries often censor sexually explicit materials, and they may define any discussion of sexuality—of which there is plenty on poly websites—as sexually explicit and thus restrict access to these sites on public access computers.
Another group of people who do not appear in these data are conservatives in multiple-partner relationships. Poly communities in the United States have a decidedly liberal tone, and those who are both conservative and poly do not have much social space to espouse their ideals publicly. Occasionally Christian polyamorists will approach online communities and are routinely badgered or shunned as illegitimate polyamorists because (for the most part) the husbands are the only ones allowed multiple wives but the wives are required to be monogamous with the husband. This type of relationship fits better with a polygynous model than a polyamorous one, but some of these husbands identify as polyamorous and approach online groups looking for additional wives—with generally poor results. Other research indicates that swingers are more politically conservative than polyamorous people, and religious polygnists also tend to be far more traditional and patriarchal than do poly folks.[14] Other nonmonogamists who openly maintain multiple-partner relationships but do not identify as polyamorous do not appear in this research. I would have been interested in talking to them, but it was challenging enough to focus on poly-identified people that expanding the study to include other versions of nonmonogamy was beyond the scope of the project.
Another group that is almost completely absent is composed of those people who used to be polyamorous but are no longer. Researchers who conduct studies while at a university (as this study was conducted at two different universities) are bound by ethics rules determined by the university Institutional Research Board (IRB) that protects respondents from abusive or exploitative research practices. Because the IRB at the first university was very nervous about studying sexual minorities in general and defined all nonheterosexuals or people in unconventional relationships as "vulnerable populations," the IRB required me to collect only pseudonyms and did not allow me to keep records of respondents' real names or contact information. While the IRB did this in an effort to protect people's identities, it also made it impossible for me to contact all of the participants in my initial study when I later decided to conduct a longitudinal analysis. The only way I could find previous respondents was to post calls for continued participation on poly Internet sites and via word-of-mouth social networks—both of which would exclude people who no longer interacted with poly communities. The second IRB allowed me to ask for contact information, and I have since been able to track very few people who initially identified as poly for a time and later decided they were no longer polyamorous. Unfortunately, many people who eventually leave a poly lifestyle appear uninterested in spending the time in interviews or possibly fear exposure as a previous polyamorist,[15] and my response rate from former polys is low. This has important implications for my conclusions, as we shall see in part II on families of polyamorists.
Similarly, children are missing from the first two waves of data collection (except for participant observation in which I observed them in community settings but was not allowed to interview them) because I could not attain permission to speak with them. Later, university officials granted me more leeway, and I was finally able to speak to children between five and seventeen years old.[16] Last, the voices of very young children are nearly absent from this book. I did not interview children under five, and interviews with children who were under eight or ten years old tended to be brief and general. Some of them were not aware that they were members of polyamorous families or were unfamiliar with the term _polyamory_ , and I did not discuss the term with them unless they brought it up. Most often, I asked general questions about how these young children felt about the adults in their lives and what kinds of things they did, rather than labeling their relationships or introducing terminology.
## Why Do They Do Polyamory?
Respondents identified six primary reasons for their participation in polyamory, including getting more needs met, more love, sexual variety, family expansion, feeling natural, and rebellion. Most people mentioned multiple reasons why they wanted to have polyamorous relationships.
### More Needs Met
By far the most common reason respondents gave for wanting multiple partners was to get more of their needs met in a more humane way. In fact, getting more needs met is such a central trope in polyamorous discussions that we will return to it in chapter 7, "Benefits of Polyamorous Family Life." Polys point out that loading all relational needs on a single relationship is a recipe for disaster. People end up with unmet needs, and others feel pressured to meet partners' needs that they would rather not have to address. For instance, Kevin and Stephanie, a white couple in their mid-thirties, had been married for seven years when Stephanie got sick of what she called Kevin's "constant whining about me depriving him of BDSM (bondage, discipline, sadomasochism). I just wasn't in to it. I had tried it a few times for him but didn't really like it. We went to a club once and it just freaked me out, but he was really excited. We didn't do anything there, but talked a lot about it for months afterwards and ultimately decided that he could go to the club himself and try it out with people who were more in to it." Initially Stephanie was not interested in dating others, and Kevin reported that it took a "frustratingly long time" for him to find dominant female partners (a rarity in high demand in many kinky social settings), so their new resolve was not tested for months. Eventually Kevin was able to "play" (engage in negotiated and sometimes scripted submissive sexual practices such as bondage or impact play, in which the submissive person may be whipped, caned, flogged, or spanked) with dominant women in the scene, though Stephanie viewed Kevin's kinky play as "his other thing, not really sex in a way because he wouldn't have intercourse with them."
Over time, however, as Kevin continued to play with some of the same women in the local kink community and Stephanie reconnected with Joe, an old boyfriend, they became what Stephanie called "real poly or truly poly, instead of poly in the abstract." Stephanie began attending local theater performances regularly with her new beau and realized she was getting needs met that she had not even known she was missing, and Kevin was "thrilled to finally be living out some of my fantasies with women who are enjoying it too." Both found that they were happier with each other as their needs previously unmet were now satiated. Stephanie commented that "I have more appreciation for him [Kevin] now that I am not feeling so much pressure, and I am happier having companionship at the theater, something Kevin always hated but has brought me and Joe back together."
Kevin told me that he and Joe "hang out occasionally and get along fine. We don't have a ton in common, but he seems like a fine guy and the three of us go out for dinner sometimes. I am fine with her seeing him, but would probably not go out of my way to hang out with him otherwise. But he's not problematic or anything, not like I dislike him or anything, we just probably won't end up being best friends because we don't click like that." Similarly, Stephanie did not spend much time with Kevin's "Dommes" (kink community lingo for a dominant woman, used to be called _dominatrix_ , but that term has fallen out of favor) because their interactions were in such specialized settings that the women rarely crossed paths with Stephanie.
### More Love
Second to getting more of their needs met, poly people mentioned a desire for, and even more importantly a _capacity_ for, more love than can be contained in a dyadic (two-person) relationship. For these people, love could come in the form of companionship, attention, conversation, doing special things together, romantic gestures (notes, flowers), sex, shared jokes, and affection. When asked if adding a partner means they love their initial partner less, many polys respond with an analogy that likens gaining a partner to having another child: The arrival of the sibling does not mean that the parents love the first child any less, but that they have a fresh wellspring of love for the new child and continue to love the first child.
### Sexual Variety
Not only did people like Stephanie and Kevin end up having different kinds of sex with different kinds of partners, they reported having more sex with different people and taking those experiences back to their primary relationship to invigorate a long-term sexual relationship with new excitement, possibilities, skills, or techniques. While the potential for sexual variety often stands out as the most titillating and important part of polyamory in the general public imagination, it is not often the most compelling reason for people who actually engage in long-term polyamorous relationships. Speaking of his motivations for poly relationships, Zack admitted:
> OK, I guess I am kind of a dirty old man, I like sex a lot. And that was more of a motivating factor at first. But over time it became abundantly clear to me that if I wanted a lot of easy sex poly was not the way to go. Waaaaaaaaay too much talking for that. If I just wanted to get laid by lots of different women all the time then I would have been a swinger. That's more what that relationship style is for. But poly is so focused on emotions and relationships that sometimes the sex falls by the wayside. But yeah, I like the sex and that is definitely part of it for me. It brings a certain fire to my relationship with Summer when I have been with someone else, and it does the same when she is with someone else too.
Like Zack, Mark reported that he initially focused on sexual variety as a driving force in initiating polyamorous relationships, "but then I grew up a little bit. In fact, over the years, I have grown up a lot and poly has been instrumental in some monumental personal growth for me. The sex is a bonus, but ironically not the main thing for me anymore. If it ever was, really."
### Family Expansion
Outside of sexual interactions, many poly folks emphasize the importance of chosen family and multiplication of support. While they often retain contact with their _biolegal_ families—or families that are connected through shared parentage and/or marriage—many polyamorists emphasize chosen family as central to their lives. Melody Lupine wanted more children, and her husband, Cristof, was satisfied with the two they already had. Part of the reason Melody was so thrilled when she and Cristof transitioned from a platonic friendship to a romantic polyamorous relationship with their close friend Quentin was that she could have another child, and a "larger family with more love to go around, outside of the framework of one man, one woman." Part II of the book addresses this idea in far greater detail.
### It Feels More Natural
For those who experience polyamory as a sexual orientation or innate characteristic, being polyamorous simply feels more natural or comfortable than monogamous relationships. For instance, in response to my query of "Why be poly?" Blake Campo responded: "Because I can not be any other way." He continued:
> Whatever it is that's supposed to make people want monogamy, I seem to have been born without it. Even as a kid, it never made sense to me. On some fundamental level, I have never, ever understood why someone would want monogamy, either personally or for a partner. The idea of limiting myself to one loving relationship, or asking a partner to limit herself, just plain doesn't work for me, and it never has.
Edward Mayfield asserted that polyamory was a
> natural state of being. As far as I'm concerned, everyone is polyamorous. It's okay, now you start there, you say, okay, what's getting in the way? Is jealousy getting in the way? Is your understanding of what proper moral behavior, is you know, there are those limitations on the behavior, but strictly speaking, everybody wants their needs to be met. Everybody, as far as I can tell, is attracted to more than one person in their lifetime, and in fact would like to be intimate with more than one person in their lifetime, and not just sexually.
People who viewed polyamory as natural often regarded it (and sometimes bisexuality) as the universal human condition that had been perverted or tamed by social controls. This rationale served not only to solidify group cohesion but also to portray those who practiced monogamy as "unnatural" and hence worthy of far greater stigma than the polyamorists themselves.
Some who saw polyamory as a natural or innate quality were less sanguine about their involvement. Lucy, a forty-six-year-old white mother of two, felt some pain in relation to the consequences of her relationship styles.
> I'm poly because I fall in love with people whether I am in another relationship or not, and I can love more than one person at a time without (mostly without) feeling jealous or worried. It was painful to feel love and passion for people I cared about while I was married, and never to be able to do _anything_ about it (except cry and write poetry) because both I and they were monogamously married. I don't want to do that to myself again. If being poly were a "choice" . . . I would have dropped it years ago, because of the judgment and rejection I have experienced from my family and friends. It would not have been worth it.
Others have cultural or social backgrounds that emphasize communality and collective living, and they feel isolated or dislocated when relegated to a dyadic unit cut off or insulated from the larger community or society. Polys in both of these groups report feeling more comfortable in poly relationships than in monogamous relationships, and some emphasize the fact that it is not a choice for them, but they are simply "hardwired this way." Polys who feel most natural in poly relationships are also more likely to see themselves as polyamorous by orientation, something I discuss in greater detail in chapter 1.
### Freedom and Rebellion
Other poly people say that polyamory is a choice for them, something they select because it fits with their desire for freedom of self-expression and rebellion against social convention. Shoshanna, a white mother of two in her mid-forties, viewed her engagement in poly relationships as " a way of being I chose . . . I'll admit that my decision to take the choice was a bit political, a bit anti-convention, a bit shit-disturbing." While Shoshanna said she was deeply invested in resisting "the social and gendered constraints of being 'normal' in my relationships," it was not out of mindless revolt, but rather,
> I use the way I live my life as a political tool to some extent, or at least, I want to . . . I resist being ordinary in part, because it traps us and perpetuates inequalities, insecurities and dysfunctions. I like deliberate choice. I like thought-out decisions. That is the real essence of the shit disturbing. I use the term ironically, actually, because I think it's pretty pathetic that taking conscious ownership of what I do and why is often seen as off the grid. I own my relationships, their functioning, my sexuality, its expression and the impact it has on everyone I am in a relationship with, including my kids and parents. I am highly principled, not just rule flaunting.
For polyamorists in this category, following social convention is stifling, and they prefer to customize their relationships to follow their own life course rather than having convention dictate the form of their relationships. They use poly relationships as a statement on social and sexual liberation.
1.
(Sheff and Hammers 2011)
2.
Polyamorists in the United States have adapted these forms of sacred sexuality; neither traditional Taoism nor Tantra advocate multiple-partner sexuality. There is no traditional form of Quodoshka; it is an amalgamation of several different Native American traditions created in the late twentieth century, practiced primarily by middle-class, white "new agers." Many polyamorists critique Quodoshka as heterocentric and monocentric; still, some have adopted it and seek to adapt it to polyamorous relationships.
3.
http://www.poly-nyc.com/
4.
http://www.kinseyinstitute.org/library/haslam.html
5.
V. Foster, P. C. Clark, M. M. Holsad, and E. O. Burgess, "Factors Associated with Risky Sexual Behaviors in Older Adults," _Journal of the Association of Nurses in AIDS Care_ 23, no. 6 (2012): 487–99. doi:10.1016/j.jana.2011.12.008
6.
Nancy Levine and Joan Silk, "Why Polyandry Fails: Sources of Instability in Polyandrous Marriages," _Current Anthropology_ 38, no. 3 (1997): 375–98.
7.
(Sheff 2005a)
8.
(Connell 2005)
9.
(Sheff 2006)
10.
(Ritchie and Barker, 2006)
11.
(Collins 2000)
12.
In this work I primarily use the term _African American_ when referring to people of African descent, though when the respondents use _black_ I do as well in order to mirror their language.
13.
For a more detailed discussion of the role of race/ethnicity, social class, and education, as well as the impact of research methods on sexuality research, see Sheff and Hammers 2011.
14.
(Gould 2000)
15.
I took scrupulous care to use neutral wording when contacting respondents, especially former respondents who might no longer identify as polyamorous. In the emails I would mention a research study the person had participated in that was associated with a specific university, but nothing regarding the content of that research.
16.
The second university renewed my IRB certification with far more lenient bookkeeping requirements that allowed me to conduct the longitudinal study, but it remained reluctant to grant me permission to talk to children under eighteen years old in poly families. After two additional years of constant revision and reapplication, the IRB summoned me (and my department chair at the time) before the entire board to justify my request to interview children. After yet more revisions and stipulations that one board member privately confided in me seemed "truly insane," the IRB allowed me to finally talk to children in poly families.
Chapter 3
# Polyamorous Communities in the United States
Polyamorous communities exist online and in person, as well as overlapping with several other communities of nonconformists. Contact among community members is especially important to those who want to form and maintain poly relationships, and those people I spoke with identified the advice and support they received from other community members as crucial to their relational successes. Community contact transmits and reinforces norms and values, provides social support, and supplies a place for polyamorists and their children to "be themselves" without having to constantly explain the presence of multiple partners or parents.
## Three Waves of Polyamory: A Select History of Nonmonogamy in the United States
Polyamory is a fairly recent addition to a litany of nonmonogamous relationships, some of which have directly influenced the evolution of polyamorous communities. I divide nonmonogamy and polyamory in the United States into three "waves" occurring in the nineteenth, twentieth, and twenty-first centuries.
### First Wave: Nineteenth-Century Transcendentalism
There were several groups of people who practiced a multiple-partner relationship style in the United States in the mid to late 1800s, most influenced by the nineteenth-century Transcendental movement. Brook Farm was an "experimental free love community" populated by "Quakers, Shakers, Mormons, and other charismatic leaders who roamed up and down the east coast preaching" a doctrine that "challenged conventional Christian doctrines of sin and human unworthiness."[1]
John Humphrey Noyes founded the Oneida community in 1848. Noyes established a system of "complex marriage" in which "each male was theoretically married to each female, and where each regarded the other as either a brother or a sister."[2] This rejection of monogamous marriage was intended to offer an alternative to "the monogamous relation [which] fostered exclusiveness and selfishness, and worked to counter communism."[3] Children similarly lived together in a communal children's house. Parents were not permitted to show special affection to their own children but were instead mandated to treat all children of the community equally.
Finally, Nashoba was a free-love community established in 1862 by Frances Wright, a wealthy Scottish immigrant. Wright formed a large communal farm, "bringing together both free blacks and whites to work and make love."[4] She opposed the racist trend at the time, and she declared "sexual passion the best source of human happiness."[5]
### Second Wave: Twentieth-Century Countercultures
The 1960s and 1970s represented an important period in the evolution of identities that allowed increasing sexual and gender latitude. Feminists included sexual issues such as the repeal of abortion laws and access to safe, legal birth control to their larger agenda of gender equity. Gays and lesbians began to question the hegemony of heterosexuality,[6] and, together with feminists, they exposed gender roles as socially constructed. Transgendered and other transgressive people began to emphasize the performative nature of gender.[7] Bisexuals further destabilized the blend of gender and sexuality by minimizing the importance of their romantic partners' genders.[8] Finally, social and economic conditions contributed to an increase in autonomy for women and sexual minorities, especially gays and lesbians. Industrialization, shrinking families, and the separation of sexuality from procreation enabled women to bear fewer children and gays and lesbians to develop urban enclaves.[9] Polyamory evolved as a direct result of the sexual revolution and intertwined with the alternative sexual forms previously discussed, especially the bisexual and free love movements. Like other aspects of the polyamorous community, the history of the movement has some points of contention.
#### Communes
One form of countercultural group was the commune. The community movement, which had declined in the United States during the late nineteenth century, reemerged in the form of communes in 1960s and 1970s. This second iteration maintained a focus on creating a chosen family for people who were "establishment dropouts, disillusioned with the dominant lifestyles in America; they are people who believe they can find a better way of life in a group living experience with like-minded persons."[10] Communes often emphasized the value of intimate relationships, personal growth, spiritual rebirth, and cooperation over competition, return to nature, and rebellion against the establishment. Many communities included some form of atypical sexuality, from celibacy to free love,[11] though only a minority of contemporary communes endorsed sexually nonexclusive relationships.[12]
Specifically polyamorous communes evolved in the late 1960s and early 1970s. John and Barbara Williamson established the Sandstone community in Los Angeles after the Kirkridge Sexuality Conferences that "served to network polyamorous clergy, researchers, writers, and artists on the East coast."[13] Sandstone was "the encounter-group oriented love community in Topanga Canyon," California, and it included such eminent counterculturalists as Betty Dodson and Sally Binford.[14]
Kerista, possibly the most influential nonmonogamous, protopolyamorous intentional community, was based in the San Francisco Bay Area between 1971 and 1991. Strassberg noted:
> During the twenty-year existence of the community, the approximately twenty-five adult members lived either in separate group marriages or in a single group marriage . . . [Kerista] was based on an experimental lifestyle that included group marriage, shared parenting, total economic sharing, a group growth process, and a utopian plan for improving life around the world by replicating their model of community living.[15]
Members owned and operated a computer sales business. During her tenure there, Ryam Nearing reported living in a community that attempted to provide emotional support for everyone. Nearing participated in seeking a Keristan vision which "started with twelve but later amped it up to twenty-four adults per family in their ideal—the goal they wanted to aim for."[16]
#### "Multilateral" Marriage and Swinging
Two more countercultural groups involved "multilateral" or group marriage and swinging. Research into these nonmonogamous relationships peaked in the early 1970s. By that time, the sexual revolution had popularized sexual experimentation, and the concepts of open and group marriages had gained notoriety. American culture was more sexually permissive than ever before, and the specter of AIDS had not yet destroyed the playful sense of sexual experimentation. Researchers such as Constantine and Constantine[17] studied those involved in "multilateral marriages," which they defined as "three or more partners, each of whom considers him/herself to be married (or committed in a functionally analogous way) to more than one of the other partners." Smith and Smith[18] compiled studies of "sexual alternatives in marriage" in an edited collection that examined such diverse topics as comarital sex (the open incorporation of extramarital sex into marital unions),[19] group sex,[20] infidelity,[21] and group marriage.[22]
Research on swinging similarly flourished in the sexually adventurous 1960s and 1970s, documenting new trends in extramarital or comarital sexual involvement.[23] Studies examined swingers' race and ethnicity,[24] social class,[25] education,[26] and political perspectives.[27] This research created a profile of a swinger as a "white, middle to upper middle class person in his or her late thirties who is fairly conventional in all ways except for her or his lack of religious participation/identification and participates in swinging."[28] Once the sexual revolution collided with the spread of AIDS and other sexually transmitted infections in the 1980s, research on sexually nonexclusive relationships dwindled. Although very few such studies were published during the 1980s and 1990s, the practice of nonmonogamous relationships endured.
#### Support Groups
Informal and organized prototypical polyamorous support groups began to spread in the 1970s, the best known of which were Family Synergy in Los Angeles and Family Tree in Boston. Inspired by Robert Heinlein's _Stranger in a Strange Land_ ,[29] Oberon Zell-Ravenheart founded the Church of All Worlds and its related Ravenheart clan, still influential in the polyamorous movement today. Individuals started organizations focused on polyamory or polyfidelity, such as Ryam Nearing's Polyfidelitous Educational Productions (PEP), a group in Denver called Beyond Monogamy that met regularly and published an edited volume, and Deborah Anapol's IntiNet. Nearing and Anapol later teamed up to create _Loving More_ magazine (which subsequently became Nearing's solo project and has since then transitioned through several editors) that published articles, poetry, and personal advertisements for, by, and about polyamorous people.
### Third Wave: Impact of the Internet
Contemporary research indicates that alternative sexual styles such as polyamory have increased with the advent of Internet technology, which facilitates communication between geographically disparate people seeking support for alternative relationships.[30] In recent years, the Internet has proved an especially important site for community building among marginalized populations. Sexual nonconformists have populated the Internet in droves, forming personal and sexual connections online.[31] The impact of the worldwide web on polyamorous (and other sexual minority) communities would be difficult to overstate. From dating or discussing jealousy to asking for advice, much polyamorous relating occurs "online." The extensive network of Internet communication has spawned an impressive number of polyamorous websites.
While polyamorous websites are too numerous to adequately list here, I have included some of the more important ones as examples of online community. Lovemore.com is _Loving More_ magazine's website. It includes not only a bulletin board but also a chat room, frequently asked questions (FAQ), stories, advice, events, "the love list" (a summary of conversations that transpired on the electronic discussion board that was emailed to list subscribers), and personal ads for those seeking others to engage in polyamorous relationships. Yahoo lists over one hundred polyamorous groups by region and interest, accessible through their "romance and relationships" section or simply by entering the key search word _polyamory_.
Alt.polyamory contains an extensive list of polyamorous information, including six different FAQ pages, a glossary of acronyms, abbreviations, and new words found on polyamorous sites, a list of polyamorous resources including fiction, nonfiction, music, movies, "poly-friendly" professionals such as mental health counselors and ministers willing to perform group marriages, art, and paraphernalia such as T-shirts and mugs. Alt.polyamory also hosts numerous topical email lists for specific subgroups including activists, parents, triads, and those seeking intentional community. Those who wish to post or read personal ads are directed to alt.personals, soc.personals, or alt.personals.poly. The "poly ring" is for members only, and it links diverse polyamorous sites across the web. PolyMatchMaker.com lists personal ads for those seeking polyamorous relationships. It, too, is open to members only, though memberships are free. Finally, numerous polyamorists' personal websites include stories of their polyamorous lifestyles, links to other pages, pictures, poetry, journal entries, artwork, information about upcoming events, and calls to activism.
Polyamorists also link to other related, but not explicitly polyamorous, websites. Janesguide.com, a guide to alternative-sex-oriented sites on the web, is a favorite among web-savvy polyamorists, as is LiveJournal.com—a free site that allows writers to create journals online and choose to make their writing available to select others or to anyone visiting the site. LiveJournal lists over one hundred relevant "community" matches and over 1,300 users interested in polyamory. Sites that contain information about swinging may overlap with polyamorous sites, and the communities share personal ads at www.altdot.com. The polyamorous presence on the web is diverse and serves as a vital component of community formation and participation.
## Characteristics of Polyamorous Communities
Polyamorous communities exist in both a geographic sense and online. Local or geographic communities vary tremendously in their level of organization and political involvement. Smaller communities tend to have very little organization and are virtually nonexistent as political entities. Larger communities seem far better organized, and members are often involved in the politics of polyamory. Community politics happen among activists and longer-term polyamorists who have known each other for many years and for whom polyamory is a central component of their identities. These longer-term polyamorists are the same people who have evolved into community leaders: some are leaders of their local groups and others leaders of the loosely defined national movement. Leaders often live in established centers of polyamorous activity, and new centers grow up around leaders who relocate to new areas. Some who used to be leaders and grew tired of the media attention and community activism have withdrawn from leadership positions.
### Geographic Distribution of Polyamory
Polyamory flourishes primarily in urban population centers. The California Bay Area is home to numerous interlocking groups and several nationally recognized leaders. Several more leaders live in Southern California, with its less organized and more cellular polyamorous communities. Boston, New York, Seattle, and Washington, D.C., are also centers for polyamorous activity.
The Hawaiian Islands, another nexus of polyamorous life, each host their own individual polyamorous communities that vary by size and degree of public acknowledgment. Jane and Sam consider themselves (and were considered so by some) national leaders, as well as leaders of a Hawaiian Island polyamorous community. They characterize their local polyamorous community as the following:
> Most of the poly folks around here are hidden. They don't identify themselves as poly, just live life with multiple relationships. The missionaries have done a good job [preaching monogamy]. Oahu is a different scene. There's a much larger population of polys there.
Both the West and East coasts of the United States emphasize polyamory and sex positivity,[32] while the Midwest hosts a more conservative community focused primarily on polyfidelity.
### Urban and Rural Distribution of Polyamory
Typical of sexual minorities in general, polyamorists who lived near large population centers seemed more successful at finding acceptance and others with whom to practice polyamory. Those in rural settings were isolated[33] and often traveled to nearby urban areas or sought support, advice, and long-distance relationships online in polyamorous cybercommunities.
Bethany and Chad, a white, polyamorous married couple in their mid-thirties and late fifties, respectively, lived in a small agricultural town in the Midwest, roughly a seven-hour drive from a large population center. They tended hogs for a large agribusiness, working long hours to support their five children (two of whom were Chad's from a previous marriage). Even so, they barely had enough money to pay the rent every month, and their second car was repossessed when they could not make the payments, leaving them with a pickup truck as the sole form of transportation for the family of seven. They could not afford Internet access to make contact with online polyamorous communities, and they subsequently suffered intense feelings of isolation, distress, and loneliness.
Internet access was financially out of reach for Bethany and Chad, so their primary contact with the polyamorous community was during an annual polyamorous campout several hundred miles from their home. Bethany confided to me that she felt "at home" during the gatherings in a way she usually did not experience in other parts of her life. "I feel like I can be myself here, not like at home. I cry all the way to [a distant town on their way home] every time we leave one of these things." While Bethany and Chad sought polyamorous community as a way to feel "at home," they were distinct from the majority of community members in many ways. Not only did they live in a rural area but also their agricultural occupations paid little, distinguishing them from the majority of other polyamorists, who were middle and upper-middle class and tended to be able to afford to socialize online frequently.
### Online Communities
While online communities provide education for members of the public, their primary function is to allow polyamorists (and other sexual minorities) to mingle in cybercommunities. Previously isolated people find others on the web, and community becomes redefined as an affective, information-based link, free of geographical boundaries. Virtual communities offer a substitute for live social gatherings as well as a way for those separated geographically to establish a cohesive community identity. Online mingling occurs in a number of forums: bulletin boards, chat rooms, email lists, and personal websites, with a variety of information including frequently asked question pages (FAQs), journal entries, photographs, or directions to homes for polyamorous gatherings.
The social structure of online polyamorous communities contains a variety of types of members and appeared to be similar to other online communities.[34] People are occasionally harsh to one another, but more often they offer support, advice, and solace. For those willing to engage in long-distance relationships, online communities offer a much larger pool of potential dating partners. People interacting with each other on poly websites expect each other to have a certain degree of web savvy—most web users have pseudonyms and often control access to particularly identifying information. Web communities are often cohesive enough to establish a collective sense of identity and thus can mutually agree to ostracize someone who violates community norms, effectively policing their own boundaries.
## Functions of Polyamorous Communities
Poly people often emphasized the vital importance of community to the health of their relationships and individual happiness. To that end, polyamorous peoples' associations fulfilled a number of functions of polyamory. Bringing people together entailed introducing them to the local or cyber community where they would find a selection of dating partners.
### Bringing People Together
Community is especially important for bringing together polys and other sexual minorities who are marginalized from society because they provide role models, a pool of potential partners, and assistance in a world where nonconformists are often targets of stigma and disdain. Community membership offers many benefits, and one of the most important is the ability to conceive of an alternative identity in which the unusual characteristic is acceptable and the support to take on that identity and maintain it. Polyamorous people routinely seek to join or create communities with like-minded others.
Some people search out or stumble upon a polyamorous subculture (usually online, but occasionally in person), and others are recruited by lovers or potential lovers hoping to involve them in poly relationships. Many people discover the idea of multiple relationships through science fiction,[35] and some establish multiple-partner relationships first and come to identify as polyamorous when they later discover the term.
### Learning the Term _Polyamory_
For some polyamorists, introduction to the term _polyamory_ was a monumental occurrence. People who were previously isolated in their desire to experience multiple partnerships talked about feeling as if a new world had opened to them when they found out about the term. For these folks, a relationship style that was previously inconceivable suddenly became plausible when they heard the term and discovered the associated community and identity. This mirrors Weinberg's[36] findings that, for some people, the "discovery of the category [bisexual] in fact existed was a turning point" in how they came to think of themselves as bisexual.
Sybil, a white woman in her late thirties, described her feelings when she first learned the term _polyamory_ from someone who had listed it in her Yahoo! profile.
> I wrote to her and asked what the heck is that? She wrote back and explained it and I was like _oh my god_! You mean you can really _do_ that? There are other people who feel this way! I didn't know people actually did that— _oh my god_! I walked around for the rest of the day just amazed, like holy shit! I didn't know anyone else who was poly yet then, but just knowing it existed made me feel so free.
Sybil explained that she had always desired relationships with multiple partners but had never done so openly because she did not think it possible within the constraints of monogamous society. She had been in few truly monogamous relationships, and most of them had ended when she was caught cheating.
Finding the term _polyamory_ was not as earth shattering for other people who were already familiar with the concept of nonmonogamy. Joya explained that "I started out feeling like I had options and I don't know where those, that came from. Perhaps being an avid sci fi reader as a young person, where people were saying your relationships did not necessarily have to look one way."
Others were already conducting polyamorous relationships when they became aware of the term. Edward Mayfield reported finding himself involved in a polyamorous relationship, and then realizing there was not only a term but also a community of people who lived their lives this way and talked to each other about it.
> We [he and his wife, Monique] met this other couple. I was infatuated by the woman, and Monique seemed to like the guy just fine and, you know, I'm scratching my head a little bit. What's happening here? And we discuss this and we actually went into a relationship. It seemed right . . . and we ended up spending a lot of time with them eventually because of some external stuff and eventually moved in with them for a time and actually stayed and found a place with them and actually moved to [another state] with them. I don't think we were calling it anything in particular but we got involved. It just seemed the right thing to do at the time.
The Mayfields had no plan, no label for their alternative relationship form. They simply realized they were in love and it "seemed the right thing to do."
Others became aware of the concept of polyamory through their social networks. Morgan Majek had heard about polyamory through a Pagan group with whom she worshipped regularly. She had not considered herself polyamorous until
> I met a man named Derek. He started coming to the meetings, and I just clicked instantly and was physically attracted and mentally attracted. He's Pagan and we were able to just talk very easily and communicate very easily. And we started hanging out with each other as friends and then we started falling in love and it just happened over about five or six months. And then we discussed it, Derek and I, because our feelings were getting really, really strong and we discussed trying to begin a relationship because he had heard of polyamory and been to the [Internet] sites and read about it. And so we decided we were gonna tell our spouses.
Morgan's initial dim awareness of polyamory suddenly became much more important once she found herself simultaneously in love with Derek and her husband, Carl. Finding a poly community with a specific word helped not only to label and solidify polyamory as a personal identity but also it became something Morgan and Edward could do or be—a poly person with an identity based on the mutual relationship style of community members.[37]
### Selection of Dating Partners
Polyamorous communities not only provide their members a group of like-minded people from whom to seek potential partners but also have useful information about those people as well. Much like gays, lesbians, and other sexual minorities,[38] polyamorists often encounter difficulty finding other polys, or monogamists willing to explore polyamory. Polyamorous communities offer members access to one another: potential partners that they can depend on to be (in varying degrees) familiar with community norms and values and open to the possibility of multiple-partner relationships.
Polyamorists often found their partners' romantic relationships with nonpolyamorous people problematic, because the nonpolyamorists' ignorance of community norms often negatively impacted the primary polyamorous relationship. These relationships were laden with all of the problems associated with newbie relationships, as well as the additional challenge of incorporating ostensibly monogamous partners. Sometimes a monogamist and a polyamorist could have a clash of the paradigms, with each attempting to get the other to convert. Other people found the lack of common norms problematic. Louise Amore discussed her qualms regarding Max's (her husband at the time) relationship with Elana:
> It was difficult for a lot of reasons. One of which was that I didn't know her. She was married and cheating on her husband, and that just didn't, she just didn't understand how important, she wasn't willing to talk to me. Whereas people who are polyamorous understand the need to talk to each other. . . . If I don't know who he's involved with it's very hard for me, because I don't trust them. . . . Max very much fell in love with this woman and it was, and because she was, I don't like her. She thinks that cheating is a game and she plays with people's lives, and it surprised me that Max would want to get involved with someone like that. . . . I didn't expect her to know our rules or to even respect them because she doesn't respect the rules of her own marriage. How's she gonna respect mine?
Not only were polyamorous community members socialized into shared norms and values such as honesty and communication, they also tended to be more of a "known quantity" as well.
Once polyamorists entered a community and were "road tested" by a number of relationships, other community members gained information about them that allowed more informed decisions regarding the possibility of dating those people. Marilyn, a white woman in her early thirties attending a bisexual coffeehouse gathering in the Bay Area, discussed what she called "partnersharing," the practice of "checking out" partners of friends. "If I see someone who has had a lot of drama in successive relationships then it gives me a red flag, not like I judge them, but it is something to look out for." She gained this valuable information through her interactions with overlapping bisexual and polyamorous communities in the Bay Area.
### Assistance
Polyamorous communities not only helped poly people find each other but they also created a community forum where people could help each other. The quality of social interaction in close-knit poly communities struck me as quite similar to people who share the same church—they may not see each other every day but interact frequently enough that they know that if something happens there is a group of people ready to help them and who they would help in turn. Because most poly people either don't practice any religion or practice an uncommon form of religion (Paganism, Buddhism, and Unitarian Universalism are the most popular), poly communities can provide them with a level of group cohesion and mutual aid that they would otherwise find hard to come by without membership in a close-knit synagogue, church, or temple.
#### Financial Assistance
Polyamorists give each other financial assistance in times of need, especially if it is a need created as a result of a polyamorous relationship. Their local poly community rallied around a couple in the San Francisco Bay Area when they were fired from a local church and lost their employee housing because their polyamorous relationships became public knowledge. Personally and financially devastated, the unemployed couple sent a message to their local polyamorous email list pleading, "We need help, please!" A list member found them free housing (temporarily) in exchange for painting the apartment. Although being outed as polyamorous cost them their jobs and home, their involvement with the local polyamorous community provided them with assistance to ease the effects of discrimination.
Poly people routinely helped each other. Melody Lupine rented a large house that became a center for people seeking connections with the local polyamorous community:
> Many people have come to live with me and they don't have much money to begin with and with the move and a big expense like that, I won't charge them the first month's rent when they're staying with me, but they know once they get back on their feet, they pay me back. So, and I've helped out. Cristof's (one of her husbands) helped me out and I've helped him out, vice versa. There's an ebb and flow with it . . . so it's taught me a lot about how to do that being in a polyamorous community.
Melody connected her generosity directly with her experiences of reciprocity within her local poly community. Things did not always work out well for Melody, however. At least once, people who were living with her moved out in the middle of the night without paying her the rent they owed. They were new to polyamory and had not yet been socialized into the community norms of generosity and honesty. The majority of the polys I met appeared more than willing to offer emotional support, and maybe even financial assistance as well depending on the circumstances.
The national polyamorous community rallied around April Divilbliss when she faced losing custody of her child because she was in a poly relationship.[39] Divilbliss lived for a time in a triadic relationship with two men, neither of whom was the father of her child. When the child's paternal grandmother discovered this, she sought legal custody of her grandchild. Divilbliss moved out of her triadic household and lived alone with her child, but the action satisfied neither her ex mother-in-law nor the courts. She reached out to polyamorous communities[40] who fueled her custody battle with financial and emotional support, but she eventually lost custody of her child.[41] The national polyamorous community collectively viewed Divilbliss's loss as a defeat any nonmonogamous family might have to endure.[42]
#### Emotional Assistance
Even more important than financial assistance, polyamorous community members provided each other emotional assistance in the form of empathy, advice, and a new frame of reference to normalize their lifestyle. Penny and Marcus, both white peace activists in their late forties, were married for twenty-three years. They were very close with Nick and Leah—another white peace activist couple in their late forties—and for fifteen years they helped to raise Nick and Leah's son, Connor. After years of spending three to five days a week together for family activities and socializing, Penny and Nick fell in love. They approached their respective spouses to discuss their feelings. Nick had erroneously anticipated that Leah would welcome Penny as his future lover, because he and Leah had been in a quad years before meeting Penny and Marcus. Instead, Penny said that "Leah freaked out. It was a huge train-wreck; we still have not recovered."
Penny explained that the peace community rallied around Leah, telling her, "Of course this level of distress is appropriate and how kind and understanding she is to put up with our [Penny and Nick's] terrible philandering." Penny complained that no one in the peace community was telling Leah, "You know, this level of jealousy is really problematic, maybe you should seek some help in dealing with this out-of-control insecurity." Feeling ostracized, Penny grieved the absence of support from her usual sources in the peace community whom she had known and socialized with for many years. She attributed this to being viewed as "The baaaaad girl!" Penny expressed her gratitude to polyamorous community members who offered their support during this difficult period. "I am so glad to find you people. I feel like less of a freak now that I know there are actually people who _do_ this. You guys are from the home planet!" Penny's portrayal of other polyamorists as "from the home planet" underlined how thoroughly estranged she felt from her customary social surroundings and how comforted she felt with like-minded people.
Support group meetings offered a forum for polys to discussed their complex relationships in a safe environment where they could expect understanding rather than condemnation. Carlie, a thirty-five-year-old white educator and mother of two, came to a support meeting and tearfully related "the disaster my life has become" since the relationship between the two men she loved degenerated into bitterness and jealousy. She responded to the sympathy and advice offered by the group, "Thanks guys! I needed to hear something else from people besides, 'well, what did you expect to happen, you little slut?'" Members of this support group, like others in polyamorous communities, offered not only advice and support but also a way to normalize the difficult life events in poly life as unpleasant but average family difficulties.
## The (Mostly Unwritten) Social Rules in Poly Communities
In sociology we frequently talk about _norms_ , or the unwritten social rules specific to a society or social group that guide and limit behaviors. When someone breaks the norms and does something unusual, they are being _deviant_. The two concepts are intertwined—norms define deviance as out of bounds, and deviance contrasts with normative behavior to highlight the differences between the two. In other words, we know what is socially correct precisely because we know what is socially incorrect.[43] When people break social rules for behavior they are subject to _stigma_ , or the negative social judgment that comes with having a difference that is "discrediting" and "spoils" the person's reputation.[44]
Poly communities have their own norms that establish guidelines for behaviors and interactions among poly folks, and newbies (people new to polyamory) learn the norms from role models who demonstrate and explain community social conventions. As is true of most groups, poly community norms are founded on idealized visions of polyamory, and some people are not always able to live up to the high standards of the ideal.
Morning Glory Zell-Ravenheart's foundational "A Bouquet of Lovers" (or "Rules of the Road")[45] provides guidelines to structure polyamorous relationships. While not formally enforced, many second-wave polyamorists read and discussed the rules of the road at support groups or social gatherings, and over time they have permeated the community so that they have become collective sensibilities.
Poly communities generally value honesty above anything else, and it is the single most important norm that underlies the others.[46] Long-term polys expect each other to be honest, and they explain the importance of the norm by linking honesty to trust, without which poly relationships do not work. Lying leads to distrust, which undermines the emotional safety that is only possible when people trust each other. Online and in support groups, polys regularly discuss ways to effectively and humanely maintain honest relationships. While those who fail to follow strict codes of honesty are stigmatized, they are not dealt with very harshly because nearly everyone has lied to someone at least once. If the liar later tells the truth and makes amends, polys will generally be understanding of the lapse, though repeat offenders are stigmatized more severely than first timers.
Another key poly community norm requires that everyone in a given relationship be bound by the same rules, regardless of gender: men and women have equal access to outside lovers within the same negotiated agreement. Generally, community members frown on relationships that allow one member access to outside lovers but deny another the same liberty.[47] A related norm involves the idea of equal power between and among relationship partners. Community rhetoric supports equality and casts power sharing as its central component. Some polyamorists successfully attain this idealized norm, and others struggle to achieve it or simply have an inequitable balance of power.
### Role Models
Consensual nonmonogamy is so uncommon in the United States that most people in poly relationships do not have easy access to relationship role models in their own families of origin or even popular media. Polyamorous communities provide that role modeling and demonstrate patterns that other families can mimic if they prove useful. Spending time with seasoned polyamorists helps newbies find the role models so essential in this potentially complex and fluid relationship style. These role models give newbies an alternative reference group,[48] teaching new relationship skills and providing a different frame of reference from conventional monogamous society.
Role models also transmitted information and relationship skills, and discovery of a polyamorous subculture improved some people's relationship skills and offered a fresh perspective on their past and current relationships. Louise Amore, a thirty-seven-year-old photographer and mother of three, said she learned new relationship tools through contact with local and online polyamorous communities. These tools helped to improve her relationship with her husband, Max, a forty-one-year-old father of three and computer programmer. Louise and Max had originally agreed to an open marriage, but "we didn't really know what we were doing." They initially attempted a "don't ask, don't tell" approach in which they could establish outside relationships as long as the other remained unaware. Louise explained that she was uncomfortable with the arrangement:
> During that time I was the only one who did see someone and I didn't like lying. Again, it just didn't seem much different than being monogamous and lying and cheating, because I still wasn't being honest about what I was doing. And so that lasted just a few weeks and I said I can't do this, and so for eight years we were monogamous. And it opened up after we found the Internet and when we found there were other people out there, we weren't the only weirdoes. And from there it progressed into—we finally, about two years ago found the polyamory groups and there was actually a word for who we were and the way we related to people.
Louise's discovery of an online polyamorous community offered her an alternative way to think about her relationship with Max, and relating with other polyamorists gave both Louise and Max the skills to support their new relationship style.
People who had established multiple-partner relationships in isolation from other nonmonogamists would sometimes rewrite their social histories once they started socializing with poly communities. Bruce, a man in his mid-forties, expressed his glee that
> I ain't no cheating, low-life horn dog no more! [laughing] I was poly the whole time. I had no idea how to ask for my needs with other partners to be met, so I did it behind their backs. Now I am honest about it and it's much better.
Discovery of the polyamorous community assisted Bruce in rehabilitating himself from "low-life horn dog" to a _poly_ person with an identity based in an ethical community. Redemption of the past[49] allowed Bruce and others like him to recast past mistakes, and also provided a welcome rationale for polyamorous activities that conflicted with the social mandates of a culture that celebrates monogamy. Poly socializing provided not only role models but also an alternative moral and emotional framework in which to understand their own actions as beneficial to themselves, their partners, and their relationships as well.
Polys who did not have access to role models often said they felt adrift, alone, and confused. Steve, a white man in his mid-forties, who had practiced polyamory intermittently since his late teens (though he did not call it polyamory until he heard the word in his early forties), recounted his teenage involvement with a group of friends engaged in what he retrospectively called "the poly experiment." Steve was the de facto leader of the group, though he said:
> I wasn't mature enough to handle that level of complexity in my first attempt at poly. There were no adult role models for us to follow; we were trying to create this thing by reading _The Harrad Experiment_.[50] A few people got majorly burned. We were in so far over our heads; we had no idea what we were doing.
Lack of community and attendant role models contributed to the failure of that particular "experiment." Steve, however, did not abandon his multiple-relational practices, and instead he refined it, partially through contact with the polyamorous community.
#### "Hubs" as Role Models
Hubs—compact geographic centers of polyamorous activity—emerged in poly communities, frequently forming around an extended family, or a couple with many lovers, with the resources to gather community and host events. When Jana Founder's moresome melded with Melody Lupine's triad, they created a combined family of eight adults and four children. Six of the adults resided within a five-block radius, and the other two visited periodically. The combined physical space, increased number of family members, and numerous ancillary lovers and friends created a hub for local community social events and support groups. The magnetic pull of the hub drew other people who were considering or practicing polyamory, who then remained in the area in part because of the quality of community available. This created a self-perpetuating cycle in which the presence of an organized community attracted others seeking polyamorous fellowship, strengthening the existing local community.
Within fifteen miles of the Founder/Lupine hub, Louise Amore created an additional social hub when she opened her home to polyamorous gatherings and organized polyamorous outings such as camping trips. Similar hubs formed in the California Bay Area around Evelyn and Mark Coach, the Wyss family, and the Ravenheart clan composed of long-term national community leaders and their many ancillary lovers.
Hub members tended to emphasize polyamory as a core aspect of their personal identities. Polyamory was a central identity characteristic to members of the Founder/Lupine hub, and similar community leaders. These leaders published books and magazines on polyamory, conducted radio and television interviews, organized conferences, managed email lists and websites, and acted as a clearing house of information for members of the general public seeking assistance or information surrounding polyamorous issues. The sheer amount of time they spent discussing and writing about polyamory made it a key component of their lives. Add to that the time they spent practicing polyamory, holding house meetings, and discussing feelings with lovers, and there was little time left for other things. The centrality of polyamory to people's identities tended to wane with members' increased social distance from the center of the hub.
When a hub family broke up it usually had a considerable impact on surrounding community members. Losing the social territory previously defined as polyamorous meant fewer places to gather, less emotional and financial support, and lower chances of meeting potential partners. Role models and friends became more difficult to find and existing relationships more difficult to maintain. People questioned the longevity of their own relationships when a hub family, which had served as a key role model for the local community, broke up. The ancillary lovers of the hub sometimes drifted away, and others considered becoming monogamous.
### Stigma Against Monogamists
Ironically, focusing on honesty and self-knowledge as key to poly identity allows polys to reverse the stigma onto monogamous people the polys see as narrow-minded or stuck, as Jeffrey, a white male in his mid-forties, put it:
> Not all people in monogamous relationships, I'm sure some are good, but in some of them people are cowering or cheating because they don't have the balls it takes to be honest about it. I know, I used to be one of them, and then I grew up (laughing). It's kinda true though, that being poly made me "put on my big boy pants" as my wife loves to say.
Jeffrey shared the common poly community view of subtly (and sometimes not so subtly) portraying monogamous people as small and grasping, too weak to face the self-awareness boot camp that poly family life can be. Poly people, in this sentiment, are more evolved, stronger, and self-realized than mere monogamists.
### Polyamorous Deviance
In the discipline of sociology, the term _deviance_ means simply being outside of the norm. Theoretically, the sociological term does not have the negative connotations it carries in conventional society, although some scholars doubt that "kinder and gentler" view of deviance.[51] While polyamorists live by a wider range of norms than that found in traditional monogamous society, sometimes they break even their own unique social rules. A number of offenses qualify as deviant among poly community members, and polys usually deal with them through gossip and subtle social pressure. Such deviance rarely erupts into open social conflict.
#### Cheating
Dating married people without the monogamous spouse's knowledge is generally severely stigmatized among poly folks, and frequently it results in social censure and a damaged reputation. Any form of deceit is largely frowned upon, as it violates the fundamental community ideals of communication and honesty. Even so, some polyamorists lie to each other with surprising regularity, and dishonesty is a constant topic of discussion in support groups and online forums. Possibly the most stigmatized forms of deception involves failure to disclose STIs or breaking safer-sex agreements. Community members often went to great lengths to educate themselves regarding STIs, and they painstakingly negotiated safer-sex agreements to protect themselves from exposure and to prevent transmission to others. Breaches in safer-sex agreements are considered especially problematic because they can pose serious risks to fluid-bonded partners.
Ironically, people in polyfidelitous relationships sometimes cheat on their spice by having sexual relationships outside of the approved group. There are a number of reasons people cheat within polyamorous relationships. A desire for drama, the adrenaline rush that accompanies the spy games of a clandestine relationship, the potential to gain power over another partner by keeping secrets, or even to avoid the inevitable complexities of their partners knowing of one another can all be motivations to cheat, for polys and infidelitous serial monogamists. The Sommers polyfidelitous quad was composed of Frank and Georgia Some and Linda and Marlin Mers, two previously monogamously married white couples who had merged to form a larger family unit, and they eventually broke up over Marlin's infidelity with a former girlfriend. The "other woman" did not wish to join the quad, claiming that she was monogamous and unable (or unwilling, depending on whose opinion you followed) to be in a polyfidelitous relationship. Linda, whom Marlin had legally married years before joining the quad, wondered aloud in a support group meeting how this woman, who was having a relationship with a "very married man," could still call herself monogamous.
The Sommers quad eventually dissolved, ostensibly because of the infidelity, though members mentioned other contributing problems. Marlin was mildly stigmatized by the polyamorous community at large and was especially harshly stigmatized by Linda, who gossiped in person and ranted online about his many alleged personal and sexual inadequacies. Even with Linda's hurt and anger, Marlin still attended polyamorous functions with Georgia and appeared to retain a fairly positive standing in the community. Some people found Linda's tirades over the top and stigmatized her for being too harsh, either by gossiping to each other about Linda or responding to her online diatribes. Other former quad members also remained involved in the community and tried to avoid each other with varying degrees of success. Among poly community members, the main method of punishing people who misbehaved was gossip, and offenders were rarely ostracized.
Joya Starr told me about a time she experienced polyamorous cheating:
> I felt exploited by being open to my partners being in other relationships but that not being reciprocated. Like they were uncomfortable with my relationships or say that they wanted to be in a trio but they continuously became involved with people who were monogamous and wanted to undermine my relationship with my partner. And that's been—more jealousy has come up around that than anything else ever for me. And I find it particularly annoying when polyamory is out as a possibility, on the table, part of how we relate to each other, but they really prefer to cheat. They want to have their relationship but not wanting to reciprocate that relationship for me so they hide that they're involved with somebody else as if I'm stupid. . . . You've got this poly surface, but with that energy of cheating inside of it. And I found out that some people just love that and it's not poly. They love that cheating stuff. . . . I think it was that balance of power where that was part of the turn-on was the hiding part.
Joya, like many community members, felt that lying and cheating in a polyamorous relationship was "not poly," although it was common enough to indicate that it is a poly practice, even if it is not poly philosophy.
#### Double Standard
Another way in which polys would break the unwritten community rules is when one person wants free access to multiple lovers themselves but finds it difficult to "allow" their partner(s) to have additional lovers as well. When the double standard is overt, people come right out and say, "I don't want you seeing anyone else, but I want to be able to see these other people." At other times it is more subtle, with one of the partners saying, "I am fine with you having other lovers," and then vetoing every candidate for one reason or another, in effect depriving their partner of other lovers while pretending otherwise. This double standard goes against the community philosophy that recommends the same rules for everyone and is wary of relationships that give one partner more latitude than another. People with a double standard were stigmatized, and such relationships often self-destructed. The double standard was not gender specific, and both men and women occasionally wished to restrain a partner from seeing others while enjoying multiple partners themselves. Melody Lupine detailed a relationship she had had with a man in which his girlfriend had a double standard regarding sexual involvement with others:
> She broke up with him and then called him and wanted him back. He had gotten into another relationship and that became real insecure for her and she didn't like it. It brought up a lot of her issues, and all of a sudden he started looking better and fears of hers came up. But what had happened, there's a double standard with her now. It's okay for her to be polyamorous, but she doesn't like that he is, and doesn't like that he's in a relationship with me.
Community members often stigmatized people who held on to the double standard; not only did it violate cherished community norms of equality and ethical treatment but also such noxious jealousy kept the person with the double standard from participating in the highly esteemed community ideal of compersion. Double-standard relationships should not be confused with poly-mono relationships, in which both partners have explicit permission to engage in outside relationships but only one of them feels the desire to do so.
## The Importance of Communication
Communication and honesty are such important aspects of polyamorous life that it is difficult to overemphasize them. Together, they are the most popular coping mechanisms that assist polyamorists in dealing with the potential difficulties in their complex relationship style. Polyamorists routinely face the possibility of jealousy, hurt feelings, and miscommunication among many partners. While monogomists experience these same difficulties in their own relationships, the increased number of people in polyamorous relationships multiplies the opportunities for miscommunication significantly. Polys developed a number of techniques to deal with these potential pitfalls. One such mechanism was _radical honesty_ , a practice of being completely honest in all situations, even when it was not "nice" or convenient.[52] Many long-term polys practiced and even sought training in Nonviolent Communication (NVC)[53] techniques, such as listening compassionately to the other person while they are speaking instead of preparing mental notes for a rebuttal, subverting the desire to argue by calmly repeating what the other person said back to them to make sure everyone shares the same understanding, and speaking in " _I_ statements."
Poly community members also value persistence and the ability to tolerate conflict, in large part because the humane practice of those traits contributes to effective communication. Morgan Majek commented, "I'm willing to work through things, so I'll just talk things to death and work through 'em." Morgan felt her relationship with her husband, Carl, had improved since they became polyamorous, primarily because
> It really opened up communication between us. Because we've been together for nine years and that was my biggest complaint about him was you don't talk to me . . . And it really opened up communication between us. So it created pain, but it really just helped us to learn how to be completely honest and communicate. And so it benefited us.
Discussing painful feelings such as jealousy or insecurity can take tremendous fortitude, and establishing schedules that allow many lovers time with one another required adept negotiation. Polyamorous communities create supportive networks in which people can learn and practice these skills.
Those who had been isolated and then found organized polyamorous communities routinely commented on how much they learned about communication (among other things) from other polyamorists. Melody Lupine had initiated a polyamorous triad in her small Midwestern town where the threesome was secluded from other polyamorists:
> And unfortunately we didn't have any pool to communicate or no support groups back then. We lived in Ohio and it was the Midwest and again this wasn't heard of, there were no people around us, any of our close friends that we approached with it had a lot of judgments about it, and actually we got ostracized from several communities because of it . . . And again we didn't have very much support and didn't know—we ended up doing a lot of hurtful things to each other, and we didn't even realize. We just didn't have the communication tools and wasn't attentive of the other's needs, and we just didn't know how to make it work and how to do it.
Melody viewed role models who had previously navigated complexities, knew "how to make it work," and could show others "how to do it" as central to the successful function of her polyamorous family and individual emotional health.
Communication is key, not only in dealing with the emotional complexities of polyamorous relationships but also in negotiating boundaries structuring those multiplistic relationships.[54] While there are plenty of role models for people in monogamous relationships, they are fairly scarce for polyamorists, who create templates for their own relationships. Negotiating safer-sex agreements, boundaries structuring relationships with a variety of partners, and even the domestic division of labor among multiple partners and coparents requires extensive communication skills.
Poly relationships with poor communication tend to self-destruct rather rapidly. Morgan Majek said of the disastrous and short-lived relationship her husband, Carl, had with his first girlfriend, Janice: "They had both been expressing [to others] how hard it was to be with one another. They don't communicate; they don't talk to each other. So it just wasn't a match." Although Janice and Carl tried to communicate and had an intense sexual connection, their relationship did not last because of the incompatibility in their communication styles. Communication is so important in polyamorous relationships that, when it fails, the malfunction often overshadows other aspects of the relationship. Even the strongest sexual connection might not be enough to keep a poly relationship together if miscommunication spoils the emotional connection.
Polys use communication to get to know each other, and they often use _relationship maps_ as a form of communication/foreplay possibly unique to polyamorous communities. Relationship maps are diagrams polys draw to explain their complex webs of relationships, generally with current and past lovers, the characteristics of the relationships (primary, secondary, fluid bonded, etc.), and lovers' lovers, when known. Vance, a twenty-seven-year-old white computer analyst, remarked:
> Every first-time date includes a map of everyone I am seeing at the time, relevant past relationships, and as much of their [his lovers'] sexual history as I can muster. Inevitably I find areas where my lovers overlap with the new person. It helps us figure out where we are and talk about [sexually transmitted] diseases.
Usually everyone involved in the courting episode draws their own map, and these discussions almost inevitably lead to disclosing the presence of any sexually transmitted infections.
## Overlap with Other Communities
There is considerable overlap between polyamorous communities and other subcultures including geeks, gamers, and science fiction fans, people with unconventional political or spiritual views, and other sexual minorities such as kinksters, bisexuals, swingers, and gays and lesbians. Some polys endorse the overlap as a paradigm shift that enclosed all forms of alternative relationships. A woman writing online, who identified herself as "Mary," stated:
> I'm very interested in all new paradigms of relationship and sex, be they poly, Tantra, Buddhism, etc. It seems to me that they share common philosophies of negotiation, trust, honesty and ethics; as opposed to the cast-iron "morality" of traditional marriage. I'm a queer friendly straight girl writer/performer and PhD student, living in Australia with my longtime bf [boy friend]. I'm not actively poly—I'm one of those people who finds just one primary relationship a pretty major thing to deal with, both emotionally and physically. (My SO [significant other] and I are open to the potential of the occasional friendly/loving fuck-buddy fling—just not multiple "intense" relationships.)
While "Mary" did not wish to expand beyond the one primary relationship that she found a "pretty major thing to deal with," she nonetheless cast herself, and was welcomed by online community members, as an ally to polyamorous people and those involved in what she viewed as natural affiliates. Polyamorous communities frequently welcome allies and do not require them to engage in multiple-partner relationships in order to be accepted into the sexually permissive environment that supports those who question a variety of cultural norms.
### Unconventional People
Polyamory appears to be especially appealing to geeks, gamers, science fiction fans, and people with unconventional religious or philosophical views. Once someone has stepped outside the mainstream, it is easier to continue considering alternatives. Usually people who become polyamorous have already done other unconventional things or held unconventional ideas: Polyamory is not usually the first step "outside of the box."
While many people play online or board games, not all are _gamers_ : Gamers are a special breed that take their play very seriously. One version of the gamer is the LARPer, or Live Action Role Player, who dresses in costume and plays a game in person, physically present with the other players in a designated space. Organizations such as the Society for Creative Anachronism (SCA) provide LARPers with opportunities to play with others in costume and battle regalia.
Science fiction fans who read or watch media that portrays alternative visions of the future are also well represented among poly communities. Heinlein's work, especially _Stranger in a Strange Land_ ,[55] is probably the most influential body of science fiction in the polyamorous community. _Stranger_ , as many polys affectionately refer to the novel, relates the story of a human man raised on Mars who returns to Earth and founds a new religion that includes nonmonogamous relationships. Rimmer's work, particularly _The Harrad Experiment_ ,[56] set on a college campus that advocated sexual sharing and nonsexual nudity among students, is similarly influential in the formation and ideas of the polyamorous community. Both novels portray groups of people living together (mostly harmoniously) and sharing sexual partners. Many polys say that they read one or both books at a time in their lives when they had not previously considered multiple-partner relationships, and they ended up using the novels as models for how to build their own expanded intimate networks.
In addition to the geeks and scifi fans, there are other people with unconventional views among the ranks of the polyamorous. Anarchists often favor nonmonogamy because of the implicit freedom of choice, and Pagans who worship multiple god/esses have a spiritual orientation that embraces multiplicity. Such strange compatriots as libertarians, feminists, and Unitarian Universalists all value polyamory for the equal/gender-neutral opportunity it provides.
### Kinksters
One of the primary overlapping communities is comprised of people who practice BD (bondage and discipline), Ds (dominance and submission), or SM (sadism and masochism), otherwise known as BDSM or more generally kinky sex. There are a number of similarities between the polyamorous and BDSM subcultures. Both are composed of members willing to challenge social norms structuring "vanilla" (the BDSM term for traditional sex) or monogamous relationships. Negotiation is centrally important to both communities, necessary to both structure relationships and to create agreements supporting the relationships. Some polyamorists practice BDSM to varying degrees, and many kinksters (people who engage in kinky sex) have multiple partners with whom they "play" or "scene." For kinksters, "playing" means engaging in episodes that involved explicitly sexual acts (up to and including penetration), or simply have a sexual tone but did not involve what others might consider sex per se. A "scene" is a sexual encounter negotiated around a specific script with detailed discussion of what will or will not happen and a "safe word" that immediately stops the scene if the submissive finds the stimulation or interaction too intense.[57]
### Bisexuals
The popularity of bisexuality among polys encourages close connections with bisexual organizations and communities. Some polys discover polyamory through their involvement in bisexual politics or organizations, and the overlapping members and politics the poly and bisexual communities share are so extensive that some people like Earl, a white man at a dinner party, think that "bisexuality and polyamory are two sides of the same coin."
### Swingers
Polys also share many practices and attitudes with swingers, and some community members as well. Like polys, swingers are predominantly white,[58] educated,[59] middle and upper-middle class people with professional or managerial employment.[60] Both groups appear to be gaining members through Internet contact.[61] Swingers and some polys distinguish between their primary partners (usually a spouse or spouselike relationship) and other sexual partners. Swingers have a code of etiquette focused on ethical treatment of spouses rather than moral guidelines, very much like poly folks.[62] Both polyamorists and swingers use honesty to normalize nonconformist sexual behavior, stigmatize people in "monogamous" relationships who cheat, talk a lot about how to manage jealousy, and each sees their common focus on truthful communication as leading to healthier relationships and better sex.[63]
Henshel argued that, because the men in her sample initiated swinging 68 percent of the time, males were the dominant force in the swinging situation.[64] I would add that swingers' focus on female bisexuality as entertaining for men and far more acceptable than the virtually prohibited male bisexuality exposes the underlying sexism and homophobia in both swing and poly communities. They even share a mascot: the Bonobo chimpanzee, whose interactions in the wild tend toward cooperation and group sex.[65]
There are, however, important differences between swingers and polyamorists. Swingers have no practice of group marriage, and the female/male couple is clearly the basic unit in swing settings. Swingers tend to be more heterosexual and far more politically conservative than polyamorists.[66] Swingers tend to live as couples, often married, and so can often pass for conventional monogamists far more easily than polyamorists with multiple spice who live or attend public functions together.
In contrast to those poly people who see their polyamorousness as an innate orientation, swingers often characterize their multiple-partner sexual practices as purely a lifestyle choice. Gould draws a distinction between recreational swingers (which he reports dominate "the Lifestyle") and utopian swingers, by far the minority. He and others in swinging settings cast the latter as more revolutionary in that utopian swingers want to change the norms relating to marriage; the recreational swingers have no such intent.[67] Polyamorists, with their desire for social change, are far more closely aligned with the minority utopian swingers.
These last two differences—viewing desire for multiple partners as innate or as a choice, and a desire or lack thereof to change traditional familial and gender roles—are the most important differences with polyamorists. Swingers' focus on choice and ability to pass make them a less cohesive group, more of a collection of individuals who share a common interest in their personal sexual satisfaction rather than social change. Polys' emphasis on innate characteristics, desire for social change, and the popularity of platonic socializing far more extensively than swingers makes polyamorists members of a community and movement. As a result, polys have more philosophical overlap with gays and lesbians, who share these crucial traits as both a movement and a community, rather than swingers.
### Lesbians and Gays
While polyamorous communities have tremendous overlaps with some sexual minorities, they also lack a significant intersection with others that initially appear to be natural affiliates. Polyamorists mirror gays and lesbians in many areas, including bearing the brunt of social stigma, the risk of losing child custody, difficulty finding partners in the general populace, and making decisions about when to come out and to whom. Even with all of these similarities, there is a glaring absence of lesbians and gay men in the mainstream polyamorous subculture. There are several potential reasons for this.
A lack of lesbians in polyamorous communities could be due to lesbian discomfort with bisexual women. Tina, a thirty-six-year-old white urban planner, reflected:
> I identified as a lesbian for so long, and I really enjoyed women's energy, just getting together and talking about things. When I actually got into the poly women's group I thought that most of them would be lesbians, and I was very surprised to find a majority of them are heterosexual or bi with a male primary partner, and I was like, that's kind of different . . . I guess I've always identified as bisexual, but I called myself lesbian for many years so I could feel more comfortable hanging out in the lesbian community. There's a lot of politics around that.
Lesbians who view bisexual women with contempt are unlikely to be comfortable in a setting so heavily populated with bisexual women, especially when the bisexual women are so highly valued in poly communities. Indeed, Rust found that the lesbians in her study held decidedly negative views against bisexual women as "sexual opportunists, fickle lovers, traitors, political cowards, or fence-sitters."[68]
Alternately, lesbians may not wish to subject themselves to male sexual advances, and the comparative dearth of available women in some poly communities could contribute to what one woman attending a community potluck termed a "feeding frenzy," when men "swarm like sharks around women who seem available." Finally, lesbians may not wish to compete with men for the potentially rare available woman who is seeking additional partners. These factors could combine to encourage lesbians to date within their own poly circles rather than brave the masculinity in mainstream poly communities.
Gay men might feel no need for additional support for multiple-partner practices. Though some gay men have a well-documented habit of establishing multiple-partner relationships,[69] most do not identify themselves as polyamorous. I observed an almost complete absence of gay men in the polyamorous communities of both the Midwest and the California Bay Area. Gay and lesbian communities in the Midwest are smaller and less developed, but the Bay Area hosts one of the largest and most well-organized gay and lesbian communities in the United States. This obvious lack of gays and lesto bians in Bay Area poly communities, compared to the large representation of bisexuals, is especially noteworthy. It is possible that nonmonogamy is so thoroughly accepted in gay male culture that most gay men feel no need for additional support from polyamorous communities. Further, it is likely that gay men in multiple-partner relationships have no desire to confront the possibility of homophobia in many polyamorous communities.
Javier, a thirty-five-year-old Mexican American man, and Christian, a thirty-nine-year-old white man, attended a national polyamory conference in California in order to reach out to the polyamorous community. They had been engaged in a successful nonmonogamous relationship for five years and wished to meet others with similar interests. Christian commented:
> Not many [men] we know identify as polyamorous but over 90 percent are what I would term "poly sexually oriented," meaning they have multiple sexual partners outside of their primary relationships or date a lot or "hook up" a lot. Many are polysexual at least. I know of very few, if any, monogamous male-male relationships, though I do know of many female-female monogamous relationships. We do not know many polyamorous gay men . . . In the male-male sex world there is on the one hand a wish to lifelong monogamy and the picket fence partnership. But reality is most male-male relationships have outside sexual encounters that run the range from shared encounters with the two partners to complete secondary, one-on-one relationships.
Their perceived isolation as polyamorous men in a gay world drew Javier and Christian to attend the national polyamorous conference. At the conference, however, both men said they felt very awkward with the primarily white, heterosexual, and bisexual crowd. While their mutual bisexuality offered some potential point of overlap with the other conference attendees, they nonetheless felt marginalized. "You just don't see folks like us around here much." Not only was there an absence of other male couples, there were hardly any people of color. As a mixed race Latino and white couple, they felt very out of place and told me they would probably not attend another poly conference.
Thaddeus, a thirty-five-year-old white musician who identified as queer, spent most of his time in what he termed the "gay subculture." He did not, however, identify this gay subculture as a community, as he did with polyamorous communities:
> [Midwestern town] is the first place I've found that has any sort of interactive polyamorous community and is something that I would call a community because they know each other. If somebody's car broke down they could call somebody else for help. The same cannot be said of what we call the gay community, which I continue to insist is not a community but a subculture. In some places there's community . . . these are places that have a very high concentration of gay people living there on a permanent basis and because they do know each other it does tend to form community, but in general, no. If you walked down the street and were hit by a car you can't count on help from somebody just because you're gay. And that's unfortunate.
Thaddeus saw more differences between gay subcultures and polyamorous communities than the lack of coherent social norms and willingness to assist each other. He also observed a profound difference in the way each group handled multiple-partner relationships:
> They [most polyamorous people] don't think of gay men as being polyamorous because we are so free sexually anyway. It's part of our subculture. What isn't part of our subculture is emotional relationships in multiple terms. Sexual relationships? Sure! That's all over the place, but emotional relationships, it's rare. You find occasional triads, you find occasional vees, one-timers or secondaries or one person with two lovers that may or may not interact. But those folks still don't think of themselves as polyamorous, they're still just calling themselves gay. Because in that subculture, pretty much, your style is your style and nobody's gonna cram monogamy down your throat there.
Thaddeus felt that he was different from most people in mainstream gay subcultures because he wanted so much emotional connection that he doubted one lover could ever fill what he saw as his vast emotional needs. While he said his high sex drive was one of the reasons he wanted to have poly relationships, the more important factor was "emotional engagement, clearly . . . I've never been satisfied with relationships that were simply sexually based."
Still, some polyamorists view gays and lesbians as natural affiliates and even patterned themselves after gay and lesbian activists. Mim Chapman, a member of the activist-oriented group in New York, wrote on the web:
> GLBT's and their allies are fighting to give male and male/female and female couples the right to enter the ark of social acceptance. Polyamorists are now taking on the second half of the Noah syndrome, the two by two bit.[70]
Gay and lesbian people have been organizing politically for more than thirty years, and while some polyamorists try to learn from the success of that identity-based movement, obstacles remain that keep polyamory from being as politically potent as the LGBT movement. For one thing, polys do not always identify themselves as a political movement, preferring to focus instead on their own emotional well-being and developing tools to navigate multiple relationships. Another reason polys are less politicized is that polyamory is such a fluid relationship style, and community members disagree about who counts as poly and why, that it is difficult to use as a base for identity politics.
Finally, although exclusively same-sex (nonbisexual) relationships are rare in most mainstream poly communities, there are lesbian polyamorous subcultures in some large cities in the United States, such as Seattle and the California Bay Area. Unfortunately, none of them volunteered to participate in the research, so they do not appear in this book. Books such as Celeste West's _Lesbian Polyfidelity_ and Marcia Munsun and Judith Stelboum's _The Lesbian Polyamory Reader_ also indicate the existence of lesbian polyamorists.
## Poly Fatigue
While poly people get many benefits from their relationships and association with other poly community members, sometimes they get tired of the drama and constant negotiation. Norman, a thirty-year-old writer, decided he was done with polyamorous relationships because
> I just can't handle this shit anymore! I mean, there are marriages that last for forty years that don't have the amount of drama you can squeeze into a four-month poly relationship. I mean, dramatic to the point of being toxic! Still, I think I will stick around the poly community [as a monogamous person] because I have found kindred souls here. There is a fellowship core of poly people that feels really good to me.
Norman, and others like him who had decided to return to a monogamous relationship style, often continued to feel accepted in the polyamorous community and chose to remain associated for social and political reasons.
Others were far more disappointed with polyamorous community interaction. After twenty-five years of polyamorous relationships, the community had not lived up to Emmanuella's expectations:
> I wanted everybody to grow the hell up and start acting like it mattered and get past the petty jealousies and talk about why it was important to open up relationships, why monogamy is such a hardship for both men and women, why the nuclear family is so damaging to children, why communal living, whether it's physically, in terms of proximity, or in terms of support, is so important. I thought those things would have occurred by now and in some faraway places randomly it is occurring somewhat, I'm told, but each one under its own principles, and polyamory is a very loose term for what most people assume are very loose people. And that is not the thing I want to be associated with. I wanted it to be the answer to nonmonogamy. I thought that it would be making more sense by now and it would be something I would be proud to introduce my children, to share with them, to say this is it. And instead my son says you're a bunch of aging hippies. I want it to be more than that, to radicalize the concept and to make it more inclusive of gays and lesbians.
Emmanuella was disillusioned by what she saw as the ultimate fate of polyamorous community politics. Rather than revolutionizing marriage, family, and relationships, she thought the polyamorous movement had squandered its potential on bickering among "a bunch of aging hippies."
Some long-term polyamorists are more optimistic, still hopeful of impacting real social change. Still others are so upset with the community and their relationships within it that they stop identifying as polyamorists and either become monogamous or begin identifying simply as "nonmonogamous." Most of them are not represented in this book.
1.
(Hutchins 2001: 72)
2.
(Muncy 1973: 160)
3.
(Muncy 1973: 168)
4.
(Hutchins 2001: 72)
5.
(Hutchins 2001: 72)
6.
(Weeks 1985)
7.
(Bornstein 1994; Butler 1990)
8.
(Udis-Kessler 1996)
9.
(D'Emilio 1983; Weeks 1985)
10.
(Stinnet and Birdsong 1978: 104)
11.
(Stinnet and Birdsong 1978: 107)
12.
(Buunk and van Driel 1989: 134)
13.
(Anapol 1997: 97); see also (Francoeur and Francoeur 1974)
14.
(Hutchins 2001: 82)
15.
(Strassberg 2003: 457)
16.
Nearing, personal communication, 2003
17.
(Constantine and Constantine 1973: 49)
18.
(Smith and Smith 1974)
19.
(Smith and Smith 1974)
20.
(Bartell 1970)
21.
(Bernard 1972)
22.
(Ellis 1970)
23.
(Bartell 1971; Breedlove and Breedlove 1964; Denfield and Gordon 1970; Fang 1976; Henshel 1973)
24.
(Bartell 1970; Jenks 1985)
25.
(Flanigan and Zingale 1991)
26.
(Gilmartin 1974; Jenks 1985; Levitt 1988)
27.
(Bartell 1970; Jenks 1986a)
28.
(Jenks 1998: 507)
29.
Robert Heinlein, _Stranger in a Strange Land_ (New York: Ace, 1987).
30.
(Bargh and McKenna 2004; Jenks 1998; Wellman et al. 1996)
31.
(Bargh and McKenna 2004)
32.
Sex positivity is an outlook that defines sexuality as a positive, life-affirming activity. Proponents define themselves in opposition to "sex negative" Victorian or repressive sexual mores, which cast sexuality as dirty, degrading, or negative.
33.
This was not the case when polyamorous groups migrated together to a rural area to establish communities.
34.
(Sproull and Faraj 1995)
35.
Robert Heinlein's 1961 novel _Stranger in a Strange Land_ was especially influential for many people who read its story of nonmonogamous relationships and envisioned creating them in their own lives.
36.
(Weinberg et al. 1995: 217)
37.
(Weeks, Heaphy, & Donovan 2001: 90)
38.
(Weeks, Heaphy, & Donovan 2001)
39.
See the Associated Press article on the Divilbliss case at http://www.polyamorysociety.org/Yahoo-Divilbliss_Article.html.
40.
http://www.polyamorysociety.org/Divilbiss_Families.html
41.
http://www.polyamorysociety.org/Divilbiss_Families_Case_Ends.html
42.
http://www.lovemore.com/april/april_divilbiss_case.htm
43.
(Durkheim 1960 [1893])
44.
(Goffman 1963)
45.
Please see http://caw.org/content/?q=bouquet.
46.
(Anapol 2012)
47.
An exception to the rule, the poly/mono relationship revolves around an explicit agreement allowing both partners equal access to outside lovers, but one chooses to remain monogamous. The monogamous partner often explains his or her abstinence from other partners through her or his monogamous relational orientation.
48.
(Merton & Rossi 1968)
49.
(Kitsuse 1962; Sartre 1969)
50.
A popular read among second-wave polyamorists, see Robert Rimmer, _The Harrad Experiment_ (New York: Prometheus Books, 1966).
51.
(Spreitzer 2004)
52.
(Blanton 1996)
53.
See http://www.cnvc.org/ for more information on Nonviolent Communication.
54.
(Anapol 2012)
55.
(Heinlein 1961)
56.
(Rimmer 1966)
57.
Ortmann & Sprott 2013)
58.
(Bartell 1970; Jenks 1985)
59.
(Gilmartin 1974; Jenks 1985; Levitt 1988)
60.
(Jenks 1985; Levitt 1988)
61.
(Gould 2000; Jenks 1998)
62.
(Gould 2000)
63.
(Jenks 2001: 171)
64.
(Henshel 1973)
65.
(Gould 2000)
66.
(Bartell 1970; Jenks 1986b)
67.
(Gould 2000)
68.
(Rust 1993: 368)
69.
(Connell 1992 and 2005; Weeks, Heaphy, & Donovan 2001)
70.
http://www.poly-nyc.com
Chapter 4
# Issues Facing Poly Relationships
## My Story
"I realize that I'm in love with Steve," I said, looking at Rick with my heart pounding and a terrible tightness in my chest. His eyes clenched in pain and he said, "I really did not want to do this tonight. I got your mom to stay with the kids tomorrow so we could talk, and I already told you that I really didn't want to process tonight before that conversation. I don't want you to be with Steve, it's not working. In fact, I think we should get married and be monogamous." In that moment I felt something break and knew that I would never really love Rick again. If we had not had children I would have gone right home and started packing because I was out of there. It scared me to death, and I buried it as soon as I realized it, spending the next five years trying to pretend it hadn't happened and searching for some way I could save my relationship with Rick. But we had been through too much, for too long, for him to suddenly shift from polyamory to marital monogamy.
* * * * *
During the fifteen years I have studied polyamory, my role as a researcher has gone from civilian (no research intent but a desire to find out about it for my own purposes), to peripheral (deciding to research the topic and "coming out" to the community as a researcher), and then complete member (having a polyamorous relationship myself) before returning once again to a peripheral role.[1] That path is actually quite common to poly relationships, in that they tend to be fluid, flexible, and changing. Relatively few of those I spoke with kept their relationships in the precise form that they were when I initially interviewed them. More often, they retained connection and continuity with the same people in their lives, but the form or specific expression of the relationship would shift over time.
Quite importantly, this fluidity in these relationships did not necessarily translate to high levels of partner attrition. Rather than cycling through and discarding a parade of partners, most respondents retained contact with significant others over long periods of time, but the form that contact took often shifted over the years. For example, in my own life this has taken the form of a twenty-three-year (to date) relationship with Rick, the father of my children. For the first fifteen years we were "together" as a couple, and for the last eight years as coparents and sometimes friends. We still know each other and communicate regularly. We remain fixtures in each other's lives—available to consult, collaborate, help, or argue—though the form of our relationship has changed over time.
## The Beginning
I was a twenty-three-year-old undergraduate student when I met Rick in 1993 at a university in Northern California. While I had been in love once before, I had very little dating or sexual experience, so I was not ready for it when I fell in love with Rick so quickly. On our first date Rick told me that he never wanted to get married or be monogamous. I thought, "Whatever, freak, you are not going to last long anyway," and said, "OK, I guess." Rick elaborated on his fantasy family with two women who loved each other and him, and how it might look like a harem but we would really be challenging patriarchal notions of marriage and monogamy with our harmonious, triadic union. My original casual attitude shifted, and as our relationship became more serious I got increasingly uncomfortable with Rick's desire for nonmonogamy.
### Unicorn Hunting and Couple Privilege
Though we didn't know it, Rick's ideal family fit the poly stereotype of female-male couples approaching the poly community to find a wife so closely that I came to think of it later as Unicorn Hunter's Syndrome. The plethora of personal ads from unicorn hunters on poly websites and the ubiquitous presence of couples cruising at poly social events hoping to meet available women make it abundantly clear that the stereotype is well grounded in reality. As my relationship with Rick deepened, he elaborated on his ideal triad in which we would find a woman we both loved equally and the three of us would live together in expanded bliss, raising children and making each other happy. Implicit in his discussion was the unexamined assumption that whatever woman we bonded with would not have other partners already but would join us in our lives without any attendant relationships of her own. Such a myopic view of a woman willing to fold herself into someone else's existing plan as if she were a mere ingredient can only exist if that fantasized woman is (at best) two-dimensional—certainly not a multidimensional human with plans and lovers of her own. Polyamorists term this dynamic _couple privilege_ —a narrow focus on the couple's needs and desires as the primary or sole determinant of a relationship at the possible expense of the unicorn's feelings, desires, or even full personhood.
Over time, the ongoing discussion with Rick expanded to include other aspects of what unicorn hunter couples often desire—a woman willing to join in the domestic life of the household, helping with cooking, cleaning, and child care. When it became clear that Rick wanted a large family and I was open to having children but also wanted an academic career, we imagined finding a female partner who might want to bear or adopt (and care for) more children. Envisioning our mythical girlfriend in a domestic role while one or both of us worked for pay was yet another symptom of a unicorn hunter couple: basically we were looking for a wife with an unconscious arrogance reminiscent of male privilege in more conventional relationships. In polyamorous lingo, we were seeking the hot bi babe.
### Hot Bi Babes
The arrangement of triadic sex between one man and two women was the most popular relationship form sought by the polyamorous men who attended support groups, frequented online chat rooms and discussion boards, and wrote personal ads in polyamorous online or print publications. Most polyamorous men did not have regular access to simultaneous sex with multiple women. Enough of them were able to occasionally fulfill that fantasy, however, to keep the hope of its occurrence alive for the others. This common fantasy seemed to appeal to a large number of polyamorous men, and it served as one of the most distinguishing features of hegemonic hypermasculinity among polyamorous men.[2]
Some polyamorous men were successful in living out the seemingly pervasive heterosexual male fantasy of having sex simultaneously with multiple women, the proverbial "girl-on-girl" scene depicted so frequently in pornography produced for heterosexual men. These men established triadic sexual relationships with two women or had sexual encounters with bisexual women and were occasionally joined by the women's female lovers. Ironically, some men found simultaneous sex with two women to be "not all that." In their fantasies, the men were the center of attention and the women related to each other and him simultaneously. In reality, however, some men reported that the women were more focused on each other and less interested in the man than he would have liked.
For example, Max Amore reported that he was less entertained by the triadic sexual encounters he had with his then-wife, Louise, and their girlfriend, Monique Mayfield.
> I guess things lasted for about five or six months with us getting together regularly. She [Monique] would come over with her kids and we would all have dinner together, watch a movie or play a game, something like that. Then the kids would take over the downstairs and the adults would go upstairs. Pretty quickly I got bored with it, but Louise and Monique were infatuated with each other and I felt like I should go along for the ride. But yeah, being with the two of them didn't turn out to be all that for me, though I think they really liked being with each other. The kids all got along great and Louise and Monique were really happy with the arrangement for a while, but it never worked out to what I really wanted it to be.
>
> It wasn't that long of a haul, less than a year. I just felt, I dunno, less attention than I wanted, less of a focus with three people than you have when it's only the two of you in the bed. Not like they [Louise and Monique] were jerks to me or anything, I just didn't, I wasn't feeling it.
Even though Louise and Max were able to establish a relationship with a bisexual woman, their triadic sexual and personal encounters left Max unsatisfied and wanting more attention.
Some men who entered a polyamorous community seeking a triad with two women changed their focus once it became clear that such a relationship was extremely difficult to find. Mark and Evelyn Coach had originally sought the HBB relationship, to no avail:
> When we started this we were sort of invested in the idea of finding the mythical hot bi babe who would come and join our relationship and what happened instead was the first person that she [Evelyn] got involved with, this lover in Seattle that she'd been involved with before who certainly—he didn't fit the hot bi babe category as we had discussed it, so at this point while we sort of had this vision of the larger family, we are no longer attached to any particular geometry of how it will occur.
Mark and Evelyn altered their expectations when it appeared their HBB was unlikely to materialize. Many polyamorists related similar stories of their unsuccessful quests for a HBB and the varied impacts that had had on their relationships. Some retained their original goal of seeking her out and looked for fifteen or more years, to no avail. Others found her, only to discover the relationship did not meet their expectations. I spoke with very few who had engaged in a lasting relationship with a HBB as she is popularly conceived, though I found far more polyaffective triads with one woman and two men.
I propose three potential explanations for this HBB phenomenon. The greater social acceptance of sex between women, stigma of bisexual men, and the scarcity of available female partners might combine to create a setting in which bisexual women become fetish objects. Much like monogamous society and swinging subcultures,[3] sex between women was more socially acceptable within polyamorous communities than was sex between men. The HBB was implicitly a woman—and not the woman already engaged in the female/male couple, but the free agent, who would be added to create the triadic sexual encounter. Sex among women, especially if there was potential for a man to "get in on it," was entertaining for most polyamorous (and many monogamous) men.
Men who engaged in sex with men, on the other hand, raised the specter of homophobia. Men interested in "getting in on" group sex with other men must have same-sex attraction—thus undermining their hegemonic heterosexual identity. Valorization of bisexual women and the stigma of bisexual men paralleled standards in general society where many heterosexual men enjoy female bisexuality more than male bisexuality.
While polyamorous people did not expressly state that men were not the focus of the HBB fantasy, it was obvious in their conversations. People either spoke explicitly of seeking women or used female pronouns in their friendly banter about desirable partners. At one party I attended, I stood in the kitchen with a group of six or seven polyamorists who were discussing their lovers. A white man in his late forties made a joke about the perfect lover everyone sought but could not seem to find. He ended with, "But she would knock all of your socks off, no doubt!" His story portrayed an idealized version of _everyone's_ elusive lover, who was a woman. The group included men and women, so a desire to have sex for all of us indicated that the idealized woman would be bisexual.
Some bisexual men, on the other hand, reported feeling constrained by potential social disapproval when considering disclosure of their sexual orientation. Sven Heartland had sought a bisexual man for seven years with whom he and Shelly could form a triad. Shelly and Sven Heartland, both white professionals with office jobs, each have a daughter from a previous marriage (Elise and Kimber), and also have a daughter together (Alice). Sven's first marriage ended in a bitter divorce when his now ex-wife discovered he was having clandestine sex with men. In an effort to avoid repeating the mistakes of his first marriage, Sven was honest with Shelly about his bisexuality from the beginning of their relationship, though he remained cautious of identifying himself as bisexual to new members of his polyamorous community:
> When I meet someone new in the community or a new person attends the support group meeting I am always careful what I say at first until I can see what they are like. I don't want a negative reaction, so I use pronouns like "they" and "we" instead of saying "he" when I am talking about me and Shelly and Adam, just in case.
The majority of bisexual women in the polyamorous community did not share Sven's caution. The women, far from concealing their sexual orientation, enjoyed the highest social status in the community. This relationship between gender and bisexuality indicated homophobia against men with which women did not have to contend.
Another possible reason for the valorization of bisexual women and the stigma of bisexual men was the scarcity of available or "free-floating" female partners. Many polyamorous women were already partnered with numerous people, and single or available women were rare at community gatherings. Single or available men seemed abundant and more willing to entertain casual sexual relationships. Scarcity would increase women's value in relationships, bisexual or not.[4]
Ironically, some elements of the polyamorous community reversed the HBB scenario, and sexualized bisexual men more so than women. Thaddeus, a thirty-five-year-old white musician who identified as queer, commented:
> I've certainly not encountered a lot of homophobia [in polyamorous communities]. I'm sure, mind you, that most of the men that I've met identify as polyamorous, um, wouldn't consider a male partner, I do know a couple that identify as bisexual, um, I know one of those who has ads in the [poly personal ads website], but you know, "I've never had a relationship with a man." They're polyamorous but they'll pursue women for relationships and men just for sex.
Clearly, some polyamorists viewed men, and not just bisexual women, as sexual objects. Joya Solarity's emphatic statement emphasized Thaddeus's assertion: "I have really intensely been attracted to bi boys; extremely, intensely attracted to bi boys . . . I just can't get enough of bi boys." Even with this intense attraction for bisexual "boys," she reported:
> And inside of that, I'm finding my attention is again drawn more toward women than before. And I feel like my connections there mean more to me and I'm finding that I want to play more with boys, just kind of play, a casual thing.
Although Joya refrained from stigmatizing bisexual men, and indeed intentionally sought their company, her relationships with them were primarily sexual in nature. Joya reserved her intense emotional connections for relationships with women.
Polyamorous ambivalence regarding male bisexuality and homosexuality was a point of contention among community members. Some in the San Francisco Bay Area, renowned for its acceptance of same-sex relationships and home of a decidedly sex-positive polyamorous community, appeared to have greater acceptance of bisexual men. While in the South Bay, I attended a "bisexual coffee night" in a local coffee shop. There was jovial conversation and flirting, and at one point, two men sitting on a couch at one corner of the circle of attendees began kissing passionately. Conversation died out as we all sat watching the men kiss. Slowly the men became aware of the silence around them and broke their embrace, followed rapidly by cheering and applause from their impromptu audience members. Clearly, the group had appreciated the erotic moment. Even in the Bay Area, however, bisexual men were not as highly valorized as were bisexual women.
Some of the men in my sample discussed their awareness of differing community standards regarding bisexual men and women and the apparent homophobia it revealed. Acknowledging the double standard, many still opined that polyamorists were far less homophobic and more tolerant of men who engaged in a wide variety of nonhegemonic activities than was the general society. Norman, a thirty-year-old African American writer, explained that polyamorous men tended to be more open-minded than monogamous men, especially when it came to men having sex with men.
> Polyamorous men are more comfortable with bi and homosexual men than straight men usually are. Even if you don't wanna have a man touch your ass, you're still cool with the fact that they like to touch other men's asses.
Nonetheless, Norman mentioned a double standard that glorified sex between women far more so than sex between men, as long as the sex between women was "entertaining" to men. His perception of lesser degrees of homophobia in polyamorous settings was inflected by his simultaneous awareness of the objectification of bisexual women. The relationship between gender and bisexuality within the polyamorous community was as complex as the relationship between polyamorous men and masculinity. Both women and men could be sexualized as sex toys, though it was far more common for women among the communities I studied. Bisexuality was an asset for polyamorous women, but fewer men viewed it as such. Some bisexual polyamorous men even reported fearing it might be a disadvantage.
## Civilian Research Role
During my second semester of graduate school I heard a National Public Radio interview with Ryam Nearing, who published _Loving More_ magazine at the time. She explained that the magazine was for people in openly conducted nonmonogamous relationships—something she called _polyamory_. My subsequent Internet search revealed that the _Loving More_ editors were based nearby and that they hosted community meetings and support groups. Rick and I began attending events such as a public discussion forum at a local library, a local support group where I could ask questions and express my fears about polyamorous relationships, and social events such as movie nights or group hikes. Many of the people profiled and mentioned in this book were people I met there.
### Marriage?
Deeply in love and looking for some definition to our relationship, when I was twenty-seven I asked Rick to marry me. He declined, reminding me that he had been clear from the beginning about his disdain for marriage as a patriarchal institution designed to give men ownership over women and the ability to hand property down to male children the men are certain that they fathered. I countered that marriage could be anything we made it, and we went around that topic for two years with no resolution. Finally Rick said that he would marry me if, in place of where the groom would usually say vows, he could turn and tell the people in attendance, "I don't want to be doing this. Marriage is a sham, and Elisabeth is forcing me to do this." Unsurprisingly, I declined. In a move I interpreted as caving in, I agreed to remain together unmarried and dropped the issue. Soon after we started trying to have a child, and ten months later our son was born.
## Peripheral Research Role
The more time I spent with the polyamorists, the more I liked them, and the more interested I became in them from a sociological standpoint. About a year and a half after my introduction to the local polyamorous community, I chose to do a study on a local poly community for my term paper in a graduate course. As part of my study, I became increasingly close with several local polyamorists, whom I found like-minded and friendly. At this point, I began to regularly attend the local monthly support group for polyamorous women.
I had forged close friendships with several women in the setting prior to considering it as a research area, and these relationships continued as I transitioned to a research role. The transition itself was sometimes challenging: occasionally I felt uncomfortable in groups, knowing that I would take notes on the interactions and that some group members might have been unaware of my status as a researcher. Fox felt similar discomfort when she felt compelled to "tread a line between overt and covert roles" in her investigation of a punk social scene.[5] Although I enjoyed socializing with polyamorists, I remained "acutely aware of differences between members and [myself]" and thus retained a peripheral membership role.[6]
### Communication, Honesty, and Sabotage
The most important poly relationship tool is honest communication. Authors, community members, and bloggers—everyone repeats the "poly mantra" of "communicate, communicate, communicate." Polyamorists agree that honesty and communication are essential to successful poly relationships because they are prerequisites for trust, and without trust polyamory will not work because people in the relationships will not feel safe or confident in their partners' abilities and willingness to be forthright.
Communication is not simply volume of contact but also techniques of how to communicate effectively. Poly folks often attempt to incorporate radical honesty and nonviolent communication in their relationship, with the outcome of using _I_ statements as opposed to _you_ or accusatory statements and listening carefully, which often means repeating back what they think the other person is saying and taking a break to cool off before reengaging if a conversation gets too heated.
In my case, the volume of communication Rick and I had obscured the fact that we were not being clear with each other. While I was well aware of my reluctance to be polyamorous and my attempts to manipulate our relationship to retain a monogamous tenor, I was not aware of it consciously at the time of how tenuous I felt Rick's connection to be and thus failed to communicate that to him. Part of that was lack of clarity on my part, and part of it was fear of his response if he perceived me to be giving him an ultimatum about being monogamous with me.
This ongoing exposure to the polys' ideas and novel forms of relationship had an impact on my relationship with Rick, and socializing regularly with polyamorists spurred many conversations between the two of us. We embarked upon a slow process of opening our relationship to outside partners and, in so doing, made a number of mistakes common to "newbies" or those who are first attempting polyamory. For instance, we decided to try a "don't ask, don't tell" (DADT) version of a relationship we had heard about from others attending polyamorous functions. Popular among newbies and swingers, long-term polyamorists generally eschew DADT relationships as doomed to fail in weirdness and unintentional deceit. As was characteristic of that type of relationship, Rick and I decided that we could each have outside lovers as long as the other person didn't have to hear about it. To that polyamorous standard, we added an additional rule that we could only see others while traveling separately, and these "road" relationships were never to interfere with the home relationship.
I saw this last rule as virtually prohibiting outside relationships, because we rarely traveled at all and, when we did, we did so as a family. The agreement, in my eyes, was a win-win situation. Rick could think of himself as a polyamorous person and revel in the freedom of the possibility, however remote, that he could have another lover. I, on the other hand, had no intention of finding us a girlfriend, and I could not readily imagine the circumstances in which Rick would do so, either. Reluctant to have a poly relationship but worn down by the constant conversation, I manipulated an impossible agreement to sabotage any hope of a poly relationship becoming real.
Such manipulation is also a common newbie mistake. All too often, couples with unresolved power issues or poor communication begin to seek a poly relationship even though one of the members does not actually wish to be polyamorous. In some cases, one partner wants to have access to multiple lovers himself or herself but is not actually comfortable with her or his partner having sex with others and so creates various impediments to that partner being able to establish additional relationships. These impediments can take the form of recurrent crises that require immediate attention each time the partner has a date scheduled and prohibit the date from actually happening. In other cases, a partner will object to each potential partner for a variety of reasons that may or may not appear to be reasonable at the time but culminate in the reality that none of the people the partner wishes to date are acceptable to the saboteur. In my case, I negotiated rules that appeared reasonable on the surface but effectively foreclosed any possibility that either Rick or I would actually be able to initiate a tryst with anyone else. All of these scenarios share a common thread of manipulation and sabotage rather than direct and clear communication—common pitfalls that routinely ensnare inexperienced polyamorists. Over the course of this phase of our relationship, I accidentally became pregnant again and had our second child.
Rick and I continued to discuss our relationship and reached another stage in the gradual opening to additional partners. As I had anticipated, we had never used the "don't ask, don't tell" arrangement. Our new agreement allowed openly conducted, same-sex relationships in our home area. Again, I agreed because I thought it incredibly unlikely that such a relationship would ever come to exist. Rick had no interest in sex with other men, and I was not seeking other partners, so I did not see how it would realistically work out for us to see others. The idea was satisfying to him and nonthreatening to me, and it allowed us each to feel some degree of comfort as we sought to figure out how to live our lives together. Rick saw himself as a polyamorist, and I viewed myself as a monogamist in polyamorous clothing. Even as I developed close ties with local community members, I did not seriously consider multiple-partner relationships as a possibility for myself.
About this time, Rick's close friends Theresa and Jonathan visited from California. During the many conversations we had that long weekend, Rick and I spent some time discussing our quasi-polyamorous relationship with them, and Theresa called several weeks later to tell me she had a crush on me and wanted to see what developed. Theresa and I included Rick and Jonathan in several phone conversations. Rick was thrilled. He thought that a quad with close friends could be another form of the ideal alternative family he sought. Theresa was uncertain if she wished to rekindle the sexual relationship she had had with Rick years ago. Jonathan was similarly ambivalent about a sexual relationship between Rick and Theresa, though he said he was "fine" with a sexual relationship between Theresa and me. Jonathan thought he might be able to become attracted to me, though, and considered triadic sexual encounters with Theresa, him, and me a strong possibility. Characteristic of long-distance poly relationships, we spent a lot of time trying to get to know each other more with phone calls and emails as we considered what shape the relationship might take, given the fact that we lived almost two thousand miles apart.
Once again, Rick and I did not test our agreement by taking additional lovers. Our discussions, however, progressed, and we decided to finally completely open our relationship to other partners. In retrospect, our reasons seem clearer to us now than they did then. I had grown exceedingly tired of the conversation and the endless consideration. Rick remained hesitant to seek a lover himself and hoped that, if he just waited long enough, I would find us a woman. So we agreed to an arrangement that allowed us to seek a girlfriend in our local area, which neither one of us intended to implement (though we did not know that of each other at the time).
### Entrée into Polyamory
During this period, I traveled to California in the hope of collecting data that were more varied from a more racially and ethnically diverse polyamorous community. While in California, I visited Theresa and Jonathan. The visit was awkward for me. I felt increasingly uncomfortable as Theresa expressed her amorous feelings for me verbally and physically. I admitted that I was not experiencing reciprocal feelings, and she retreated, feeling hurt and rejected. I left Jonathan and Theresa's house feeling confused, upset, and more convinced of my lack of desire for multiple partners. If I did not wish to explore a sexual relationship with these wonderful people whom I already loved as friends, I reasoned, there must be no one other than Rick who would spark my interest. I thought, after all, I must be monogamous. I enjoyed the interviews and had a great time socializing with community members, but once again I refrained from sexual interaction.
I called Rick at home and explained my lack of desire for Theresa and all the other suitors in California and across the years. At that point we had been discussing our potential poly relationship several times a week for almost ten years, and I had long since wearied of the conversation. I told him that I was probably never going to be actively polyamorous myself, but I would consider a poly/mono relationship in which he would be polyamorous and I would remain monogamous. After even more discussion, he created an online profile and began attempting to date.
While Rick and I moved with glacial slowness toward polyamory and discussed every aspect of it in excruciating detail, others rush in, and still others evolve into the relationship. In chapter 2 I discussed different ways of thinking about and being polyamorous, and my findings are that those with a polyamorous sexual/relational orientation frequently established open relationships in their youths and came to identify as poly later. More often, people would have monogamous relationships first and then become polyamorous either by a purposeful decision or through happenstance. Because Rick and I exemplify an extreme example of purposeful decision and I discuss that in such length throughout the chapter, here I focus on relationships formed via happenstance.
The most successful poly relationships that begin with happenstance usually include people exploring the possibility to mutually open the relationship prior to anyone acting as if she or he was in an open relationship. People who begin acting as if they are in open relationships and then try to negotiate their boundaries in retrospect—have affairs first and then confess or get caught and then try to discuss opening their relationships—often have disastrous results. Because honesty is crucial to the successful function of poly relationships, they often implode when founded on dishonesty. In the case of polyamory, it is generally much better to ask for permission first than forgiveness afterward.
There are some significant exceptions to this rule among my respondents. The Bayside triad with Cal, John, and Sara began with an affair between Sara, legally married to Cal, and John, in the process of divorcing someone else. Sara reported that her affair with John had been going on for barely a month, and she knew Cal was going to find out about it because
> he did the bills, and there was this new number all over my cell phone bill with like, hours of conversations in the middle of the night. So of course he was going to figure it out, how could he not? I was kind of deer-in-the-headlights, frozen waiting for the disaster to start, and it never came.
>
>
>
>
> Cal: We worked it out. We had never been super invested in monogamy, so it was not as big of a deal for us as it might have been for some other people.
>
> The first thing I did was to talk to John about it. I called him up, because I had his number you see (laughing), and asked him to meet me in [a neighboring part of town].
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> John: I was stunned, I thought he wanted to meet so he could knock my block off. Instead, we sit there having coffee and talking about Sara. It was really weird at first, but I slowly came to realize over the next few, the next couple of times we saw, I met him, that he really was OK with it and wanted the best for Sara.
We discussed the evolution of their triad for some time, with the three explaining how they evolved to a more comfortable state together. Eventually I asked, "So why is monogamy not a big deal for you?" and they responded:
> Sara: Even from the beginning, we had trysts with other people. I remember at our wedding I was taking the best man, Cal's good friend, up to the roof to have sex with him and when we got there, Cal and one of the bridesmaids were already there. He was like, oops, pulling up his pants. We all laughed about it and went back downstairs.
>
>
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> Cal: It is also where we live. The Bay Area is really free, sexually open compared to most places, so it was easier and lots of people just kind of went with it. Our friends didn't think it was a big deal either, well, some of our friends. Others were more monogamous, but even so . . .
>
>
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> John: It actually had been a big deal for me, and my ex-wife. Neither of us were "faithful" [air quotes] and we both used it against the other, so we weren't really monogamous but we fought about it a lot.
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> Elisabeth: So Sara, if monogamy is not a big deal, why not just tell Cal that you wanted to have an open relationship? Why not just be direct from the get go?
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> Sara: I have wondered that myself, and I have to say I don't really have a good answer for that. It just kind of happened, I didn't plan it, I didn't even really know it was fully happening until it kind of already started, I just kind of found myself doing it. Really, I wish I had a better answer for you, but there it is.
This relationship, begun by happenstance and not negotiated, was one of the few that I observed continuing beyond the initial attempts to slide from monogamy to polyamory.
In addition to having a flexible attitude toward monogamy—something that is crucial for polyamorous relating—one of the main reasons that the Bayside triad was able to make the transition from affair to triad was because of the emotional health of Sara and Cal's relationship. Sara's deception regarding her relationship with John was an isolated incident, Cal had loved and felt loved by Sara for many years, and he was confident enough in himself not to be completely undone by his wife's desire for another man. If this was simply another in a long string of lies and cheating from Sara, or had Cal been more jealous or emotionally excitable, things would have most likely turned out quite differently. As it was, Sara told the truth the vast majority of the time, and as a result Cal trusted her, and Cal was an extraordinarily calm and thoughtful person. In fact, John sees Cal's equanimity as advantageous to his relationship with Sara, even beyond its calming effect when Cal did not "knock his block off." John reported that
> Sara and I are both kind of high strung sometimes, a little more reactive, a little bit wooooaaaaa occasionally. Cal though, he is solid. He can calm us both down and help us communicate. I can think of at least two different times we would have broken up if Cal hadn't been there to help us talk it out. He keeps us grounded, helps us work it out. He really is an amazing guy.
Cal's calm personality, both men's ability to accept open relationships, and John's reciprocal willingness to connect emotionally with Cal enabled the three to become friends, rather than rivals or enemies. The men clearly admired each other and loved Sara deeply.
For her part, Sara reported being
> completely blissed out. Things are good and have pretty much always been good with Cal, we are really good together, and that is super stable. John, on the other hand, is a little flighty, more intense, and as kinky as a cheap garden hose. So I get to explore new areas of myself with him, try new things that maybe Cal is not that into, and just be a different person with him . . . And the sex is great. It takes both of 'em to keep up with me [laughing]. Just the other week we were out for afternoon coffee. The girls [Sara and Cal's sixteen-year-old daughter, and John's eighteen-year-old daughter] were out somewhere, doing their thing, and we had "slept late" [air quotes] and walked down the block to get some coffee. We were cuddled up in the booth together, both men kind of petting me or touching me, and the guy at the table across the way said "You look like you're happy" and I had to smile because I really was. So we must have just been radiating our satisfaction, for this guy to notice it, but yeah, things are great, they are really really good. I am grateful every day for the wonderful men in my life.
Cal, John, and Sara found that a relationship begun in dishonesty can still thrive if other elements of it—honesty in most areas, trust, love, a mutual commitment to work it out—are stable and functioning positively.
### The Veto Trap
Eventually Rick responded to two online ads—the first time either of us initiated a search for an additional partner rather than simply discussing it. Rick's Internet dating was unsuccessful, and his pursuit of it was lukewarm at best. During this phase, I requested that we institute a "veto policy" in which either one of us could ask the other to stop seeing someone if we felt that the other person was a threat to our family. At that point we both thought the veto was to help me feel more comfortable with any girlfriends with whom Rick might become emotionally intimate.
Characteristic of newbie mistakes and couple privilege, many poly relationships have foundered on the impact of the veto. In our case, I was the one who pushed to include veto power in our agreement because I wanted Rick to keep his ultimate loyalty with me and the kids. In other cases, men are the ones who stipulate or use the veto, and for still other relationships both partners occasionally implement veto power. Among most long-term polyamorists, however, the existence and use of the veto is a red flag signaling couple privilege, insecurity, and poor relationship skills. In our case, the veto was initially a symptom of my insecurity with polyamory and my place in the relationship with Rick. At root, the assumption that the couple relationship should supersede all others and be protected at all costs is definitional to couple privilege, and so the veto becomes emblematic of the couple's ultimate ability to preserve itself.
Rigid rules in general can in fact be a sign of couple privilege in that they usually function to preserve the sanctity of the couple relationship. Tacit Campo commented that
> I talk to people all the time about relationship structures, and every time it comes down to things like veto power or rules-based relationships I always hear the same defense: "Well, if all the people involved are happy, what's wrong with it?" Thing is, I don't know that I've _ever_ seen anyone with this approach to relationship who is actually happy. "Not feeling jealous right at this moment in time as long as the rules are followed" isn't the same thing as "happy."
Rather than a veto power and other rules, practiced polyamorists tend to rely on each others' good judgment, ability to work through difficulties, and positive regard for each other. Phoenix and Zack found that, after many years together:
> Zack: We didn't need a lot of rules with each other. We know each other well enough and trust each other to have our best interest at heart, so rules are pretty much moot at this point. We simply lead our lives in accordance with what works best for us and our relationship, and that tends to be a fairly consistent thing. If something comes up we talk about it, but usually there is rather little we need to negotiate now. We've been doing this long enough that we know what we're doing.
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> Phoenix: A lot of it is just common sense. We don't have to have a lot of rules about who sleeps where and what kinds of sex we can have because we are aware of each other's needs and boundaries. And we know we practice safer sex anyway, so we don't have to rehash it all the time.
While he disavowed rules in general, Tacit later acknowledged that there were some rule-ish kinds of things that could structure poly relationships.
> Common rules that I have seen in real-world healthy poly relationships among happy people . . . tend to be few . . . The ones I tend to see include:
>
> * Communicate openly.
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> * Be honest.
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> * Negotiate safer-sex arrangements and make sure everyone is in the loop about changes to sexual status.
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> * Be compassionate.
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> * Take responsibility for yourself and your actions, including the unintended consequences of your actions.
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> * Be flexible.
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> * Don't be a dick.
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>
Rather than an externalized list of detailed rules, these experienced polys used an internally directed approach, applying their ethical perspectives to the situation of the moment. Above all, "Don't be a dick/Be compassionate" to your partners and other people is the primary ethic underlying these poly structures, and the others flow from there. As Cliff put it, "It's not about controlling the behaviors of other people with rules, it's about how not be a dumbass."
## Complete Membership
Gradually I took on a complete group membership role, meaning that I had a polyamorous relationship and thus became a community "insider." It is only in retrospect that I see this transition; at the time everything was so murky that I did not identify a specific moment or incident in which Rick and I became polyamorous or I adopted a poly identity.
### Friendship with Steve
My friendship with Steve began during this time, over a year after I had interviewed him. At that time we had acknowledged mutual intellectual interests during the interview and discussed seeing each other socially, but neither pursued it. Steve heard that I had returned home through the polyamorous grapevine, emailed me, and we discussed spending time together socially. He lived nearby and came over one evening for a spur-of-the-moment walk with me, Rick, and the kids. We quickly began seeing each other several times a week. I was fascinated by his sharp mind and charmed by his wit, and I was well accustomed to becoming friends with people related to my research and thought that my relationship with Steve would fall into the well-worn groove created by my numerous platonic relationships with local polyamorists.
Steve began to drop by to help me in the morning, a time of day that had proved particularly onerous for me as I tried to get myself ready for work and the kids fed and dressed. Even more important than his assistance, Steve's companionship was wonderful for me. Recent difficulties with other friends had left me in search of additional friendship and support, and my friendship with Steve met those needs, rapidly becoming increasingly emotionally intimate. Initially, Rick and Steve enjoyed spending time together. Rick and I would have Steve over for dinner, walks, park outings, movies, and late-night conversations. Our time together as a group usually included our infant and toddler, who quickly came to see Steve as another one of his adult friends (the infant was too young to articulate the concept of a friend). Within several weeks, however, Rick and Steve began to irritate each other. Steve arrived for his morning visits after Rick had already departed for work, so Steve and I spent far more time together independently than the three of us did as a group. Rick became increasingly alarmed at the growing bond between Steve and me.
Roughly two weeks after we began seeing each other as friends, Steve said that he had fallen in love with me. "I didn't mean to," he reported. "It just happened." I was shocked. I was just getting to know him and already he was in love with me? I wondered if it were true, and why he moved so quickly. I told him I liked him a lot and would like to continue to spend time with him, but I did not, even now, think of myself as a polyamorous person and was really seeking a platonic friendship. Steve was confused by my lack of polyamorous identification and pointed out that Rick and I were in an open relationship. I replied that yes, Rick and I were in an open relationship, but I had never intended to become romantically involved with others. Steve's confusion remained, but he agreed to a continued friendship. Rick was displeased by Steve's admission and relieved that I did not return Steve's feelings.
### Rick's Polyamorous Relationship
At this same time, Rick had a short fling with Joya, a good friend of ours from the local polyamorous community. We had socialized regularly with her and her son, and our children had become friends. Joya also provided paid childcare for us for several months, watching our children so I could write. She and I routinely attended numerous women's support group meetings together. Joya was also homeless (or as she preferred, a "postmodern Gypsy") for much of this time, staying with various lovers, friends, or her ex-husband. She would sleep on our couch once a month or so, sometimes more. She had a key to our house, and she knew she was free to come in whenever she wished.
Very late one night, Joya dropped by on her way home from work to pick up something she had left at our house. The house was dark, and she thought everyone was asleep. She let herself in to silently retrieve her belongings and depart. Instead, she found Rick and I snuggled on the floor in a pile of pillows with a bottle of wine. Joya joined us on the cushions where we snuggled and chatted. I became uncomfortable as the snuggling took on a decidedly more erotic tone. My stomach began to ache, and I realized that I was just not comfortable having a threesome that night, if at all.
I excused myself, saying I was exhausted because it was three a.m. and I had been up since six a.m. the previous day. Rick came into our bedroom to see if I was okay. I replied that I was fine, I was tired, and I did not want to have sex with Joya. He should, I informed him, go back into the living room and have sex with her. She was clearly open to it, and I wanted Rick to finally try polyamory so we could finally settle the issue.
Rick was hesitant, but after some discussion, he returned to Joya in the living room. I briefly anticipated lying awake, nervously listening or feeling uncomfortable. Then I immediately went to sleep and did not awaken until I felt Rick and Joya join me later. I moved over and the three of us slept for the rest of the night.
I awoke last the next day, amazed to find myself nonchalant about the whole incident. Joya, off to work hours before, had left a note asking me to call her and tell her how I was feeling. Rick called from work to check in, and I informed him of what I found to be a surprising lack of reaction. I kept waiting for the onset of insecurities and jealousy I had feared for these many years, and they never came. I surprised myself by feeling just fine about it, and I began to think that I might be able to do this, after all. Granted, Joya had been a close friend for several years and I trusted her. I knew that she respected me and would never try to get Rick to leave me for her. After that night, Rick and I continued to socialize with Joya. Rick and Joya only had one more sexual encounter while I was out of town. After that, neither Rick nor Joya put much effort into arranging additional sexual liaisons, and both expressed to me that they were less interested in each other as individual lovers than as components of a trio that included me.
Joya already knew Steve through connections in the local polyamorous community, and she began to socialize with him at the house Rick and I shared with our children. Joya, Steve, Rick, and I spent a few late nights together, sitting around the kitchen table discussing polyamorous life and our relationships. During one of these late-night kitchen conversations I expressed some dissatisfaction with a long-term communication pattern in which Rick and I routinely engaged. Steve and Joya held views similar to mine, and Rick began to feel "ganged up on" when they both intoned their opinions. After a series of negative communication, the relationship between Rick and Joya deteriorated to almost nothing.
Initially, the relationship between Joya and Steve was cautious. As they came to know each other, they established a friendship and ultimately united in their common opposition to Rick. Concurrently, Rick had a series of negative experiences that changed his mind regarding the desirability of polyamorous relationships. Joya's anger and rejection mystified and alarmed him. In California, Jonathan discovered Theresa's ongoing adultery and filed for divorce. Rick worried that the introduction of the idea of multiple partners into their relationship had spurred Theresa's infidelity, and that the two might still be married were it not for our discussions of polyamory. Most importantly, Rick did not like Steve and was becoming increasingly uncomfortable with my friendship with him. The combination of these factors spurred Rick's rising unease with the practice of polyamory.
### I Fell in Love with Steve
Soon after the night of the argument in the kitchen, Rick asked me to make an appointment with him so we could discuss our relationship during the day, with no children present. I agreed and we arranged for my mother to spend time with the children that coming Saturday so Rick and I could talk. Coincidentally, Rick, Steve, and I had arranged a few weeks before to go dancing with several other polyamorous friends that Friday night, the night before the scheduled discussion. Rick asked that we refrain from serious conversation regarding our relationships that night and just "have fun," and I agreed.
That Friday evening during the dancing I felt a significant shift and realized I was in love with Steve. Steve could tell something had just happened and wanted to know what it was and so, even though Rick had requested a night free of serious conversation, the three of us left the dance club and took a walk. Rick requested some time alone with me before I told them both what was on my mind and, characteristic of couple privilege that relegates secondary partners to a secondary class in service of maintaining the couple relationship, we asked Steve to give us a moment. Steve went away for fifteen minutes, and Rick reminded me of our conversation date for the next day. He confessed that he had been planning to ask me to abandon polyamory in favor of a monogamous relationship. Initially I was stunned, aghast that both events (his desire to quit polyamory and mine to engage in it) should happen at once. Then I became irate, livid at what I perceived to be Rick's constant harassment of me to engage in polyamory for ten years and then his rapid reversal as soon as it was a man with whom I wanted to have a relationship rather than the woman Rick had envisioned all these years. Steve returned and I told him I loved him, but that Rick did not want us to be together. Steve was very angry with Rick and yelled at him briefly before I stepped between them. In our continued discussions over the next several days Rick explained that he was not necessarily requesting that I forgo polyamory completely, only a relationship with Steve specifically, whom he found increasingly problematic. I replied that Steve was the only person I had been even remotely interested in over the many years of interaction with numerous potential suitors, and that I was not interested in polyamory in the broad sense but in a relationship specifically with Steve.
Rick effectively "vetoed" Steve, asserting his desire that Steve and I discontinue any form of relationship. While I had initially been the one to request a veto policy in order to protect my relationship with Rick, when the time came that it was my relationship that was being vetoed I balked, arguing that he had bullied me into polyamory in the first place. Those agreements I had made with him were not valid, I argued, because they were made under duress and thus not binding. Rick felt betrayed by my refusal to honor such a foundational element of our agreement, especially when he had agreed to it at my behest. I felt betrayed by what seemed to me his willingness to sacrifice years of my emotional comfort in order to have access to outside partners, only to suddenly reverse his position once it appeared that it was I, and not he, who might have multiple lovers.
### 180-Degree Turn About
Rick and I had switched places: now I wanted to have a poly relationship with both him and Steve, and Rick wanted to be monogamous. Common to other newbie disasters, the way the relationship ended up working out was far different than either of us had imagined, and we had a hard time adapting to it. With what in retrospect appears to be eerie prescience, Joya Starr told me in her initial interview in 1997:
> It is a poly phenomenon that often the woman in the couple is kind of reluctant and is dragged kicking and screaming into poly. Then when the man is done with his experimentation, the woman often finds that it suits her character and stays with it. It is almost like acquiring a skill, once she's got it, it becomes part of how she wants to live her life. It can be real confronting when the man wants to become involved in the poly lifestyle and then finds out that it is really much easier for a woman to establish relationships, and not only do they establish them easier, they tend to get more intimate and deeper faster, cause that is what women are good at. Speaking generally, women like that kind of stuff. So the men can become very uncomfortable.
Little did I know how accurate her prediction would be at the time and how closely my relationships with Steve and Rick would mirror the cliché "poly phenomena."
### Things Fall Apart
Rick and I sought couple's counseling and found it effective in improving our communication, and I sought personal counseling. I tried to maintain a platonic friendship with Steve in order to give the men a chance to work out an amicable relationship, but when it became clear that they were never going to be able to share me in a friendly fashion, I "broke up" with Steve. At the end of that discussion, I kissed him goodbye, which turned out to be a big mistake. Previously untested sexual tension bubbled to the surface and we "made out" with the fervor of those who thought they would never see each other again.
By establishing an amorous relationship with someone from my research setting, I made the transition to a complete membership role characterized by full immersion in the scene as a "native" who shared a "common set of experiences, feelings, and goals" with those in my setting.[7] Ironically, it was at this transition to complete membership status that I disengaged almost completely from the local polyamorous community. I was overwhelmed with emotional discord and spent so much time in personal and couple's counseling and intense personal discussion with both men that I had no time left for community interaction.
Steve and I stayed broken up for two months, during which time I tried to forget about him and polyamory. I could not put him out of my mind, though, and six weeks into the breakup I initiated discussions with Rick regarding my desire to reunite with Steve. Rick and I had not yet reached agreement when I ran into Steve in public and began seeing him again. That time it lasted for a month. I could not tolerate the tension of being pulled between the men, so I again broke up with Steve.
During this tumultuous and painful period, my greatest concern was for my children. Rick and I tried to shield them from the fighting by having the discussions only late at night when they were asleep, a strategy that was proving increasingly difficult to maintain. We were exhausted by our late-night discussions, and our discord occasionally spilled over into daylight hours. Although the children were young enough to remain oblivious to the majority of the content of the arguments, they were still uncomfortable because they could feel the tension and could tell the adults were upset.
In the midst of this turmoil, I saw an adjunct job listing at university in a town where my sister and brother-in-law resided. I accepted the adjunct job, Rick and I sold our house, and we moved away from the area we had recently shared with Steve. While I wanted to be near my sister and the job offered the potential for a professional future, these incentives would not normally have been sufficient to spur me to quit my job, sell my house, and move my family to another state. The added impetus of my overwhelming desire to see Steve and the negative impact that had on my relationship with Rick drove me to drastic action.
### Relationship Drama
The dramatic, almost soap-operatic tone my relationships took on at that point is characteristic of the worst of polyamory. Other respondents and their children routinely mentioned the drawbacks of emotional pain, complexity, and dramatic relationships as negative aspects of polyamory. Especially potent at first before newbies have had a change to practice managing jealousy or negotiating safer-sex agreements, the complexities inherent in this challenging relational style magnify relationship strengths and weaknesses, creating higher highs and lower lows for those involved.
This is not to say that all poly relationships are drama laden—much like monogamous relationships, there is tremendous variety in how people handle their multiple-partner interactions. In our case specifically, our high expectations and inability to shift our agreements with changing life and relationship circumstances made it very difficult to work things out smoothly. In other relationships, there is a marked lack of drama. Some people, especially those who have practiced for many years and established long-term poly relationships, have honed communication skills and crafted agreements that are low drama and low maintenance. They simply function, with normal family issues and no particular upheavals.
Both the Campo and Wyss families are exemplary of calm, nondramatic poly families. In each case, family members have gone through significant transitions with each other and the form of the family has shifted over time. Things happen in their family lives—the death of a spouse in a traffic accident, a traumatic brain injury, cross-country moves, divorce—and they use their skills to deal with it. They are not constantly freaked out; their family lives flow smoothly without recurrent drama. Tacit Campo explained the easy functioning of his family:
> You keep asking me how I deal with the complexity but I am really not finding my relationships all that complex. It is what it is, and it works well for all of us. The primary worry in our lives is financial because the farm is in such dire straits, that is where the tension comes from. And we deal with it as best we can, but things are good in many other ways. There is a marked absence of the drama and complexity I hear about in other poly relationships, I think because of how we handle ourselves and the fact that no one had to be talked into becoming poly.
A few adept polyamorists were able to manage these myriad pitfalls, but such navigation took skill reliably developed only through practice. The majority of polyamorists had at least some relationships filled with pain and drama, and they did not tend to continue with those relationships. Most often, they took what they learned about relationships, themselves, and their partners and went on to form other relationships with new people or deepen existing relationships. I do not discuss drama in depth here because drama is such a constant theme throughout the book that I am certain readers will have many opportunities to understand it in context.
## Withdrawal from the Poly Community
My disastrous relationship, moving to a new place, and the fact that I had completed the first wave of data collection all combined to propel me out of contact with the poly community. After our initial slow-motion disaster with polyamory, both Rick and I were wary of polyamory and focused instead on trying to strengthen our relationship with each other. For several years I did not collect any data on polyamory, though I continued speaking on the research findings at academic conferences. Audience members would routinely ask the same questions—"What about the children? How does this relationship style affect the children?" While I could provide data from my years of participant observation and watching poly families interact, I could not answer questions about the children's thoughts, feelings, or experiences because I was not allowed to ask them questions. Over time I began to wonder how my former respondents were doing and decided to see if I could find out, embarking on the Longitudinal Polyamorous Family Study.
### Picking Up the Pieces
For the next five years Rick and I tried doggedly to heal the rift in our relationship, attending couples counseling and both working hard to nurture positive feelings for each other. While I missed Steve keenly, I refrained from contacting him at that point and instead invested myself in my work at a university and my family. I sought to deal with my lingering angst with personal counseling, meditation, exercise, antidepressants, aromatherapy, chocolate, and pretending to be content. Things did slowly improve, and over a period of about three years Rick and I fought less and felt better about each other. I kept looking for ways to fall in love with Rick again, change my desire to leave the relationship, or at least manage to endure it more comfortably. Try as I might nothing seemed to work, and I became increasingly despondent. Three years into that cycle it became clear to me that our progress toward a more intimate and positive relationship had ground to a halt, and we stagnated in that steady state for two more years.
### Splitting Up with Rick
By the time the children were five and seven, I could no longer tolerate my distressingly leaden relationship and informed Rick that I wanted to split up. We had both tried as hard as we could and, while our relationship was tolerable for him, I was miserable, and it did not appear to be getting any better. Several months later I moved out, and we began to share custody of the children, who stayed with him most of the time but came to me regularly. Although the fact that we had tried so hard, told each other the truth, and refrained from cheating on each other made it easier to continue to trust each other and collaborate as coparents, it was still a painful time. I tried to reestablish contact with Steve at that point, but he wanted nothing to do with me and I don't blame him, after the way Rick and I used couple privilege to try to protect our relationship at Steve's direct emotional expense.
### Coparenting, Amicable Divorce, and Relationship Continuity
Characteristic of some polys whose romantic relationships implode but whose friendships are able to weather the transition, Rick and I focused on maintaining positive relations with each other so we could coparent with the least conflict possible. At times we have been good friends again, supporting each other and coming to each other's assistance in times of need. Other times we have fought because our issues did not magically resolve with the end of our romantic relationship. It is easier now, and ultimately we both want what is best for the other. This positive regard allows us to mingle comfortably at family events or school functions, and we both continue to interact with each other's extended families.
Melody Lupine reported a similar experience with maintaining positive relationships with both of her ex-husbands from her triad:
> We stayed really good friends and one of the things that polyamory really benefitted from was we didn't need to hire attorneys, we didn't do the battle, the custody thing, it was all agreed upon and we, I actually wrote up the divorce papers. We wanted what was best for the kids . . . and so I actually saw a real benefit from polyamory in the sense that I still loved this person and that's why we had gotten together in the triad. It wasn't about, just because something doesn't work out or things don't go right that you just hate this person and then, I think polyamory really gave me a view of, just because some things don't work out doesn't mean we have to become enemies and so we kept a good friendship through it. Definitely a positive related to polyamory, a benefit.
In fact, Melody was able to maintain such a positive relationship with Quentin that he routinely attended family functions such as birthdays and holiday dinners:
> Quentin and I had a tumultuous time when we fought for custody of Zane and I moved out here. It got pretty ugly. We are in a loving relationship right now; he is very supportive. He even brought this up just this week with my birthday, saying how remorseful he was that things didn't work out with us. Anyway, life is interesting, that we can even express that to each other. So I know that even though there may be things you go through, you can still have this connection and keep it. Even though Quentin and I aren't in this romantic or sexual or intimate relationship, we still love each other, that love is still there . . . When we had a birthday party for Zane, we all went out to a family dinner. Quentin's part of the family, we go to all of Zane's soccer games together. My new partner went to his birthday party, and he's like "oh, you and Quentin sit together" and then we took pictures together and he's like "Wow, he's your ex!" When Pete graduated, Quentin went to the graduation. He's going to be at Joyce's graduation. He's family, very much so.
Melody related her ongoing connection with Quentin to their mutual willingness to work things out, something that was currently lacking in her relationship with Cristof, even though their initial divorce had gone well. I asked her how things had changed with her and Cristof, and she stated the following:
> Elisabeth: So your relationship with Cristof, it sounds like, at some point, it went from you two having a relationship to your relationship being around his contact with the kids. Am I reading that correctly?
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> Melody: Yeah, when he got into a monogamous relationship, that is when things shifted in a lot of ways. With us, with the kids. You know, he came into this monogamous relationship that no longer included me, which was fine; I was fine with that. "Let's just still have a connection." But then it became about the kids, and when I would confront him about "Hey, your kids over here" then it got tenuous between us. As he got further and further away from his kids, now it's like, we don't talk at all.
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> Melody: No. Because as the kids turned eighteen, both of them, I told him, when both of them turn eighteen, the relationship between you and your dad is between you two. When you were under eighteen I was the mediator in between, but now whatever you guys want to make it, that's up to you.
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> Melody: He was different. He moved out here. He kept saying, "I don't care how rough it is, I still want to stay connected to you and Zane." I think we had to go through that ugly custody battle thing to really get through some of our stuff. After that, he moved out here and kind of looked at what his priorities were. We keep a lot of personal distance, but it still works, absolutely.
Cristof's decision to pour all of his time and resources into his "new" family contrasted sharply with Quentin's decision to move to be closer to his child and coparent.
### Now What?
For two years I barely dated, focusing on my work, children, and playing roller derby. I was very clear about my relationship priorities: My children were (and remain) my "primaries," meaning that I consider them first when I make decisions, allocate resources such as time or money, or face any significant life issues. Emotionally, my secondary relationship was with roller derby, and dating came in a distant third. Characteristic of both the polyamorous tendency to tailor relationships outside of conventional roles and the propensity for some single parents to prioritize their relationships with their children over dating, I invested myself most deeply in work, playing a sport, and platonic relationships with family and friends.
When I eventually did begin to date, I found that poly people regularly contacted me online, and the dating website routinely suggested matches with people who ended up being polyamorous. Unsure of what I wanted or how I felt, I did not (and still do not) identify as poly, but I did not demand or offer sexual exclusivity. I eventually dated several poly people, and I had fun with them but did not identify it as a polyamorous relationship on my end because I was not emotionally invested, and I was certainly not in love. My dates may have seen me as polyamorous, and even possibly defined me as a tertiary or even secondary partner, but we never had a "define the relationship" talk. In complex and shifting relationships, it is quite possible that different people will define the same relationship in quite different ways.
After five years of single life, sometimes dating and sometimes not, I established a deep emotional connection with "Ann." Reminiscent of my attitude when I was in love with Rick, I felt no desire to see other people and transitioned some other relationships to platonic friendships. Ann had not asked me to do so, and I am not sure if she even knew—the impetus to be sexually exclusive came from my own experience of emotional intimacy with Ann and lack of desire for sexual contact with others. While Ann and I were both aware of polyamory and thought that we might consider the possibility at some future point, we agreed easily to monogamy and maintained that throughout our relationship. Eventually we broke up for reasons completely independent of this research.
For the last year and a half I have been in a "monogamish" relationship with Kira, a woman I adore. We are monogamous in practice, with the flexibility to allow things to spontaneously happen with other people—the "make out with someone in a bar" pass. Like my previous experiences, I am finding that being in love with someone significantly dampens not only my desire for, but also even my awareness of, other people. While in theory either of us could have another partner (or at least a fling) at any point, neither of us are particularly interested in seeking other relationships. In a life that sometimes feels too busy with children and work, I have a limited amount of time and attention for relationships and want to focus on nurturing the one I already have.
In light of this experience, I have concluded that I both believe in and can practice polyamory, but I am not polyamorous by orientation. Because my sex drive is insufficient to meet even Kira's needs at this point, I always have a lot to do, I enjoy time alone, and above all I am devoted to my beloved. I am in no rush to establish additional relationships—or even make out with anyone in a bar. I am, however, happy to have the freedom to do so if I wish, and I am glad that Kira and I have agreed to such flexibility. Because I seem to prefer monogamy when I am emotionally invested, I do not identify as polyamorous. If there is one thing I have learned in my study of poly folks, it is that people and life circumstances change over time, and frequently things end up turning out quite differently than anyone would have anticipated. If at some point in the future my relationship status changes, then my identification may change as well.
Melody Lupine made a similar transition to monogamy after having been in polyamorous relationships for many years, though hers was far more intentional, where mine was experimental and accidental. Melody reported that she
> realized that I had not yet experienced a conscious monogamous relationship. When I was married to my husband [Cristof] for seventeen years it was very much unconscious and was doing it more out of healing and repeating patterns from my family of origin and learning stuff, and so now what I wanted to create was what I considered a conscious monogamous relationship because I haven't experienced that yet. And so that wasn't very much in line with polyamory, so it was very hard having this background of polyamory then to make a shift back into monogamy.
While Melody had already begun to grow increasingly "disheartened with pursuing a polyamorous lifestyle in the sense that I was looking for one committed relationship," she shifted more to conscious monogamy when she was diagnosed with a serious illness. In a support group session for women with life-threatening illnesses, Melody had an epiphany that the significant role she played in the national polyamorous community was not working for her because
> polyamory was no longer congruent with where my life was going and so I realized that I no longer wanted to be the spokesperson for the polyamorous community because I wasn't finding it working in my own life and that I had to represent a population who wasn't always doing poly the way I believed poly. Many people that were just cheating around and having affairs and saying "well, I'm poly," so after being so prominent in national leadership for about four or five years I was just, I realized it was not something I wanted to do any more.
Instead, Melody said that she wished to create a relationship based on conscious monogamy:
> I think there is a level of depth and connection that I wanted to experience. When I say a conscious monogamous that wasn't there in the, just because I'm a different person than I was then and again its just what I am choosing to experience at this time in my life. I just want to take this relationship with this one person, see how much I can learn and grow and go deep with this one person. I know when you try to include more than one, I have unlimited love, so I have experience. But time and energy are limited resources, so with that, the more relationships you bring in, the less time and energy you have for each of them. I'm not saying that polyamory is bad or wrong, its just that right now that's not what's working in my life and what I choose to do, but its nice to know that it is a choice, because before it wasn't. Now I can choose monogamy, I can choose polyamory, I can choose to be heterosexual, I'm at choice. My choices can change.
One of the choices Melody saw as remaining a possibility was the potential for a flexible vision of monogamy:
> I still consider myself polyamorous in my _views_ , I am very poly _oriented_. I'm open, I'm not jealous. I can be open to sharing my partner under the right circumstances. Actually, in this conscious monogamy we have an agreement: For our sexual learning, like, my partner right now has never been with two women and him. I told him, with agreements, I would totally go into that situation to give him that experience. If that would be growing and learning for him, then I am not jealous. I am not afraid to enter that as a one-time experience. I would embrace that with a sister, as long as it didn't take anything away from the relationship and added something to it.
After coming to realize what I call the _polyamorous possibility_ , or that it is possible to maintain happy and healthy, openly conducted, nonmonogamous relationships, Melody retained inflections of those options, as well as the poly ethics of self-growth and rejection of jealousy.
Melody is unique among my respondents for a number of reasons, and especially germane to this section are her prominence in the national polyamorous community in the United States and her willingness to participate in the follow-up study. Of the fifteen respondents I was able to find from my first wave of data collection, Melody was the only person who had become monogamous and still chose to respond.
## Jealousy and Compersion
Jealousy is a significant issue in polyamorous relationships, and poly community members spend quite a bit of time and effort thinking and talking about jealousy. _Compersion_ is a word polyamorists made up to describe the opposite of jealousy, or the feeling poly people get when they observe their partners happily in love with someone else. While some people anticipate being jealous, others are surprised by their own reactions.
Because polyamory can be so complex and surprising, many people anticipate their multiple-partner relationships working out differently in their imaginations than the actual relationships do in real life. In my case, this translated into my virtual certainty that I would be uncomfortable and jealous if Rick had other partners, and his virtual certainty that he would not experience jealousy of me and my imagined girlfriend. What actually happened surprised us both: I was not at all jealous of Rick's tryst with Joya, and Rick was quite jealous of my interactions with Steve.
James Majek discussed the common polyamorous event that a secondary partner is not jealous of the primary partner but becomes jealous if his or her partner establishes a relationship with another secondary. When Morgan began dating Nash, James reported that he
> had a meltdown . . . Well, I had just the classic jealousy reaction. No anger, just freaking out thinking or feeling like I was going to lose this person [Morgan]. I think it's the standard fear that if someone else gets into this other person's life, that they're going to find out that this other person is better and I'm going to lose.
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> James: See, that's what's interesting to me is that I never once had the slightest bit of jealousy around Clark because I see him as an integral part of the relationship that I have with Morgan because it's not just her, it's everything that comes with her. It's him; it's her kids. So to me, I want them [Morgan and Clark] to be doing well because when they're doing well, Morgan and I are doing well . . . with Clark and Melissa, as opposed to Morgan and Nash, I had just been with Melissa for so much longer. We've been married about ten, but together about eighteen years now. To me, there was a very mature relationship already in place. I didn't feel like this was a huge threat because we were talking about it seriously and openly . . . It's a kind of relationship that I consider rock solid and stable, difficult at times, but we really know each other. My relationship with Morgan is younger, more uncertain. If all of this happened within two years of Melissa and I being together, I think I would have freaked out because it's too uncertain. Here all of a sudden someone else comes along, what the fuck is this going on now? With Morgan it was "OK, I'm not settled in what we have yet. I don't know where this is going." So to have someone else coming in scared the shit out of me in a way it didn't with Melissa.
Characteristic of polyamorous community experience and rhetoric, James identified differing types of and reasons for jealousy, connecting it with uncertainty and fear of loss. What provoked jealousy with Morgan was unproblematic with Melissa because James felt more secure in the durability of his marriage and the clarity of his communication with Melissa than he did in his relationship with Morgan. In this and many other instances, jealousy proves situational and dependent on the specific tenor and interactions of the people involved.
Polyamorists also identify specific scenarios that can provoke jealousy and detail strategies to manage those situations. One recurrently problematic scenario involves one partner in an open couple with children establishing a relationship with a new person. Common poly community wisdom identifies New Relationship Energy or NRE as a booby trap of jealousy. NRE is the exiting, almost effervescent feeling one gets at the beginning of a positive relationship in which everything the other person says is new and fascinating, before reality and mundanity tarnish the exciting glow of perfect possibility. Unless they anticipate and compensate for the phenomena, people in the thrall of NRE are likely to neglect their longer-term partners in favor of the excitement of a new relationship. Colette—a white project manager in her late thirties—described how she might react if her husband, Bruce, became enamored with someone else:
> Yeah, I could totally see being jealous if he hooked up with someone he really dug and they ended up having lots of dates and I took care of the kids all the time. I love him and I want him to be happy, and I really do want to be poly, but I just don't want to be stuck at home eating mac and cheese with the kids while he is out eating steak with his new flame. Right now we are both kind of dating a little bit but neither of us is serious with anyone else. For the dating, though, we make sure things even out, so if I am home with the kids in a week when he has a date after work then he makes sure to take them somewhere on the weekend so I can take a break or have lunch with a friend or whatever, date or not, just I have some time for myself, and vice versa. It is not tit for tat, not like I get a date and he gets a date, but more like we are both aware of how much personal time and time with the kids we each get. So if I ever did end up eating mac and cheese with the kids while he is at [local restaurant] with a hottie then it would be OK because I can have my own hottie, or a weekend in Vegas with the girls, or whatever, just, he recognizes that my time is important too. I know things are even in the long run so I don't really keep track.
In that same interview, Colette mentioned her confidence in the durability of her relationship and the couple's love for each other as a "secure bond" as additional reasons that she did not tend to feel jealousy. She and Bruce negotiated accommodations so that the event that Colette imagined would provoke jealousy—being "stuck at home eating mac and cheese with the kids while he is out eating steak with his new flame"—was defused by the explicitly acknowledged equity that deemed both partners equally worthy of personal time for dating or other things.
Discussing his strategy for dealing with NRE in his long-term relationship with Summer, Zack Phoenix (the person who actually coined the term _NRE_ ) said:
> I think of it as compensating for the wind in archery. If you are aiming for a target 50 yards away and the wind is blowing from the left, you angle your bow in to the wind a little bit so the arrow flies in an arc and still hits the target, even though the wind has been pushing on it. The same thing with a long-term partner, if you have this exciting new person you have just started seeing and you are really excited about them, then make sure to make a date night with your person over here, the one who might not be as exciting in the moment, not quite as new and shiny, but you love her and you have deep ties with her that need to be nurtured too. So bring her flowers, take her out to dinner, go away for the weekend together. Just don't get so blinded by the new that you forget to take care of the long-term relationships in your life.
In addition to overcompensating for the potential to ignore a long-term partner, polys also council each other to make plans or seek emotional support when a partner is out on a date. Billy Holestrom experienced the discomfort of remaining at home alone while his wife, Megan, and her boyfriend Jack went to a concert. Prior to their departure Billy had assured Megan and Jack that he was fine at home, he had a lot to do, and they should go have fun. Megan and Jack hesitated, offered to find a babysitter and to give Billy the spare ticket they had purchased prior to finding out that Billy did not want to attend the concert. After receiving Billy's reassurances that he was fine staying at home with Megan and Billy's daughter, Ariel, Jack and Megan left for the show and were gone for hours. Meanwhile, Billy put Ariel to bed and
> proceeded to have a minor meltdown, moping around the house feeling so sorry for myself while they were out having fun. Wah wah (melodramatic crying sound). It wasn't that I didn't want them to have fun, it was just that I was lonely and bored and did not want to deal with the mountains of laundry that I had on my agenda for that evening. And the fact that they had invited me and made it clear that I could come with them only made it worse, I couldn't even blame them for my being upset at them going out without me. So from then on I either go with them or plan to have something of my own to do that is more fun than laundry while they're out having a good time. It works better that way, I get a lot less upset and they don't worry about me brooding at home while they're trying to have fun.
Poly folks also report using introspection, counseling, and conversations with supportive friends as tools to investigate the underpinnings of jealousy. People who express problems with jealousy on poly websites or in support groups are routinely instructed to seek the issue beneath the jealousy that is provoking such an uncomfortable emotion. Those discussions routinely evolve to include significant focus on insecurity, determining one's needs, and asking for those needs to be met. Among long-term poly folks, the presence of jealousy signals the need for some accommodations or strategies to manage the specific situation, rather than a signal that the situation provoking the jealousy should stop immediately because it is making someone uncomfortable.
## Gender
While being extremely unusual in some regards, the polyamorous families I studied were surprisingly gender conforming in many ways. When there was a full-time parent it was most often a woman, and it tended to be women who were the family schedulers and managers. Women in these families did most of the cooking and cleaning and child care. Not that men were completely uninvolved in household and child care—certainly the poly fathers involved in this study were at least as involved with their children as mainstream monogamous men, some of them were excellent cooks, and a few of the families had full-time dads at home.
Some of this gender conformity was due to financial constraints: Men still make, on average, more money than women, and these families were no exception. It makes sense for the lower-earning parent to stay home with the children, even when there are multiple parents. Other poly families reported that desire to parent or practicality of childbearing and nursing were decisive in determining who stayed home with children. While Leah Tree changed her schedule in response to bearing the trio's son Will far more than her husbands Bjorn and Gene changed their schedules, she did it because she wanted to breastfeed Will:
> Leah: I spend more time parenting because I am nursing, and my more flexible job makes that easier. I can work from home, and we even have a nanny who comes to the house to take over with Will. I work fewer hours with basically no impact on my post-doc because there is no set job schedule I have to meet. I can coordinate the lab from home and write papers. This is a choice for me because there is not a financial cost or cost to my career to be home so I can nurse . . . all that said I also get a tremendous amount of support from the men. Especially at the beginning the men did everything, and Gene especially has been able to take flex time and be home one day per week, as well as taking three months off when Will came. So even though I spend a bit more time with Will, but it is not a big inequality and often related to nursing.
>
>
>
>
> Gene: Leah is also more likely to be around on the weekends with Will and Bjorn and I are more likely to go do something for a few hours, but we arrange our schedules so that when Will is awake we can spend time with him. I can often work at home, and I usually get up really early to have time to work before others are awake, and then I can play with Will when he wakes up so Leah can pump and I get to be with Will before I go to work.
>
>
>
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> Bjorn: We have all done a great job of supporting each other, getting as much access to Will as we can, and balancing our personal needs so everyone gets what they need. We all do everything we can, given the limitations of our jobs and commutes.
>
>
>
>
> Leah: Other than the nursing thing, the parenting and family life in general is fairly gender equitable. We each cook one night a week, and the dads usually do bath time because I spend all day with Will. Each parent feeds Will and I pump so Bjorn and Gene can give him bottles. The men put Will to bed and I do the naps throughout the day.
>
>
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> Gene: Usually one of us gets up to get Will in the morning and get him ready for the day—whoever is sleeping alone has the monitor and gets up to deal with Will. The dads sleep through the monitor unless Will is really screaming, but Leah hears the monitor, every little sound.
>
>
>
>
> Leah: Now the monitor is wherever I am not, except if Gene goes out on a date then I take the monitor.
The inequity of Leah hardly ever being on baby-monitor duty all night does not seem to bother "the dads" because, as Bjorn said, "We like to have the time with Will." Upon further inspection, this seemingly traditional gender division of labor is revealed as a carefully thought-out balancing act in which each of the adults contributes what appears to them to be an effort roughly equivalent to that of the other two, and each of the three is satisfied with how much time they get with the baby and alone for personal needs.
## Unpredictability
The prevalence of divorce and extramarital birth indicate that people in serially monogamous relationships have unpredictable relationship outcomes, such as committing to a relationship for life and then changing their minds once it becomes clear it is not working out. Considering how difficult it can be for two people to manage a dyadic relationship, it is no wonder that some—especially younger or less seasoned—polyamorous relationships can face a higher degree of unpredictability. My findings indicate that the larger the relationship and the more people involved, the more likely it is to have partners change over time. In addition to the complexity that accompanies multiple-partner relationships, the relative lack of cultural role models demonstrating consensually nonmonogamous relationships means that poly people must either construct their own patterns or find role models in polyamorous communities. Such self-direction can provide freedom and unpredictability.
In my own case, Rick had a very clear idea of what kind of poly relationship he wanted and how that should work out. I was less clear about what I wanted and did not harbor my own plan for how things would work out except that it would involve my continued relationship with Rick. In both cases, the relationship ended up being quite different from what either of us expected. Rick expected a woman with whom we could both partner, and I found a man. I expected to feel jealous and insecure, Rick expected that he would not feel jealous, and both of us were wrong.
While some people like Rick plan and envision their poly relationships in fantastic detail, in other cases poly relationships happen _to_ people, simply evolving from their existing social lives in a way that no one had anticipated. Those who plan, seek, and avidly envision a poly relationship prior to engaging in one may build an image of what the relationship will be, which might be profoundly different than they had anticipated.
### Complexity and Drama
Emotions and issues can be magnified in poly relationships, which are thus more likely to produce extremes—the highs of being crazy in love with not only one person but two or more people, at the same time, who also love each other and have had a healthy child together, can be overwhelming. The lows can be equally extreme—having multiple relationships break up at the same time or a fluid-transfer accident that involves transmitting a sexually transmitted infection or unplanned pregnancy.
All of this emotional intensity and constant communication can become wearing for some people. Kristine, a white poly woman, student, and world traveler, confessed to me that she was feeling quite exhausted by poly relationships for a time.
> I haven't seen many successful poly relationships. I've seen poly relationships where one partner tries to be poly, because they don't want to lose the other partner. I've seen women living in fear for years—afraid their husbands will leave them for someone younger, cuter, perkier . . . I've seen botox, and tears. I've heard people who had been in the poly community leave because they could no longer handle the stress and uncertainty, and I've heard people refer to poly as a graduate program in love. It can be a grueling fast lane to self-awareness, a long trip through hell, and moments of complete ecstasy. I don't advocate it. It isn't enlightened, it is complicated . . . and I've seen it bring joy, and pain.
Even though Kristine had previously been in a vee/sometimes triad with Evelyn and Mark Coach and was seeing two other people at the time of the interview (one of whom was engaged to be married to another woman), she was still not sure if she wanted to be polyamorous.
> I still wonder whether poly is worth the time, energy, and bandwidth, or whether it is just too rough on my self-esteem and emotions. The processing gets tiresome, and at the end of the day, wouldn't it be nice to simply be chosen, and cherished, and be the "most" whatever he loves me for being the "most" of? Who knows, maybe once I had my own primary, and felt safe, I would want to open things, maybe as a couple I'd be open to finding other couples. Two men one woman was lovely [grin]. But one thing is for sure—I would have to think long, and hard, and have some serious talks about commitment [before having a child in a polyamorous relationship]. The CHILD would have to be our partnership's first priority, before I would do poly and children, because NO child will do well as an afterthought.
While Kristine reported that her poly relationships had spurred tremendous personal growth for her at various points, their high-maintenance communication and emotional styles came at such a high cost that she was not sure if the growth was worth it to her.
Experienced polyamorists know that polyamorous relationships with those new to polyamory, termed _newbies_ or _first timers_ , can potentially end catastrophically. Because of this risk, many long-time polys shy away from dating new polyamorists, who are more likely to revert to monogamy once the inevitable complexities of polyamorous relationships begin to take a toll.
Emmanuella Ruiz related her concerns regarding her new lover, whom she called "theoretically poly":
> Poly in the abstract, no experience, no context. And in fact has a jealous streak a mile wide, something he didn't recognize until he met me and started having to face the reality of polyamory, so yeah. For him I think it is still very much in the abstract. I think he is a man of one heart. I do not think he is poly, I don't. Some people love one person. I think he's, I know he is polysexual. He's more than happy to be involved sexually with others, but he won't love more than one person, and the idea that I might love someone else threatens him.
While Emmanuella said she was definitely in love with him, she worried that their relationship might not make it because "I _am_ poly; it is just what I am." She was concerned that they might not be able to maintain a long-term romantic relationship if he could not accept what she viewed as her "relational orientation." In that case, he might be a fly-through while she remained polyamorous.
Nori, a forty-nine-year-old small business owner with an adult child spoke of the regret she felt because she was in love with Lindsey, her girlfriend of a year and a half. Although Nori had been clear from their first meeting that she was polyamorous, Lindsey still wished them to be monogamous:
> In the beginning of our relationship the poly thing was just a conversation and you'd talk about it and it seemed, like, OK, because it was in the distance and nothing very real was happening. And then the more we talked about Jorge the more she just didn't want to hear anything about him, didn't want to meet him, nothing. And it's just gotten worse. So now I'm having a great big bout of, oh my God, don't get involved with anybody who's not poly, who doesn't come from that orientation from the center!
Tacit Campo reiterated this point with tongue planted firmly in cheek when he said, "Poly newcomers tend to make the same mistakes, and I'd like to think that at this point in life, I'm ready to explore the exciting world of advanced, sophisticated relationship mistakes that waits beyond that field."
Even practiced polyamorists who fall in love with "monos" occasionally forget community wisdom and attempt to convert someone who was formerly monogamous to become polyamorous—sometimes successfully and sometimes with disastrous results. Poly people in smaller communities or more rural areas are more likely to attempt to convert a monogamous person to polyamory, though such attempts are less common in places with larger communities and many more potential partners who already identify as poly.
## Stigma
In this section I discuss my own experiences of poly-related stigma and relate them to the stigma respondents report encountering, using the concepts of _sex negativity_ and the _polyamorous possibility_ to explain the fear, suspicion, and hypersexualization that polys can experience in their social environments. I discuss respondents' experiences of stigma in relationship to families in far greater detail in chapter 7.
I experienced the stigma of polyamory, both personally and professionally. "Margaret," a close friend since childhood, stopped socializing with me because, she told me years later, she was "sick to death of always hearing about the poly thing. It seemed like it always came up in conversation, every single time I hung out with you and Rick." Other than Margaret, I have not been aware of any other friends rejecting me for attempting polyamory. My family has been similarly supportive, especially my mother and sister.
Professionally, the stigma of studying a sexual minority group—and worse (in some people's eyes) becoming one of "them" myself—was far more damaging than was the stigma in my personal life. Two separate Institutional Research Boards required gymnastic application processes with extremely time-consuming and absurd restrictions about the research records I was allowed to retain and the informed consent procedures I was required to follow.[8] While my open-minded dissertation chair was extremely helpful and provided excellent mentoring, other members of my dissertation committee were less able to compartmentalize their prejudice and acted unprofessionally during and after my dissertation defense.
Publishing has also been challenging. Journals related to sexualities have always been receptive to research on polyamory but are unfortunately ghettoized as second rate in the political world of academic sociology. Mentors like the chair of my department encouraged me to publish in "more diverse" (read mainstream) journals that spoke to the "core of the discipline" in order to gain tenure and be promoted to an associate professor. Unfortunately, editors and reviewers at many of the more mainstream journals had negative reactions to the topic itself, and I was routinely required to offer additional explanations and documentation that editors did not request from colleagues studying other topics.[9]
Finally, it was extremely challenging to identify grants appropriate to fund my research, something that became increasingly important as the worsening economic climate at the end of the 2000s forced universities across the United States to slash budgets and require faculty to find external resources to fund their own salaries, teaching assistants, and contribute financially to the university. Most grants were designed to support the study of mainstream families, and those dedicated to sexual minorities usually targeted gay and lesbian families. Additionally, grants targeted to "alternative" families often focused on abuse, molestation, drug use, and other negative family outcomes—issues I did not see appearing in my data. If respondents had reported these issues I certainly would have pursued them, but I was not about to manufacture them in order to get grants. Failing to secure external funding contributed significantly to my inability to gain tenure, and the narrowness of funded grant topics made it almost impossible for me to attain that funding. Sex negativity contributed to narrowing that field of potential topics and cast research related to sexual minorities as fringe enough to be irrelevant. As we shall see in chapter 10, this research has tremendous relevance for other families beyond those of polyamorists.
### Sex Negativity, the Polyamorous Possibility, and Fear
This general attitude of fear, disdain, and suspicion, coupled with the relegation of sexuality to a sphere apart from legitimate social discourse, is what some scholars have termed _sex negativity_.[10] It was evident during my dissertation defense, dealings with IRBs, and editors' and reviewers' comments: sexuality is at once dismissed as a valid topic while simultaneously being held to different and higher standards than other topics. Were I to study monogamous, heterosexual families I would be viewed as a _family_ researcher because I focus on family issues such as division of labor, parenting, interactions with institutions, and relationships with families of origin. In other words, my research does not focus on sexuality per se in that I am not asking my respondents about sexual positions or the mechanics of group sex. This study has focused on intimate partner and family relationships, much the same way other family studies have done. But the fact that my respondents are sexual/relational nonconformists brands the entire family style as hypersexual and me as a _sexuality_ researcher. Such one-dimensional thinking that can see no farther than the sexual relationship is further evidence of sex negativity.
Among forms of sexual nonconformity, polyamory is unusual in that it could potentially be appealing to everyone who desires to have intimate relationships with other people. Most people are heterosexual, and it is readily apparent that not every one experiences same-sex sexual attraction or desire.[11] In other words, not everyone has the capacity or desire to be gay, lesbian, or bisexual. However, most people in marriages or other long-term relationships, regardless of sexual orientation, have had the experience of being attracted to someone else besides their partner. As such, almost everyone has the potential to be polyamorous in a way that they do not have the same potential to be gay. Once people become aware of the potential to negotiate openly conducted, nonmonogamous relationships, whether or not they actually wish to engage in them, they have realized the _polyamorous possibility_ , and they can never unthink it again. They may reject the idea or decide to explore it further, but the potential for themselves or their partner to initiate discussion of a polyamorous relationship exists in a way it had not before they became aware of polyamory as a social option. Because the polyamorous possibility is potentially open to everyone, it is more threatening to mainstream society than are bisexual, gay, or lesbian relationships that require the (rarer) presence of same-sex desire.
It is this almost universal potential that links the polyamorous possibility so strongly to fear in some people's minds. This fear can be especially potent for those with unresolved infidelity issues in their own lives, as I experienced with my friends and committee members: Margaret told me about the deep pain she experienced when her mother and mother-in-law were both abandoned (respectively) by her father and father-in-law for younger women, and she expressed feelings of profound insecurity when she considered any scenario in which her husband might date someone else. One of the women on my dissertation committee had a nasty divorce several years earlier when she discovered her then-husband cheating on her with another woman. It is possible that journal editors or members of IRBs had similar personal issues that were enflamed by hearing about polyamory as well. Alternately, it is also possible that they were simply put off by the topic, or that they did not approve of my scholarship, writing, or conclusions. The specific tone of the reviews and recurrent nature of the negative feedback signals a deeper, institutionalized issue of sex negativity. Every writer gets critiques, but not every critique is so defensive and vitriolic in tone.
Respondents reported being held in suspicion when others in their social environments learn of their polyamorous relationships. Jana Founder reported that she was at a party for her son's graduation from kindergarten when another mom in attendance realized that Jana was polyamorous:
> It was ridiculous, she clung on to her husband with both hands and never took her eyes off me, as if I might just steal him out from under her nose. And my dance card is full, I have no room and no desire for any more partners. I am in no way a threat to her relationship, but clearly she saw me as a threat, much more so than my husband who is also polyamorous and was there with me. She never really looked at him, but kept her eyes glued to me in a way that made me really uncomfortable. It was creepy, really. Obviously something was up for her, she must have some insecurity in her relationship or maybe she cheated on him or whatever, but it came across as somehow my fault or something. It was weird. We had never met before but she zoned right in on me once she knew, I was on her radar.
That feeling of suspicious surveillance was common among polys that came out or were outed in conventional, ostensibly monogamous social settings.
### Buffers to Discrimination
One major difference between lesbigays and polyamorists is that the mainstream public is relatively oblivious to polyamory, with poly people remaining virtually invisible to society at large. Whether they embrace, despise, or are indifferent to lesbigays, almost everyone in the United States today is aware of the existence of lesbians, gay men, and (to a lesser extent) bisexuals. The same cannot be said of polyamorists, and this affords them a measure of protection from social stigma that is not as readily available to lesbigay people.
Because many in polyamorous relationships can legally marry in ostensibly monogamous, heterosexual dyads, they have different relationships with marriage than do most people in same-sex relationships. While lesbigays may also elect to marry someone of another sex in a similarly ostensibly monogamous and heterosexual dyad, it requires a far greater effort to maintain a closeted gay life than it would for polys with other-sex partners—a configuration that makes them socially intelligible as heterosexual couples with "close friends." This ability to remain closeted almost effortlessly is a resource to which many people in same-sex relationships do not have access and thus functions as a form of (often misattributed) heterosexual privilege that provides an easy buffer against effects of stigma against sexual nonconformists.
### Is It Worth the Effort?
So was the challenge of my own poly relationship worth it to me? In retrospect, I think that if Rick and I had not tried polyamory we would not have split up. I certainly loved him enough to tolerate his wacky ideas, and he loved me. We were good companions with a lot in common who shared similar parenting styles and did not fight about money. Maybe it would have been best for us to eschew the whole mess. It certainly would have been better for Steve, who was extremely hurt by the entire episode. Alternately, Rick and I still might have split up even if we had never tried polyamory. Our miscommunication and power differences would have remained problematic for us regardless of our interactions with others. While it is possible that remaining monogamous (or even possibly poly in theory) would have placed less stress on our relationship issues, it is also possible that something else altogether would have happened. Polyamory did not create our problems, but our nonmonogamous relationships certainly exacerbated the problems we already had and created some new ones.
In addition to the pain and drama I experienced in what I came to think of as "the poly debacle" with Rick and Steve, I ended up learning a lot about myself and gaining some relationship skills that have been useful. I would not be the person I am today without those experiences. I am far more aware of my own motivations and behavior patterns, have much better boundaries, and am able to articulate my needs and emotions. Many poly people mention the potential for polyamory to spur self-knowledge and personal growth as a motivation for, element of, and outcome of polyamorous relationships. We will explore this idea in greater depth in chapter 8.
### Impact on My Children
The children were so young during our family's brief and ill-fated poly episode that they were not aware of the implications of the adults' interactions at the time, and they had not established enough of a bond with Steve to miss his presence keenly the way they might have if they had been older. Rick and I were quite social, so our children were used to friends coming over and then going away: Steve's presence and absence blended in to the recurring parade of adults that passed through their social lives.
The tension that filled our lives, from a low background noise in daily family life to loud arguing, had a greater impact on the children than did Steve's presence or absence. Tension was a problem during "the poly debacle" and afterward as we tried to put our lives back together. Their parents' splitting up has been hard on my children, and they have felt many of the same fears and anxieties that most children feel when their parents separate. The overall positive tone that has characterized my coparental relationship with Rick helps to ameliorate some of their anxieties, and we have successfully modeled conflict resolution and ethical treatment. Less of the time, we have also modeled petty childishness and indignant outrage. The children have learned to consider boundaries, accept differences of opinion, flex with changing social circumstances, and gained a broader understanding of a wider variety of people as direct and indirect results of Rick and I experimenting with polyamory.
It has been challenging to know how exactly to discuss polyamory and our separation with our children. Several years ago I overheard the older child tell the younger child, "Mom and dad split up because mom had a boyfriend that dad didn't like." I was stunned and could not conceive of a way to tell them it was actually far more complicated than that, so I let the comment pass unremarked. They have accompanied me on research trips several times and overheard me discuss my research enough to know something of the broader story, and their understandings of it probably changed as they have matured. I am hoping they will eventually ask me what happened, but in the meantime I have not seen a good opportunity to discuss it that would not come across as bashing their dad.
I would have to say that, overall, polyamory has negatively affected my children and my life: My gut feeling is that it would have been better for all of us, and especially the children, if Rick and I had stayed together as a monogamous couple and never tried polyamory. Once we tried it and reacted as we did to a circumstance we had not anticipated, I was unable to trust Rick and move on, which made it impossible for me to remain in a romantic relationship with him. Polyamory can be high stakes—"playing with fire," as one respondent put it. For many it is well worth the risk, and for others like myself it is not. A different relationship with different people would have turned out quite differently, obviously, so I do not see my own example as an indictment of polyamory, any more than any other breakup is an indictment of monogamy.
1.
(Adler and Adler 1987)
2.
(Connell 1987 and 2005)
3.
(Jenks 1986a; Gould 2000)
4.
(Collins 1992; Guttentag and Secord 1983)
5.
(Fox 1987: 341); see also (Adler 1985; Henslin 1972)
6.
(Adler and Adler 1987: 39)
7.
(Adler and Adler 1987: 67)
8.
I am not challenging the need for Institutional Research Boards or the validity of what they do. While Institutional Research Boards were created in response to a direct need and they serve a vital function of protecting "human subjects" from potential harms associated with academic research, their sex negativity and legal paranoia has severely impeded me and other sex researchers who have similarly faced outlandish restrictions from which people studying more conventional topics have been exempt.
9.
This statement is based on my informal conversations with peers in which I would ask them about the reviews they received from journals and the kinds of documentation the journals required them to produce. Generally, their reviews had a decidedly less personal tone, and none of them had ever been asked to furnish evidence of IRB approval. In contrast, my reviews were personally scathing to a noticeably larger degree than were my colleagues', and several journals requested evidence of IRB approval when considering my submissions.
10.
(Rubin 1992: 150). See also Michel Foucault, _The History of Sexuality_ (New York: Vintage, 1990) and Jeffery Weeks, _Sex, Politics, and Society_ (Essex, UK: Longman Group, 1981).
11.
(Laumann, Gagnon, Michael, & Michaels 1994)
Part Two
# Polyamorous Families with Children
Chapter 5
# Children in Poly Families
Children's experiences depended in large part on their ages. Overall, though, the children seemed remarkably articulate, intelligent, self-confident, and well adjusted. While they dealt with the usual issues of childhood—difficulty sharing toys, middle school social issues, awkwardness—the children in these families appeared to be thriving with the plentiful resources and adult attention their families provided. These findings mirror those of other studies that included children in polyamorous[1] or "group marriage"[2] families.
## Age Differences
Although I only interviewed children five years old and older, I was able to ask parents about their very young children and observe very young children in poly families. Logan Tex, father of two small children, said of his seventeen-month-old son, Pip:
> He recognizes Amelia, our good friend who rents the other half of our duplex, but he doesn't see her as much as he sees our girlfriend Rhiannon. My mom and her partner come down all the time, and my dad, too. We also have friends we met in our homebirth class we spend a lot of time with, we've never had to hire a babysitter. Rhiannon or his grandmothers have probably spent the most time with him. Given how he reacts when people walk in the door he seems excited to see them, about eight or ten people he reacts to that way.
At about a year-and-a-half old, Pip responded positively to the adults who routinely spent time with him, regardless of their level of sexual involvement (or lack thereof) with his parents. Developmentally, children that small are not even aware of sexuality, much less socially developed enough to understand adult sexual relationships. As such, adults who were consistently present and available to meet the children's needs would blend in to a montage of loving caretakers indistinguishable (to the infant or toddler) on the basis of their relationships to each other, and important because of their relationship to the child.
When I was waiting to interview her older brother, I chatted with three-year-old ("and a half," she pointedly reminded me) Kassie while she and Vanessa (another preschooler visiting for a play date with Kassie) played on the living room floor. When I asked the girls what they were playing, Kassie explained that the figures laid out before her were going on a trip and described the figures as "Mommy, Daddy, Hercules [the family dog], my Dave, and Bessie [a plastic horse]." I asked her about "my Dave," whom I knew to be her parents' boyfriend, and she responded that "he comes on the fun." Then she gestured toward Vanessa and said, "She don't gots a Dave. She just gots a mommy and a daddy." It was clear from Kassie's story, her facial expression, and the sympathetic tone of her voice that she viewed Vanessa's lack of a Dave as a clear disadvantage. Kassie also did not question or problematize Dave's presence in her life; he was simply "my Dave," there to go "on the fun." Among the children old enough to respond, three distinct age groups stood out: young children (five to eight years old), tweens (nine to twelve years old), and teenagers (thirteen to seventeen years old).
### Young Children
Children between five and eight years old often did not notice that their families were any different from other families, and instead they took their family form for granted. Like those in monogamous families, young children in poly families responded to their environments with a self-centeredness characteristic of their developmental stage. Young children tended to view the adults in their lives through the lens of what the adult did for or with the child, and they placed much less emphasis on how the adults were related to each other. Rather than mentally categorizing a parent's partner as _mom's boyfriend_ , young kids in poly families were far more likely to think of that person as _willing to_ _be dressed-up_ or _bringer of_ _ice cream_. As an important consequence, the presence or level of sexual interaction among the adults was simply not germane to these young children's experiences or conversations.
This became abundantly clear to me when I chatted with seven-year-old Marni Ballard and her five-year-old brother Milo. They lived in a rambling, ranch-style home in the Pacific Northwest with their extended family, composed of Hillary, their mother; Geoff, their father; Jake, Hillary's boyfriend; Barbara or Grammy, Hillary's mother; and Garth or Papa, Barbara's boyfriend. Hillary, Geoff, and Jake shared the upper portion of the house with Marni and Milo, and Garth and Barbara lived in the separate apartment that occupied the lower level of the large structure. As I sat at the dining room table with Marni and Milo and asked them about their relationships with Jake and their parents, they would occasionally look at me with confusion. They described their interactions with Jake, their mom, and dad, and during a conversation about who put them to bed, Marni asked me, "Why do you just keep asking us about them? Papa and Grammy read to us before bed and stuff, too."
Clearly, my focus on the _polyamorous_ aspect of their family overlooked other parts of the family that were just as, or more, important to the children. What mattered to the children was that they had five loving and attentive adults caring for them, taking them places, picking them up from school, and putting them to bed at night. The children did not categorize this wealth of attention by the sexual relationships among the adults: It made no more difference to Marni and Milo that Jake was their mother's boyfriend than it did that Papa was Grammy's boyfriend because the children interacted with both Jake and Papa as trusted adults who cared for them. The children did not factor in the sexuality among the adults because it was simply not germane to their relationships with the adults. The smorgasbord of love was available to the children regardless of the adults' sexual relationships, or lack thereof.
### Tweens
Children between nine and twelve years old, or "tweens," were more aware that their families were different from many of their friends' families, and they were increasingly aware of how the adults interacted with each other than were their younger compatriots. Like other kids their age, they knew the adults had sex and preferred to know as little about it as possible. They also knew that other families were frequently different from their own families, and that this information could sometimes be upsetting to adults. Children in this category began to actively think about how to explain their families at school, to their peers, and to other adults. As they realized their families were different from many of their peers, tweens asked their parents questions about the family and considered what they would tell their peers if and when someone else noticed.
Inevitably, some of their peers did notice the extra adults, and tweens in poly families had to explain their families far more often than their younger siblings. Adam and Michelle Hadaway began attending the same school when the Hadaway quad coalesced and Michelle's family came to live with Adam's family. I asked what Adam's peers thought when his family suddenly expanded.
> Elisabeth: If people at school ask what's up with your family, what do you tell them?
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> Adam: This kid Lawrence, in my same grade, he'll be like, "How are you related?"
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> Elisabeth: You?
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> Adam: Me and Michelle, cause we are in the same grade. But we just don't say anything. They'll just keep on guessing, and they'll never get it right.
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> Elisabeth: When you just don't say anything, how do you get away with that? Without responding at all?
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> Adam: They just get so confused, they quit. It's only middle school.
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> Elisabeth: I'm sorry I keep asking you about this, it's just hard for me to wrap my mind around that you don't have to offer some kind of explanation. Like, it's an unusual family, so when people ask you and you just don't say anything, how is it that you can get away without offering any sort of explanation?
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> Adam: They'll just think they know what's going on and they stop asking. But they probably don't.
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> Elisabeth: So they kind of make things up themselves, and you don't correct them?
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> Adam: Exactly.
Adam and Michelle's peers noticed that there was something unusual about their families, but they were not quite able to figure out what was happening with the many siblings who suddenly began attending the same school. Even so, Adam did not see it as problematic, and indeed he seemed amused by his peers' confusion. Among peers with divorced and remarried parents, adopted children, and single parents, an addition or subtraction of siblings has many possible explanations and blends in to the social background of constantly shifting relationships among serial-monogamous families. In this case, Adam did not find his unconventional family problematic and was able to pass with ease as one of his many peers with more conventionally blended families, though as we shall see in chapter 9, some of his siblings had more problematic experiences managing that information with their extended family members.
### Teenagers
Teenagers from thirteen to seventeen were generally establishing an increasing level of independence and an identity formed outside of their families, more invested in exploring their own social relationships and sexualities than were their younger brethren. This had a number of consequences, including the teens having to explain their families in more complex social settings and being less focused on their parental relationships as their own social relationships eclipse familial bonds in emotional urgency. Teens in poly families often consider whether they want to have poly relationships themselves, or if they would prefer monogamy—something I discuss in greater detail later in this chapter. Generally, the teens in the study were like other teens in that they were more involved in their own social lives than their parents' social lives. In an interview when she was fifteen years old, Kethry Wyss explained why her poly family was
> yeah, whatever, no big deal. I have my own stuff going on. I'm still swimming [competitively on a team], have a lot of friends, and we're really into anime and make costumes and go to cons and stuff. It is a big DIY [do it yourself] thing, so Mama helps us sew so my friends know her, it's not like I hide them or anything, I still love them, I'm just doing my own thing and not so much in their business. I can't wait to be able to drive! Then I can go to practice, whatever, don't have to wait to carpool and they don't have to come pick me up.
Kethry continued from there with a litany of the fun things she could do once she could drive herself. Her specialized social environment—a diverse magnet high school in the California Bay Area—contributed significantly to her ability to see her poly family as "no big deal" because many of her peers were adopted or had single, divorced, or gay parents and so her own multiple parents were unremarkable. In addition to being fully engaged in her own life and "not so much in their business" when it came to the adults, Kethry later explained how she felt she could trust her parents and talk to them about anything. She had both the emotional and social distance developmentally characteristic of teens busy establishing their own identities, but she also had the support of devoted parents she felt she could trust.
## Coming Out
As far as sexual minority families go, poly families are not nearly as visible or recognizable as lesbian or gay families, so these families were often able to exercise wide latitudes when deciding to come out or not.
### Children Coming Out to Peers
For the most part, children did not have to deal with coming out to strangers, classmates, coaches, or teachers. The popularity of _serial monogamy_ —a cycle of coupling monogamously/marriage, breaking up/divorce, and coupling monogamously with someone else/remarriage—in the United States makes it commonplace for children to have multiple parents. Now that stepparents are standard social fare, kids from poly families with several parental figures simply blend in. Unless poly family members intentionally highlight and explain their family structure, they are rarely called upon to justify their "extra" members.
If they choose to come out, children in poly families do so selectively, revealing family details only to those they know and trust, or those who ask politely in low-risk or need-to-know situations. Sebastian, a white high school student and Nash Majek's younger son, reported that he blended in among the varieties of families his peers inhabited.
> Sebastian: No um surprisingly a lot of people in my school have like kind of like this situation with parents not really, not as far as I know not like polyamorous, but like parents issues.
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> Nash: Like divorced.
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> Sebastian: Yeah.
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> Elisabeth: So they'll have multiple parents like divorced and remarried?
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> Sebastian: Yeah, some people do.
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> Elisabeth: So do you feel like you're different from your peers?
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> Sebastian: No, not really, it's kind of normal to some people, if they don't have that happen to them because there are other people that have had it happen to them.
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> Elisabeth: It meaning? What's it?
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> Sebastian: Like just like the divorce issues or just, just parental issues . . .
Like many of his peers, the various adults in Sebastian's life blended in with his social surroundings, and he was rarely called upon to explain them.
In a sea of single parents, divorced, remarried, and cohabitating adults with children they bore or adopted, poly families may appear to be just another blended family. Their status as sexual minorities, however, presents different issues than their peers in (ostensibly) monogamous and heterosexual families. I asked Sebastian:
> Elisabeth: Do you ever feel like you need to hide it from your peers or like you can't talk to them about it?
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> Sebastian: No, but I haven't told anybody about the polyamorous thing.
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> Elisabeth: How come?
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> Sebastian: Never really came about.
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> Elisabeth: If someone asked you about it, what would you say?
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> Sebastian: I would go ahead and tell—well it depends on the person. 'Cause there I know some, some kids are there at my school are like they have major Christian families or other beliefs that don't think it's OK to have multiple partners. I would only lie to them.
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> Elisabeth: Only lie to the Christians?
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> Sebastian: Yeah. Well, like that are strict about that. Some are flexible that they don't really mind that don't meddle with other people's business. But some are like . . .
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> Nash: Have you had a friend over when Morgan was there?
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> Sebastian: Um, I don't think so, not that I can recall.
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> Nash: I don't think so, but I know his brother Beck has.
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> Elisabeth: Do you know how that went with his brother?
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> Nash: Um, by all appearances . . . so this friend that um his brother has is one who has been a friend for some time and uh . . . you know knows that I am married to Marcy and has met her. Um . . . And so he saw you know Morgan at our house and that she was spending the night, and I think Beck told him that she was my girlfriend. And as far as I know he was completely oblivious.
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> Elisabeth: Mmmhmmm. How do you know Beck told him?
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> Nash: I think I heard him on the phone when we were discussing plans for his friend possibly coming over that um . . . I am pretty sure he used the term that "my dad's girlfriend was here" or that my dad was going to go do, go out, you know, I'm pretty sure he used the word _girlfriend_.
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> Elisabeth: Did your son say anything about the way his friend reacted?
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> Sebastian: Um . . . this certain friend, he has parents who are divorced, so he would probably be like "okay." He really wouldn't care.
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> Nash: That is exactly how he stated it. Okay, whatever.
It was clear from this exchange that Sebastian and most likely Beck did not care about their father's polyamorous relationship, rarely had to think about what to say or how to hide the information, and did not feel excluded from their peer groups.
Characteristic of this lack of tension around coming out, Zane Lupine—a seventeen-year-old white male high school student—said that he had never felt uncomfortable about being in a polyamorous family.
> If a friend came over and both of my dads were there I would just tell them straight out, that's my other dad. If they were a little weirded out, I didn't take it as anything. I was never ashamed about it or never thought it was weird. I never really had to come out about it. If someone wanted to ask, they could ask but I wasn't going to just like . . . it wasn't something I had to announce, because it wasn't something embarrassing for me that I was trying to hide. If someone wants to know they can know, but I'm not going to go screaming about it. It's just not what I do about anything in my life really. I just don't ever talk about that kind of stuff that much. I guess if it came up, but it doesn't come up that much. I just never think to bring it up because it's normal to me. . . . It just obviously was not a big deal.
While the Lupine family originally lived in a small town in the Midwest, they had moved to a more liberal college town near a larger city when Zane was a child, so he was attending a fairly large, liberal, and diverse (for the area) public high school in an upper-middle-class, suburban neighborhood.
Other children, however, had far more difficulty dealing with their polyamorous family identities. Cole Cypress, a fifteen-year-old white male high school student, related how tremendously awkward and uncomfortable he felt when deciding to disclose his family formation to peers at school.
> Cole: It's always been very hard to explain to my friends at school, especially since I go to a private school. And I just feel, my friends would just be like, who's that person that's picking you up? And I'll just be like, that's Bettina, my mom and dad's girlfriend. It's kinda hard when you say that. . . . And then for a while I would do "family friends" and then when I got into about eighth grade they finally realized that I had a lot of people coming to pick me up and it was kind of weird to have so many family friends and some of the closer people, the people that were closer to me, I started explaining it in the best way I could. And it was really hard, but after they knew it was kind of a load off, you know. It's kind of weird to live with a secret, something you can't tell any of your friends cause they wouldn't understand.
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> Elisabeth: So when you said it was kind of hard . . . telling them was hard?
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> Cole: Yeah.
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> Elisabeth: Or keeping it a secret was hard?
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> Cole: Both. And trying to fit in, especially since I was not one of those kids that was the cool kid. I was never popular, I never had a lot of friends, especially in my earlier years. It was only in eighth grade when I think I actually made any real type of friends. But yeah, it was hard because, like when Bettina first came to pick me up at school one day I just tried to kind of explain how she wasn't my new mom but she was like my dad's girlfriend or something and you know it was just kinda weird.
In this case, Cole's discomfort with revealing his unconventional family life magnified his shyness and difficulty making friends. For Cole and other members of poly families, polyamorous relationships can magnify preexisting personal and relational issues. The same way a piece of metal will break along a fault line when stressed, the intensity and complexity of poly relationships place stress on issues in relationships that already existed but might not have come to a head in the same way if the relationship or family were monogamous. Long-term polys attempt to face these challenges head-on, using them as opportunities for growth or to explore personal boundaries. Others find that the added stress makes the issues too disadvantageous to pursue and decide that a polyamorous relationship form is not for them.
Other researchers have also reached similar conclusions in their research. In her book _Border Sexualities, Border Families in Schools_ , Dr. Maria Pallotta-Chiarolli found that polyamorous and bisexual families in Australia used three primary strategies to present their unconventional families at school: 1) _passing_ as a conventional family by staying quiet about their differences and attempting to blend in; 2) _bordering_ regular and unconventional society by straddling both worlds, moving between them as needed or desired (what Pallotta-Chiarolli calls a _mestizaje_ );[3] or 3) _polluting_ by infiltrating a formerly monogamous social setting and openly displaying their social unconformity.[4] Many of these families used different strategies at different times, or blended characteristics of each as the situation demanded. Sebastian Majek had no problem passing as a member of a conventional family, and Adam Hadaway easily navigated the border, even in the face of his peers' scrutiny, simply by evading their questions. While Adam and Sebastian—both attendees at suburban public high schools—managed these interactions with ease, Cole Cypress's smaller and more tightly knit private school intensified peer surveillance and made it more difficult for him to pass. Cole's attempt to border met with mixed results, and he was pushed somewhat unwillingly into the role of polluter. In chapter 9 I return to Pallotta-Chiarolli's concept of polluting when I discuss respondents' strategies for dealing with stigma.
### Parents Coming Out to Children
Polyamorous parents came out to their children in a variety of ways, and at a variety of points in their relationships, depending on the age of the children, the past and current familial configurations, and factors external to the family. Sometimes parents come out at different times or in different ways to their various children, tailoring the timing and information to the children's needs and the family situation.
If a child is born into a polyamorous family, parents will often wait until the children ask something about the family and then give age-appropriate information in direct response to a request. Marcus, the middle child in the Amore family, estimated that he was seven or eight years old when he first asked his parents, Louise and Max, about their family.
> They never paraded their partners around in front of us. They never tried to hide it, but they never threw it in our faces. They kept it private. . . . We were in the car and they were talking about their other partners, and . . . [when] I questioned them on this, they simply said they were polyamorous. I don't remember the exact words, but it was all simple enough. They were patient enough and helped me to understand it.
In contrast, Louise came out as polyamorous directly to Marcus's older brother Dave because, as Dave bluntly put it, "I was an eavesdropper." He continued:
> Dave: She told me when I got a little bit older. I think it was around ten or eleven, not real sure. I think part of the reason she told me is because, when I was that age I was an eavesdropper. I could be down in the basement and I could hear what's going on, on the top floor.
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> Elisabeth: How so?
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> Dave: I just have really good hearing. The vents sometimes helped, depending on which room you're trying to hear. But I just have really good hearing. I also have really good sight. I can read lips. Things like that help.
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> Elisabeth: So when she talked to you about it, what did she say?
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> Dave: I don't know. I'm not really sure. She basically explained the situation. She cares for this person; this person . . . I'm like, "Okay, whatever makes you happy."
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> Elisabeth: It sounds like you weren't freaked out about that.
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> Dave: Yeah, I wasn't freaked out. I'm like, whatever. I'm gonna go play with my toys and go hang out with my friend Matt. Which I'm still friends with . . . I'm like whatever makes you happy. I'm going to go do something now. Go play with my friends. It didn't really occur to me that it was that unusual, I guess.
At the time, Dave did not see it as unusual and in fact wondered a little why his mother was "making kind of a big deal out of it." It was normal life for Dave and his siblings, so their poly family did not require much explanation.
When people who are already parents become polyamorous, they are more likely to have a coming-out conversation with their tween or teen children. Sebastian reported that his parents came out to him in a conversation much like the one Dave's parents had with him:
> Um, it kind of took me by surprise I guess. They just kind of called me into the room and they just kind of told me. Like, we both love other people, and I was just like, okay. So it wasn't . . . it kind of surprised me because I hadn't really thought about it before. But I heard some . . . there were some clues about it, I forgot what, but . . . I just kind of . . . it took me by surprise, I just deal with it. I can't really care after a while.
Other tweens and teens knew something was going on but avoided the subject, did not want to know any more about it, and refused to discuss it. Elise Heartland said that
> they didn't really sit us down and have a big talk or anything. We were sitting around watching TV and something came on, some gay guy or something, and I joked around, something about Sven, and I was like, well, are you bi? And he didn't really say anything so then I was like, it's fine, I don't really need to know. And something about Adam came up and it was like, I gotta go, I gotta get outta here. I need to go to my room. Really, it's just like, even knowing you guys have sex is enough, but I don't need to know anything beyond that, so why would I go out of my way to ask more about something I don't want to know anyway?
Much like children in monogamous, blended, or single-parent homes, children in poly families generally preferred not to know about their parents' sex lives.
## Sexuality
Although it is somewhat awkward to point it out, any study of children in sexual minority families has to address the _sexual_ aspect of the sexual minority (as if heterosexuality is not a form of sexuality as well, or as if most sexual abuse and incest didn't happen in heterosexual families). But the point became clear the more I spoke to poly families: none of my respondents had sex with or in front of their children. None of the children reported it, and none of the adults mentioned it either. One significant exception is a child who was molested by a parent's partner, which I discuss in far greater depth in chapter 7. I looked long and hard for the families with difficulties like custody suits in which children were most likely to be at risk, and found very few. Like in any family study, those who are molesting their children are far less likely to volunteer for research than are families who feel they have nothing to hide. Additionally, members of stigmatized groups often feel compelled to present the most positive image possible of their families in order to forestall any potential critique of poor or inappropriate care—something Pallotta-Chiarolli terms "passing as perfect"[5] and saw in her own study.
### Children's Awareness of Parental Sexuality
It is not that these families were perfect, it is just that they were well within the range of usual experience and difficulties present in average family lives. When I asked adults about how they interacted in front of the children, they reported showing affection in an appropriate manner publicly and saving actively sexual interactions for the bedroom. I did not ask children explicitly about their parents' sexual interactions (focusing instead on the family's social interactions as a unit), but children would volunteer statements like Nolan Hadaway's, when he said: "I see them kiss and hug but nothing more mushy than that. Once I saw Pops swat Mom on the rear and she got him with the dishtowel, you know, like PG, and not even PG-13."[6]
Children reported being aware that their parents had sex but not wanting to think about it or talk much about it at all. In a tandem interview, best friends Heather Majek and Alice Heartland (both thirteen years old at the time, girls who attended the same middle school) each relayed their experiences growing up in a poly family. When I asked about their parents' public displays of affection, each made a face and gagging noises:
> Alice: Well, they usually, let's just say that I know what they are going to do, once they send me downstairs.
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> Heather: Mine doesn't. They usually go upstairs, into their room, like they are "watching movies" (air quotes).
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> Alice: They'll be like, it's 9 o'clock, your bedtime. And I'm happy to leave! Cause I don't wanna be there watching them. Weird! Even though they don't just do it right there, in front of your face. . . . It's like awk-ward! Yeah. Because I never thought about Adam or Richie in that way. I don't like to think about it. I don't want to think about it. Yeah, it's just not . . . It's kind of weird. "Oh, there's my friend Adam up there. Wonder what they're doing." You know. It's really awkward.
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> Heather: I mostly see them kiss hello or goodbye, it's not like if we go to a movie they're making out or anything.
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> Alice: It wasn't as awkward as when I was younger. I didn't even know at all what they were doing. I was like, "Oh they just want me to go to bed. Whatever." It was no big deal at all. But now I understand and it's weird, but they love each other so I don't care.
While Alice and Heather were aware that their parents had sex and were fairly grossed out by it, the children were neither exposed to nor traumatized by it.
Elise Heartland's experience contrasted with her younger sister Alice's. Piqued by curiosity about Adam (Sven and Shelly's—Elise's parents—boyfriend) and unbeknownst to Elise, after high school one day some inquisitive friends had invaded Shelly and Sven's privacy while they were hanging out at Elise's house and discovered information regarding their sex lives. In an interview with Shelly and Sven, Elise reported:
> You guys don't know this but I got a lot of shit about Adam, I mean a lot. Because my friends all knew. They knew before I did.
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> Sven: Really?
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> Elise: They figured it out before I did. I was just like, he's over a lot, he's their good friend, they have friends over all the time. And it was finally like Jared, one of my good friends now but he was kind of an ass at the time, was like, "You know they do it—Svennie and little Adammie are getting it on." And I was just like whatever, you guys don't know what you're talking about. And my friends would spend the night and Adam would be like, "Suzanne, do you need a ride home or something?" and she would be like, he would give my friends rides home and it wasn't a big deal. We were in high school, we just laughed about it. And it wasn't a big deal. My excuse was always just, my parents are freaks you know, no big deal. My friends liked you guys up until the end of junior year and I would have people over and I remember one time it was like Jackie, Brad, and Jared went upstairs and found a bunch of stuff and it was like, oh shit.
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> Elisabeth: A bunch of stuff like what?
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> Sven: Porn.
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> Elise: Sex toys. Yeah (hands clapping, laughing). So I got in trouble for that and I didn't even tell them to do that. Because I didn't even know that they did that. And they were just giving me so much shit about Adam and about them and I felt like I was always needed to hard-core defend them [Shelly and Sven], or I had no association with them, like they are freaks, they do what they want. I couldn't like be one or the other. I couldn't be accepting of it, either way, because people are really closed-minded in high school. But now like when I see Jared or talk to my other friend online, if they say something I would just be like yeah, whatever.
Elise Heartland found her peers' awareness of her parents' sexuality alternately painful and tedious, a source of teasing even though her friends actually liked her parents. Even so, she was able to navigate her peers' ribbing and retain relationships with her friends over time. While initially Elise tried doggedly to pass as a member of a "normal" family—even to herself—her friends' discovery of her parents' collection of sex toys and pornography destroyed that fiction forever and forced Elise to transition to a new phase Pallotta-Chiarolli would call _bordering_ , or managing her friends' knowledge of her family's unconformity in a monogamous social environment. Eventually Elise rebelled against what she saw as her friend's "asshole streak" and "closed-mindedness" and became what Pallotta-Chiarolli would see as a polluter, challenging the assumed dichotomous nature of families and her friends' short-sightedness. In fact, Elise asserted that polyamory had become a litmus test in her life, to some degree:
> If people can't deal with it, then I don't want to hang out with them anyway. I just can't get along with people so closed-minded, I don't wanna waste my time with assholes. Sometimes if I'm not sure about somebody I'll ask a question about gay marriage or something, or _Will and Grace_.[7] And if they're weird about that, I don't even bother with them. I don't need to be around anybody I have to hide my family from or I can't be myself around.
Generally, poly parents kept their sexuality private, and when the children did become aware of it, everyone involved actively sought to shield the kids from any specific knowledge of what went on behind closed doors.
### Children's Ideas about Their Own Sexualities
Children's ideas about their own sexuality and future or current relationships varied primarily by age. Young children had no concept and did not tend to focus on or even understand sexuality. I did observe some play among young children, and I noticed on more than one occasion that when they would play family or house, they had a very wide interpretation of marriage that incorporated partners of the same gender or groups. It appeared to me that the normalcy of the poly family had permeated to the level of preschoolers' play. Tweens were aware of sexuality and felt varying degrees of comfort discussing it, from red-faced refusal and stuttering or simply confusion, to eloquent monologues about the pros and cons of polyamory and monogamy. Some teens had more definite opinions regarding their current and potential future relationships. For instance, teens such as Elise Heartland were certain they would not become polyamorous in the future. In response to my question about her thoughts regarding polyamory as a possibility for her, Elise responded:
> Would I ever be poly? Nooo-oooo, no way. I need way too much attention for that. I want to be the center of his attention, you know? Not sharing him with other girls. I watched my mom share Sven's attention for all of these years and it looked so hard sometimes. No, I want someone all to myself. Definitely not going to be poly.
Like her sister Elise, Alice Heartland thought it highly unlikely that she would establish a polyamorous relationship.
> Elisabeth: When you look forward into your life, do you see yourself being monogamous, polyamorous, how if at all do you see yourself?
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> Alice: Just one person.
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> Elisabeth: How come?
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> Alice: I don't know. I think there would be jealousy. Like I'm going out with my other boyfriend today. I don't think that would make anybody really happy. So I think that's just why I'd prefer to stick with one person. I think there would be jealousy between, like well why aren't you going out to dinner with me, you know?
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> Elisabeth: So you imagine other partners being jealous of your partners. Do you imagine yourself being jealous of other people?
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> Alice: Yeah.
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> Elisabeth: You're concerned about your own jealousy and their jealousy.
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> Alice: Yeah.
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> Elisabeth: So anything else besides jealousy make you wanna steer clear of—
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> Alice: Not really. Well you like who you like, there's not really anything you can do about it. If it happens it happens I guess, but I don't really think so.
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> Elisabeth: You're not gonna look for it on purpose?
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> Alice: Yeah, I'm probably just gonna stick with one person for any relationships.
While Alice feared the high likelihood of jealousy within a poly relationship, she remained open to the potential for someone she liked to draw her into a poly relationship.
Heather Majek did not actively reject polyamory, but rather she simply accepted monogamy as the default that guided her relationships.
> Elisabeth: Okay, so did you say, Heather, on your demographic form—You're dating, you cutie.
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> Heather: Hee.
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> Elisabeth: You, how's that going?
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> Heather: Good.
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> Elisabeth: How long have you been dating? Your form said you are heterosexual, so probably a dude, a boy?
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> Heather: "A dude." (laughs)
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> Elisabeth: A dude. How's it going?
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> Heather: Um, good I guess.
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> Elisabeth: Just the one dude?
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> Heather: Yeah. I don't cheat.
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> Elisabeth: You don't cheat. So seeing more than one dude would be cheating?
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> Heather: Yeah.
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> Elisabeth: How come?
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> Heather: I don't know.
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> Elisabeth: That's just the way it is.
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> Heather: Yeah.
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> Elisabeth: Okay. Did you and the dude talk about it?
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> Heather: Hmm?
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> Elisabeth: Like, decide that you were just going to see each other. Or was it just assumed?
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> Heather: It's just what everybody does, in junior high I guess.
Rather than making an explicit choice to be monogamous, Heather Majek fell into the general cultural trope of monogamy by default. Her parents' polyamorous relationship did not exert enough sway over her thought process to supersede the weight of the monogamous culture at large or the more immediate social circle of her junior high school. Clark and Morgan Majek refrained from proselytizing to their children and did not encourage them to become polyamorous, and in the absence of that pressure Heather had chosen the path of least cultural resistance at thirteen years old, with no serious thought to what she might want from a future relationship or family.
Zane Lupine took a flexible approach to his own potential to be polyamorous or not. When explaining why he was in a monogamous relationship with his girlfriend of two years, Ekaterina, Zane reported:
> Zane: I just like the one that I'm with, so I don't really need anyone else. I guess if I felt that I needed more variety then I'd just be open about it and get more variety. But I don't feel that need for it or want it.
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> Elisabeth: Did you and Ekaterina ever talk about OK, are we going to be polyamorous, or are we going to be monogamous?
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> Zane: No, it's kind of like the classic—it's ninth grade so it was less mature I guess. "Would you go out with me?" kind of thing. It started where it was just that. We were boyfriend and girlfriend. Neither of us were expecting it to last very long—like most high school relationships, you know. But then it just did and we became like best friends. We've been together for like two years. There's never been a question of what we are, it's just kind of happened.
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> Elisabeth: And is she the same age as you, seventeen now? So she was fifteen when you got together, you were both fifteen? And now you're both seventeen?
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> Zane: Yeah.
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> Elisabeth: Do you have an agreement to be monogamous with each other?
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> Zane: Yeah, because if we cheated on each other it would probably be over I guess. We'd probably still be best friends, but we wouldn't have the relationship as boyfriend/girlfriend.
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> Elisabeth: How come?
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> Zane: I don't know, I haven't ever really looked at it any other way. I don't think she has either. That's just kind of how it started and how its been going.
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> Elisabeth: You don't have to discuss the rules, you just know?
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> Zane: Yeah, that's kind of how it is.
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> Elisabeth: Have you ever talked about potentially being polyamorous in the future?
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> Zane: No, I've never even thought of that. I don't really think too much about the future. That's I think, kind of a bad thing to do. I don't want to focus too much on the future, and just kind of keep going with it. I don't know, maybe sometime it will come up. Maybe if we keep on after high school, more stuff will be brought up, I don't know. I just know I don't get bored with her, that's why she's my best friend. I can always hang out with her, that's the reason it has lasted so long. I haven't gotten sick of her, she's always new to me. We always have something to talk about.
Zane was not interested in pursuing polyamory himself, at least at the moment, and he was not eager to make plans for the future. True to his mother Melody's Buddhist influences, Zane focused on staying in the moment and being open to what comes. True to society's influence, he automatically adopted social standards of monogamy that influenced so effortlessly that he and Ekaterina did not even need to discuss rules structuring their relationship.
Alternately, some people raised in poly families felt that monogamy would be far too stifling and they would definitely construct their own poly families. At fifteen years old, Marcus Amore had not established any serious romantic relationships, but he foresaw himself most likely being in a polyamorous relationship once he began dating:
> I seriously want to try for polyamory. I have never really had issues of jealousy myself, however, not being in anything that I could consider a true romantic relationship, I'm not sure how jealousy would relate to that for me. . . . It has to do with freedom. I don't like the idea of restricting myself to a single relationship and I don't want to bind her to a single relationship. I know that much for certain. If I did have issues with jealousy and could not be in a poly relationship, then I could not allow our relationship to continue because I would not want to restrict her in that way.
Marcus's older brother Dave wasn't certain he would always be polyamorous, but he felt more comfortable in a poly relationship than in a monogamous relationship. Discussing his two close female friends Annabeth and Kari, Dave described how they ended up in a polyamorous quad with Kari's boyfriend Kaden.
> Dave: Yes. I was in love with both of them. At first, it was really challenging. I wasn't sure quite what to do about it. Me, Kari, Kaden and Annabeth all sat down and talked about it. We ended up having a quad relationship. Annabeth dated me and Kaden, and I dated Annabeth and Kari.
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> Elisabeth: And Kaden dated both Kari and Annabeth as well?
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> Dave: Yes.
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> Elisabeth: How'd that go?
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> Dave: It was actually really nice for the first few months. One of the challenges that went wrong in that relationship, aside from the fact that we were all still young and all still into high school drama and bullshit—
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> Elisabeth: Young meaning like seventeen?
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> Dave: This was a year or two ago.
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> Elisabeth: So seventeen-ish.
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> Dave: Yeah, I was actually seventeen about to turn eighteen that summer. Two years ago, then. That went really well. One of the problems was, I didn't establish any boundaries. That ended up hurting the relationship, as well as a number of other things.
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> Elisabeth: How did the lack of boundaries hurt the relationship?
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> Dave: One of the things that I didn't know at the time, when I first went into the relationship is, I don't have a problem being polyamorous . . . The only time I really have a problem is when I have jealousy come up and it's very hard for me to deal with, is when I'm not being distracted by work or by being with one of my partners. So I'm sitting at home going ughn! (Brooding sound) And just steaming.
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> Elisabeth: So you were saying that lack of boundaries and jealousy . . .
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> Dave: Yes. I didn't establish any boundaries. One of the things I learned later on is that, while I don't mind someone dating someone else that I'm dating, it does bother me if I don't have anything else to do or distract me. Whether it's work or anything else. That's something that's bothered me. I've gotten better about it, but at that time it was something that was really bad for me. That was probably just one of many things. It just started a chain of events and caused the relationship to fall apart.
Even though the relationship was imperfect and lasted no longer than many other high school romances, Dave felt that it had been a good thing for him and for the other quad members. They all continued to be friends, and Annabeth and Kari continued a sexual relationship after breaking up with both Dave and Kaden at different times.
It came as no surprise that the teens in polyamorous families were often undecided about their potential future sexual partners or relationship styles. None of them reported feeling pressured to become polyamorous in the future or feeling that their choices were constrained. Even more striking, these teen's automatic acceptance of monogamy indicated the social strength of that convention, even among those raised in families that practiced nonmonogamy.
## Who Do Children See as Parents?
Four primary factors contributed to whether or not children saw adults as parents or not. The first and most obvious was a biological connection: all of the children know who their biological parents are. In her book _Pregnancy and Polyamory_ ,[8] Jessica Burde describes some polyamorous families who either intentionally avoid knowing the biological parentage or accidentally become pregnant in a situation where multiple men might be the biological father, so there are clearly a number of ways to approach childbearing in polyamorous families. The second most likely factor influencing a child's likelihood to view someone as a parent is the child's age at the relationship onset: the younger the child, the more likely the adult will move into a parental category in that child's mind. Third, children are more likely to see adults who share a living space with them as parental figures, so cohabitational partners are more likely to become parental figures than are noncohabitational partners. Fourth, it depends in large part on how long the person has been around the child and how much time they spend together. Those children who know an adult for many years and/or regularly interact with an adult over time are more likely to view that person as a parental figure than they are a relative newcomer or someone who is present infrequently.
Children reported viewing the adults in their lives as akin to aunts/uncles, older siblings, or friends far more often than they thought of them as parents. In a conversation with me and her parents Shelly and Sven Heartland, Elise explained that she did not see Adam—Shelly and Sven's former boyfriend—as a father figure.
> Elise: No, not at all. But he was a really cool guy and a great friend and stuff. But since he was so young and I felt like a lot of the time he was more on my level than on their level or something . . .
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> Shelly: But I think Alice viewed him as more of a parental figure because she was so young.
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> Sven: She even called him daddy number two.
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> Shelly: She would listen to him like a parent, where Elise, she was too old for that.
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> Elise: And Adam came along when I was like sixteen, or fifteen, or something, so first of all it was like, "What is going on here?" Like, this is a little weird. But when I got used to it, it was like fine and stuff.
For Elise and many older children, parents' partners were often trusted adults the young people could rely upon for advice, attention, and assistance, but the partners were not often parental figures themselves.
In contrast, Kethry Wyss views all of the Wyss quad members as parents. Born into a poly family and cared for by the entire quad for fourteen years, Kethry did not distinguish between her biological parents and her social parents: they were all simply parents to her. This assumption created such a clear substratum in how she spoke about her parents that I do not have a single quote exemplifying her attitude toward her parents _as_ parents per se. Kethry's silence on the topic of who counted as a "real" parent and her discussion of all four Wyss adults as her parents clearly indicated that she viewed them all as parental figures, regardless of biological or legal connection. They had all lived with and cared for her from infancy, and that is what made them parents.
Marni and Milo Ballard held a fairly amorphous view of the many adults in their lives who cared for them. Rather than distinguishing strongly between adult roles, the children seemed comfortable with adults who could occupy numerous roles. Marni reported that her parents' partner Jake was like a papa—the term Marni and Milo use for their grandfather—or a nanny, sometimes like an uncle.
> Marni: Once at dinner I asked what are we having for dinner? And he [Jake] said slugs!
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> Elisabeth: So he jokes with you?
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> Marni: Yeah. He is like a papa only funner. Or a nanny.
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> Elisabeth: What makes him like a papa or a nanny?
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> Marni: Because a nanny stays with children all the time and plays with them, and is like a parent only parents who can't control their kids [adults laughing in background], they're the nanny that comes to play with them . . . A papa is funny, I have a papa downstairs too. Jake is like an uncle but he's like a papa and like a nanny . . . Papa and Grammy live downstairs. Grammy kisses me and cuddles with me a lot, gives me gumball things . . . and Auntie Stacia comes to pick us up from school and sometimes she brings us treats.
The list of important adults in Marni and Milo's lives included people who loved and spent time with them, regardless of their biological or legal connections to the children or the other adults. In fact, their own family of origin was complex enough that their polyamorous family members simply blended in with the other chosen kinship relationships around them. Milo described his Papa this way:
> Papa is not really Mommy's dad. Grammy actually married a different grandpa, another grandpa, we have two grandpas, and Grammy married Grandpa instead of Papa and Grandpa is actually Mommy's dad. But we like to think that Papa is Mommy's dad. Even though he wasn't the daddy that made her, he helped her grow up.
With such a flexible understanding of parentage already, Jake's quasi-parental/uncle/papa/nanny relationship made complete sense to Milo and Marni.
Although most of the children I interviewed did not usually see their parents' partners as parental figures themselves, they did often see them as family members nonetheless. Marcus Amore reflected on his mother's partner Valentino:
> I guess for me, Val is most comparable to an uncle. I certainly don't see him as my father. Despite my real father's jokes about that, I never will. He is an adult male figure that I can look up to, but he is not my father and he has never tried to be. He has tried to be my friend and a member of my family, but never has he tried to replace anyone.
Expanding on what made Valentino a member of his family, Marcus said:
> It is a matter of attachment. He is very important to all of us; we all love him. He cares a great deal about us. We talk. If we need something, we can go to him about it. He'll help us when he can. That's what makes family, people you can count on. I consider my friends more than just that; I consider them my family. So in this respect, another advantage to being in a poly family is how I can view things as a result of this. You could say I have a very large, extended family, and I love that. I have many people I can talk to and connect to and go to for help. Generally I do not need to look beyond my own household for that, but if I ever do, there they are.
Like Marcus, other kids in poly families also had significantly expanded definitions of family that went beyond the scope of traditional biolegal definitions and included what scholars have usually termed _chosen kin_.
Speaking with me a few months after he moved across several states to live with Louise, Valentino agreed that he was emotionally close to Louise's children but did not see himself as their father.
> Elisabeth: Have you taken on a parental role with Louise's children?
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> Valentino: Yes and no. I've developed a friendship with her children, who are just great people. Her sons and I play online games together, we go head to head against each other, that kind of thing. Her daughter's great, she and I have private chats sometimes where she confides in me a few things, and that's been nice. So probably just in the beginning, not too in-depth yet. I guess right now you can call it just more establishing a friendship with her [Louise's] family.
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> Elisabeth: So I'm interpreting you to say that it's gone pretty smoothly—is that true?
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> Valentino: Yeah.
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> Elisabeth: Have you and Louise discussed any kind of shared responsibility for the children?
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> Valentino: We've discussed that we need to discuss it [laughing]. Right now my role, until there is a full-blown family meeting, is that these are her children. This is her house. These are their rules. My involvement with the children as far as traditional guidance—I always leave that up to Louise. The schoolwork is to be done, Louise is the one. I simply ask them, you know, how their schoolwork is coming along. Any permissions to be going out with certain friends, they always have to ask Louise. You know, I treat them with respect and we can be friendly, but when it comes to making decisions that is up to Louise; I am not their guardian. "Louise is your mother, go check with her."
I asked if it ever became frustrating not being able to enforce anything as a parent, and Valentino responded:
> Yeah, sometimes. Like, Louise told Mina to clean up after the dog in the back yard, not a fun job. And it was a forest, she couldn't really see what she needed to pick up. I said, "Would you please excuse us?" I wanted to go over her head and enforce that, but I couldn't. They're her kids. Discipline and all that stuff, I couldn't do it. So I simply asked Mina, "Are you going to be able to do this or not?" She says, "Yeah, but . . ." and unfortunately I got a little perturbed to the point where I said, "Yes or no—are you really gonna do this?" "Well, no."—"Thank you for telling me." And so I went over her, I went ahead and just did it. It probably wasn't the best way, but I just needed to know . . . I told Louise about it later and she said she was going to go ahead and have a chat with the kids. That's why it's another thing we added to the list of what we need to talk about.
Like many other poly families, the Amores used family meetings and other smaller group discussions to navigate the complexities of multiple parents and children sharing a domicile. They also exemplified the propensity for both children and adults to assign parental status to only those partners who entered the family when the children were young and have continued interaction over time that usually included cohabitation. In the absence of those factors, children and adults in poly families constructed relationships with chosen kin more likely to take on roles like those of aunts, uncles, cousins, or friends.
1.
(Pallotta-Chiarolli 2010)
2.
(Constantine & Constantine 1973)
3.
A Spanish term meaning mixed or messy, Latina feminists such as Gloria Anzaldua (1987) and Cherrie Moraga (1981) used the term _mestizaje_ to describe the multiplistic intersecting natures of sexuality, racial and ethnic identity, gender, religion, politics, and culture. The metaphors of the borderlands, slipping through the cracks between cultures, and mixing disparate elements to become an amalgam that challenges a simplistic bifurcated or dualistic reality are central to the concept of the mestizaje.
4.
(Pallotta-Chiarolli 2010: 26)
5.
(Pallotta-Chiarolli 2010: 214)
6.
In the United States films are rated G for general audiences including small children; PG for parental guidance suggested for small children; PG-13, which is not recommended for children under thirteen; R, which is restricted to anyone under eighteen years old who is not accompanied by an adult; NC-17, which means that no one under eighteen is allowed entrance into the movie; and X, which is pornographic. By saying that the action was PG, Nolan meant that it was very tame and shocking or inappropriate only to very small children or extremely modest people.
7.
_Will and Grace_ was a popular television show that aired in the United States during the late 1990s and early 2000s and prominently featured several gay characters.
8.
(Burde 2013)
Chapter 6
# Adults in Poly Families
The families who participated in the development of this book, on the whole, felt satisfied with their family lives, and cast polyamory as having a positive impact on themselves and their children. This optimistic tone could result from the reality that poly families are good for the people who live in them, and that the people in these families generally have race and class privilege so their lives are just easier on those fronts than people who don't have those privileges. It could also result from a group that feels judged by conventional society trying to make their unconventional choices appear as positive as possible in order to defuse possible criticisms.[1] Overall, these families seemed to work quite well for the people who continued to live in them.
## Family Forms
The most common form of poly family seems to be an open couple with children (two people in a long-term relationship who often live together and have additional sexual relationships) and their attendant constellation of kin, both biolegal and chosen. Open-couple families appear to identify as family for longer periods than do larger groupings, which are rarer and experience greater membership fluidity. Some have children from previous relationships, others have children from their poly familial unions, and still others remain child free/child less and identify themselves as members of poly families composed of adults. While some people actively seek poly relationships for years and consciously construct a chosen family, others become polyamorous by spontaneously forming a relationship first and then coming to identify later as polyamorous.
## Polyamorous Family Issues
Polyamorous families experience a number of relationship issues, including relations with biolegal families, marriage and commitment, and divorce. Because they have such a tremendous amount in common with families of other sexual minorities, and especially lesbians, bisexuals, and gays, in this chapter I include them as a comparison group and term them _lesbigays_.
### Relations with Biolegal Families
Relationships with biolegal family members varied dramatically, from close and loving to severely strained or estranged. Similar to families of people who come out as gay, some families reacted especially negatively at first and then became more accepting over time. At one end of the spectrum, some of my respondents were at ease being "out" with their families of origin regarding their polyamorous relationships. For example, Louise Amore was comfortable being candid with her mother, JP, because
> my mom is poly too. She doesn't call herself that, but she has been my whole life. She was very open about her sexuality and we talk about our sex lives together all the time. . . . She doesn't judge me for anything, she's one of my best friends!
Key polyamorous ideals like communication and honesty cultivated the sense of intimacy Louise perceived between herself and JP, whose ostensible status as a potential polyamorist herself further reinforced their bond. Louise and JP's comfort with being candid with each other mirrored that of lesbigays who were also at ease being candid about their sexual orientations with their families of origin.[2]
The Tree triad—composed of Bjorn, Gene, and Leah, all with PhDs and academic or high-level industrial jobs—found that relationships with their families of origin that had been slightly strained by the triad's inception as a family became warmer as all six of the parents came to view themselves as grandparents to the triad's son, Will. While none of the parents outright rejected any of the Tree triad members, some did express significant concern over their adult children's well-being and fear that the unconventional lifestyle would potentially harm them emotionally.
Initially, this sense of dismay was significantly heightened for the triad's parents when Leah became pregnant and the triad refused to disclose to anyone which of the men was the biological father. The Trees felt strongly that they were a single family, and that distinguishing between the two men to designate one of them as the "real" father was fundamentally against their relational orientation. This refusal to identify the biological father was initially frustrating to the grandparents, but once Will was born their collective level of acceptance rose. Tree family members reported the following:
> Gene: We each have parents and so they are all equal grandparents of Will. No one came out to visit [for the birth] because we did not want anyone out, we wanted to deal with the initial weeks on our own and then they could come. We have enough manpower and wanted to get ourselves established first. The mothers were relieved; it is the first grandson on both dad's sides, even though Leah's brother has a son so I guess it is not really their first, but even so everyone is super excited about him [Will]. A month ago we had all the grandparents to a house on the East Coast and spent the weekend trading Will around and playing tennis.
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> Elisabeth: So all of the grandparents are cool with you now?
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> Leah: Yeah, there is no animosity with each other or with us—they have gotten over all the weirdness, and having Will sealed the deal with me and Bjorn's mom. They were still a little tense when we told them, there was a five-second beat of oh, my, well, of course, congratulations. Now they are soooo into him and his mom emails me all the time.
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> Bjorn: We visit with each grandparent every three or four months, and we Skype all the time. We have been to the East Coast and they have come here, so all of the grandparents have been able to spend time with Will fairly regularly.
The fact that "the grandparents" became (at least ostensibly) more accepting of the Tree's unconventional family style once they had a child is due both to the triad's patience and willingness to endure parental disapproval and the irresistible power of an infant to coax a reluctant grandparent's adoration. This increase of acceptance after a sexual-minority family has a child mirrors other researchers' findings in studies of the families of gays and lesbians, who reported that their own parents often accepted them or their partners to a greater degree once they had a child.[3]
Even in the absence of an irresistible infant, some families accepted their polyamorous family members with open arms. When Megan Holstrom's father died her husband, Billy, prepared to go to the funeral with her. Megan's boyfriend Jack, who lived in a neighboring state, responded to the news of his beloved's father's death with instant action. Billy reported that
> Jack just would not stay home. He said, you need support and I am coming to the funeral, you can't stop me. So we told her [Megan's] mom and she said, "You love each other?" We said yeah. "He is good to you?" Yeah. "Then that's great, as long as you love each other it's fine with me." Jack went to the funeral and helped fold the flag, which I thought was a huge acceptance on the part of the family because they had never met him and accepted him right away as a part of the funeral party.
In this case, Megan's family accepted both Billy and Jack as her partners with no further scrutiny beyond investigating their emotional commitment to each other, even integrating Jack into the funeral party.
The Wyss family has experienced a wide range of acceptance and rejection from biolegal family members. Kiyowara Wyss's experience with her grandmother's eightieth birthday party was at the positive end of that spectrum. The party was a major event for Kiyowara's mother, Suka, and her extended family members who were in attendance from various states in the United States and Japan. It was also the first such event the entire quad attended as a family unit. Because of their appearance as two heterosexual couples, the Wyss quad expected the true nature of their relationships to remain unrecognized. Kiyowara reported that, during the party, she was
> focused on my grandmother's birthday. You know, I didn't feel a need to make a statement about "We're here together" or anything. And I couldn't believe that, my mom was up on stage thanking everyone for coming and she called us all up and she said, "I want to introduce you to my children" and that was it. Everybody knows that me and my sister are her only _biological_ children, so some of them had no idea what she was talking about. But now we're all her kids and that was that! I was really touched, for her, you know, to do that, it really meant a lot.
Kiyowara thought that her mother's public acknowledgement of all the spice as her children was Suka's way of recognizing the legitimacy of Kiyowara's unions. Suka's public acceptance of the quad facilitated friendly contact between herself and the quad, as well as their interactions with Suka and Kiyowara's extended family.
In the Wyss quad's case, Suka's acceptance waxed, waned, and ultimately proved to be firmly rooted in the quad's ostensible heterosexual relationships. Over time Suka became quite ill and moved in with the quad to recover from a hospitalization. She was in pain, bewildered, had trouble breathing, and had to be monitored around the clock. Because she was already Kethry's full-time parent at the time, Loretta agreed to care for Suka as well. In an effort to manage the considerable caretaking demands, Loretta sought assistance from many state and federal agencies and was scrupulously forthcoming with the various social workers, home health aides, and assistants regarding the adults' polyamorous relationships. Suka, however, frequently tried to conceal the sexual relationship between Loretta and Kiyowara by telling the host of personal and medical assistants that the two were sisters. Loretta suspected that Suka's initial ostensible acceptance might have provided a cover for her veiled discomfort and homophobia that emerged as her flagging health became increasingly problematic. Like Suka, Kiyowara's extended biolegal family was similarly ambivalent, happy to accept Loretta's role as a full-time caregiver and the Wyss family's continual financial gifts (including purchasing two different homes for Suka), but unwilling to grant the Wyss family recognition as genuine family members at Suka's funeral.
Similar to the Wyss family, the Southern triad had a spectrum of relationships with their biolegal families, from affable to horrid. Triad members were Earl, Tom, and Melinda, all in their early forties. Tom and Melinda were married for eleven years and had two children when they formed a triad with their longtime friend, Earl. Each member of the triad invited their parents to the commitment ceremony that marked their eventual coalescence as a family unit. Earl reported that his parents were "thrilled . . . they'd given up on ever having grandkids when I came out to them as gay, so to have two ready-made grandkids put them into grandparent heaven!" While Melinda's parents were accepting, they were notably less enthusiastic than Earl's. The most politically and religiously conservative of all the Southern triad's biolegal kin, Tom's parents not only rejected the triad's invitation to the ceremony, they then rebuffed further contact with that entire family (including their grandchildren).
Things changed, however, four years later when Tom's father was diagnosed with cancer. Tom's mother called him to let him know his father was in the hospital, and she said "life was too short to hold this kind of a grudge." His father consented to speak to Tom, and while he was happy to be "patching things up," Tom's parents' initial rejection still hurt. "Things can't ever be the same again once your parents have told you that you aren't their son anymore."
Some respondents came from complex families of origin, so their poly families were not particularly shocking to their biolegal and chosen kin. Logan Tex was characteristic of respondents who came from unconventional backgrounds and became polyamorous. Logan was in an open-couple relationship with his wife, Melina, and their girlfriend Rhiannon. At the time of the interview, Melina and Logan had two small children, an infant and a toddler named Pip less than two years apart. Logan's parents had split up when he was a child, so his family of origin included his mother, Jess, her wife, Paula, his father, Nick, and his wife, Erin—all of whom took parental roles to some degree in Logan's life—as well as a variety of siblings from several combinations of parents and their exes. When I asked him what his family of origin thought of his poly family, Logan replied:
> My moms seem generally supportive of whatever I do. Paula, my biological mom's partner, feels Rhiannon might be trying to steal me away and doesn't like her in the middle of my family. Jess, my bio mom, is fairly distant with Rhiannon. It's hard to get a read on her. My dad seems to be envious that I can have the life I do—his wife does not like him being with other women, but I think he would [do so] happily if the relationship allowed for it.
The fact that Erin, Jess, and Paula all reacted slightly negatively to Rhiannon—a common reaction among mothers-in-law to their son's girlfriends who retain sexual ties to other partners—is ironically conventional, considering that Jess and Nick had raised their children in a decidedly unconventional lifestyle. Logan links his upbringing directly to his contemporary poly family:
> I think my childhood heavily influenced how I am. But I am still figuring out how, exactly. I was raised, somehow, to not have much regard for traditional ways of doing things—divorce, gay parents, parents that still talked and were friends postdivorce all played in to that. I learned that relationships can evolve, which means that the risk of a romantic relationship is less, since you can keep something if the romance part goes . . . I was raised on a hippie commune and those adults are still my friends, not just people my parents knew. I liked that and would like to give that to my kids. One of the features of that is that is they have been friends for decades, which means committing to being friends. As far as romantic relationships go, it would be important to me to maintain friendships even if the romantic part ends.
For Logan, bonds outside of conventional marital life can be enduring and outlast divorce, supersede romance, and become lifelong commitments among friends—definitional of both chosen kinship and polyaffectivity. While his mothers allow for and themselves express wide social variation, they remain somewhat uncertain about their son's poly family. Logan himself has mixed feelings about conventional relationships himself, as we shall see in the next section.
### Marriage and Commitment
People in same-sex relationships seem far more interested in attaining legalized same-sex marriage than do polyamorists, who appear to be significantly less personally or politically devoted to plural marriage.[4] My findings indicate that respondents do not mention marriage as a central concern, and when they do, some do so disparagingly. Those poly people who wish to marry can do so as pairs, and the tendency toward hetero and bisexuality among polys makes it possible for them to (ostensibly) meet requirements for heterosexuality. Because most of the people in this book are white and middle-class professionals, their race and class privileges offer some protection against discrimination,[5] making the rights associated with legal marriage less important for polys than they would be to others with fewer social privileges. Such access grants polys greater social maneuverability than those in recognizably same-sex relationships, a latitude that is reflected in polys' views of marriage. Some reject marriage as inherently flawed; others are married but do not see it as very important; and still others view marriage as profoundly important in shaping their relationship structures and interactions.
#### Commitment Ceremonies
Poly folks expressed a variety of views about marriage and commitment ceremonies. Like some lesbigay couples, polyamorists occasionally formalize their commitments with public ceremonies that acknowledge the group as a family unit. For some, ceremonially announcing that they are "fluid bonded" (a negotiated safer-sex agreement that allows people to share bodily fluids only with specific lovers who have been tested for STIs) signals their lasting pledge to their partners and communities at large. One trio of two women and a man who had dated for several years gleefully informed the attendees at their ceremony/party that marked their fluid bonding that "we are a family now!" Other polys choose alternative forms of union such as handfasting, a Pagan ritual in which people are ceremonially bound wrist to wrist with soft cord for three days and thereafter considered to be married.
Occasionally large and stable families like the Wysses deal with the lack of official recognition by creating corporations or trusts to manage taxes, child custody, medical power of attorney, inheritance, and joint property ownership. As scholars documenting lesbigay's attempts to secure similar legal rights find, such arrangements require extensive legal documentation in an attempt to address every foreseeable contingency, from the division of property in case of "divorce" to the assurance of continued custody of children if both biological parents die.[6] The high cost of this legal documentation makes this route almost impossible for anyone without significant financial resources for such extensive legal preparation.[7]
#### Marriage
Because many polys can legally marry as ostensibly monogamous, heterosexual couples, they have different relationships with marriage than do most lesbigays. While lesbigays may also choose to marry someone of another sex in a similarly ostensibly monogamous and heterosexual couple, it requires a far greater effort to maintain a closeted gay life than it would for polys with other-sex partners—a configuration that makes them socially recognizable as heterosexual couples with "close friends." This ability to remain closeted almost effortlessly is a resource many lesbigays cannot have, and so it functions as a form of (often misattributed) heterosexual privilege that provides a buffer against the effects of stigma against sexual nonconformists.
Few poly people talk about legal plural marriage at all, and even fewer identify it as an important goal. Some polys avoid or even ridicule monogamous marriage as an ill-conceived experiment. Joya Starr told me: "I think [marriage] is an institution, and that's fine if you want to be institutionalized." Others scorned people in monogamous marriages as "coasting" or "on automatic pilot." Thaddeus, a forty-one-year-old musician, cast marriage as detrimental to the health of relationships: "The thing that ruins their marriage was a piece of paper saying that they were married . . . There wasn't communication, that these were things that they certainly couldn't talk about because they felt stuck." Polyamory provides Joya and Thaddeus a vantage point from which to critique monogamous families and relationships, much like those who oppose same-sex marriage because they contest all marriage or advocate decoupling social benefits from relationship status.[8]
Like the majority of polyamorists who have participated in research, Joya and Thaddeus were both white, well educated, and middle class—with access to the privileges that allow them to focus on rebellion against the patriarchal norms of conventional families. Their socioeconomic status and cultural cache provide the kind of security that is scarce for lesbigay and/or working-class people. The larger and more diverse lesbigay community has a broader range of people, and the social privileges that attend legal marriage can be far more important to those who have few other privileges. The more scarce the privileges, the more precious each becomes. Mainstream polyamorists' myriad privileges allow them to downplay or forgo marriage in favor of rebellion precisely because they are so well endowed in other areas.
In some cases, legally married polys cast their marriages as inconsequential. Phoenix and Zack, a white couple in their early sixties, date their relationship from its inception over thirty years ago, rather than the date of their actual legal marriage, which Phoenix sees as "pretty much just a piece of paper. We did it so he could get health insurance—at the courthouse." Many legally married polys mention it only in passing and do not identify it as important in their interviews, but they are still able to avail themselves of its advantages and secure benefits that remain unavailable to their counterparts in same-sex relationships. This near-universal poly disinterest in legalizing multiple-partner marriage, or even investing heavily in conventional marriage, stands in sharp contrast to the significance many lesbigays accord same-sex marriage.
In rare instances, legal marriage plays a significant role in shaping partners' expectations of each other. For example, the Hadaway quad members had complex attitudes toward marriage. The quad is composed of two legally married couples and their ten children (five from each couple), with sexual relationships between the women and both men independently, but not between the men. Its members, all in their early forties, include: Gwenyth, a full-time homemaker; her legal husband, Mitch, a real-estate broker; Tammy, a part-time assistant to both Mitch and Gwenyth; and her legal husband, Phil, an electrician and technician. Each couple had been together for almost fifteen years when the women, both pregnant with their fifth child, met in an Internet parenting chat room and began an online relationship that was mostly friendship with, Tammy reported, an undercurrent of "strange intensity." After meeting in person with their spouses and eventually establishing "cross-coupled" sexual relationships between Gwenyth and Phil and Tammy and Mitch, the four decided that Phil and Tammy would move from their neighboring state to live near Mitch and Gwenyth. Shortly after arriving, Phil had a nervous breakdown, partially in response to the tremendous stress of working in the Gulf Coast region of the southern United States after hurricane Katrina had devastated New Orleans and the surrounding areas. Phil reported that "it had been coming for a long time," and Mitch opined that Phil was "finally able to let go once he knew there was someone else there to take care of his family." Tammy and Phil subsequently moved in with Mitch and Gwenyth, blending their households and nine of their children (Tammy and Phil's eldest daughter moved to her own apartment).
Tammy reported that Phil expected her to make him breakfast every day before he left for work—even though Gwenyth was already up getting the children ready for school—specifically because she was his wife and "that is the kind of thing a good wife does." Phil expressed dismay at what he saw as Tammy's waning devotion. "She used to do it when it was just her and me, but now that we live with them it's like she's not really my wife anymore. At least not the way she used to be." Similarly, Mitch considered his relationship with Gwenyth to be his priority, not only because they had been together for many years but also because they were married and thus should have primary allegiance to each other. Gwenyth reported feeling hurt by Phil's "fixation" on having Tammy do things for him. "I like spending that time with you and you don't appreciate it at all. It doesn't matter that we're not married, I still love you and can make your lunch!" She rejected legal marriage as the overriding relational structure, saying, "I don't recognize any primary-secondary, we're all on the same level," regardless of legal marital status. Even within this family, members did not necessarily agree on its terms. While this is possibly true of any marriage in which partners have differing views on the nature, function, or dynamics of their relationships, it can be even more pronounced in poly families. Retention of significant elements of monogamous or other patriarchal familial types can potentially impair adaptability, as the attempt to graft on elements of the previous form inevitably chafe against the new form. The quad experienced growing pains as they attempted to redefine their roles and relationships to each other, stretching their abilities to adapt to changing relational configurations and precipitating various crises and conflicts over mundane issues of daily life.
In contrast to the Hadaways who framed the issue as emotional and traditional connections between and among quad members, Logan Tex explicitly cited the privileges associated with marriage. When I asked him why he and Melina married, he responded:
> We decided to do it legally because it makes things a lot easier, legal things around the kids and being incapacitated and things like that. We were toying with the idea of writing our own nonmarriage contract with nothing about monogamy but we were lazy and took the easy route and just signed the state thing. But that means that there will on some level be some disparity with us and Rhiannon, even if she ends up living with us. I do not see us having another ceremony where we invite our friends and family to get married to Rhiannon. We have given a privilege to our own relationship over other relationships that is not particularly polyesque in a way, but on a philosophical level it is interesting that we chose this.
>
>
>
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> Elisabeth: How does Rhiannon feel about this?
>
>
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> Logan: Hard to say. She says she feels good about her connection to us as a family, but I think eventually she will want something deep with someone and thus her relationship with us will get a little more distant as she gives priority to someone else. Or she will become fully integrated with us. It's a weird position for her to be in and I can't imagine exactly what it's like.
Logan's assumption that, if their relationship progressed, Rhiannon would "end up living with us" and "become fully integrated with us" was based in couple privilege, something he acknowledged as "not particularly polyesque" but preferable to him and Melina. Well aware of Rhiannon's potential dissatisfaction, Logan knew that it might mean changing or losing his relationship with Rhiannon.
Logan and Rhiannon had begun dating roughly two years earlier when Melina was pregnant with Pip. Melina and Logan spent some time talking about their relationship and decided that if Logan was going to find a girlfriend:
> It was only going to get harder after we had the baby, so if I want a girlfriend I should find one now. Rhiannon is so much more of a girlfriend than I was really looking for. I was after someone to frolic with but she has become really important to us and much more integrated into our lives than we had anticipated. She and Melina really clicked too. Melina was very pregnant when they first met and it was good for them to meet before the baby was born. Having the baby coming definitely put things on a timetable.
Even with the unexpected and slightly rushed beginning, over the next two years things had gone well with the Texes. Logan explained that "Rhiannon independently loves Pip, likes to spend time with him for the sake of spending time with him. We agree that it seems nice all around." While Rhiannon was subject to the disadvantages of being the secondary relationship, outside the protective circle delineated by couple privilege, she clearly got enough of her needs met in the relationship to stay in for at least two years, and possibly more.
## Divorce
Polyamorists' various views on marriage parallel their similarly diverse relationships with divorce. Some of my respondents selected polyamory as an alternative to divorce, while others became poly subsequent to divorce from monogamous marriages. Still others divorced and retained sexual and/or cohabitational relationships with their "exes" after dissolving their legal unions. Most similar to lesbigay families, some members of disbanded polyamorous families did not have access to legal divorce.
### Become Poly Instead of Divorcing
Some people transition to poly families rather than divorce. Typically this happens when one of the partners is discovered engaging in an adulterous affair or confesses a transgression to their spouse, and those involved choose extramarital relationships for both partners rather than divorce. Claire and Tim, a Mexican American woman and a white man both in their mid-thirties and married for nine years, decided to become polyamorous instead of divorcing when Claire learned of Tim's extramarital affair. Claire articulated feeling betrayed by Tim's initial deception but, while she did not want to be the "dupe who stays at home with the kids while he is out screwing around," she was not willing to end their relationship. Claire and Tim reconsidered the meaning and stability of their union, and they ultimately chose to open their relationship to outside lovers. Claire reported greater personal satisfaction and equality in her marriage since she has outside relationships as well, in part, she thought, because Tim no longer took her for granted as much. By agreeing to alter the definition of their relationship, Claire and Tim simultaneously reformed the power dynamic from a traditional familial structure rife with power imbalances to one that Claire thought "leveled the playing field." Poly families' flexibility permits them to adjust to shifting family circumstances, allowing families to outlast the crisis moment and reposition themselves to accommodate changes in structure and form, fostering an adaptable kinship network.
### Polyamorous after a Divorce
Some people whose previous marriages ended because of cheating will begin a new relationship with the explicit intention of creating a polyamorous family. Sven Heartland's divorce resulted from his lying and hiding his sexual relationships with men from his now ex-wife, so Sven vowed to himself to be honest in future relationships to avoid making the same mistake again. When he met Shelly, Sven was forthright about his bisexuality from the beginning of their relationship. Initially shocked by Sven's suggestion to add a boyfriend to their family, Shelly eventually became more accepting of polyamory, though she remained somewhat dubious at times. "I never would have considered it before I met Sven, but I would rather be involved with these guys than have him taking so much energy and time away from the family to be with them."
For several years Shelly and Sven dated men with limited success. Ultimately they met and fell in love with Adam, a thirty-five-year-old white computer systems support provider with whom they established a triadic relationship. While the triad seemed to coexist peacefully for several years and all three members reported being happy together, the relationship eventually began to experience some difficulties. Shelly was more attracted to Adam than he was to her, and she occasionally felt some tension around this imbalance of desire. After almost four years together, Adam broke up with Shelly and Sven, who eventually began dating other men again. The flexibility of a poly family allowed Sven to be honest with Shelly and meet his need for sex with men while still retaining his familial connection with his wife and children. The frank dialogue characteristic of this and other poly families[9] similarly set the stage for Shelly to verbalize her needs and openly negotiate a safer-sex agreement.
### Divorced but Still Lovers
Some polys divorced but continued their relationships much as they had prior to the divorce. Melody Lupine's triad was characteristic of this tendency to create new familial patterns. She had already had two children with Cristof, her legally wed husband, and she intentionally became pregnant with a third child when Quentin, her additional (extralegal) husband, expressed the desire for a child. Both Cristof and Quentin accompanied Melody in the delivery room when she gave birth to Zane, her second son. Though the triad specified paternity and expressed their intent to coparent, officials insisted on listing Cristof as the father on the birth certificate because state law stipulated that a married woman's husband is the legal father of any child she bears, regardless of evidence to the contrary. Melody said:
> We told everybody Quentin is the father. I'm married to Cristof, and Cristof's name had to be put on the birth certificate, legally, because we were married. Even though we said no, this is who is and this is who it isn't. And they were just like, we don't care. You're married, his name goes on. Quentin was outraged.
In order to clarify Quentin's relationship with his infant son and Melody's relationship with both men, the triad decided that a legal divorce was in order. Ironically, a social system designed to support families in this case actually encouraged divorce through its lack of flexibility. The Lupine triad's relational adaptability allowed them to outlast the legal marriage by negotiating a flexible arrangement to suit their kinship needs. Melody was optimistic about the impact the divorce had on the family, and she felt it set a good example for her children, who saw their parents remaining connected during a congenial divorce:
> They get to see that a divorce or break-up doesn't have to be this destructive, I hate this other person, I have to choose between mom and dad, I have to hear them arguing, they don't talk to each other. Children take on so much stress and trauma from divorce where parents pit one against the other. That didn't happen.
As society grows ever more complex and social changes we have been experiencing for some time already continue, this ability to maintain friendly contact through changes in family life and structure is becoming increasingly important. By deemphasizing biolegal connections and embracing a broader definition of family, both polys and lesbigays demonstrate the resilience of polyaffectivity and chosen kinship.
### Lack of Access to Legal Divorce
While divorce and its polyamorous proxy of separation exert a mixed impact on polyamorous people and their children, the lack of access to official divorce can sometimes be as difficult as a divorce itself. The Mayfield quad, composed of Alicia, Ben, Monique, and Edward, all in their late thirties or early forties at the time, was together for eleven years before breaking up. Ben, Monique, and Edward had all been employed during their term in the quad, but Alicia's back injury prevented her from performing paid labor. Instead, she cared for their home and Monique and Edward's biological children, who were five and seven years old when the quad coalesced as a family. When the quad disbanded, Alicia had no access to the usual recourses available to women whose monogamous legal marriages end. Without legally recognized relationships to any other quad members except her soon-to-be-ex husband, formalized access to the children she had cared for during the last eleven years, or the legally recognized ability to seek the alimony traditionally awarded to homemakers who divorce a wage earner, Alicia was in a difficult position indeed. Although legal protections would not have shielded Alicia from the emotional impact of the family's dissolution, they would at least have allowed her visitation of the children she reared and financial compensation for the years she spent raising them and maintaining the household to facilitate the waged work of her spice. Lack of official recognition of her polyamorous family contributed to Alicia's personal and financial devastation. No marriage means no divorce, and in many cases, no mediated negotiation of custody and property issues. Legal divorce is clearly far from perfect, but it does provide some protections for nonbiological parents and homemakers that are unavailable to people in relationships that are not legally recognized. For both polyamorists and lesbigays who wish to marry or divorce, legal recognition remains a double-edged sword: it constrains the forms families are able to take, but the lack of those protections can be costly for those whose relationships are not recognized by the legal system.
## When Is a Poly Relationship a Success? A Failure? Over?
Although most families have divorced members in their kinship networks, conventional wisdom still defines a marriage or long-term relationship that ends in any other outcome besides death as a _failure_. Children of divorce are said to come from "broken homes"[10] and their parents have "failed marriages" that mark them as personal, relational, and often financial failures.[11] These cultural norms define "successful" relationships as monogamous and permanent in that the two people involved remain together at all costs. In this worldview, sexual fidelity is fundamental to the successful relationship and functions as both a cause and a symptom of relationship success.
Polyamorists, in contrast, define the ends of their relationships in a number of ways in addition to success or failure. Many poly people view their relationships as fundamentally based on personal choice, and if the relationship became unhealthy or intolerable, violated boundaries, or no longer met the participants' needs, then the correct response was to modify or end the relationship. Tacit Campo said:
> If you are in a relationship or several relationships then you _choose_ to do that, every day, whether you recognize it or not. You can stay because you consciously make that decision or you can just stay because you are on automatic pilot, but that is a choice too.
This consciously engaged choice means that polyamorous people acknowledge their own responsibility for their relationships, with little or no social pressure (from the polyamorous paradigm at least) to either stay together or break up. As a result, poly people ultimately define their relationships as both voluntary and utilitarian, in that they are designed to meet participants' needs. Clearly it is easier to focus on self-responsibility when the people in question are financially self-supporting and do not have children whose lives would be affected by parental separation. Given the framework of those familial and social constraints, poly people attach diverse meanings to the ends or transitional points of relationships.
In my research, three primary definitions of the ends of relationships stood out: success or failure, shifting interests and needs, and change or transition. While each category is distinct, they are not mutually exclusive and often overlap. Fewer of the poly people I interviewed defined their relationship ends in terms of failure, and many more emphasized their shifting needs and interests, and especially the fluid nature of relationships over time.
### It Is Really Over: Success and Failure
Some polyamorous relationships last until one of the partners dies, and in that sense they meet the conventional definition of "success" because the family members did not separate from each other during their lives. The Wysses began as a sextet of three couples and evolved significantly over time, losing partners to death and divorce. The original sextet was composed of three legally married couples—Loretta and Albert, Kiyowara and Patrick, and Margret and Tim—who conglomerated into a cohabitational family with older children from previous relationships. After two years of love, fighting, and conciliation, Margret divorced the entire family, including legally divorcing Tim. The resultant group had only just restabilized when Tim was killed in an automobile accident. Even though the surviving spice lost their husband to death, they did not frame it as a "successful" end. Instead of using a success/failure characterization, the Wyss quad emphasized the joy they had with Tim when he was alive, the pain they felt at his death, and how the relative invisibility of their poly widowhood compounded their sense of loss because the monogamous culture at large did not define them as widow/ers.
About the same time Tim was killed in the accident, Kiyowara became pregnant with Albert's child and bore the quad's daughter Kethry. Fourteen very full years later, the Wyss quad became the Wyss triad when Patrick divorced Kiyowara (legally), Albert and Loretta (socially). Kiyowara characterized the relationship as a success even though it ended:
> I am glad we are coparenting and not married. . . . I certainly can't call it a failure; it was a twenty-year marriage. And I am glad his current choices are not my problem. Any time a relationship ends there is a tendency to view it as a failure. I was very clear that a relationship that had good times and lasted twenty years was not a failure, it just ended. End does not mean fail. That totally invalidates anything good that came out of it. I had a lot of people remind me that it is not a personal failure just because something had run a full cycle and came to its end.
Kiyowara redefined the end of the relationship with Patrick from failure to relief from dealing with his choices and continued contact as coparents. Friends in her poly community "reminded" her that it was not failure but rather the end of a cycle, supporting her redefinition. Such reinforcement allowed these alternate meanings to take on more social gravity and ultimately become solidified as poly social norms that accept the ends of relationships and encourage former lovers to remain friends.
For others, the end of a poly relationship kept the taint of failure in the conventional sense. Although poly community norms encourage people to remain friends with former lovers, some relationships end with such acrimony that former lovers find remaining friends to be neither desirable nor feasible. People whose relationships ended with infuriated distance were more likely to see the end of the relationship as a failure, both in the conventional sense of ending sexual and intimate relations and as a _poly_ failure in that they broke community norms dictating continued friendly contact with former lovers as friends.
Jessica, a forty-three-year-old woman and registered nurse, had been in a triad when she was in her mid-thirties with Mira and James, a married couple with two young children. For about a year and a half the triad spent five to seven nights a week together, often at the couple's home engaged in family activities such as making dinner, doing dishes, and bathing and putting the children to bed. When the triad broke up, Jessica reported feeling like they had failed because
> at the beginning we said that if we were going to be like a family then I would stay connected to the girls, no matter what happened with us [the adults]. And for that time I was definitely, not quite a second mom, but at least an auntie who was around all the time . . . But then when we broke up, I just realized they [Mira and James] were not who I wanted to spend time with and it was awkward to call them or try to talk to the girls. Mira was especially weird on the phone and . . . eventually I just kind of stopped calling, and now it has been years since I have seen them. So I guess in that way it feels like a failure, because we didn't stay connected like we had planned to.
In Jessica's view, the end of the triad was a failure not only because the adults stopped interacting but also because she lost contact with the children she had lovingly cared for over a year and a half.
Because poly relationships can have multiple adults involved, relationships between some members can end while they continue between others. In these cases, some of the people involved may define it as a failure but others may not. Morgan and Clark Majek's family was characteristic of this tendency for some adults to maintain contact even though others stop seeing each other. Morgan and Clark, both white and middle class, met in college and married in their mid-twenties. After several happy years of marriage and the birth of their daughter, they attempted to form a quad with another female/male couple. Six months later it was clear to everyone that the quad was not working, and while they no longer stayed in contact Morgan reported that "I learned a lot from that initial experience so I don't think of it as a failure—it was a learning experience."
Later, when Morgan was pregnant with their second child, she and Clark established another quad with James and Melissa, a couple who had been married for almost ten years. Melissa and James's marriage had been in crisis before, and they had separated for almost six months several years earlier but had reunited prior to meeting Morgan and Clark. James and Morgan fell in love, and Clark and Melissa investigated a relationship but realized, as Clark reported, "we did not have the right chemistry." Melissa was sometimes close to Morgan and Clark and at other times quite distant, but Morgan, Clark, and James established an intimate emotional connection. For five years James, Morgan, Clark, and their two children spent three to six days per week together and shared many family events.
Eventually James and Morgan's relationship soured and, with hurt feelings on both sides, they stopped seeing each other. Clark, however, reported that he and James maintained friendly relations:
> Oh yeah, we get to see him all the time. Either we drive down to [a town about forty-five minutes away] or he comes up here. Actually, usually we go down there, probably every other week or so. I actually get along with James better than Morgan does right now, so it makes sense for me to take [the kids] down to see him. I know the kids miss him a lot so I definitely put effort in to getting them together. I still like him, too, so it is nice for me to see him, though I don't think I would do it nearly as much if it weren't for the kids.
While James and Morgan's relationship fit one definition of failure because they no longer saw each other, the rest of the family maintained a successful relationship with James, if success is defined as remaining in contact. This flexible definition allows for polyaffective relationships in which children can stay in contact with adults who are important to them, even if the adults are no longer in sexually intimate relationships with their parents. In that sense, this expansion of options that allows polys to define the relationships as successful (even though they have "failed") also sustains family connections.
### Moving Apart: Diverging Interests and Needs
Some polys like Angela, a thirty-two-year old white woman in the IT industry, emphasized the idea that they were no longer relating to former partners the same way (or possibly at all), but rather:
> moving apart without blame—people change over time and what worked before no longer does, or what was once interesting to everyone is now boring to some of us who are now interested in this new thing. Like [my ex-husband] Mike with his whole anime thing, that holds no interest for me, absolutely none . . . and he has no interest in crafting, which has become really important to me and takes up a lot of my time. There is no judgment or shame for changing from the people we were when we met at SCA[12] all those years ago, we are just not who we used to be and don't fit together as well anymore.
Like Angela, people in this category emphasized divergent interests and decreasing time spent with partners who had formerly shared more interests as the key factors that influenced how they defined their shifting relationships. Poly people often have full lives and hectic schedules, so time is at a premium, and how people "spend" it frequently indicates their relational allegiances. If partners spend a lot of time doing different things, then they may develop divergent social lives, resulting in less overlap in social circles and decreasing importance for some relationships as others increase in intimacy and time together. This shift is not necessarily failure; for some it is simply change.
Some poly people discussed the shifting definitions of relationships as they ended or changed once they were no longer meeting participants' needs. If communication and renegotiation did not address the lack, and the relationship remained unsatisfying or defective despite attempts to address the problems, then poly people either reconfigured their expectations or ended the relationship in that form. Jared, a forty-six-year-old divorced father of two and a health care professional, linked his recent breakup with a girlfriend to the fact that the relationship was no longer meeting needs for either of them.
> When I first started dating Janice we were pretty much on the same page with our needs. She has a primary who is out of town a lot and wanted a close secondary, and I am not ready for a primary but wanted a close secondary, so it was great that way for a while. Then she started dating Erika and Mark and began spending more and more time with them to the point that I only got to see her, from two or three nights a week sometimes down to every other week or something. That just wasn't enough for me—I didn't need to move in with her or anything, but twice a month? I mean, come on. So when it became clear that she needed more freedom and I needed more intimacy, we split.
Characteristic of the many poly people who identified the ability for multiple relationships to meet a variety of needs as one of the primary reasons they became polyamorous, Jared and Janice had begun dating to meet their needs for companionship and sex. When the amount or kind of companionship—or any other basic motivator for the specific relationship—no longer met their needs, people like Jared reported "moving on to other relationships that will meet my needs better, at least I hope." Polys in this category often saw the relationship as ending or at least changing dramatically to something far less than it had been previously. Even so, it was not a failure as conventionally defined—rather acceptance that people change and no one need be at fault.
### Not Really the End: Changes and Continuity
For some polys, simply no longer having sex did not signal the end of a relationship, but rather a shift to a new phase. In these cases, the emphasis of the relationship changed to a nonsexual interaction, but the emotional and social connections remained continuous. JP Amore—a sixty-eight-year old woman with five children, eight grandchildren, and one great-grandchild—had been married eight times, four of them to her first husband, Richard, with whom she retained an emotionally intimate, nonsexual relationship. Reflecting on her long and varied relationship with Richard, which began in high school when they "got pregnant and got married immediately—both of us were virgins and we got pregnant on our first time, imagine that!" JP reported that
> we have a tremendous closeness. We've always been able to talk. Intellectual connection, spiritual connection. Just a very intimate relationship. We've got all of this history together, grandkids, a great-grandchild even! I went to Houston not too long ago, and we celebrated the fiftieth anniversary of our wedding. We got to celebrate all of it!
While JP harbored no illusions that Richard was perfect, stating that he has a "multifaceted personality, a wonderful person on one hand, and a male chauvinist controlling jerk on the other," she was able to preserve the positive aspects of the relationship and celebrate a fiftieth wedding anniversary with her long-time companion, even though they had both been married to other people over the years. Their relationship overflowed the boundaries of conventional marriage, and their emotional continuity overshadowed the fact that they no longer had sex.
True to form in poly communities who shape language to reflect their relationships,[13] some polys reject or redefine the concept of the "ex." Laszlo, a man in his mid-thirties, commented that
> the notion of _ex_ is ill-defined unless you have a social context, like (serial) monogamy where at least some "privileged" relationship statuses are single-person-only exclusive. That is, if you don't _have_ to "break up" to be with someone else, then attempting to categorize _all_ of the people from your past relationships as "ex-pickrelationshiplabel" is kinda goofy/nonsensical . . . I can see using the "ex" label structure for relationships that were abusive and continued contact would be unhealthy, but if instead they're still-or-once-again a friend, why focus on what they aren't anymore instead of what they are right now?
While Gabrielle, a woman in her mid-forties, was clear that "I am not best buddies with all of my exes, not by any stretch," she nonetheless asserted that
> I have other former lovers that I suppose ex would be _a_ term for. But, I don't think of them as exes. We were lovers and now we're friends, and ex just seems kind of a weird way to think of someone I'm close to and care about. The real difference here, I think, is that the changes in relationship tended to have a much more gentle evolution rather than "official" breakups.
Rather than an "official breakup," the relationship went through a transition and entered a new phase. Emphasizing the present and continuing existence of the relationship, Gabrielle and Laszlo had room to define former lovers as friends with whom they remained close and caring.
As in most relationship styles, this varies by relationship and depends on how people handle transitions. Sorcia, a Native American woman in her mid-thirties, commented:
> Of course, it depends on the person. Of my former triad—one parent is . . . not even on the remotest of friendly terms with the other two of us. On the other hand, my ex-wife and I are still good friends. We do the holidays together with the kids, get together regularly for dinner and generally weather our ups and downs. We consider each other to be family. She moved in with a boyfriend last fall and one of her pre-reqs was being OK with our familial connection. It's turned out much better than I ever expected and it's pretty cool.
People in poly relationships create a range of relationship outcomes and a wide array of meanings from which to select. Some follow a conventional pattern of alienation when a sexual relationship ends, while others forge views that define former partners as continued intimates, or "chosen kin."[14]
Shifting the crux of the relationship from sexuality to emotional intimacy can foster more connected and cooperative coparenting because it allows for continued and cooperative relationships among adults. While Michael, a father in his mid-fifties, and his coparent divorced fifteen years ago, they continued to cohabit for six years afterward, and
> we have stayed in frequent contact, taking vacations together (sometimes with our other lovers), continuing to raise our kids in close concert, and recently undertook a major multiyear project together (though we were on opposite coasts). She recently told me that she was thinking about her best friends in the whole world, and of the four people she identified, one was me and another was my long-term nesting partner.
Michael reported that his nonsexual relationships had been crucial to his life and well-being, and that being in poly relationships allowed him the unique opportunity to not only remain emotionally intimate in a cooperative, coparenting relationship but also "being free _not_ to have sex with your intimate partner(s)."
> I have these amazing relationships that were once sexual, and in the monogamous world, if I stayed as close as I am with these women, it would be likely to cause substantial stress, or at least some negative social pressure. And each of my emotionally intimate relationships can be sexual or not, sometimes shifting one way or another, without damaging our basic relationship. In a monogamous world, if I stopped being sexual with my primary partner, this would either be a major source of distress, or might end the relationship entirely. As a poly person, I don't feel uniquely responsible to meet my partner's sexual needs. If it best serves our intimacy not to be sexual, either temporarily or permanently, then we can do that without any other _necessary_ consequences.
Michael emphasized the changing nature of relationships over time, as sexual interest waxed and waned due to the vigor of youth, having children, shifting circumstances, and passage along the life course.
> Over the years, I've had two lovers, both previously _very_ sexually assertive, who found that menopause made sex less interesting and less enjoyable for them. They suspect that this may change back at some point, when their hormones settle down, but in the meantime, sex is pretty much off the table for them with all their lovers. This didn't change our connection at all, though. We still sleep (sleep!) together from time to time, do naked cuddling, and have intense, intimate conversations. We just don't have sex, as it is usually conceived of.
Regardless of whether this relationship phase was truly the end of their sexual connection or simply a hiatus, Michael's long-term relationships with his partners continued despite changing sexual and relational circumstances.
1.
See Maria Pallotta-Chiarolli, _Border Sexualities, Border Families in Schools_ (Lanham, MD: Rowman & Littlefield, 2010) for more discussion of her respondents, who also attempted to appear to be perfect to forestall any potential criticism by seeming to be above reproach.
2.
(Baptist and Allen 2008)
3.
(Goldberg 2010)
4.
(Aviram 2007)
5.
(Sheff and Hammers 2011)
6.
(Christensen 1996–1997)
7.
(Adam 2003)
8.
(Card 2007; Emens 2004; Polikoff 1993)
9.
(Sheff 2011)
10.
(Fagan 1999)
11.
W. Rubenstein, "We Are Family: A Reflection on the Search for Legal Recognition of Lesbian and Gay Relationships," _Journal of Law & Politics_ 8, no. 89 (1991).
12.
The Society for Creative Anachronism is an "international organization dedicated to researching and recreating the arts and skills of pre-17th century Europe" that hosts gatherings across the United States, http://www.sca.org/, accessed October 24, 2012.
13.
(Ritchie and Barker 2006)
14.
(Weston 1991)
Chapter 7
# Benefits of Polyamorous Family Life
People in polyamorous families identified a variety of benefits that accompanied their family style. These included honesty and emotional intimacy among family members and a number of benefits associated with increased resources that come with multiple-adult families, such as more money, accommodating disabilities, personal time for parents, more attention for children, and abundant role models for children. Some children identified their parents' ability to remain friendly as advantageous, and many polys discussed the advantages of an expanded chosen family.
## Honesty and Emotional Intimacy among Family Members
Parents emphasized honesty with their children as a key element of their overall relationship philosophy and parenting strategy. Poly parents routinely use honesty in a variety of discussions, ranging from their own shortcomings or mistakes to age-appropriate answers to questions about sexuality. Polyamorous parents often characterize honesty as the primary factor that cultivates emotional intimacy because, as Brad (a white father of two) commented, "the kids get to see us as real people too." He continued:
> We [Brad's wife and their boyfriend] make mistakes, and we cop to them. We tell them what is really happening in our lives, and they do the same with us. Of course there is a line—we don't tell them anything about our sex lives or adult relationship details, but we tell them the most truth we can and still remain in the parental role.
Evelyn and Mark Coach, both white, middle-class professionals, were in a polyamorous marriage for eighteen years and similarly focused on being truthful with Martine, their older daughter from Mark's previous marriage, and Annabelle, the daughter of their union. Mark asserted:
> We're just very straight with the kids and I just don't know any other way to be. Whatever Martine asks I always answer it completely straight. Annabelle, too, but just in a different way. Something that is easier for her to understand, whereas I give Martine the longer version.
Similarly, Alexander, a white machinist/mechanic and father of two, emphasized honesty. He and his wife, Yansa, an African American health care provider, told their adolescent daughter, Chantal (from Alexander's previous marriage), the truth about everything, including sex. Alexander detailed Chantal's reaction to seeing a movie scene with women kissing:
> My daughter goes, "Ooooo, that's disgusting!" And . . . Yansa says, "How can that be disgusting? Every woman you know is like that." And you could see the gears grinding in her head and finally one of them engages and she goes, "But you mean, you are?" And Yansa's like, "Yes." And then Chantal stopped for a little while and another gear engaged and it was like, "You mean my mother?" Yansa goes, "Yes." And then she decided uh, yeah, it's not all that bad.
Such candor about sexuality contributes to a sex-positive environment where children feel comfortable asking questions that might seem taboo in other settings. Some parents reported that they, and their children, became sources of sex education for entire peer groups of adolescents. Kay, a white woman with five children who identified as bisexual/queer/pan-sexual, commented that
> my older kids' friends come to us a lot for, you know, since they know we have this open relationship and we're poly and I'm bisexual. I've had a lot of their friends ask me about their relationships or how to come out, or handle multiple relationships, or how to even manage some of their friendship relationships when everyone isn't getting along. Also about birth control and things like that, things that they feel like they can't talk to their own parents about.
Kay celebrated her ability to offer candid, sex-positive advice because "these kids see me as a relationship expert." True to polyamorous form, Kay used honesty as one of her most valued relationship tools in order to foster emotional intimacy with her own children and their friends.
Even when parents are not immediately honest with their children, if they are honest later it can still contribute to emotional intimacy. Mandy, a white college student and worker in the hospitality industry, said that her parents had been in a polyamorous relationship when she was growing up in a small town in the Midwest. They never discussed it with her, but Mandy reported that she
> knew something was going on. And I was always like OK, Mom, I know you are hiding something from me. Just come out with it. And it took them a while but once I went away to college she told me the truth and now we are a lot closer. Almost like more of a friend relationship, a lot more emotionally intimate than we used to be. And we are a lot closer than she is with my sister, who just couldn't handle it. I mean, she lives in a small town and trains horses, and I moved to this big city to go to college and I work in the city. I am just more open-minded than she is. And my mom knew she could tell me the truth, and I'm glad she did because now we are a lot closer now that there are no more secrets.
Marcus Amore linked honesty and choice to significant personal advantages associated with poly families.
> One of the main advantages is knowing you have choices. Understanding that I have a choice and that I do not have to conform to society, being able to decide for myself. . . . The freedom of choice is in many ways the definition of being human in my opinion. So because I've always been presented with the freedom of choice rather than anything about trying to follow a societal norm—and this was open to me because of [my parents'] honesty—I feel that I have had the freedom and as such, all those choices led to a positive life for me.
Throughout his interview, Marcus elaborated on his reasons for feeling lucky to have grown up in a polyamorous family, such as a relaxed atmosphere without the tension of trying to hide anything, the freedom to think anything without any topics being off limits, and his ability to "make some very good friends, and very true friends who do not abandon me just because I'm different or anything like that." Most importantly, he expressed a deep conviction that freedom, honesty, and choice pervade the true human condition, and he felt that being in a polyamorous family allowed him unique access to this compelling humanity:
> In my opinion, this is what sets humans apart from many other species on the planet. Not our advanced technology, not our "superiority" over them. Frankly, it's our freedom. Unfortunately, I feel that it's also a major lack of that. Not so much in appearance as it is in mentality. That is causing a lot of our problems. The people who have done the most for the world have been very free thinkers. So, I believe that the freedom of choice, the freedom of thought are the best things about being human. Unfortunately, I feel that that's absent in a lot of people who simply try to conform to society. So I feel like that's the advantage. That's what I enjoy about my family.
Similarly, Kethry Wyss thought that she was closer to her parents because of their acceptance of and engagement with her—something she saw in sharp contrast with her peers' parents.
> My parents are aware of my life. We have a good dialog, there is nothing I would keep from them. We are just very open people; there is no need to hide anything. There is nothing I could do that would cause my parents to freak out and ground me. They might be worried about me, but they would not freak and send me to a mental institution like one of my friends' parents did. They did not understand what she was doing so they sent her to an institution in Nevada and did not tell her. Some of my friends, things are bad for them at home, they can't and don't want to talk to their parents. It is kinda sad; they don't think they can trust their parents. In a lot of cases they are right to not trust their parents, that they can't tell their parents things is legitimate, they should not tell their parents stuff. I can tell my parents things, there is really nothing I should hide from my parents.
In this case, Kethry thought that her poly family was far more advantageous to her than a conventional monogamous family would be because it encouraged parents and children to be honest with each other. In that same interview Kethry elaborated on her parents' involvement and investment in her life:
> My friends all want my parents to be their parents because my parents are cool, they go to concerts and stuff, they take me cool places and do cool things with me. I am always taken aback when my friends say they have never even been to a concert, and I have been going since I was a wee one. Lots of steampunk, goth, industrial. My mom (Loretta) took me to _The Rocky Horror Picture Show_ , for goodness sake. I have the rockstar parents! My friends are Facebook friends with my parents. My parents don't control what I say on Facebook, and my mom posted a picture of us going to _Rocky Horror_ —it was fun. . . . [Good parenting] is being willing to listen to your children, to really listen, and to not shun them for being interested in something. I got into anime, and my mama (Kiyowara) has helped me sew costumes and takes me thrift shopping for costume pieces. My other friends' mom just does not understand it and it keeps my friend from being as involved. My mama also helps me dye my hair—pink, red, blue, black, maybe purple next. She also helped my friend dye his hair green cause his mom wouldn't. She was OK with him dying his hair but had never dyed hair before so mom helped him dye his hair green.
Kethry and Marcus both saw their family lives and relationships with their parents as fostering positive, authentic connections. Polyamory, with its emphasis on communication and honesty, helps children in poly families to feel connected to their parents in a way Kethry and Marcus did not see among their peers in monogamous families. In that way, poly families are especially advantageous for children who value emotional intimacy with their parents. Adults and children in poly families as a whole are optimistic about their familial styles and the impact multiple-partner relating has on their lives, prizing especially what they saw as tremendous emotional intimacy with family members.
## Shared Resources
Poly parents routinely mention the ability of multiple partners to meet a variety of familial needs as one of the most important benefits to polyamorous family life. From shared income to increased personal time for adults and more attention for children, having numerous adults in the family allows members to distribute tasks so that (ideally) no one person has to take the brunt of family care.
### Money
Pooling financial resources frequently results in more money for everyone. Larger family units are often able to keep a parent at home because they have multiple adults doing waged work. The Wyss quad, for example, was able to afford a stay-at-home parent for their daughter Kethry's entire childhood, even in the notoriously expensive California Bay Area. As a computer programmer with a stable income, Albert was the family's primary economic support. Cycling through self-employment, professional managerial positions, and college attendance, each of the other three adults took primary parenting responsibility at different times, and they shared parenting and transit duties once Kethry was in high school. The assurance of a predictable income granted the quad the flexibility of rotating the position of full-time parenthood, enabling other adults to be selective when looking for work, in establishing businesses, and in pursuing higher education.
The Wysses, however, also experienced the negative side of shared income when two of their three workers lost jobs in an economic downturn, leaving Albert the sole wage earner. Albert reported that "it felt like a lot of pressure . . . everyone was counting on me and it made me really nervous. What if I lost my job too?" Other single-wage-earner families face similar fears, but fewer have the flexibility of multiple reserve wage earners to get jobs and simultaneously retain a full-time parent. While these larger groupings require a lot of food, large houses, and multiple cars, their pooled resources grant greater flexibility and save money on expenses such as child care and separate dwellings.
### Social Stimulation (and Loot)
Children in these families also see the family's pooled resources as an advantage. Zane Lupine reminisced about the glee of having such a large extended family when it was time to receive gifts as a child. "When I was young I guess more presents at Christmas. More people. Just more people in general. I liked it as a little kid, cause I liked having people around. And great loot for birthdays and Christmas, with three parents and so many grandparents."
Adam, Jonathan, and Zoe Hadaway agreed that the increased resources were great for the children in the family, and to the parents as well.
> Jonathan: They're probably not as bored as they used to be. They have a lot more people to actually talk with.
>
>
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> Elisabeth: Who used to be bored?
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> Adam: Our parents. One of them would be at work and the other would be at home with four-plus kids. It just became aggravating sometimes. It's a little better for all of them to just be there for each other, I think . . . and we are all more open-minded, too.
>
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> Zoe: Yeah, technically we are an alternative family. Having that on our side, we're not against the alternative lifestyle. Which makes, in my opinion, if you're open to other kinds of life, that's going to open so many doors to you. That's going to make so many more opportunities for friendships. Through friendships you get contacts and through contacts you get financial opportunities that you may not be able to pass up. That's my mind-set.
Overall, poly family members said sharing resources was the single most important advantage to their family style. In addition to the more general financial and personal resources, they identified other advantages such as accommodating family members with disabilities and allowing parents to have more personal time for themselves.
### Accommodating Disabilities
Several respondents mentioned how useful it was to have multiple partners when dealing with disabilities, either their own or their children's. Heidi Ballard reported that she occasionally had "anxiety severe enough that it made going places uncomfortable, and sometimes even staying home alone can be uncomfortable." While she did not require "someone constantly holding my hand, I don't need a babysitter," Heidi did experience significant angst dealing with her occasionally debilitating anxiety. She explained how her polyamorous relationship had improved her life and made it easier to manage her periods of extreme anxiety:
> And then when Jason moved in it came as such a relief. 'Cause he works at home a lot of the time, and just having someone there during the day when I was home with the kids, I could hear him in the other room or just chat for a few minutes, it brought my anxiety level down quite a bit. And if I'm not up for going out but don't want to be alone at home, we can still do things or get groceries or whatever because there's someone to be home with me and someone to go do whatever needs to be done. It just works out better that way. George has more freedom to come and go or stay late at work or do something on the way home because Jason is here, I'm not alone with the kids in the house.
With the accommodations her polyamorous family was able to provide, Heidi's anxiety felt much more manageable for the entire family. George reported that he "just feel[s] better knowing Jason is here with Heidi and the kids, that she's more comfortable, I don't feel so pressed to rush home." Jason was similarly enthusiastic about the arrangement, saying, "I like spending time with Heidi, so it works great for me too."
Dillon, son of the Kenmore quad, had what his mother, Natalie, termed a "cognitive processing disorder" that resulted in speech delays and interaction issues like avoidance of eye contact. His disability wasn't clearly diagnosed yet because Dillon was only six years old at the time of the interview and had not yet had some of the learning or cognitive tests the Kenmores planned to schedule for the future. A quiet and gentle child, Dillon played with action figures while I sat on the floor next to him and asked him questions.
> Elisabeth: So what do you think of living with Iris and William too [in addition to parents Dax and Natalie]?
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> Dillon: Good.
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> Elisabeth: What's good about it?
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> Dillon: There's always someone here [pause] just in case.
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> Elisabeth: In case of what?
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> Dillon: If I run in the street or, or, or can't find my way home, or [pause] they will always come get me.
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> Elisabeth: Has that happened, do you ever run away accidentally?
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> Dillon: No.
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> Elisabeth: But if it did, they would come find you?
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> Dillon: Yeah, they will always come.
When he said this, Dillon radiated calm and smiled up at his father, Dax, sitting in a chair to his left. While he had some difficulty formulating the words to describe how his behavior might be unpredictable even to himself, it was abundantly clear to me that the four adults' combined attention provided Dillon with a safety net that made his amorphous disability less frightening and more manageable for him and for the adults in his life.
### Personal Time
It is clear that, in general, polyamorists perceive themselves to be happier when they are getting more of their needs met,[1] and they are able to get a wider range of needs met through multiple partners. This same dynamic appears to extend to nonsexual familial relationships as well. When the Wyss quad had Kethry, their ability to distribute parenting meant that Patrick Wyss could parent full time and "retain my sanity." After spending all day with a rambunctious toddler who "did better when she stayed home, [because she had] major fits in public for a little while," Patrick felt harried and claustrophobic. Patrick reported that when Kiyowara or Albert arrived home one or both would "take over with Kethry and I would split, go ride my bike in the foothills for an hour or two . . . It saved me, I never could have done it without it." The ability to leave Kethry with others allowed Patrick to meet his need for time away from a demanding toddler. For the Wyss quad, this made a very challenging period in the parenting cycle much easier than it would have been with only two (or fewer) parents.
The Tree triad, composed of Bjorn, Gene, and Leah—all professionals or academicians in their mid-thirties—similarly found multiple parents to be invaluable when caring for their infant son, Will. Leah said that everyone got more sleep because there were more people to take night shifts:
> As far as having Will goes, it has made a huge difference to have multiple parents, and multiple grandparents. Gene and Bjorn took turns staying at the hospital or going home for a full night's sleep just after Will was born, and being able to be well rested has made a world of difference not only then but Will's whole life so far. The difference has not been quite as big for me because I was there at the hospital full time after I had Will, but overall spreading out the parenting has been great. Despite only getting four or five hours of sleep for five days or so right when we had Will, we were all really calm and excited about meeting this new little person. We were lucky enough to have a hospital room all to ourselves so we ended up with an extra bed in the room so daddy or papa could get a real sleep while they were at the hospital with us [Leah and Will], which helped a lot.
Bjorn agreed with Leah, putting the poly parenting experience in context with some of his other friends who had children around the same time:
> It has been amusing having monogamous friends who first asked questions about polyamory, like "isn't that complicated or a lot of work?" Then as we all got to child-rearing age our friends have changed their tune or seemed a little jealous and talk about how wonderful it would be to have more parents—single parents want more help and the couples want another parent as well. It's a full-time job for the three of us; we can't even imagine doing it with fewer adults. Our friends tease us but wish they had more help too.
In addition to more time to themselves and spreading parenting around, Gene said that having a baby in a polyamorous situation seemed to help him get dates with other women.
> Before we got pregnant I was only in one other relationship that ended and I chose not to look for another knowing that I wanted to devote all my energy and flexibility to that [having a baby]. Once I determined what life with a kid was like and got things more under control (or as much control as things ever are once you have a kid), I went back to dating again. It's going great, the people that I have been dating have been really interested in hearing about and meeting Will. No complications. If anything, it is something that people find interesting—the kind of people I would be interested in dating see it as a bonus, intrinsically appealing and interesting—"How do you do that?" So it has led a couple of people to contact me. Women are reaching out to me on my online dating profile and they have reached out to me seeing Will as something interesting.
Not only was the new father still fully engaged in life in a way he saw his monogamous counterparts trying and mostly failing to achieve but also being in a polyamorous family actually made Gene more appealing to women who then wanted to date him.
### Attention for Children
Another important advantage poly people mention is the considerable attention available to children when families with multiple adults pool their resources. Many parents say that their children's lives, experiences, and self-concepts are richer for the multiple loving adults in their families. Joya Starr said polyamory was beneficial for her son Gideon because
> there's more attention for the kids . . . It takes five adults to raise a kid and one of those adults is just around to take care of mom. And let me tell you, a happy mom is a good mom. If mom gets enough sleep then everyone is in much better shape.
Having multiple adults in the household benefitted both children and adults, Joya observed, because happy and well-rested parents provided better care for children. Not only did children get more attention from a wider variety of adults, but adults who were able to support each other (ideally) parented more effectively.
Some respondents connected this increased attention with a feeling of community. Emmanuella Ruiz identified the connection to chosen family her poly family provided as crucial to her children's well-being:
> It gives my children a sense of community. They've not had reliable grandparents. They don't have cousins or the typical biological extended family. But they have a big, happy, productive, healthy family nonetheless, and it is a chosen family. They know each person's relationship to them the same way they would know if they were first or second cousins, aunts or uncles . . . The sense of extended community is the most important thing in respect to my children.
Emmanuella saw her children as gaining both a community in lieu of their unreliable grandparents and a sense of how to construct chosen relationships that contributed to a healthy sense of intimacy.
In addition to a sense of extended community contributing to children's well-being, some polys feel that their assistance to poly parents provided what one woman at a party jokingly referred to as "fresh horses," referring to the Pony Express message service operating from 1860 to 1861 in the Western United States that provided riders with a new mount every ten miles to get the messages to their destinations more rapidly.[2] Kristine saw herself as relief for Mark and Evelyn Coach in the demanding task of parenting their (then) elementary-aged daughter, Annabelle:
> Of course, I had it easier, in a way. It was easier to make her a priority when I was there, not being a full-time parent, because she really is a handful. The time I had with her could be just with her, focusing on being with her and not trying to do other things at the same time.
This allowed for Evelyn and Mark to do paid work, complete household tasks like making dinner or cleaning, and have a few rare moments of personal time. It also provided Annabelle with the undivided attention of an adoring adult who was not harried by thoughts of needing to do something else. Sharing child care tasks and parental resources proved beneficial in a variety of ways to both children and adults in poly families.
### Role Models for Children
One of the major advantages poly parents mention is the plentiful positive role modeling available to children in poly families. These role models include ethical considerations like honesty, a willingness to meet other's needs, and careful communication and negotiation. Perhaps most importantly, parents emphasize the relationships between their children, partners, and friends as sources of personal role modeling through life examples and advice. Melody Lupine lived for seven years in a triad with two men: Cristof, her husband of eighteen years, and Quentin, the couple's long-time friend-turned-lover whom Melody considered her husband. Melody noted that Quentin functioned as a positive example for her son, Pete (the biological child of Melody and Cristof):
> Quentin is another male role model in Pete's life. He has his dad and that's his dad, but here is another man in his life or other men in his life and this is what they do and their acceptance of him. And so which I think is very beneficial for a young man to have those different role models and know that, Pete knows that he could go to them at any time for anything if he needed something, he knew that they were available.
The availability of multiple adults not only provided a broad range of role models but it also gave children in poly families access to nonparental trusted adults with whom to discuss things the children might not wish to tell their parents.
Zoe Hadaway similarly saw multiple adults as a positive source of attention and role models:
> I think having four parents is the best thing about being in a poly family. You know, some people at school are like "I hate that my parents are divorced and I hate that I have all these extra parents now!" and I'm like, well, minus the divorce part, I can't say that having more parents is all that bad. I always feel like if Gwenyth, my biological mother, is busy then I can go to Tammy or I can go to Phil. Or to my dad. I don't just have two options now; I have four. It sort of makes trying to find advice a little bit easier. You know that each parent has their own specific forte of advice. Like Phil's mechanical. Dad's business. Mom's in the house, as is Tammy. Tammy's more artsy and technology. Mom's very clean oriented, organization. It's nice, you get lots of different things from the different parents.
For Zoe, growing up in a polyamorous family provided a wealth of role models and advice from the expanded number of parents.
Similarly, Cole Cypress explained how more numerous authority figures provided him with a greater diversity of parental options, avenues for support, and a profusion of role models. Cole said that he was a "wild thing" when he was in sixth and seventh grade, and while his parents would "scream bloody murder at me," he appreciated how his parent's girlfriend Bettina "had a different way of going about her business with me. She would react very calmly and know exactly what to do and the exact right punishment. And it would still be really hard for me, but it helped me learn my lesson better. And it felt more fair." Cole elaborated on an incident in which he got in trouble at school and Bettina created a creative punishment for him:
> There was one incident where I took, well, part of it was an accident and part of it kind of wasn't. I accidentally forgot two pocket knives in my backpack. But the fact that didn't make it an accident was that I started kinda showing them off, but then I got caught with them, and I got punished for it. At school they took the knives away and gave me a very strict warning, which was pretty traditional at the private school. And my parents heard about it and they of course were very upset. But Bettina of course had a different way of punishing me . . . she made me build a dog house using only the pocket knives. And after about three days of that she made me stop because I was injuring my hands because the knives kept slipping. But I never used a pocket knife after that. And of course my parents wanted to ground me for a month and take away all of my privileges and stuff like that. But Bettina would always step in and calm them down and say this stuff really politely and quietly.
Cole felt that he had benefitted from Bettina's alternate way of handling both family conflict and discipline, as well as gaining greater social interaction and diversity of role models.
## Parents' Ability to Remain Friendly
Some children whose families have experienced divorce but stayed in social contact said they valued their parents' ability to remain friends, even after a divorce. Speaking of her parents' divorce, Kethry Wyss expressed mixed feelings but ultimately felt that the divorce had not changed things much for her at all:
> It was a bit of a change now that there are two houses instead of one, but not really that big of a deal. The first thing I said when they told me they were getting divorced was, "now the shouting will stop." It got better immediately after they decided to divorce. By the time everything was through the court system they were back to being friends, Mama (Kiyowara) and Poppa (Patrick) were friends again. . . . In terms of being a kid of a divorce, I was dealt a really good deal in terms of being with Mama and Poppa—even when they were fighting they weren't out of control. They would take a deep breath and even walk away for a bit if they needed to cool down. They were really rational about their fights, as rational as you can be in that situation. Dealing with the courts and everything they became friends again and they can still hang out in the same room. Watching as Lucia (Patrick's new wife) has struggled with her ex-husband, about their son Evan and that whole debacle, as well as some of Nina and Paton's issues with their kids and their ex-spouses—I was dealt a very very good hand with parents who were able to become friends again afterwards. They are still friends. The other divorces, just, some, like, these kids I know with their stepmother and their dad, their stepmother is not a very nice person. . . . it was hard to watch from the sidelines to see how the daughters and the mom were taking it.
While children in other families who divorce may experience cooperative parenting after a divorce, Kethry saw her peers and their parents struggling far more than her parents did. She reported that she was able to see Patrick regularly, even though he had moved out of the quad's house:
> Usually, like last year [I saw my dad] four times a week or so, he would pick me up from school on Tuesdays and Fridays and then I would ride the train home on Thursday and he would pick me up and take me to the game, bring me back afterwards. We would play Dungeons and Dragons role-play for like three hours, 7 to 10. We still play, we both have characters and we have been gaming with a group of people for the last three years. We have also gone to a gaming convention together, and I game with my friends as well . . . On Saturdays Daddy and Poppa and I go to Costco and come back and watch _Doctor Who_ or _Torchwood_. I see Poppa several days every week. I hang out with him and watch TV at his house . . . Special things we do regularly together. He and Mama are friends with Mommy and Daddy too, so it is not like it is awkward visitation or anything.
The fact that Kethry was routinely able to see Patrick several days a week was clearly advantageous to her—she spoke with relish of the time she spent with him—and to Patrick himself, because he made a significant amount of time to spend with his daughter every week. This ability to retain positive relationships is advantageous for parents who wish to stay connected with their children after divorce, as well.
## Building Chosen Kinship Networks
Fundamental to poly families, the option to build relationships outside of conventional frameworks is a hallmark of polyamory. While the sexual relationships polys establish with each other get the most attention from the media—in part because they distinguish polyamory from monogamy and friendship, and in part because they are the most sensational aspect of the family—they are not the only or even most important aspect of poly relationships. Respondents note that the emotional or affective elements of their relationships are what make poly families really work or not. Much like heterosexual families, poly families spend far more time hanging out together, doing homework, making dinner, carpooling, folding laundry, and having family meetings or relationship talks than they do having sex. Sexuality is not the heart of the family: without positive emotional relationships, a sexual relationship alone is often insufficient to sustain a complex, long-term relationship. Polyaffectivity, or the nonsexual emotional ties that bind people in poly families together, is far more important to the overall family connection than is any sexual connection between or among adults.
### Children's Active Construction of Kinship Networks
Some children told me they actively constructed their own chosen family networks, either by establishing friendships with people they met through their parents or going beyond their parent's social networks and establishing connections with people they felt like they could trust. In some cases children from different families would meet each other at various poly events (the summer campout was a popular meeting spot for kids) and form lasting bonds, and in other cases their connections would be with others outside of the poly community. Zina Campo said that she found her mother Lexi's polyamorous relationships to be advantageous on a number of levels, one of the most important is discussed in the following:
> Zina: It has actually brought me one of my closer friends, too, so, because of the social networking my mother does and has relationships with other people, one of the people who she's friends with is actually my friend's mom, so, it's really cool. Most of the people she's in relationships with are just kind of too cool, I like being around them. One of them even introduced me to someone I think is totally awesome, even though she's in her thirties now, we have become really good friends. . . . I met her because she was a partner of one of my mom's partners and she and my mom were possibly going to be partners at some point or something, but, I don't know. But she's really awesome, it's really cool. When my mom says that she's visiting I get really excited and jump up and down like yay, I get to see her!
>
>
>
>
> Elisabeth: So what do you two do together, how do you connect?
>
>
>
>
> Zina: Go shopping, go get coffee or something, or walk our dogs together. Pretty much anything that we can. Friend stuff. . . . Mom has brought another one of her partners home besides Blake, and one of his partners, and the awesome friend who is one of her partners and one of my friends now, and she brought home one of her friends who is just a friend, and probably more that I can't remember right now. She brings home people occasionally, kind of spread out, so it is kind of a special occasion. I like it when she brings people home with her, living out in the middle of nowhere it's kind of cool when anyone comes over at all, so um it's also really cool that I get to meet people that she is interested in or friends with or maybe partners or whatever, and I get to see them through, I tell her if I approve or not. It may not change what she does about it, it's usually in a positive way 'cause I usually approve of them.
Zina Campo not only established enduring friendships with people she met in her polyamorous community but she also took an active role in creating her own family by screening her mother's partners and telling her mother "if I approve or not." In so doing, she experienced significant personal and emotional advantages to being in a poly family, and she came to conceive of and possibly even exert a degree of control over the construction of her family that children in many conventional families would not even consider. These findings confirm Riggs's conclusions that children in sexual minority families, and especially polyamorous or same-sex families, often act as agents who intentionally cocreate their families in unconventional ways.[3]
### Family Expansion
For these families, an important part of creating new forms of family is investing themselves in relationships outside of the biological or legal connections usually used to determine family status—what scholars have called _chosen kin_ or _chosen family_.[4] An only child, Cole Cypress felt that being in a poly family had provided him with a wider range of family relationships with his parents' partner Bettina's children, who took on sibling roles with Cole in his family. Cole said that
> I learned a lot from the kids, too. Because I've always been an only child, I've always wanted a brother. It was always an older brother. Or a younger sister. But I ended up getting two older sisters and an older brother. And I actually, at some points I was really close with the brother, Caz, Bettina's son . . . whenever my teachers asked me why I would swear so much I would always always blame Caz. I would always blame him because he would swear tons. He went to a public school. He was raised in a society where, Bettina raised him to swear smart but still he sweared a lot. And he would always act out and be a smart alec. I always acted a lot like him, I always looked to him as an older brother. At some points he was my idol, you know. The only difference is that, by seventh grade, he was the cool kid that could get away with anything. He was still doing well in school and I was the uncool kid that was swearing and getting in trouble and not going anywhere . . . He was always a huge football player, that's a big reason of why he got out of stuff was because he was athletic and I wasn't. He was a huge linebacker, defensive end or whatever he's doing right now. And I always looked up to that and he always loved playing football. He was always really athletic; he was always at the gym. He was the first to take me to the gym, and showed me the ropes. And now he's, I think he got a full scholarship down at [Prestigious University], uh, on a football scholarship, even though he's pretty smart.
From a big brother useful as a role model and a pal to "show him the ropes" to a handy scapegoat for taking the blame for misbehavior, Cole forged a brotherly relationship with Bettina's son, Caz. Cole saw this unusual opportunity for an only child to establish siblingesque relationships as an advantage associated with his parent's polyamorous relationships.
### Otherfathering
While the father has been considered the most important or "real" parent in some periods of history,[5] contemporary society in the United States is firmly in the mother's camp when it comes to assigning primary parenting roles. The fact that child care remains a "working mother's issue" speaks volumes about the amount of responsibility society expects fathers—working or not—to take for their children's daily maintenance. Even in multiple-partner families, it is generally the multiple wives who help care for each other's children, and the children caring for each other, as opposed to numerous adult men present in the family to care for the children. Scholars use the term _othermothering_[6] to describe the care work women do for children with whom they share no biological or legal connection. Although Collins used the term in reference to women of African descent or with an Afrocentric worldview caring for each other's children directly by providing meals and clothing or indirectly through offering advice or support,[7] othermothering has some flexibility to be used for men who take on the same role, but it is not generally applied to men.
I argue that polyamory provides men with a unique opportunity to form lasting relationships with children who are not their biological progeny—to become _otherfathers_. Very few of the polyamorous otherfathers are of African descent, so the term does not have racial parity with Collins's use of othermothers,[8] but the emotional and caretaking intent is similar in that each cares for children who are not their own. For instance, Warren Bien considers himself a father of three girls, though only two of them are biologically and legally related to him. He and his ex-wife, Julie, had two daughters—Rebecca and Callie—during their polyamorous marriage, and Julie had another daughter (Macy) with her then-boyfriend, now-husband Andrew during that same time. While Macy is not Warren's biological child, she is his daughters' sister through their mother and, as Warren put it, "a child of my heart. I love her like she is my own, and she calls me her papa bear." Warren said that "my partner Estella and her legal husband Devon are thinking about having kids, and they have asked me what role I would expect to play if they do. I would see myself as a coparent, at least as active as I am in Macy's life. I wouldn't have any legal or genetic tie to any of their kids, but I would still want to be their papa bear."
Similarly, James Majek reported that he had "come as close as I will ever get" to fatherhood through his association with his then-girlfriend Morgan's children, Heather and Brady:
> I am hard-pressed to come up with a negative. I love those kids and they love me. I will never forget being out camping with them at the poly campout, and Morgan and Clark decided to go for a little walk and it was just me and Heather. I said, "Yeah, I'll watch her." So they go off and take their walk and I am making Heather a sandwich. They hadn't been gone for maybe five minutes when she just sits there and she looks at me and says, "James," and I said, "Yes, Heather?" and she said, "I love you." It was that moment that just knocked me out. And I said, "Well, I love you very much, too." We have a bond. I mean, she has said to me "you are like my other dad" or "you are like my uncle." Every time I have been honored. . . . Brady is special because he is just four now, and Morgan and I got together when she was pregnant which is a story in and of itself. That is the closest, I am sure, I will ever come to being a dad. I was there through the entire pregnancy, before she even showed and then right through the birth. I was there rocking him to sleep, feeding him at three o'clock in the morning, changing diapers, making sure Morgan could sleep when Clark was down hanging out with my wife, I would be up here. I was here as much as I could be. And it was precious, and it was beautiful, and fantastic and that kid will have a place in my heart that I can't describe.
The fact that James (with Clark's help) made the effort to continue his relationship with Heather and Brady two years after breaking up with their mother attested to his enduring emotional connection with the children. While Brady was too young for me to interview at the time, I was able to ask his older sister, Heather, how she thought he felt about James. Heather responded:
> Yeah, Brady really loved James, he was around all the time until like two years ago. I love James, too, I miss him now that he is not here nearly as much. But we get to see him sometimes. Not enough, but at least sometimes. Daddy takes us to see James, every other weekend or so we drive down to [a town about forty-five minutes away] to meet him for lunch and we play games and stuff. It's nice to see him, but it's not the same as when they were all together and we got to see him all the time, all all the time.
Clark, Morgan's husband and father to Heather and Brady, commented that he would routinely take the children to see James:
> Oh yeah, we get to see him all the time. Either we drive down to [a town about forty-five minutes away] or he comes up here. Actually, usually we go down there, probably every other week or so. I actually get along with James better than Morgan does right now, so it makes sense for me to take Heather and Brady down to see him. I know the kids miss him a lot so I definitely put effort in to getting them together. I still like him, too, so it is nice for me to see him, though I don't think I would do it nearly as much if it weren't for the kids.
The fact that Clark maintained a more congenial relationship than Morgan did with James was characteristic of poly men who would help their children remain in contact with their wife or partner's former boyfriends. This happened most commonly among men who had not established a sexual relationship with each other, something I think allowed them to move more easily beyond the romantic stage of the relationship without the hurt feelings that their wives or girlfriends might harbor in relationship to their ex-lovers. It was also more common among men who respected each other and treated each other ethically—men who felt their female partner's ex-boyfriends had lied to them or mistreated their partners were much less likely to attempt to keep an ongoing relationship alive.
Even fathers connected by biology, legal ties, or long attentive association helped each other maintain contact with children after splitting up with the children's mothers. Patrick Wyss was part of the Wyss quad for fourteen years (and two years previously as part of a moresome) before moving out to live with Lucia and her son, Evan. Albert Wyss reported that, due to some logistical constraints and some personal taste:
> Neither Loretta nor Kiyowara arrange to see him [Patrick] regularly except for Kethry's school things, but he and I have lunch every Saturday with Kethry at Costco and then come home to watch British comedy TV. When I see Patrick it's because we are doing something with Kethry. I don't do things with him without Kethry. I have nothing massively against Patrick. I was not as emotionally charged about the whole thing when we were in the breakup. It was sad we could not get that to work out . . . Some of me hanging out with him is shared interest—we both like _Doctor Who_ and BBC-type things that I have liked forever and am indoctrinating my child into, and Patrick is fond of as well. We like to watch the people shopping [at Costco] and comment on their fashion faux pas, or Patrick will talk about what is going on with his art installations and things from his art classes . . . it is enjoyable to go out and chat with Kethry and Patrick about what's going on. I see a little more about what Kethry is up to as well, because when she is chatting to Patrick she talks more about what happens in the role-playing game they both play. When she goes off to play I ask how it went and I get five- or ten-sentence summaries without a lot of detail, but when she talks to Patrick it is more involved because they both play.
At other points it was clear that Albert treasured his connection with Kethry and went out of his way to "meet her where she is," tailoring his schedule on vacation to fit Kethry's and wanting to know what she thought about and how she felt. Patrick similarly went to great lengths to see Kethry regularly, picking her up at the train station on her way home from school several days a week and playing role-playing games with her regularly. Patrick's connection with Kethry enhanced Albert's connection with Kethry because he got a glimpse of an expanded understanding of his daughter that he would otherwise not have seen. In other words, poly men can help each other deepen their relationships with their children and sustain contact in a way that appears to be quite difficult for serial-monogamous relationships that break up and in which are still overwhelmingly women who are single parents.[9]
It seemed in retrospect to the remaining triad that Patrick had slowly withdrawn from the quad, symbolized by his "stuff" being contained within his own bedroom—separate from the other three who shared a bed—and the outdoor workshop, but his stuff was notably absent from the other rooms, as evidenced by how little they changed when he moved out. Kiyowara had married Albert and Loretta independently, and the three had gotten rings together to symbolize their family connection, but Patrick did not marry Loretta or Albert, even though Patrick and Loretta had been sexually involved: The only nonpair was Patrick and Albert, both heterosexual men. Albert concluded: "We are still friends who see each other occasionally but no sexual interaction. Maybe it is easier to break up and still be friends if you weren't sexual, um. Hard to say, not enough data points."
### Cohusbands
Far less familiar than co-wives or sister-wives, co-husbands (or even more awkwardly, brother-husbands) forge a new category for men rarely seen in any society. Men in poly families who share a relationship with the same woman defy the strict demands of mainstream masculinity that require "real" men to have exclusive sexual access to "their" women.[10] Far from being rivals, some men in poly families have deeply supportive, emotionally intimate relationships with each other. Characteristic of men who do "dude things" together, the Majek men collaborated on many major home repairs. Once when I arrived to interview they family, Clark, Nash, and James were all up on a scaffold out front, painting the house. Another time I found Nash and Clark laying paving stones to create a backyard walkway. Cooperating on work provided a familiar masculine territory for the men to establish and strengthen relationships, as well as making the house look great. Clark told me, "I'm thrilled to have the help, I learn a lot from these guys and it's too much work for one person. It's fun to hang out with them too, have a beer afterwards and check out the awesome stuff we did."
Summer and Zack Phoenix shared a house with Summer's lover Jared. Usually Zack and Summer shared the master bedroom and Jared slept in his own attached in-law apartment, but when Summer would travel out of town for work, sometimes the men would sleep together. Zack told me, "Sometimes when we are both really missing her we cuddle up in the big bed and talk about her or what happened during our day or whatever, and fall asleep. It's nice and comforting to have him there. We eat together and hang out, we're family." When Summer is gone, Jared and Zack keep each other company and avoid getting lonely.
Bjorn and Gene Tree have a similarly close relationship, enough so that when they are out in public together with their infant son Will they are routinely mistaken for a gay couple. Bjorn said:
> It's really funny to watch how people react to us in different combinations. When I'm with Leah and Will people don't really give us a second look except at the baby, but when I'm with Gene and Will I definitely get the feeling people think we are a couple by the way they react to us. We get more looks and more comments—usually smiles, sometimes "oh how cute," gay dads nodding to us at the playground. This is the Bay Area so gay couples are pretty common, so I guess in that way we kind of blend in, except that we're both straight. But we don't necessarily tell other people that, we just let them assume we are a couple with our son. Because we kind of are—definitely with our son at least, even if we're not a couple.
Sharing parenting, emotional intimacy, time, and their mutual love for Leah gave Gene and Bjorn an uncommon relationship that benefitted both of their lives. Outside of sports and activity-directed buddy relationships, mainstream men in the United States do not have a very wide range of emotional or relationship options,[11] and poly relationships provide men with new avenues to establish emotionally intimate, mutually supportive relationships with each other.
## Polyaffectivity
All of these relationships we have discussed in this section of the chapter characterize polyaffectivity. Polyaffectivity differs from "regular" friendship in that the people involved see it as far more important than mainstream society usually views relationships that are not biologically, legally, or even sexually connected. People who love each other on that level hesitate to call themselves "just friends" because their friendships are among the most important relationships in their lives: the fact that they do not have sex does not mean they are "just" friends. Polyaffective relationships can develop the devotion and degree of seriousness that most people associate only with marriage (or at minimum an ongoing sexual relationship). Not all nonsexual relationships in poly situations are polyaffective: the participants must consider each other to be significant relationships to qualify as polyaffective. That is, people associated through poly relationships who are acquaintances or casual friends do not possess the emotional intimacy or expectation of mutual support that is present in polyaffective relationships. Polyaffectivity has a number of significant implications, as I'll discuss in greater detail later. It allows for a much wider variety of relationships and a far broader base of support than does a more conventional relationship that relies more heavily on the sometimes tenuous bonds of romantic love.
1.
(Sheff 2005)
2.
http://officialponyexpress.org/pony-express-quick-facts.html, accessed April 29, 2013.
3.
(Riggs 2010)
4.
See the foundational _Families We Choose: Lesbians, Gays, Kinship_ by Kath Weston (1991), as well as _No Place Like Home: Relationships and Family Life among Lesbians and Gay Men_ by Christopher Carrington (1999) or _Same Sex Intimacies: Families of Choice and Other Life Experiments_ by Jeffrey Weeks, Brian Heaphy, and Catherine Donovan (2001).
5.
(LaRossa 1997)
6.
(Collins 2000; James 1993: 47)
7.
(Collins 2000); See also Wanda Thomas Bernard and Candace Bernard, "Passing the Torch. A Mother and Daughter Reflect on Experiences Across Generations," _Canadian Women's Studies les cahiers de la femme_ 18, nos. 2, 3 (Summer/Fall): 46–50.
8.
(Collins 2000)
9.
(Mather 2010)
10.
(Connell 2005). See also Sheff 2006 for more information on polyhegemonic masculinity.
11.
Ibid.
Chapter 8
# Difficulties in Polyamorous Families
While there are many advantages to a poly household, children and parents also describe a variety of disadvantages, including dealing with social stigma, children's emotional pain with the loss of a treasured adult after a breakup, household crowding, family complexities, and too much supervision. Importantly, the difficulties poly families face are the same difficulties facing other complex families in the United States today. Household crowding, partners breaking up, family drama, and even the most heinous family problem of child molestation are things that happen in nonpoly families as well. None of the problems the poly folks discussed were isolated to only people in polyamorous families.
## Stigma
In the discipline of sociology, the term _stigma_ refers to "an attribute that is deeply discrediting,"[1] a personal characteristic that society has deemed undesirable and thus marks the stigmatized person as tainted or spoiled. Stigma always exists in social context and can change dramatically from one setting, historical era, or subculture to another. Fifty years ago, decorative tattoos on women were considered taboo—a scandalous rarity worthy of harsh whispers and social exclusion or pity. Now tattoos are so popular in mainstream U.S. culture that they have become commonplace, unremarkable on women in many age groups. While stigma against people with tattoos has waned to a large extent, other stigmas such as those against people of color or sexual minorities prove more durable. If increased public acceptance of same-sex marriage and neutral or positive media portrayals of same-sex relationships are any indication, mainstream public opinion in the United States appears to be shifting toward greater acceptance of people perceived as gay, lesbian, or (to a somewhat lesser extent) transgendered. Even so, homophobia, sexual prejudice, and sex negativity—all reciprocal symptoms and causes of stigma against sexual minorities—remain important social forces. Coming out or being exposed as a sexual minority can still result in alienation from family and friends,[2] physical attack or harassment,[3] loss of a job or custody of a child,[4] public degradation, and incarceration.[5]
One of the primary disadvantages facing poly families is the stigma associated with being sexual minorities. I have found that their social privileges and a comparatively low level of public awareness that allows/forces poly people to remain invisible provides mainstream polyamorists some protection from the effects of stigma. Nonetheless, poly families experience and fear a variety of stigma-related issues including social rejection, fear that their children will be negatively affected, their children's experiences of stigma, institutional vulnerability resulting from stigma, and the leverage that vulnerability gives disgruntled teens.
### Social Rejection
While Melody Lupine's triad with Cristof and Quentin had never been fully embraced by portions of their social circle, even those who had accepted the triad became increasingly intolerant when Melody intentionally became pregnant with Quentin's child while still married to Cristof. Melody remembered that friends expressed discomfort and
> judgments, how could you do that, it's immoral and you know, how could you do that to Cristof. And that baby's gonna grow up being so confused. They thought it was worse than cheating, that you have a baby with someone else while you're married to somebody was just beyond, just unfathomable to people. And even some polyamorous people were pretty judgmental about it. . . .
Breaking such an important norm as bearing solely the husband's children while married was more than some of the Lupines' associates would tolerate, and they rejected Melody and her family. While the triad and their children paid for their nonconformity, there were some advantages as well. It gave Melody the opportunity to have the third child she had wanted (which Cristof did not wish to father), and Quentin a "second chance" at parenting now that his older children were grown.
### Rejection from Family Members
Poly people can lose not only friends but relationships with family members as well. Baldwin Omni, a middle-class white man in his early sixties, experienced the social backlash of stigma when his adult children Rosaline and Wade, wife Nadia, and Nadia's extended family rejected him for becoming polyamorous. At our initial interview, Baldwin identified himself as being in a poly/mono relationship, albeit "not the typical one." He continued:
> From 1974 to 1980 my wife and I had an open marriage in the sense that we sometimes played with others, always with each other's knowledge, sometimes in their presence, occasionally with their participation. We didn't really "date"; these were generally people we or one of us already knew and were curious about. It wasn't equivalent to poly, cause amory wasn't part of the equation. Any intimate encounters did not affect the underlying friendships and make them more of a relationship. We also attended a few swinging parties and decided that was too casual for us.
Then in 1980, when pregnant with their first child, Baldwin said that Nadia: "became much more conservative, said 'I don't want to do this any more, I feel it's wrong' and so I became totally mono for twenty-six years." Baldwin, deeply dismayed about the turn of events, reported that "had she not been pregnant I might have left her." Nadia had a son they named Wade, and three years later Baldwin and Nadia had a daughter named Rosaline, and they appeared for all intents and purposes as a conventional family.
Baldwin reported that "the kids and I got along great when they were little," but as they aged Nadia and Baldwin disagreed on how to discipline the children and they eventually "solidified into dad as the enforcer and mom as the nurturer." Further alienating Baldwin from the children, he had what he described as the "job from hell" and "came home frustrated and angry every day. The kids would hear the front door and go hide in their rooms, deal with me as little as possible. There was also a lot of frustration and resentment over Nadia's increasing involvement in her church." This left what Baldwin characterized as "a lot of baggage" in his relationship with his children, "predisposing them to think poorly of me in relationship to the poly," although he admitted, "I definitely left something to be desired as a father."
In 2006, Nadia initiated what Baldwin termed a "remarkable conversation" in which she
> thanked me for my years of loving, faithful partnership and said that she felt somewhat guilty for "inhibiting who you have always been" . . . she said, "I changed but you didn't, I never expected you to follow my new standards of morality, but I was glad you did." Still, it bothered her that it inhibited me. She also recognized that anything else that happened back in the old days didn't affect my commitment to her then, and offered that if I wanted to have some discreet and safe adventures that would be OK, but she would prefer not to know about them at all.
Baldwin "took her up on the offer" and began a series of forays into the dating world, making sure to be discreet and careful of sexually transmitted infections. Dating new people brought new experiences, and Baldwin experienced significant personal growth and some sexual adventures.
Eventually Baldwin established a relationship with Abigail and began seeing her once a week. Nadia oscillated between trying to accept his relationships and feeling terrible about them. At one point Nadia encouraged Baldwin to take one of his dates to their timeshare condo at Lake Tahoe and that he should "be sure to take her to our favorite romantic bistro on a pier overlooking the lake for dinner. I'd say that was pretty compersive!" Alternately, Baldwin thought Nadia began to drink more and was quite upset with him regularly, eventually saying that she
> accepted and allowed but did not approve of my relationships. From her Christian faith perspective, poly is wrong. I pointed out that I have accepted, even grown to support her participation in a faith practice that I feel is wrong, because of how much it's a part of her life and how much it nourishes her. There's also the issue that she has been having an intimate poly relationship with Jesus Christ for the last thirty years and when she goes away for a weekend or a week or whatever for a church retreat or conference its like going away with her lover—except that there isn't physical sex, but it might even be more intimate than that. She bristled, and said sharply that she was "just exercising her faith." Oh really? Every morning she would wake up one-half hour before my alarm went off and go upstairs to the den to pray out loud and "be with The Lord." When she came down, she brought me a mug of coffee in bed. I sometimes asked, "May I please have a hug to reconnect with you after you've been with Him?," but she would just sit up in bed and read a tract published by her church . . . She's also nervous about the social issue, if her friends or family find out that her husband is basically having an ongoing affair with another woman with her consent, how will they judge her for allowing that?
Although Baldwin had been "excruciatingly discreet," eventually his children found out that he was having polyamorous relationships. Baldwin reported that, when he took a date on an overnight trip to a hot springs, Rosaline asked where he was and Nadia
> outed me, saying I was there with this woman. Nadia thought Rosaline would be OK. Nope. She totally disapproved, and wondered how Mom could tolerate being with "such a man." Rosaline told Wade, who actually came over and wanted to physically pummel me, but his mother stopped him at the door. She told him it was consensual; he could not accept that. He said to his mother that I am dead to him because of what I "did to her." Nadia told him we had an agreement and it was consensual, but it had no impact. Nor did the fact that our wedding vows, which we wrote to deliberately exclude the "forsaking all others" part, and also did not include the "til death do us part."
Baldwin's children began to "harass their mother, putting a lot of pressure on Mom to leave me because of the poly. She tried to tell them that we have a non-typical relationship, but it didn't matter. They acted like Dad was pressuring Mom into something and she caved in." Later, Baldwin mentioned that "while Nadia would prefer it was not happening, she has consented to it voluntarily."
Over two years Baldwin continued dating, having a serious relationship with Abigail and less serious relationships with several other women. Baldwin said that Nadia was
> struggling with it, feeling somewhat jealous, trying to handle it. She is intellectually OK with it, but emotionally it's very difficult, particularly the societal mores and her church's belief that anything like poly is wrong, our kids' disapproval, and some "primal jealousy" as she calls it. Still, she knows I am committed to her/us, and she said she has to handle it somehow as neither of us wants to split.
While Nadia struggled with her angst over Baldwin's polyamorous relationships, things went from bad to worse in Baldwin's relationships with his children, until neither child would speak to him or come over to their parents' house if Baldwin was home. Family holidays became tense, and Baldwin reported that
> I went to Thanksgiving with Nadia at my sister-in-law's, and both kids were there. Not one word was spoken between us. About a week before Wade outed me to Nadia's cousin, who usually has a Thanksgiving dessert and coffee thing. Wade explained why he was so upset and refused to talk to me, and Nadia's cousin sent a message back through Wade to Nadia and me that I was uninvited from the Thanksgiving dessert party. I don't know what Wade said, but I can be sure it wasn't flattering. He is the one who told Nadia "poly is just Dad's bullshit intellectual justification for fucking around."
Even with all of his assertions that Nadia was allowing him to be polyamorous voluntarily, by December she decided that she could not tolerate the relationship style.
> She said poly just wasn't working for her. She initially reserved the right to change her mind. It destroyed my relationship with both kids and caused a strain on Nadia. She said she wants me to move out. This is all being handled very amicably, and one evening I was crying and saying I was sorry, and Nadia said, "For what, being who you have always been?" Then another night she was sobbing in the car, and said, "I'm really going to miss you, but this is the only way for both of us." We are hoping that once we are no longer living in a minefield, having the same fight over and over for three decades, we can be friends. So part of me is feeling like "bad dog, bad dog!" and I'm being exiled from the home I've lived in for twenty-seven years, but most of the time I feel like I'm being given the freedom to be fully myself and happier than I have ever been.
Baldwin experienced significant social censure for his engagement in polyamorous relationships, and while he gained a sense of freedom and joy, it came at the expense of his relationships with his wife and children. Although so many marriages break up after years when the husband wants to experiment sexually with younger women that it has become cliché, few of them do so with the honesty and integrity that Baldwin and Nadia brought to bear on the end of their marriage.
## Potential to Inflict Pain on Children
Parents in poly families were painfully aware that their children have or may face the difficult chore of managing the stigma of their parents' unconventional relationships, and some parents expressed remorse about the pain their relationships have caused their children. Joya Starr recounted her sadness over the challenges her polyamorous lifestyle created for her then six-year-old son Gideon, when
> he started going to school and they were asking "Who's your mommy, who's your daddy?" And he's able to identify us biologically without a problem. But for him it felt like—why are they only asking about those people? Like those are the only important people? . . . Now he knows this information about mom being poly and whatnot can actually really scare and freak people out. And having him be so young and having to manage that amount of responsibility for how adults and other kids relate to him, I can sometimes feel regret . . . And I wish that I was in a more stable trio for him so that he had this solid place to come from instead of like this multiple relating, my marriage didn't work kind of thing.
While Joya was keenly aware of the difficulty her son faced in relation to her polyamorous lifestyle, true to what she saw as her polyamorous nature, her ideal solution was a more stable poly family, rather than a monogamous one.
Melody Lupine similarly reported a deep conflict between her role as editor of a polyamorous magazine and a parent of children who wished to
> be normal. The website needs some new pictures and I am the logical choice, with my kids even better for the site. But for my kids? Definitely not! I would never ask them to put their pictures on the web—I am not sure if I can even put my own picture on the website. What if one of their friend's parents sees it and then it hurts my kids somehow? That would be terrible! I have to walk a fine line, decide each time to come out or not depending on the impact on my kids.
In weighing the needs of the magazine versus the needs of her family, Melody prioritized her children's perceived emotional well-being and used a picture of herself alone.
### Children's Experiences of Stigma
Children in poly families were also aware of the potential for stigma, and occasionally they had direct experiences of it. While Zoe Hadaway had not experienced negative reactions to coming out as being a member of a poly family, she imagined that she would and was thus extremely careful about those with whom she discussed her family. "The disadvantage socially is, you don't know how people are going to react until after you tell them. And that, their reaction, is how they feel about that problem—it makes or breaks the decision of whether or not to tell them." Silas Hadaway reported that he did not often invite his friends over after school because he was reluctant to explain the multiple adults living in his household: "I guess it kind of keeps me from having friends, sometimes, but I am kinda shy anyway, and even before we all moved in I didn't have friends over all the time, or very much at all."
Overall, the children I spoke with did not report experiencing many significant experiences of stigma. There are at least three explanations for this relative dearth of experiences of stigma. First, this could be because of the people who chose to volunteer for the research and their generally optimistic view of polyamorous families. Second, the race and class privileges that provide many of these children with social advantages may shield them from some of the effects of stigma. Third, the fact that poly families are still relatively unknown makes them more difficult to recognize and can protect them from the stigma of being recognizable sexual minorities.
### Family Vulnerability
People in poly families were also aware that the stigma of being a sexual minority made them more vulnerable to accusations of poor parenting or questionable family situations from relatives, neighbors, teachers, and Child Protective Services (CPS, sometimes called Division of Family and Children Services or DFCS) officials. Social stigma places polyamorists at a disadvantage, because disclosure or discovery of what Goffman would call their "discrediting status" can give a boss, angry teenager, or extended family member leverage to use against the poly person. In some cases, the presence of polyamorous relationships made interaction with CPS officials even more frightening for parents who had come to the authorities' attention for other, nonpoly reasons. Evelyn Coach remembered the pain she felt when her daughter, Annabelle—six years old at the time, struggling with urinary tract infections due to an undiagnosed physical anomaly, and recently diagnosed with ADHD—was taken in to protective custody on a Friday afternoon before a three-day weekend.
> Our child was summarily removed from our care without question and placed in foster care. I was told nothing, other than that I could bring her medications and a toy or anything else she might need. We had no idea what the source of the issue was, whether there was _real_ abuse that had happened—like at the hands of a stranger or someone in the neighborhood—or whether someone had misconstrued our [poly] lifestyle in some way. On Tuesday when I spoke with someone by phone, they told me that the source of the call was someone at school and that the quick action was because they suspected my husband Mark of sexually abusing our daughter. At least then we could relax about whether something had actually happened to Annabelle! But now we had to be prepared for the possibility of Mark facing a court battle to somehow prove his innocence. It took them until Wednesday to have someone talk directly to Annabelle, at which time they immediately determined that no, she was obviously NOT an abused child.
By Thursday social services allowed the Coachs to come in and explain themselves, and the social worker "lectured [us] for our in-house nudism." The social worker was "at least somewhat understanding of some of our choices" and explained that Annabelle had made a comment to a teacher that her father "played doctor to her," by which Annabelle meant that Mark had put ointment on her rash. Evelyn said that "the officials had put together that comment, her toileting issues, and her ADHD issues, made some assumptions about her responses to questions like 'did he touch you down there?' and decided that she fit the profile of an abused child" and removed her from the Coach's custody.
Annabelle's removal from her parents not only traumatized all three of them significantly but also interfered with her schooling and new medication. Most significantly, it singled Annabelle out from her peers at school and created a lasting tension and fear in the Coach household.
> The incident has colored our family ever since. Mark took to wearing clothing around the house, albeit resentfully. We changed the policies on a large event we host, to separate kids from nudity, and keeping even our own child off site during the part allowing adult, nonsexual nudity. We lived in fear of the next time someone would report us for something out of ignorance. My husband became depressed and anxious. My own depression and anxiety worsened. And my daughter, formerly bold and uninhibited, developed social anxiety with strangers, which persists to this day.
The Coachs dealt with the trauma by becoming even more "out" as polyamorous activists, offering public education to "help ensure that others would not need to go through the same terror and fear of being 'outed' or misunderstood for making a choice to have more loving adults present in their lives, and the lives of their children." While Evelyn is grateful for the perspective the experience provided and the insight it gave her into other's experiences, the thought of that episode still sickens her even years later. "I wouldn't wish that experience on my own worst enemy."
Evelyn, Annabelle, and Mark were all traumatized by the incident. Annabelle referred hesitantly to the episode in an interview when she was eleven years old but was reluctant to discuss it further:
> Elisabeth: Do people in your school know that your parents are poly? No? Why not?
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> Annabelle: Kind of . . . not something to talk about at school.
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> Elisabeth: How come?
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> Annabelle: [pause, then in a low voice] It's just not really [pause] something to talk about at school.
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> Elisabeth: So who can you talk about it with?
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> Annabelle: The cat. (laughter)
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> Evelyn: [to Annabelle] It's okay to tell her the stuff that happened that we had to be a little more close-mouthed about it. No? OK.
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> Elisabeth: We'll talk about something else then. So you have friends in the Pagan community who know you're poly. Are some of them poly too?
It was obvious from Annabelle's demeanor that she was still uncomfortable discussing it, and she seemed nervous remembering what had happened five years earlier. Clearly, the experience had been negative for her.
While Mark and Evelyn were able to regain custody of Annabelle in less than a week, Warren Bien's three daughters Rebecca, Callie, and Macy were placed in foster care when they were removed from their mother Julie's house and lived at a foster home for months before Warren was able to regain custody of the children. Warren and Julie married in the early 1990s and became polyamorous almost immediately. Over the years they had two daughters—Rebecca and Callie—and dated a variety of people. Eventually Julie established a serious relationship with Andrew, a friend of the family who had moved to the area to find a job. Warren said that Andrew was "interested in doing family things with us, since he did not have any relatives in the area. He started spending more time with us, and he and Julie developed an interest in each other." In 2000 Andrew moved in with Julie, Warren, and the girls, and Andrew and Julie later had a daughter named Macy who Warren also considered his daughter. Warren reported that
> at first it went well, and after a while I also had Estella, one of my partners, move in and the family moved together to a different state. At that point it started to involve other things besides poly. My oldest daughter said that her now stepfather Andrew started molesting her in 2002. That lasted for about three years, and I was not aware of it at the time . . . In late 2005 I decided my marriage was not salvageable; Julie and I had not communicated about anything but child care in two years. I did not feel I was important to her beyond a paycheck and I did not think that was going to change, so I moved out with Estella and her husband, Devon. Julie has since become extremely emotionally and financially dependent on Andrew. Then, in 2008, Estella and Rebecca were talking about the time we all lived together and Rebecca tells Estella that Andrew had molested her during that time. On her ninth birthday Andrew took Rebecca out to a movie and stopped at a truckers' motel on the way back and had oral sex with her. Rebecca is not sure how many times it happened over the next three years, but there were times when the two of them would go out on the property of our forty-acre wooded lot and there were other occasions when he would go into her bedroom at night.
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> Elisabeth: Why didn't she say anything at the time?
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> Warren: Andrew made quite an effort to ensure that she enjoyed the experience so she was not sure if it qualified as rape. She didn't know who should could talk to that would believe her and do something about it. Even at that point, her mother was getting more and more dependent on Andrew, and I spent a lot more time out of the house at work than I did at home so I was there but not as much of the time as I probably should have been. Most of the time that I was there and awake the girls were at school. At one point I was working sixteen-hour days.
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> Elisabeth: So what happened when Rebecca told Estella?
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> Warren: Estella called me at work and asked me to come home. When Rebecca told me what happened, I immediately called the sheriff's office and they sent a deputy out to take her statement that evening and I called the DFACS hotline soon after he left. At this time Estella and I were living with her husband, Devon, and Andrew and Julie were living in the same house that we had moved out of, maybe twenty or twenty-five minutes away. Callie lived with Julie and had regular visits with me, and Rebecca had been living with me since July of 2007 when I had moved from an apartment to a house that was large enough for her to have a bedroom. Macy was living with her mom [Julie].
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> Elisabeth: What did you do after the whole story came out?
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> Warren: I believed Rebecca and took action. I called Julie and said I wanted custody of Callie as well, and at first it seemed as though she was going to be reasonable about it . . . but over the course of the next week she decided that Rebecca was not telling the truth, which absolutely thrilled Rebecca [sarcastically]. My opinion is that Julie can't afford emotionally or financially to believe that her husband is a child molester.
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> Elisabeth: How has that worked out with the courts?
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> Warren: In early July 2008 the DCS asked that all three of the girls be placed in safe custody while they investigated what was going on in both households. That is still ongoing [as of April 2009]. Julie has limited supervised visitation with the two younger girls, without her husband who has been barred from all contact with any of the kids, and Julie has also been barred from visiting with Rebecca. Rebecca has expressed some interest in seeing her mom, but Rebecca's therapist thinks it would be bad for her to see her mom . . . I have had much more extensive visitation in my home, unsupervised, with all three girls, since November, which was a pleasant surprise. That has gone very well and the girls are happy to be here on the weekends and enjoy spending time with me . . . They say they want to come live with me all the time . . . I think it is very likely that the older two girls at least will be living here at the end of the school year. Julie is still living with someone who is on trial for the rape of a child, so the youngest daughter will probably not move back in with her mom. The judge lets her come over here on the weekends, so she might be able to come when the older girls move back in as well. I want her to move in with me with the other two, I think of her as my daughter and she calls me her papa bear.
Warren said his divorce had nothing to do with polyamory, that he and Julie had communication issues that predated their polyamorous relationships. The transition from a household of five adults and three children to two separate households had been a challenge for everyone, but the adults remained unaware of Andrew's molestation of Rebecca that had already occurred. Once the information came out, Warren asked Callie about possible molestation, and she had responded that "Andrew had never touched her."
In an effort to "encourage the courts to place the kids with me," Warren and his lawyer decided that he should live separately from Estella and Devon so "the court was not looking at a poly household." Having to separate from his lover Estella and her husband Devon, whom Warren loved like a brother, was devastating to Warren at a time when he needed his support system around him. Warren said that the court case had "interfered with my relationships but not ended them."
When I asked why the girls went into foster care and were not placed with him in the first place, Warren responded:
> I briefly had custody of Callie but while the hearing was going on to determine if temporary custody should be continued or not, Julie made some allegations in the courtroom specifically regarding Devon. She said that she believed that Rebecca had been molested, but she did not believe that it was her husband that was doing it, and while she did not say so specifically it was clear she was implying that it had been Devon. The judge said that he was not in a position to determine the truth of that matter, so instead of placing the kids in either household where they may or may not be at risk, he was going to place them elsewhere . . . I am pretty sure I would have gotten the kids right away if I had not been poly.
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> Elisabeth: Do you think the molestation would not have happened if your family had not been poly?
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> Warren: I don't think that is accurate. Whether Julie and I had been sexually open before it occurred or not, we still would have had other friends and while it might have been more difficult for Andrew to have access to my daughter if he had not been in a sexual relationship with Julie, I am fairly certain he still would have been a friend of the family and I think it is likely that he would have had access to her [Rebecca] anyway. So he [Andrew] caused the molestation, not polyamory.
Child molestation has in recent years been revealed to be a major social issue, so unfortunately widespread that it is not confined only to sexual minorities, people with unconventional lifestyles, or children who attend Catholic churches. Even so, it was extremely disadvantageous to this child in this poly family who was molested by one of her parent's partners—a terrible turn of events no parent would want to subject their children to.
#### Teenagers Have Leverage Against Poly Parents
Because authorities can be suspicious of anything out of the ordinary, the stigma of being in an unconventional family can make some poly parents vulnerable to attempted blackmail from disgruntled teenagers. The Holstrom family was composed of Billy and Megan, a married couple with a daughter, Ariel, of their union; a son, Nolan, from Megan's previous marriage; and a son, Simon, from Billy's previous marriage; Jack (Megan's partner); Sabine (Billy's partner); and Tad (Sabine's husband). Things were going along smoothly until Nolan "blew his stack," as Billy reported. Nolan had run into Rex, his biological father, while visiting relatives in another state, and Billy said that Nolan
> used it as ammunition against us when he got back. Ammunition for anything he could use to hurt his mother [Megan]. We never found out why he wanted to hurt her before he got us kicked out of family therapy. He thinks we kept him from his dad even though his dad was in the federal penitentiary in jail for selling firearms to an undercover agent . . . He was in jail for . . . over half of Nolan's life when Nolan found him.
In an attempt to interrupt a negative behavior cycle in which Nolan was "being ugly and violent, getting in trouble at school, hanging with the wrong crowd," Megan had "sent him [Nolan] away from that element to change his scene." Unfortunately, taking a break from the "wrong crowd" did not significantly alter Nolan's behavior, and Billy reported that when Nolan returned:
> We knew something was up but we didn't know what. He caused a big scene, decided he was going to be big and bad and take a swing at me so I had him arrested for assault, terroristic threats, and cruelty to children because all of this happened in front of Ariel. He spent the night in jail and did an arraignment the next morning, and this is where poly enters the story. Through all of this we had tried to fix it on our own; we went to family therapy but he would not go to individual counseling, he was not in it to fix whatever problem there was. So before we go into the courtroom the public defender asks if there is anything we need to say but once we get in the courtroom and the judge sentences him [Nolan] to eighteen months supervised, so then the judge asks, "What is this I hear about you having someone extra in your relationship? Don't you know that adultery is against the law?" I [Billy] said that it was not illegal to love more than one person, and the judge said, "Adultery is illegal in [this state]!" and he tells us we have to go to counseling and we will be investigated by DFACS and we need to reevaluate our relationship choices to make sure we are doing what is best for our family. So instead of putting the responsibility on Nolan for his own behavior, the cause was legally shifted to us where we were the bad guys.
Over the next several months the family was under DFACS surveillance, with home visits and case managers visiting Ariel at school. Billy reported telling the DFACS case manager:
> Anything that happens in our bedroom stays there. Have you ever had a sleepover? That is about how much the kids get to see, everything beyond that happens in the bedroom. That was good, she seemed to understand poly . . . So the case went on and she interviewed us, Jack, Megan, Megan's mom, and some of Ariel's teachers. After about a month we had not heard anything so we called her and asked what was happening. She said, "You didn't hear? I certified you as a perfectly functioning family and Nolan sounds like he needs to work on himself." It turns out Nolan got a girl pregnant, lied to us and lied to her, and then she lied and said she lost the baby even though she was still pregnant. We didn't know any of this until we went to court, and that's when it all fell into place what Nolan's problem had been—he'd gotten somebody pregnant and didn't know how to talk to us so he was weird and angry.
While the DFACS case manager appeared to understand the polyamorous dynamic in the Holstrom family, Nolan's probation officer took a decidedly different approach to the family. Billy reported:
> The juvenile justice system on the whole has a negative attitude toward poly or any alternative lifestyle. The probation officer would say things like "the way you people are," or "the lifestyle that you participate in is not normal, that is not a normal household." . . . We didn't hide anything, and it became apparent that because we were different, she judged us negatively . . . Now Nolan lives with a friend whose mom is unemployed, in the middle of a divorce, has five teenagers, but his probation officer thinks that is still a better place for him than what she called "THAT household with all that STUFF going on." Nolan swears up and down that all of his problems are because we are poly, that he just can't live that way. I told him you don't have to live that way, you just have to accept people who are different from you. We thought we had raised an open-minded, accepting child only to have it blow up in our faces. Our other son has no problems at all with it; he would go to the moon and back for us. He hasn't said anything about it; he sees nothing wrong with people living their lives the way they want to live. Not his cup of tea right now, but whatever we do is fine with him . . . and it is not even like Jack lives with us, he lives several hours away in [a city] and comes up to visit two or three weekends a month. That is the other thing that puzzled us, with Jack not being here all the time, he [Nolan] was always angry. If it was really about the poly, then why be so angry all the time and not just when we are doing poly activities? Now we only hear from Nolan when he needs money, and I told him to call his dad for that because his dad owes him a quarter of a million dollars in back child support he never paid. Nolan said that was not fair, and I said we don't have any money so ask your bio dad.
Frustrated with their son's refusal to take responsibility for himself and angry at the juvenile justice system that judged them so harshly, Billy and Megan felt the impacts of institutional stigma that aggravated their existing family problems.
After the investigation ended the Holstrom family "did not hear a peep from DFACS since," and Billy and Megan began to relax because it appeared DFACS would not remove their youngest child, Ariel, from the home. Ariel's teachers and social support circle all reported to DFACS that Ariel was doing fine and the Holstrom family seemed to be free of further surveillance. Even so, Billy reported:
> This actually has Megan and I talking about a legal divorce, though there is no statute of limitations on adultery. If we divorce it would prevent anything like this from happening again because if we were divorced we would no longer be committing the crime of adultery in [this state]. We would be committing fornication and cohabitation, bigamous cohabitation where two people who are assumed to be married living in the same house but having sexual relations with other people. The third party can also be charged with adultery and bigamy in [this state] . . . At this point they could also go after Jack, who lives in [a neighboring state] and comes to visit regularly. The penalties are different for each crime and in each state. In [Billy and Megan's home state] you can be prosecuted on a felony level for bigamous cohabitation, but fornication and regular cohabitation are misdemeanors if I remember correctly . . . We [Billy and Megan] would divorce only as a legal protective measure, not because we are unhappy together. More of an annulment than a divorce, I guess.
In order to protect their family from the legal ramifications of nonmonogamy, Billy and Megan felt they had to dissolve their legal marriage—something other poly families have reported as well. In their case, outdated laws against fornication, adultery, and cohabitation selectively enforced against polyamorists and other sexual minorities—but not generally applied to heterosexual "vanilla" relationships—made Billy and Megan Holstrom more vulnerable to their teenage son's angst and attempts at manipulation. This level of vulnerability to authorities that might misunderstand or misjudge poly families based on the stigma against sexual nonconformists has been disadvantageous for polyamorists and other sexual minorities.
## Children Become Attached to Partners Who Leave
While the presence of numerous adults attending to children in polyamorous families may provide an atmosphere of love and caring, it also sets the stage for children to become attached to adults who are related to them through the potentially tenuous bonds of a polyamorous relationship. Many parents reported their children's attachment to partners who eventually left the relationship, much to the children's chagrin. Joya Starr remembered her son Gideon's misery after the departure of one of her boyfriends, a man who had been the boy's treasured friend, and how Gideon had asked her, "I know why you guys are breaking up, but why does he have to break up with me too?"
Shelly and Sven Heartland formed a triad with Adam when their daughter Alice was a small child, and Alice became very attached to Adam over time. Even though he did not live with them, Adam spent almost all of his free time with the Heartlands and became quite close to the whole family. When Adam broke up with Shelly and Sven, Alice was heartbroken. Shelly reported that, even two or three years later:
> Alice will still call him and leave him messages and then when he doesn't call back that doesn't make her very happy 'cause when, she doesn't really understand why, even though he's not around any more, why he won't kind of respond to her, and he's pretty much cut off all ties with her.
While the entire family mentioned missing Adam at one point or another in their interviews, it seemed to be especially poignant for Alice, who loved Adam as a father.
Alice's sister Elise said she noticed a difference in Alice after Adam left and when Rich, Sven's new boyfriend, arrived. Elise said:
> Like when Richie came in to the picture and I was like talking to Alice about it, 'cause I don't know if she totally understands everything, and I was just like "Hey, how do you like Rich and stuff?" and she was like "He's no Adam." So she is still like totally attached to him [Adam] and stuff.
Similar to Alice, Cole related his pain at missing his parent's former girlfriend Bettina:
> The only thing I really regret is that, now that Bettina's gone, and her kids, that was the hard part. Just having them leave. Because they were such a huge influence in my life. I wasn't always regretting them, a lot of times they were really helpful. They got me out of a lot of situations, and I learned a lot from the kids. . . I miss her [Bettina] a great deal. And I wish I could see her more but she's moved away. I wish I could just take her out to lunch, see how she's doing. . . I always thought she would be here to help me through high school, but she's not. I made it through my first year without her, and, well, um, but I think about her a lot. I really miss her.
Cole said that he missed not only Bettina, his parents' ex-girlfriend, but also her children, with whom he had established friendships bordering at some points on sibling relationships. He found it especially painful when Bettina did not attend his bar mitzvah, an important rite of passage for young Jewish men who spend years preparing to be called to recite portions of the Torah and are then ushered into religious adulthood.
> Cole: She wanted to come to my bar mitzvah. I always knew I would have one, and I always knew she would be there. I thought my parents had sent an invite, but it turns out they didn't. And I never confronted my parents, they never found out I was angry. I asked her what happened and she said I never got an invite, when was it, how was it kind of thing, you know, a few days after my bar mitzvah. And I realized she wasn't there, and after that, I never really talked to her, pretty much after that.
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> Elisabeth: Why did you stop talking to her after that?
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> Cole: I guess maybe part of it was that I got a cue from my parents that, maybe it was a false cue, but again I got a feeling from my parents that they didn't want me talking to her. That they didn't want me as part of her, as part of my life any more. I don't think that's what happened, I just think that they honestly forgot or that, you know, maybe there was some harsh breakup feelings but I don't think they have them now.
Two years later, Cole asked his parents about failing to invite Bettina to the bar mitzvah, and they assured him it had been an oversight, that if he had wanted her to come he should have told them.
While some of the children I interviewed keenly missed beloved adults who had exited their parents' lives, many of them did not. For instance, Zane did not feel damaged by his parents' partners leaving.
> I mean, I liked a lot of them, I was friends with them, but I was so used to them not being permanent that it was fine. I was glad I met them and I was happy to spend time with them but it was never like, I never thought too much about it. It was not a big deal. One specifically I remember, I was in sixth grade or something, I'd do a bunch with them. I'd skateboard with them, I'd ski with them, it was kind of just like an older friend. It wasn't that big of a deal that he left, I wasn't too bummed out. Just someone to hang out with, I guess. They were on and off again for a really long time, I can't even remember the time span. It was like pretty long, but it was always on and off. And he's still a friend, he's still around sometimes. He's cool, I like him. It wasn't like I avoided being friends with them or anything, I just kind of eventually warmed up to him. Like I said, it never was really that big of a deal. I don't know why, it never really crossed my mind as a bad thing that they were leaving.
Like Zane, many others did not see their parents' partners exit on a permanent basis but rather retained friendly contact with the family over time.
Partners could also become attached to children, and then feel upset when the relationship with the parents dissolved and the partner no longer spent as much time with the child of their former partners. When Kristine broke up with Mark and Evelyn, she missed their daughter, Annabelle, keenly.
> While I was with Evelyn and Mark, there was a lot of love and it rips me apart to have had to walk away from Annabel. She became a priority for me, and often I spent more time with her than either of her parents. . . I remember weeks taking care of her [Annabelle], helping her with her homework, spending time with her. One time I was down with food poisoning, and I still spent an hour-and-a-half brushing out her hair of knots and tangles.
While Kristine felt loved when she was with the Coach family and thought that Evelyn and Mark were "extremely loving parents, really devoted to their daughters," Kristine was not sure that polyamory was a positive thing for the children or the family as a whole.
> But I don't honestly believe that either Annabel or Martine benefited from Evelyn and Mark being poly, other than more adults riding herd on Annabel. Would they have benefited from them being monogamous? I don't know the marriage would have lasted as long, but there might have been fewer distractions.
I was unable to talk with Martine, but Annabelle reported to me that she felt she had gained numerous advantages from being in a poly family and did not identify the disadvantages Kristine found so distracting.
In this instance with the Coach family, the complexity of the adult relationships did not appear to translate to complexity in the child's relationship. In other cases, children see adult relationships in a worse light than do the adults engaged in them. Elise Heartland watched her mother
> go through hell with Adam. I mean you guys loved each other and obviously it was a good thing for you in some ways, but from where I stood it looked like a lot of pain.
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> Shelly: But you didn't see us all the time, there are whole parts of the relationship you didn't see.
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> Elise: Of course. But the parts I did see, it looked like a lot of pain and work for not that much fun for you. Sven seemed to do better with it, but Adam loved him that way too. I couldn't handle that, that would piss me off. There is no way, I need more attention than that. Like if I was with somebody and he was with somebody else I would be like really annoyed, that I'm going to go get another boyfriend.
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> Shelly: I wasn't really angry with Adam, that was just the way it was.
Like everything else, the disadvantages and complexities have widely varying impacts on different members of the family, each can view the situation quite differently, and the same member can shift views over time.
Cole Cypress had a complex relationship with polyamory, and found it both good and bad at different times and in different ways. In chapter 7 Cole discussed taking the pocket knives to school and getting in trouble, and I asked him about how, if at all, that related to his family.
> Elisabeth: When you were having your episode of being very angry and upset, acting up at school with the knives and stuff, did that have anything to do with polyamory?
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> Cole: I'm sure it did. I'm sure then was probably one of my bigger episodes of just like, you guys forced your lifestyle upon me, I never had any real friends because of you. And I just had a lot of hatred and that was the perfect scapegoat. Polyamory. I wasn't getting good grades and uh, there was also an issue with me swearing in school, cause this is of course a private school. And whenever my teachers ask me why I swear so much I would always always blame Bettina's son Caz . . .
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> Elisabeth: So you mentioned that you felt like you never had any friends because your parents were polyamorous?
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> Cole: I never, I feel like I never had any friends—I felt like I never had any friends just because I couldn't explain it to them. I couldn't really open up to them. It was 'cause of polyamory. I felt like all of my life was centered, centered around it now, and if I ever told the story about my life it would involve somebody from the poly community and I'd have to go into a whole explanation of polyamory and so I stayed away from people.
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> Elisabeth: Was that painful?
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> Cole: Yes. It was, yeah, yeah. It was extraordinarily painful just to (breathes deeply) just to live a life uh, you know, without friends. Uh, I would always, I was like a pariah, or an outcast in a class of forty-seven students or so.
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> Elisabeth: Specifically because of the polyamory?
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> Cole: Well, I felt that way. It wasn't though. Because I was acting out and because nobody wanted to be my friend. I wouldn't talk to people, I would swear and get in trouble and talk back to teachers, I didn't get good grades. I didn't like people that much. People didn't like me.
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> Elisabeth: Because you felt different?
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> Cole: Yeah.
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> Elisabeth: And you acted different? And the difference was based in the polyamory?
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> Cole: Yeah, it was based on that and just, you know, trying to live up to Caz, live up to his behavior.
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> Elisabeth: So that was a big factor.
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> Cole: Yeah, pretty much.
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> Elisabeth: How is it now?
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> Cole: Well in eighth grade I kinda got my act together. I wasn't living with Bettina anymore, I didn't see Caz or Bettina's daughters like at all. And I started talking to people and I started opening up, and I got, how do I say this, I started just trying to be more friendly and I wasn't a smart alec to people. I also liked my teachers a whole lot more and I could adapt to the environment a lot easier.
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> Elisabeth: Do you think that shift was specifically because you weren't living in a poly household anymore?
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> Cole: Yeah, I think a lot of it had to do with that. I think I just kinda got a life, uh more of a life. I had a lot of time to think and I had a lot of time to go out and meet more people, attend birthday parties, and you know just, I had a lot more fun. Eighth grade was probably, if it wasn't my best school year, it was either my best school year or this school year.
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> Elisabeth: Best how so?
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> Cole: I have friends now. And I can talk to people. And I get along with my teachers, I'm getting better grades. Let's see, I do better with my parents. But I'm still relatively well connected to the poly community. I don't know, part of it, I think a big part of it was her getting out and I think a big part of it was just getting my act together.
Cole had mixed feelings about polyamory—having to keep the secret or manage the information in such an enclosed social environment as a small private school clearly heightened his social awkwardness and made his tenuous relationships with peers even more problematic. At the same time, Cole felt that he had gained many personal advantages from his relationships with Bettina and Caz, and he valued his ongoing association with his local poly community. Perhaps most significantly, he acknowledged his own part in the difficulties, calling polyamory "the perfect scapegoat" for the consequences of his "smart alecy" behavior. In fact, throughout his interview, Cole struck me with his self-insight and willingness to own his part of a difficult situation. Characteristic of children in polyamorous families, he was articulate, reflective, and confident that his thoughts and ideas were important.
## Household Crowding
In some families, children felt crowded and wanted more space. Zane Lupine remembered feeling crowded when growing up with his triadic poly family. "It was overwhelming sometimes, six people in a three-bedroom house. The adults shared a room, me and my brother shared a room, and my sister got her own room. Physically, I just wanted my own room and more privacy."
Zoe Hadaway felt a similar distress over a lack of privacy and had attempted to segment a portion of the large basement that housed the older children in the Hadaway family. Each adult couple had a large bedroom on the third floor, and the younger children shared rooms on the second floor, which left the large, semifinished basement for the older children—half for the girls and the other half for the boys. While her younger sister Michelle's bed was against the enclosed side of the staircase, Zoe had placed her bed in the corner and hung sheets, tapestries, and scarves as walls that delineated and screened the space from sight. Zany sighed, "Yeah, I think it is more fun for the little kids, to always have someone around to play with. But I am sick of always having everyone around, always having to share the bathroom, never having any privacy at all, even in the bathroom!"
Later, the family moved to a different house in the same neighborhood and redistributed the space. Several factors combined to allow Zoe to have her own bedroom, again in the basement, but this time:
> It has a door! I keep it shut most of the time. If I want to see people I go up and there they are. Otherwise, I'm in here and nobody can bother me. At least they're not supposed to. Before when I could never get any time alone it was a big problem. I could never have a private phone conversation, they were always wearing my clothes, going through my stuff. It sucked. Now they still try to go through my stuff but I'm going to put a lock on my door so they can't get in and take my stuff.
The negative effects of crowding appear to become increasingly acute as children age, and the teenagers seemed especially dismayed by their lack of privacy and space. For Jonathan Hadaway, it wasn't so much about privacy as lack of quiet. "It's really loud all the time with all of these kids," he said. "I can't sleep. They're all up too early."
## Family Complexity
For some people in polyamorous families the complexity of the relationships could be quite disadvantageous at times. These families experienced a number of complexities such as dealing with previous partners, jealousy and tension among siblings, difficulties adjusting to new parenting styles, adult drama, and difficulties managing information with families of origin.
### Previous Partners
In monogamous relationships that end, people usually do not have to deal with someone close to them continuing to date someone with whom they have just ended a relationship. For Shelly and Sven Heartland, however, this was not the case. Initially they had formed a triad with Adam, but eventually:
> Adam and I [pause] broke up, I guess you could call it that, while he and Sven were still together. So he [Adam] and I were just friends for the last like, two years of the relationship. It was a hard [breakup]. It was harder for me I'd say because I was really attached to him. And I just knew the whole dynamic was really going to change once it went from being a triad to just a vee. And it was hard, I mean, it was like any other breakup. Painful . . . He was primarily gay and had some bisexuality, enough that at the beginning it was OK. But as time went on he couldn't, he just didn't feel that way, you know, about me and all. And so we broke up and they [Adam and Sven] continued their relationship and we [Adam and Shelly] were just friends, and we adjusted OK. . . . It was hard because it was such a rejection, and then on the other hand I knew how much he cared about me, it was just on a different level, so, it was hard. And it was hard for him [Adam] too. I mean, we would both be crying when we were talking about it. I think he really wanted to make that part work too, but you can't make yourself feel something you don't really feel.
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> Elisabeth: Did it put a strain on your marriage?
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> Shelly: No.
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> Sven: No.
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> Elisabeth: How was it for you, Sven?
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> Sven: It is kind of probably what ended up leading to the breakup—my own breakup with Adam. It was stressful. I mean, we had good times, not that long ago we went on a Fourth of July ATV [all terrain vehicle, recreational riding] thing and it was fun, but it is just different. Because you start off being a triad and then you're not . . . The breakup [between Adam and me] started because it was not working with Shelly, and it just tumbled down from there, it kept getting worse and worse and worse until, attitudes . . . it was like a coin toss. He would come over and it was like, you never knew if he was going to be in a good mood and enjoyable or whether he was going to sit on the couch and be all pissy and grouchy.
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> Shelly: And everything changed. We went from him [Adam] coming and joining our family to them [Adam and Sven], it just being, if Sven did go down to [Adam's house], then he is away from everyone else.
While Adam's moodiness had been a factor previously when the triad was still together, it became more marked once he and Shelly broke up and there was more tension in the relationship. Going from a romantic to platonic relationship can be challenging for many people, but attempting to maintain a friendship when your ex-boyfriend is still dating your husband can be even more difficult.
### Jealousy and Tension among Siblings
The intricacies of polyamorous families are complicated not only for adults but for the children in those families as well. Routine family challenges like jealousy among siblings can sometimes become even thornier when intensified by complex poly family dynamics. Zane Lupine reflected on his relationships with Pete and Joyce, his elder siblings:
> There is jealousy I guess, between my brother and sister and I. Because we have different dads, you know, there's always been that tension. Especially after their dad is not really that active in their life anymore, and my dad moved out here so he could be with us. There's just kinda always been a problem. But it's been a lot better, cause everyone's just kinda grown up and gotten over it.
Like many other families, children in poly families can have issues with their siblings, and these can be compounded by the effects of mixed parentage. In Zane's case, he shared the same mother (Melody) with his sister Joyce and his brother Pete, but Zane's father is Quentin and Pete and Joyce's father is Cristof. Mirroring experiences of other blended families with half- and step-siblings in serial monogamous families, the Lupine siblings felt some tension over the varying degrees of effort and number of resources each father was willing to contribute to the family.
Like Zane Lupine, Zoe Hadaway felt some friction with her siblings in her large, blended family. Prior to moving in together, before their parents formed the Hadaway family, Zoe and Michelle had been good friends. Combining households, however, took a toll on the girls' friendship. Zoe said:
> We were best buds at the time and now we're sisters and so I don't know. Sort of like you can be best friends with somebody, then you move in together and you're like, that's a fault, that's a fault . . . We're still friends, but not like we used to be. Like it's like every single little thing we would tell each other before, and now it's just like you are really getting on my nerves and I am going to go shut my door in your face and you just have to go away.
In this instance, the parents' decision to move in together took precedence over their daughters' need for space to balance their friendship, which was strained by the intensity of interaction among siblings.
### Adjusting to New Parenting Styles
Becoming accustomed to various parenting styles sometimes proved difficult for children in larger or blended poly families. Alvin, Jonathan, and Zoe Hadaway reported that their two fathers had quite distinct parenting styles.
> Zoe: Yeah, both of the fathers are very dominant personalities. For kids who didn't grow up under them, when we were younger we believed almost everything we heard. We were very naive. Phil and Dad are, for the most part, the exact same person with completely polar opposite skills and personalities. They're both very smart alecy and they have quick reaction times. Mitch is mathematical and
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> Alvin: (interrupting) logical.
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> Zoe: Yeah, he's the logic stuff. Phil's very hands on, and Dad's [Mitch] not at all. He has no domestic talent whatsoever, besides that he's a very big leader. They fit easily into a leadership role. So the fact that I grew up in an academic-based household, where the head honcho [Mitch] was all business, "You make your money, do what you love, not because of the money but because you love it," and Phil was always, "We need to do this, don't question me. We just need to do it, we're going to fix it and make it work and . . .
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> Jonathan: There is no "because" from Phil, just do it. With Mitch there is a because, he explains . . .
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> Zoe: You could question him [Mitch] and he wouldn't be happy about it, but he would try to be logical with it.
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> Alvin: You could ask him questions, if you asked him a question he would answer it.
As the Hadaway children saw it, Phil and Mitch's clashing parenting styles wasn't the only source of conflict for the two men.
### Adult Drama
In some polyamorous families, all of the extra communication and relational intensity translated to frequent arguments, and that was true for Mitch and Phil in the Hadaway quad.
> Zoe: They just don't get along.
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> Jonathan: They have clashing personalities.
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> Alvin: Neither one would try to hurt the other; they just don't get along. I don't think it's as much of a problem between them, Jonathan is exaggerating, but . . .
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> Zoe: I would compare them to brothers, brothers that don't really get along. They get along sometimes, but they're always at each other's necks.
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> Alvin: It's back and forth all the time.
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> Jonathan: There's two alpha males living in the same household. Sharing a family.
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> Alvin: It's never like they're just helping each other, it's never like they are beating each other up either.
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> Elisabeth: Tension?
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> Alvin: Usually.
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> Elisabeth: Uncomfortable?
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> Alvin: Somewhat. But they kind of joke around with it, to make it seem more comfortable. They joke around when they are talking. The house, when the moms want to do something with it, he's like, "OK, if you think that's the right way to go, I'll be right with ya." If both moms agree then the dads will go along with them. Actually, it's usually the moms who have the final say for real.
Even with the joking around, there remained some family tension when the four adults would be fighting. Referring to her younger sibling Silas, Zoe reported that
> he's finally reaching that age where he sees how hard the adult relationships can be on each other, and on the other adults. I remember crossing that line, where you finally start recognizing that the adults are really fighting. Even if you're not witness to the fights, you can tell when the adults are really not happy with each other.
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> Alvin: Dad won't sit with Mitch at cards, and the moms will do their things in silence. Tammy does her thing.
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> Zoe: Tammy is just completely quiet . . . she retires quickly. And when Mom's [Gwenyth] upset, she . . .
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> Alvin: the moms will get upset and . . .
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> Zoe: she tries to be as normal as possible.
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> Alvin: But then she melts down at other times.
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> Jonathan: It's there, that things are off.
Monogamous and blended families sometimes experience tension as well, though in this case the sheer number of people involved in a polyamorous family can create additional tension simply by volume of interactions.
Zane Lupine also reported that his parents argued regularly:
> Zane: I guess the argument aspect could be a disadvantage. Cause there's more issues. More people, more issues. That's just how it goes.
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> Elisabeth: So did you have to deal with a lot of arguing when you were growing up?
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> Zane: Yeah, I'd say. Definitely not as much as some people do. I guess a decent amount. But whose parents don't, really, I guess.
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> Elisabeth: How did you handle it?
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> Zane: I used to think like, a lot of it was my fault, I'd worry about that. But of course they'd reassure me that it wasn't so eventually I just got fine with it. Of course it was a bummer, but it was never that big of a deal.
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> Elisabeth: So when it was happening what would you do?
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> Zane: Just go to my room and do my own thing. Just space it out.
Like many tweens and teens, Zane retreated to his private space to allow the adults to have their conflict unobserved. He distracted himself with music and homework or reading, and he left the adults to argue it out.
## Families of Origin
Dealing with families of origin can present disadvantages to both adults and children. With four quite different sets of parents/grandparents, the Hadaway family had a wide range of families of origin with many differing beliefs. Gwenyth's mother, whom the children called Omi, lived in the area and became suspicious of her daughter's seemingly permanent "houseguests" as the couples grew closer. Eventually those suspicions boiled over into outright accusations after the two couples moved together to a larger home. Omi would see all of the grandchildren on what Jonathan called "a rotation, every weekend someone different goes to her house and gets to spend the weekend with her. There are nine kids, so a full rotation takes about two months to go through. But that's probably going to be on hiatus for a while, cause last weekend was so bad." When I asked what was so bad about last weekend, Zoe and Jonathan related the story of how Omi came to find out the quad was romantically involved. Zoe said:
> She was like, "I know what goes on over there and I think it's disgusting!" And I was like, "Well, that's your opinion!" and I stomped out of the room. But she would not drop it. It just got worse and worse. Finally she was like, "OK, we're going to dinner now. Come on, you said you wanted to go before." And I was like, "Actually I think I am just going to go home now." And she was like, "Don't do that." And I said, "I just want to go home now."
After a very awkward dinner in which "we didn't really eat much" and Omi "rambled on," trying to soothe Zoe's hurt feelings, Omi finally took Zoe home. Zoe reported that, on the way back to the Hadaway house:
> Eventually, Omi was like, "Don't tell your mom about this, OK? Don't tell her you're upset with me." Not that I had to. I walked in the door and mom said, "How was your trip?" and I said, "It was fine." And she said, "What's wrong?" I didn't say anything, I just came downstairs and put my bag away. By that time I finally broke down. Mom came down and said, "What happened" so I told her and it was bad. When I went back upstairs Omi was upstairs and Mom was still downstairs, and she [Omi] slipped out the door without another word. She knew that if Mom was down there talking to me, I was going to tell the truth.
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> Jonathan: She [Omi] was like, "I need to go. She's [Gwenyth] going to be mad at me."
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> Elisabeth: So when it happened the first time, she got information out of you? She kept pestering you?
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> Jonathan: Like I said, she kind of has a way of just talking, she's getting information kind of without my knowing it.
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> Elisabeth: What kind of information did you give her?
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> Jonathan: The thing is, she asked a question I didn't answer, which kind of told her it was a yes.
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> Elisabeth: What was the question?
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> Jonathan: "Do you think Mitch and Tammy have a relationship?" Actually, I think it was Zoe who she asked, "Why do Mitch and Tammy always sit in the front seats together?"
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> Zoe: Yeah, that was what she asked me. And then I came home and she asked me this. What was I supposed to do with that? Now she knows. It's like, "Jonathan, you have to go tell the parents. You have to let them know that she knows." Yeah. I was like, "You have to go tell her, Jonathan. If you don't tell her I will." And the way [the house was configured] with this great vaulted ceiling, the entire thing echoed . . . so whatever I said standing around in the doorway with Jonathan echoed upstairs and the parents heard it. They were like, "What's going on?" . . . So I went upstairs and it was like, "Omi knows. We need some damage control going on here."
The weight and intensity of keeping the family secret and managing older relatives' reactions to their unconventional family style was, in this case, emotionally painful and stressful for Zoe and Jonathan. Having to bear bad news and strategize about "damage control" can be burdensome for adults, and it is especially uncomfortable for children and young adults who have less control than their elders and may be unable to leave a difficult situation when they wish and end up having to sit through an awkward dinner.
In other cases, parents in polyamorous families fear negative consequences to their family complexity that never actually come to pass. When I first interviewed Shelly Heartland in 1999, she expressed concern that her daughter, Elise, might have to keep the secret about her mother's polyamorous relationship from her father, Shelly's ex-husband. Shelly said:
> What a terrible burden it could be if she [Elise] feels like she can't talk to her dad. He's kind of childish and unpredictable, he might be fine with it, or he might make a big deal out of it and use it to sue for custody. He was an ass during the divorce, wanted custody of Elise just because he knew it would hurt me, and then didn't really even want to take her once he had gotten the legal right. Just because he was an ass to me and a mediocre dad most of the time doesn't mean that she shouldn't feel comfortable telling him the truth. So yeah, I worry that my poly relationships could make her have to keep secrets or other things like that that could be a pain in the neck for her. But we are concerned, because he is kind of volatile and I don't know how he would react with him, how he, knowing that she was a minor and we are seeing someone else.
When I interviewed Elise, Shelly, and Sven together in 2007, Elise dispelled the misconception that it was painful for her to hide her mother's poly relationships from her father. I asked Elise if she had any problem managing the information about her polyamorous family with "the school, with your friends, or your bio dad?," and she responded:
> Well, he [Elise's biological father] really didn't know anything, really, the less he knows about my life the better. I lived out there for a year because things weren't going so well here and I thought I wanted to live in California. But let's just say I really, I guess it was a good eye-opener, but I don't think I would ever live with him again. So just the less he knows, the less questions he has, you know, he already has in mind what kind of a person I am and he's not going to change his mind about that . . . I think seriously if I tried to explain it to him he just wouldn't get it. Seriously, he's really naïve and closed-minded, and I seriously, when I go back there if we ever talk about anything I hear enough about the normal stuff he hates about you guys [Shelly and Sven], why add it to the fuel? He's just a bitter, hating man, very unhappy . . . When he found out I was moving back [to Shelly and Sven's place in another state] he flipped out. And at the airport, when he knew I was going back, he was like, he threw down my bags and was like "good luck" and left, that was it. He didn't even say I love you, nothing like that. So I don't know why I would say anything to him about Adam or my mom, why give him any more ammunition against us?
Rather than laboring to keep her mother's secret, Elise aligned herself with Shelly and Sven—"us"—against her father, whom she characterized as a "bitter, hating man." In this case, Shelly's fear of Elise having to keep her polyamorous relationships a secret becoming burdensome for Elise ended up being unfounded.
## Too Much Supervision
Some, especially older, children from poly families expressed frustration at the degree of supervision they received from the numerous adults in their lives. Not only did such surveillance hamper their plans to sneak out at night or skip school but also children found that it was extremely difficult to maintain a coherent lie when dealing with multiple parents. Marcus Amore found that multiple adults in the same household made things more difficult when he attempted to lie to his parents. While he felt that "I probably didn't even need to lie to them, they were willing to allow me to do a lot of things as it was," Marcus lied to his parents about walking to the local strip mall with his friends, but without adult supervision. Marcus explained:
> When mom asked me what I did that afternoon I remembered to say that I got dropped off after practice and did my homework. Later, dad asked where I had gotten the [chewing] gum and I said I went to the grocery store with mom the other day and got some gum. Jim was sitting right there and said, "That's weird, I don't remember you coming with us." Mom, Dad, and Jim talked about it and could not remember when or where I would have gotten gum, so they asked me again and I tried to lie but it fell apart right away. They saw right through me, but I guess at eight or ten or whatever I was I was a pretty bad liar and having the three of them to cross check stories made it even worse. So it made it hard to lie, but it wasn't that big of a deal cause I hardly ever lied anyway, I just didn't need to.
Multiple adults providing supervision for children makes it more difficult for those children to do the kinds of things children do when adults are not actively watching them.
Thinking back on her "rebellious phase," Elise Heartland reported:
> Sometimes it was a huge drag—I couldn't get away with _anything_. I mean, anything! The 'rents [her mother, father, and their partners] were always around, so if I tried to ditch school or pretend I went to practice [for the high school color guard] but went to hang out with my friends instead, someone would always find out. And if I tried to say I was somewhere else, somewhere I wasn't really, they would poke holes in my story. I would tell mom one thing and try to remember what I had said to her when Adam (Elise's parents' boyfriend) asked me how my day was, things like that. And they would talk to each other, so if I couldn't keep my story straight they would figure it out pretty quick. So yeah, that part sucked, but in other ways it was good to have so many people around, it kept me from getting into more trouble in high school.
Elise found the amount of adult attention she received to be both positive and negative for her. She liked it when there was always someone to pick her up or make her dinner, but she did not like the degree of supervision that kept her from "getting away with anything."
For Elise Heartland, too much supervision coincided with family complexity and produced an irritating level of adult intervention in her teenaged life. For Elise's father, Sven, he saw multiple adults in the household as an advantage because "Like if Shelly and I are arguing and I am being unreasonable about it, the third person can step in and say 'Sven, you are being unreasonable about this.'" Elise responded:
> Elise: Yeah, but if you are a kid then you can sometimes have three people ganged up against you, telling you that, you know, you are being unreasonable, that really sucks.
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> Shelly: That's an advantage for us.
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> Elise: No, but if that extra person isn't your parent it really just pisses you off.
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> Shelly: I can see that, that is a valid.
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> Elise: (interrupting) I mean I used to fight with Adam a lot, like we used to, like "You don't have anything to say about that." But most of the time he was really nice and like really reasonable and I never had an issue with him, but if it was really serious about like us getting punished or something and then he would say something about it, it would just push me over the edge. It was like, "You don't even live here!" . . . Like when [Adam] would try to change from being a friend to like "I'm going to be on your parents' side now" and I would be like "What the hell?"
While Shelly and Sven saw Adam's additional interaction as helpful to their relationship with each other and their parenting, it proved to be disadvantageous for Elise when it translated to "three people ganged up against you."
In general, adults found the complexities of poly family life most difficult to deal with in terms of their increased vulnerability to authorities who might misunderstand their families and take their children away. Children also found family complexity challenging, especially when dealing with nosy relatives or peers at school who would not be misled with simple misdirection and continued to ask about the family. Like the other difficulties facing polyamorous families, these challenges are similar to the troubles that many families face, rather than being confined only to polyamorous families.
1.
(Goffman 1963)
2.
(Barker 2005; Califia 2000)
3.
(Keres 1994; Wright 2006)
4.
(Dalton 2001; Hequembourg 2007; Klein and Moser 2006; Ridinger 2002)
5.
(Attias 2004; White 2006)
Chapter 9
# Overcoming Obstacles
People in polyamorous families use a variety of strategies for dealing with the disadvantages we discussed in chapter 8. These strategies focus on communication, emotional protection, stigma management, and being socially selective.
## Communication
Unsurprisingly, communication and honesty are chief among the practical polyamorists' tools for navigating their complex family lives. Poly families have frequent discussions, family meetings, smaller group chats, and one-on-one talks. If schedules are tight (as they often are), then the family meeting is a necessity because it can be otherwise virtually impossible to have everyone just wind up at home at the same time, awake, and able to focus on a serious conversation because they have no other pressing responsibilities in the moment. Many use the family calendar as a means of communication, because it specifies who is doing what and when and what people should expect from each other. Time carved out to be together routinely includes heartfelt discussions, and polys' first reaction to any issue is often to discuss it to see what family members think.
The Tree polyaffective triad consciously communicated about anything they could foresee as even potentially becoming problematic. In the hope that they could anticipate and defuse problems before they became bigger issues, the three would talk about their boundaries, feelings, and needs routinely. Such frequent discussion intentionally focused on dealing with issues when they were small and less charged, before they "blew up into big problems" as Leah put it, and this was a common strategy among polys. By being honest about their feelings and needs, addressing small irritations before they become larger, and communicating with each other frequently, the Trees and other polys use communication to check in frequently and keep everything moving smoothly in the relationship.
Shelly and Sven Heartland found communication useful when dealing with the departure of their beloved triad member Adam, whom the Heartland daughters Alice and Elise had come to adore. When Adam left and Alice was very sad, Shelly reported how the family dealt with it:
> We just kept talking to her [Alice] and saying, you know, we are not together in that same way and we still care about him and I am sure he still cares about us but he needed to make a life down there in [a larger town about sixty miles away] and he just can't really come up here any more. I just tried to explain as best we could.
Families used conversations to make sure the children felt that they could ask questions and express their feelings, so that the kids knew it was OK to miss someone. The pervasive expectation of honesty meant that children often felt comfortable asking their parents questions and expected a degree of candor that children in monogamous families might not have dared to ask.
## Emotional Protection
Parents' efforts to protect their children from the potential emotional pitfalls of polyamorous relationships took a number of forms, including carefully screening potential partners, training children in the realities of life, and prioritizing their relationships with their children above their romantic relationships. Poly people who are not parents also talked about using caution when dating parents, or simply not dating anyone with children if they are not prepared to establish a relationship with the child.
### Screening Potential Partners
To counter the potential for their children to be hurt when partners leave, many poly parents use extreme caution when introducing new people to their children. In addition to taking the time to get to know potential partners through conversations, interactions, and observations, practiced polyamorists often check in to potential partners' relationship histories and ask other community members what they know of that person. Usually parents only meet potential or new partners outside of the home for extended periods of time until the parents are certain they can confidently introduce the new people to their children. Even once they meet the children, new partners are most often understood as friends and routinely blend in to the family's larger friendship network, especially at first. Usually the relationship only stands out to the children as significant beyond ordinary friendship if it lasts over a long period of time and reaches a level of significant or sustained intensity of emotion and interaction. In other words, polyamorists are not only quite selective about whom they date but also when and how they introduce people to their children.
Once they have introduced their partners, poly folks often encourage long-term partners to establish independent relationships with the children, relationships that occasionally outlast the sexual connections among the adults. Emmanuella Ruiz required her partners to establish a lifelong commitment to her children prior to being considered part of the family unit:
> I bring people into my life and there's a point at which I allow them the honor of being part of my family and I have great expectations from that and I expect the expectations of my children not to be dashed within that. So people are not allowed to come and go . . . I tell people if you get close to my kid, stay close to my kid. If you make a promise to my kid, it'd better be forever. So I'm very cautious about telling my children who is family and who is not. This person is mama's boyfriend and this person is family. So they know who they can trust . . . It's been going on for over a decade and it's working for all of them.
Emmanuella's caution and high expectations appeared to be effective in retaining emotional ties and ongoing supportive relationships among her extended chosen family.
### Training in Real Life
Poly parents also talk about the importance of teaching their children how to deal with the end of relationships as a valuable component of emotional protection. Rather than make futile attempts to avoid loved ones' departure, these parents seek to protect their children's emotional well-being by teaching them how to deal with loss as an inevitable feature of life. In discussing the impact of her divorce on her children, Melody Lupine commented:
> It happens in everyone's life. The kids are learning that people come and go, but they're okay. And that it does not have to be this big thing . . . there's sadness but there's also joy when people come in or come back and that it can fluctuate, when people leave it does not mean forever.
Mark Coach similarly used communication as a tool to help his children deal with complex life issues, including those related to loss or polyamory.
> I believe strongly that it is not a parent's job to protect their kids from the world. It is their job to prepare them for the world. To be effective adults in that world. Yes, that means you have to do what you can to insure basic survival and health. But preventing them from experiencing the pains life brings, that is robbing them of an opportunity to learn. It is much better to let them experience pain _and support them through it_. When I was fourteen my maternal grandmother died. My grandparents (from my point of view then) had been married since the beginning of time. They looked like they had the perfect loving, supportive relationship. (In retrospect I had no basis for making that kind of assessment, but I didn't know that then.) And yet, for all that perfection, she died, and he was left alone. That kind of loss, be it because of death or because of natural changes in relationship, is a part of life. I will have to deal with the loss of relationships. My children will have to deal with the loss of relationships. It is better to work through things, to support each other as a family, to experience loss and be supported through it, to learn how to ask for what they need, to learn how to share that pain with others who are being supportive, than to hide them from the pains they will have to deal with later.
Rather than what he felt certain would be a foolish and useless attempt at shielding his children from pain or change, Mark chose to communicate openly, to "experience the loss and be supported through it" as a way to help his children grow into "effective adults in the world." Poly parents express concern that attempting to insulate children from the inevitable loss of relationship that routinely accompanies life would actually be a disservice. Helping children develop the skills to manage loss or transition in many types of relationships, these parents hope, will provide more effective protection.
### Prioritize Children
Poly parents routinely reported taking excruciating care to check out their partners, get to know people over time, meet people outside of the home, and prioritize the children in all things. Parents in the study discussed organizing their social lives and living spaces around their children. Logan Tex considered his eighteen-month-old son Pip and infant in every aspect of his relationships:
> Because I have my kids, I am more committed to Rhiannon [Logan's girlfriend] than I would be otherwise, because of Pip's relationship to her I am less likely to just walk away if things get hard. That being said, I wouldn't keep the relationship because of my son if things were really bad. It is a terrible mistake to stay in a bad relationship for the kids, baggage inherited from my dad.
Logan explained how children suffer in "bitter, awful homes" when their parents are in bad relationships that really should end but instead "hang on to the bitter end" so they can stay together for the children. It would be better to break up, Logan said, because "kids are sensitive to the moods of people around them, and if they grow up in an environment where people are always angry with each other, that would be damaging to them." For Logan and other poly parents (as well as many serial monogamists), consideration for their children looms large in their own relationship choices, whether that means choosing to stay with partner in part to sustain their relationships with children or choosing to leave partners who would negatively affect children either directly or indirectly.
If poly parents found that a relationship was having a negative impact on their child, they often ended the relationship. Claire Morgan, a mother of one daughter, Renate, was firm in her conviction that Renate was the most important relationship in her life: "It's a hard limit I have: Nobody gets between me and my child." The extent of that limit was tested, Claire reported, when her partner of over a year, May, began to get increasingly irritable with Renate and became more controlling with Claire, until:
> About three weeks before everything truly came crashing down around us, May and I went out to dinner, and she demanded that I give her full parental control over my daughter. She insisted that my health couldn't handle it [because I have a chronic illness] and that my daughter was out of control. That was one of those bucket-of-water-in-the-face moments, when you realize that someone is in a different reality than you are. Because, Renate, people love her . . . she's always gotten along well with people of all ages. People have pulled us aside to tell us how well behaved she is . . . But she and May were increasingly butting heads every time they turned around, and May demanded full control over Renate but there was no way in hell . . .
As Claire refused to punish Renate for misbehavior Claire thought May had made up in order to manipulate the situation, May became increasingly sullen until "she wouldn't even acknowledge Renate as a human for over a week." That was the ultimate relationship violation for Claire, who decided to leave as a result of May's mistreatment of Renate. Collaborating with her boyfriend Sylvester to "run interference" and shield Renate from May, Claire immediately began to collect the financial resources that would allow them to move out. May, however, "beat me to it" and kicked the three out of the house they had been helping to pay for but were not on the lease.
Claire was mystified by the whole experience, because she had been quite careful when she established the relationship, and especially before she moved in with May. Claire said:
> This all happened despite the fact that we were very careful. I had known her for several years and we had spent more than a year very consciously working towards blending our households and making her a part of our family. Even so, we obviously didn't know enough about her and didn't realize she had a narcissistic personality disorder. It wasn't until after we moved in together that things got increasingly strained . . . I still don't know what I could have done to avoid that. That could have happened in a monogamous relationship. I knew her in the community we shared, I'd known her for seven years before I moved in with her. There wasn't any kind of drama in our relationship. The only clue I had was that she talked a lot about an ex-partner who was supposedly stalking her. This person had left town, and I didn't know anyone who knew her or anything. Now I know, after doing some legal research, that she had basically done the same thing to another couple she lived with ten years before. She was arrested for domestic violence in that case, but she wasn't convicted so it wouldn't have come up even if I had done a criminal background check. Which I never thought I had to do. I do now, though, just to be sure. I'm very cautious, so much so that it kind of creeps me out. I don't like feeling this suspicious.
Similar to other poly parents, Claire normalized her relationship difficulties by equating them with the same kinds of difficulties monogamous families face and pointing out how poly families are better because they provide more resources—in this case, protection:
> Like I said, it could have happened to people in a monogamous relationship too, I don't think it had anything to do with polyamory. Actually, having Sylvester helped us out a lot, he was something of a protection against May. I think she would have been worse and lot more likely to get more physical without him.
Considering the rarity of polyamorous families and the frequency of intimate partner violence (IPV), obviously most IPV happens between heterosexual, (ostensibly) monogamous people because they are the bulk of the population. While IPV is obviously a problem in some poly relationships, polys can use their contacts with multiple partners and social networks to ameliorate some of its impacts when it does happen. Because isolation is so often linked with one partner's ability to control another and perpetuate IPV,[1] people in poly relationships might be less likely to experience IPV because they are more connected to more people and thus potentially much harder to isolate. Multiple partners provide additional social resources that can help polys leave abusive relationships.
Claire's suggestion that Sylvester's presence kept May from escalating into violence deserves some additional consideration. It makes sense that people might hesitate to abuse their partners—something society has deemed deeply undesirable—in front of other people simply for the well-deserved shame of it. The presence of another person may not only embarrass the potential abuser but also may shift the balance of power if that person might be able to physically intervene on behalf of the partner who is being attacked. In case caution and slow negotiations fail to produce a reliably safe home environment, the strategies poly people use to deal with life events also have the potential to be benefits as well.
### Nonparental Partners Caution
Poly parents were not the only ones who took great care with family interactions, and people who were partnered with parents routinely mentioned carefully monitoring or curtailing their relationships with poly parents in order to protect the children of those parents. After becoming close to Annabelle, Evelyn and Mark Coach's daughter, Kristine decided that she would no longer date parents:
> It's just too painful for me, for them [Evelyn and Mark], for the girls, especially Annabelle. I tried not to get too involved, but I just fell in love with Annabelle, she's a really special spirit. Then I tried to stay for her and for them, but that didn't work either for a variety of reasons. So now we're split and I hardly ever see Annabelle any more. When I do it's nice for me, I have no hard feelings towards Mark or Evelyn and I love to see Annabelle, but I can tell that they [Evelyn and Mark] are still tense around me so it is a little weird still. Yeah, I don't come over as much because of that, and because of that I have given up dating parents, at least for a while.
In order to protect her own and Annabelle's emotions, Kristine limited the time she spent with the Coach family, and she anticipated avoiding poly relationships with parents for the foreseeable future.
Like Kristine, Drew, a white man I spoke with at a poly potluck who appeared to be in his early to mid-twenties, told me he was not interested in dating people with children:
> I'm too young for that. No one in my group has kids, so it's not that big of a deal, not like it's hard to avoid, but even if they were around I think I'd steer clear because I just don't even wanna start with that yet. Monogamous, poly, whatever, I'm just not ready to be that guy yet with anybody and it would just make dating so much more, uh, yaaaaaaa [claw hands] or something, like a whole nother person depends on this too. I'm just not ready for that at all. I'll definitely have kids later, but for now I really wanna, you know, check out what's out there.
In Drew's case, his youth meant he was not ready to take on an even quasi-parental role in any relationship, regardless of its level of sexual exclusivity.
Another way nonparent partners tried to protect the children in their lives was to attempt to foresee and avoid potential pitfalls. Some poly people were painfully aware of the potential for their attentions to the children of their partners to be misinterpreted and were hypervigilant in their attempts to make sure it was clear that they were not interacting sexually with the children. When Heather began to grow out of the small-child phase, James Majek reported that he was very careful to be excruciatingly appropriate with her:
> I see myself as a very important adult in her [Heather's] life and I'm very careful. For example, simple things like, we've talked very early on—Morgan, Clark, and I—it's been almost four years back now so Heather would have been six years old. I said I would not give her a bath, not because anything is going to happen, but because she might say the wrong thing to someone like "Oh yeah, that's James, he gave me a bath last night." Very, very careful; I will not give her a bath, and if I step out of the shower and she's around I'm putting on a towel because I don't want anything to screw this up.
James and other poly people in families with children—painfully aware of the vulnerable situation he and the Majeks occupied as polyamorous people with children—were extremely careful to prevent any misinterpretations of their interactions with the children. Aware that any misstep could "screw this up," James took special care to avoid even the appearance of impropriety.
## Stigma Management
Stigma threatens poly families from a variety of sources, among them adults' and children's peers; legal, medical, and educational institutions; and the parents of the children's friends. Poly people's strategies for stigma management include honesty, rejection of the stigma, normalizing the situation, and shifting stigma through education and patience.
### Honesty
Logan Tex explicitly connected honesty and parenting with rejecting stigma: "Hiding our life would teach our kids that even close people are not what they seem, or that feeling shame for being who you are is appropriate somehow." By demonstrating self-acceptance and trustworthiness, Logan hoped to undermine the stigma associated with polyamory and provide his children with positive alternatives to counter any negative self-concept they might develop in reaction to conventional social expectations.
Honesty also reinforces the highly prized emotional intimacy between parents and children, an intimacy that parents use to shield their children from potential negative impacts of stigma. Many poly parents reason that, if they are consistently truthful, the children will trust them. Jonathan, a white father of three daughters in his mid-forties, believed that
> if I want them to deal in a forthright way with me, and everyone else in their lives, then I have got to demonstrate integrity by telling them the truth. It is an important thing, as a father, to be able to talk to them as much as they will talk to me. To let them be as much of who they are and love them for it, and show them who I am too.
For Jonathan, candid self-revelation is a marker of integrity, a way to establish trust with his beloved children, and the key to emotionally intimate relationships in which everyone was allowed to be (ideally) "as much of who they are" as possible. Rebuffing stigma, these parents offer their children an alternative view, based on a loving, authentic family with integrity. Families thus become havens of acceptance and sources of support, providing members with intimacy and positive role models to combat the harmful effects of stigma.
### Rejection of Stigma
One of the most common strategies I found among poly families was the rejection of stigma, or refusing to identify with the perceived negative judgments of conventional society. Characteristic of this tendency, Tyler Warren—a white male high school student—refused to internalize negative social judgments about his family. When I asked him how things went when he interacted with peer's parents or other authority figures, Tyler replied that it generally went quite well, and that most of the time he had no need to navigate adults' negative assessments of his polyamorous family. Speaking of "regular people" who might negatively judge polyamory as immoral or somehow wrong, Tyler questioned the legitimacy of other people's judgments about his family, actions, or beliefs.
> You prove it to me how it's wrong. I'm doing fine, I don't need to prove anything to you, you're the one with the problem. If you have a problem with me or my family, show me exactly how it is wrong. It's up to you. Because I look around and I see love, I see caring, I see people living together pretty happily. Most of the time (laughing). What's wrong with that? If you have a problem with it, that shit's on you, not me. Innocent until proven guilty, isn't it?
Rather than taking on the shame he saw conventional society trying to project on him and his family, Tyler refused to absorb the stigma that others may "throw at me, screw them! Who are they to say? They lie all the time—like they can judge." Zane Lupine similarly rejected conventional society's judgments about his family:
> I've always been like, if they have a problem with that, that's their problem. I'm not going to hide it just so they can be comfortable. That's who we are. It's not something to hide really. It's fine with me. If it's not fine with them, then we shouldn't hang out.
In addition to defending his family from stigma, Zane used another common strategy I observed among youth in polyamorous families: normalizing the situation.
### Normalizing the Situation
The majority of the time, life in poly families is boringly mundane, composed of homework, dinner dishes, and mowing the lawn. In such cases, it is not difficult to normalize the polyamorous family because the daily details of family life are normal in that they are the same things every other family does. Small children reflexively normalized the situation, meaning that their families seemed to them to be the de facto norm. True to their developmental stages, the young children between five and eight years old generally did not problematize their family forms but rather took them for granted as the norm. Older children were more likely to recognize the differences between their families and the more conventional families of their mainstream peers. For instance, Zane Lupine affirmed that his polyamorous family had been quite good for him.
> I think I've turned out fine, so I'd just use myself as an example of a successful relationship. Anything can go wrong in any relationship. Anything can go wrong in any raising of a child. If you have two persons in a relationship, that doesn't mean they're going to have a good childhood either. You can't guarantee it's going to be good for the kids, but you can't say it's going to be any worse.
In a way, Tyler and Zane's mutual rejection of stigma from conventional sources is simply age appropriate—it is a developmental hallmark of teenage years to question elders' assumptions and rebel against established norms. Both young men also grew up involved in poly communities, and their shared disdain for the judgment of conventional society was also influenced by their lifelong association with poly communities in which their families' style was celebrated and supported. Both spent large portions of their youth socializing with other poly families, where they did not have to hesitate or consider how to introduce their multiple parents. Their normalcy was taken for granted, and Zane and Tyler were able to compare their mundane poly family lives with the hellish tableau conservatives envision for multiple-partner families.[2] In normalizing their familial experiences, poly family members like Tyler and Zane engaged in what Pallotta-Chiarolli termed "polluting," in that they challenged the legitimacy of conventional or monogamous relationships "through noncompliance, personal agency, outing, resistance, and politicization by polluting the school world with one's bisexual and polyamorous existence."[3]
A very popular strategy among polyamorists wishing to reject stigma is to point out that the disadvantages associated with polyamory are not unique, and that monogamous relationships experience the same kinds of jealousies and complexities, only with the added layer of lying about it. In monogamous families parents argue, blended siblings crowd each other and have complex jealousies or rivalries, and people who used to be lovers struggle (sometimes unsuccessfully) to figure out how to coparent. Most significantly, divorced parents involved in serial monogamous relationships have similar issues when people they are dating build relationships with their children and then leave. Logan Tex summarized a common poly response:
> We don't tell single parents OK, that's it, no more dating for you ever. But we do tell them to be careful, don't just invite people into your house and then kick them out again. But the model of denying yourself everything for the kid is not a good option either. Single parents, if they don't get to date to find a relationship, that is a sacrifice. Many humans look for a partnership with people, and to say that because you have kids you should not look for that or you will end up as a damaged parent. Because they can't celebrate their full selves, they are not going to be the best parent . . . the better place you are in personally, the better for you as a parent, and the better for the kid.
Logan not only normalized parents who date by pointing out the high number of single parents that society allows to date but also went a step beyond to imply that a fulfilled parent is superior to the self-denying, "damaged" parents who deny themselves romantic relationships and "can't celebrate their full selves."
While it is true that serial monogamous relationships experience disruption and single parents who date monogamously often break up with one partner and date another, what remains unclear is whether or not those things happen more often or more intensely with polyamorous relationships. Unfortunately, I am unable to answer that, and to date there are no statistics on the longevity of polyamorous relationships. As I have discussed throughout the book, my findings show both substantial romantic turnover (sexual relationships come and go) and significant relationship consistency (people remain connected outside of a sexual context) among poly folks.
### Shifting Stigma
Some poly families were able to endure the stigma they faced calmly enough to educate the people who were stigmatizing them. For instance, Melody Lupine (Zane's mom) sought family counseling to deal with issues in her relationship with her then-husband Cristof. Initially the therapist was
> so judgmental. Oh, you know, "She [Melody] wants out of the relationship so she's getting involved with this other man and she's just prolonging it." And we're like, "No no no you don't understand. We're doing this thing called polyamory blah blah blah," and he's like, "Well, how about this other guy, is he willing to come in?" And we were like "Yeah, he's the one who suggested we come here." And he was like "What? OK, well, bring him in." He sat with all three of us and he was like, "Hmmm," he was baffled, literally scratching his head. "Well, bring in the kids" because he thought he would find pathology in the kids. So he interviewed the kids separately and then together and finally came back to the adults and he said, "You guys are on to something here and I want to learn everything I can about it." And he said, "If you work with me I'll work with you." And I thought that was huge of him and we, I ended up working with him for five years and he was such a wonderful man.
By reacting calmly and taking the time to teach their family counselor, Melody's family was able to navigate the initial stigma, neutralize it with education, and get the access to the counseling they needed. Their counselor gained a deeper understanding of polyamory and helped Melody to discover some profound personal revelations.
## Surrounded by Support
Many poly people are extremely selective about their friends and limit their social circles to those who will, if not actively approve of, at least accept their polyamorous families. In some of the families' social settings, their romantic configuration was irrelevant. Nonfriend classmates at school or colleagues at work, acquaintances, and random people the family members interact with in public settings do not need to know the details of poly family life and are routinely left to make sense of the poly family on their own. People with closer emotional and social ties are more likely to require (or be viewed as worthy of) an explanation, and poly family members tailor their interactions to the person and the setting. Deciding whom to tell and whom not to tell, people in poly families often blur their familial relationships with some people and explain them to varying degrees with others. In this section, we discuss the process of being socially selective in reference to stealth and secrecy, selective disclosure, and intentional socializing.
### Stealth and Secrecy
Remaining closeted as a member of a poly family, or what some call passing,[4] was not difficult for the majority of the poly people I interviewed. Over and over I heard that, unless the person in question actively pointed out the fact that they were involved in a polyamorous family, others would simply make assumptions that explained the presence of various people in their lives and did not rely on polyamory to do so. As Adam Hadaway and Zina Campo's experiences illustrate, simply refusing to answer questions in some settings was all it took to derail peers' questions regarding family forms. In other cases, peers would not even know to ask questions because the poly family members would never appear at school, work, or other social settings that would possibly require explanation.
Dave Amore used a "don't ask, don't tell" policy with his former girlfriend Annabeth's parents, who would most likely not have approved of Dave's poly family or the quad Dave and Annabeth had previously established with their friends.
> Annabeth grew up in a Catholic family, though they're not necessarily practicing Catholics. She's not much of a person to believe in many religions, I think she's more agnostic than anything else. Her family had no idea about the quad, although I actually know her dad quite well and her stepmother. They don't know that I come from a poly family or that Annabeth and I have been in a poly relationship. It's never really come up in discussion; it's never been an issue. Which is ironic, because I've been on several shows and Annabeth's actually told her parents about them. So, they probably "know" but they choose to ignore it. They've never discussed it with me, or with Annabeth as far as I know.
In this instance, it was not difficult for Dave and Annabeth to obscure the polyamorous nature of Dave's family, or Annabeth's relationship with Dave and the other quad members, even though Dave had appeared on several national television talk shows discussing his poly family. Annabeth's family tacitly agreed to ignore the things they did know and avoid asking questions or finding out any additional information.
Children in poly families intentionally seek out peers with whom they can safely be honest about their families. In many instances that meant peers who were adopted, had parents who were divorced, or had demonstrated their ability to be open-minded. Mina Amore said that she was honest about her family with her good friend, but not the friend's family:
> I guess I'm not necessarily what you would call normal, but who cares? Normal is boring. Some of my friends who seem really normal are actually super cool. My best friend, she's not normal either cause she's a Christian and her whole family really live like Christians, follow _all_ of the rules that lots of people who call themselves Christian don't really follow. And we're Pagans, but she doesn't care. Her mom and Grandpa would _freak_ if they knew about the whole Pagan poly thing—I mean _lose it_! But she doesn't care at all. We just do our own thing and are cool with each other. We're really different [from each other], but we don't care what other people think so we are kinda the same too.
For Mina and many other adolescents in poly families, normalcy appears to be overrated. Even things that may appear to be the norm, like Christianity, can become unconventional when practiced with fervor. The stigma that would otherwise accompany being outside of social norms does not necessarily translate as a disadvantage to those children (some tweens and many teens) who do not value conformity and who seek out peers with a similar disregard of convention.
Parents usually allow their children to use their own discretion when it comes to disclosing information about the poly family to the children's peers and other members of their social circles, at least to the greatest degree possible. Younger children who are possibly even unaware that they are members of poly families rarely have to consider issues related to disclosure, but elementary schoolers, tweens, and teens all have to make decisions about how to talk about their families to other people. Commenting on her children's social lives, Sara Bayside said:
> We let them take the lead, however they want to talk about it, they do. Their friends are pretty sophisticated, so I don't think it is generally a big issue for them. But we just keep our mouths shut about it, unless they [the Bayside daughters] ask us to say something, it just doesn't even generally come up that we're even in a position to deliver that kind of information to their friends. In general they run their own social lives, at seventeen and nineteen we are way past the play-date stage.
The Baysides' parenting strategy—allow their children as much freedom as possible within age-appropriate limits and provide backup when necessary—proved workable for them in many situations, poly related or not. That same strategy is popular among other poly parents as well, which is no surprise given the community emphasis on freedom and age-appropriate self-responsibility.
### Intentional Socializing
Many poly families, especially those who live near large urban areas, intentionally socialize with other poly families to find friends and create community settings in which their families are unremarkable. As discussed in chapter 3, poly people intentionally seek community in order to get advice and support, look for partners, and find role models for themselves and their children who help to normalize poly families. Children told me they met other kids from poly families at poly campouts, movie nights, support group meetings, and dinner parties. By socializing with the same families over the years, these children are able to establish friendships with peers who do not judge their families and who immediately understand the relationships among the adults.
Even outside of poly social settings, children from poly families intentionally seek open-minded peers with whom they can become friends. Zina Campo reported that "I don't have to deal with it that much because I live out here on the farm with my dad, and mom and Blake live in [nearby town] most of the time, so it just doesn't come up most of the time." When I asked her if she had a hard time making friends while keeping the secret of her family's unconventional relationships, Zina responded: "Not really. Not at all. If they can't handle it I don't want to be their friend anyway." Zina and the majority of her peers in poly families seemed to easily control the flow of information with their peers and peers' parents, and Zina commented that "Only the cool ones get to know."
Zina devised a strategy that allowed her to talk about her friends from the poly community by saying, "Oh, those are just my mother's weird friends, they do whatever they want." Leaving the precise kind of weirdness vague worked well because "when I'm talking to my friends if an uncomfortable question ever comes up, that's one of the beauties of being twelve and thirteen, any bit of the conversation can just go away in like thirty seconds if you change the subject, so yeah, I don't have to deal with it hardly ever." This strategy allowed Zina to both maintain her privacy and respect her father's wishes for discretion with her "pool friends" from the local swim team that her father coached, as well a way to talk about the interesting things she did during the weekends she spent in town with her mother and their shared social network.
1.
(Johnson 2008)
2.
(Associated Press April 23, 2003; Kurtz 2003a, 2003b; Lithwick 2004)
3.
(Pallotta-Chiarolli 2010: 26)
4.
(Pallotta-Chiarolli 2010)
# Conclusion
Implications for Serial Monogamy and Policies
Given the increasing diversity of families and the rising popularity of multiple-partner relationships,[1] as a society we must come to grips with a wider range of family formations and different sexual realities. Understanding polyamorous families provides us insight into how a range of families adapt to shifting social conditions. Some popular media pundits talk about poly families as if they are the sign of the death of civilization,[2] but they are mistaken. Rather than serving as the point of no return down the slippery slope of social decline to complete moral chaos, polyamorous families demonstrate an innovative flexibility that allows them to flourish in a complicated family style.[3] Society itself has become increasingly complex, with people living longer and more diverse lives—relationships must keep up with the changing social landscape. In this complex age, polyamorists have two very broad choices: 1) either learn the relationship skills, communication techniques, and at least a degree of self-awareness that it takes to have an ongoing poly relationship, or 2) leave the lifestyle. The learning process is ongoing, and the polys who choose option one are constantly doing just that—choosing and rechoosing, learning and practicing, never finished tinkering with ever finer points of relationship maintenance and emotional intimacy. Persistent polyamorists spend so much time working on, talking about, and perfecting their relationship expertise that they have developed a range of useful skills that can be helpful to serial monogamists.
While poly relationships are not for everyone, their strategic innovations can be, because they offer insight into the unique ways people in serial monogamous relationships can deal with their own blended families that mix multiple parents and children from past and current relationships. The tactics poly families have created, tested, refined, and practiced can work for monogamous families who divorce, and their experiences coparenting can illuminate how blended families of all types might deal with multiple parents, regardless of how or if they are sexually connected. This ability to adapt becomes increasingly important as divorce dilutes the expectation of permanent connections with spouses and in-laws, while simultaneously creating multiple-parent blended families.
## Stability and Flexibility
Families change. They have for the entire history of the human race, and they continue to do so. Contemporary shifts in families are simply a continuation (and acceleration) of endless variation, not the abrupt shift from a previously homogenous and unchanging form of family that some conservatives claim.[4] While the current prevalence of divorce and single parenthood appear unprecedented, wars, accidents, and pestilence have left many single parents across history.[5] Death used to end marriages when life spans were shorter, and longer life spans means more time for changes.[6] In an age when change is rapid and pervasive, poly families are especially flexible and resilient, adept at surfing the slippery and unpredictable currents of postmodern family life. Most significantly, these families have developed ways to stay connected to significant others among the fluidity, and they continue relationships even as they shift form over time. By allowing their relationships to flex with changing circumstances, polys are able to preserve ongoing links with significant others and coparents.
Poly relationships may not last in the traditional sense of permanently staying in the same form. Instead, some poly relationships appear more durable than many monogamous relationships because their flexibility allows them to meet shifting needs over time in a way that monogamous relationships—with their abundant norms and requirements of sexual fidelity—find more challenging. While the familiar and well-explored structure of monogamy can promote a comforting predictability, it can also limit the choices available to people in monogamous relationships. This is not to say that there are no relationship innovators among monogamous people—feminists and others have a long history of creating alternative definitions that provide meanings outside of a patriarchal framework.[7] The scarcity of role models frees people in polyamorous relationships to craft new relationships and innovate alternative roles that better suit their life circumstance in the moment. Polyamorous relationships provide the flexible and plentiful relationship choices that conventional monogamous relationships, with their firmly defined roles and well-explored models, cannot.
Not only are poly families able to provide continuity across changing circumstances, they also provide an ethical foundation to help guide their actions. In other words, their fluidity does not mean they are lacking a moral compass or that they are consequently adrift in a sea of social chaos: These families have clear guidelines that prioritize honesty, compassion, freedom, and self-responsibility, forming an ethical framework to guide interactions and decision making. These foundational ideas provide stability for children and adults. Unconventional and frequently shifting, reliant on ethics rather than a conventional or religious morality, poly families provide members with significant stability while they flex to adapt to changing life circumstances. This flexibility and willingness to explore alternatives makes some poly families uniquely resilient.
Family resilience researchers emphasize positive communication skills and the cohesion of family network connections as key elements that help families effectively weather crises. With multiple partners, intricate schedules, and households that blend numerous parents and children, polyamorous families face tremendous logistical and emotional complexities as they deal with the various challenges associated with any family life, magnified both by the number of people involved and the potential for discrimination based on their status as sexual minorities. Poly families' intentional expansion of familial roles and the self-examination required by such deliberate innovation makes family members focus on cultivating the specific competencies required to navigate complex familial circumstances and interactions. If resilience is "normal development under difficult circumstances,"[8] then poly families provide a unique perspective on potentially difficult (or at minimum complex) circumstances. Strategies that allow families to adapt to changing circumstances are quite effective in contemporary society and would certainly prove useful to help more conventional blended families survive life in such a rapidly changing society.
### Strategies for Monogamous Resilience
That is not to say that poly families are free of disadvantages—in fact they face the same disadvantages that other families face. Stigma, breakups, drama, teenage children who freak out, spouses who die—these are issues associated with contemporary family life. The advantages, however, significantly outweigh the disadvantages for most poly people who continue to engage in poly relationships. While the disadvantages are generally those associated with being in a blended family and occur culturewide, the advantages are specifically polyamorous. These advantages have helped long-term poly families to become uniquely resilient.
Poly people have established several strategies that can help people in families blended through serial monogamy to navigate multiple parentage and ongoing relationships with former partners. Divorce does not have to be so hellishly destructive, and if people are able to interact compassionately then they can have more resilient relationships that nurture positive coparenting even after they are no longer involved sexually.
## Poly Strategies for Success in Serial-Monogamous Relationships
* Be honest early and often so you can trust each other. Communicate freely and frequently, and be honest with each other. Without those things, relationships will break down due to lack of trust, suspicion, and misunderstanding.
* Be willing to negotiate new ways to be; try alternatives. If the way things are going is not working for the relationship or family as a whole, be willing to consider doing things differently, and not just differently from how you have usually done them, but differently even than most people do them. Be creative, and brainstorm a wide range of alternatives to make things better for family members and accommodate changing circumstances.
* Don't give up too soon. Put the best effort you can into making things work, and try your hardest. Don't give up, or as one respondent told me, "Don't mistake a bad situation or a rocky phase for a bad relationship." Consider alternatives you have not tried yet, and keep trying new things if what you are doing doesn't work. Find people you trust and ask for their support and advice. Get counseling. Give it your best effort, and assume the people who love you are trying everything they can think of to help things work out.
* Don't stay too long. If you have tried everything and the relationship still doesn't work in its current form, change it or end it before you hate each other. Do this before anyone lies to or cheats on each other, because it is much easier to trust someone and continue to coparent with them if they have not mistreated, lied to, or cheated on you, or if you have not done those things to them. It is important to change the relationship and end that phase in such a way that you can still trust each other and work together in the future.
* Redefine success. Monogamy no longer means what it used to mean. Even though most people who marry vow to be together "until death do we part," the popularity of divorce means that for many people that does not happen. These families do not have to be "broken homes"—outcomes for children and adults depend on how people handle themselves during and after a divorce much more so than the divorce itself. If you can successfully coparent and amicably attend family functions with ex and current lovers who are coparents, then that is success. Simply staying together is not necessarily success—it is the tone of the relationship that determines the degree of success, not just whether people remain in sexual contact with each other (or at least do not have sex with others) over the years. Success can be defined as meeting peoples' needs for a specific period of time. In such a rapidly shifting society, it is inevitable that things will change, and it does not mean people have failed. If people are able to accept change and handle transitions ethically, they will still be able to trust each other and coparent.
* Prioritize the children. Adults who become emotionally intimate with children should distinguish their parental relationships with their children from their romantic or emotional relationship with their coparent. All parents should consider and act on the basis of the children's best interest, and if the romantic relationship between the coparents changes or dissolves, then the coparents must prioritize their relationship with the child over any anger or desire for revenge they may feel. Ending a sexual phase of a relationship with a coparent does not mean that the relationship with the child must end or even suffer. In other words, adults' relationships to children should not depend on the adults' sexual interactions—adults can commit to cooperative coparenting without having a sexual relationship.
* Prepare children for the reality of postmodern family life. Many things in this society are in a constant state of flux, and learning to adjust to shifting circumstances is an important life skill. Be very slow and careful about introducing new people to children, do so in an age-appropriate manner, and clarify the status of the new person. (Is this person family or not? What should children expect from this person?) Let children know that people will come in and out of their lives—that is reality. Teaching children how to deal with inevitable fluidity is more effective than mourning the fact that relationships are impermanent. Help children to actively construct their own families, independent of adult whims. Provide them with the transportation, emotional, and financial support that enables them to see people that the children define as important—even if it makes the adults who used to have sexual relationships uncomfortable—so that children and young adults can establish lasting relationships with support structures outside of their parents' sexual interactions.
Underlying all of these strategies is the assumption that partners are negotiating as equals, which means women must have a framework that allows them access to economic independence, if not actual financial independence in the moment.
## Polyaffectivity
Polyamory, the most flamboyant version of poly identity, is explicitly sexual in that it centers on being open to multiple sexual partners. A quieter version of poly identity, polyaffectivity appears to be more durable and flexible—able to supersede, coexist with, and outlast sexual interactions. Relationships that have such a multitude of options for interaction and define emotional intimacy as more significant than sexual intimacy allow poly people to craft a wide selection of possible outcomes. Most importantly, they allow polys to establish significant nonsexual relationships and maintain relationships over time even if their sexual content changes.
This expanded choice has two primary implications for poly relationships: graceful endings and extended connections between adults. Once a relationship can end without someone being at fault, the social mandate for couples to stay together and fixed in the same relational form at all costs can relax. As stigma subsides, the resulting decrease in shame and blame simultaneously diminishes the need for previous lovers to stay together until they have exhausted their patience and sympathy for each other, and possibly lied to or betrayed each other in the process. Once it becomes clear that the relationship no longer meets participants' needs or works for people who have grown apart, accepting the change and shifting to accommodate new realities can contribute to more graceful endings and transitions. If adults are able to amicably end one phase of their relationship, it increases the chances they will be able to make the transition to a new phase characterized by continued connection, communication, and cooperation. As one poly person stated, "Don't drag it out until the bitter end, disemboweling each other along the way. Split up while you can still be friends, _before_ anybody does something horrible they will regret later."
A central component of polyaffectivity is removing sexuality as the hallmark of "real" intimacy. If sexuality can be shared among more than two people, and emotional intimacy can outlast or supersede sexual intimacy, then nonsexual relationships can take on the degree of importance usually reserved for sexual or mated relationships. That is, friends and chosen family members can be as _or more_ important than a spouse or sexual mate. This extrasexual allegiance is fundamental to my concept of polyaffectivity and to the durability of these relationships that can flex and last over time.
Expanding important adult relationships beyond sexual confines, whether they be former sexual partners or polyaffective partners with whom there was never sexual interaction, provides people with more templates for interaction and choices in how to define relationships. One of the primary reasons to define the end of a relationship as failure is that it negatively impacts children. Bitter interactions among beloved adults are painful for children, and they aggravate the other emotional and financial disadvantages associated with divorce. Children don't care if their parents have sex, and they generally would rather not think about it at all. What matters to children is that they can have all of their parents at holiday and graduation dinners and that everyone is able to interact cordially. Ongoing positive interaction among adults is advantageous for the children in poly (and other) families because it means more support, harmonious family time, shared resources, and less money spent on lawyers.
This does not mean that no one in poly relationships gets hurt or mistreated in a breakup—poly people lie, betray, and cheat each other just like everyone else. But the existence of alternatives allows for relationships to end in one phase and begin in another, or continue across many versions that may or may not include sexuality. Expanding possible meanings, redefining success, deemphasizing continued parental sexual interaction, and focusing on cooperative coparenting provides options that can be advantageous for parents and children.
### Allows Expanded Family
Polyaffectivity allows families to expand, and most significantly, it allows men more ways to be connected to families. Many families are in crisis; that is real. The predicament is worst for the children whose fathers act as merely sperm donors but provide little or no ongoing emotional, practical, or financial support. The crisis is among couples who attempt to sustain the weight of extremely high personal and social expectations on the narrow base of sexual attraction. It is abundantly clear that simple romantic attraction is inadequate to sustain a family over such long periods with life's complex and shifting circumstances. Expanding the base of family to a wider support, spread among more people who are connected by more than the potentially tenuous bonds of romantic affection, would provide a much more secure foundation for adults and children.
#### Spice
Multiple spouses need not be sexually connected, and indeed it appears as if the requirement that everyone involved must have sex with everyone else in the group (often accompanied by the expectation that everyone will love each other equally) can create unnecessary tension. The most lasting triads appeared to be women who were sexually connected with two men who were not themselves lovers but had significant positive regard for each other. Cowives or sister-wives have been common in many cultures across history and even recently on national television in the United States,[9] but cohusbands or brother-husbands are so rare that even the phrases themselves sound strange. Polyamory creates the opportunity for men to be spice in a way that they generally have not had access to in monogamy or polygamy, which is usually practiced as polygyny (one man with multiple wives) rather than polyandry (one woman with multiple husbands).[10]
#### Otherfathering
Just as revolutionary as allowing men to create emotional relationships outside of the stifling confines of traditional masculinity, otherfathering allows men to remain attached to children with whom they have developed emotional relationships. Although most of the people who bemoan "the crisis of the family" place it in the context of mothers who work for pay outside of the home rather than for free inside the home,[11] feminist scholars view it as a crisis of men in families who see true fatherhood as optional beyond sperm donation, as men sire children and then abandon them. Legislation pursuing "deadbeat dads" has attempted to make the men shoulder some of the financial responsibility for the children they fail to care for by garnishing wages and seizing tax returns. In the United States, we have taken a very punitive approach in our efforts to fix the problem of the absent dad, with much less focus on helping dads be better fathers. In other words, it is pretty clear that the stick approach has limited effectiveness, and I believe we should also try using the carrot as well. I am not saying we should repeal child-support enforcement legislation—on the contrary, we clearly need ways to make selfish or vengeful parents contribute to the children they create. But relying only on punishment and failing to encourage or recognize positive parenting and the attention of other loving adults who want to support children over time is lopsided and short-sighted. By recognizing and supporting the relationships men create with children, society can help them to sustain those connections over time and encourage men to remain connected to families and especially to children who love them.
Social changes have enlarged women's options and opportunities far more in the last sixty years than they have for men. Even though men remain firmly in control of finance, government, law enforcement, and industry in the United States and around the world, women have nonetheless made significant strides toward new opportunities and greatly expanded personal options since the 1950s. The same has not been true for men, and they remain almost as bound by the expectations of traditional masculinity as they were sixty years ago. Not only have their clothes barely changed, but their personal, emotional, and expressive options have not expanded on par with women's. This is due in part to women catching up with the options that men already had, and in part to the rigor mortis that seems to have set in around conventional masculinity. As a society we give lip service to approving of stay-at-home dads and men who are able to cry, but it is a grudging and shallow approval that stops well short of actually valuing gentle, empathetic, and kind men. Contemporary ideals of masculinity in the United States remain firmly rooted in aggression, and it is the chiseled features of the muscular hero able to escape the building moments before it detonates that capture our imaginations, not the devotion of a man to his children against all odds. Movies about relationships are still chick flicks, and movies about men are really about action—as long as something explodes they are in tediously familiar territory.
The problem is that this familiar territory is stifling and cramped, far too limited to accommodate the true range of men that exist and the choices they make. While laws focus on punishing men for their (admittedly grievous) failings and popular culture celebrates a very limited, pale, and muscular version of masculinity, poly men and people like them are quietly expanding the boundaries of what it means to be a man in a family today. Allowing women into positions previously reserved for men has expanded their choices, but it has not made positions traditionally held by women (like those associated with caring for a family, including the cooking and cleaning) become more valuable. As a society we take it for granted that women will care for their children, and we are still amazed and overjoyed when men change a diaper.
Many men in poly families take _real responsibility_ for their children and continue to nurture supportive relationships with children—even once they no longer have sex with the children's mother. That is precisely what society has expected from women all along—that women prioritize their relationships with their children over their sexual relationships with men. Women who fail to do this are severely stigmatized and are branded whores and bad mothers. Applying that same standard to men is revolutionary and worthy of social and legal attention, because these men establish relationships with their children (even children with whom they do not share a genetic link) independent of ongoing sexual relationships with their mothers.
Expanding men's social and legal options can help them stay connected to families and supporting families in general by valuing the functions and parts of family usually viewed as women's work. The more men do what previously was women's work, the more valued it will be. If the real responsibilities of caring for children continue to fall only to women, then too many children will remain in poverty and emotional need. It is only when men are also as deeply responsible for children as are women that the kids will have the benefits of a wide base of support. It would be better for children, and for the many men who are unable or unwilling to live up to the requirements of conventional masculinity, if we could allow men's personal, emotional, and relationship choices to expand as much as we have allowed women's to increase.
## Policy Implications
On the most basic level, public policies should help the people who live in the society that the policies regulate. That means assisting the people who actually exist, rather than bemoaning the fact that they are not the people who used to exist or lawmakers might prefer to exist. At minimum, policies should not hurt families or hinder their attempts to cope with crises. Extending the same marriage benefits to people in same-sex and polyamorous families (and others) as those available to people in heterosexual couples would be a good start in supporting contemporary families. Even though many polyamorists and some lesbians and gays scorn multiple or same-sex marriage, in the interest of full citizenship and equality they still deserve to have the opportunity if they decide they want to marry. Encoding second-class citizenship into marital laws only further alienates already disenfranchised sexual minorities and perpetuates institutionalized homophobia. Worse yet, it hurts the children from those families.
Rather than evidence of decline, polyamorous families are a symptom of a society that is constantly and rapidly changing. It is clear that the singular family model of the monogamous, heterosexual couple linked by romance alone, precariously perched on their unwavering emotional, sexual, and financial investment in each other, does not work for everyone. As numerous divorce studies illustrate, for at the least 40 percent to 50 percent of all marriages that experience a "disruption,"[12] one size no longer fits all, and blended, serial monogamous, same-sex, and polyamorous families are here to stay. Policies and laws should catch up to serve society as it actually is, rather than punishing people who do not fit the white, middle-class, heterosexual, monogamous model of sixty-five years ago. For some people, family forms with broader foundations are more flexible and resilient, better able to meet the complex needs of diverse contemporary lives. It is wiser for some, especially sexual minorities, to invest their long-term emotional and financial care and parenting arrangements in relationships with friends, siblings, or platonic coparents.[13] Policies and laws should legally recognize and serve these families.
Are same-sex or polyamorous marriages truly so terrifyingly powerful that their mere presence could obliterate heterosexual, monogamous marriage? I think not. Marriage based on monogamous, heterosexual couples is and will probably continue to be a very popular form of relationship in some regions of the world. Because the majority of the population is heterosexual,[14] it is clearly better suited to more people than same-sex marriage. Monogamous marriages can also be less complex, offer more plentiful conventional role models, and earn greater social approval, making them more appealing for many people than the potentially more complex and high-maintenance polyamorous family.[15]
### Policy Recommendations
One thing this research makes abundantly clear is that our social and institutional framework can no longer shape itself to serve the legally married heterosexual couple with nondisabled children, a dad who earns enough to support the entire family, and a mom who "doesn't work" (meaning she provides free home and child care). Certainly some people marry and maintain one paid worker and one stay-at-home parent who (at least for a time) leaves the paid workforce in order to take care of the children, but these families are no longer the majority. Families with a full-time earner and a full-time parent are the statistical minority, but current policies are still designed to serve families as if everyone with children has a full-time caretaker at home. Shifting policies to better serve families as they are, rather than how we imagine them to have been decades ago, will not undermine heterosexual, dyadic families—they will still be covered under health insurance policies, inheritance laws, and hospital regulations.
Refocusing laws, regulations, and policies on children, rather than the adults to whom they are related, would far better serve the diverse families that actually live in the United States today. If a child is in poverty, lacks health care, clothes, or food, that child should get assistance regardless of the status of the child's parents. Rather than concerning themselves with who lives in the home or if the child's mother has a marital/sexual/any relationship with the child's father, policies should be based on what is best for the child. The monogamous heterosexual family is no longer the sole family form, and policies need to evolve beyond partisan squabbling about religion and sexuality to providing for children who will be the future of our society, regardless of their parents' sexual relationships or lack thereof.
In addition to focusing policies and laws on children's needs, this research highlights a need to shift laws regulating child custody and adults' relationships. Currently, there can be only two legal parents of any child, and if someone else seeks to become a parent, then one of the other parents must relinquish parental rights prior to the new parent being legally recognized.[16] Rather than making it more difficult for adults to attach to children, policies should allow multiple adults to be legal parents to the same child in order to distribute the extensive emotional, practical, and financial needs associated with raising a child among multiple adults. By officially recognizing multiple parents, policies can help to attach more adults more securely to children. As this research has repeatedly demonstrated, pooling resources allows families to meet a wide variety of children's and adults' needs, and far too often children's needs go unmet.
Obviously, multiple parents can introduce additional complexities, which would require courts and policymakers to create new ways to manage the complexity. I suggest assigning parents to specific portions of the child's life that best fit the specific parent's strengths: educators should be in charge of the children's education, medical professionals should be the chief decision makers when it comes to medical issues, and the primary caretaking parent should have the most say in disciplinary situations. With rights come responsibilities, and these additional parents would be responsible for supporting the children financially as well as emotionally and physically.
Second, policies should expand to recognize multiple levels of relationships among adults. Currently, most laws and regulations recognize a very limited range of relationships—primarily biological relatives or legally married couples. To more effectively support contemporary families, laws should recognize connections between and among significant others like siblings, cohusbands, or lifelong friends who function as family members. By recognizing adults' relationships, laws can support connections between and among adults who are attempting to care for children or each other. In a society with a crumbling social safety net, the more ways in which people can care for each other the better. Polyamorous families provide examples and innovations that can help serial-monogamous families navigate postmodern family life, and as a society we should attend to their innovative ideas that can prove useful for other families as well.
1.
(Anderlini-D'Onofrio 2005; Bergstrand & Sinski 2010; Rubin 2001)
2.
(Kurtz 2003a, 2003b); Stanley Kurtz, "Here Come the Brides : Plural Marriage Is Waiting in the Wings," _Weekly Standard_ 11, no. 15 (December 26, 2005); (Lithwick 2004)
3.
(Goldfeder and Sheff 2013)
4.
Stephanie Coontz, _Marriage, a History: From Obedience to Intimacy, or How Love Conquered Marriage_ (New York: Viking, 2005).
5.
Stephanie Coontz, _The Way We Really Are: Coming to Terms with America's Changing Families_ (New York: Basic Books, 1997).
6.
Stephanie Coontz, _The Way We Never Were: American Families and the Nostalgia Trap_ (New York: Basic Books, 1992; new edition with new introduction, 2000).
7.
Foundational to this genre is Gayle Rubin's 1984 "Thinking Sex: Notes for a Radical Theory of the Politics of Sexuality," in Carole Vance, ed., _Pleasure and Danger: Exploring Female Sexuality_ (New York: Routledge & Kegan Paul). See also the more recent S. Lee, _Erotic Revolutionaries: Black Women, Sexuality, and Popular Culture_ (Lanham, MD: Hamilton Books, 2010).
8.
(Fonagy et al. 1994)
9.
The reality television show _Sister Wives_ , aired on TLC, chronicles the lives of the polygynist Fundamentalist Latter-Day Saints (FLDS, also called Fundamentalist Mormons) Brown family with a husband, four wives, and many children as they attempt to navigate life in the contemporary Western United States.
10.
(Levine & Silk 1997)
11.
(Popenoe 1996)
12.
(Cherlin 2010: 405)
13.
(Muraco 2006; Oswald 2000, 2002; Weston 1991)
14.
(Laumann et al. 1994)
15.
(Sheff 2011)
16.
In some cases, lesbian couples choose to have the egg of one partner extracted, fertilized, and implanted in the other partner, so both women and the man who provided the sperm are biologically related to the child. Specialized cases like that allow for multiple parentage, but the vast majority of localities allow only two legal parents at any point.
# Appendix A
List of Recurring Families
**Amore** : | Louise, Max, and Valentino; sons Dave and Marcus, daughter Mina; Louise's mother JP
---|---
**Ballard** : | Geoff, Hillary, and Jake; daughter Marni, son Milo
**Bayside** : | Calbair, John, and Sara; two grown daughters**
**Bien** : | Warren, Andrew,** Devon,** Estella,** Julie;** daughters Rebecca,** Callie,** and Macy**
**Campo** : | Samuel, Blake, and Lexi; daughter Zina; Lexi's mother Dia and father Brian
**Coach** : | Evelyn, Kristine, Mark, Marshall, and Regan; daughters Annabelle and Martine**
**Cypress** : | Howard and Josephine; son Cole
**Founder:** | Jana, George,* Michelle,* Mike,* and Sam;** son Zachariah*
**Hadaway** : | Gwenyth, Mitch, Phil, and Tammy; ten children including
_Daughters_ : Amelinda; Bunny, Candace, Lily, Michelle, and Zoe
_Sons_ : Adam, Jonathan, Nick, Silas
**Heartland** : | Adam, Richie,** Shelly, and Sven; Daughters Alice, Elise, and Kimber*
**Holstrom** : | Billy, Jack,* Megan,* Sabine,* and Tad;** daughter Ariel,* sons Nolan* and Simon**
**Kenmore** : | Dax, Iris, Natalie, and William; son Dillon
**Lupine** : | Cristof,* Melody, and Quentin;* daughter Joyce, sons Pete* and Zane
**Majek** : | Clark, James, Morgan, and Nash; daughter Heather, sons Brady,* Beck,** and Sebastian
**Mayfield** : | Alicia, Ben,* Monique, and Edward; daughters Josie and Kate*
**Omni** : | Baldwin and Nadia;** daughter Rosaline,** son Wade**
**Phoenix** : | Jared, Summer, and Zack
**Rivers** : | Rebecca; daughters Clarabelle and Eliz
**Ruiz:** | Emmanuella; three grown children
**Starr** : | Joya; son Gideon*
**Taylor** : | Ada and Jasper; daughter Octavia and son Xander
**Tex** : | Logan, Melina,** and Rhiannon;** two children, Pip (17 mos) and infant**
**Tree** : | Bjorn, Gene, and Leah; son Will**
**Warren** : | Dani, Lex, and Mike; son Tyler
**Wyss** : | Albert, Kiyowara, Loretta, Lucia, Patrick, and Fred; daughter Kethry, son Evan*
*Denotes a family member with participant observational data but no interview.
**Denotes a family member with no interview or participant observational data.
Please note that any respondent whose name is bolded and appears at the beginning of the listing, out of alphabetical order, is the sole respondent from that family. In such cases, I list the respondent first to be clear that they are constructing their entire data line, and other family members did not participate. Most of the time I was able to interview and/or observe multiple members of the same family, and in those cases their names appear in alphabetical order.
# Appendix B
Research Methods
While I will be using more academic language in this section, I encourage everyone to read it even if they are not academicians. Please do not be afraid of the language—these are simply words that describe ideas, and readers are capable of grasping these concepts if they allow themselves a bit of time and patience with themselves. Thinking critically about research results presented as "facts" in news media, radio talk shows, and online discussions means being able to put that research in context to understand its potential strengths and weaknesses. Critical thinking also means a more informed, thoughtful, and meaningful public dialogue—something we are clearly lacking in large sections of our cultural sphere. In this book, I have made every opportunity to put the respondents and myself in social context so readers can think critically about the research findings. What follows are more details about the research methods I used in the Polyamorous Family Study.
## The Study
This book is based on a fifteen-year qualitative, ethnographic study of people in polyamorous relationships that came to focus increasingly on families with children. I collected the data in three waves using several different research methods. A very common strategy among ethnographic researchers, I began with _participant observation_ —basically hanging out with poly people, chatting with them, and watching how they interact with each other in their "native" social settings of public meetings, support groups, "meet-ups" in local restaurants, group hikes, movie nights, and "potluck" dinner parties hosted in private homes. I also used _content analysis_ , in which I documented common themes in the poly people's books, magazines, and blogs that I read. Using the Internet allowed me to _read discussions and interact with people on poly list-serves_. One specific list-serve, Polyfamilies, was already focused on what I was studying, and it proved quite responsive to my questions. Routine interaction with that list-serve evolved into a kind of ongoing, slow-motion focus group with some people participating a lot, others commenting occasionally, and many simply following the discussion or "lurking." Some of those conversations are documented in this book, always with the original author's permission. Finally, I used in-depth _interviews_ that began with routine questions about how the person defined polyamory, how they initially got involved, and their past and current relationships. From there, each interview followed whatever the interviewee felt was most important to discuss. People who participated in the second two waves of data collection also filled out demographic _questionnaires_ that asked questions about their age, race, sexual orientation, occupation, (dis)abilities, and gender identity.
### Three Waves of Data Collection
Wave one (1996–2003), my dissertation research, included participant observation with roughly three hundred people and forty in-depth interviews (twenty women, twenty men) with adults who identified as poly, with one sample in the Midwest and another in the California Bay Area. During this phase I attended a wide variety of poly events, especially frequenting a monthly poly women's support group and attending two national conferences sponsored by the Loving More organization. In what turned out later to have a significant impact on the follow-up study, the Institutional Research Board (IRB, a committee at every research university in the United States that oversees a professor's research to make sure it is safe for the people who volunteer to participate) at the University of Colorado decided that the interviewees would be safer from being accidentally exposed as sexual minorities if I did not keep any identifying information on them. The only records the IRB allowed me to keep from the original study were the pseudonyms the interviewees had chosen.
The second wave of data collection (2007–2009) did not begin until four years later, in part because the IRB at Georgia State University discussed it in many of their committee meetings and they required frequent (sometimes opposing) revisions to the research protocol (a document that describes the methods I planned to use and how I would protect respondents' identities) before they would allow me to begin the research again, keeping contact information this time. The greatest obstacle to continuing my research was the IRB's reluctance to allow me to talk to children, so I deleted that component of the research plan and was finally able to meet their requirements.
Once I got IRB approval, I posted messages on as many of the poly websites and list-serves that would allow me, asking people who had participated in the first wave of research to email me so I could follow-up with them. This initial Internet call resulted in seventeen previous respondents emailing me, with fifteen of them consenting to interviews. I also interviewed their partners and one adult child, expanding the sample to include an additional thirty-one adults. Interviews in this phase focused more on managing family life and interacting with social institutions such as schools, medical establishments, and child welfare agencies.
The fact that less than half of the first sample responded again in the follow-up study means that the perspective of those people who did not participate is missing from the research. Because I used poly community connections to look for people (the only way I could think of that was realistically within my budget and allowable under university research guidelines), and the people most likely to stay connected to the poly community are those who still live a poly lifestyle, this research emphasized continued involvement in poly families and community relationships. It is highly likely that some of those people who did not participate dropped out of the poly community because their poly relationships did not work out. With the exception of Melody Lupine, their perspectives are missing from the data.
While I was collecting the second wave of data, I was also revising my research protocol and meeting with the IRB officials at Georgia State University in an effort to get their permission to talk to children—a feat that took three grueling years and resulted in research protective practices that one IRB committee member privately confided he thought were "ludicrous and paranoid." As past academic abuses indicate, it is important to protect people who participate in research, and IRBs across the nation serve a valuable purpose. In this case, however, sex negativity (fear of and disdain for sexuality or anything related to it) clouded the Board's judgment, made the process far more difficult and time-consuming than it needed to be, and significantly hampered my research process and output.
As soon as I received permission to interview children, I began the third wave of data collection (2010–2012), which included twenty-two children and thirty-eight new adults, as well as former respondents participating in follow-up interviews. The third wave focused primarily on children and their important adults, and interviews continued on the same themes of family life and interactions with social institutions. I also relied more heavily on participant observation in the third wave of the study, watching small children too young to participate in interviews and observing family interactions between kids and adults, and among children.
### Data Analysis
In quantitative research, data usually comes in the form of numbers, and findings often appear as percentages. Qualitative research (such as this study) produces data in the form of words, and the researcher's goal is to find trends and patterns in what the people said, where they contradicted each other, and the many different ways in which they experience or understand their lives. To analyze the data from this study, I used a modified form of grounded theory that began with _inductive data-gathering_ methods, where the questions I asked grew out of the environment I studied (rather than deductive research that begins with a hypothesis and then checks to see if it is true). Once I got started, I used _constant comparative methods_ to analyze the interview data and my field notes with a process that included (1) reading transcripts and generating initial coding categories (broad categories such as _interaction with peers_ or _experiences of jealousy_ ), (2) identifying and relating similar ideas and the relationships between and among categories, (3) adjusting these analytical categories to fit emergent theoretical concepts, (4) collecting additional data to verify and/or challenge the validity of those concepts, and (5) probing these data for the boundaries and variations of common themes. This process allowed me to constantly refine my questions and understand the nuances of complex ideas that I had first begun to grasp in the early portions of the research.
While many researchers use the methods described above, I added my own twist by sending drafts of what I wrote to respondents so they could see how I was using their words. This gave them an opportunity to correct anything I had gotten wrong (which happened only rarely and tended to concern details such as whose birthday it was or whose parent said what), to update information, and to include additional thoughts. This combination of constant comparative methods and a feminist research framework that empowers respondents to actively participate in and shape the research process was available to me only because email allowed such easy communication and document sharing.
### The Sample
The total sample for the study is 131 interviewees and five hundred participants observed. Some respondents I interviewed or interacted with only once, and others I interviewed up to four times and interacted with over fifteen years. Race was the most homogeneous demographic characteristic, with 89 percent of the sample identifying as white. Socioeconomic status was high among these respondents, with 74 percent in professional jobs. Fully 88 percent reported some college, with 67 percent attaining bachelor's degrees and 21 percent completing graduate degrees.
Defining polyamorous families is challenging, not only because social scientists and members of the public disagree on the definition of families but also because poly community members dispute the precise boundaries of what it means to be polyamorous. For this study, I included people who self-identified as polyamorous, and in this book I focus on those who identify as members of poly families. There are many respondents without children or connections to families with children who do not appear in this portion of the research, but they are more evident in some of my earlier publications on polyamorous women (2005) and polyamorous men (2006).
### Names
As with most research, I used _pseudonyms_ (fake names) for people in the study to help protect their identities. Many of the participants selected their own first names, though I eventually changed some of the most unusual ones to more conventional names when several different editors mentioned that the eccentric names were becoming a distraction from the content of what the people were saying. As the book evolved, it became clear that keeping track of all of the different people in each family was becoming increasingly confusing, so I made up last names for each family group. In reality, all family members very rarely share the same last name.
## A Final Note to Readers and Respondents
While I made every attempt to explain these families in as great a detail as possible, it is very difficult to squeeze fifteen years of research into a single book. Of necessity, most of what people told me has been left out. In order to give the respondents an opportunity to add important information and possibly tell an entirely different side of the story that does not appear here, I am collaborating with the Woodhull Sexual Freedom Alliance on the Family Matters project. Woodhull has offered a special section of the Family Matters website for respondents of the Polyamorous Family Study, which you can find at http://www.familymattersproject.org/. If you participated in the research and would like to elaborate on something in the book or provide a different perspective, please email me at dr.elisabeth.sheff@gmail.com and I will give you instructions on how to access that part of the site. If you are a reader who wants to follow the people in the book, please visit the _Polyamorists Next Door_ section of the Family Matters website, and consider posting about your own family while you are there.
# Notes
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# Index
A
Activism, poly _see also_ Poly Pride Day NYC
Being out as activist,
Age _see also_ children, polygeezers, teens
Adult children reject poly parent, 1.1-1.2
Age-appropriate answers, importance of, 1.1-1.2
Influence on makeup of poly communities,
Older people, Polygeezers,
Teens, 1.1-1.2
Tweens, 1.1-1.2
Very young children missing from the sample,
Young children, 1.1-1.2
Young people and hookup culture,
Autonomy
As element of choice,
As source of power,
Primary relationship with self,
B
Benefits of poly family life _see also_ autonomy, children, chosen kinship, community, emotional intimacy, emotional protection, freedom, family poly, honesty, parenting, resources, sexuality
Accommodating disabilities, 1.1-1.2
Attention for children, 1.1-1.2
Easier coparenting after divorce,
Easier to get dates, makes you more interesting,
Family expansion, 1.1-1.2
Honesty cultivates emotional intimacy, 1.1-1.2
More loot,
More sleep with new baby,
Multiple partners provide protection from Intimate Partner Violence, 1.1-1.2
Parents ability to remain friendly after divorce, 1.1-1.2 ,
Role models for children, 1.1-1.2
Shared resources, money,
Shared resources, parental attention and expertise,
Shared resources, time, 1.1-1.2
Smooth function of poly family _see also_ children comfortable in family, 1.1-1.2 , , 3.1-3.2 , ,
Social stimulation, 1.1-1.2 , ,
Bisexuality _see also_ gay, gender, lesbian, lesbigay, sexual minority
As advantageous to women in poly communities, ,
As entertaining to men, ,
Coffee night in the Bay Area,
Invisibility of,
Overlap with poly community,
Threat of male bisexuality, , 2.1-2.2 , 3.1-3.2
BDSM _see also_ kinkster, kinky
Definition of, ,
Differences in desire for, 1.1-1.2
Break-up _see also_ divorce, relationships
Still friends with polyaffective partner, ,
C
Celibacy
While married,
Change _see also_ continuity
Acceptable, not a signal of failure, 1.1-1.2 , , 3.1-3.2 ,
In poly families, , 2.1-2.2 , 3.1-3.2 , 4.1-4.2 , 5.1-5.2 , 6.1-6.2
Needs change over time, 1.1-1.2
Over lifespan things change, 1.1-1.2
Parents prepare kids to deal with change,
Stability and flexibility, 1.1-1.2
Turns out better than anticipated,
Cheating _see also_ emotional intimacy, honesty, lying,
As deviant in poly communities, 1.1-1.2
And polyfidelity, , 2.1-2.2
In poly relationships, 1.1-1.2
Redefinition of to polyamory, 1.1-1.2
Resulting in divorce,
Seeing more than one person is cheating for teens and tweens,
Children _see also_ age, teens
Access to non-parental trusted adults _see also_ benefits, communication, 1.1-1.2 ,
Adjusting to new parenting styles, 1.1-1.2
Adults leave, not that big of a deal to child, 1.1-1.2
Adult children reject poly parent, 1.1-1.2
Agency, ability to make things happen,
Attention, get a lot of it,
Attitude towards parents' divorce,
Awareness of family difference, poly child, 1.1-1.2
Awareness of family difference, peers, 1.1-1.2
Awareness of parental sexuality, 1.1-1.2
Babies and toddlers, 1.1-1.2
Blend in with other families, adopted or divorced _see also_ passing,
Categorizing adults by relationship to child, 1.1-1.2
Comfortable in poly family, not a big deal _see also_ family poly smooth function of, , 2.1-2.2 , , , ,
Coming out, to extended family,
Coming out, to peers, 1.1-1.2 , 2.1-2.2
Custody _see also_ Child Protective Services, legal issues, parenting, 1.1-1.2
Difficult to lie to poly family, 1.1-1.2
Discomfort in poly family, 1.1-1.2 , 2.1-2.2
Experiences of poly families age-dependent, 1.1-1.2
Experiences of stigma, 1.1-1.2
Focus benefits/custody decisions on children, regardless of adults, 1.1-1.2
Grossed out by parental sexuality _see also_ sexuality parental,
Importance of when adults making relationship decisions,
Independence,
Learn from parents' partners, 1.1-1.2
Make friends in poly community, 1.1-1.2 , 2.1-2.2
Metaphor of more children create more love, same with partners,
More important to mother than work, 1.1-1.2
Multiple parents gang up on kid, 1.1-1.2
My Dave who comes on the fun,
Overall low levels of stigma, reasons for, 1.1-1.2
Peers teasing regarding poly family, 1.1-1.2
Rejection of judgmental peers, 1.1-1.2
Seek open-minded peers,
Sibling complexity, 1.1-1.2
Singled out from peers, 1.1-1.2
Still close to dad after divorce,
Take family form for granted,
Unaware of parental sexuality, ,
Young child feels safe in poly family, 1.1-1.2
Child molestation _see also_ children, difficulties
All too common in mainstream society,
In a poly family, 1.1-1.2
Mistakenly attributed to poly family, 1.1-1.2
Child Protective Services (CPS)
Children removed from poly families, 1.1-1.2
Effect of stigma,
Explanation of,
Parent calls when daughter reports molestation,
Social worker says poly family is OK,
Social worker fairly understanding,
Social worker hostile,
Under surveillance, 1.1-1.2
Chosen kinship _see also_ divorce, polyaffective, otherfather, othermother, spice
Cherished adults who had been kin leave, 1.1-1.2
Children create their own networks, 1.1-1.2
Definition of,
Non-parental adults caution with poly families with kids, 1.1-1.2
Non-parent adults who children trust and can talk to, 1.1-1.2
Parents use extreme caution when considering new partners, 1.1-1.2
Polyaffective family members, 1.1-1.2
Shifting status, peer or parent?, 1.1-1.2
Social siblings, only child gets a brother, 1.1-1.2 , 2.1-2.2
Class _see also_ people of color, race, whiteness,
Expense of legal paperwork prohibitive to poor people,
Lack of Internet access impedes poly life, 1.1-1.2
Privilege gives buffer to stigma,
Privilege makes polyamory more easily accessible, , , 3.1-3.2
Privilege provides access to fast and private Internet, 1.1-1.2
Rarity of working class people in poly communities,
Surveillance in housing as deterrent,
Cohusband _see also_ family, gender, men, polyhegemonic masculinity, 1.1-1.2
Awkwardness of term, ,
In polyaffective triad, 1.1-1.2
Should be legally recognized,
Uncommon,
Commitment
Ceremonies, 1.1-1.2
Financial trust as form of,
Fluid bonding as form of,
Communication _see also_ emotional intimacy, families, honesty, strategies
And manipulation, 1.1-1.2
Attempt to foresee and defuse future problems, 1.1-1.2 , 2.1-2.2
Contributes to family resilience, 1.1-1.2
Family tool, 1.1-1.2 , 2.1-2.2 , 3.1-3.2
Financial and emotional dependency muddle communication, 1.1-1.2
Goes better than anticipated, 1.1-1.2
Importance of, , 2.1-2.2 , 3.1-3.2
Link to honesty,
Non-violent communication NVC,
Poly mantra communicate,
Poor communication relationship destruct,
Relationship maps, 1.1-1.2
Willingness to communicate,
Community
Age of people in community,
Assistance to members, 1.1-1.2
Beneficial to children, more attention, 1.1-1.2
Bringing people together, 1.1-1.2
Characteristics of, , 2.1-2.2
Dating partners, checking out background,
Dating partners, finding, 1.1-1.2
Desire for,
Deviance, specific to poly community, 1.1-1.2
Emotional assistance to members, 1.1-1.2
Financial assistance to members, 1.1-1.2
Forms around "hub" families, 1.1-1.2
Functions of, 1.1-1.2
Learning the term polyamory, 1.1-1.2
Liberal tone of,
Online, 1.1-1.2 ,
Overlap with other communities, 1.1-1.2
Provides alternatives,
Role models available in community, 1.1-1.2 ,
Rural polys miss community, 1.1-1.2
Social norms specific to poly communities, 1.1-1.2 ,
Stigma against monogamists, 1.1-1.2
Teaches relationship skills and boundaries, 1.1-1.2 , 2.1-2.2
Complexity
And drama, 1.1-1.2
Communication as tool to deal with complexity, , 2.1-2.2
Creates differing definitions of the same situation,
Difficulty in raising children, , 2.1-2.2
Family related, 1.1-1.2
In poly families, 1.1-1.2 , 2.1-2.2
Is it worth it?, 1.1-1.2
Makes monogamy challenging, 1.1-1.2
Previous partners, 1.1-1.2
Teen experiences family as both positive and negative _see also_ family
poly, teens, 1.1-1.2
Counseling _see_ therapy
Conservative people, rarity of in poly communities,
Continuity _see also_ change
Across many years, same people in relationship, 1.1-1.2
Amid change, 1.1-1.2
Communication facilitates,
Flexibility provides stability,
Polyaffectivity provides long-term flexibility, 1.1-1.2
Stay together no sex, 1.1-1.2
Couple privilege _see also_ unicorn hunters,
As myopic, 1.1-1.2 ,
Related to marriage,
Related to veto, 1.1-1.2
Sympathetic to difficult position for secondary partner, 1.1-1.2
Custody of children _see also_ children, legal issues, parenting
Divilbliss case, loss of custody,
D
Demographic characteristics _see also_ class, race, research
Of mainstream poly communities in US,
Of research sample,
Department of Children and Family Services _see_ Children and Family Services (CPS)
Deviance _see also_ cheating, community, lying, stigma
Definition of, ,
Difficulties of poly families _see also_ children, legal issues, parenting, stigma
Adjusting to new parenting styles, 1.1-1.2
Adult drama, effect on family, 1.1-1.2
Biolegal family rejection and tension, 1.1-1.2 , 2.1-2.2
Buffered by race and class privilege,
Children become attached to adults who leave, 1.1-1.2
Children removed by CPS, 1.1-1.2
Children singled out from peers, 1.1-1.2
Child molestation, _see_ child molestation, stigma, sexuality
Complexity _see also_ complexity, 1.1-1.2 , 2.1-2.2
Conflict, 1.1-1.2
Emotional pain, children, 1.1-1.2 , 2.1-2.2 , 3.1-3.2
Families of origin get upset, 1.1-1.2
Fear of emotional pain for children, , 2.1-2.2
Fear of Department of Children Services taking child again,
Household crowding see also teen, families poly, 1.1-1.2
Lack of privacy,
Live separately to get kids back from foster placement,
Makes it harder for kids to socialize with peers, , 2.1-2.2
Seems worse to teen than to mother, 1.1-1.2
Social rejection, 1.1-1.2
Stigma _see also_ stigma, 1.1-1.2
Teens have leverage against poly parents _see also_ Child Protective Services, stigma, teens, 1.1-1.2
Tension,
Too loud, can't sleep,
Too much supervision, children can't get away with anything, 1.1-1.2
Vulnerability to authorities because of stigma _see also_ Child Protective Services, stigma, teens, 1.1-1.2
Disability
Child with cognitive or learning disability, 1.1-1.2
Head injury,
Paralyzing anxiety,
Prevents paid work, places at economic disadvantage,
Symptoms mistaken for sexual abuse, interferes with treatment, 1.1-1.2
Divorce,
_see also_ break-up family, marriage, relationships
Annulment,
Because of polyamory,
Congenial, 1.1-1.2 , 2.1-2.2 , ,
Doesn't change anything, , , 3.1-3.2
Doesn't have to define the relationship, rejection of term "ex", 1.1-1.2
Means failure, or does it?,
Not because of polyamory,
No access to legal divorce, 1.1-1.2
Policies related to poly and divorce, 1.1-1.2
Poly after divorce, , ,
Poly as alternative to divorce,
Relief of not having to deal with that person's choices any more, 1.1-1.2
Staying friends afterwards, 1.1-1.2 , 2.1-2.2
Still lovers,
To avoid adultery laws,
Unfriendly, not congenial, ,
Don't ask don't tell DADT
And poly/mono couples, ,
And safer sex,
As manipulation,
In teen relationship,
Double standard
Against bisexual men, ,
Stigmatized among polys, 1.1-1.2
E
Ethics
As relationship guideline,
Poly families provide ethical guidelines,
Emotional intimacy _see also_ benefits, honesty, polyaffective
Between children and parents, 1.1-1.2 , 2.1-2.2 , , 4.1-4.2
Between men, 1.1-1.2
During tough times,
Heart of poly family, not sexuality,
Importance of, 1.1-1.2
Mixed feelings about poly family _see also_ emotional intimacy, 1.1-1.2
More important than sex, , 2.1-2.2 , 3.1-3.2
Teen feels close to her family, 1.1-1.2
Trying to feel OK about polyamory,
Emotional protection _see also_ emotional intimacy, parents prioritize children, strategies
Delay polyamory after accident,
Involves training in loss, not trying to prevent the inevitable, 1.1-1.2
Parents successfully shield child from emotional turmoil _see also_ poly family smooth function,
Parents prepare kids to deal with change,
Provide children with emotional support,
Strategy to deal with possible disadvantages of poly family, 1.1-1.2
F
Families, biolegal _see also_ chosen kin, polyaffective, 1.1-1.2
Accept children who come out as poly,
Being "out" to, 1.1-1.2
Definition of,
Legally recognized,
Set the stage for person to become poly as an adult, 1.1-1.2
Reject children who come out as poly,
Families, idealized version, 1.1-1.2
Families, gay _see_ families, lesbigay
Families, lesbigay _see also_ bisexual, gay, lesbian, lesbigay, sexual minorities ,
More easily recognizable than poly families,
Parallels with polyamory, commitment ceremony, 1.1-1.2
Parallels with polyamory, emphasis on chosen kinship,
Slippery slope to polyamory,
Families, polyamorous
Careful selection of members, 1.1-1.2
Changes in over time, , 2.1-2.2 , 3.1-3.2 , 4.1-4.2 , 5.1-5.2 ,
Chosen kin, inclusion, 1.1-1.2 ,
Chosen kin, support family members in crisis, 1.1-1.2
Complexity, 1.1-1.2 , 2.1-2.2
Conflict, 1.1-1.2 , 2.1-2.2
Different family members provide different role models, 1.1-1.2
Expansion of family more realistic for some people than monogamy, 1.1-1.2
Expansion of family through polyaffectivity, 1.1-1.2
Forms of, 1.1-1.2
Goes better than parents had feared it might, 1.1-1.2
Goes well at first, then falls apart,
Institution blames poly family for hardships,
Live separately to get kids back from foster placement,
Men get more choices in poly families _see also_ gender, otherfather, 1.1-1.2
Mixed feelings about poly family _see also_ emotional intimacy, teens, 1.1-1.2
More easily closeted than lesbigay families,
More stable base than romantic love, 1.1-1.2
Multiple partners provide protection from Intimate Partner Violence, 1.1-1.2
Not responsible for molestation, 1.1-1.2
Research on,
Resilience in,
Satisfaction in, ,
Smooth function of _see also_ children comfortable in family, 1.1-1.2 , , 3.1-3.2 , ,
Family resilience theory,
Fly-through
Definition of,
Freedom
As essence of humanity,
Choice as essential to relationships, 1.1-1.2 , 2.1-2.2
Essential to poly identity, 1.1-1.2
Form of self-expression,
Part of poly worldview,
Result from becoming poly, 1.1-1.2
From stifling social convention,
Friendship _see also_ polyaffective
Importance of,
Funeral
Partner meets family, included in ceremony, 1.1-1.2
Poly family marginalized during,
G
Gay _see also_ bisexual, lesbian, lesbigay, sexual minorities
Community more developed than poly community,
Fewer rights than polys,
Identity absorbs bisexuals,
Overlap with poly, ,
Polyaffective dads mistaken as gay couple, 1.1-1.2
Primarily gay, triadic relationship including woman doesn't work, 1.1-1.2
Rarity of gay men in poly communities, , , 3.1-3.2
Gender,
And cohusbands, 1.1-1.2
And the hot bi babe, 1.1-1.2
As performative,
Cisgendered population,
Conformity and complexity among polys, 1.1-1.2 , 2.1-2.2
Equality in polyamory, , , , 4.1-4.2 , , 6.1-6.2
Expansion of choices for women, not as much for men,
Impact on durability of triad,
Inequality in polyyny,
Men bond with household repairs, 1.1-1.2
Men get more diverse options in poly relationships, , 2.1-2.2 , , 4.1-4.2
Polyhegemonic masculinity, 1.1-1.2
Traditional expectations linked with legal marriage _see also_ marriage, monogamy,
Grandparents _see also_ biolegal family
As babysitters,
Become a grandparent through poly relationship,
Become more accepting of polyamory, 1.1-1.2
Nosy about poly family, 1.1-1.2
H
Heterosexuality
Dominant in general society, 1.1-1.2
Dominant in poly community,
Invisibility of,
Privilege give access to marriage,
History of polyamory in U.S., 1.1-1.2
Communes, 1.1-1.2
First wave, 1.1-1.2
Nashoba community,
Oneida community,
Second wave, 1.1-1.2
Support groups,
Third wave, 1.1-1.2
Homebirth,
Homophobia
And double standard, 1.1-1.2
And stigma,
Against bisexual men,
From mom, 1.1-1.2
Institutionalized, 1.1-1.2
Low levels of in poly communities?,
Underlying poly and swing communities, , 2.1-2.2
Worse against men, , 2.1-2.2
Honesty _see also_ emotional intimacy,
As community norm,
As protective mechanism to keep relationship healthy,
Builds emotional intimacy with adult child, 1.1-1.2
Creates trust, 1.1-1.2
Positive role model for children,
Provides clear choices,
Radical honesty, ,
Relationships implode without it,
With children, 1.1-1.2
Hot Bi Babes _see also_ bisexual, gender, homophobia, women
Definition of, 1.1-1.2
Three explanations,
I
Institutional Research Board _see also_ research,
Integrity _see_ ethics, honesty
Intimate network
Definition of,
Example of, ,
Fragility of, 1.1-1.2
J
Jealousy _see also_ cheating, emotional intimacy
Absent from some situations that could provoke it, 1.1-1.2
Factors affecting jealousy, 1.1-1.2
From secondary partners, 1.1-1.2
In open couples,
In poly/mono couples, , ,
Lack of for teen interested in poly,
Lack of in poly/mono family, ,
Not working out like anyone expected, , 2.1-2.2
Of wife's relationship with religion,
Significance of,
Veto in response,
K
Kerista, 1.1-1.2 , 2.1-2.2
Kinksters _see also_ BDSM, kinky, 1.1-1.2
Kinky _see also_ BDSM, kinkster
Definition,
L
Language
Compersion,
Ex,
Propensity to coin new words, , , ,
Evolution of the term polyamory,
Fluid bonding,
Frubbly,
Lovestyle,
Metamour,
NRE,
OSO,
Polyactive,
Polyaffectivity,
Polyamorous possibility, ,
Polyamory, learning the term, 1.1-1.2
Polyfidele,
Polygeezers,
Relationship maps, 1.1-1.2
Spice,
Swolly, ,
Unicorn,
Useful terms,
Use of poly as umbrella term, ,
Legal issues, poly families _see also_ children custody, Child Protective Services, divorce
Adultery illegal in this state,
Custody of children, Divilbliss case,
Divorce to avoid adultery laws,
Expand men's legal options,
Focus benefits/custody decisions on children, regardless of adults, 1.1-1.2
Increase number of legal parents, 1.1-1.2
Judge hostile to polyamory,
Laws and policies need to catch up with social evolution, 1.1-1.2
New ways to deal with custody, 1.1-1.2
Probation officer hostile to poly family,
Lesbian _see also_ bisexual, gay, lesbigay, sexual minority
Existence of lesbian poly community,
Fewer rights than polys,
Rarity of in poly communities, , 2.1-2.2
Lesbigay _see also_ bisexual, family lesbigay, gay, lesbian, sexual minority
Community larger and more diverse than poly community, 1.1-1.2
Movement paved the way for poly legal advocacy,
Predecessor of polyamorous identity,
Lying _see also_ cheating, emotional intimacy, honesty
Poly community stigma against,
Usually toxic to poly relationships,
M
Marriage _see also_ monogamy, polyfidelity, polygamy
Extend marriage benefits to all, policy, 1.1-1.2
In crisis prior to polyamory,
In poly relationships,
Legal documentation required to simulate some marital rights, 1.1-1.2
Multilateral marriage,
Multiple marriages in lifetime,
Poly attitude towards, disdain,
Poly does not strain marriage even though poly challenging, 1.1-1.2
Popular for many people, 1.1-1.2
Provides definition to relationship,
Provides privileges, ,
Rejection of legal marriage as central to relationship, ,
Significant to definition of relationship, 1.1-1.2
Men _see also_ gender, otherfather, parenting fathering, polyhegemonic masculinity
Changing expectations of HBB, 1.1-1.2
Conventional (hegemonic) masculinity, , , 3.1-3.2 , 4.1-4.2
Egalitarian,
Entertained by female bisexuality,
Fantasy of hot bi babe, 1.1-1.2
Heterosexuality, importance of, 1.1-1.2
Poly men are different, 1.1-1.2
Relationships with children _see also_ otherfathering,
Social justice seekers,
Monogamy _see also_ cheating, marriage, polyamory, polygamy
As an orientation,
As automatic pilot,
As a mindset makes polyamory difficult,
As suffocating, ,
Conscious monogamy, 1.1-1.2
Conversion to polyamory,
Default for tweens and teens _see also_ sexuality, 1.1-1.2
Difficult to sustain on sexual attraction alone, 1.1-1.2
Doesn't work for everyone,
Doing monogamy, behavior, ,
Has some of the same problems as polyamory, 1.1-1.2
How to know when you are monogamous,
In response to pregnancy,
Monogamous privilege, ,
Poly seems weird at first, makes more sense when have baby,
Popular for many people, 1.1-1.2
No ability or desire to be monogamous,
Rejection of, ,
Stigma against monogamists, 1.1-1.2
Mothers _see also_ parenting
Accepting of adult poly children, 1.1-1.2 , 2.1-2.2 , 3.1-3.2
Already a lesbian so poly less shocking,
Taken for granted that mothers will prioritize kids over sexuality,
Uncomfortable with adult children's polyamory, 1.1-1.2 , , 3.1-3.2
Moresomes
Anchors social community as hub, 1.1-1.2
Definition of,
Example of,
Grew from open couple, 1.1-1.2
Levels of sexual interaction,
N
Nesting/Non-nesting
Alternative ways to structure relationships, 1.1-1.2
In contrast to primary/secondary,
Rejection of hierarchy,
Newbie
Cliché mistakes, veto, , 2.1-2.2
New relationship energy NRE _see also_ language, jealousy,
Dealing with NRE, 1.1-1.2
Nonmonogamies See also cheating, swinging, polyamory, polygamy,
Diversity of nonmonogamies missing from this book,
Multilateral marriage,
Norms see also community social norms
Blended families now the norm,
Specific to poly communities,
O
Open couple
Dating as a couple,
Dating individually,
Definition of,
Most common poly family form,
Prevalence of, ,
Otherfather _see also_ children, chosen kinship, gender, parenting, polyaffective, men
And polyaffective relationships, 1.1-1.2
Definition of, 1.1-1.2
Different dads provide different role models, 1.1-1.2
Molests daughter, 1.1-1.2
Practice of, , 2.1-2.2 , 3.1-3.2 , 4.1-4.2 , 5.1-5.2
Relationship with kids independent of relationship with mom,
Spice,
Stay in touch with children after romantic relationship with mom ends, 1.1-1.2
Taking real responsibility for children, 1.1-1.2
Takes in "child of his heart" when she was in foster care, 1.1-1.2
Othermother _see also_ children, chosen kinship, parenting
Definition of, 1.1-1.2
Miss partner's child after breakup, 1.1-1.2
Practice of, , 2.1-2.2
Support family members in crisis, 1.1-1.2
P
Passing
Blending in,
Easy for poly families to pass as monogamous, , ,
Explaining Mom's "weird friends",
Polyaffective dads mistaken as gay couple so blend in San Fran, 1.1-1.2
Remain silent, don't bring up poly family, , 2.1-2.2 , 3.1-3.2
To gain monogamous privilege,
Parenting _see also_ children, chosen kinship, otherfather, othermother
Allow children as much freedom as possible within age range,
Allow children to come out to peers on their own terms, 1.1-1.2
Better parenting when well rested _see also_ benefits, ,
Children adjusting to new parenting styles, 1.1-1.2
Coming out to children, 1.1-1.2
Coparenting after divorce, congenial, ,
Coparenting, general,
Easier to date as poly parent, it makes you interesting,
Fathering, 1.1-1.2 , , 3.1-3.2 , 4.1-4.2 , , 6.1-6.2 , , 8.1-8.2
Fresh horses metaphor for shared parenting,
Honesty with children, emotional intimacy, 1.1-1.2
Increase number of legal parents, 1.1-1.2
Live in fear of Child Protective Services taking child again,
Mothering, , 2.1-2.2 , 3.1-3.2 , 4.1-4.2 , 5.1-5.2
New partners bring new parenting skills, 1.1-1.2
Part-time parenting is easier,
Poly parents are cool,
Prepare kids to deal with change,
Prioritize children over other relationships, 1.1-1.2
Provide children with emotional support, 1.1-1.2 ,
Rambunctious toddler,
Seek family counseling to deal with issues,
Sleeping with new infant and multiple parents,
Strategies vary by child, 1.1-1.2
Use extreme caution when considering new partners, 1.1-1.2 , 2.1-2.2
Whose time is spent parenting? _see also_ fathering, gender, mothering, 1.1-1.2
Who do children see as a parent, , 2.1-2.2
Wife outs husband as poly to adult children,
People of color _see also_ class, race, whiteness
Continued existence of racism,
Deterrents to poly identity, 1.1-1.2
Disdainful of unconventional sexuality,
Poly in action but not identity, 1.1-1.2
Poly in identity but not attending mainstream meetings,
Privilege _see also_ class, heterosexual, people of color, race, whiteness
Linked to poly rebelliousness, 1.1-1.2
Policy implications _see also_ legal issues,
Expand legal recognition of chosen kinship, 1.1-1.2
Extend marriage benefits to all, 1.1-1.2
Focus benefits on children, regardless of adults,
Increase number of legal parents, 1.1-1.2
What are public policies for?,
Polyaffective _see also_ chosen kin, otherfather, othermother, ,
Between men, 1.1-1.2 , 2.1-2.2
Break-up easier when no sexual interaction?,
Contrast with polyamory, 1.1-1.2
Definition of, ,
Dethrone sexuality as ultimate relationship signifier, 1.1-1.2
Differs from friendship,
Expanded family _see also_ chosen kin, 1.1-1.2
Flexibility allows kids to stay connected _see_ parenting, otherfathering, , 2.1-2.2
Flexibility allows relationship continuity, 1.1-1.2
Relationships among coparents,
Supersedes sexuality, 1.1-1.2
Triads,
Polyamorous communities _see_ community
Polyamorous people
Estimated number of,
Rebellious streak _see also_ Privilege,
And religion, ,
Social suspicion of, 1.1-1.2
Summary of characteristics common to, 1.1-1.2 ,
Polyamorous possibility,
Definition and explanation, 1.1-1.2
Polyamory
As an orientation, , ,
As a belief/worldview, ,
As a form of sacred sexuality, 1.1-1.2
As a lifestyle,
As a movement,
Biologically based, 1.1-1.2
By happenstance,
Contrasted with swinging, 1.1-1.2
Definition of, ,
Doing polyamory,
Drama, relationship, 1.1-1.2
Geographic distribution of, , ,
Idea of, rather than practice of, , ,
Lack of drama, 1.1-1.2
Polyfidelity
Definition of,
And family,
Origin of term see also language, ,
And sexually transmitted infections,
And sexual exclusivity,
Polygamy _see also_ marriage, monogamy, nonmonogamies
Polygyny, , , ,
Polyandry, , ,
Polyhegemonic masculinity _see also_ gender, men,
Among cohusbands, 1.1-1.2
As a quiet alternative to mainstream masculinity, 1.1-1.2
Polysexuality
As distinct from polyamory,
Definition of,
Sexual variety,
Poly/mono
Child's attitude towards,
Definition of,
Example of, , , 3.1-3.2 , 4.1-4.2
Not working out, 1.1-1.2
Two paradigms,
Poly fatigue,
Grow up already, 1.1-1.2
Tired of drama, 1.1-1.2
Poly Pride Day NYC,
Poly singles,
Pregnancy _see also_ children, marriage, monogamy, parenting, polyamory
And otherfathering, 1.1-1.2
And paternity, 1.1-1.2
And social rejection for paternity, 1.1-1.2
Catalyst for marriage,
Catalyst for monogamy,
Catalyst for polyamory, 1.1-1.2
Diversity in, 1.1-1.2
Impact on budding relationship,
Impact on poly quad family,
Put off finding relationship during pregnancy,
Refuse to disclose biological father, 1.1-1.2
Rejection when pregnant with other man's child (not legal husband), 1.1-1.2
Speeds up the timetable of secondary relationship, 1.1-1.2
Teen son gets girl pregnant, 1.1-1.2
Primary partners
Changing status over time, 1.1-1.2
Definition of,
Rejection of as organizing principle, 1.1-1.2
Pseudonyms _see also_ research
Description of,
Explanation of,
Q
Quads
Attempt goes poorly, 1.1-1.2
Changes in quad over time, 1.1-1.2 , 2.1-2.2 , 3.1-3.2
Clearly not working,
Cliché of instability, 1.1-1.2
Definition of,
Example of, ,
Formed from friendship group, 1.1-1.2 ,
Formed when couple meets couple, , 2.1-2.2 , 3.1-3.2 , 4.1-4.2
Instability of, , ,
Levels of sexual interaction,
Transition from quad to triad, 1.1-1.2
Transition to quad from moresome,
Who counts as primary? _see also_ primary partners,
R
Race _see also_ class, people of color, whiteness,
Disadvantages of racism makes poly less likely, , 2.1-2.2 ,
Fear of disdain from family or other people of color, , ,
Fear of tokenism,
Feeling out of place, 1.1-1.2
Less racism in poly community?,
Predominance of whiteness, ,
Stereotype of hypersexuality, disadvantage, ,
Surveillance,
White privilege as buffer to stigma,
Ravenheart, 1.1-1.2
Relationships _see also_ divorce, marriage
And choice,
End means change, no judgment, 1.1-1.2 ,
End means failure, 1.1-1.2 ,
End means relief, , 2.1-2.2
End means success or failure?, ,
End means transition, no end, 1.1-1.2 ,
End, three primary poly definitions,
Flexibility is good,
Graceful endings,
Persistent polyamorists, 1.1-1.2 , 2.1-2.2
OK to leave bad relationship,
Success/failure, 1.1-1.2 ,
Voluntary and utilitarian, poly,
Relationship guidelines, poly _see also_ rules structuring poly relationships
Allow change,
Be kind,
Communicate,
Gender equity,
List of common relationship guidelines, 1.1-1.2
No rules,
Safer sex,
Self-growth,
Tell the truth,
Religion _see also_ community characteristics, polyamory and religion, sacred sexuality
Avoid telling girlfriend's religious parents about polyamory,
Real Christians are not common or normal,
Relationship with G*d at least as intimate as a lover,
Religion at odds with polyamory in family, 1.1-1.2
Unconventional religion connection to polyamory, , ,
Research _see also_ IRB, pseudonyms, stigma
And critical thinking,
Data analysis, 1.1-1.2
Data collection, 1.1-1.2
Entrée as researcher,
Ethnographer's path, 1.1-1.2 , 2.1-2.2 , 3.1-3.2 , 4.1-4.2 , 5.1-5.2 , 6.1-6.2 , 7.1-7.2 , , 9.1-9.2 , 10.1-10.2
Impact of IRB on sample, , 2.1-2.2
Limitations of, 1.1-1.2 ,
Note to respondents and readers,
Sample, , 2.1-2.2
The study, 1.1-1.2
Resources _see also_ benefits, families poly, parenting
Multiple partners provide protection from Intimate Partner Violence,
Role Models _see also_ benefits, community role models
Different family members provide different role models, 1.1-1.2
"Hub" families as role models, 1.1-1.2
Lack of role models painful, 1.1-1.2
Science fiction as role model, , ,
Rules structuring poly relationships _see also_ relationship guidelines
Establishing ground rules,
Polyfidelity,
Rejection of rules as necessary, ,
Rigid rules can be a sign of couple privilege, 1.1-1.2
Rules of the Road, Morning-Glory Ravenheart,
Same rules apply to everyone,
Use judgment instead of rules, 1.1-1.2
Veto,
S
Sacred sexuality
Connection with polyamory, 1.1-1.2
Safer sex
Among intimate networks,
And cheating,
And communication,
Discussion of sexuality with teens, 1.1-1.2
Making agreements,
Sandstone poly commune,
Secondary partners
Definition of,
More committed relationship because of child,
Self-growth
Introspection helps deal with jealousy,
Relationship guideline,
Serial monogamy _see also_ monogamy, strategies for serial-monogamous resilience
Definition of, ,
Differing expectations for single parents vs. poly parents, 1.1-1.2
Provides camouflage for children in poly families, 1.1-1.2
Redefine success,
Single moms,
Sexuality _see also_ bisexual, BDSM, gay, kink, lesbian, lesbigay, polyaffective, sacred sexuality, sexual minority
As less important than anticipated,
As less important than other things _see also_ parenting, polyaffective, , , , 4.1-4.2
Children's awareness of parental sexuality, 1.1-1.2 ,
Complicates things,
Decrease in desire over time,
Developing a sense of one's own sexuality as a teen _see also_ teens,
Irrelevant to young children,
Parental sexuality gross to child, , 2.1-2.2
Parental sexuality irrelevant/no big deal to child _see also_ children, parents, , ,
Parental sexuality private from children,
Parental sexuality uncomfortable to child, 1.1-1.2
Sexual abuse (actual) leads to CPS taking child, 1.1-1.2
Sexual abuse (misconstrued) leads to CPS taking child, 1.1-1.2
Sex-positive environment encourages knowledge for teens, 1.1-1.2
Teens doing polyamory, 1.1-1.2
Teens not sure about sexuality, too soon, 1.1-1.2 ,
Teens reject polyamory, 1.1-1.2
Young people and hookup culture,
Sexual exclusivity _see_ monogamy, polyfidelity, serial monogamy
Sexual minority _see also_ bisexual, gay, kinky, lesbian, lesbigay, poly
Dangers of exposure, 1.1-1.2
Stigma, 1.1-1.2
Sexual orientation, general _see also_ bisexual, gay, heterosexual, lesbian, and lesbigay
,
Siblings
Adult quasi-siblings, _see also_ polyaffective, spice, , 2.1-2.2
Crowd each other _see_ difficulties crowding
Explaining new siblings to peers, 1.1-1.2
Jealousy, 1.1-1.2
Love,
Only child gets poly siblings _see also_ chosen kin, 1.1-1.2 ,
Role models, 1.1-1.2
Tension, 1.1-1.2
Size of group
Changes over time,
Difficulty of dating as intimate network,
Larger is rarer, ,
Poly geometry,
Spice
Killed in car accident, 1.1-1.2
Polyaffective connections durable,
Stability _see_ continuity
Statistics not available, 1.1-1.2
Stigma
Against mothers who prioritize sex,
Against polys,
Against sexual minorities,
And sex negativity,
Children's experiences of, 1.1-1.2
Community as buffer, , 2.1-2.2
Dangerous for people of color,
Dangerous for poor people,
Definition of, 1.1-1.2 ,
End of relationship stigma subsides,
From biolegal family members, 1.1-1.2
Impact of stigma, 1.1-1.2
Makes family disband to get kids back from foster placement,
Makes holiday parties tense,
Neutralize stigma with education, 1.1-1.2
Normalizing the situation, 1.1-1.2
Overall low levels of stigma, 1.1-1.2
Poly-related, 1.1-1.2
Professional impact on Elisabeth Sheff, 1.1-1.2
Rejection of stigma, 1.1-1.2
Role models help to combat, 1.1-1.2
Seek support, 1.1-1.2
Shifting stigma, 1.1-1.2
Vulnerability to authorities because of stigma _see also_ Child Protective Services, teens, 1.1-1.2
Strategies for overcoming obstacles, poly families
Attempt to foresee possible pitfalls, 1.1-1.2 , 2.1-2.2
Communication, 1.1-1.2
Disregard convention as unimportant, 1.1-1.2
Emotional protection _see also_ emotional intimacy, 1.1-1.2
Honesty, 1.1-1.2
Intentional socializing, 1.1-1.2
Non-parental adults caution with poly families with kids, 1.1-1.2
Normalizing the situation, 1.1-1.2
Prioritize children, 1.1-1.2
Refuse to date parents, 1.1-1.2
Rejection of stigma, 1.1-1.2
Screening potential partners, 1.1-1.2
Seek support, 1.1-1.2
Shifting stigma, 1.1-1.2
Stigma management, 1.1-1.2
Training in real life, 1.1-1.2
Strategies for serial-monogamous resilience _see also_ monogamy, serial monogamy
Be honest,
Don't stay too long,
Prepare kids to deal with change,
Prioritize the children,
Negotiate, 1.1-1.2
Redefine success,
Stick with it,
Swinging, ,
Differences from polyamory, 1.1-1.2
Differing motivations from polyamory,
Overlap with poly community, 1.1-1.2
Research on,
Similarity with polyamory,
T
Teens, 1.1-1.2
Aligns with poly parents against bio dad, 1.1-1.2
Assaults step-father,
Blame personal problems on polyamory,
Different family members provide different role models, 1.1-1.2
Difficulties managing information with grandparent, 1.1-1.2
Disgruntled teen attempts to blackmail poly parents, 1.1-1.2
Doing polyamory, 1.1-1.2
Easy managing information with father,
Emotional distance from family, 1.1-1.2
Emotional intimacy with family, 1.1-1.2 ,
Experiences of poly families age-dependent, 1.1-1.2
Fighting with mom's girlfriend, 1.1-1.2
Freak out and take it out on family, 1.1-1.2
Focus on their own lives, 1.1-1.2
Have leverage against poly parents _see also_ Child Protective Services, difficulties, stigma,, 1.1-1.2
Lack of privacy, crowding, 1.1-1.2
Mixed feelings about poly family _see also_ emotional intimacy, family poly, 1.1-1.2
Normalizing the situation, 1.1-1.2
Normal is boring,
Not sure about sexuality, too soon to say, 1.1-1.2 ,
Polyamory is perfect scapegoat for personal awkwardness, 1.1-1.2
Rejection of stigma, 1.1-1.2
Reject polyamory, 1.1-1.2 , 2.1-2.2
Separate social spheres,
Too much supervision, can't get away with anything, 1.1-1.2
Too loud, can't sleep,
Tertiary partners
Definition of,
Therapy
Change therapist's mind about poly, 1.1-1.2
Kicked out of family therapy,
Seek therapy to deal with family issues, 1.1-1.2 , , ,
Useful as a relationship tool, ,
Time _see also_ change, continuity
As indicator of a relationship's importance, 1.1-1.2 , 2.1-2.2 , , , 5.1-5.2 , , , , 9.1-9.2 , 10.1-10.2 , 11.1-11.2 , , , 14.1-14.2
As limited resource, , , , ,
Personal time, , , , 4.1-4.2
Poly is time consuming _see also_ complexity, difficulties,
To heal from head-injury _see also_ disability, emotional protection, 1.1-1.2
Too early, can't sleep,
Whose time is spent parenting?, 1.1-1.2
Triads _see also_ hot bi babes
And couple privilege _see also_ unicorn hunters,
And paternity,
And parenting, 1.1-1.2
As ideal relationship form, ,
As less exciting than men expected _see also_ hot bi babe, 1.1-1.2
Definition of,
Differing outcomes with triad members break up, 1.1-1.2
Formed through cheating, 1.1-1.2
Formed with couple and friend, 1.1-1.2 , , , ,
Long-lasting, , ,
Mostly one woman with two men,
Polyaffective, , 2.1-2.2 , 3.1-3.2
Primarily gay, triadic relationship including woman doesn't work, 1.1-1.2
Rejection of hierarchy in triad, 1.1-1.2
Transition from quad to triad, 1.1-1.2
Transition from triad to vee _see also_ change, 1.1-1.2
U
Unconventional people
Anarchists,
Gamers,
Geeks,
Gravitate to polyamory, 1.1-1.2
Pagans,
Science fiction fans,
Unicorn hunters
Community response to,
Definition of, ,
Myopia of, 1.1-1.2
Unicorn
Description of cliché, , , 3.1-3.2
Relationship becomes far more than originally intended,
Unpredictability _see also_ change over time, complexity, 1.1-1.2
V
Vees
Definition of,
Triad becomes a vee _see also_ change, 1.1-1.2
Veto _see_ couple privilege, jealousy, newbie mistakes
W
Whiteness _see also_ class, people of color, race
Continued existence of racism,
Dominant in mainstream poly communities,
Makes polyamory more easily accessible, , 2.1-2.2
Otherfathers usually white even though term has African roots,
Poly as nasty white thing, 1.1-1.2
Privilege as buffer to stigma, 1.1-1.2
Why do polyamory?,
Family expansion, 1.1-1.2
Feels more free, choice, 1.1-1.2
Feels more natural _see also_ polyamory as a sexual orientation, 1.1-1.2
Involuntary,
More love,
More needs met, 1.1-1.2 ,
Women
Advantages in poly community,
Bisexuality,
Hot bi babe, 1.1-1.2
Freedom to define boundaries,
Leaders in poly community,
Social class,
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 1,578
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Q: Using AJAX to Post Data and Return Result from PHP processor.php Its been asked a million different ways but this is specific to this code. This is not a duplicate post.
index.html
<html>
<head>
<script>
function ajax_post(){
// Create our XMLHttpRequest object
var hr = new XMLHttpRequest();
// Create some variables we need to send to our PHP file
var url = "process.php";
var fn = document.getElementById("s").value;
var vars = "s="+fn;
hr.open("POST", url, true);
// Set content type header information for sending url encoded variables in the request
hr.setRequestHeader("Content-type", "application/x-www-form-urlencoded");
// Access the onreadystatechange event for the XMLHttpRequest object
hr.onreadystatechange = function() {
if(hr.readyState == 4 && hr.status == 200) {
var return_data = hr.responseText;
document.getElementById("status").innerHTML = return_data;
}
}
// Send the data to PHP now... and wait for response to update the status div
hr.send(vars); // Actually execute the request
document.getElementById("status").innerHTML = "processing...";
}
</script>
</head>
<body>
<h2>Ajax Post to PHP and Get Return Data</h2>
Domain: <input id="s" name="s" type="text"> <br><br>
<input name="myBtn" type="submit" value="Submit Data" onclick="ajax_post();"> <br><br>
<div id="status"></div>
</body>
</html>
process.php
<?php
if($_GET["s"])
{
echo '<table>';
echo '<thead>';
echo '<tr>';
echo '<th width="200">CLASS</th>';
echo '<th width="150">HOST</th>';
echo '<th width="150">TYPE</th>';
echo '<th width="150">TTL</th>';
echo '<th width="150">TARGET</th>';
echo '</tr>';
echo '</thead>';
echo '<tbody>';
$result = dns_get_record($_GET['s'], DNS_ALL, $authns, $addtl);
foreach($result as $key=>$value){
echo '<tr>';
echo '<td>' . $value['class'] . '</td>';
echo '<td>' . $value['host'] . '</td>';
echo '<td>' . $value['type'] . '</td>';
echo '<td>' . $value['ttl'] . '</td>';
echo '<td>' . $value['target'] . $value['ip'] . '</td>';
echo '</tr>';
}
echo '</tbody>';
echo '</table>';
}
?>
The Problem
I can't get process.php to return the echo? If you access process.php?s=google.com DIRECTLY then it works fine and echo's the response without a hitch. Figures it probably something small but I can't figure it our can someone please point out my mistake please? Thanks!
DEMO
Form: http://main.xfiddle.com/14eb8a38/ajax/index.html
PROOF: http://main.xfiddle.com/14eb8a38/ajax/process.php?s=google.com
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,049
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\section{Introduction}
Galaxies are generally complex systems that can consist of several structural
components, such as bulges, stellar and gaseous discs, spiral arms, bars, and
shells. The formation of a galaxy itself and its substructures is a complex
process and can require a number of subsequent events ranging from ``secular
evolution'' (e.g., orbital instability, formation of new stars, quenching) to
``interactions'' (e.g., mergers, accretions, ram pressure stripping); see for
example \citet{Kormendy+04, Naab+17} for reviews.
The detailed study of the various structural components, their
morphology, their kinematics, and their stellar populations can shed
light on the various processes that contributed to their
formation. Because multiple structural components can co-exist in the same
region of a galaxy, such as a small disc engulfed in a large bulge, it is
necessary to remove their mutual contamination from the observations
in order to study them separately. This ``decomposition'' has
been widely applied in the past by fitting analytic functions to the
galaxy light in order to reconstruct the images of the various
components (e.g., \citealt{Kormendy77, Peng+02, Erwin+15}).
The decomposition can be done also on spectroscopic data, by taking
the different kinematic properties into account. This involves
decomposing the galaxy's line-of-sight velocity distribution (LOSVD)
into multiple kinematic components (e.g. \citealt{Rubin+92,
Kuijken+93}), and, more recently, decomposing the observed spectrum
into multiple spectral components (\citealt{Coccato+11},
\citealt{Katkov+11a}, \citealt{Johnston+12}). Key requirements for the
kinematic and spectroscopic decomposition are a relatively good
spectral resolution and high signal-to-noise,
which are fundamental to properly sample the shape of the LOSVD and
separate the various kinematics and spectral components (e.g.,
\citealt{Fabricius+14, Coccato+14, Coccato+15}).
Recently, we started an observational campaign aimed at studying the detailed
structure of the LOSVDs of the bulges in nearby galaxies, exploiting the
Integral field unit and the superb spectral resolution of the VIRUS-W
spectrograph ($\sigma_{\rm instr} \sim 15$ km s$^{-1}$\ at 5200 \AA) of the McDonald
Observatory. The main goal of the survey is to study the kinematic signatures
of the various components, such as kinematically decoupled cores, and/or a
detailed characterization of the orbital structure that, for example, can be
complex in the case of triaxial potential or bars.
In \citet{Fabricius+14} we studied the spiral galaxy NGC\,7217.
Previous works, including our own, hinted at the existence of two
counter-rotating stellar components \citep{Kuijken+93,
Fabricius+12}. However, the higher spectral resolution of VIRUS-W
data show that the stars in NGC\,7217 are co-rotating. By studying
the LOSVD, we identified a kinematically hot component (that we
associated with the bulge) and a kinematically cold component (that we
associated with the disc). A spectral decomposition analysis allowed us
to study the properties of the bulge and the disc in NGC\,7217
independently. Our measurements suggest that NGC\,7217 is in the
process of regrowing a disc inside a more massive and higher
dispersion stellar spheroid.
The same analysis was carried on the S0 galaxy NGC\,4191 \citep{Coccato+15}. In
this latter case, we discovered two stellar counter-rotating components, as
suggested by previous studies \citep{Krajnovic+11}, and proposed gas accretion
along two filaments as their formation mechanism.
In this paper, we focus on another galaxy in our sample that has been
claimed to host a stellar counter-rotating component
\citep{Zeilinger+01}: NGC\,3521. NGC\,3521 is a late-type spiral galaxy
with a mixed barred and inner ring morphology, classed as SAB(rs)bc by
the RC3 catalog \citep{rc3}.
One of the first kinematic studies of NGC\,3521 is from \citet{Burbidge+64};
the authors measured the rotation curve of the ionized gas out to $\sim 70$\,kpc
and estimated a total enclosed mass of $8\cdot10^{10}$ M$_{\odot}$\ and an upper
limit of the stellar mass of $2\cdot10^{10}$ M$_{\odot}$. More recent measurements
estimate the stellar and H{\small{I}}\ masses of NGC\,3521 to be $5\cdot10^{10}$ M$_{\odot}$\
and $8\cdot10^9$ M$_{\odot}$, respectively \citep{Leroy+08, Walter+08}.
\citet{Elson14} studied the kinematics and distribution of the H{\small{I}}\ halo
surrounding NGC\,3521 out to 25 kpc and found evidence for a secondary H{\small{I}}\
component. This component, called ``anomalous'' by the author, is distributed
on a thick disc and it rotates more slowly than the main H{\small{I}}\ component.
Moreover, the spatial distribution of the ``anomalous'' component coincides
with the inner regions of the stellar disc where the star formation rate is
highest; this suggests a link between the stellar feedback and the gas in the
halo, probably driven by galactic fountains. \citet{Elson14} found no evidence
of gas accretion from outside the galaxy system.
\citet{Zeilinger+01} and \citet{Fabricius+12} reported the presence of
a stellar component that is counter-rotating with respect the main
body of the galaxy. The measured velocity separation between the two
stellar components is about 200 km s$^{-1}$. This kinematic decoupling was
interpreted as a ``projection effect induced by the presence of a bar
component seen almost end on'' \citep{Zeilinger+01}. The retrograde
orbits allowed by the bar potential and the viewing angle caused the
bi-modal distribution of line of sight velocities. This
``internal'' origin of the observed kinematic decoupling is
consistent with the lack of evidence for gas accretion or
interaction with other galaxies (although it does not prove that
accretion or interaction did not take place).
The purpose of this paper is to study the kinematics and stellar populations of
the various kinematic and structural components in NGC\,3521, exploiting the
high spectral resolution and sensitivity, and the integral field mode of the VIRUS-W
spectrograph. With the aid of spectral decomposition techniques, we
isolate the contribution of the various structural components from the observed
spectrum and determine their kinematics, age, and metallicity independently in
order to get clues on their formation mechanisms. The advantage of this
approach is that we minimize the mutual contamination of these components in
the regions where their light contribution is comparable.
Moreover, as done for other spiral galaxies with detected stellar
counter-rotation, we test whether or not the presence of a population of
counter-rotating stars is confirmed (e.g. as in NGC\,4191, \citealt{Coccato+15})
or not (e.g., as in NGC\,7217, \citealt{Fabricius+14}) when repeating the
observations with higher spectral resolution.
The paper is structured as follows. In Sections \ref{sec:fors1} and
\ref{sec:observations} we describe the observations and the data
reduction, in Section \ref{sec:kinematics} we describe the kinematic
measurements, the identification of two kinematically distinct
components, and their association with the galaxy structural
components. In Section \ref{sec:populations} we measure the properties
(age, metallicity, and $\alpha$-enhancement) of the stars of both
components. We describe and discuss the results in Section
\ref{sec:discussion}. In this paper, we adopt a distance to NGC\,3521
of 8.1 Mpc (as in \citealt{Fabricius+12}), which corresponds to a
linear scale of 40 pc/arcsec.
\section{Photometric observations and data reduction}
\label{sec:fors1}
We exploit the ESO and {\it Spitzer} archive images of NGC\,3521.
ESO data were
taken on April $16^{\rm th}$ 1999 with FORS1 at the Very Large
Telescope for the program 63.N-0530 (P.I. Kudritzki). The galaxy was
observed in the 3 ESO ``Bessel'' filters B, V, and I with a single
exposure of 300 seconds in each filter. Data reduction (bias and flat
fielding) was performed with the standard ESO FORS pipeline run under
the EsoReflex \citep{Freudling+13} environment and adapting the
available FORS-imaging workflow to handle observations from 1999.
Despite the fact that the exposures were saturated in the centre and the reduced
data are not flux calibrated due to the lack of a standard star
observation, these FORS1 data are deep enough to highlight the
peculiar morphology of the outskirts of NGC\,3521. The left panel of
figure \ref{fig:fors1} shows some extended and faint structures
that depart from an axisymmetric light
distribution (see regions marked with the labels A, B, and C in
Fig. \ref{fig:fors1}).
The \textit{Spitzer} IRAC1 (3.6 \micron) image was taken as part as the
\textit{Spitzer} Infrared Nearby Galaxies Survey
\citep{kennicutt+03}. The combined mosaic image, with a
pixel scale of 0.75 arcsec, was produced as part of the SINGS Fifth
Data
Delivery\footnote{http://irsa.ipac.caltech.edu/data/SPITZER/SINGS/doc /sings\_fifth\_delivery\_v2.pdf}
and was retrieved from NED. We performed a final background
subtraction of the image by measuring median values in approximately
50 $20 \times 20$-pixel boxes well outside the galaxy and computing
the mean of those values.
Because NIR data are less affected by dust absorption, they are
particularly useful to highlight the inner structure of the
galaxy. The right panel of Fig. \ref{fig:fors1} shows the inner
spiral structure of the galaxy and the isophotal contours. Despite
the RC3 classification as SAB, we found no evidence of a bar. The
same structures observed in the FORS image are visible in the outer
isophotes of the Spitzer image, as deviations from an symmetric
elliptical profile.
The presence of such structure is an indicator that at least one
merging event contributed to the growth of the galaxy in its past.
\begin{figure}
\psfig{file=image_1.ps,width=8.5cm}
\psfig{file=image_2.ps,width=8.5cm}
\caption{VLT-FORS2 B-Band (top left) and Spitzer-IRAC1 3.6
\micron\ (top right and bottom) images of NGC 3521. The field of
view in the top panels is $6.\arcmin4 \times 6.\arcmin8$. The
field of view in the bottom panel is
$125\arcsec\times 70\arcsec$. Color cuts are selected to
highlight faint outer regions in the FORS2 image and the inner
disc and spiral structure in the Spitzer image. Contour plots are
superimposed to the Spizer image. The three red labels mark the
position of asymmetric features, that are recognizable as
``bumps'' in the light distribution (FORS image) and distortions
of the isophotes (IRAC image). The red rectangle shows the
VIRUS-W field of view.}
\label{fig:fors1}
\end{figure}
\section{Spectroscopic observations and data reduction}
\label{sec:observations}
The observations of NGC\,3521 were carried on May 29 and May 30 2011 using the
VIRUS-W Integral Field Unit (IFU) Spectrograph \citep{Fabricius+12} at the 2.7
m Harlan J. Smith Telescope of the McDonald Observatory (Texas, US). The
observed field of view is $105\arcsec \times 55\arcsec$ and it is mapped with
267 fibers of 3\farcs2 diameter on the sky, resulting in a filling factor of 1/3.
Three dithered exposures are sufficient to cover the field of view. We took a
second set of exposures offset by half a fiber diameter with respect to the
first, thereby sub-dithering the observation to increase the spatial resolution.
The individual exposure time was 1800\,s.
Off-set sky exposures of 300\,s each were interleaved with the target
exposures.
The VIRUS-W instrument was configured in the
high-resolution mode, covering the 4850 -- 5480 \AA\ wavelength range
with a sampling of 0.19 \AA\ pixel$^{-1}$ and a resolving power of
$R \sim 8700$, corresponding to an instrumental dispersion of
$\sigma_{\rm instr} \sim 15$ km s$^{-1}$\ at the centre of the wavelength
range.
Data were reduced using the {\tt FITSTOOLS} package of
\citet{Gossl+02} and the {\tt Cure} pipeline, which was originally
developed for the HETDEX project \citep{Snigula+14}. We
followed the prescriptions of \citet{Fabricius+14}. The final reduced
datacube was spatially binned to increase the signal-to-noise ratio
($S/N$) using the implementation for Voronoi Tesselation of
\citet{Cappellari+03} for a total of 83 binned spectra. We
measured\footnote{The computation was done directly on the binned
spectra by using the {\tt der\_snr} tool available at:
http://www.stecf.org/software/ASTROsoft/} $40 < S/N < 130$ per pixel
in the spatial bins where the stellar population analysis is performed,
and $20 < S/N < 40$ per pixel elsewhere.
\section{Kinematics and spectral decomposition}
\label{sec:kinematics}
The stellar and ionized-gas kinematics of NGC\,3521 were initially
measured non parametrically in each spatial bin by fitting the
observations with a series of stellar templates. The fitting procedure
recovered the full shape of the line-of-sight velocity distribution
(LOSVD) by exploiting the maximum penalized likelihood method (MPL) of
\citet{Gebhardt2000b} as implemented by \citet{Fabricius+14} to
include \hbeta, [O{\small III}] $\lambda\lambda$ 4959, 5007 \AA, and [N{\small I}] $\lambda\lambda$ 5190, 5200 \AA\ ionized-gas emission lines in the
fitting process. All the emission lines are set to share the same
kinematics. MPL determines the LOSVD in a non-parametric way. The
best fitting spectrum is obtained by weighted linear combination of a
number of stellar templates that represent the galaxy stellar
populations; the templates are convolved by the same LOSVD. Therefore,
the algorithm uses the same LOSVD for the different stellar
components.
Then, the recovered LOSVD is modeled as the contribution of two
separate components by fitting two Gaussian functions with independent
velocity, velocity dispersion, and amplitude. Specifically, we used a
Monte Carlo Markov Chain algorithm based on {\tt pymc} to determine
the maximum likelihood combinations of amplitude, mean velocity, and
velocity dispersion for each of the two Gaussians in each bin.
The knowledge of the kinematics of the two stellar components allowed
us then to disentangle their contribution to the observed spectra in
each spatial bin. We applied a spectroscopic decomposition technique
\citep{Coccato+11} by using the Python implementation of the Penalized
Pixel Fitting (pPXF) code by \citet{Cappellari+04, Cappellari+17}. In
contrast with the MPL method, pPXF parametrizes the LOSVD with a
Gaussian-Hermite function. It can handle multiple LOSVDs so that each
stellar component is formed by a linear combination of templates and
convolved with its LOSVD. In the fitting process, we used the
previously determined kinematics as initial starting parameters in the
fit. This strategy allowed us to obtain the best fitting stellar
spectra for the bulge and the disc for each spatial bin, minimizing
the degeneracy between the fitting parameters and the stellar
components. The kinematics obtained with the pPXF are compatible
within errors to those determined via the LOSVD Gaussian
decomposition.
We show in Figure \ref{fig:spectral_decomposition} the
LOSVD and the decomposition for one Voronoi bin along the photometric
major axis.
The output of the spectroscopic decomposition for each
Voronoi-binned galaxy spectrum are: i) the kinematics (velocity and
velocity dispersion of the 2 components); ii) the best-fitting stellar
templates for each component obtained as linear combination of the
stars in the input library; iii) the best-fitting stellar models
of the stellar components, i.e., the best-fitting stellar templates
convolved for kinematics that account also for the contribution of
multiplicative polynomials; and iv) the kinematics and best-fit
model of the ionized gas component.
The best-fitting stellar templates are used in Section
\ref{sec:populations} to study the age and metallicity of the two
stellar components, whereas the best-fitting stellar models are used in
Section \ref{sec:kinem_results} to study their light distributions.
\begin{figure}
\psfig{file=spectral_decomposition.ps,width=9cm,bb=75 360 393 692}
\caption{Kinematic and spectral decomposition in one spatial bin of
NGC\,3521. The best fit LOSVD is parameterized by the sum of two
Gaussian functions (upper panel). The kinematics obtained from the
LOSVD parametrization is used in the spectral decomposition code to
extract the spectra of the two components from the observations
(lower panel). Blue and red represent the kinematically cold and
kinematically hot components, respectively. Purple shows the
emission lines and green the stellar best fit models.}
\label{fig:spectral_decomposition}
\end{figure}
\begin{figure*}
\vbox{
\hbox{
\psfig{file=v_bulge.ps,width=6cm,bb=55 340 300 540 }
\psfig{file=v_disk.ps,width=6cm,bb=55 340 300 540}
\psfig{file=v_gas.ps,width=6cm,bb=55 340 300 540}}
\hbox{
\psfig{file=s_bulge.ps,width=6cm,bb=55 340 300 540}
\psfig{file=s_disk.ps,width=6cm,bb=55 340 300 540}
\psfig{file=s_gas.ps,width=6cm,bb=55 340 300 540}}
}
\caption{The two-dimensional velocity (upper panels) and velocity dispersion
(lower panels) fields of the kinematically hot (bulge), the kinematically cold
(disc), and the ionized-gas components in NGC\,3521. North is up, East is left.}
\label{fig:kinematics}
\end{figure*}
\subsection{The extended stellar template library}
\label{sec:library}
To ensure that the kinematic results are not biased by the differences
of the intrinsic properties of the galaxy's stellar population to
those of the template star, it has become common practice to include a
multitude of template stellar spectra in the kinematic extraction
covering a representative region of temperature and metallicity. In
addition to the line of sight velocities, the spectral decomposition routine
also determines an optimal set of weights for the linear combination of
templates that ultimately becomes the best fitting model. Because we
decompose the observed stellar spectra into different components, we
derive stellar population parameters directly from those models rather
than the input data themselves. This further strengthens the need to
include a large set of different stellar types with varying
metallicities and alpha element over-abundances, to prevent a biasing
of the derived stellar population parameters.
From previous work, we already possess a library of spectra for 30
giant stars (A -- M type) observed with VIRUS-W which are also part of
the ELODIE high resolution (R $\simeq$ 42000) library of stellar
spectra \cite{Prugniel+04}. We use these to determine the difference
in spectral resolution between the ELODIE high resolution spectra and
the VIRUS-W spectra, and to then convolve the ELODIE spectra to the
spectral resolution of VIRUS-W.
\begin{figure}
\psfig{file=dsigma_of_lambda.ps,width=6cm}
\caption{
The spectrally dependent broadening function between the 30 VIRUS-W template
spectra and the corresponding ELODIE spectrum. Each point represents the mean
values of the dispersion that we computed across the 30 spectra and error bars
represent the RMS in dispersion. The line corresponds to the linear function
that we use to model the spectrally dependent differential broadening.
}
\label{fig:dsigma_of_lambda}
\end{figure}
The spatial variation of the VIRUS-W's spectral resolution is
negligible ($< 3$\%, \citealt{Fabricius+12}). However, it changes as a
function of wavelength, varying from 17 kms$^{-1}$ around 4800 \AA\ to
14 kms$^{-1}$ at 5400 \AA. To account for this, we derive a
differential broadening function, which measures as a function of
wavelength the difference in instrumental FWHM between the stars in
the ELODIE library and those observed with VIRUS-W. We initially
divide the spectral range into 12 equally spaced and 50 \AA\ wide
sub-regions and measure the differential broadening separately in
each. For this, we convolve the ELODIE spectrum with a Gaussian and
vary the centroid and the width of the Gaussian until we obtain
minimal residuals between the ELODIE spectrum and the corresponding
VIRUS-W spectrum. We use the standard \textsc{scipy} least\_squares
routine for the optimization. We repeat this process for each of the
30 spectra that the VIRUS-W library and the ELDOIE library have in
common and then compute the mean and the RMS values in differential
dispersion for each wavelength bin. We find that the change of the
FWHM of the differential broadening with wavelength is well
represented by a first order polynomial (see Figure
\ref{fig:dsigma_of_lambda}); we then use this linear trend as model
for the differential broadening. We then convolved all of the spectra
of giant stars in ELODIE to the VIRUS-W resolution (as function of
wavelength), by using the linear model of the differential
broadening. The ELODIE high resolution contains spectra of 220 giant
and subgiant stars which is an impractically large number of templates
to use in our kinematic extraction. To fairly sample ages,
metallicities and $\alpha$ over-abundances, we use the published
values for the LICK indices to locate the stars in the four
dimensional space of H$\beta$, Mg\,$b$, $[\mbox{MgFe}]'$, and
$\langle{\rm Fe}\rangle$. We split this space into one \AA\ wide bins
and select one star per bin, resulting in a total of 72 stars
(broadened to the VIRUS-W resolution) that we use in the further
analysis.
$[\mbox{MgFe}]'$ and $\langle{\rm Fe}\rangle$ are defined by \citet{Thomas+03,
Gorgas+90}, respectively as:
\begin{eqnarray}
\label{xx}
[\mbox{MgFe}]' & = & \sqrt{\mbox{Mg\,$b$} \cdot \left( 0.82 \cdot {\rm Fe}_{5270} + 0.28 \cdot {\rm Fe}_{5335}\right)}\\
\langle{\rm Fe}\rangle& = & 0.5 \cdot ({\rm Fe}_{5270} + {\rm Fe}_{5335}),
\end{eqnarray}
where ${\rm Fe}_{5270}$ + ${\rm Fe}_{5335}$ are iron Lick indices.
\section{Kinematics and light distribution}
\label{sec:kinem_results}
We found clear evidence of the presence of two distinct stellar
kinematic components in NGC 3521. Our result is supported by two
different methods: the non-parametric LOSVD recovery (MPL) and a
parametric fitting (pPXF). Contrarily to the previous studies
\citep{Zeilinger+01, Fabricius+12}, we found that the two stellar
components in NGC 3521 co-rotate with respect to each other and with
respect to the ionized-gas. The main reason for the disagreement is
that previous data had poorer spectral resolution ($\geq 40$ km s$^{-1}$).
This generated spurious signals in the Fourier space when recovering
the shape of the LOSVD, which was misinterpreted as signal of
counter-rotation.
On the basis of their kinematic properties, we can classify the two
components as kinematically ``cold'' and kinematically ``hot'', as
shown in Figure \ref{fig:kinematics}. As their names suggest, the
``cold'' component has large rotational velocity (up to about 150
km s$^{-1}$\ along the major axis) and low velocity dispersion (about 50
km s$^{-1}$\ over the observed field of view), whereas the ``hot'' component
has low rotational velocity (up to about 50 km s$^{-1}$\ along the major
axis) and large velocity dispersion (about 120 km s$^{-1}$\ over the observed
field of view). Both the ``hot'' and ``cold'' stellar components show
peculiar kinematic features in the central 3\arcsec, e.g. an apparent
counter-rotating structure in the ``hot'' component velocity field,
irregular velocity field and high values of velocity dispersion in the
``cold'' component. These features are probably artifacts from the
spectral decomposition due to the very small velocity difference
between the stellar components, and not real features.
The ionized gas rotates in the same direction as the ``cold'' stellar
component. Its rotational amplitude is about 150 km s$^{-1}$\ along the major
axis and its velocity dispersion is about 25 km s$^{-1}$\ over the observed
field of view. The ionized-gas velocity dispersion peaks in the
central $\pm 2''$ ($\sigma \sim 100$ km s$^{-1}$); it is not clear if this is
due to unresolved rotation or to the presence of an AGN (NGC 3521 is
classified as LINER, \citealt{Goulding+09}).
It is natural to associate the cold and hot stellar components with
the disc and the bulge of NGC\,3521, despite their identification
being based on kinematics rather than photometry. In order to further
justify this association, in Section \ref{sec:kinem_phot_results} we
study the light contributions of the two kinematic components, as
returned by the spectral decomposition code, and compare them with those of
the bulge and disc as derived from photometry.
\subsection{Light distribution of bulge and disc}
\label{sec:kinem_phot_results}
In the following sections we study the light distribution of the two
kinematic stellar components identified in Section
\ref{sec:kinem_results}. We follow two complementary approaches: we
study the two-dimensional images (\ref{sec:light_2d}) and the
azimuthally averaged radial profiles (\ref{sec:light_1d}) of NGC 3521
and its structural components.
\subsubsection{Two-dimensional distribution}
\label{sec:light_2d}
Figure \ref{fig:decomposition_model} shows a two-dimensional map of
the light contribution of the two stellar components as derived from
the spectral decomposition. The map was created in the following
fashion. First we collapse the VIRUS-W data cube along the spectral
direction to obtain an image on the galaxy. We then multiply the
integrated flux in each pixel by the relative amplitude that we
obtained from the spectral decomposition code for a given
component. Both the collapsed image and the reconstructed image of the
host stellar component are asymmetric with a higher surface brightness
to the North-Eastern side of the centre as compared to the
South-Western side. On the other hand the reconstructed image of the
cold component does not show such asymmetry.
\begin{figure}
\psfig{file=decomposition_model_2.ps,width=8.4cm}
\caption{Reconstructed images of the respective two kinematic
components of NGC3521 and a model to explain the asymmetry. The top
and middle panels are the reconstructed images of the cold and hot
components, respectively, as they would appear if we could observe
them separately. The bottom panel shows a projected image of a model
spheroid that we obtained by a spherical deprojection of the
photometric bulge by \citet{Fabricius+12}, adding a dust screen
corresponding in inclination and position angle to the disc of
NGC3512 and reprojecting this model onto a 2D image.}
\label{fig:decomposition_model}
\end{figure}
This is easily explained by dust obscuration of the light that
originates from the part of the bulge that is located behind the near
side of the disc. The disc itself would -- at least if assumed to be
thin -- appear symmetric in this model because the line of sight
always penetrates about the same amount of disc light before reaching
the dust screen. On the other hand, the bulge is three dimensional:
the side further away will always be more dust obscured than the
closest side, creating an asymmetric image of the bulge.
To better understand this, we executed a simple spherical
decomposition of the bulge light using the decomposition parameters of
\citet{Fabricius+12} with an effective surface brightness of 15.5 mag
arcsec$^{-2}$, an effective radius of $8.5"$ and a S\'{e}rsic index of
3.7. To model the dust obscuration we then add a infinitely large and
infinitely thin plane through the centre of the spheroid to this model
with an inclination angle of 72.7$^{\circ}$ as derived for NGC3521 by
\citet{Bagetakos+11}. We found that an obscuration to 80\% (in the
sense that 80\% of the light behind the disc is blocked), gives a good
match with the observations. We finally reproject the image onto a two
dimensional plane.
The result is shown in the bottom right of
Fig. \ref{fig:decomposition_model}. There is reasonable qualitative
agreement between the reconstructed image of the high velocity
dispersion component and the screen spheroid, suggesting that the two
components of the spectral decompositions do indeed correspond to
physical counter parts.
\subsubsection{Radial surface brightness profiles}
\label{sec:light_1d}
Figure\ \ref{fig:photometry} shows the bulge/disc surface brightness
profiles as measured by \citet{Fabricius+12} with the -2.5 log of the
median counts measured in elliptical annuli on the hot/cold
components. From Fig. \ref{fig:photometry} we see a qualitative
correspondence, albeit not optimal, between the kinematic and
photometric components. The surface brightness radial profile of the
kinematically cold component well matches that of the disc. The
kinematically hot component has a steeper profile and is less luminous
than the kinematically cold component, in agreement to the bulge
profile. However, the spectral decomposition suggests a somewhat
slower drop-off as function of radius than the photometric
decomposition. Part of these differences can be attributed to the
difference between the VIRUS-W (optical) spectral range and the
near-IR photometric data.
\subsubsection{Conclusions about the light distribution}
Driven by the similarities described in Sections \ref{sec:light_2d}
and \ref{sec:light_1d}, we refer to the cold and hot components as
``disc'' and ``bulge'' in the rest of the paper. However, the
differences highlighted in these sections prevented us from
using the relative light contributions from the photometric
decomposition as priors for the spectroscopic decomposition.
\begin{figure}
\psfig{file=photometry.ps,width=8.7cm}
\caption{Comparison between i) the surface brightness radial
profiles of the best-fitting photometric model (black line), bulge
(red line) and disc (blue line) components in NGC\,3521 as
determined by photometric decomposition (from
\citealt{Fabricius+12}), and ii) the surface brightness as
measured on the images of the two kinematic components determined
by the spectroscopic decomposition (filled circles, this work).
These images are obtained by first collapsing the VIRUS-W
datacube along the wavelength direction and by then multiplying
each of the resulting pixel values by the amplitude of
the two respective components from the spectral decomposition.
The black points represent the combined contribution of the two
kinematic components; their photometric zeropoint is determined
to minimize the scatter with the best-fitting photometric model
(black line).}
\label{fig:photometry}
\end{figure}
\begin{figure*}
\hspace{.2cm}
\vbox{
\hbox{
\psfig{file=wmap_light4.ps,width=7.1cm}
\psfig{file=wmap_mass4.ps,width=7.1cm}}
\hbox{
\psfig{file=disk_wmap_light4.ps,width=7.1cm}
\psfig{file=disk_wmap_mass4.ps,width=7.1cm}}
\hbox{
\psfig{file=bulge_wmap_light4.ps,width=7.1cm}
\psfig{file=bulge_wmap_mass4.ps,width=7.1cm}}
}
\caption{Contribution in light (left panels) and mass (right panels)
of individual stellar populations to the total galaxy (top panels),
disc component (middle panels), and bulge component (bottom panels).
The histogram values and the gray scales in the upper panels
correspond directly to the weights that we obtained from pPXF using
the SSP models.}
\label{fig:sfh}
\end{figure*}
\section{Stellar populations}
\label{sec:populations}
In this section we describe the procedures adopted to study the
properties of the stellar populations of the bulge and disc in NGC
3521 independently. The results will be presented and discussed in
Section \ref{sec:discussion}. First, we exploit the high
signal-to-noise spectrum obtained by adding all the individual spectra
of each component to derive the luminosity-weighted and the
mass-weighted contributions of the multiple stellar populations that
are present in each component (Section \ref{sec:sfh0}). The analysis
is done by integrating the spectra over the entire radial range
(Section \ref{sec:sfh}) and by considering 3 radial bins to highlight
possible trends with radius (Section \ref{sec:sfh_radial}). Second, we
exploit the information of multiple absorption line indices to derive
the luminosity-weighted simple stellar population (age, metallicity,
and $\alpha$-enhancement) in each spatial bin in order to obtain the
two-dimensional maps of the properties of the bulge and disc (Section
\ref{sec:ssp}).
\subsection{Multiple stellar populations in the disc and the bulge of NGC\,3521}
\label{sec:sfh0}
In this section we identify the presence of multiple stellar
populations in NGC\,3521, by analyzing the observed spectrum and the
spectra of the bulge and disc as determined via the spectroscopic
decomposition. The analysis is done both globally (Section
\ref{sec:sfh}) and in 3 radial bins (Section \ref{sec:sfh_radial}).
In order to minimize the effects of degeneracy from the spectral
decomposition, we considered only the spectra in the bins where the
absolute velocity difference is higher than 50 km s$^{-1}$. Indeed, we find
that if we decompose the spatial bins where the velocity difference
between the two components is small, the models of the two components
are degenerate: the same total best model is reached when exchanging the
templates from one component to the other. By selecting only bins with
high velocity separation, the degeneracy is minimized by the different
position of the spectral features of the two stellar components.
\subsubsection{Integrated properties}
\label{sec:sfh}
The analysis is divided into 2 parts: first we analyse the observed
spectra of NGC 3521, secondly we analyse the bulge and the disc
separately.
Therefore, we have 3 spectra to fit: one for the entire system, one
for the disc, and one for the bulge.
The fit is performed using the pPXF routine with a set of simple
stellar population (SSP) models from \citet{Vazdekis+10} and Gaussian
functions to remove emission lines as templates. The SSP templates
span a wide range of ages (from 0.5 to 14 Gyrs), and they include
sub-solar ([Z/H]$=-0.25]$), solar ([Z/H]$=0.06]$), and super-solar
([Z/H]$=+0.40]$) metallicities \footnote{This choice is justified by
the range of values obtained in Section \ref{sec:ssp}}. We used SSP
models and not the ELODIE stars because, contrarily to Section
\ref{sec:kinematics}, we are interested in finding the stellar
population and not a precise fit to the kinematics.
The pPXF fitting routine computes the best-fitting stellar template as a
linear combination of all the individual SSP models in the library,
assigning a weight to each template. We divided each SSP model by its
median flux so that the weight assigned by pPXF to each SSP indicates
the contribution to the total light. By removing this normalization,
the weights indicate the contribution of that SSP model to the total
mass.
Figure \ref{fig:sfh} shows the distributions of best-fitting weights
as a function of age and metallicity for the observed spectrum (top),
the disc component (middle) and the bulge component (bottom). The left column
shows the contribution to the total light while the right column
shows the contribution to the stellar mass. We clearly identify 3
main components, an old population ($\geq7$ Gyr), an intermediate-age
population ($\approx$ 3 Gyr), and a young population ($\leq$1 Gyr). It
is beyond the scope of the paper to interpret the full star formation
history of NGC 3521 and study in detail how mass built up over
time. This, in theory, can be derived from the shape of the histograms
in the right panels of Fig. \ref{fig:sfh}. However, the fitting
procedure is sensitive to certain features in the spectra that do not
vary linearly with ages or with $\log$(age). For example, for age < 1
Gyr, the light (and therefore the features in the observed spectrum)
is dominated by blue massive stars, while for age > 5 Gyr, the light
is dominated by stars in the red giant branch. Therefore, one has to
be cautious in how to interpret the shape of the histograms in terms
of mass assembly over time. We therefore limit our analysis to 3 age
bins: $<1$ Gyr, 2--5 Gyrs, and $> 7$ Gyrs.
\subsubsection{Radial profiles}
\label{sec:sfh_radial}
We repeated the analysis of multiple stellar populations by grouping
the spectra of the bulge and disc in three radial bins. A finer radial
description is beyond the signal-to-noise of our data. The aim is to
study the radial variation of the contribution of the young,
intermediate, and old stellar populations to the mass of the bulge and
disc. This result is shown in Figure \ref{fig:sfh_radial}. The mass
fraction of the bulge and disc is dominated by the intermediate-age
component. The contribution of the young component to the disc mass
increases with radius, whereas it is negligible in the bulge.
\begin{figure}
\vbox{
\psfig{file=disk_mass_percentage_distribution.ps,width=8.5cm}
\psfig{file=bulge_mass_percentage_distribution.ps,width=8.5cm}
}
\caption{Radial distribution of the contribution of each stellar
population to the mass fraction of the disc (upper panel) and bulge (lower
panel). }
\label{fig:sfh_radial}
\end{figure}
\subsubsection{Computation of the errors in the contribution of each
stellar population}
\label{sec:sfh_errors}
In order to compute the errors on the contribution to the light and
mass of each stellar population (Figs \ref{fig:sfh} and
\ref{fig:sfh_radial}, we used a boot-strapping Monte Carlo approach.
The fit of each spectrum was repeated 100 times using the best-fit
model with the addition of random noise as the input spectrum. The added
noise is consistent with the observed signal-to-noise of the fitted
spectrum. Each of these 100 realization gave a different result; the
standard deviation of the results gives us the estimate of the error
due to the signal-to-noise.
\subsection{Spatially-resolved simple stellar population}
\label{sec:ssp}
In this section we exploit the information of several absorption line
indices to infer the luminosity-weighted simple stellar population
properties (SSP) and their spatial distribution. This is done both for
the bulge and the disc. Because we fit simple stellar population
models to components that are know to host multiple stellar
populations (see Section \ref{sec:sfh}), the results are biased towards the
most ``important'' component. According to \citet{Serra+07}, the
luminosity-weighted stellar SSP age derived for multiple populations
will be biased towards the youngest and most luminous component. On
the other hand, the luminosity-weighted SSP metallicity is less
affected by this bias than the age, and therefore it traces better the
properties of the main bulk of stars.
For each spatial bin, we determined the best fitting linear
combination of ELODIE spectra (broadened to the spectral resolution of
VIRUS-W) for each of the kinematic components. After convolving these
best-fitting spectra to the LICK spectral resolution (FWHM=8.4 \AA),
we measured the line strengths of the indices \hbeta, Mg {\it b}, Fe{\small 5270}, and
Fe{\small 5335}. The derived line strengths are then corrected using a linear
calibration to the Lick system that we obtained by comparing the
values measured on the stars in the stellar library that are in common
with the sample of \citet{Worthey+94}. We then fit stellar population
models by \citet{Thomas+03} that account for variable
$\alpha$-elements abundance ratio to the measured indices. We did not
include Fe{\small 5015}\ in our procedure because it was the index that deviated
most from the best fit. The distribution of absolute normalized
residuals of Fe{\small 5015}\ was about 4 times higher than those of the other
indices. Moreover, contrary to the other iron indices, the inclusion
of Fe{\small 5015}\ in the fit systematically increased the global metallicity by
0.2 dex.
For visualization purposes, in Figure \ref{fig:diagnostic_plots} we
show a combination of some of the measured indices in the \hbeta\
vs. [MgFe]'\ and Mg {\it b}\ vs $\langle$Fe$\rangle$\ parameter spaces, and compare them with
the prediction of simple stellar population models.
The two-dimensional maps of ages, [Z/H], and [$\alpha$/Fe] are shown
in Figure \ref{fig:ssp_maps} and their radial profiles computed along
concentric ellipses oriented as the galaxy photometric major axis are
shown in Figure \ref{fig:radial_profiles}. Similarly to what we did in
Section \ref{sec:sfh}, we highlight in
Figs. \ref{fig:diagnostic_plots}-\ref{fig:radial_profiles} the spatial
bins where the absolute velocity difference between the two stellar
components is higher than 50 km s$^{-1}$.
\begin{figure*}
\psfig{file=diagnostic_plots.ps,width=18cm,clip=}
\caption{Equivalent width of some of the absorption line indices
measured in the bulge (blue) and disc (red) spectra produced by the
spectral decomposition. Open symbols are the measurements in all the
spatial bins, filled symbols are the measurements in those spatial
bins were the velocity separation between disc and bulge is larger
than 50 km s$^{-1}$. Predictions from single stellar population models by
\citet{Thomas+03} are superimposed.}
\label{fig:diagnostic_plots}
\end{figure*}
\begin{figure*}
\psfig{file=ssp_maps1.ps,width=18cm,clip=,bb=10 385 588 515}
\psfig{file=ssp_maps2.ps,width=18cm,clip=,bb=10 385 588 515}
\psfig{file=ssp_maps1.ps,width=18cm,clip=,bb=10 10 588 30}
\psfig{file=ssp_maps1.ps,width=18cm,clip=,bb=10 342 588 385}
\caption{Two-dimensional maps of age (left), metallicty (middle) and
$\alpha$-enhancement for the disc (upper panels) and bulge (lower
panels) of NGC\,3521. Only spatial bins outside the central 5'' and
where the velocity separation between the two components is larger
than 50 km s$^{-1}$\ are shown.}
\label{fig:ssp_maps}
\end{figure*}
\begin{figure}
\psfig{file=radial_profiles.ps,width=8cm,clip=}
\caption{Values of age, metallicity and $\alpha$-enhancement for the
disc (blue symbols) and bulge (red symbols) of NGC\,3521. Only bins
where the velocity separation between the two components is larger
than 50 km s$^{-1}$\ are considered. Values computed in each individual
bin and plotted against the semi-major axis of the ellipse passing
on that bin. Semi-major axis are computed considering ellipses with
position angle $-19^{\circ}$ and ellipticity $e=0.35$
\citep{Fabricius+12}. Small open symbols represent the values in
individual bins, filled circles represent the mean value computed
on elliptical annuli. Error bars of filled symbols are the standard
deviations of the values within a give elliptical annulus. The
continuous lines are the linear fit to the mean values.}
\label{fig:radial_profiles}
\end{figure}
\section{Discussion and conclusions}
\label{sec:discussion}
The spectroscopic decomposition clearly revealed the presence of two
stellar kinematic components in NGC\,3521, which we identified with
the bulge and the disc, and one ionized-gas component. The stars in
the bulge and disc co-rotate and are characterized by different
morphology, kinematics (velocity and velocity dispersion), stellar
population content, and equivalent width of the absorption lines.
The integrated fit of multiple stellar populations
(Sect. \ref{sec:sfh0}) reveals the presence of three populations of
stars that are highlighted in the upper panels of Fig.
\ref{fig:sfh}:
\begin{itemize}
\item a young ($\leq$1 Gyr) and metal poor population ([Z/H]$ \lesssim 0$);
\item an intermediate-age ($\approx$ 3 Gyr) and metal rich component
([Z/H]$ \gtrsim 0$), which dominates the light and mass of NGC
3521;
\item an old population ($\geq7$ Gyr) that spans a large range of
metallicity values ($-0.25 \lesssim $[Z/H]$ \lesssim 0.4$).
\end{itemize}
The presence of a young component (age $\leq 1$ Gyr, that we are able
to associate mainly with disc stars, see Fig. \ref{fig:sfh}) is
compatible with the independent detection of active star formation in
NGC\,3521 ($\log \Sigma_{SFH}/[{\rm M}_{\odot} yr^{-1} kpc^{-2}]$ =
-2.583, \citealt{Calzetti+10}).
Exploiting the spectral decomposition, we managed to remove the mutual
contamination of bulge and disc, and therefore we could investigate
how these stellar populations distribute among the bulge and the disc
of NGC\,3521. As shown in the middle and bottom panels of Figure
\ref{fig:sfh}, the light of the disc is dominated by the young and
intermediate populations, whereas the bulge is dominated by the
intermediate population. The mass of the disc is dominated by the
intermediate and old components, whereas the mass of the bulge is
dominated by the intermediate component. The stars of the old
population that are associated to the disc are metal rich
(Fig. \ref{fig:sfh}, middle panels), whereas the stars of the old
population that are associated to the bulge are metal poor
(Fig. \ref{fig:sfh}, bottom panels).
Figure \ref{fig:sfh_radial} indicates that the different populations
of stars have a distribution in the disc and bulge that depends on
radius. Indeed, the percentage of young stars in the disc increases
with radius. On the other hand, the young stars in the bulge are
concentrated only in the centre, and their (negligible) contribution
to the bulge mass decrease with radius.
The equivalent width of the spectral indices and the
luminosity-weighted simple stellar populations of the bulge are
different from those of the disc (Figures \ref{fig:diagnostic_plots},
\ref{fig:radial_profiles}). With respect to the luminosity-weighted
simple stellar population, the main difference between bulge and disc
is seen in their ages. The disc is younger and has a negative age
radial gradient (from $1.7 \pm 0.5$ Gyr in the innermost elliptical
bin down to $0.7 \pm 0.2$ Gyr in the outermost elliptical bin). The
bulge is slightly older with an average age of $1.9 \pm 0.4$ Gyr (see
Fig. \ref{fig:radial_profiles}).
Both components have very similar luminosity-weighed metallicity and
$\alpha$-enhancement, within the uncertainties. Their metal content is
super-solar and has a mild negative radial profile (from $0.4 \pm 0.1$
in the innermost elliptical bin down to $0.2 \pm 0.2$); their alpha
enhancement is sub-solar and nearly constant towards the entire
observed radial range ($\langle
$[$\alpha$/Fe]$\rangle \pm -0.14 \pm 0.03$).
The results of Sections \ref{sec:sfh0}, and \ref{sec:ssp} are
consistent with each other, at least from a qualitative point of
view. The range of luminosity-weighted ages as determined via the SSP
fit in Section \ref{sec:ssp} (disc $\leq 2$ Gyr, and bulge $\approx 2$
Gyr) is of the same order as the mean age of the luminosity-dominant
stellar populations measured in Section \ref{sec:sfh} ($\approx 3$
Gyr, Fig. \ref{fig:sfh} left panels). Moreover, the range of mean
luminosity-weighted metallicities ($0.2 \leq {\rm [Z/H]} \leq 0.4$,
Fig. \ref{fig:radial_profiles}) is consistent with the metallicity of
the main bulk of stars ($0 \leq {\rm [Z/H]} \leq 0.4$,
Fig. \ref{fig:sfh} right panels). The negative age gradient of the
disc (Fig. \ref{fig:radial_profiles}, upper panel) is a consequence of
the mild radial increase of the contribution of young stars to the
mass of the disc (Fig. \ref{fig:sfh_radial}, upper panel).
From the combined information of the integrated multiple stellar
populations analysis (Sect. \ref{sec:sfh}), and the spatially-resolved
simple stellar population analysis (Sect. \ref{sec:ssp}), we can infer
the following formation scenario for NGC\,3521.
\begin{table}
\begin{center}
\begin{tabular}{l c c c}
\hline
Component & $\Delta$Age & $\Delta$[Z/H] & $\Delta$[$\alpha$/Fe] \\
& [Gyr kpc$^{-1}$] & [dex kpc$^{-1}$] & [dex kpc$^{-1}$] \\
\hline
Disc & --0.8$\pm$ 0.4 & --0.3 $\pm$ 0.2 & 0.1 $\pm$ 0.1 \\
Bulge & 0.1 $\pm$ 0.1 & --0.27 $\pm$ 0.04 & --0.1 $\pm$ 0.1 \\
\hline
\end{tabular}
\end{center}
\caption{The measured radial gradient of the luminosity-weighted simple stellar population parameters for the disc and bulge components in NGC\,3521.}
\label{tab:gradients}
\end{table}
The galaxy formed with multiple formation episodes, the first
occurring $> 7$ Gyrs ago from metal rich material. A second episode of
star formation occurred about 3 Gyr ago; this episode involved both
components, but it affected mainly the bulge. Because the
intermediate-age population is also the most luminous, the age radial
profiles shown in Figure \ref{fig:radial_profiles} refer to this
second episode.
The third episode of star formation occurred $\leq 1.5$ Gyr ago and
added mass mainly to the disc component. This population of stars dominates
the light of the disc component (although their mass contribution is
small), therefore the radial age gradient shown in Figure
\ref{fig:radial_profiles} describes these stars. This radial profile
suggests that the formation of the young disc stars started right after the
formation of the intermediate-bulge population of stars in the
innermost elliptical bin. Because the mass contribution of the
youngest disc star is very small ($\sim 8$\% of the disc stellar mass,
Fig. \ref{fig:sfh}), the radial metallicity gradient shown in Figure
\ref{fig:radial_profiles} describes the intermediate and old disc
stars.
The metallicity of the stars associated with the youngest disc component
ranges from solar to sub-solar (Fig. \ref{fig:sfh}), therefore
they cannot originate from recycled material of previous generation of
stars (either from disc or bulge), because the latter is are more metal
rich. The material must come from outside the disc and the bulge of
NGC\,3521. On the other hand, there is no evidence of gas accretion
onto NGC\,3521 from the intracluster medium \citep{Elson14}. The most
probable scenario is that the origin of this young component is
material coming from the so-called ``anomalous'' gas halo surrounding
NGC\,3521 detected by \citet{Elson14}. The authors also suggest that
the interplay between the stellar feedback and star formation is
regulated by galactic fountains that transfer material from and to
the halo, which is consistent with our interpretation.
The disc of NGC\,3521 has luminosity-weighted properties that are
similar to those of other spiral galaxies for which multiple
age-components are detected (e.g., \citealt{Morelli+15}).
However, the majority of the disc light in the sample of
\citet{Morelli+15} is dominated by stars older than 7 Gyrs (Figure 2
in their paper), whereas the light of the disc of NGC 3521 is
dominated by stars younger than 7 Gyrs.
The bulge of NGC\,3521 is consistent with those of other bulges in
spirals where an attempt was made to minimize the contamination from
the disc \citep{Morelli+08, Morelli+12}. The majority of the bulges in
those studies have intermediate ages and super solar metallicity,
positive age gradients, and negative metallicity and $\alpha$
gradients, similar the bulge of NGC 3521.
The properties of NGC\,3521 differ from those of the majority of S0
galaxies in which spectral decomposition was attempted to isolate the
contribute of disc and bulge \citep{Johnston+12, Tabor+17}. The
bulges in those S0 samples have intermediate ages, as the majority of
the stars in the bulge of NGC 3521. However there is evidence that the
bulges in those samples are younger than the corresponding discs,
unlike NGC 3521.
We cannot rule out merging as an alternative scenario opposite to the
in-situ formation of stars as builder of the structural components in
NGC\,3521, at least for the old and the intermediate
populations. Indeed, as discussed in Section \ref{sec:fors1}, the
galaxy shows deviations from a smooth axi-symmetric structure with
spiral arms. These structures could be the relics of the past merger
history of NGC\,3521 that contributed to the build-up of its
components. However, it is difficult to associate these non
axi-symmetric structures to the youngest disc component in NGC\,3521.
Indeed, they extend perpendicularly to the disc and are apparently
detached from the blue spiral structure that trace the youngest
component (Figure \ref{fig:fors1}). Therefore, if a merger indeed
contributed to the formation of the stellar components in NGC 3521, it
is more likely that it directly contributed to formation of the old
and/or the intermediate-age populations. The same merger could also be
the mechanism that allowed NGC 3521 to acquire the ``anomalous'' HI
cloud, from which the youngest stars in the disc have originated.
In summary, we propose the following formation scenario. The bulge and
disc components of NGC\,3521 formed a long time ago from stars of
similar age; the formation was followed by a rejuvenation process (or
accretion of younger stars via a merger) that involved both
components, but mainly the bulge. Then, very recently, the disc of NGC
3521 accreted material from the surrounding gas halo, which could have
been acquired during a past merger, and formed a new generation of
stars in an inside-out manner.
NGC 3521 is yet another good example of how it is possible to recover
the formation and mass assembly of bulges and discs via the
spectroscopic decomposition approach independently, especially for
cases of fainter bulges embedded in luminous discs, or vice-versa. The
next natural step for this kind of investigation is to study a
representative sample of galaxies to understand what are the preferred
formation channels.
\section*{Acknowledgements}
We wish to thank Michael Opitsch for useful discussion during the
preliminary phase of this work. LC thanks the Department of
Physics and Astronomy of the Padova University for hospitality while
this paper was in progress.
|
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| 5,688
|
HUSKER VOLLEYBALL: Nebraska is No. 10 In NCAA Tourney, Will Face Campbell In First Round
Nebraska volleyball setter Nicklin Hames (Courtesy of Scott Bruhn/NU Communications)
LINCOLN–(NU Athletics Nov. 28)–The Nebraska volleyball team was selected as the No. 10 overall seed for the NCAA Tournament Sunday night and will host the NCAA first and second rounds at the Bob Devaney Sports Center this coming weekend.
The Huskers, making their 40th straight NCAA Tournament appearance, will open the tournament on Friday, Dec. 3 at 7 p.m. against Campbell (21-9), champions of the Big South Conference. The 4:30 p.m. match will feature Florida State (19-9) of the Atlantic Coast Conference and Kansas State (15-12) of the Big 12 Conference. Friday's winners will meet on Saturday, Dec. 4 at 7 p.m.
Louisville, Texas, Pitt and Wisconsin received the top four seeds from the NCAA Tournament Committee. Should those teams all advance past the first weekend, they would host an NCAA Regional the following weekend. Nebraska is in the same region as Texas along with No. 7 national seed Kentucky and No. 15 Washington. The Huskers would have an opportunity to host an NCAA Regional only if they advance past the opening two rounds and Texas and Kentucky were upset in the first or second round.
Tickets to the first and second rounds in Lincoln are available for purchase online at Huskers.com/tickets as of Sunday night. Tickets will be available by phone at 800-8-BIG RED beginning at 8 a.m. Monday based on availability. Streaming and broadcast information will be announced later this week.
Friday will be Nebraska's first meeting in program history with Campbell, which earned its first NCAA Tournament berth. Kansas State was part of NU's Husker Invitational tournament field to open the 2021 season, and Nebraska leads the all-time series with the Wildcats, 84-4. The Huskers last faced Florida State in 2014 at the AVCA Showcase and lead the all-time series with the Seminoles, 2-1.
The Huskers, who finished the regular season at 21-7 overall, took second in the Big Ten Conference this season. Nebraska carries a streak of nine straight NCAA Regional Final appearances and owns five national championships (1995, 2000, 2006, 2015 and 2017). It is the most recent Big Ten program to win a national championship (2017).
NU also ranks second in NCAA history in postseason wins (118-34) and winning percentage (.776) over 39 previous tournament appearances.
The 2021 season marks the 36th time that Lincoln has hosted NCAA first- and second-round competition.
The 2021 Husker squad looks to return to the Final Four for the 16th time in program history. This year's NCAA Championship is set for December 16-18 at Nationwide Arena in Columbus, Ohio.
NCAA FIRST AND SECOND ROUND SCHEDULE
First Round – Friday, Dec. 3
4:30 p.m. – Kansas State vs. Florida State
7 p.m.* – Campbell vs. Nebraska
*or 30 minutes following the conclusion of the 4:30 p.m. match, but no sooner than 7 p.m.
Second Round – Saturday, Dec. 4
7 p.m. – First Round Winners.
|
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Home News/Ent One Direction, Demi Lovato, Little Mix and Maisie Williams to present Christmas...
One Direction, Demi Lovato, Little Mix and Maisie Williams to present Christmas Day shows on BBC Radio 1
Flavourmag Team
BBC Radio 1 today announced that some of the hottest names in the world including One Direction, Demi Lovato and Little Mix will host special Christmas Day shows on the station this year.
From 1-7pm, One Direction, The Vamps, Troye Sivan, Demi Lovato, Little Mix and Maisie Williams will take to Radio 1's airwaves to host'Radio 1's Superstar Playlist' – hour long shows presented and curated by the stars – featuring their favourite tracks, including a few Christmas favourites to get Radio 1 listeners in the festive spirit.
Radio 1's Superstar Playlist schedule:
1-2pm: Radio 1's Superstar Playlist with One Direction
2-3pm: Radio 1's Superstar Playlist with Demi Lovato
3-4pm: Radio 1's Superstar Playlist with Troye Sivan
4-5pm: Radio 1's Superstar Playlist with Little Mix
5-6pm: Radio 1's Superstar Playlist with Maisie Williams
6-7pm: Radio 1's Superstar Playlist with The Vamps
One Direction say: "The best present this Christmas will be coming round to Alice Levine¹s and sharing the day with the Radio 1 listeners."
The Vamps say: "We're so excited to give the BBC Radio 1 DJ's a run for their money by taking over the decks on Christmas Day playing our favourite choooons and chatting about reeeally important things like snow, turkey vs beef debate and presents. Come party with us at 6pm. The playlist is mega!"
Rhys Hughes, Head of Programmes, Radio 1 and 1Xtra said: "I'm absolutely delighted to announce such a stellar line up for Radio 1 on Christmas day. Along with the Christmas No.1 on The Official Chart with Greg James, it should be a great listen, bringing the nation together on Christmas Day."
Prior to 'Radio 1's Superstar Playlist', for the first time ever on a Friday, Greg James will be announcing the Christmas Number 1 live on The Official Chart (10am-1pm). This will be the first time the Christmas Number 1 has been announced on a Friday since moving from its Sunday slot, and Greg's first time announcing it. The three-hour show will include a look back at all of the Christmas Number 1's since the year 2000 and the Official Christmas Chart, helping listeners celebrate the family festivities.
Greg James says: "It's the biggest chart of the year, and it's worth delaying my entire family's Christmas celebrations for! I love being the Official Chart host, and I'm so excited that I'll be announcing the Christmas Number 1 live on Christmas Day."
Listeners at home will be able to enjoy the show live on Radio 1, via mobile and the BBC iPlayer Radio app.
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{"url":"https:\/\/www.lesswrong.com\/posts\/zcPLNNw4wgBX5k8kQ\/decision-theory?commentId=yjncGDysPXEGE5Jgz","text":"## LESSWRONGLW\n\nYou can, for a certain value of \"can\". It won't have happened, of course, but you may still decide to act contrary to how you act, two different outcomes of the same algorithm.\n\nThis confuses me even more. You can imagine act contrary to your own algorithm, but the imagining different possible outcomes is a side effect of running the main algorithm that takes $10. It is never the outcome of it. Or an outcome. Since you know you will end up taking$10, I also don't understand the idea of playing chicken with the universe. Are there any references for it?\n\nYou don't know that it's inaccurate, you've just run the computation and it said $5. Wait, what? We started with the assumption that examining the algorithm, or running it, shows that you will take$10, no? I guess I still don't understand how\n\nWhat if you see that your algorithm leads to taking the $10 and instead of stopping there, you take the$5?\n\nis even possible, or worth considering.\n\nThis map from predictions to decisions could be anything.\n\nHmm, maybe this is where I miss some of the logic. If the predictions are accurate, the map is bijective. If the predictions are inaccurate, you need a better algorithm analysis tool.\n\nThe map doesn't have to be identity, decision doesn't have to reflect prediction, because you may write an algorithm where it's not identity.\n\nTo me this screams \"get a better algorithm analyzer!\" and has nothing to do with whether it's your own algorithm, or someone else's. Can you maybe give an example where one ends up in a situation where there is no obvious algorithm analyzer one can apply?","date":"2020-05-30 03:33:08","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\": 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\": 0.6262272596359253, \"perplexity\": 540.1379467807778}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347407001.36\/warc\/CC-MAIN-20200530005804-20200530035804-00312.warc.gz\"}"}
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{"url":"https:\/\/icml.cc\/Conferences\/2021\/ScheduleMultitrack?event=11888","text":"Timezone: \u00bb\n\nRobust Generalization of Quadratic Neural Networks via Function Identification\nKan Xu \u00b7 Hamsa Bastani \u00b7 Osbert Bastani\n\nA key challenge facing deep learning is that neural networks are often not robust to shifts in the underlying data distribution. We study this problem from the perspective of the statistical concept of parameter identification. Generalization bounds from learning theory often assume that the test distribution is close to the training distribution. In contrast, if we can identify the true'' parameters, then the model generalizes to arbitrary distribution shifts. However, neural networks are typically overparameterized, making parameter identification impossible. We show that for quadratic neural networks, we can identify the function represented by the model even though we cannot identify its parameters. Thus, we can obtain robust generalization bounds even in the overparameterized setting. We leverage this result to obtain new bounds for contextual bandits and transfer learning with quadratic neural networks. Overall, our results suggest that we can improve robustness of neural networks by designing models that can represent the true data generating process. In practice, the true data generating process is often very complex; thus, we study how our framework might connect to neural module networks, which are designed to break down complex tasks into compositions of simpler ones. We prove robust generalization bounds when individual neural modules are identifiable.","date":"2022-01-18 12:33:39","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\": 0.6714472770690918, \"perplexity\": 487.6624439812597}, \"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-2022-05\/segments\/1642320300849.28\/warc\/CC-MAIN-20220118122602-20220118152602-00415.warc.gz\"}"}
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{"url":"https:\/\/blender.stackexchange.com\/tags\/interpolation\/hot","text":"# Tag Info\n\n10\n\nThere is no way to customize them. The only way out is to not use the premade interpolations. Stay in Bezier interpolation, and use \"free\" handles to create your own bounces:\n\n8\n\nAfter some digging through the Blender source code, I found the answer. Short answer: They're just B\u00e9zier curves. Long answer: They're normal B\u00e9zier curves with certain restrictions placed on the positions of the handles (the red circles). If the first handle is to the left of the second handle, as in the following image, then the curve is evaluated as a ...\n\n8\n\nAn easier way to roll a ball downhill is to use rigid bodies. This way, the active objects are subject to the scene gravity. For this, you will want two rigid bodies, one active and one passive. The active one will be the ball, the passive will be slope. Add rigid body physics to each object in the physics tab: These are the settings for the ball: These ...\n\n7\n\nIt's right under your nose ;-) They are a member of the \"Bezier Spline\" family called an \"F-curve,\" shortened from \"Ferguson's Parametric Cubic Curves.\" The F-curve is actually a direct derivative of the Catmull-Rom Spline (names you should recognize from other CGI algorithms). As stated in the introductory research an f-curve: \"attempts to fix the ...\n\n7\n\nTo modify the vectors that define the curve, you can press the V key (or select Key > Handle type) That will allow you to align the handles into a straight line, making a smoother transition between two vectors.\n\n6\n\nI'll post another answer since this one will actually solve your problem (I hope). Use this script to set the interpolation of each and every one of your \"Image Texture\" nodes to \"Closest\" : import bpy for mat in bpy.data.materials: if not mat.node_tree: continue for node in mat.node_tree.nodes: if node.type == '...\n\n6\n\nLet's look at what the two mean and create examples. Our example image will be this (enlarged) $2\\times2$ pixel grid: Linear: In spaces between other pixels, there is a linear gradient of the two mixing colors: Cubic: In spaces between other pixels, there is a cubic (also called Ease) gradient between the two colors: When we compare the linear and cubic ...\n\n6\n\nSelect your keyframes in the Graph Editor and you can use Shift + Ctrl + M or use the menu Key -> Add F-Curve Modifier -> Stepped Interpolation. The options for the modifier are in the N-panel.\n\n5\n\nUse offset factor of follow path constraint. Use the offset factor of follow curve constraint. Test script, select the curve in object mode and run script. import bpy context = bpy.context scene = context.scene curve_obj = context.object spline = curve_obj.data.splines[0] bpy.ops.mesh.primitive_ico_sphere_add(size=0.05, location=(0, 0, 0)) sphere = ...\n\n5\n\nNo, you can't adjust the bounce interpolation. However, there are manual ways to progress from your state. Select the two encasing keyframes and press ShiftO (2.7x) or ShiftAltO (2.8x) to sample frames between them, reset their interpolation type to Bezier afterwards. You can now manipulate them as you wish. Note, that this is also important as the down ...\n\n5\n\nThe gradient is linear but not in the colourspace you are expecting - as evidenced by the following : Here the Attribute node is making the Vertex Color (Col) available to the shader. This is split into separate RGB components and compared with the Generated 'X' coordinate (which runs from 0.0 at the left to 1.0 at the right of the cube). The gradient is ...\n\n5\n\nIn general, you cannot access data from some other sample than the one you're evaluating. Not in Blender. (This is someplace where I wish we had partial derivatives to work with in shaders....) So you can see the vertex color or location of a sample, but not of two samples, not of some sample and some particular vertex, etc. One reason vertex color doesn'...\n\n4\n\nThat's not too hard actually: I just move the UV coordinates around. Create a seamless background image. Left and right borders should match. UV-Map it on a plane that fills your background. Use the Node Editor for the material and create this node setup: The texture coordinates are provided to a Vector Math node. There, I am adding both vectors together, ...\n\n4\n\nMake sure you are on the shader editor. In 2.9.12 the interpolation options are there:\n\n3\n\nThe first tip is that pressing T will apply the interpolation to every selected keyframe. First, to simplify the graph editor, hide curves you aren't working on. It would appear that you want to do this to every third keyframe, so select all won't help. We have the common selection options in the graph editor. B to box select keyframes C to circle select ...\n\n3\n\nThe problem is that you're using Quaternion (WXYZ) Rotation, which is the default setting for bones in Pose Mode. Set the Rotation method to XYZ Euler and animate the Y Rotation from 0\u00b0 to 360\u00b0. You should also set the last keyframe with 360\u00b0 on frame 101 instead of 100 if you want a looped animation from frames 1-100, because 0\u00b0 = 360\u00b0 and the animation ...\n\n3\n\nYou might find it more intuitive to use an RGB Curve node once you've normalized the value using Map Range.\n\n2\n\nIn 3D animation tools I've used, the key difference is that animation curves as shown in the graph editor are not actually 2D B\u00e9zier splines. They're 1D splines, parameterized by time (plotted along the X axis in the graph editor). When you use splines to create curves in 2D or 3D, the spline has a parameter t that runs from 0 to 1 along its length, and the ...\n\n2\n\n@eromod's hint was the solution. I created a huge ovoid (an icosphere that I compressed on one axis for an additional hump on the other axis), aligned it carefully manually, extended the original top surface in top direction, applied a boolean modifier with \"Intersect\" operation, and did a bit afterwork on the mesh. Please leave a comment if there's ...\n\n2\n\nYou are not supposed to be changing Extrapolation mode, that is used for \"extending\" or repeating an animation after the last keyframe, as far as I know. You are supposed to change the handle types, instead. Select all your keyframes with A and then press V Set Keyframe Handle Type > Vector\n\n2\n\nIt's not exactly what you're asking for, but maybe it can help : there is an (integrated) addon, Simplify Curves. It reduces the number of keyframes, preserving the general shape of the curve. Activate the addon (search \"simplify\" and click the checkbox next to \"Add Curve: Simplify Curves\") Select your F-curves type Spacebar then \"simplify\", select \"...\n\n2\n\nThe settings in the N region only apply to the active keyframe (usually the one selected last). To change the interpolation for all selected keyframes, press Header > Key > Interpolation Mode in the graph editor header, or press T while hovering over the graph editor.\n\n2\n\nAs far as I know you can't modify the default setting easily but you can use a simple node group to force the behaviour wherever you want to use your image texture node : Related : https:\/\/blender.stackexchange.com\/a\/168033\/86891\n\n2\n\nWhile can't see how you do it, here is how i do it: You frame all the wanted keyframes including their handles Either go in the menu on top or use the context menu with RMB to change the interpolation. Here also a visual step by step: Should this not work with your version, please explain where it differs from this.\n\n2\n\nyou can get the result you want by plugging the generated texture coordinates from the vertex positions of your cube in the Texture Coordinate node instead of the Geometry node. Here's the result :\n\n2\n\nIn the Dope Sheet you can select one or more keyframes, right click on it and select the interpolation of the segments following the selected keyframes. You can choose between ease in, ease out, ease in and out, and then choose between various interpolation rates (linear, sinusoidal, quadratic, cubic, ....., bounce, overshoot, elastic). In 2D grease pencil ...\n\n1\n\non this picture there are two nodes with UV parameters\n\n1\n\nIn general - each vertex is usually used to store only one color. In your example one vertex consists from three colors. So the only one way to interpolate is to set average color.\n\n1\n\nIf you only need grayscale data to be used in shader nodes, here's an ugly workaround - use alpha channel instead. In your script I set the alpha values to 0.25 and 0.75 respectively and using a greater than 0.5 node visually confirmed that the alpha channel is interpolated lineary. Perhaps other attributes could be used as well, like UV. I'm afraid a ...\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2021-10-26 03:36:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.34583550691604614, \"perplexity\": 1791.8262656106392}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323587794.19\/warc\/CC-MAIN-20211026011138-20211026041138-00413.warc.gz\"}"}
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Q: Can a python unittest class add an assert statement for each test method in a parent class? I have a unit test class that is a sub-class of python's unittest:
import unittest
class MyTestClass(unittest.TestCase):
run_parameters = {param1: 'on'}
def someTest(self):
self.assertEquals(something, something_else)
Now I want to create a child class that modifies, say run_parameters, and adds an additional assert statement on top of what was already written:
class NewWayToRunThings_TestClass(MyTestClass):
run_parameters = {param1: 'blue'}
# Want someTest, and all other tests in MyTestClass to now run
# with an additional assert statement
Is there someway to accomplish this so that each test runs with an additional assert statement to check that my parameter change worked properly across all my tests?
A: Yes there is but it may not be a good idea because:
*
*assertions are hidden behind difficult to understand python magic
*assertions aren't explicit
Could you update your methods to reflect the new contract, and expected param?
Also if a single parameter change breaks a huge amount of tests, where it is easier to dynamically patch the test class then it is to update the tests, the test suite may not be focused enough.
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"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/math.stackexchange.com\/questions\/805279\/am-i-wrong-2\/805290","text":"# Am I wrong ? (2)\n\nLet $X=C[0,1]$ be the space of real continous functions on $[0,1]$. $X$ is a Banach space with the two norms $$|f|_\\infty=\\sup_{s\\in[0,1]}|f(s)|$$ and $$|f|_2=\\left(\\int_0^1|f(s)|^2\\right)^\\frac{1}{2}$$ The canonical map $(X,|.|_\\infty)\\to (X,|.|_2):f\\mapsto f$ is a bounded bijective operator since $$|f|_2\\leq |f|_\\infty.$$ Then by the \"Bounded inverse theorem\" the inverse of this map is also continuous and we have a constant $c>0$ such that $$|f|_\\infty\\leq c|f|_2.$$ I feel something is wrong in this.\n\nThe problem is that $X$ is not a Banach space when equipped with the second norm (the $L^2$ norm). The continuous functions are dense in $L^2$. They aren't closed, so they aren't a Banach space.","date":"2020-04-10 10:31:28","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\": 0.8812835216522217, \"perplexity\": 55.27208665518153}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371893683.94\/warc\/CC-MAIN-20200410075105-20200410105605-00223.warc.gz\"}"}
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{"url":"http:\/\/www.ufs.edu.pl\/hussar_doc\/phipade.html","text":"phipade - Evaluate phi functions using (6,6)-Pad\ufffd approximations.\n\nSYNOPSIS:\n[phi_1, phi_2, ..., phi_k] = phipade(z, k);\n\nDESCRIPTION:\nThis function evaluates phi functions needed in exponential\nWe define the phi functions according to the integral representation\n\n\\phi_k(z) = \\frac{1}{(k - 1)!} \\int_0^1 e^{z (1-x)} x^{k-1} dx\n\nfor k=1, 2, ...\n\nPARAMETERS:\nz - Evaluation point. Assumed to be one of\n- 1D vector, treated as the main diagonal of a diagonal matrix\n- sparse diagonal matrix\n- full or sparse matrix\nk - Which phi function(s) to evaluate.\nIndex (integer) of the (highest) phi function needed.\n\nRETURNS:\nphi_k = \\phi_k(z)\n[phi_1, phi_2, ..., phi_k] = DEAL(\\phi_1(z), \\phi_2(z), ..., \\phi_k(z))\n\nNOTES:\nWhen computing more than one phi function, it is the caller's\nresponsibility to provide enough output arguments to hold all of the\n\\phi_k function values.\n\nFor efficiency reasons, phipade caches recently computed function\nvalues. The caching behaviour is contingent on the WANTCACHE\nfunction and may be toggled on or off as needed.","date":"2018-08-16 08:40:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.6420937180519104, \"perplexity\": 14509.524181426836}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221210559.6\/warc\/CC-MAIN-20180816074040-20180816094040-00551.warc.gz\"}"}
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{"url":"https:\/\/math.stackexchange.com\/questions\/2468327\/substitution-in-integration-about-log\/2468330","text":"# Substitution in Integration about log\n\nA question on my book: $$\\int{\\frac{1}{x(\\log{x})^n}dx}$$where n is an integer.\n\nWhat I did:\nLet u=$\\frac{1}{\\log{x}}$, then $\\frac{du}{dx}=x, dx=\\frac{du}{x}.$\n\nNow the integral becomes $$\\int{\\frac{1}{xu^n}dx}=\\int{\\frac{1}{xu^n}\\times {\\frac{du}{x}}}$$\n\nI don't know what to do next. Should I change another u? Note: if possible, please use substitution to solve the problem.\n\n\u2022 $\\dfrac{\\mathrm d u}{\\mathrm d x}=-\\dfrac1{x\\log^2x}$, not $\\;\\dfrac1{\\log' x}$. \u2013\u00a0Bernard Oct 11 '17 at 22:10\n\nConsider the substitution $u=\\log x$ and so $du=\\frac{1}{x}dx$. Hence,$$\\int \\frac{1}{x(\\log x)^n}\\,dx=\\int\\frac{(\\log x)^{-n}}{x}\\,dx=\\int u^{-n}\\,du.$$ Then, for $n\\geq 2$, we get$$\\int u^{-n}\\,du=\\frac{u^{1-n}}{1-n}=\\frac{(\\log x)^{1-n}}{1-n}.$$ For $n=1$, we get$$\\int u^{-1}\\,du=\\log u=\\log(\\log x).$$\n\u2022 Not true if $n=1$. \u2013\u00a0Mark Viola Oct 11 '17 at 23:02\nCall $u=\\log x$, then $du = dx\/x$ and your integral becomes\n$$\\int\\frac{dx}{x\\log^n x} = \\int\\frac{du}{u^n} = \\cdots$$","date":"2019-05-25 21:17:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9907993674278259, \"perplexity\": 648.158084111301}, \"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-2019-22\/segments\/1558232258451.88\/warc\/CC-MAIN-20190525204936-20190525230936-00028.warc.gz\"}"}
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var playdrawr = (function(){
var dims = getViewportDimensions();
var canvas_elem = document.querySelector('canvas');
var height = dims.height;
var width = dims.width;
function init() {
var bg = document.getElementById('bg');
bg.setAttribute('width', width)
bg.setAttribute('height', height);
}
return {
init: init
};
})();
window.onload = playdrawr.init;
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 3,021
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We investigate leukocyte migration processes in zebrafish embryos during inflammation, which as part of the innate immune system form the first line of response to infection and trauma. Fluorescence microscopy has been used to generate time-lapse images that follow migrating leukocytes during inflammation. Image processing, including object detection and tracking, allows statistical analysis of the leucocyte trajectories as a function of time and space. With this information we can link intracellular signalling dynamics to tissue-level processes. Immune response mechanisms span both scales and automated image analysis will allow us to elucidate this intrinsically multi-scale problem. Zebrafish embryos, because of their transparency, provide a powerful model organism to study the spatio-temporal aspects of the innate immune response in vivo; in particular for the inference of mathematical multi-scale models it has probably unique advantages. We will illustrate the power of the model system in describing multi-scale dynamics by coupling image processing with explicit mathematical models for leukocyte chemotaxis.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 1,833
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.. spelling::
wt
.. index:: frameworks ; wt
.. _pkg.wt:
wt
==
- `Official <https://github.com/emweb/wt>`__
- `Hunterized <https://github.com/hunter-packages/wt>`__
- `Example <https://github.com/ruslo/hunter/blob/master/examples/wt/CMakeLists.txt>`__
- Added by `Casey <https://github.com/caseymcc>`__ (`pr-1655 <https://github.com/ruslo/hunter/pull/1655>`__)
Wt is a web GUI library in modern C++.
.. literalinclude:: /../examples/wt/CMakeLists.txt
:language: cmake
:start-after: # DOCUMENTATION_START {
:end-before: # DOCUMENTATION_END }
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 9,973
|
{"url":"https:\/\/stats.stackexchange.com\/questions\/336797\/simulating-data-for-linear-regression","text":"# Simulating data for linear regression\n\nI am trying to simulate two data set for multiple linear regression. I want one data which is independent and identically distributed and the other is not. So far, I have done the following:\n\nx1 <- rnorm(10000,11,.5)\nx2<-rnorm(10000,5,95)\nx3 <- rnorm(10000,5,.5)\n\nb1 <- 0.1\nb2 <- 0.9\nb3 <- 0.6\nsigma <- 0.4\n\neps <- rnorm(x1,10000,sigma)\ny <- b1*x1 + b2*x2 + b3*x3 +eps\nY<-as.matrix(y)\nX<-cbind(as.matrix(x1),as.matrix(x2),as.matrix(x3))\n\n\nDoes this satisfy the iid case? How to generate data that is not iid? For non iid case I am thinking of having autocorrelation.\n\n\u2022 check my edited answer for more related discussions. \u2013\u00a0Haitao Du Mar 26 '18 at 13:44\n\nDoes this satisfy the iid case?\n\nYes.\n\nHow to generate data that is not iid?\n\nFor example, data is generated from a mixture of Gaussian. When generate samples, we first sample the \"membership\", the according to the membership, we sample different Gaussian distribution.\n\nIn addition, Note that, linear regression does not have the assumption on the distribution on $X$, only on the residual to be normal distribution ed with constant variance, because of Gauss\u2013Markov theorem.\n\nIn other words, your simulation satisfy linear regression assumptions and the simulation has stronger assumptions on distribution of $X$.\n\nIn your comment, you asked how to generate not IID on residuals, I think you have a confusion on the assumptions. IID is on data, and normal distribution is on residuals.\n\nRelated topics\n\nRegression when the OLS residuals are not normally distributed\n\nWhy is the normality of residuals \"barely important at all\" for the purpose of estimating the regression line?\n\nAssumptions of multiple regression: how is normality assumption different from constant variance assumption?\n\nWhat is a complete list of the usual assumptions for linear regression?\n\nHow does linear regression use the normal distribution?\n\n\u2022 How can I generate the data which violates this assumption on residuals? \u2013\u00a0Waqas Mar 26 '18 at 13:21\n\u2022 I was referring to your comment \"In other words, your simulation satisfy linear regression assumptions and the simulation has stronger assumptions on distribution of X\". I just wanted residual not normal, I did not use the word iid and residuals. Apologies for creating confusion \u2013\u00a0Waqas Mar 26 '18 at 13:32\n\nTo generate correlated independent variables, you could do something like this:\n\nx1 <- rnorm(10^5,0,1)\nx2 <- rnorm(10^5,0,1)\nx2 <- 0.4*x1 + sqrt(1-0.4^2)*x2\nx1 < 11 + 0.5*x1\nx2 < 5 + 95*x2\n\nThere are ways to do something comparable with 3 or more variables, as well...though I will stop here for now.\n\nFor autocorrelation, a similar strategy can be used:\n\neps <- rnorm(10^5,0,1)\nfor(i in 2:10^5) eps[i] <- 0.15 * eps[i-1] + sqrt(1-0.15^2)*eps[i]\neps <- sigma * eps\n\n\nHope this helps with your simulation.","date":"2020-02-26 04:33:08","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\": 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\": 0.8128241300582886, \"perplexity\": 1642.350576896946}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875146186.62\/warc\/CC-MAIN-20200226023658-20200226053658-00400.warc.gz\"}"}
| null | null |
{"url":"https:\/\/electronics.stackexchange.com\/questions\/66303\/understanding-a-circuit-label-in-icsp-programmer-for-atmega","text":"# Understanding a circuit label in ICSP programmer for atmega\n\nI am completely new in electronics. I wanted to make a programmer (or burner) to burn my program in my Atmel16a microcontroller. I found a circuit on the internet:\n\n(source)\n\nEverything is clear, except what is the +5V in the ICSP? Is it same as VCC? Do we have to externally supply 5V to this circuit?\n\nAnd there are two diodes in the bottom. Are they zener diodes? Do they have special names or characteristics?\n\nThe diodes at the bottom are indeed zeners. They have 5.1V breakdown voltage. The reason they are there is to make it so you can actually program your chip instead of zapping it with 12V (see below).\n\nIt should be possible to replace 1N4148 diode with another switching diode. What that part of the circuit is used for is for lowering the RESET line which is necessary during the ISP programming. As jippie suggests, it is there to protect the transistor: pin 3 on the DB9 is hte RS-232 TX pin, which may be as low as -15V, which is lower than the emitter voltage, which is 0 because it is tied to ground. The maximum emitter\/base voltage for BCS549 is 5V, whereas in this case, without the diode it could be as much as 15V.\n\nThe +5V in this case is Vcc. The reason you have to supply +5V to the chip being programmed is because if the chip is not powered, it cannot be programmed. And yes, you have to supply the +5V yourself, since RS-232 does have a +5V connection. Neither can RS-232 pins be used to provide +5V in a simple fashion since RS-232 does not necessarily use +5 voltage levels.\n\nWith some extra programming and extra components, one might be able to make the free DB-9 pin (pin 9, the ring indicator) provide +5V, however, this will make your circuit and programmer life more complicated, not simpler.\n\nOn the simplicity of this circuit (responding to some edited-out bits): most other ISP programmers use an IC (such as another ATmega) internally (see USBtinyISP, USBasp, etc). Your programmer has only nine components (not counting the connectors) which is a pretty simple circuit. You could consider a parallel programmer such as this one or this one or many other variations on this theme you might find on Google if you need a simplest circuit. The simplest way to make an ISP programmer is to buy one: Ebay has a ton of them quite cheaply.\n\n\u2022 So, does the circuit now looks like this? : postimg.org\/image\/6zll7ta1v \u2013\u00a0cipher Apr 20 '13 at 17:39\n\u2022 That's one way. Another way is to power the chip itself: like this. It doesn't really matter whether you connect +5 on the programmer PCB or on the chip PCB. \u2013\u00a0angelatlarge Apr 20 '13 at 17:44\n\u2022 Ok! One last question: Can we use normal 1N4007 instead of 1N4148 diode? and can we use any NPN transistor instead of the specified one ? \u2013\u00a0cipher Apr 20 '13 at 17:50\n\u2022 Ok. But i would have been extremely grateful, if you answered the above question \u2013\u00a0cipher Apr 20 '13 at 17:59","date":"2021-08-06 02:44:38","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\": 0.3739677369594574, \"perplexity\": 1737.6405848870256}, \"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-2021-31\/segments\/1627046152112.54\/warc\/CC-MAIN-20210806020121-20210806050121-00029.warc.gz\"}"}
| null | null |
{"url":"https:\/\/zhuanzhi.ai\/topic\/2001405336420235","text":"### \u6700\u65b0\u5185\u5bb9\n\nIn this paper, we study the cooperative card game, The Crew: The Quest for Planet Nine from the viewpoint of algorithmic combinatorial game theory. The Crew: The Quest for Planet Nine, is a game based on traditional trick-taking card games, like bridge or hearts. In The Crew, players are dealt a hand of cards, with cards being from one of $c$ colors and having a value between 1 to $n$. Players also draft objectives, which correspond to a card in the current game that they must collect in order to win. Players then take turns each playing one card in a trick, with the player who played the highest value card taking the trick and all cards played in it. If all players complete all of their objectives, the players win. The game also forces players to not talk about the cards in their hand and has a number of \"Task Tokens\" which can modify the rules slightly. In this work, we introduce and formally define a perfect-information model of this problem, and show that the general unbounded version is computationally intractable. However, we also show that three bounded versions of this decision problem - deciding whether or not all players can complete their objectives - can be solved in polynomial time. \\end{abstract}\n\n### \u6700\u65b0\u8bba\u6587\n\nIn this paper, we study the cooperative card game, The Crew: The Quest for Planet Nine from the viewpoint of algorithmic combinatorial game theory. The Crew: The Quest for Planet Nine, is a game based on traditional trick-taking card games, like bridge or hearts. In The Crew, players are dealt a hand of cards, with cards being from one of $c$ colors and having a value between 1 to $n$. Players also draft objectives, which correspond to a card in the current game that they must collect in order to win. Players then take turns each playing one card in a trick, with the player who played the highest value card taking the trick and all cards played in it. If all players complete all of their objectives, the players win. The game also forces players to not talk about the cards in their hand and has a number of \"Task Tokens\" which can modify the rules slightly. In this work, we introduce and formally define a perfect-information model of this problem, and show that the general unbounded version is computationally intractable. However, we also show that three bounded versions of this decision problem - deciding whether or not all players can complete their objectives - can be solved in polynomial time. \\end{abstract}\n\nTop","date":"2021-10-26 01:59:58","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\": 0.473600298166275, \"perplexity\": 588.7348703049435}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323587794.19\/warc\/CC-MAIN-20211026011138-20211026041138-00558.warc.gz\"}"}
| null | null |
Self-reported changes in drug use behaviors and syringe...
Self-reported changes in drug use behaviors and syringe disposal methods following the opening of a supervised injecting facility in Copenhagen, Denmark Kinnard, Elizabeth N; Howe, Chanelle J; Kerr, Thomas; Skjødt Hass, Vibeke; Marshall, Brandon David Lewis
Background: In Denmark, the first standalone supervised injecting facility (SIF) opened in Copenhagen's Vesterbro neighborhood on October 1, 2012. The purpose of this study was to assess whether use of services provided by the recently opened SIF was associated with changes in injecting behavior and syringe disposal practices among people who inject drugs (PWID). We hypothesized that risk behaviors (e.g., syringe sharing), and unsafe syringe disposal (e.g., dropping used equipment on the ground) had decreased among PWID utilizing the SIF. Methods Between February and August of 2013, we conducted interviews using a survey (in English and Danish) with forty-one people who reported injecting drugs at the SIF. We used descriptive statistics and McNemar's test to examine sociodemographic characteristics of the sample, current drugs used, sites of syringe disposal before and after opening of the SIF, and perceived behavior change since using the SIF. Results Of the interviewed participants, 90.2% were male and the majority were younger than 40 years old (60.9%). Three-quarters (75.6%) of participants reported reductions in injection risk behaviors since the opening of the SIF, such as injecting in a less rushed manner (63.4%), fewer outdoor injections (56.1%), no longer syringe sharing (53.7%), and cleaning injecting site(s) more often (43.9%). Approximately two-thirds (65.9%) of participants did not feel that their frequency of injecting had changed; five participants (12.2%) reported a decrease in injecting frequency, and only two participants (4.9%) reported an increase in injecting frequency. Twenty-four (58.5%) individuals reported changing their syringe disposal practices since the opening of the SIF; of those, twenty-three (95.8%) reported changing from not always disposing safely to always disposing safely (McNemar's test p-value
Self-reported changes in drug use behaviors and syringe disposal methods following the opening of a supervised injecting facility in Copenhagen, Denmark
Kinnard, Elizabeth N; Howe, Chanelle J; Kerr, Thomas; Skjødt Hass, Vibeke; Marshall, Brandon David Lewis
Supervised injecting facility; Drug consumption room; People who inject drugs; Injection drug users; Harm reduction; Risk behaviors; Syringe disposal; Denmark
Medicine, Faculty of; Non UBC
Harm Reduction Journal. 2014 Oct 28;11(1):29
10.1186/1477-7517-11-29
Kinnard et al.; licensee BioMed Central Ltd.
12954_2014_Article_337.pdf -- 329.97kB
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,946
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Located in Broward County, Florida, the city of Hallandale Beach is a well-designed area. It has racetracks, parks, gaming centers, golf courses and dance as well as fitness clubs. More so, you can avail sightseeing tours to rejuvenate your boring weekends. With several business centers, the city has a lot of career opportunities for the youth. The beach has clear waters and is a popular spot for tourists with many neighboring food joints.
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This Packers and Movers is a fully licensed, insured and bonded Company that provides complete insurance coverage. They value your sentimentstowards your belongingsand make an all-out effort to impart full-value protection services during the move. Full-value protection plan is devised to safeguardthe quality and value of your personal goods and protect those goods from being mistreated in unforeseen circumstances during the move. Their services are inclusive of moving services, receiving and set-up services, packing and crating services, storage services, loading and unloading services, and most importantly, emergency services.
The team has an impressive professional expertise in handling household moving, apartment moving, as well as condos.Special care is taken in the craft of packing, loading, unloading, and setting up the goods, to guarantee maximum protection against all odds.They provide a vast suite of services and genuine solutions to make your much-dreaded home moving process calm and easy. They are world class residential movers that aid and abet you at every possible step of your house moving process.
These services include handling moves for professional businesses, offices, government buildings, hotels, restaurants, and many more. Their responsible team endeavors to incorporate every smart techniqueto make the office relocation process a highly secure and pleasant experience for you and your employees. The packing, moving and storage process is efficiently strategized and executed by professionals. Their warehousing service and prompt team co-ordination makes every effortfully catering to the needs of your business as well as concerns regarding your budget.
SMS Moving renders top-notch services and easy solutions for moving delicate or relatively fragile items such as your piano, pool table, grandfather clocks, mirrors, artwork and antiques, automobiles, furniture and several other priceless possessions. Moving such items requires a great sense of responsibility as well as careful handling and specific paraphernalia. This dedicated team focuses on a quality-centric approach that is supported by the wide range of services they offer. These services include safe and secure warehousing, padded van lift-gated transportation, reverse logistics which refers to the process of moving items from their final destination for the purpose of recapturing their value, or rather proper disposal in case some items are useless to you, simplified supply chain logistics, trade show support, methodical local trucking and amazing freight management.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,470
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import React from 'react';
import withRoot from 'docs/src/modules/components/withRoot';
import MarkdownDocs from 'docs/src/modules/components/MarkdownDocs';
import markdown from 'docs/src/pages/guides/minimizing-bundle-size/minimizing-bundle-size.md';
function Page() {
return <MarkdownDocs markdown={markdown} />;
}
export default withRoot(Page);
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,113
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/**
* @module zrender/core/util
*/
// 用于处理merge时无法遍历Date等对象的问题
var BUILTIN_OBJECT = {
'[object Function]': 1,
'[object RegExp]': 1,
'[object Date]': 1,
'[object Error]': 1,
'[object CanvasGradient]': 1,
'[object CanvasPattern]': 1,
// For node-canvas
'[object Image]': 1,
'[object Canvas]': 1
};
var TYPED_ARRAY = {
'[object Int8Array]': 1,
'[object Uint8Array]': 1,
'[object Uint8ClampedArray]': 1,
'[object Int16Array]': 1,
'[object Uint16Array]': 1,
'[object Int32Array]': 1,
'[object Uint32Array]': 1,
'[object Float32Array]': 1,
'[object Float64Array]': 1
};
var objToString = Object.prototype.toString;
var arrayProto = Array.prototype;
var nativeForEach = arrayProto.forEach;
var nativeFilter = arrayProto.filter;
var nativeSlice = arrayProto.slice;
var nativeMap = arrayProto.map;
var nativeReduce = arrayProto.reduce;
/**
* Those data types can be cloned:
* Plain object, Array, TypedArray, number, string, null, undefined.
* Those data types will be assgined using the orginal data:
* BUILTIN_OBJECT
* Instance of user defined class will be cloned to a plain object, without
* properties in prototype.
* Other data types is not supported (not sure what will happen).
*
* Caution: do not support clone Date, for performance consideration.
* (There might be a large number of date in `series.data`).
* So date should not be modified in and out of echarts.
*
* @param {*} source
* @return {*} new
*/
function clone(source) {
if (source == null || typeof source != 'object') {
return source;
}
var result = source;
var typeStr = objToString.call(source);
if (typeStr === '[object Array]') {
result = [];
for (var i = 0, len = source.length; i < len; i++) {
result[i] = clone(source[i]);
}
}
else if (TYPED_ARRAY[typeStr]) {
result = source.constructor.from(source);
}
else if (!BUILTIN_OBJECT[typeStr] && !isDom(source)) {
result = {};
for (var key in source) {
if (source.hasOwnProperty(key)) {
result[key] = clone(source[key]);
}
}
}
return result;
}
/**
* @memberOf module:zrender/core/util
* @param {*} target
* @param {*} source
* @param {boolean} [overwrite=false]
*/
function merge(target, source, overwrite) {
// We should escapse that source is string
// and enter for ... in ...
if (!isObject(source) || !isObject(target)) {
return overwrite ? clone(source) : target;
}
for (var key in source) {
if (source.hasOwnProperty(key)) {
var targetProp = target[key];
var sourceProp = source[key];
if (isObject(sourceProp)
&& isObject(targetProp)
&& !isArray(sourceProp)
&& !isArray(targetProp)
&& !isDom(sourceProp)
&& !isDom(targetProp)
&& !isBuildInObject(sourceProp)
&& !isBuildInObject(targetProp)
) {
// 如果需要递归覆盖,就递归调用merge
merge(targetProp, sourceProp, overwrite);
}
else if (overwrite || !(key in target)) {
// 否则只处理overwrite为true,或者在目标对象中没有此属性的情况
// NOTE,在 target[key] 不存在的时候也是直接覆盖
target[key] = clone(source[key], true);
}
}
}
return target;
}
/**
* @param {Array} targetAndSources The first item is target, and the rests are source.
* @param {boolean} [overwrite=false]
* @return {*} target
*/
function mergeAll(targetAndSources, overwrite) {
var result = targetAndSources[0];
for (var i = 1, len = targetAndSources.length; i < len; i++) {
result = merge(result, targetAndSources[i], overwrite);
}
return result;
}
/**
* @param {*} target
* @param {*} source
* @memberOf module:zrender/core/util
*/
function extend(target, source) {
for (var key in source) {
if (source.hasOwnProperty(key)) {
target[key] = source[key];
}
}
return target;
}
/**
* @param {*} target
* @param {*} source
* @param {boolen} [overlay=false]
* @memberOf module:zrender/core/util
*/
function defaults(target, source, overlay) {
for (var key in source) {
if (source.hasOwnProperty(key)
&& (overlay ? source[key] != null : target[key] == null)
) {
target[key] = source[key];
}
}
return target;
}
function createCanvas() {
return document.createElement('canvas');
}
// FIXME
var _ctx;
function getContext() {
if (!_ctx) {
// Use util.createCanvas instead of createCanvas
// because createCanvas may be overwritten in different environment
_ctx = util.createCanvas().getContext('2d');
}
return _ctx;
}
/**
* 查询数组中元素的index
* @memberOf module:zrender/core/util
*/
function indexOf(array, value) {
if (array) {
if (array.indexOf) {
return array.indexOf(value);
}
for (var i = 0, len = array.length; i < len; i++) {
if (array[i] === value) {
return i;
}
}
}
return -1;
}
/**
* 构造类继承关系
*
* @memberOf module:zrender/core/util
* @param {Function} clazz 源类
* @param {Function} baseClazz 基类
*/
function inherits(clazz, baseClazz) {
var clazzPrototype = clazz.prototype;
function F() {}
F.prototype = baseClazz.prototype;
clazz.prototype = new F();
for (var prop in clazzPrototype) {
clazz.prototype[prop] = clazzPrototype[prop];
}
clazz.prototype.constructor = clazz;
clazz.superClass = baseClazz;
}
/**
* @memberOf module:zrender/core/util
* @param {Object|Function} target
* @param {Object|Function} sorce
* @param {boolean} overlay
*/
function mixin(target, source, overlay) {
target = 'prototype' in target ? target.prototype : target;
source = 'prototype' in source ? source.prototype : source;
defaults(target, source, overlay);
}
/**
* @param {Array|TypedArray} data
*/
function isArrayLike(data) {
if (! data) {
return;
}
if (typeof data == 'string') {
return false;
}
return typeof data.length == 'number';
}
/**
* 数组或对象遍历
* @memberOf module:zrender/core/util
* @param {Object|Array} obj
* @param {Function} cb
* @param {*} [context]
*/
function each(obj, cb, context) {
if (!(obj && cb)) {
return;
}
if (obj.forEach && obj.forEach === nativeForEach) {
obj.forEach(cb, context);
}
else if (obj.length === +obj.length) {
for (var i = 0, len = obj.length; i < len; i++) {
cb.call(context, obj[i], i, obj);
}
}
else {
for (var key in obj) {
if (obj.hasOwnProperty(key)) {
cb.call(context, obj[key], key, obj);
}
}
}
}
/**
* 数组映射
* @memberOf module:zrender/core/util
* @param {Array} obj
* @param {Function} cb
* @param {*} [context]
* @return {Array}
*/
function map(obj, cb, context) {
if (!(obj && cb)) {
return;
}
if (obj.map && obj.map === nativeMap) {
return obj.map(cb, context);
}
else {
var result = [];
for (var i = 0, len = obj.length; i < len; i++) {
result.push(cb.call(context, obj[i], i, obj));
}
return result;
}
}
/**
* @memberOf module:zrender/core/util
* @param {Array} obj
* @param {Function} cb
* @param {Object} [memo]
* @param {*} [context]
* @return {Array}
*/
function reduce(obj, cb, memo, context) {
if (!(obj && cb)) {
return;
}
if (obj.reduce && obj.reduce === nativeReduce) {
return obj.reduce(cb, memo, context);
}
else {
for (var i = 0, len = obj.length; i < len; i++) {
memo = cb.call(context, memo, obj[i], i, obj);
}
return memo;
}
}
/**
* 数组过滤
* @memberOf module:zrender/core/util
* @param {Array} obj
* @param {Function} cb
* @param {*} [context]
* @return {Array}
*/
function filter(obj, cb, context) {
if (!(obj && cb)) {
return;
}
if (obj.filter && obj.filter === nativeFilter) {
return obj.filter(cb, context);
}
else {
var result = [];
for (var i = 0, len = obj.length; i < len; i++) {
if (cb.call(context, obj[i], i, obj)) {
result.push(obj[i]);
}
}
return result;
}
}
/**
* 数组项查找
* @memberOf module:zrender/core/util
* @param {Array} obj
* @param {Function} cb
* @param {*} [context]
* @return {Array}
*/
function find(obj, cb, context) {
if (!(obj && cb)) {
return;
}
for (var i = 0, len = obj.length; i < len; i++) {
if (cb.call(context, obj[i], i, obj)) {
return obj[i];
}
}
}
/**
* @memberOf module:zrender/core/util
* @param {Function} func
* @param {*} context
* @return {Function}
*/
function bind(func, context) {
var args = nativeSlice.call(arguments, 2);
return function () {
return func.apply(context, args.concat(nativeSlice.call(arguments)));
};
}
/**
* @memberOf module:zrender/core/util
* @param {Function} func
* @return {Function}
*/
function curry(func) {
var args = nativeSlice.call(arguments, 1);
return function () {
return func.apply(this, args.concat(nativeSlice.call(arguments)));
};
}
/**
* @memberOf module:zrender/core/util
* @param {*} value
* @return {boolean}
*/
function isArray(value) {
return objToString.call(value) === '[object Array]';
}
/**
* @memberOf module:zrender/core/util
* @param {*} value
* @return {boolean}
*/
function isFunction(value) {
return typeof value === 'function';
}
/**
* @memberOf module:zrender/core/util
* @param {*} value
* @return {boolean}
*/
function isString(value) {
return objToString.call(value) === '[object String]';
}
/**
* @memberOf module:zrender/core/util
* @param {*} value
* @return {boolean}
*/
function isObject(value) {
// Avoid a V8 JIT bug in Chrome 19-20.
// See https://code.google.com/p/v8/issues/detail?id=2291 for more details.
var type = typeof value;
return type === 'function' || (!!value && type == 'object');
}
/**
* @memberOf module:zrender/core/util
* @param {*} value
* @return {boolean}
*/
function isBuildInObject(value) {
return !!BUILTIN_OBJECT[objToString.call(value)];
}
/**
* @memberOf module:zrender/core/util
* @param {*} value
* @return {boolean}
*/
function isDom(value) {
return typeof value === 'object'
&& typeof value.nodeType === 'number'
&& typeof value.ownerDocument === 'object';
}
/**
* If value1 is not null, then return value1, otherwise judget rest of values.
* @memberOf module:zrender/core/util
* @return {*} Final value
*/
function retrieve(values) {
for (var i = 0, len = arguments.length; i < len; i++) {
if (arguments[i] != null) {
return arguments[i];
}
}
}
/**
* @memberOf module:zrender/core/util
* @param {Array} arr
* @param {number} startIndex
* @param {number} endIndex
* @return {Array}
*/
function slice() {
return Function.call.apply(nativeSlice, arguments);
}
/**
* @memberOf module:zrender/core/util
* @param {boolean} condition
* @param {string} message
*/
function assert(condition, message) {
if (!condition) {
throw new Error(message);
}
}
var util = {
inherits: inherits,
mixin: mixin,
clone: clone,
merge: merge,
mergeAll: mergeAll,
extend: extend,
defaults: defaults,
getContext: getContext,
createCanvas: createCanvas,
indexOf: indexOf,
slice: slice,
find: find,
isArrayLike: isArrayLike,
each: each,
map: map,
reduce: reduce,
filter: filter,
bind: bind,
curry: curry,
isArray: isArray,
isString: isString,
isObject: isObject,
isFunction: isFunction,
isBuildInObject: isBuildInObject,
isDom: isDom,
retrieve: retrieve,
assert: assert,
noop: function () {}
};
module.exports = util;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 457
|
{"url":"https:\/\/brilliant.org\/problems\/attack-the-series\/","text":"# Attack the series\n\nCalculus Level 3\n\n$\\sum_{n=2}^{\\infty} \\dfrac{1}{\\ln(\\ln(n))}$\n\nDetermine whether the series above is convergent or divergent.\n\n\u00d7","date":"2020-02-18 05:11:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 1, \"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\": 0.7862313389778137, \"perplexity\": 13183.469595740502}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875143505.60\/warc\/CC-MAIN-20200218025323-20200218055323-00544.warc.gz\"}"}
| null | null |
"use strict";
const Koa = require("koa");
const bodyParser = require("koa-bodyparser");
const { synchronize } = require("misc");
const { AsyncEventEmitter } = require("emitter");
const crypto = require("crypto");
class GithubEventEmitter extends AsyncEventEmitter {
constructor(log = false) {
super();
// Ignore logs if no log function set
this.log = typeof log === "function" ? log : () => null;
synchronize(this, "emit");
}
async _setupWebServer(port, webhookSecret) {
const app = new Koa();
app.use(bodyParser());
app.use(async (ctx) => {
const header = ctx.request.header;
const body = ctx.request.body;
if (webhookSecret) {
const hash = crypto.createHmac("sha1", webhookSecret).update(body).digest("hex");
if (!header["X-Hub-Signature"] || header["X-Hub-Signature"] !== hash) {
throw Error("Missing or incorrect signature in GitHub webhook event");
}
}
// We need a body with an action and a header with an event type
if (body && header && header["x-github-event"]) {
switch (header["x-github-event"]) {
case "ping":
this.emit("ping", body);
break;
case "pull_request":
switch (body.action) {
case "opened":
await this.emit("pull_request_opened", body);
break;
case "synchronize":
await this.emit("pull_request_updated", body);
break;
case "closed":
await this.emit("pull_request_closed", body);
break;
default:
await this.emit("pull_request_unknown", body);
}
break;
case "push":
await this.emit("push", body);
break;
case "pull_request_review":
await this.emit("pull_request_review", body);
break;
default:
await this.emit("unknown_event", { type: header["x-github-event"], body: body });
}
} else {
await this.emit("malformed_event", { header: header, body: body });
}
ctx.response.status = 200;
});
this.server = app.listen(port);
return new Promise((resolve, reject) => {
this.server.on("listening", () => {
this.server.removeListener("error", reject);
resolve(`Webhook server up on port ${port}`);
});
this.server.once("error", reject);
});
}
async teardownWebServer() {
this.log("verbose", "Tearing down web hook server");
this.server.close();
}
async start(port) {
return await this._setupWebServer(port);
}
async dispose() {
await this.teardownWebServer();
this.removeAllListeners();
}
}
module.exports = GithubEventEmitter;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,247
|
{"url":"https:\/\/www.physicsforums.com\/threads\/trivial-power-series-question.580698\/","text":"# Trivial power series question\n\nHow do I find the power series for $z^7$?\n\nI can't remember.\n\nRelated Calculus and Beyond Homework Help News on Phys.org\nMark44\nMentor\nHow do I find the power series for $z^7$?\n\nI can't remember.\nWhat would be the coefficient for the constant term?\nWhat would be the coefficient for the z term?\nWhat would be the coefficient for the z2 term?\n.\n.\n.\n\nDick\nHomework Helper\nIf you mean the power series expansion around a point that's not 0, say around z=1, then write z^7=((z-1)+1)^7 and expand that.\n\nIf you mean the power series expansion around a point that's not 0, say around z=1, then write z^7=((z-1)+1)^7 and expand that.\nAround zero. I have been looking through Rudin and Rosenlicht but I don't see an example of what I am looking for.\n\nDick\nHomework Helper\nAround zero. I have been looking through Rudin and Rosenlicht but I don't see an example of what I am looking for.\nThen what are you looking for? z^7 IS the power series for the function f(z)=z^7 around z=0.\n\nThen what are you looking for? z^7 IS the power series for the function f(z)=z^7 around z=0.\n$$f(z) = \\frac{1}{2\\pi i}\\sum_{n = 0}^{\\infty}\\left[\\int_0^{2\\pi}\\frac{u^7}{u^{n + 1}}du z^n\\right] = z^7.$$\n\nI want to show the above equality. I know since u^7 is analytic in the unit disk, g will be the same as f. But is there a way to show that without stating this?\n\nDick\nHomework Helper\n$$f(z) = \\frac{1}{2\\pi i}\\sum_{n = 0}^{\\infty}\\left[\\int_0^{2\\pi}\\frac{u^7}{u^{n + 1}}du z^n\\right] = z^7.$$\n\nI want to show the above equality. I know since u^7 is analytic in the unit disk, g will be the same as f. But is there a way to show that without stating this?\nWell, the integral is zero unless n=7, isn't it? The integral of u^k is zero unless k=(-1). And I'm assuming you actually meant a contour integral around the origin, not u=0 to u=2pi.\n\nLast edited:\nWell, the integral is zero unless n=7, isn't it? The integral of u^k is zero unless k=(-1).\nu is a complex number and u^7 is analytic and continuous.\n\nFor all z inside of C (C the unit circle oriented counterclockwise),\n$$f(z) = \\frac{1}{2\\pi i}\\int_C \\frac{g(u)}{u-z} du$$\nwhere $g(u) = \\bar{u}$ is a continuous function and $f$ is analytic in C. Describe $f$in C in terms of a power series.\n\n$\\displaystyle f(z) = \\frac{1}{2\\pi i}\\int_C \\frac{\\bar{u}}{u-z} du$ I am confused with what I am supposed to do. I know it says describe $f$ in terms of a power series.\n\nOnly difference I am dealing with u^7 not the conjugate.\n\nDick\nHomework Helper\n$\\int_C \\frac{u^7}{u^{n + 1}}du=\\int_C u^{7 - n - 1}du$. That's zero unless the exponent is -1 which happens when n=7.\n\n$\\int_C \\frac{u^7}{u^{n + 1}}du=\\int_C u^{7 - n - 1}du$. That's zero unless the exponent is -1 which happens when n=7.\nOk, I understand now, thanks.\n\nOk so if the function was 1\/u, we would have u^{-n-2}. This one would always be 0 then?\n\nDick\nHomework Helper\nOk so if the function was 1\/u, we would have u^{-n-2}. This one would always be 0 then?\nSure, but f(u)=1\/u doesn't have a power series expansion around 0 with only positive powers. You'd need a Laurent series instead of a power series to represent it.\n\nSure, but f(u)=1\/u doesn't have a power series expansion around 0 with only positive powers. You'd need a Laurent series instead of a power series to represent it.\nIf we haven't done Laurent Series yet, how should I handle it then?\n\nDick\nHomework Helper\nIf we haven't done Laurent Series yet, how should I handle it then?\nYou don't. f(z)=1\/z doesn't have a power series expansion around z=0. f(z) has to be analytic at z=0 to have a power series expansion. 1\/z isn't analytic at z=0.\n\nYou don't. f(z)=1\/z doesn't have a power series expansion around z=0. f(z) has to be analytic at z=0 to have a power series expansion. 1\/z isn't analytic at z=0.\nEvaluating the sum and the integral for it yields f(z) = 0. Is that correct to put down after evaluating f(z) since all the terms are 0?\n\nDick\nHomework Helper\nEvaluating the sum and the integral for it yields f(z) = 0. Is that correct to put down after evaluating f(z) since all the terms are 0?\nAll of the terms in your series are zero, yes. But that still doesn't make f(z)=1\/z=0. I'm not sure you are paying attention here.\n\nAll of the terms in your series are zero, yes. But that still doesn't make f(z)=1\/z=0. I'm not sure you are paying attention here.\nI understand what you are saying but I am trying to solve for\n$$f(z) = \\frac{1}{2\\pi i}\\sum_{n = 0}^{\\infty}\\left[\\int_0^{2\\pi}\\frac{\\frac{1}{u}}{u^{n + 1}}du z^n\\right].$$\nSince Laurent series are out and all the terms are 0, what else could f(z) be?\n\nDick\nHomework Helper\nI understand what you are saying but I am trying to solve for\n$$f(z) = \\frac{1}{2\\pi i}\\sum_{n = 0}^{\\infty}\\left[\\int_0^{2\\pi}\\frac{\\frac{1}{u}}{u^{n + 1}}du z^n\\right].$$\nSince Laurent series are out and all the terms are 0, what else could f(z) be?\nI am trying to tell you that the series you are quoting is NOT valid for all functions f(z). f(z) has to be analytic at z=0 to apply that. f(z)=1\/z is NOT analytic at z=0. I've already told you this.\n\nI am trying to tell you that the series you are quoting is NOT valid for all functions f(z). f(z) has to be analytic at z=0 to apply that. f(z)=1\/z is NOT analytic at z=0. I've already told you this.\nBy the integral transform theorem, if you put a continuous function g(u) into f(z), you get out an analytic function. If g is analytic, you get the same function. So f(z) has to equal something.\n\nDick\nHomework Helper\nBy the integral transform theorem, if you put a continuous function g(u) into f(z), you get out an analytic function. If g is analytic, you get the same function. So f(z) has to equal something.\nI'm not quite sure why this is so difficult. 1\/u is continuous on the contour. And yes, you get an analytic function out. It's f(z)=0. Now you say \"If g is analytic, you get the same function.\". g(u)=1\/u ISN'T analytic at u=0. So the function you get out f(z)=0, ISN'T the same as the function you put in f(z)=1\/z.\n\nI'm not quite sure why this is so difficult. 1\/u is continuous on the contour. And yes, you get an analytic function out. It's f(z)=0. Now you say \"If g is analytic, you get the same function.\". g(u)=1\/u ISN'T analytic at u=0. So the function you get out f(z)=0, ISN'T the same as the function you put in f(z)=1\/z.\nI understand that. I was verifying that f(z) = 0 is correct. You kept saying you don't understand what I telling you.\n\nDick\nHomework Helper\nI understand that. I was verifying that f(z) = 0 is correct. You kept saying you don't understand what I telling you.\nApologies if I'm misunderstanding. But I'm just saying 0 isn't the power series representation of 1\/z because it doesn't have one. And f(z)=1\/z is not equal to 0. That's all. So, ok. Yes f(z)=0. I somehow thought you were trying to represent 1\/z as a power series. Sorry.\n\nApologies if I'm misunderstanding. But I'm just saying 0 isn't the power series representation of 1\/z because it doesn't have one. And f(z)=1\/z is not equal to 0. That's all. So, ok. Yes f(z)=0. I somehow thought you were trying to represent 1\/z as a power series. Sorry.\nNo problem. I know f(z) = 1\/z\\neq 0 but when I said f(z) I meant,\n\n$$f(z) = \\frac{1}{2\\pi i}\\sum_{n = 0}^{\\infty}\\left[\\int_0^{2\\pi}\\frac{\\frac{1}{u}}{u^{n + 1}}du z^n\\right].$$\n\nDick\n$$f(z) = \\frac{1}{2\\pi i}\\sum_{n = 0}^{\\infty}\\left[\\int_0^{2\\pi}\\frac{\\frac{1}{u}}{u^{n + 1}}du z^n\\right].$$","date":"2020-10-23 20:58:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 2, \"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\": 0.8980088829994202, \"perplexity\": 539.3393159507875}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107865665.7\/warc\/CC-MAIN-20201023204939-20201023234939-00617.warc.gz\"}"}
| null | null |
Q: DDD: Where should I set modified date and modified by? Repository or application service? Where should I set fields like CreatedDate, CreatedBy, ModifiedDate, ModifiedBy? Should I pass current user context to repository and set it there or maybe better way is to set it in application service (but then it must be done in each API method rather than only in Add/Update in repository)?
A: It depends on your domain.
If values like CreatedDate, CreatedBy... are for tracking or logging purpose, then I'd place them in the Infrastructure (Repository).
On the other hand, if these values belong to my domain for any reason, then I'd place them in the domain layer.
Example: imaging that in a Banking Transfer context, a customer could cancel a transfer only until 24 hours after submit it for settlement. Then the domain needs CreateTransferDate to satisfy the invariants.
Another option could be a listener that consume all domain events and save historial time data of what happen.
A: CreatedDate, CreatedBy, ModifiedDate and ModifiedBy are typically concepts that have no real domain value, but they are more like a technical concept; and that's why I typically don't set these in application or domain layer, but in repository layer. After all, the repository layer is a technical layer. In a domain I don't care about this kind of information. Unless of course, it's part of the ubiquitous language, but mostly it's not.
Also, if ModifiedDate would be part of the domain/application, you would have to set it with each manipulation you do, which would be very tedious and error-prone. If you do it in the repository, it's easier, because you do it with every update.
So I would ask the question: does business care about it?
A: I prefer setting them in the repository. Add one parameter in existing add/update method named int operatorId or something similar.
If you put the code in application service, you need to always repeat yourself. And if you forget to set some values, exceptions will be thrown when changes are saved (sometimes even worse, no exceptions but dirty data).
A: We always set entity state inside application services. Repositories are for saving data, they should not contain any logic. The domain entity is created/modified in the application service, therefore that is where you set these fields. The creation/modification dates should not reflect when the entity was saved to the database, but rather when the entity was actually saved/modified.
Also, we find that we usually have to map our domain entities to data layer entities to be more friendly with the underlying database technology. We often use tools like automapper help us automate these mappings. Having fields in an entity that do not exist in its data entity equivalent adds unnecessary complexity
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,090
|
package org.skife.gressil;
public enum DaemonStatus
{
// for status
STATUS_RUNNING(0), STATUS_DEAD(1), STATUS_NOT_RUNNING(3), STATUS_UNKNOWN(4),
// for stop
STOP_NOT_RUNNING(7), STOP_GENERAL_ERROR(1), STOP_SUCCESS(0);
private final int exitCode;
public int getExitCode() {
return exitCode;
}
DaemonStatus(final int code) {exitCode = code;}
/*
* 0 program is running or service is OK
* 1 program is dead and /var/run pid file exists
* 2 program is dead and /var/lock lock file exists
* 3 program is not running
* 4 program or service status is unknown
* 5-99 reserved for future LSB use
* 100-149 reserved for distribution use
* 150-199 reserved for application use
* 200-254 reserved
*/
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,384
|
Residents who wish to purchase a yard waste cart may fill out an online form. Carts will be delivered to your home upon completion of your order, and are yours to keep. The City will provide maintenance of the carts.
The City of Lawrence sells 95-gallon yard waste carts to be used for weekly yard waste collection days. The carts can be purchased by any household that receives Lawrence Solid Waste collection services. The yard waste carts are designed with vents to prevent strong odors from stored yard waste materials such as grass, leaves, and woody debris. These brown carts are for yard waste collection only; no trash, please.
Cost: $60 each+tax – This can be a one time payment, six monthly payments of $10+tax, OR 12 monthly payments of $5+tax – billed to your City Utility account.
For information about the City's yard waste program including collection schedule and types of set-out containers allowed, visit our Curbside Collection of Yard Waste page.
Yard waste carts have a capacity of 330 lbs. (equivalent to six/seven yard waste bags).
Easy mobility and wide opening for easy fills.
Weatherproof and more durable than paper yard waste bags.
Carts are safer, easier and more efficient for residents and city crews.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,592
|
Q: Printing full symbolic link information, starting from a root directory We are interested in printing all the symbolic link information, starting from a root directory. We would like the information to be printed in the following format:
symbolic name -> actual name [Notice the same line]
We have tried the following and it gives us the symbolic name and actual name, but on different lines:
find . -type l -print -exec readlink -f {} \;
A: If using GNU find:
find . -type l -printf '%p -> %l\n'
The %p format is the file name and %l the target of a symbolic link (or blank for other file types)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,493
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{"url":"https:\/\/robokofumihisoku.rolf-luettecke.com\/curves-that-fill-a-three-dimensional-space-book-27473zv.php","text":"Curves that fill a three-dimensional space ...\nShare\n\n# Curves that fill a three-dimensional space ... by Ruth Otilia Peterson\n\n\u2022 \u00b7\n\nPublished .\nWritten in English\n\n### Subjects:\n\n\u2022 Curves.,\n\u2022 Curves, Cubic.\n\n## Book details:\n\nThe Physical Object\nPagination3 p.l., 62 [1] leaves,\nNumber of Pages62\nID Numbers\nOpen LibraryOL16882531M\n\npoints in three-dimensional space. Geometry of Curves. Before a discussion of surfaces, curves in three dimensions will be covered for two reasons: surfaces are described by using certain special curves, and representations for curves generalize to representations for surfaces. Curves. Ex Describe the curve ${\\bf r}=\\langle t\\cos t,t\\sin t,t\\rangle$. Ex Describe the curve ${\\bf r}=\\langle t,t^2,\\cos t\\rangle$. Ex Describe the curve ${\\bf r}=\\langle \\cos(20t)\\sqrt{1-t^2},\\sin(20t)\\sqrt{1-t^2},t\\rangle$ Ex Find a vector. In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an n-dimensional unit hypercube).Because Giuseppe Peano (\u2013) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example . A Three-Dimensional Hilbert Curve 26 Problems 29 Chapter 3. Peanos Space-Filling Curve 31 Definition of Peanos Space-Filling Curve 31 Nowhere Differentiability of the Peano Curve 34 Geometrie Generation of the Peano Curve 34 Proof that the Peano Curve and the Geometrie Peano Curve are the Same 36 Ces\u00e4ros.\nWe argue that the properties that make Hilbert's curve unique in two dimensions, are shared by structurally different space-filling curves in three dimensions. These include several curves that have, in some sense, better locality properties than any generalized Hilbert curve that has been considered in the literature before. here are curves and surfaces in two- and three-dimensional space, and they are primarily studied by means of parametrization. The main properties of these objects, which will be studied, are notions related to the shape. We will study tangents of curves and tangent spaces of surfaces, and the notion of curvature will be introduced. \u201cBader\u2019s book provides an introduction to the algorithmics of space-filling curves. The book has many color illustrations and can be used as a textbook and as reference monograph for research.\u201d (Luiz Henrique de Figueiredo, MAA Reviews, April, ) \u201cThis is a gentle introduction to space filling s: 2. Space Curves. A vector-valued function $$\\vec r(t)$$ whose values are three-dimensional functions traces out a space curve, a curve in three-dimensional space. For example, $\\vec r(t) = \\left\\langle t, \\frac{t^3}{18},\\frac{t^2}{3} \\right\\rangle$ traces out a space curve called a twisted cubic.","date":"2021-04-20 13:33:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"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\": 0.6878761053085327, \"perplexity\": 995.6078421734284}, \"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-2021-17\/segments\/1618039398307.76\/warc\/CC-MAIN-20210420122023-20210420152023-00184.warc.gz\"}"}
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\section{Introduction}
Stochastic optimization has been extensively studied in the machine learning community~\cite{DBLP:conf/icml/Zhang04,rakhlin2011making,shamir2013stochastic,duchi2009efficient,luo1992convergence,mangasarian1999successive,hsieh2008dual,DBLP:journals/jmlr/Shalev-ShwartzT11,DBLP:journals/corr/abs-1207-4747,DBLP:journals/siamjo/Nesterov12,DBLP:journals/corr/abs-1211-2717,DBLP:journals/jmlr/ShaiTong13,shalev2012proximal}. In general, at every step, a traditional stochastic optimization method will sample one training example or one dual coordinate uniformly at random from the training data, and then update the model parameter using the sampled example or dual coordinate. Although uniform sampling simplifies the analysis, it is insufficient because it may introduce a very high variance of the sampled quantity, which will negatively affect the convergence rate of the resulting optimization procedure. In this paper we study stochastic optimization with importance sampling, which reduces the stochastic variance to significantly improve the convergence rate. Specifically, this paper focus on importance sampling techniques for Proximal Stochastic Gradient Descent (prox-SGD) (actually general proximal stochastic mirror descent)~\cite{duchi2009efficient,DBLP:conf/colt/DuchiSST10} and Proximal Stochastic Dual Coordinate Ascent (prox-SDCA)~\cite{shalev2012proximal}.
For prox-SGD, the traditional algorithms such as Stochastic Gradient Descent (SGD) sample training examples uniformly at random during the entire learning process, so that the stochastic gradient is an unbiased estimation of the true gradient~\cite{DBLP:conf/icml/Zhang04,rakhlin2011making,shamir2013stochastic,duchi2009efficient}. However, the variance of the resulting stochastic gradient estimator may be very high since the stochastic gradient can vary significantly over different examples.
In order to improve convergence, this paper proposes a sampling distribution and the corresponding unbiased importance weighted gradient estimator that achieves minimal variance. To this end, we analyze the relation between the variance of stochastic gradient and the sampling distribution. We show that to minimize the variance, the optimal sampling distribution should be roughly proportional to the norm of the stochastic gradient.
To simplify computation, we also consider the use of upper bounds for the norms. Our theoretical analysis shows that under certain conditions, the proposed sampling method can significantly improve the convergence rate, and our results include the existing theoretical results for uniformly sampled prox-SGD and SGD as special cases.
Similarly for prox-SDCA, the traditional approach such as Stochastic Dual Coordinate Ascent (SDCA)~\cite{DBLP:journals/jmlr/ShaiTong13} picks
a coordinate to update by sampling the training data uniformly at random \cite{luo1992convergence,mangasarian1999successive,hsieh2008dual,DBLP:journals/jmlr/Shalev-ShwartzT11,DBLP:journals/corr/abs-1207-4747,DBLP:journals/siamjo/Nesterov12,DBLP:journals/corr/abs-1211-2717,DBLP:journals/jmlr/ShaiTong13,shalev2012proximal}.
It was shown recently that SDCA and prox-SDCA algorithm with uniform random sampling converges much faster than a fixed cyclic ordering~\cite{DBLP:journals/jmlr/ShaiTong13,shalev2012proximal}.
However, this paper shows that if we employ an appropriately defined importance sampling strategy, the convergence could be further improved. To find the optimal sampling distribution, we analyze the connection between the expected increase of dual objective and the sampling distribution,
and obtain the optimal solution that depends on the smooth parameters of the loss functions.
Our analysis shows that under certain conditions, the proposed sampling method can significantly improve the convergence rate. In addition, our theoretical results include the existing results for uniformly sampled prox-SDCA and SDCA as special cases.
The rest of this paper is organized as follows. Section~\ref{sec:related} reviews the related work. Section~\ref{sec:prelim} presents some preliminaries. In section~\ref{sec:imp-sample}, we will study stochastic optimization with importance sampling. Section~\ref{sec:application} lists several applications for the proposed algorithms. Section~\ref{sec:experiment} gives our empirical evaluations. Section~\ref{sec:conclusion} concludes the paper.
\section{Related Work}
\label{sec:related}
We review some related work on Proximal Stochastic Gradient Descent (including more general proximal stochastic mirror descent) and Proximal Stochastic Dual Coordinate Ascent.
In recent years Proximal Stochastic Gradient Descent has been extensively studied~\cite{duchi2009efficient,DBLP:conf/colt/DuchiSST10}. As a special case of prox-SGD, Stochastic Gradient Descent has been extensively studied in stochastic approximation theory~\cite{kushner2003stochastic}; however these results are often asymptotic, so there is no explicit bound in terms of $T$. Later on, finite sample convergence rate of SGD for solving linear prediction problem were studied by a number of authors~\cite{DBLP:conf/icml/Zhang04,DBLP:conf/icml/Shalev-ShwartzSS07}. In general prox-SGD can achieve a convergence rate of $O(1/\sqrt{T})$ for convex loss functions, and a convergence rate of $O(\log T/T)$ for strongly convex loss functions, where $T$ is the number of iterations of the algorithm.
More recently, researchers have improved the previous bound to $O(1/T)$ by $\alpha$-Suffix Averaging ~\cite{rakhlin2011making}, which means instead of returning the average of the entire sequence of classifiers, the algorithm will average and return just an $\alpha$-suffix: the average of the last $\alpha$ fraction of the whole sequence of classifiers. In practice it may be difficult for users to decide when to compute the $\alpha$-suffix. To solve this issue, a polynomial decay averaging strategy is proposed by~\cite{shamir2013stochastic}, which will decay the weights of old individual classifiers polynomially and also guarantee a $O(1/T)$ convergence bound.
For Proximal Stochastic Dual Coordinate Ascent~\cite{shalev2012proximal}, Shalev-Shwartz and Zhang recently proved that the algorithm achieves a convergence rate of $O(1/T)$ for Lipschitz loss functions, and enjoys a linear convergence rate of $O(\exp(-O(T)))$ for smooth loss functions. For structural SVM, a similar result was also obtained in \cite{DBLP:journals/corr/abs-1207-4747}. Several others researchers~\cite{mangasarian1999successive,hsieh2008dual} have studied the convergence behavior of the related non-randomized DCA (dual coordinate ascent) algorithm for SVM, but could only obtain weaker convergence results. The related randomized coordinate descent method has been investigated by some other authors~\cite{DBLP:journals/jmlr/Shalev-ShwartzT11,DBLP:journals/siamjo/Nesterov12,richtarik2012iteration}. However, when applied to SDCA, the analysis can only lead to a convergence rate of the dual objective value while we are mainly interested in the convergence of primal objective in machine learning applications. Recently, Shai Shalev-Shwartz and Tong Zhang has resolved this issue by providing a primal-dual analysis that showed a linear convergence rate $O(\exp(- O(T)))$ of the duality gap for SDCA with smooth loss function~\cite{DBLP:journals/jmlr/ShaiTong13}.
Although both prox-SGD and prox-SDCA have been extensively studied, most of the existing work only considered the uniform sampling scheme during the entire learning process. Recently, we noticed that~\cite{needell2014stochastic} Deanna Needell et. al. considered importance sampling for stochastic gradient descent, where they suggested similar or the same sampling distributions. Strohmer and Vershynin~\cite{strohmer2009randomized} proposed a variant of the Kaczmarz method (an iterative method for solving systems of linear equations) which selects rows with probability proportional to their squared norm. It is pointed out that, this algorithm is actually a SGD algorithm with importance sampling~\cite{needell2014stochastic}. However, we have studied importance sampling for more general composite objectives and more general proximal stochastic gradient descent, i.e., proximal stochastic mirror descent which covers their algorithms as special cases. Furthermore, we have also studied prox-SDCA with importance sampling, which is not covered by their study. In addition, Xiao and Zhang~\cite{xiao2014proximal} have also proposed a proximal stochastic gradient method with progressive variance reduction, where they also provide importance sampling strategy for only smooth loss functions, which is the same with ours. Because our analysis is based on the basic version of stochastic gradient (mirror) descent, the convergence rate is worse than the linear rates in SAG~\cite{roux2012stochastic} and SVRG~\cite{xiao2014proximal} for smooth strongly convex objective functions. However, our main concern is on the effectiveness of importance sampling, which could be applied to many other gradient based algorithms.
We shall mention that for coordinate descent, some researchers have recently considered non-uniform sampling strategies~\cite{nesterov2012efficiency,lee2013efficient}, but their results cannot be directly applied to proximal SDCA which we are interested in here. The reasons are several-folds. The primal-dual analysis of prox-SDCA in this paper is analogous to that of ~\cite{DBLP:journals/jmlr/ShaiTong13}, which directly implies a convergence rate for the duality gap. The proof techniques rely on the structure of the regularized loss minimization, which can not be applied to general primal coordinate descent. The suggested distribution of the primal coordinate descent is propositional to the smoothness constant of every coordinate, while the distribution of prox-SDCA is propositional to a constant plus the smoothness constant of the primal individual loss function, which is the inverse of the strongly convex constant of the dual coordinate. These two distributions are quite different. In addition, we also provide an importance sampling distribution when the individual loss functions are Lipschitz. We also noticed that a mini-batch SDCA~\cite{DBLP:conf/nips/ShaiTong} and an accelerated version of prox-SDCA~\cite{DBLP:conf/icml/Shalev-Shwartz014} were studied recently by Shalev-Shwartz and Zhang. The accelerated version in~\cite{DBLP:conf/icml/Shalev-Shwartz014} uses an inner-outer-iteration strategy, where the inner iteration is the standard prox-SDCA procedure.
Therefore the importance sampling results of this paper can be directly applied to the
accelerated prox-SDCA because
the convergence of inner iteration is faster than that of uniform sampling.
Therefore in this paper we will only focus on showing the effectiveness of importance sampling for the unaccelerated prox-SDCA.
Related to this paper, non-uniform sampling in the online setting is related to selective sampling, which can be regarded as a form of online active learning which has been extensively studied in the literature~\cite{DBLP:conf/colt/Cesa-BianchiCG03,DBLP:conf/nips/CavallantiCG08,DBLP:conf/icml/Cesa-BianchiGO09,DBLP:conf/icml/OrabonaC11,DBLP:journals/ml/CavallantiCG11}. Similar to importance sampling in stochastic optimization, selective sampling also works in iterations. However the purposes are quite different.
Specifically, selective sampling draws unlabeled instances uniformly at random from a
fixed distribution and decides which samples to label --- the goal is to reduce the number
of labels needed to achieve a certain accuracy; importance sampling considered in this paper
does not reduce the number of labels needed, and the goal is to reduce the training time.
\section{Preliminaries}
\label{sec:prelim}
Here, we briefly introduce some key definitions and propositions that are useful throughout the paper (for details, please refer to~\cite{borwein2006convex} ). We consider vector functions: $\phi: \R^d\rightarrow\R$.
\begin{definition}
For $\sigma\ge 0$, a function $\phi: \R^d\rightarrow\R$ is $\sigma$-strongly convex with respect to (w.r.t.) a norm $\|\cdot\|$, if for all $\u,\v \in \R^d$, we have
\begin{eqnarray*}
\phi(\u) \ge \phi(\v) + \nabla\phi(\v)^\top(\u-\v) + \frac{\sigma}{2}\|\u-\v\|^2 ,
\end{eqnarray*}
or equivalently, $\forall s\in[0,1]$
\begin{eqnarray*}
\phi(s\u+(1-s)\v)\le s\phi(\u) + (1-s)\phi(\v) - \frac{\sigma s(1-s)}{2}\|\u-\v\|^2 .
\end{eqnarray*}
\end{definition}
For example, $\phi(\w)=\frac{1}{2}\|\w\|_2^2$ is $1$-strongly convex w.r.t. $\|\cdot\|_2$.
\begin{definition}
A function $\phi: \R^d\rightarrow\R$ is $L$-Lipschitz w.r.t. a norm $\|\cdot\|$, if for all $\u,\v \in\R^d$, we have
\begin{eqnarray*}
|\phi(\u)-\phi(\v)|\le L \|\u-\v\|.
\end{eqnarray*}
\end{definition}
\begin{definition}
A function $\phi: \R^d\rightarrow\R$ is $(1/\gamma)$-smooth if it is differentiable and its gradient is $(1/\gamma)$-Lipschitz, or, equivalently for all $\u,\v\in\R^d$, we have
\begin{eqnarray*}
\phi(\u) \le \phi(\v) + \nabla\phi(\v)^\top(\u-\v) + \frac{1}{2\gamma}\|\u-\v\|^2 .
\end{eqnarray*}
\end{definition}
For example, $\phi(\w)=\frac{1}{2}\|\w\|_2^2$ is $1$-smooth w.r.t. $\|\cdot\|_2$.
\begin{prop}
If $\phi$ is $(1/\gamma)$-smooth with respect to a norm $\|\cdot\|_P$, then its dual function $\phi^*$ is $\gamma$-strongly convex with respect to its dual norm $\|\cdot\|_D$, where
\begin{eqnarray*}
\phi^*(\v)=\sup_\w(\v^\top\w-\phi(\w)) ,
\end{eqnarray*}
and the dual norm is defined as
\begin{eqnarray*}
\|\v\|_D=\sup_{\|\w\|_P=1}\v^\top\w .
\end{eqnarray*}
\end{prop}
For example, the dual norm of $\|\cdot\|_2$ is itself. The dual norm of $\|\cdot\|_1$ is $\|\cdot\|_\infty$. The dual norm of $\|\cdot\|_p$ is $\|\cdot\|_q$, where $1/q+1/p=1$.
\begin{definition}
Let $\psi:\R^d\rightarrow\R$ be a continuously-differentiable real-valued and strictly convex function. Then the Bregman divergence associated with $\psi$ is
\begin{eqnarray*}
\B_{\psi}(\u,\v)=\psi(\u)-\psi(\v) - \langle\nabla\psi(\v),\u-\v\rangle,
\end{eqnarray*}
which is the difference between the value of $\psi$ at $\u$ and the value of the first-order Taylor expansion of $\psi$ around $\v$ evaluated at $\u$.
\end{definition}
Throughout, $\psi$ denotes a continuously differentiable function that is $\sigma$-strongly convex w.r.t. a norm $\|\cdot\|$, so that $\Bpsi(\u,\v)\ge \frac{\sigma}{2}\|\u-\v\|^2$.
\begin{definition}
A function $f:\R^d \rightarrow \R$ is $\mu$-strongly convex with respect to a differentiable function $\psi$, if for any $\u,\v$, we have
\begin{eqnarray*}
f(\u) \ge f(\v) + \langle \nabla f(\v), \u-\v \rangle + \mu \Bpsi(\u,\v).
\end{eqnarray*}
\end{definition}
For example, when $\psi(\w)=\frac{1}{2}\|\w\|_2^2$, we recover the usual definition of strongly convexity.
\begin{definition}
A function $f:\R^d \rightarrow \R$ is $(1/\gamma)$-smooth with respect to a differentiable function $\psi$, if for any $\u,\v$, we have
\begin{eqnarray*}
f(\u) \le f(\v) + \langle \nabla f(\v), \u-\v \rangle + (1/\gamma) \Bpsi(\u,\v).
\end{eqnarray*}
\end{definition}
\section{Stochastic Optimization with Importance Sampling}
\label{sec:imp-sample}
We consider the following generic optimization problem associated with regularized loss minimization of linear predictors. Let $\phi_1,\phi_2,\ldots,\phi_n$ be $n$ vector functions from $\R^d$ to $\R$. Our goal is to find an approximate solution of the following optimization problem
\begin{eqnarray}\label{eqn:primal-objective}
\min_{\w\in\R^d} P(\w):=\underbrace{\frac{1}{n}\sum^n_{i=1}\phi_i(\w)}_{f(\w)}+ \lambda r(\w),
\end{eqnarray}
where $\lambda >0$ is a regularization parameter, and $r$ is a regularizer.
For example, given examples $(\x_i,y_i)$ where $\x_i\in\R^d$ and $y_i\in\{-1,+1\}$, the Support Vector Machine problem is obtained by setting $\phi_i(\w)=[1-y_i\x_i^\top\w]_+$, $[z]_+=\max(0,z)$, and $r(\w)=\frac{1}{2}\|\w\|_2^2$. Regression problems also fall into the above. For example, ridge regression is obtained by setting $\phi_i(\w)=(y_i-\x_i^\top\w)^2$ and $r(\w)=\frac{1}{2}\|\w\|_2^2$, lasso is obtained by setting $\phi_i(\w)=(y_i-\x_i^\top\w)^2$ and $r(\w)=\|\w\|_1$.
Let $\w^*$ be the optimum of~\eqref{eqn:primal-objective}. We say that a solution $\w$ is $\epsilon_P$-sub-optimal if $P(\w)-P(\w^*)\le \epsilon_P$. We analyze the convergence rates of the proposed algorithms with respect to the number of iterations.
\subsection{Proximal Stochastic Gradient Descent with Importance Sampling}
In this subsection, we would consider the proximal stochastic mirror descent with importance sampling. Because proximal stochastic mirror descent is general version of proximal stochastic gradient descent (prox-SGD), we will abuse SGD to replace stochastic mirror descent.
If we directly apply full or stochastic gradient descent to the optimization problem~\eqref{eqn:primal-objective}, the solution may not satisfy some desirable property. For example, when $r(\w)=\|\w\|_1$, the optimal solution of the problem~\eqref{eqn:primal-objective} should be sparse, and we would like the approximate solution to be sparse as well. However, if we directly use stochastic (sub-)gradient descent, then the resulting solution will not achieve sparsity~\cite{duchi2009efficient}.
To effectively and efficiently solve the optimization problem~\eqref{eqn:primal-objective}, a well known method is the proximal stochastic (sub)-gradient descent. Specifically, Proximal Stochastic Gradient Descent works in iterations. At each iteration $t=1,2,\ldots$, $i_t$ will be uniformly randomly draw from $\{1,2,\ldots,n\}$, and the iterative solution will be updated according to the formula
\begin{eqnarray}\label{eqn:psgd}
\w^{t+1} = \arg\min_{\w}\left[\langle \nabla \phi_{i_t}(\wt), \w\rangle + \lambda r(\w) + \frac{1}{\eta_t}\Bpsi(\w,\wt) \right] .
\end{eqnarray}
where $\Bpsi$ is a Bregman divergence and $\nabla \phi_{i_t}(\wt)$ denotes an arbitrary (sub-)gradient of $\phi_{i_t}$. Intuitively, this method works by minimizing a first-order approximation of the function $\phi_{i_t}$ at the current iterate $\wt$ plus the regularizer $\lambda r(\w)$, and forcing the next iterate $\w^{t+1}$ to lie close to $\wt$. The step size $\eta_t$ is used to controls the trade-off between these two objectives.Because the expectation of $\nabla \phi_{i_t}(\wt)$ is the same with $\nabla f(\wt)$, i.e., $\E[\nabla \phi_{i_t}(\wt)|\wt]=\frac{1}{n}\sum^n_{i=1}\nabla \phi_i(\wt)=\nabla f(\wt)$, the optimization problem~(\ref{eqn:psgd}) is an unbiased estimation of that for the proximal gradient descent.
We assume that the exact solution of the above optimization~(\ref{eqn:psgd}) can be efficiently solved. For example, when $\psi(\w)=\frac{1}{2}\|\w\|^2_2$, we have $\Bpsi(\u,\v)=\frac{1}{2}\|\u-\v\|_2^2$, and the above optimization will produce the $t+1$-th iterate as:
\begin{eqnarray*}
\w^{t+1}=\prox_{\eta_t\lambda r}\left(\wt-\eta_t\nabla \phi_{i_t}(\wt)\right),
\end{eqnarray*}
where $\prox_h(\x)=\arg\min_{\w}\Big(h(\w)+\frac{1}{2}\|\w-\x\|_2^2\Big)$. Furthermore, it is also assumed that the proximal mapping of $\eta_t\lambda r(\w)$, i.e., $\prox_{\eta_t\lambda r}(\x)$, is easy to compute. For example, when $r(\w)=\|\w\|_1$, the proximal mapping of $\lambda r(\w)$ is the following shrinkage operation
\begin{eqnarray*}
\prox_{\lambda r}(\x)=\sign(\x)\odot[|\x|-\lambda]_+,
\end{eqnarray*}
where $\odot$ is the element wise product, which can be computed in time complexity $O(d)$.
The advantage of proximal stochastic gradient descent is that each step only relies on a single derivative $\nabla\phi_{i_t}(\cdot)$, and thus the computational cost is $1/n$ of that of the standard proximal gradient descent. However, a disadvantage of the method is that the randomness introduces variance - this is caused by the fact that $\nabla \phi_{i_t}(\wt)$ equals the gradient $\nabla f(\wt)$ in expectation, but $\nabla \phi_i(\wt)$ varies with $i$. In particular, if the stochastic gradient has a large variance, then the convergence will become slow.
Now, we would like to study prox-SGD with importance sampling to reduce the variance of stochastic gradient. The idea of importance sampling is, at the $t$-th step, to assign each $i\in\{1,\ldots,n\}$ a probability $p_i^t\ge 0$ such that $\sum^n_{i=1} p_i^t=1$. We then sample $i_t$ from $\{1,\ldots, n\}$ based on probability $\p^t=(p_1^t,\ldots,p_n^t)^\top$. If we adopt this distribution, then proximal SGD with importance sampling will work as follows:
\begin{eqnarray}\label{eqn:iprox-sgd}
\w^{t+1}=\arg\min_{\w}\Big[\langle(n p_{i_t}^t)^{-1} \nabla\phi_{i_t}(\wt), \w \rangle +\lambda r(\w)+\frac{1}{\eta_t}\Bpsi(\w,\wt)\Big],
\end{eqnarray}
which is another unbiased estimation of the optimization problem for proximal gradient descent, because $\E[(np_{i_t}^t)^{-1}\nabla \phi_{i_t}(\wt)|\wt]=\sum^n_{i=1}p_i^t(np_i^t)^{-1}\nabla\phi_i(\wt)=\nabla f(\wt)$.
Similarly, if $\psi(\w)=\frac{1}{2}\|\w\|_2^2$, the proximal SGD with importance sampling will produce the $t+1$-th iterate as:
\begin{eqnarray*}
\w^{t+1}=\prox_{\eta_t\lambda r}\left(\wt-\eta_t(np_{i_t}^t)^{-1} \nabla\phi_{i_t}(\wt)\right).
\end{eqnarray*}
In addition, setting the derivative of optimization function in equation~(\ref{eqn:iprox-sgd}) as zero, we can obtain the following implicit update rule for the iterative solution:
\begin{eqnarray*}
\nabla\psi(\w^{t+1})=\nabla\psi(\wt)-\eta_t(np_{i_t}^t)^{-1}\nabla\phi_{i_t}(\wt)-\eta_t\lambda\partial r(\w^{t+1}),
\end{eqnarray*}
where $\partial r(\w^{t+1})$ is a subgradient.
Now the key question that attracted us is which $\p^t$ can optimally reduce the variance of the stochastic gradient. To answer this question, we will firstly prove a lemma, that can indicates the relationship between $\p^t$ and the convergence rate of prox-SGD with importance sampling.
\begin{lemma}\label{lemma:primal-gap-t}
Let $\w^{t+1}$ be defined by the update~(\ref{eqn:iprox-sgd}). Assume that $\psi(\cdot)$ is $\sigma$-strongly convex with respect to a norm $\|\cdot\|$, and that $f$ is $\mu$-strongly convex and $(1/\gamma)$-smooth with respect to $\psi$, if $r(\w)$ is convex and $\eta_t\in(0,\gamma]$ then $\w^{t+1}$ satisfies the following inequality for any $t\ge 1$,
\begin{eqnarray*}
\E[P(\w^{t+1})-P(\w^*)] \le\frac{1}{\eta_t}\E[\Bpsi(\w^*,\wt) -\Bpsi(\w^*,\w^{t+1})] -\mu\E\Bpsi(\w^*,\wt) + \frac{\eta_t}{\sigma}\E\V\left( (np_{i_t}^t)^{-1} \nabla\phi_{i_t}(\wt)\right),
\end{eqnarray*}
where the variance is defined as $\V( (np_{i_t}^t)^{-1} \nabla\phi_{i_t}(\wt))=\E \| (np_{i_t}^t)^{-1} \nabla \phi_{i_t}(\wt)- \nabla f(\wt)\|_*^2$, and the expectation is take with the distribution $\p^t$.
\end{lemma}
\begin{proof}
To simplify the notation, we denote $\g_t=(n p_{i_t}^t)^{-1} \nabla\phi_{i_t}(\wt)$. Because $f(\w)$ is $\mu$-strongly convex w.r.t. $\psi$, and $r(\w)$ is convex, we can derive
\begin{eqnarray*}
P(\w^*)\ge f(\wt)+\langle\nabla f(\wt), \w^* -\wt \rangle + \mu\Bpsi(\w^*,\wt)+\lambda r(\w^{t+1})+\lambda\langle\partial r(\w^{t+1}), \w^{*}-\w^{t+1} \rangle.
\end{eqnarray*}
Using the fact $f$ is $(1/\gamma)$-smooth w.r.t. $\psi$, we can further lower bound $f(\wt)$ by
\begin{eqnarray*}
f(\wt)\ge f(\w^{t+1}) - \langle \nabla f(\wt), \w^{t+1}-\wt \rangle - (1/\gamma)\Bpsi(\w^{t+1},\wt).
\end{eqnarray*}
Combining the above two inequalities, we have
\begin{eqnarray*}
P(\w^*)\ge P(\w^{t+1})+ \langle\nabla f(\wt)+\lambda \partial r(\w^{t+1}), \w^*-\w^{t+1} \rangle + \mu\Bpsi(\w^*,\wt) - (1/\gamma)\Bpsi(\w^{t+1},\wt).
\end{eqnarray*}
Considering the second term on the right-hand side, we have
\begin{eqnarray*}
\langle\nabla f(\wt)+\lambda \partial r(\w^{t+1}), \w^*-\w^{t+1} \rangle&=&\langle\nabla f(\wt)+[\nabla\psi(\wt)-\nabla\psi(\w^{t+1})]/\eta_t -\g_t, \w^*-\w^{t+1} \rangle\\
&=&\frac{1}{\eta_t}\langle\nabla\psi(\wt)-\nabla\psi(\w^{t+1}) , \w^*-\w^{t+1} \rangle + \langle \g_t- \nabla f(\wt), \w^{t+1}-\w^* \rangle.
\end{eqnarray*}
Combining the above two inequalities, we get
\begin{eqnarray*}
&&P(\w^*)-P(\w^{t+1})-\mu\Bpsi(\w^*,\wt)- \langle \g_t -\nabla f(\wt) , \w^{t+1}-\w^* \rangle\\
&&\ge \langle\nabla f(\wt)+\lambda \partial r(\w^{t+1}), \w^*-\w^{t+1} \rangle - (1/\gamma)\Bpsi(\w^{t+1},\wt) - \langle \g_t -\nabla f(\wt) , \w^{t+1}-\w^* \rangle\\
&&=\frac{1}{\eta_t}\langle \nabla\psi(\wt)-\nabla\psi(\w^{t+1}), \w^*-\w^{t+1} \rangle - (1/\gamma)\Bpsi(\w^{t+1},\wt).
\end{eqnarray*}
Plugging the following equality (Lemma 11.1 from~\cite{DBLP:books/daglib/0016248})
\begin{eqnarray*}
\Bpsi(\w^*,\w^{t+1}) + \Bpsi(\w^{t+1},\wt)-\Bpsi(\w^*,\wt) = \langle \nabla\psi(\wt)-\nabla\psi(\w^{t+1}), \w^*-\w^{t+1}\rangle,
\end{eqnarray*}
into the previous inequality gives
\begin{eqnarray*}
&&P(\w^*)-P(\w^{t+1})-\mu\Bpsi(\w^*,\wt)- \langle \g_t -\nabla f(\wt) , \w^{t+1}-\w^* \rangle\\
&&\ge \frac{1}{\eta_t}\left[\Bpsi(\w^*,\w^{t+1}) + \Bpsi(\w^{t+1},\wt)-\Bpsi(\w^*,\wt)\right] - (1/\gamma)\Bpsi(\w^{t+1},\wt)\\
&&\ge \frac{1}{\eta_t}\left[\Bpsi(\w^*,\w^{t+1}) -\Bpsi(\w^*,\wt)\right],
\end{eqnarray*}
where $\eta_t\in(0,\gamma]$ is used for the final inequality. Re-arranging the above inequality and taking expectation on both sides result in
\begin{eqnarray*}
\E[P(\w^{t+1})-P(\w^*)]\le\frac{1}{\eta_t}\E[\Bpsi(\w^*,\wt) -\Bpsi(\w^*,\w^{t+1})] -\mu\E\Bpsi(\w^*,\wt)- \E\langle\g_t -\nabla f(\wt) , \w^{t+1}-\w^* \rangle.
\end{eqnarray*}
To upper bound the last inner product term on the right-hand side, we can define the proximal full gradient update as $\wh^{t+1} = \arg\min_{\w}\left[\langle \nabla f(\wt), \w\rangle + \lambda r(\w) + \frac{1}{\eta_t}\Bpsi(\w, \wt) \right]$, which is independent with $\g_t$. Then we can bound $- \E \langle\g_t -\nabla f(\wt) , \w^{t+1}-\w^* \rangle$ as follows
\begin{eqnarray*}
-\E \langle\g_t -\nabla f(\wt), \w^{t+1}-\w^* \rangle&=& -\E \langle \g_t -\nabla f(\wt), \w^{t+1}-\wh^{t+1}+\wh^{t+1}-\w^* \rangle\\
&=& -\E \langle \g_t -\nabla f(\wt), \w^{t+1}-\wh^{t+1}\rangle - \E \langle \g_t -\nabla f(\wt), \wh^{t+1}-\w^* \rangle\\
&\le& \E \| \g_t -\nabla f(\wt)\|_* \|\w^{t+1}-\wh^{t+1}\|- \E \langle \g_t -\nabla f(\wt), \wh^{t+1}-\w^* \rangle\\
&\le &\E \frac{\eta_t}{\sigma}\|\g_t- \nabla f(\wt)\|_*^2- \E \langle \g_t -\nabla f(\wt), \wh^{t+1}-\w^* \rangle\\
&= &\E \frac{\eta_t}{\sigma}\|(np_{i_t}^t)^{-1} \nabla \phi_{i_t}(\wt) - \nabla f(\wt)\|_*^2= \frac{\eta_t}{\sigma}\V\left((np_{i_t}^t)^{-1} \nabla \phi_{i_t}(\wt)\right),
\end{eqnarray*}
where, the first inequality is due to Cauchy-Schwartz inequality, the second inequality is due to Lemma~\ref{lem:nonexpansive}, and the last equality is because of $\E[ \langle \g_t -\nabla f(\wt), \wh^{t+1}-\w^* \rangle|\wt]=0$. Finally, plugging the above inequality into the previous one concludes the proof of this lemma.
\end{proof}
From the above analysis, we can observe that the smaller the variance, the more reduction on objective function we have. In the next subsection, we will study how to adopt importance sampling to reduce the variance. This observation will be made more rigorous below.
\subsubsection{Algorithm}
According to the result in the Lemma~\ref{lemma:primal-gap-t} , to maximize the reduction on the objective value, we should choose $\p^t$ as the solution of the following optimization
\begin{eqnarray}\label{eqn:optimization-psgd}
\min_{\p^t,p_i^t\in[0,1],\sum^n_{i=1}p_i^t=1}\V((np_{i_t}^t)^{-1}\nabla\phi_{i_t}(\wt))\Leftrightarrow \min_{\p^t,p_i^t\in[0,1],\sum^n_{i=1}p_i^t=1}\frac{1}{n^2}\sum^n_{i=1}(p_i^t)^{-1}\|\nabla\phi_i(\wt)\|_*^2.
\end{eqnarray}
It is easy to verify, that the solution of the above optimization is
\begin{eqnarray}\label{eqn:distribution-psgd}
p_i^t=\frac{\|\nabla \phi_{i}(\wt)\|_*}{\sum^n_{j=1}\|\nabla\phi_{j}(\wt)\|_*},\quad \forall i\in\{1,2,\ldots,n\}.
\end{eqnarray}
Although, this distribution can minimize the variance of the $t$-th stochastic gradient, it requires calculation of $n$ derivatives at each step, which is clearly inefficient. To solve this issue, a potential solution is to calculate the $n$ derivatives at some steps and then keep it for use for a long time. In addition the true derivatives will changes every step, it may be better to add a smooth parameter to the sampling distribution. However this solution still can be inefficient. Another more practical solution is to relax the previous optimization~\eqref{eqn:optimization-psgd} as follows
\begin{eqnarray}\label{eqn:relax-psgd}
\min_{\p^t,p_i^t\in[0,1],\sum^n_{i=1}p_i^t=1}\frac{1}{n^2}\sum^n_{i=1}(p_i^t)^{-1}\|\nabla\phi_i(\wt)\|_*^2\le \min_{\p^t,p_i^t\in[0,1],\sum^n_{i=1}p_i^t=1}\frac{1}{n^2}\sum^n_{i=1}(p_i^t)^{-1}G_i^2
\end{eqnarray}
by introducing
\begin{eqnarray*}
G_i\ge\|\nabla \phi_i(\w^t)\|_*,\quad \forall t.
\end{eqnarray*}
Then, we can approximate the distribution in equation~(\ref{eqn:distribution-psgd}) by solving the the right hand side of the inequality~\eqref{eqn:relax-psgd} as
\begin{eqnarray*}
p_i^t=\frac{G_i}{\sum^n_{j=1}G_j},\quad \forall i\in\{1,2,\ldots,n\},
\end{eqnarray*}
which is independent with $t$.
Based on the above solution, we will suggest distributions for two kinds of loss functions - Lipschitz functions and smooth functions. Firstly, if $\phi_i(\w)$ is $L_i$-Lipschitz w.r.t. $\|\cdot\|_*$, then $\|\nabla \phi_i(\w)\|_*\le L_i$ for any $\w\in\R^d$, and the suggested distribution is
\begin{eqnarray*}
p_i^t=\frac{L_i}{\sum^n_{j=1}L_j},\quad \forall i\in\{1,2,\ldots,n\}.
\end{eqnarray*}
Secondly, if $\phi_i(\w)$ is $(1/\gamma_i)$-smooth and $\|\wt\|\le R$ for any $t$, then $\|\nabla \phi_i(\w^t)\|_*\le R/\gamma_i$, then the advised distribution is
\begin{eqnarray*}
p_i^t=\frac{\frac{1}{\gamma_i}}{\sum^n_{j=1}\frac{1}{\gamma_j}},\quad \forall i\in\{1,2,\ldots,n\}.
\end{eqnarray*}
Finally, we can summarize the proposed Proximal SGD with importance sampling in Algorithm~\ref{alg:ipsgd}.
\begin{algorithm}[htpb]
\caption{Proximal Stochastic Gradient Descent with Importance Sampling (Iprox-SGD)} \label{alg:ipsgd}
\begin{algorithmic}
\STATE {\bf Input}: $\lambda\ge 0$, the learning rates $\eta_1,\ldots,\eta_T>0$.
\STATE {\bf Initialize}: $\w^1=0$, $\p^1=(1/n,\ldots,1/n)^\top$.
\FOR{$t=1,\ldots,T$}
\STATE Update $\p^t$;
\STATE Sample $i_t$ from $\{1,\ldots,n\}$ based on $\p^t$;
\STATE Update $\w^{t+1}=\arg\min_{\w}\left[\left\langle(n p_{i_t}^t)^{-1} \nabla\phi_{i_t}(\wt), \w \right\rangle +\lambda r(\w)+\frac{1}{\eta_t}\Bpsi(\w,\wt)\right]$;
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsubsection{Analysis}
This section provides a convergence analysis of the proposed algorithm. Before presenting the results, we make some general assumptions:
\[
r(\textbf{0})=0,\quad \text{and}\ r(\w)\ge0,\ \text{for all} \; \w.
\]
It is easy to see that these two assumptions are generally satisfied by all the well-known regularizers.
Under the above assumptions, we first prove a convergence result for Proximal SGD with importance sampling using the previous Lemma~\ref{lemma:primal-gap-t}.
\begin{thm}\label{thm:psgd}
Let $\wt$ be generated by the proposed algorithm. Assume that $\psi(\cdot)$ is $\sigma$-strongly convex with respect to a norm $\|\cdot\|$, and that $f$ is $\mu$-strongly convex and $(1/\gamma)$-smooth with respect to $\psi$, if $r(\w)$ is convex and $\eta_t =\frac{1}{\alpha+\mu t}$ with $\alpha \ge 1/\gamma - \mu$, the following inequality holds for any $T\ge 1$,
\begin{eqnarray}\label{eqn:psgd2}
\frac{1}{T}\sum^T_{t=1} \E P(\w^{t+1})- P(\w^*)\le \frac{1}{T}\left[\alpha\Bpsi(\w^*,\w^1) + \E \sum^T_{t=1}\frac{V_t}{\sigma (\alpha+\mu t)}\right],
\end{eqnarray}
where the variance is defined as $V_t =\V[ (np_{i_t}^t)^{-1} \nabla\phi_{i_t}(\wt)]=\E \| (np_{i_t}^t)^{-1} \nabla \phi_{i_t}(\wt)- \nabla f(\wt)\|_*^2$, and the expectation is take with the distribution $\p^t$.
\end{thm}
\begin{proof}
Firstly it is easy to check $\eta_t\in(0, \gamma]$. Because the functions $\psi$, $f$, $r$ satisfy the the assumptions in Lemma~\ref{lemma:primal-gap-t}, we have
\begin{eqnarray*}
\E[P(\w^{t+1})-P(\w^*)] \le\frac{1}{\eta_t}\E[\Bpsi(\w^*,\wt) -\Bpsi(\w^*,\w^{t+1})] -\mu\E\Bpsi(\w^*,\wt) + \frac{\eta_t}{\sigma}\E\V\left[ (np_{i_t}^t)^{-1} \nabla\phi_{i_t}(\wt)\right].
\end{eqnarray*}
Summing the above inequality over $t=1,\ldots,T$, and using $\eta_t=1/(\alpha +\mu t)$ we get
\begin{eqnarray*}
&& \sum^T_{t=1} E P(\w^{t+1})-\sum^T_{t=1}P(\w^*) \\
&&\le \sum^T_{t=1}(\alpha+\mu t)\E\left[\Bpsi(\w^*,\wt) -\Bpsi(\w^*,\w^{t+1})\right] -\mu \sum^T_{t=1}\E\Bpsi(\w^*,\wt) + \E\sum^T_{t=1}\frac{V_t}{\sigma(\alpha+\mu t)}\\
&&= \alpha\Bpsi(\w^*,\w^1) - (\alpha+\mu T)\Bpsi(\w^*,\w^{T+1}) + \E \sum^T_{t=1}\frac{V_t}{\sigma(\alpha+\mu t)}\le \alpha\Bpsi(\w^*,\w^1) + \E \sum^T_{t=1}\frac{V_t}{\sigma(\alpha+\mu t)}.
\end{eqnarray*}
Dividing both sides of the above inequality by $T$ concludes the proof.
\end{proof}
\begin{cor}
Under the same assumptions in the Theorem~\ref{thm:psgd}, if we further assume $\phi_i(\w)$ is $(1/\gamma_i)$-smooth, $\|\wt\|\le R$ for any $t$, and the distribution is set as $p_i^t=\frac{R/\gamma_i}{\sum^n_{j=1} R/\gamma_j}$, then the following inequality holds for any $T\ge 1$,
\begin{eqnarray*}
\frac{1}{T}\sum^T_{t=1} \E P(\w^{t+1})- P(\w^*)&\le& \frac{1}{T}\left[\alpha\Bpsi(\w^*,\w^1) + \frac{(\sum^n_{i=1} R/\gamma_i)^2}{\sigma \mu n^2}\left(\frac{\mu}{\alpha+\mu}+ \ln(\alpha+\mu T)-\ln(\alpha+\mu)\right)\right]\nonumber\\
&=& O\left(\frac{(\sum^n_{i=1}R/\gamma_i)^2}{\sigma \mu n^2}\frac{\ln (\alpha +\mu T)}{T}\right).
\end{eqnarray*}
In addition, if $\mu=0$, the above bound is invalid, however if $\eta_t$ is set as $\sqrt{\sigma\Bpsi(\w^*,\w^1)}/(\sqrt{T}\frac{\sum^n_{i=1}R/\gamma_i}{n})$, we can prove the following inequality for any $T\ge 1$,
\begin{eqnarray*}
\frac{1}{T}\sum^T_{t=1} \E P(\w^{t+1})- P(\w^*)\le 2\sqrt{\frac{\Bpsi(\w^*,\w^1)}{\sigma}} \frac{\sum^n_{i=1}R/\gamma_i}{n}\frac{1}{\sqrt{T}}.
\end{eqnarray*}
\end{cor}
{\bf Remark:} If $\psi(\w)=\frac{1}{2}\|\w\|^2_2$ and $r(\w)=0$, then $\B_\psi(\u,\v)=\frac{1}{2}\|\u-\v\|^2_2$, and the proposed algorithm becomes SGD with importance sampling. Under these assumptions, it is achievable to get rid of the $\ln T$ factor in the convergence bound, when the objective function is strongly convex. However, we will not provide the details for concision. For more general Bregman divergence, it is difficult to remove this $\ln T$ factor, because many properties of $\frac{1}{2}\|\u-\v\|^2_2$ are not satisfied by the Bregman divergence, such as symmetry.
In addition, it is easy to derive high probability bound using existing work, such as the Theorem 8 in the~\cite{DBLP:conf/colt/DuchiSST10}. In this theorem, the high probability bound depends on the variance of the stochastic gradient, so our sampling strategy can improve this bound, since we are minimizing the variance. However, we do not explicitly provide the resulting bounds, because the consequence is relatively straightforward.
\begin{proof}
Firstly, the fact $\phi_i(\w)$ is $(1/\gamma_i)$-smooth, and $\|\wt\|\le R$ for any $t$ implies $\|\nabla \phi_i(\wt)\|_*\le R/\gamma_i$. Using this result and the distribution $p_i^t=\frac{R/\gamma_i}{\sum^n_{j=1}R/\gamma_j}$, we can get
\begin{eqnarray*}
V_t=\E \| (np_{i_t}^t)^{-1} \nabla \phi_{i_t}(\wt)- \nabla f(\wt)\|_*^2\le \E \| (np_{i_t}^t)^{-1} \nabla \phi_{i_t}(\wt)\|_*^2=\frac{1}{n^2}\sum^n_{i=1}\frac{1}{p_i}\|\nabla \phi_i(\wt)\|_*^2\le \left(\frac{\sum^n_{i=1} R/\gamma_i}{n}\right)^2,
\end{eqnarray*}
where the first inequality is due to $\E\|\z-\E\z\|^2=\E\|\z\|^2-\|\E\z\|^2$. Using the above inequality gives
\begin{eqnarray*}
\sum^T_{t=1}\frac{V_t}{\sigma (\alpha+\mu t)}&\le& \left(\frac{\sum^n_{i=1} R/\gamma_i}{n}\right)^2 \sum^T_{t=1}\frac{1}{\sigma (\alpha+\mu t)}\\
&\le& \left(\frac{\sum^n_{i=1} R/\gamma_i}{n}\right)^2\frac{1}{\sigma}\left[\frac{1}{\alpha+\mu}+\int^T_{t=1}\frac{1}{ \alpha+\mu t}\right]\\
&\le& \left(\frac{\sum^n_{i=1} R/\gamma_i}{n}\right)^2\frac{1}{\sigma \mu}\left[\frac{\mu}{\alpha+\mu}+ \ln(\alpha+\mu T)-\ln(\alpha+\mu)\right]
\end{eqnarray*}
Plugging the above inequality into the inequality~(\ref{eqn:psgd2}) concludes the proof of the first part.
To prove the second part, we can plug the bound on $V_t$ and the equality $\eta_t=\sqrt{\sigma\Bpsi(\w^*,\w^1)}/(\sqrt{T}\frac{\sum^n_{i=1}R/\gamma_i}{n})$ into the inequality~(\ref{eqn:psgd2}).
\end{proof}
\textbf{Remark.} If the uniform distribution is adopted, it is easy to observe that $V_t$ is bounded by $\frac{\sum^n_{i=1}(R/\gamma_i)^2}{n}$, and the Theorem 1 will results in $\frac{1}{T}\sum^T_{t=1}\E P(\w^{t+1})- P(\w^*)\le O\left(\frac{\sum^n_{i=1}(R/\gamma_i)^2}{\sigma \mu n}\frac{\ln (\alpha +\mu T)}{T}\right)$ for strongly convex $f$, and $\frac{1}{T}\sum^T_{t=1}\E P(\w^{t+1})- P(\w^*)\le 2\sqrt{\frac{\Bpsi(\w^*,\w^1)}{\sigma} \frac{\sum^n_{i=1}(R/\gamma_i)^2}{n}}\frac{1}{\sqrt{T}}$ for general convex $f$. However, according to Cauchy-Schwarz inequality,
\begin{eqnarray*}
\frac{\sum^n_{i=1}(R/\gamma_i)^2}{n}/\left( \frac{\sum^n_{i=1}R/\gamma_i}{n}\right)^2 =\frac{n\sum^n_{i=1}(R/\gamma_i)^2}{(\sum^n_{i=1}R/\gamma_i)^2} \ge 1,
\end{eqnarray*}
implies importance sampling does improve the convergence rate, especially when $\frac{(\sum^n_{i=1}R/\gamma_i)^2}{\sum^n_{i=1}(R/\gamma_i)^2} \ll n $.
\begin{thm}\label{thm:comid}
Let $\wt$ be generated by the proposed algorithm. Assume that $\psi(\cdot)$ is $\sigma$-strongly convex with respect to a norm $\|\cdot\|$, $f$ is convex, and $r(\w)$ is $1$-strongly convex, if $\eta_t$ is set as $1/(\lambda t)$ for all $t$, the following inequality holds for any $T\ge 1$,
\begin{eqnarray}\label{eqn:comid}
\frac{1}{T}\sum^T_{t=1} \E P(\wt)- P(\w^*)\le \frac{1}{T}\left[\lambda\Bpsi(\w^*,\w^1)+\frac{1}{\lambda\sigma}\sum^T_{t=1}\frac{1}{t}\E\|\frac{\nabla \phi_{i_t}(\wt)}{np_{i_t}^t}\|_*^2\right],
\end{eqnarray}
where the expectation is take with the distribution $\p^t$.
\end{thm}
\begin{proof}
The fact $r(\w)$ is $1$-strongly convex implies that $\lambda r(\w)$ is $\lambda$-strongly convex. Then, all the assumptions in the Corollary~6 of~\cite{DBLP:conf/colt/DuchiSST10} are satisfied, so we have the following inequality,
\begin{eqnarray*}
\sum^T_{t=1}[\frac{1}{n p^t_{i_t}}\phi_{i_t}(\wt)+\lambda r(\w^{t+1})-\frac{1}{n p^t_{i_t}}\phi_{i_t}(\w^*)-\lambda r(\w^*)]\le \lambda \Bpsi(\w^*,\w^1) +\frac{1}{\lambda\sigma}\sum^T_{t=1}\frac{1}{t}\|\frac{1}{n p^t_{i_t}}\nabla \phi_{i_t}(\wt)\|_*^2
\end{eqnarray*}
which is actually the same with the last display in the page 5 of~\cite{DBLP:conf/colt/DuchiSST10}. Taking expectation on both sides of the above inequality and using $r(\w^1)=0$ concludes the proof.
\end{proof}
We will use the above Theorem to derive two logarithmic convergence bounds.
\begin{cor}
Under the same assumptions in the Theorem~\ref{thm:comid}, if we further assume $\phi_i(\w)$ is $(1/\gamma_i)$-smooth, $\|\wt\|\le R$ for any $t$, and the distribution is set as $p_i^t=\frac{R/\gamma_i}{\sum^n_{j=1} R/\gamma_j}$, then the following inequality holds for any $T\ge 1$,
\begin{eqnarray*}
\frac{1}{T}\sum^T_{t=1} \E P(\wt)- P(\w^*)\le \frac{1}{T}\left[ \lambda \Bpsi(\w^*,\w^1) +\frac{\left(\sum^n_{i=1}R/\gamma_i\right)^2}{\lambda\sigma n^2} (\ln T + 1)\right] = O\left(\frac{\left(\sum^n_{i=1}R/\gamma_i\right)^2}{\lambda\sigma n^2} \frac{\ln T}{T} \right).
\end{eqnarray*}
Under the same assumptions in the theorem~\ref{thm:comid}, if $\phi_i(\w)$ is $L_i$-Lipschitz, and the distribution is set as $p_i=L_i/\sum^n_{j=1}L_j$, $\forall i$.
\begin{eqnarray*}
\frac{1}{T}\sum^T_{t=1} \E P(\wt)- P(\w^*)\le\frac{1}{T}\left[ \lambda \Bpsi(\w^*,\w^1) +\frac{\left(\sum^n_{i=1}L_i\right)^2}{\lambda\sigma n^2} (\ln T + 1)\right] = O\left(\frac{\left(\sum^n_{i=1}L_i\right)^2}{\lambda\sigma n^2} \frac{\ln T}{T} \right).
\end{eqnarray*}
\end{cor}
\begin{proof}
If $\|\phi_i(\wt)\|_* \le G_i$ and $p^t_i$ is set as $\frac{G_i}{\sum^n_{j=1} G_j}$ for any $i$, we have
\begin{eqnarray*}\label{eqn:expectation-squared-variable}
\E\|\frac{\nabla \phi_{i_t}(\wt)}{n p^t_{i_t}}\|_*^2 =\frac{1}{n^2}\sum^n_{i=1}\frac{\|\nabla\phi_i(\wt)\|_*^2}{p_i}\le\frac{(\sum^n_{i=1}G_i)^2}{n^2}.
\end{eqnarray*}
Since the same assumptions in the Theorem~\ref{thm:comid} hold, plugging the above inequality into the inequality~(\ref{eqn:comid}), and using the fact $\sum^T_{t=1}\le 1 + \ln T$ results in
\begin{eqnarray}\label{eqn:comid-bound-G}
\frac{1}{T}\sum^T_{t=1} \E P(\wt)- P(\w^*)\le\frac{1}{T}\left[ \lambda \Bpsi(\w^*,\w^1) +\frac{\left(\sum^n_{i=1}G_i\right)^2}{\lambda\sigma n^2} (\ln T + 1)\right] = O\left(\frac{\left(\sum^n_{i=1}G_i\right)^2}{\lambda\sigma n^2} \frac{\ln T}{T} \right).
\end{eqnarray}
When $\phi_i(\w)$ is $(1/\gamma_i)$-smooth, $\|\wt\|\le R$ for any $t$, we have $\|\nabla \phi_i(\wt) \|_* \le R/\gamma_i$. Plugging $G_i=R/\gamma_i$ into the inequality~(\ref{eqn:comid-bound-G}) concludes the proof of the first part.
When $\phi_i(\w)$ is $L_i$-Lipschitz, we have $\|\nabla\phi_i(\wt)\|_*\le L_i$. Plugging $G_i=L_i$ into the inequality~(\ref{eqn:comid-bound-G}) concludes the proof of the second part.
\end{proof}
{\bf Remark:} If the uniform distribution is adopted, it is easy to observe that $V_t$ is bounded by $\frac{\sum^n_{i=1}(R/\gamma_i)^2}{n}$ for smooth $\phi_i(\cdot)$, and bounded by $\frac{\sum^n_{i=1}(L_i)^2}{n}$ for Lipschitz $\phi_i(\cdot)$. So the Theorem 2 will results in $\frac{1}{T}\sum^T_{t=1}\E P(\wt)- P(\w^*)\le O\left(\frac{\sum^n_{i=1}(R/\gamma_i)^2}{\lambda\sigma n} \frac{\ln T}{T} \right)$ for smooth $\phi_i$, and $\frac{1}{T}\sum^T_{t=1}\E P(\wt)- P(\w^*)\le O\left(\frac{\sum^n_{i=1}(L_i)^2}{\lambda\sigma n} \frac{\ln T}{T} \right)$ for Lipschitz $\phi_i$. However, according to Cauchy-Schwarz inequality,
\begin{eqnarray*}
\frac{\sum^n_{i=1}(R/\gamma_i)^2}{n}/\left( \frac{\sum^n_{i=1}R/\gamma_i}{n}\right)^2 =\frac{n\sum^n_{i=1}(R/\gamma_i)^2}{(\sum^n_{i=1}R/\gamma_i)^2} \ge 1,\quad \frac{\sum^n_{i=1}L_i^2}{n}/(\frac{\sum^n_{i=1} L_i}{n})^2 = \frac{n\sum^n_{i=1}L_i^2}{(\sum^n_{i=1}L_i)^2}\ge 1
\end{eqnarray*}
implies importance sampling does improve the convergence rate, especially when $\frac{(\sum^n_{i=1}R/\gamma_i)^2}{\sum^n_{i=1}(R/\gamma_i)^2} \ll n $, and $\frac{(\sum^n_{i=1}L_i)^2}{\sum^n_{i=1}(L_i)^2} \ll n $.
\begin{thm}\label{thm:comid-convex}
Let $\wt$ be generated by the proposed algorithm. Assume that $\psi(\cdot)$ is $\sigma$-strongly convex with respect to a norm $\|\cdot\|$, $f$ and $r(\w)$ are convex, if $\eta_t =\eta$, the following inequality holds for any $T\ge 1$,
\begin{eqnarray}\label{eqn:comid-convex}
\frac{1}{T}\sum^T_{t=1} \E P(\wt)- P(\w^*)\le \frac{1}{T}\left[\frac{1}{\eta} \Bpsi(\w^*,\w^1)+\frac{\eta}{2\sigma}\sum^T_{t=1}\E\|\frac{\nabla \phi_{i_t}(\wt)}{np_{i_t}^t}\|_*^2\right],
\end{eqnarray}
where the expectation is take with the distribution $\p^t$.
\end{thm}
\begin{proof}
Given the above conditions, all the assumptions of the Lemma 1 in the page 3 of~\cite{DBLP:conf/colt/DuchiSST10} are satisfied. So the following inequality holds for any $T\ge 1$,
\begin{eqnarray*}
\sum^T_{t=1}[\frac{1}{n p^t_{i_t}}\phi_{i_t}(\wt)+\lambda r(\wt)-\frac{1}{n p^t_{i_t}}\phi_{i_t}(\w^*)-\lambda r(\w^*)]\le \frac{1}{\eta} \Bpsi(\w^*,\w^1) +\lambda r(\w^1)+\frac{\eta}{2 \sigma}\sum^T_{t=1}\|\frac{1}{n p^t_{i_t}}\nabla \phi_{i_t}(\wt)\|_*^2,
\end{eqnarray*}
which is actually the same with the inequality in the Theorem 2 in the page 4 of~\cite{DBLP:conf/colt/DuchiSST10}. Taking expectation on both sides of the above inequality and using $r(\w^1)=0$ concludes the proof.
\end{proof}
\begin{cor}
Under the same assumptions in the Theorem~\ref{thm:comid-convex}, if we further assume $\phi_i(\w)$ is $(1/\gamma_i)$-smooth, $\|\wt\|\le R$ for any $t$, and the distribution is set as $p_i^t=\frac{R/\gamma_i}{\sum^n_{j=1} R/\gamma_j}$, then when $\eta_t$ is set as $\sqrt{2\sigma\Bpsi(\w^*,\w^1)}/(\frac{\sum^n_{i=1}R/\gamma_i}{n}\sqrt{T} )$, the following inequality holds for any $T\ge 1$,
\begin{eqnarray*}
\frac{1}{T}\sum^T_{t=1} \E P(\wt)- P(\w^*)\le \sqrt{ \Bpsi(\w^*,\w^1) \frac{2}{ \sigma}} (\frac{\sum^n_{i=1}R/\gamma_i}{n})\frac{1}{\sqrt{T}}.
\end{eqnarray*}
Under the same assumptions in the theorem~\ref{thm:comid-convex}, if $\phi_i(\w)$ is $L_i$-Lipschitz, and the distribution is set as $p_i=L_i/\sum^n_{j=1}L_j$, $\forall i$, then when $\eta_t$ is set as $\sqrt{2\sigma\Bpsi(\w^*,\w^1)}/(\frac{\sum^n_{i=1}L_i}{n}\sqrt{T} )$, the following inequality holds for any $T\ge 1$,
\begin{eqnarray*}
\frac{1}{T}\sum^T_{t=1} \E P(\wt)- P(\w^*) \le \sqrt{ \Bpsi(\w^*,\w^1) \frac{2}{ \sigma}} (\frac{\sum^n_{i=1}L_i}{n})\frac{1}{\sqrt{T}}.
\end{eqnarray*}
\end{cor}
\begin{proof}
Under the same assumptions in the Theorem~\ref{thm:comid-convex}, if $\|\phi_i(\wt)\|_*\le G_i$ and $p_i^t$ is set as $\frac{G_i}{\sum^n_{j=1}G_j}$ for any $i$, then we have
\begin{eqnarray*}
\E\|\frac{\nabla \phi_{i_t}(\wt)}{n p^t_{i_t}}\|_*^2 \le\frac{(\sum^n_{i=1}G_i)^2}{n^2}.
\end{eqnarray*}
Plugging the above inequality and $\eta_t = \sqrt{2\sigma\Bpsi(\w^*,\w^1)}/(\frac{\sum^n_{i=1}G_i}{n}\sqrt{T} )$, into the inequality~(\ref{eqn:comid-convex}) gives
\begin{eqnarray}\label{eqn:comid-convex-bound-G}
\frac{1}{T}\sum^T_{t=1} \E P(\wt)- P(\w^*)\le \sqrt{ \Bpsi(\w^*,\w^1) \frac{2}{ \sigma}} (\frac{\sum^n_{i=1}G_i}{n})\frac{1}{\sqrt{T}}.
\end{eqnarray}
When $\phi_i(\w)$ is $(1/\gamma_i)$-smooth, $\|\wt\|\le R$ for any $t$, we have $\|\nabla \phi_i(\wt) \|_* \le R/\gamma_i$. Plugging $G_i=R/\gamma_i$ into the inequality~(\ref{eqn:comid-convex-bound-G}) concludes the proof of the first part.
When $\phi_i(\w)$ is $L_i$-Lipschitz, we have $\|\nabla\phi_i(\wt)\|_*\le L_i$. Plugging $G_i=L_i$ into the inequality~(\ref{eqn:comid-convex-bound-G}) concludes the proof of the second part.
\end{proof}
{\bf Remark:} If the uniform distribution is adopted, it is easy to observe that $V_t$ is bounded by $\frac{\sum^n_{i=1}(R/\gamma_i)^2}{n}$ for smooth $\phi_i(\cdot)$, and bounded by $\frac{\sum^n_{i=1}(L_i)^2}{n}$ for Lipschitz $\phi_i(\cdot)$. So the Theorem 2 will results in $\frac{1}{T}\sum^T_{t=1}\E P(\wt)- P(\w^*)\le \sqrt{ \frac{2\Bpsi(\w^*,\w^1)\sum^n_{i=1}(R/\gamma_i)^2}{\sigma nT}}$ for smooth $\phi_i$, and $\frac{1}{T}\sum^T_{t=1}\E P(\wt)- P(\w^*)\le \sqrt{ \frac{2\Bpsi(\w^*,\w^1)\sum^n_{i=1}(L_i)^2}{\sigma nT}}$ for Lipschitz $\phi_i$. However, according to Cauchy-Schwarz inequality,
\begin{eqnarray*}
\frac{\sum^n_{i=1}(R/\gamma_i)^2}{n}/\left( \frac{\sum^n_{i=1}R/\gamma_i}{n}\right)^2 =\frac{n\sum^n_{i=1}(R/\gamma_i)^2}{(\sum^n_{i=1}R/\gamma_i)^2} \ge 1,\quad \frac{\sum^n_{i=1}L_i^2}{n}/(\frac{\sum^n_{i=1} L_i}{n})^2 = \frac{n\sum^n_{i=1}L_i^2}{(\sum^n_{i=1}L_i)^2}\ge 1
\end{eqnarray*}
implies importance sampling does improve the convergence rate, especially when $\frac{(\sum^n_{i=1}R/\gamma_i)^2}{\sum^n_{i=1}(R/\gamma_i)^2} \ll n $, and $\frac{(\sum^n_{i=1}L_i)^2}{\sum^n_{i=1}(L_i)^2} \ll n $.
\subsection{Proximal Stochastic Dual Coordinate Ascent with Importance Sampling}
In this section, we study the Proximal Stochastic Dual Coordinate Ascent method (prox-SDCA) with importance sampling.
Prox-SDCA works with the following dual problem of~\eqref{eqn:primal-objective}:
\begin{eqnarray}
\max_{\theta} D(\theta) := \frac{1}{n}\sum^n_{i=1}-\phi_i^*(-\theta_i)-\lambda r^*(\frac{1}{\lambda n}\sum^n_{i=1}\theta_i).
\end{eqnarray}
We assume that $r^*(\cdot)$ is continuous differentiable; the relationship between primal variable $\w$ and dual variable is
\begin{eqnarray*}
\w =\nabla r^*\left(\v(\theta) \right),\quad \v(\theta)=\frac{1}{\lambda n}\sum^n_{i=1} \theta_i.
\end{eqnarray*}
We also assume that $r(\w)$ is 1-strongly convex with respect to a norm $\|\cdot\|_{P'}$, i.e.,
\begin{eqnarray*}
r(\w + \Delta \w)\ge r(\w) + \nabla r(\w)^\top\Delta\w + \frac{1}{2}\|\Delta\w\|^2_{P'},
\end{eqnarray*}
which means that $r^*(\w)$ is 1-smooth with respect to its dual norm $\|\cdot\|_{D'}$. Namely,
\begin{eqnarray*}
r^*(\v+\Delta \v)\le h(\v; \Delta \v),
\end{eqnarray*}
where
\begin{eqnarray*}
h(\v;\Delta\v):= r^*(\v)+\nabla r^*(\v)^\top \Delta\v + \frac{1}{2}\|\Delta\v\|^2_{D'}.
\end{eqnarray*}
At the $t$-th step, traditional Proximal Stochastic Dual Coordinate Ascent (prox-SDCA) will uniformly randomly pick $i\in\{1,\ldots,n\}$, and update the dual variable $\theta^{t-1}_i$ as follows:
\begin{eqnarray*}
\theta^t_{i}=\theta^{t-1}_i + \Delta \theta^{t-1}_i,
\end{eqnarray*}
where
\begin{eqnarray}\label{eqn:update-psdca}
\Delta \theta^{t-1}_i &=& \arg\max_{\Delta \theta_i}\left[-\frac{1}{n}\phi^*_i(-(\theta_i^{t-1}+\Delta \theta_i))-\lambda\left(\frac{1}{\lambda n}\nabla r^*(\v^{t-1})^\top\Delta\theta_i+\frac{1}{2}\|\frac{1}{\lambda n}\Delta \theta_i\|^2_{D'}\right)\right] \nonumber\\
&=& \arg\max_{\Delta \theta_i}\left[-\phi^*_i(-(\theta_i^{t-1}+\Delta \theta_i)) - (\w^{t-1})^\top\Delta \theta_i - \frac{1}{2\lambda n}\|\Delta \theta_i\|^2_{D'} )\right],
\end{eqnarray}
where $\v^{t-1}=\frac{1}{\lambda n}\sum^n_{i=1}\theta^{t-1}_i$, which is equivalent to maximizing a lower bound of the following problem:
\begin{eqnarray*}
\Delta\theta_i^{t-1}=\arg\max_{\Delta \theta_i}\left[-\frac{1}{n}\phi^*_i(-(\theta_i^{t-1}+\Delta \theta_i))-\lambda r^*(\v^{t-1}+ \frac{1}{\lambda n}\Delta \theta_i) \right].
\end{eqnarray*}
However, the optimization~\eqref{eqn:update-psdca} may not have a closed form solution, and in prox-SDCA we may adopt other update rules $\Delta \theta_i = s(\u- \theta^{t-1}_i)$ for an appropriately chosen step size parameter $s>0$ and any vector $\u\in \R^d$ such that $-\u\in\partial\phi_i(\w^{t-1})$. When $r(\w)=\frac{1}{2}\|\w\|^2$, the proximal SDCA method is known as SDCA.
Now we will study prox-SDCA with importance sampling, which is to allow the algorithm to randomly pick $i$ according to probability $p_i$, which is the $i$-th element of $\p\in\R^n_+$, $\sum p_i=1$. Once we pick the coordinate $i$, $\theta_i$ is updated as traditional prox-SDCA. The main question we are interested in here is which $\p=(p_1,\ldots,p_n)^\top$ can optimally accelerate the convergence rate of prox-SDCA. To answer this question, we will introduce a lemma which will state the relationship between $\p$ and the convergence rate of prox-SDCA with importance sampling.
\begin{lemma}\label{lemma:dual-ascent-psdca}
Given a distribution $\p$, if assume $\phi_i$ is $(1/\gamma_i)$-smooth with norm $\|\cdot\|_P$, then for any iteration $t$ and any $s$ such that $s_i= s/(p_i n)\in [0,1],\quad \forall i$, we have
\begin{eqnarray}\label{eqn:dual-ascent-ipsdca}
\E[D(\theta^t)-D(\theta^{t-1})]\ge \frac{s}{n}\E[P(\w^{t-1})-D(\theta^{t-1})]-\frac{s}{2\lambda n^2}G^t,
\end{eqnarray}
where
\begin{eqnarray*}
G^t = \frac{1}{n}\sum^n_{i=1}(s_i R^2-\gamma_i(1-s_i)\lambda n)\E\|\u^{t-1}_i-\theta^{t-1}_i\|^2_D,
\end{eqnarray*}
$R=\sup_{\u\not=0}\|\u\|_{D'}/\|\u\|_D$, and $-\u^{t-1}_i\in \partial\phi_i(\w^{t-1})$.
\end{lemma}
\begin{proof}
Since only the $i$-th element of $\theta$ is updated, the improvement in the dual objective can written as
\begin{eqnarray}\label{eqn:stochastic-dual-ascent-ipsdca}
&&n[D(\theta^t)-D(\theta^{t-1})]\nonumber\\
&&=\left[-\phi^*_i(-\theta^t_i)-\lambda n r^*(\v^{t-1} +\frac{1}{\lambda n}\Delta \theta^{t-1}_i)\right]-\left[-\phi_i^*(-\theta^{t-1}_i)-\lambda n r^*(\v^{t-1})\right]\nonumber\\
&&\ge \underbrace{\left[-\phi^*_i(-\theta^t_i)-\lambda n h(\v^{t-1} ;\frac{1}{\lambda n}\Delta \theta^{t-1}_i)\right]}_{A_i}-\underbrace{\left[-\phi_i^*(-\theta^{t-1}_i)-\lambda n r^*(\v^{t-1})\right]}_{B_i}.
\end{eqnarray}
By the definition of the update, for all $s_i\in[0,1]$ we have
\begin{eqnarray}\label{eqn:Ai}
A_i&=&\max_{\Delta\theta_i}-\phi_i^*(-(\theta^{t-1}_i+\Delta\theta_i))-\lambda n h(\v^{t-1} ;\frac{1}{\lambda n}\Delta \theta_i)\nonumber\\
&\ge& -\phi_i^*(-(\theta_i^{t-1}+s_i(\u_i^{t-1}-\theta_i^{t-1})))-\lambda n h(\v^{t-1}; \frac{s_i}{\lambda n}(\u_i^{t-1}-\theta_i^{t-1})) .
\end{eqnarray}
From now on, we drop the superscript $(t-1)$ to simplify our discussion. Because $\phi_i$ is $(1/\gamma_i)$-smooth with $\|\cdot\|_P$, $\phi_i^*$ is $\gamma_i$-strongly convex with $\|\cdot\|_D$, and we have that
\begin{eqnarray*}
\phi_i^*(-(\theta_i+s_i(\u_i-\theta_i)))=\phi_i^*(s_i(-\u_i)+(1-s_i)(-\theta_i))\le s_i\phi_i^*(-\u_i)+(1-s_i)\phi_i^*(-\theta_i) - \frac{\gamma_i}{2}s_i(1-s_i)\|\u_i-\theta_i\|^2_D .
\end{eqnarray*}
Combining the above two inequalities, we obtain
\begin{eqnarray*}
&&\hspace{-0.3in}A_i\\
&&\hspace{-0.3in}\ge -s_i\phi_i^*(-\u_i) - (1-s_i)\phi_i^*(-\theta_i)+\frac{\gamma_i}{2}s_i(1-s_i)\|\u_i-\theta_i\|^2_D-\lambda n h(\v; \frac{s_i}{\lambda n}(\u_i-\theta_i))\\
&&\hspace{-0.3in}= -s_i\phi_i^*(-\u_i) - (1-s_i)\phi_i^*(-\theta_i)+\frac{\gamma_i}{2}s_i(1-s_i)\|\u_i-\theta_i\|^2_D-\lambda n r^*(\v)-s_i\w^\top(\u_i-\theta_i)-\frac{s_i^2}{2\lambda n}\|\u_i-\theta_i\|^2_{D'}\\
&&\hspace{-0.3in}\ge -s_i\phi_i^*(-\u_i) - (1-s_i)\phi_i^*(-\theta_i)+\frac{\gamma_i}{2}s_i(1-s_i)\|\u_i-\theta_i\|^2_D-\lambda n r^*(\v)-s_i\w^\top(\u_i-\theta_i)-\frac{s_i^2}{2\lambda n}R^2\|\u_i-\theta_i\|^2_{D}\\\\
&&\hspace{-0.3in}=\underbrace{-s_i(\phi_i^*(-\u_i)+\u_i^\top\w)}_{s_i\phi_i(\w)}+\underbrace{(-\phi_i^*(-\theta_i)-\lambda nr^*(\v))}_{B_i}+\frac{s_i}{2}(\gamma_i(1-s_i)-\frac{s_i R^2}{\lambda n})\|\u_i-\theta_i\|^2_D + s_i (\phi_i^*(-\theta_i)+\theta_i^\top\w),
\end{eqnarray*}
where we used $-\u^{t-1}_i\in \partial\phi_i(\w^{t-1})$ which yields $\phi_i^*(-\u_i)=-\u_i^\top\w - \phi_i(\w_i)$. Therefore
\begin{eqnarray}\label{eqn:bound-A-minus-B}
A_i-B_i\ge s_i\Big[\phi_i(\w) +\phi_i^*(-\theta_i)+\theta_i^\top \w +\left(\frac{\gamma_i(1-s_i)}{2}-\frac{s_i R^2}{2\lambda n}\right)\|\u_i-\theta_i\|^2_D \Big].
\end{eqnarray}
Therefore, if we take expectation of inequality~\eqref{eqn:bound-A-minus-B} with respect to the choice of $i$ and use the fact $s_i = s/(p_i n)$, we obtain that
\begin{eqnarray*}
\frac{1}{s}\E_t [A_i-B_i] = \frac{1}{n}\sum^n_{i=1}\frac{1}{s_i}[A_i-B_i]\ge\frac{1}{n}\sum^n_{i=1}\Big[\phi_i(\w)+\phi_i^*(-\theta_i)+\theta_i^\top \w +\left(\frac{\gamma_i(1-s_i)}{2}-\frac{s_i R^2}{2\lambda n}\right)\|\u_i-\theta_i\|^2_D\Big].
\end{eqnarray*}
Next note that with $\w=\nabla r^*(\v)$, we have $r(\w)+r^*(\v)=\w^\top\v$. Therefore
\begin{eqnarray*}
P(\w)-D(\theta) &=& \frac{1}{n}\sum^n_{i=1}\phi_i(\w) + \lambda r(\w) - \left(-\frac{1}{n}\sum^n_{i=1}\phi_i^*(-\theta_i)-\lambda r^*(\v)\right)\\
&=&\frac{1}{n}\sum^n_{i=1}\phi_i(\w)+\frac{1}{n}\sum^n_{i=1}\phi_i^*(-\theta_i)+\lambda\w^\top\v\\
&=&\frac{1}{n}\sum^n_{i=1}(\phi_i(\w)+\phi_i^*(-\theta_i)+\theta_i^\top\w).
\end{eqnarray*}
Taking expectation of inequality~\eqref{eqn:stochastic-dual-ascent-ipsdca} and plugging the above two equations give
\begin{eqnarray*}
\frac{n}{s}\E[D(\theta^t)-D(\theta^{t-1})]\ge \frac{1}{s}\E[A_i-B_i]\ge \E[P(\w^{t-1})-D(\theta^{t-1})]- \frac{G^t}{2\lambda n},
\end{eqnarray*}
where the equality $G^t = \frac{1}{n}\sum^n_{i=1}(s_i R^2-\gamma_i(1-s_i)\lambda n)\E\|\u^{t-1}_i-\theta^{t-1}_i\|^2_D$ is used. Multiplying both sides by $s/n$ concludes the proof.
\end{proof}
For many interesting cases, it is easy to estimate $R=\sup_{\u\not=0}\|\u\|_{D'}/\|\u\|_D$. For example, if $p>r>0$, then $\|\w\|_p\le\|\w\|_r\le d^{(1/r-1/p)}\|\w\|_p$ for any $\w\in\R^d$.
\subsubsection{Algorithm}
According to Lemma~\ref{lemma:dual-ascent-psdca}, to maximize the dual ascent for the $t$-th update, we should choose $s$ and $\p$ as the solution of the following optimization
\begin{eqnarray*}
\max_{s/(p_i n)\in[0,1], \p\in\R^n_+,\sum^n_{i=1}p_i=1} \frac{s}{n}\E[P(\w^{t-1})-D(\theta^{t-1})]-\frac{s}{n^2}\frac{G^t}{2\lambda}.
\end{eqnarray*}
However, because this optimization problem is difficult to solve, we choose to relax it as follows:
\begin{eqnarray*}
&&\max_{s/(p_i n)\in[0,1], \p\in\R^n_+, \sum^n_{i=1}p_i=1} \frac{s}{n}\E[P(\w^{t-1})-D(\theta^{t-1})]-\frac{s}{n^2}\frac{G^t}{2\lambda}\\
&&\ge\max_{s/(p_i n)\in[0,\frac{\lambda n \gamma_i}{R^2+\lambda n \gamma_i}], \p\in\R^n_+, \sum^n_{i=1}p_i=1} \frac{s}{n}\E[P(\w^{t-1})-D(\theta^{t-1})]-\frac{s}{n^2}\frac{G^t}{2\lambda}\\
&&\ge \max_{s/(p_i n)\in[0,\frac{\lambda n \gamma_i}{R^2+\lambda n \gamma_i}], \p\in\R^n_+, \sum^n_{i=1}p_i=1} \frac{s}{n}\E[P(\w^{t-1})-D(\theta^{t-1})].
\end{eqnarray*}
where the last inequality used $G^t = \frac{1}{n}\sum^n_{i=1}(s_i R^2-\gamma_i(1-s_i)\lambda n)\E\|\u^{t-1}_i-\theta^{t-1}_i\|^2_D\le 0$, since $s_i=s/(p_i n)\le \frac{\lambda n \gamma_i}{R^2+\lambda n \gamma_i}$. To optimize the final relaxation, we have the following proposition
\begin{prop}\label{prop:distribution-ipsdca}
The solution for the optimization
\begin{eqnarray*}
\max_{s, \p}\ s\quad s.t.\ s/(p_i n)\in[0, \frac{\lambda n\gamma_i}{R^2+ \lambda n\gamma_i}],\ p_i\ge 0,\ \sum^n_{i=1}p_i =1,
\end{eqnarray*}
is
\begin{eqnarray}\label{eqn:distribution-ipsdca}
s=\frac{n}{n+\sum^n_{i=1}\frac{R^2}{\lambda n \gamma_i}},\ p_i = \frac{1+\frac{R^2}{\lambda n\gamma_i}}{n + \sum^n_{j=1}\frac{R^2}{\lambda n \gamma_j}}.
\end{eqnarray}
\end{prop}
We omit the proof of this proposition since it is simple. Given that $\phi_i$ is ($1 /\gamma_i$)-smooth, $\forall i\in\{1,\ldots,n\}$, the sampling distribution should be set as in~\eqref{eqn:distribution-ipsdca}. Although $s$ can be set as in~\eqref{eqn:distribution-ipsdca}, it can also be optimized by maximizing some terms in the analysis, such as $A_i$, or the right hand side of inequality~\eqref{eqn:Ai}, or inequality~\eqref{eqn:bound-A-minus-B}, which can all guarantee the dual ascent $\E[D(\theta^t)-D(\theta^{t-1})]$ is no worse the the one
obtained by setting $s$ as in \eqref{eqn:distribution-ipsdca}.
When $\gamma_i=0$, the above distribution in the equation~\eqref{eqn:distribution-ipsdca} is not valid. To solve this problem, we combine the facts
\begin{eqnarray*}
P(\w^{t-1})-D(\theta^{t-1})\ge D(\theta^*)-D(\theta^{t-1}):=\epsilon^{t-1}_D,
\end{eqnarray*}
where $\theta^*$ the optimal solution of the dual problem $\max_\theta D(\theta)$,
\[
D(\theta^t)-D(\theta^{t-1})=\epsilon^{t-1}_D - \epsilon^t_D,
\]
and the inequality~~\eqref{eqn:dual-ascent-ipsdca}, to obtain
\begin{eqnarray}
\E[\epsilon^t_D]\le(1-\frac{s}{n})\E[\epsilon^{t-1}_D]+\frac{s}{2\lambda n^2}G^t.
\end{eqnarray}
According to this inequality, although every $\gamma_i=0$, if we further assume every $\phi_i$ is $L_i$-Lipschitz, then
\begin{eqnarray}\label{eqn:Gt-bound}
G^t = \frac{1}{n}\sum^n_{i=1}(s_i R^2-\gamma_i(1-s_i)\lambda n)\E\|\u^{t-1}_i-\theta^{t-1}_i\|^2_D\le \frac{4R^2s}{n^2}\sum^n_{i=1}\frac{1}{p_i}L_i^2,
\end{eqnarray}
where we used $s_i=s/(n p_i)$, $\|\u^{t-1}_i\|\le L_i$ and $\|\theta^{t-1}_i\|\le L_i$, since $-\u^{t-1}_i,-\theta^{t-1}_i\in\partial \phi_i(\w^{t-1})$.
Combining the above two inequalities results in
\begin{eqnarray}\label{eqn:dual-ascent-lipschitz}
\E[\epsilon^t_D]\le (1-\frac{s}{n})\E[\epsilon^{t-1}_D]+\frac{s}{2\lambda n^2}\frac{4R^2s}{n^2}\sum^n_{i=1}\frac{1}{p_i}L_i^2.
\end{eqnarray}
According to the above inequality, to minimize the $t$-th duality gap, we should choose a proper distribution to optimize the following problem
\begin{eqnarray*}
\min_{\p\in\R^n_+,\sum^n_{i=1}p_i=1}\sum^n_{i=1}\frac{1}{p_i}L_i^2,
\end{eqnarray*}
for which the optimal distribution is obviously
\[
p_i=\frac{L_i}{\sum^n_{j=1}L_j}.
\]
Because $s_i=s/(np_i)\in[0,1]$, the above distribution furthermore indicates
\[
s\in\bigcap^n_{i=1}[0, np_i]=\left[0, \frac{n L_{min}}{\sum^n_{j=1}L_j}\right]:=[0, \rho]
\]
where $L_{min}=\min\{L_1,L_2,\ldots,L_n\}$ and $\rho\le 1$.
In summary, the prox-SDCA with importance sampling can be summarized as in the algorithm~\ref{alg:Iprox-SDCA}.
\begin{algorithm}[ht]
\caption{Proximal Stochastic Dual Coordinate Ascent with Importance Sampling (Iprox-SDCA)} \label{alg:Iprox-SDCA}
\begin{algorithmic}
\STATE {\bf Input}: $\lambda> 0$, $R=\sup_{\u\not=0}\|\u\|_{D'}/\|\u\|_{D}$, norms $\|\cdot\|_D$, $\|\cdot\|_{D'}$, $\gamma_1,\ldots,\gamma_n > 0$, or $L_1,\ldots,L_n\ge 0$.
\STATE {\bf Initialize}: $\theta^0_i=0$, $\w^0=\nabla r^*(0)$, $p_i= \frac{1+\frac{R^2}{\lambda n\gamma_i}}{n+\sum^n_{j=1}\frac{R^2}{\lambda n\gamma_j}}$, or $p_i=\frac{L_i}{\sum^n_{j=1}L_j}$, $\forall i\in\{1,\ldots,n\}$.
\FOR{$t=1,\ldots, T$}
\STATE Sample $i_t$ from $\{1,\ldots,n\}$ based on $\p$;
\STATE Calculate $\Delta \theta^{t-1}_{i_t}$ using any of the following options (or achieving larger dual objective than one of the options);
\STATE \textbf{Option I:}
\STATE $\Delta \theta^{t-1}_{i_t} =\arg\max_{\Delta \theta_{i_t}}\left[-\phi^*_{i_t}(-(\theta_{i_t}^{t-1}+\Delta \theta_{i_t})) - (\w^{t-1})^\top\Delta \theta_{i_t} - \frac{1}{2\lambda n}\|\Delta \theta_{i_t}\|^2_{D'} \right] $;
\STATE \textbf{Option II:}
\STATE Let $\u$ be s.t. $-\u\in \partial \phi_{i_t}(\w^{t-1})$;
\STATE Let $\z= \u-\theta^{t-1}_{i_t}$;
\STATE Let $s_{i_t}=\arg\max_{s\in[0,1]}\left[-\phi^*_{i_t}(-(\theta_{i_t}^{t-1}+s\z)) - s(\w^{t-1})^\top\z- \frac{s^2}{2\lambda n}\|\z\|^2_{D'} \right]$;
\STATE Set $\Delta\theta^{t-1}_{i_t} = s_{i_t}\z$;
\STATE \textbf{Option III:}
\STATE Same as Option II, but replace the definition of $s_{i_t}$ as follows:
\STATE Let $s_{i_t}=\frac{\phi_{i_t}(\w^{t-1})+\phi^*_{i_t}(-\theta^{t-1}_{i_t})+(\w^{t-1})^\top\theta^{t-1}_{i_t}+\frac{\gamma_{i_t}}{2}\|\z\|^2_D}{\|\z\|^2_D(\gamma_{i_t}+R^2/(\lambda n))}$;
\STATE \textbf{Option IV (only for Lipschitz losses):}
\STATE Same as Option III, but replace $\|\z\|^2_D$ with $4L^2_{i_t}$;
\STATE \textbf{Option V (only for smooth losses):}
\STATE $\Delta \theta^{t-1}_{i_t}=\frac{n}{n+\sum^n_{i=1}\frac{R^2}{\lambda n \gamma_i}}\left(-\nabla\phi_{i_t}(\w^{t-1})-\theta^{t-1}_{i_t}\right) $;
\STATE $\theta^{t}_{i_t}=\theta^{t-1}_{i_t}+\Delta\theta^{t-1}_{i_t}$;
\STATE $\v^t=\v^{t-1}+\frac{1}{\lambda n}\Delta\theta_{i_t}^{t-1}$;
\STATE $\wt = \nabla r^*(\v^t)$;
\ENDFOR
\end{algorithmic}
\end{algorithm}
{\bf Remark:} It is easy to check the first three options can do no worse than the Option IV for Lipschitz losses and Option V for smooth losses.
Specifically, option I is to optimize $A_i$ directly. Option II only requires choosing $\Delta\theta_i=s(\u^{t-1}_i-\theta^{t-1}_i)$ and then chooses $s$ to optimize a lower bound of $A_i$, i.e., the right hand side of inequality~\eqref{eqn:Ai}. Option III is similar with option II, and chooses $s$ to optimize the bound on the right hand side of~\eqref{eqn:bound-A-minus-B}. As a result, all the first three options can do no worse than choosing the optimal $s$ for the Lemma~\ref{lemma:dual-ascent-psdca}. Option IV replace $\|\z\|^2_D$ with its upper bound $4L^2_{i_t}$, so that the inequality~\eqref{eqn:dual-ascent-lipschitz} is still valid. Option V is similar with options II and III, and chooses $s=\frac{n}{n+\sum^n_{i=1}\frac{R^2}{\lambda n \gamma_i}}$ as in the Proposition~\ref{prop:distribution-ipsdca}, so that $G^t\le 0$.
\subsubsection{Analysis}
In this subsection we analyze the convergence behavior of the proposed algorithm. Before presenting the theoretical results, we will make several assumptions without loss of generality:
\begin{itemize}
\item for the loss functions: $\phi_i(0)\le 1$, and $\phi_i(\w)\ge 0,\ \forall \w$, and
\item for the regularizer: $r(0)=0$ and $r(\w)\ge 0,\ \forall \w$.
\end{itemize}
Under the above assumptions, we have the following theorem for the expected duality gap of $\E[P(\wt)-D(\theta^T)]$.
\begin{thm}
Assume $\phi_i$ is ($1/\gamma_i$)-smooth $\forall i\in\{1,\ldots,n\}$ and set $p_i=(1+\frac{R^2}{\lambda n\gamma_i})/(n+\sum^n_{j=1}\frac{R^2}{\lambda n\gamma_j})$, for all $i\in\{1,\dots,n\}$. To obtain an expected duality gap of $\E[P(\wt)-D(\theta^T)]\le \epsilon_P$ for the proposed Proximal SDCA with importance sampling, it suffices to have a total number of iterations of
\begin{eqnarray*}
T \ge (n + \sum^n_{i=1}\frac{R^2}{\lambda n\gamma_i})\log\left((n+\sum^n_{i=1}\frac{R^2}{\lambda n\gamma_i})\frac{1}{\epsilon_P}\right).
\end{eqnarray*}
\end{thm}
\begin{proof}
Given the distribution $\p$ and step size $s$ in equation~\eqref{eqn:distribution-ipsdca}, since $\phi_i$ is ($1/\gamma_i$)-smooth, according to Lemma~\ref{lemma:dual-ascent-psdca}, we have
\begin{eqnarray*}
(s_i R^2- \gamma_i(1-s_i)\lambda n)\le 0,
\end{eqnarray*}
and hence $G^t\le 0$ for all $t$. By Lemma~\ref{lemma:dual-ascent-psdca}, this yields
\begin{eqnarray}\label{eqn:dual-ascent-psdca}
\E[D(\theta^t)-D(\theta^{t-1})]\ge \frac{s}{n}\E[P(\w^{t-1})-D(\theta^{t-1})].
\end{eqnarray}
Furthermore since
\begin{eqnarray*}
P(\w^{t-1})-D(\theta^{t-1})\ge D(\theta^*)-D(\theta^{t-1}):=\epsilon^{t-1}_D,
\end{eqnarray*}
where $\theta^*$ the optimal solution of the dual problem and $D(\theta^t)-D(\theta^{t-1})=\epsilon^{t-1}_D - \epsilon^t_D$, we obtain that
\begin{eqnarray}\label{eqn:convergence-psdca}
\E[\epsilon^t_D] \le \left(1-\frac{s}{n}\right)\E[\epsilon^{t-1}_D]\le \left(1-\frac{s}{n}\right)^t\E[\epsilon^0_D].
\end{eqnarray}
In addition, since $P(0)=\frac{1}{n}\sum^n_{i=1}\phi_i(0)+\lambda r(0)\le 1$ and
\begin{eqnarray*}
D(0)=\frac{1}{n}\sum^n_{i=1}-\phi_i^*(0)-\lambda r^*(0) = \frac{1}{n}\sum^n_{i=1}-\max_{\u_i}(0-\phi_i(\u_i))-\max_{\u}(0-r(\u))=\frac{1}{n}\sum^n_{i=1}\min_{\u_i}\phi_i(\u_i)+\lambda r(\u)\ge 0,
\end{eqnarray*}
we have $\epsilon^0_D\le P(0)-D(0)\le 1$. Combining this with inequality~\eqref{eqn:convergence-psdca}, we obtain
\begin{eqnarray*}
\E[\epsilon^t_D]\le (1-\frac{s}{n})^t \le \exp(-\frac{st}{n})=\exp(-\frac{t}{n+\sum^n_{i=1}\frac{R^2}{\lambda n \gamma_i}}).
\end{eqnarray*}
where the equality $s=\frac{n}{n+\sum^n_{i=1}\frac{R^2}{\lambda n \gamma_i}}$ in equation~\eqref{eqn:distribution-ipsdca} is used.
According to the above inequality, by setting
\begin{eqnarray*}
t\ge\left(n+\sum^n_{i=1}\frac{R^2}{\lambda n \gamma_i}\right)\log(\frac{1}{\epsilon_D}) ,
\end{eqnarray*}
the proposed algorithm will achieve $\E[\epsilon^t_D]\le \epsilon_D$. Furthermore, according to inequality~\eqref{eqn:dual-ascent-psdca}
\begin{eqnarray*}
\E[P(\wt)-D(\theta^t)]\le \frac{n}{s}\E[\epsilon^t_D-\epsilon^{t+1}_D]\le \frac{n}{s}\E[\epsilon^t_D],
\end{eqnarray*}
by setting
\begin{eqnarray*}
t\ge(n+\sum^n_{i=1}\frac{R^2}{\lambda n \gamma_i})\log\left((n+\sum^n_{i=1}\frac{R^2}{\lambda n\gamma_i})\frac{1}{\epsilon_P}\right),
\end{eqnarray*}
we will obtain $\E[\epsilon^t_D]\le \frac{s}{n}\epsilon_P$ and $\E[P(\wt)-D(\theta^t)]\le \epsilon_P$.
\end{proof}
\textbf{Remark:} If we adopt uniform sampling, i.e., $p_i=1/n$ $\forall i$, then we have to use the same $\gamma$ for all $\phi_i$, which should be $\gamma_{min} = \min\{\gamma_1,\ldots,\gamma_n\}$ according to the analysis. Once replacing $\gamma_i$ with $\gamma_{min}$, this theorem will recover the conclusion in the theorem 1 of \cite{shalev2012proximal}, i.e., $T\ge (n+\frac{R^2}{\lambda \gamma_{min}})\log\left((n+\frac{R^2}{\lambda \gamma_{min}})\frac{1}{\epsilon_P}\right)$ . However,
\begin{eqnarray*}
\frac{n+\frac{R^2}{\lambda \gamma_{min}}}{n+ \sum^n_{i=1}\frac{R^2}{\lambda \gamma_i n}}=
\frac{n\lambda \gamma_{min}+R^2}{n\lambda \gamma_{min}+ \frac{R^2}{n}\sum^n_{i=1}\frac{ \gamma_{min}}{ \gamma_i }}\ge 1,
\end{eqnarray*}
implies importance sampling does improve convergence, especially when $\sum^n_{i=1}\frac{ \gamma_{min}}{ \gamma_i } \ll n $.
For non-smooth loss functions, the convergence rate for Proximal SDCA with importance sampling is given below.
\begin{thm}
Consider the proposed proximal SDCA with importance sampling. Assume that $\phi_i$ is $L_i$-Lipschitz and set $p_i=L_i/\sum^n_{j=1}L_j$, $\forall i\in\{1,\ldots,n\}$. To obtain an expected duality gap of $\E[P(\bar{\w})-D(\bar{\theta)}]\le\epsilon_P$ where $\bar{\w}=\frac{1}{T-T_0}\sum^T_{t=T_0+1}\w^{t-1}$ and $\bar{\theta}=\frac{1}{T-T_0}\sum^T_{t=T_0+1}\theta^{t-1}$, it suffices to have a total number of iterations of
\begin{eqnarray*}
T\ge \max(0, 2\lceil \frac{n}{\rho} \log(\frac{\lambda n}{\rho 2 R^2(\sum^n_{i=1}L_i)^2/n^2})\rceil) -n/\rho+\frac{20 R^2(\sum^n_{i=1}L_i)^2}{n^2\lambda\epsilon_P},
\end{eqnarray*}
where $\rho=\frac{n L_{min}}{\sum^n_{i=1}L_i}$. Moreover, when
\begin{eqnarray*}
t\ge \max(0, 2\lceil \frac{n}{\rho} \log(\frac{\lambda n}{\rho 2 R^2(\sum^n_{i=1}L_i)^2/n^2})\rceil)-2n/\rho+\frac{16 R^2(\sum^n_{i=1}L_i)^2}{n^2\lambda\epsilon_P} ,
\end{eqnarray*}
we have dual sub-optimality bound of $\E[D(\theta^*)-D(\theta^t)]\le\epsilon_P/2$.
\end{thm}
\begin{proof}
Since $p_i=L_i/\sum^n_{j=1}L_j$, the inequality~\eqref{eqn:Gt-bound} indicates $G^t\le G$ where $G=4 R^2(\sum^n_{i=1} L_i)^2/n^2$. Lemma~\ref{lemma:dual-ascent-psdca}, with $\gamma_i=0$, tells that
\begin{eqnarray}\label{eqn:duality-gap-lipschitz}
\E[D(\theta^t)-D(\theta^{t-1})]\ge \frac{s}{n}\E[P(\w^{t-1})-D(\theta^{t-1})]-(\frac{s}{n})^2\frac{G}{2\lambda},
\end{eqnarray}
for all $s\in[0, \rho]$, which further indicates
\begin{eqnarray*}
\E[\epsilon^t_D]\le(1-\frac{s}{n})\E[\epsilon^{t-1}_D]+(\frac{s}{n})^2\frac{G}{2\lambda},
\end{eqnarray*}
for all $s\in[0, \rho]$. Expanding the above inequality implies
\begin{eqnarray}\label{eqn:prox-sdca-lipschitz-i}
\E[\epsilon^t_D]\le (1-\frac{s}{n})^t\epsilon^0_D+(\frac{s}{n})^2\frac{G}{2\lambda}\sum^t_{\tau=1}(1-\frac{s}{n})^{\tau-1},
\end{eqnarray}
for all $s\in[0,\rho]$.
We next show that the above yields
\begin{eqnarray}\label{eqn:duality-gap-ipsdca}
\E[\epsilon^t_D]\le\frac{2 G}{\lambda(2n/\rho+t-t_0)},
\end{eqnarray}
for all $t\ge t_0=\max(0, \lceil \frac{n}{\rho} \log(\frac{2\lambda n}{\rho G})\rceil)$. Indeed, let us choose $s=\rho$, then at $t=t_0$, we have
\begin{eqnarray*}
\E[\epsilon^t_D]\le (1-\frac{\rho}{n})^t\epsilon^0_D+\frac{\rho^2 G}{n^2 2\lambda}\frac{1}{1-(1-\rho/n)}\le\exp(-\rho t/n)+\frac{\rho G}{2\lambda n}\le \frac{ G}{\lambda n/\rho}=\frac{2 G}{\lambda(2n/\rho+t_0-t_0)}
\end{eqnarray*}
where the first inequality used the inequality~\eqref{eqn:prox-sdca-lipschitz-i}, the second inequality used the facts $(1-\frac{\rho}{n})^t\le \exp(-\rho t/n)$ and $\epsilon^0_D\le 1$, and the third inequality used the fact $t_0 \ge \lceil \frac{n}{\rho} \log(\frac{2\lambda n}{\rho G})\rceil$. This implies that the inequality~\eqref{eqn:duality-gap-ipsdca} holds at $t=t_0$. For $t> t_0$ we can use inductive argument. Suppose the claim holds for $t-1$, therefore
\[
\E[\epsilon^t_D]\le (1-\frac{s}{n})\E[\epsilon^{t-1}_D] + (\frac{s}{n})^2\frac{G}{2\lambda}\le(1-\frac{s}{n})\frac{2 G}{\lambda(2n/\rho+t-1-t_0)}+(\frac{s}{n})^2\frac{G}{2\lambda}.
\]
Choosing $s=\frac{2 n}{(2n/\rho+t-1-t_0)}\in[0,\rho]$ yields
\begin{eqnarray*}
\E[\epsilon^t_D]&\le&\left(1-\frac{2}{2n/\rho+t-1-t_0}\right)\frac{2 G}{\lambda(2n/\rho+t-1-t_0)} +\left(\frac{2}{2n/\rho+t-1-t_0}\right)^2\frac{G}{2\lambda}\\
&=&\frac{2 G}{\lambda(2n/\rho+t-1-t_0)}(1-\frac{1}{2n/\rho+t-1-t_0})\\
&=&\frac{2 G}{\lambda(2n/\rho+t-1-t_0)}(\frac{2n/\rho+t-1-t_0-1}{2n/\rho+t-1-t_0})\\
&\le& \frac{2 G}{\lambda(2n/\rho+t-1-t_0)}(\frac{2n/\rho+t-1-t_0}{2n/\rho+t-t_0})\\
&=& \frac{2 G}{\lambda(2n/\rho+t-t_0)}.
\end{eqnarray*}
This provides a bound on the dual sub-optimality. We next turn to bound the duality gap. Summing the inequality~\eqref{eqn:duality-gap-lipschitz} over $t=T_0+1,\ldots,T$ and rearranging terms we obtain that
\[
\E\left[\frac{1}{T-T_0}\sum^T_{t=T_0+1}(P(\w^{t-1})-D(\theta^{t-1}))\right]\le \frac{n}{s(T-T_0)}\E[D(\theta^T)-D(\theta^{T_0})]+\frac{s G}{2\lambda n}.
\]
Now if we choose $\bar{\w}$, $\bar{\theta}$ to be either the average vector or a randomly chosen vector over $t\in\{T_0+1,\ldots,T\}$, the the above implies
\[
\E[(P(\bar{\w})-D(\bar{\theta}))]\le \frac{n}{s(T-T_0)}\E[D(\theta^T)-D(\theta^{T_0})]+\frac{s G}{2\lambda n}.
\]
If $T\ge n/\rho+T_0$ and $T_0\ge t_0$, we can set $s=n/(T-T_0)\le \rho$ and combining with~\eqref{eqn:duality-gap-ipsdca} we obtain
\begin{eqnarray*}
\E[(P(\bar{\w})-D(\bar{\theta}))]&\le& \frac{n}{s(T-T_0)}\E[D(\theta^T)-D(\theta^{T_0})]+\frac{s G}{2\lambda n}\\
&\le&\E[D(\theta^*)-D(\theta^{T_0})]+\frac{ G}{2\lambda(T-T_0)}\\
&\le& \frac{2G}{\lambda(2n/\rho+T_0-t_0)}+\frac{ G}{2\lambda(T-T_0)}.
\end{eqnarray*}
A sufficient condition for the above to be smaller than $\epsilon_P$ is that $T_0\ge\frac{4G}{\lambda\epsilon_P}-2n/\rho+t_0$ and $T\ge T_0+\frac{G}{\lambda\epsilon_P}$. It also implies that $\E[D(\theta^*)-D(\theta{T_0})]\le\epsilon_P/2$. Since we also need $T_0\ge t_0$ and $T-T_0\ge n/\rho$, the overall number of required iterations can be
\begin{eqnarray*}
T_0\ge\max\{t_0, 4G/(\lambda\epsilon_P)-2n/\rho+t_0\},\quad T-T_0\ge \max\{n/\rho, G/(\lambda\epsilon_P)\}.
\end{eqnarray*}
Using the fact $a+b\ge \max(a, b)$ concludes the proof of this theorem.
\end{proof}
{\bf Remark:} When we replace all the $L_i$ with $L_{max}=\max\{L_1,\ldots,L_n\}$, the above theorem will still be valid, and the sampling distribution becomes the uniform distribution. In this case we will recover Theorem 2 of~\cite{shalev2012proximal}, i.e., $T\ge \max(0, 2\lceil n \log(\frac{\lambda n}{2R^2 L_{max}^2})\rceil) -n+\frac{20 R^2(L_{max})^2}{\lambda\epsilon_P}$. However, the ratio between the leading terms is
\begin{eqnarray*}
\frac{(L_{max})^2}{(\sum^n_{i=1}L_i)^2/n^2}= (\frac{n}{\sum^n_{i=1}L_i/L_{max}})^2\ge 1 ,
\end{eqnarray*}
which again implies that the importance sampling strategy will improve convergence, especially when $(\sum^n_{i=1}\frac{L_i}{L_{max}})^2\ll n^2$.
\section{Applications}
\label{sec:application}
There are numerous possible applications of our proposed algorithms. Here we will list several popular applications. In this section, we will set $\psi(\w)=\frac{1}{2}\|\w\|_2^2$, so that the stochastic mirror descent is stochastic gradient descent.
\subsection{Hinge Loss Based SVM with $\ell_2$ Regularization}
Suppose our task is to solve the typical Support Vector Machine (SVM):
\begin{eqnarray*}
\min_{\w} \frac{1}{n}\sum^n_{i=1} [1-y_i\w^\top\x_i]_+ +\frac{\lambda}{2}\|\w\|_2^2.
\end{eqnarray*}
Assume that $X=\max_i\|\x_i\|_2$ is not too large, such as for text categorization problems where each $\x_i$ is a bag-of-words representation of some short document. To solve this problem we can use two different proximal SGD or proximal SDCA with importance sampling as follows.
\subsubsection{Proximal SGD with Importance Sampling}
We can set regularizer as $r(\w)=0$, and the loss function as $\phi_i(\w)=[1-y_i\w^\top\x_i]_+ +\frac{\lambda}{2}\|\w\|_2^2$, which is $\lambda$-strongly convex, so that
\[
\prox_{\lambda r}(\x)=\arg\min_{\w}\left(0+\frac{1}{2}\|\w-\x\|_2^2\right)=\x,
\]
and
\[
\nabla \phi_i(\w)= - \sign([1-y_i\w^\top\x_i]_+) y_i\x_i+\lambda\w .
\]
Using $P(\w^*)= D(\theta^*)$, we find that the optimal solution of SVM satisfies $\|\w^*\|_2\le 1/\sqrt{\lambda}$. So we can project the iterative solutions into $\{\w\in\R^d|\|\w\|_2\le 1/\sqrt{\lambda}\}$ using Euclidean distance, while the theoretical analysis is still valid. In this way, we have $ \|\nabla\phi_i(\w)\|_2\le \|\x_i\|_2+\sqrt{\lambda}$. According to our analysis, we should set
\[
p_i=\frac{\|\x_i\|_2+\sqrt{\lambda}}{\sum^n_{j}(\|\x_j\|_2+\sqrt{\lambda})}.
\]
\subsubsection{Proximal SDCA with Importance Sampling}
We can set $r(\w)=\frac{1}{2}\|\w\|_2^2$, and loss function as $\phi_i(\w)=[1-y_i\w^\top\x_i]_+$ which is $\|\x_i\|_2$-Lipschitz. According to our analysis, the distribution should be
\begin{eqnarray*}
p_i=\frac{\|\x_i\|_2}{\sum^n_{i=1}\|\x_i\|_2}.
\end{eqnarray*}
Furthermore, it is easy to get the dual function of $\phi_i$ as
\begin{displaymath}
\phi^*_i(-\theta)= \left\{ \begin{array}{ll}
-\alpha & \textrm{$\theta=\alpha y_i \x_i$, $\alpha\in [0,1]$}\\
\infty& \textrm{otherwise}
\end{array} \right.
\end{displaymath}
Given the dual function, using $\theta_i=\alpha_i y_i \x_i$, the options I in the algorithm produces a closed form solution as
\begin{eqnarray*}
\Delta\theta_i=\max\left(-\alpha_i,\min\left(1-\alpha_i,\frac{1-y_i\x_i^\top\w^{t-1}}{\|\x_i\|_2^2/(\lambda n)}\right)\right)y_i\x_i.
\end{eqnarray*}
\subsection{Squared Hinge Loss Based SVM with $\ell_2$ Regularization}
Suppose our interest is to solve the task of optimizing squared hinge loss based Support Vector Machine (SVM) with $\ell_2$ regularization:
\begin{eqnarray*}
\min_{\w} \frac{1}{n}\sum^n_{i=1}\left([1-y_i\w^\top\x_i]_+\right)^2 +\frac{\lambda}{2}\|\w\|_2^2.
\end{eqnarray*}
\subsubsection{Proximal SGD with Importance Sampling}
Firstly, using the inequality $P(\w^*)=D(\theta^*)$, we can get $\|\w^*\|_2\le 1/\sqrt{\lambda}$. So we can project the iterative solutions into $\{\w\in\R^d|\|\w\|_2\le 1/\sqrt{\lambda}\}$ using Euclidean distance, while the previous theoretical analysis is still valid.
If we set $r(\w)=0$ and $\phi_i(\w)= \left([1-y_i\w^\top\x_i]_+\right)^2+\frac{\lambda}{2}\|\w\|_2^2$ so that $\prox_{\lambda r}(\x)=\x$ and
\begin{eqnarray*}
\nabla \phi_i(\w)=- 2[1-y_i\w^\top\x_i]_+ y_i\x_i+\lambda\w.
\end{eqnarray*}
Because $\|\nabla \phi_i(\w)\|_2\le 2(1+\|\x_i\|_2/\sqrt{\lambda})\|\x_i\|_2+\sqrt{\lambda}$, according the previous analysis, the optimal distribution for this case should be
\begin{eqnarray*}
p_i=\frac{2(1+\|\x_i\|_2/\sqrt{\lambda})\|\x_i\|_2+\sqrt{\lambda}}{\sum^n_{j=1}[2(1+\|\x_j\|_2/\sqrt{\lambda})\|\x_j\|_2+\sqrt{\lambda}]}.
\end{eqnarray*}
\subsubsection{Proximal SDCA with Importance Sampling}
If we set $r(\w)=\frac{1}{2}\|\w\|^2$, which is $1$-strongly convex with $\|\cdot\|_{P'}=\|\cdot\|_2$, we have $\phi_i(\w)=\left([1-y_i\w^\top\x_i]_+\right)^2$, which is $(2\|\x_i\|_2^2)$-smooth with respect to $\|\cdot\|_{P}=\|\cdot\|_2$. As a result, the optimal distribution for proximal SDCA with importance sampling should be
\begin{eqnarray*}
p_i=(1+\frac{2\|\x_i\|_2^2}{\lambda n})/(n+\sum^n_{j=1}\frac{2\|\x_j\|_2^2}{\lambda n}),
\end{eqnarray*}
where we used the fact $R=\sup_{\u\not=0}\|\u\|_{D'}/\|\u\|_D=1$.
It can be derived that the dual function of $\phi(\cdot)$ is
\begin{displaymath}
\phi^*_i(-\theta)= \left\{ \begin{array}{ll}
-\alpha + \alpha^2/4 & \textrm{$\theta=\alpha y_i \x_i$, $\alpha\ge 0$}\\
\infty& \textrm{otherwise}
\end{array} \right.
\end{displaymath}
Plugging the above equation into the update, we can observe that option I in the algorithm has the following closed-form solution:
\begin{eqnarray*}
\Delta \theta_i =\max\left(\frac{1-y_i\w^\top\x_i-\alpha_i/2}{1/2+\|\x_i\|_2^2/(\lambda n)},\ -\alpha_i \right)y_i\x_i.
\end{eqnarray*}
\subsection{Squared Hinge Loss Based SVM with $\ell_1$ Regularization}
Suppose our interest is to solve the task of optimizing squared hinge loss based Support Vector Machine (SVM) with $\ell_1$ regularization:
\begin{eqnarray*}
\min_{\w} \frac{1}{n}\sum^n_{i=1}\left([1-y_i\w^\top\x_i]_+\right)^2 + \lambda \|\w\|_1.
\end{eqnarray*}
where $\ell_1$ regularization is introduced to make the optimal model sparse, which can alleviate the effect of the curse of dimensionality, and improve the model interpretability.
\subsubsection{Proximal SGD with Importance Sampling}
Using the inequality $P(\w^*)\le P(0)$, we obtain the fact that $\|\w^*\|_2\le \|\w^*\|_1\le 1/\lambda$. So we can project the iterative solutions into $\{\w\in\R^d|\|\w\|_2\le 1/\lambda\}$ using Euclidean distance, while the previous analysis is still valid.
If we set regularizer as $ r(\w)=\|\w\|_1$, then loss function is $\phi_i(\w)= \left([1-y_i\w^\top\x_i]_+\right)^2$, so that the proximal mapping is
\[
\prox_{\lambda r}(\x)=\sign(\x)\odot[|\x|-\lambda]_+,
\]
and
\[
\nabla \phi_i(\w)=- 2 \left([1-y_i\w^\top\x_i]_+\right)y_i\x_i.
\]
Because $\nabla\phi_i(\w)\le 2(1+\|\x_i\|_2/\lambda)\|\x_i\|_2$, according to the previous analysis, we should set
\[
p_i=\frac{(1+\|\x_i\|_2/\lambda)\|\x_i\|_2}{\sum^n_{j=1}(1+\|\x_j\|_2/\lambda)\|\x_j\|_2}.
\]
\subsubsection{Proximal SDCA with Importance Sampling}
Let $\w^*$ be the optimal solution, which satisfies $\|\w^*\|_2\le 1/\lambda$. Choosing $\delta=\lambda^2\epsilon$ and
\[
r(\w)=\frac{1}{2}\|\w\|_2^2+\frac{\lambda}{\delta}\|\w\|_1,
\]
which is $1$-strongly convex with respect to $\|\cdot\|_{P'}=\|\cdot\|_2$.
Consider the problem
\[
\min_\w \widehat{P}(\w) =\left[\frac{1}{n}\sum^n_{i=1}\phi_i(\w)+\delta r(\w)\right],
\]
then if $\w$ is an $\epsilon/2$ approximated solution of the above problem , it holds that
\[
\frac{1}{n}\sum^n_{i=1}\phi_i(\w)+\lambda\|\w\|_1 \le \widehat{P}(\w)\le \widehat{P}(\w^*)+\frac{\epsilon}{2}\le \frac{1}{n}\sum^n_{i=1}\phi_i(\w^*)+\lambda\|\w^*\|_1+\epsilon,
\]
which implies $\w$ is an $\epsilon$ approximated solution of the original optimization problem.
When we adopt Proximal SDCA with importance sampling to minimize $\hat{P}(\w)$, we have $\phi_i(\w)=\left([1-y_i\w^\top\x_i]_+\right)^2$,
which is $(2\|\x_i\|^2)$-smooth with respect to $\|\cdot\|_P=\|\cdot\|_2$. As a result, the optimal distribution for Proximal SDCA with importance sampling should be
\begin{eqnarray*}
p_i=(1+\frac{2\|\x_i\|_2^2}{\lambda n})/(n+\sum^n_{j=1}\frac{2\|\x_j\|_2^2}{\lambda n}).
\end{eqnarray*}
Plugging $\phi^*_i(-\theta)= -\alpha + \alpha^2/4$, $\theta=\alpha y_i \x_i$, $\alpha\ge 0$ into the algorithm, we can observe that the option I produces the following closed-form solution:
\begin{eqnarray*}
\Delta \theta_i =\max\left(\frac{1-y_i\w^\top\x_i-\alpha_i/2}{1/2+\|\x_i\|_2^2/(\lambda n)},\ -\alpha_i \right)y_i\x_i.
\end{eqnarray*}
Finally it is easy to verify that
\begin{eqnarray*}
\nabla r^*(\v)=\sign(\v)\odot[|\v|-\frac{\lambda}{\delta}]_+.
\end{eqnarray*}
\section{Experimental Results}
\label{sec:experiment}
In this section, we evaluate the empirical performance of the proposed algorithms.
\subsection{Experimental Testbed and Setup}
To compare our algorithms with their traditional versions without importance sampling, we focus on the task of optimizing squared hinge loss based SVM with $\ell_2$ regularization:
\begin{eqnarray*}
\min_{\w} \frac{1}{n}\sum^n_{i=1}\left( [1-y_i\w^\top\x_i]_+\right)^2 +\frac{\lambda}{2}\|\w\|_2^2.
\end{eqnarray*}
We compared our Iprox-SGD with traditional prox-SGD (actually Pegasos~\cite{DBLP:conf/icml/Shalev-ShwartzSS07}), and Iprox-SDCA with prox-SDCA (actually SDCA~\cite{DBLP:journals/jmlr/ShaiTong13}). For Iprox-SGD, we adopt the method in the subsection 5.2.1, while for Iprox-SDCA, we adopt the method in the subsection 5.2.2.
\begin{table}[h]
\begin{center}
\vspace{-0.2in}
\caption{Datasets used in the experiments.}\label{tab:datasets}
\begin{tabular}{|c|c|c|c|} \hline
{ Dataset} &{Dataset Size} & {Features}
\\\hline\hline
ijcnn1 & 49990 & 22 \\
kdd2010(algebra) & 8407752 & 20216830 \\
w8a & 49749 & 300
\\\hline
\end{tabular}
\vspace{-0.1in}
\end{center}
\end{table}
To evaluate the performance of our algorithms, the experiments were performed on several real world datasets, which are chosen fairly randomly in order to cover various aspects of datasets. All the datasets can be downloaded from LIBSVM website\footnote{\url{http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/}}. The details of the dataset characteristics are provided in the Table~\ref{tab:datasets}.
To make a fair comparison, all algorithms adopt the same setup in our experiments. In particular, the regularization parameter $\lambda$ of SVM is set as $10^{-4}$, $10^{-6}$, $10^{-4}$ for ijcnn1, kdd2010(algebra), and w8a, respectively. For prox-SGD and Iprox-SGD, the step size is set as $\eta_t = 1/(\lambda t)$ for all the datasets.
Given these parameters, we estimated the ratios between the constants in the convergence bounds for uniform sampling and the proposed importance sampling strategies, which is listed in the table~\ref{tab:ratio}.
\begin{table}[h]
\begin{center}
\vspace{-0.2in}
\caption{Theoretical Constant Ratios for The Experiment Datasets.}\label{tab:ratio}
\begin{tabular}{|c|c|c|c|c|} \hline
Constant Ratio & ijcnn1 & kdd2010 & w8a
\\\hline\hline
$\frac{n\sum^n_{i=1}(G_i)^2}{(\sum^n_{i=1}G_i)^2}$ (for SGD) & 1.0643 & 1.4667 & 1.9236\\
$\frac{n\lambda \gamma_{min}+R^2}{n\lambda \gamma_{min}+ \frac{R^2}{n}\sum^n_{i=1}\frac{ \gamma_{min}}{ \gamma_i }}$ (for SDCA) & 1.1262 &1.1404 & 1.3467 \\
\hline
\end{tabular}
\vspace{-0.1in}
\end{center}
\end{table}
These ratios imply that the importance sampling will be effective for SGD on kdd2010 and w8a, but not very effective for ijcnn1, which will be verified by later empirical results. In addition, these ratios imply the importance sampling will accelerate the minimization of the duality gap for all the datasets, which will also be demonstrated by later experiments.
All the experiments were conducted by fixing 5 different random seeds for each dataset. All the results were reported by averaging over these 5 runs. We evaluated the learning performance by measuring primal objective value ($P(\w^t)$) for SGD algorithms, and duality gap value ($P(\w^t)-D(\theta^t)$) for SDCA algorithms. In addition, to examine the generalization ability of the learning algorithm, we also evaluated the test error rate. Finally, we also reported the variances of the stochastic gradients of the two algorithms to check the effectiveness of importance sampling. Finally, for Iprox-SGD and Iprox-SDCA, the uniform sampling is adopted at the first epoch, so that the performance is the same with SGD and SDCA there, respectively.
\subsection{Evaluation on Iprox-SGD}
The figure~\ref{fig:ISGD} summarized experimental results in terms of primal objective values, test error rates and variances of the stochastic gradients varying over the learning process on all the datasets for SGD and Iprox-SGD.
\begin{figure}[htp]
\begin{center}
\includegraphics[width=2.1in]{SGD-ijcnn1-primal.eps}
\includegraphics[width=2.1in]{SGD-ijcnn1-test.eps}
\includegraphics[width=2.1in]{SGD-ijcnn1-variance.eps}
{\scriptsize \makebox[2.1in]{(a)~primal objective value on {\bf ijcnn1}}~\makebox[2.1in]{(b)~test error rate on {\bf ijcnn1}}~\makebox[2.1in]{(c)~variance on {\bf ijcnn1}}}
\end{center}
\begin{center}
\includegraphics[width=2.1in]{SGD-kdda-primal.eps}
\includegraphics[width=2.1in]{SGD-kdda-test.eps}
\includegraphics[width=2.1in]{SGD-kdda-variance.eps}
{\scriptsize \makebox[2.1in]{(d)~primal objective value on {\bf kdd2010}}~\makebox[2.1in]{(e)~test error rate on {\bf kdd2010}}~\makebox[2.1in]{(f)~variance on {\bf kdd2010}}}
\end{center}
\begin{center}
\includegraphics[width=2.1in]{SGD-w8a-primal.eps}
\includegraphics[width=2.1in]{SGD-w8a-test.eps}
\includegraphics[width=2.1in]{SGD-w8a-variance.eps}
{\scriptsize \makebox[2.1in]{(g)~primal objective value on {\bf w8a}}~\makebox[2.1in]{(h)~test error rate on {\bf w8a}}~\makebox[2.1in]{(i)~variance on {\bf w8a}}}
\end{center}
\caption{Comparison between Pegasos with Iprox-SGD on several datasets. Epoch for the horizontal axis is the number of iterations divided by dataset size.}
\label{fig:ISGD}
\end{figure}
First, the left column summarized the primal objective values of Iprox-SGD in comparison to SGD with uniform sampling on all the datasets. On the last two datasets,, the proposed Iprox-SGD algorithm achieved the fastest convergence rates. Because these two algorithms adopted the same learning rates, this observation implies that the proposed importance sampling does sampled more informative stochastic gradient during the learning process. Second, the central column summarized the test error rates of the two algorithms, where Iprox-SGD achieves significantly smaller test error rates than those of SGD on the last two dataset. This indicates that the proposed importance sampling approach is effective in improving generalization ability. In addition, the right column shows the variances of stochastic gradients for the Iprox-SGD and SGD algorithms, where we can observe Iprox-SGD enjoys much smaller variances than SGD on the last two dataset. This again demonstrates that the proposed importance sampling strategy is effective in reducing the variance of the stochastic gradients. Finally, on the first dataset, the proposed Iprox-SGD algorithm achieved comparable convergence rate compared with traditional prox-SGD, which indicates that Iprox-SGD may degenerate into the traditional prox-SGD when the variance of training dataset is significantly small.
\subsection{Evaluation on Iprox-SDCA}
The figure~\ref{fig:ISDCA} summarized experimental results in terms of duality gap values, test error rates and variances of the stochastic gradients varying over the learning process on all the datasets for SDCA and Iprox-SDCA.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.1in]{SDCA-ijcnn1-dualgap.eps}
\includegraphics[width=2.1in]{SDCA-ijcnn1-test.eps}
\includegraphics[width=2.1in]{SDCA-ijcnn1-variance.eps}
{\scriptsize \makebox[2.1in]{(a)~duality gap value on {\bf ijcnn1}}~\makebox[2.1in]{(b)~test error rate on {\bf ijcnn1}}~\makebox[2.1in]{(c)~variance on {\bf ijcnn1}}}
\end{center}
\begin{center}
\includegraphics[width=2.1in]{SDCA-kdda-dualgap.eps}
\includegraphics[width=2.1in]{SDCA-kdda-test.eps}
\includegraphics[width=2.1in]{SDCA-kdda-variance.eps}
{\scriptsize \makebox[2.1in]{(d)~duality gap value on {\bf kdd2010}}~\makebox[2.1in]{(e)~test error rate on {\bf kdd2010}}~\makebox[2.1in]{(f)~variance on {\bf kdd2010}}}
\end{center}
\begin{center}
\includegraphics[width=2.1in]{SDCA-w8a-dualgap.eps}
\includegraphics[width=2.1in]{SDCA-w8a-test.eps}
\includegraphics[width=2.1in]{SDCA-w8a-variance.eps}
{\scriptsize \makebox[2.1in]{(g)~duality gap value on {\bf w8a}}~\makebox[2.1in]{(h)~test error rate on {\bf w8a}}~\makebox[2.1in]{(i)~variance on {\bf w8a}}}
\end{center}
\caption{Comparison between SDCA with Iprox-SDCA on several datasets. Epoch for the horizontal axis is the number of iterations divided by dataset size.}
\label{fig:ISDCA}
\end{figure}
We have several observations from these empirical results. First, the left column summarized the dual gap values of Iprox-SDCA in comparison to SDCA with uniform sampling on all the datasets. According to the dual gap values on all the datasets, the proposed Iprox-SDCA algorithm converged much faster than the standard SDCA, which indicates that the proposed importance sampling strategy make the duality gap minimization more efficient during the learning process. Second, the central column summarized the test error rates of the two algorithms, where the test error rates of Iprox-SDCA is comparable with those of SGD on all the dataset. The results indicate that SDCA is quite fast at the first few epochs so that the importance sampling does not improve the test accuracy, although importance sampling can accelerate the minimization of duality gap. In addition, the right column shows the variances of stochastic gradients for the Iprox-SDCA and SDCA algorithms, where we can observe Iprox-SDCA enjoys a bit smaller variances than SDCA on all the dataset. However the improvement is too small so that the test error rate is not significantly reduced. This might due to that SDCA is a kind of stochastic variance reduction gradient method~\cite{DBLP:conf/nips/Johnson013}.
\section{Conclusion}
\label{sec:conclusion}
This paper studied stochastic optimization with importance sampling that reduces the variance. Specifically we considered importance sampling strategies for Proximal Stochastic Gradient Descent and for Proximal Stochastic Dual Coordinate Ascent. For prox-SGD with importance sampling, our analysis showed that in order to reduce variance, the sample distribution should depend on the norms of the gradients of the loss functions, which can be relaxed to the smooth constants or the Lipschitz constants of all the loss functions; for prox-SDCA with importance sampling, our analysis showed that the sampling distribution should rely on the smooth constants or Lipschitz constants of all the loss functions. Compared to the traditional prox-SGD and prox-SDCA methods with uniform sampling, we showed that the proposed importance sampling methods can significantly improve the convergence rate of optimization under suitable situations. Finally, a set of experiments confirm our theoretical analysis.
\section*{Appendix}
\begin{lemma}\label{lem:nonexpansive}
Let $r$ be a convex function, and $\psi$ be a $\sigma$-strongly convex function w.r.t. $\|\cdot\|$. Assume
\begin{eqnarray*}
\uh = \arg\min_{\w}\left[\langle \g_u , \w\rangle + \lambda r(\w) + \frac{1}{\eta}\Bpsi(\w, \z) \right],\quad \vh = \arg\min_{\w}\left[\langle \g_v, \w\rangle + \lambda r(\w) + \frac{1}{\eta}\Bpsi(\w, \z) \right],
\end{eqnarray*}
then, we have
\begin{eqnarray*}
\|\uh-\vh\|\le \frac{\eta}{\sigma}\|\g_u -\g_v\|_* .
\end{eqnarray*}
\end{lemma}
\begin{proof}
Firstly, using the definition of Bregman divergence and the optimality of $\uh$ we have
\begin{eqnarray*}
\mathbf{a}=\nabla\psi(\z)-\nabla\psi(\uh)-\eta \g_u \in \eta \lambda\partial r(\uh).
\end{eqnarray*}
Similarly, we also have
\begin{eqnarray*}
\mathbf{b}=\nabla\psi(\z)-\nabla\psi(\vh)-\eta \g_v \in \eta \lambda\partial r(\vh).
\end{eqnarray*}
Since $\eta \lambda r(\w)$ is convex, its subdifferential is a monotone operator, so we have
\begin{eqnarray*}
0\le \langle\mathbf{a}-\mathbf{b}, \uh-\vh\rangle = \langle\nabla\psi(\vh)-\nabla\psi(\uh)+\eta \g_v-\eta \g_u, \uh-\vh\rangle.
\end{eqnarray*}
Re-arranging the above inequality and using the $\sigma$-strongly convexity of $\psi$, we can get
\begin{eqnarray*}
\langle\eta \g_v-\eta \g_u, \uh-\vh\rangle \ge \langle\nabla\psi(\uh) - \nabla\psi(\vh), \uh-\vh\rangle\ge \sigma \|\uh-\vh\|^2.
\end{eqnarray*}
Using Cauchy-Schwartz inequality for the left-hand side results in
\begin{eqnarray*}
\eta\|\g_v-\g_u\|_*\|\uh-\vh\| \ge \sigma \|\uh-\vh\|^2.
\end{eqnarray*}
Dividing both side by $\sigma\|\uh-\vh\|$ concludes the proof.
\end{proof}
{
|
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{"url":"https:\/\/math.stackexchange.com\/questions\/3193001\/help-with-elementary-proof-about-r-in-the-real-numbers","text":"# Help with elementary proof about r in the real numbers\n\nIf $$r<0$$, there exists no $$x \\in \\mathbb{R}$$ such that $$x^2 = r$$.\n\nI'm thinking I need to prove by contradiction assuming there does exist an $$x$$ such that $$x^2=r$$, but I'm having trouble finding the next step. I'm very new at proof writing so any help would be appreciated.\n\n\u2022 I would think you would just need to prove the 3 cases (a) x < 0; (b) x = 0; and (c) x > 0. \u2013\u00a0dan post Apr 19 at 0:04","date":"2019-07-16 00:49:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 5, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6432996392250061, \"perplexity\": 116.63223233546996}, \"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-2019-30\/segments\/1563195524290.60\/warc\/CC-MAIN-20190715235156-20190716021156-00132.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.physicsforums.com\/threads\/at-what-time-was-the-yeast-added.53743\/","text":"# At what time was the yeast added?\n\n1. Nov 22, 2004\n\n### mborn\n\nHi, I had this story in my Physics book\n\nThe police was baffled by what seemed to be the prefect nurder of a girlwho had been found, apparently suffocated, in her kitchen. The girl had been making bread in her kitchen, whose dimensions were 6,10, 10 m. she had formed the dough into a ball of volume 1\/6 cubic m and turned away to wash some dishes. The criminal added a special virulent strain of yeast to the bread. As a result the bread immediately started to rise in volume, triple every 4 min. before long the dough filled the room, stopping the clock at 3:48 and squashing the girl into the wall. By the time the police came the next day, the yeast worked itself and the dough returned to its normal size.\n\nat what time was the yeast added?\n\nMy answer is 3:20 is it true?\n\nM B\n\n2. Nov 22, 2004\n\n### Tide\n\nI got 3:18. Maybe we rounded differently?\n\n3. Nov 22, 2004\n\n### Gokul43201\n\nStaff Emeritus\nI too (like tide) get t ~ 29 min 50 sec\n\n4. Nov 22, 2004\n\n### mborn\n\nOk!,\nSince the answer is not there in my book, I say I solved it as a geometric series, am I right?\n\n5. Nov 22, 2004\n\n### Tide\n\nI am not sure what you mean when you say you \"solved it as a geometric series.\" Please explain. :-)\n\n6. Nov 24, 2004\n\n### mborn\n\nWell,\na geometric series goes like this\na, ar, a(r^2), a(r^3), ...\nGeneral term take the following look\nar^(n-1)\nnow we have the initial volume (1\/6 cubic meter) which is the first term, a.\nthe nth term is the final volume 6*10*10. therefore we need to know how mnay terms are there to reach 600 starting with 1\/6\naccordingly;\n600=(1\/6) * 3^(n-1) and solve for n-1\nthe time taken to reach the volume 600 is (n-1) * 4 min\nsince it was found that the clock stopped at 3:48\ntherefore, t0=3:48 - (n-1)*4\n\nM B\n\n7. Nov 24, 2004\n\n### Tide\n\nThe question reduces to how you solved that last equation for n. I think it might be clearer if you set up the problem as follows:\n\nSince the volume triples every four minutes you can write the volume as\n\n$$V = V_0 \\times 3^{t\/4}$$\n\nwhere t is the elapsed time. You can solve for t using logarithms:\n\n$$t = 4 \\frac {\\ln V\/V_0}{\\ln 3}$$\n\n8. Nov 25, 2004\n\n### mborn\n\nsame as my answer without rounding.\n\nThanks\n\nM B","date":"2017-02-27 16:35:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.34904515743255615, \"perplexity\": 2804.567942262024}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-09\/segments\/1487501172902.42\/warc\/CC-MAIN-20170219104612-00593-ip-10-171-10-108.ec2.internal.warc.gz\"}"}
| null | null |
Q: Vertex Winding Order in DX11 I'm trying to draw a simple square with dx11, but the order of the indices of each triangle determines whether or not it shows up. I set the cull mode to none in the rasterizer state, but it doesn't seem to change anything.
If I specify the vertices of the first triangle to be 0, 1, 2 instead of 2, 1, 0 then the triangle doesn't show up. So my question is, do I need to order the vertices of a triangle in a specific way regardless of the cull mode?
P.S. I'm drawing a triangle list and not a strip.
UINT indices[] = {2, 1, 0,
1, 3, 0};
MeshVertex vertices[] =
{
{ Vector3(1.0f, 1.0f, 0.0f)}, //Top Right
{ Vector3(-1.0f, -1.0f, 0.0f)}, //Bottom Left
{ Vector3(1.0f, -1.0f, 0.0f)}, //Bottom Right
{ Vector3(-1.0f, 1.0f, 0.0f)} //Top Left
};
mesh = new Mesh(vertices, indices, 4, 6, shader);
A:
So my question is, do I need to order the vertices of a triangle in a
specific way regardless of the cull mode?
** Define/Draw your vertices in clockwise order, and don't change the default cull mode.**
To determine whether a triangle can be displayed, you have to considering 2 factors
*
*The front face, or you call it winding order, by default Direct3D treat a vertices defined in clockwise order as a front face, and all faces other than front faces are back faces.
*The culling mode, by default, Direct3D will cull the back faces(the faces which vertices defined in counter-clockwise order).
You can set the front face and culling mode in D3D11_RASTERIZER_DESC structure
The reason why you didn't see difference when you change the culling mode to NONE is because both triangle were in the same order and D3D11_CULL_NONE means didn't cull any faces.
If triangle 1 is in clockwise order and triangle 2 is in counter clockwise order, you will only see one triangle when you set the cull mode to D3D11_CULL_FRONT or D3D11_CULL_BACK, and see both when you set it to D3D11_CULL_NONE.
|
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| 7,459
|
Dauricine is a plant metabolite, chemically classified as a phenol, an aromatic ether, and an isoquinoline alkaloid. It has been isolated from the Asian vine Menispermum dauricum, commonly known as Asian moonseed, and the North American vine Menispermum canadense, commonly known as Canadian moonseed. Scientists Tetsuji Kametani and Keiichiro Fukumoto of Japan are credited with being the first to synthesize dauricine in 1964, using both the Arndt-Eistert reaction and Bischler-Napieralski reaction to do so. Dauricine has been studied in vitro for its potential to inhibit cancer cell growth and to block cardiac transmembrane Na+, K+, and Ca2+ ion currents.
References
Calcium channel blockers
Phenols
Norsalsolinol ethers
Benzylisoquinoline alkaloids
|
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This RUST electricity guide was created by our content partner Quick Electric. He provides simplified, easy to follow video tutorials around RUST's electricity system. Each of Quick Electric's videos walks through the necessary components and their interactions. You can follow up each video with a detailed schematic that shows all of the components and how they're integrated into the greater system.
In this video, Quick Electric applies some of his previous video's topics into an actual RUST base design to see them in action.
Be sure to browse Quick Electric's past videos and don't forget to subscribe to his channel to receive notifications on his latest videos.
Hey everybody, Rust Quick Electric here with you today to showcase a video that is outside the norm in which there won't necessarily be much wiring, but it will be a demonstration of the practical use of many of my prior video tutorials. I come to you today with a base that showcases the use of practical electronic devices in the base, since many people seem to have a hard time distinguishing what actually might be useful in a wipe and what is not.
Now this base showcases, while it is not a professionally built base, it is meant more so to demonstrate the practical use of different builds in a wipe. So this build video will showcase the automatic anti-door camp turret, the minicopter garage, automatic lights for your base, as well as a small new addition, but it implements the exact same use of the minicopter garage, and that would be a boat garage.
Now for many wipes, the minicopter garage has been an absolute necessity for me. The amount of distance you can travel on the map, be it that from one corner to another on a large map takes approximately one hundred to one hundred fifty low grade, that's a just outrageous amount of travel for minimal cost. And the access of always having a boat.
Now while this build is just too small for the rib, it is perfect for the rowboat and having access to any cargo ship immediately available rather than having to search the coastline is very very useful. You can do it solo with the sensors so it allows you to pilot and land the minicopter as well as the boat without assistance.
So let's get onto it. The space, while structurally sound, is not that impressive. I would suggest many other fantastic base builders such as Drizzza or plenty others that have many options. I tend to find people that have bases that actually know a little thing or two about making them before I perform any buildings.
But this one is one that I've created for now. It implements several base designs, such as honeycomb, boat, and minicopter garages. You'll see as we enter in through the front doors, we have a switch for the anti-door camp turrets. Now none of these have power right now because power management is a huge thing, and with a two turbine base, it's really hard to accumulate power and have a constant flow running.
So you'll see here that we just have more furnaces in that other garage, but in the boat garage, we come up to our main power supply. Now this is assumed that you just don't have that many people, and wind turbines are extremely expensive, so this kind of display uses two wind turbines connected through combiners that power into a large battery which is charging, as well as two solar panels with a route combiner as well as a large battery for outputting to an excessive amount of lights.
This can be done with a small battery, but when you start chaining together eight or nine lights, you want the output of one hundred versus ten. Now with this main power source here, this is our on/off, and you'll see that we have the three outputs, one being the minicopter garage, one being the boat garage, and one being the front door anti-camp turrets.
Now you always want the kill switch with your main power source. You could direct line it in, but it's going to be a constant drain through here since it will be using a splitter. You could use a branch, but for means of convenience, a splitter is perfect for this scenario because I have three specific outputs. Now you'll see with two wind turbines the drain is minimal. It's about one drain per two seconds. And this does not always need to be on, it will typically only be on anything that you're going to be exiting the base, and most people do not leave the base for more than two hours or so at a time, I'm speaking from my personal experience.
With a four hour battery life you will have more than enough battery life, especially since it's only draining one every two seconds or one and a half second. So we're going to leave this one for now as we showcase the different functionalities of this, the first being the auto-turret, the anti-door camp. So once your main power source is on in your base, you'll see that flipping the switch takes use of the simple anti-door camp auto turret which is just the three door controllers with garages as well as a splitter, which once again could be improvised for branches. That's simple on/off switch will activate that, make sure no one is camping your door, and then you can exit your base and leave comfortably.
Next up is the minicopter auto garage. Which when a person is detected, once it's on, so we'll see here that we have our minicopter here, it turns on. We can activate the timer for thirty second timer, at which point it will be thirty seconds that this timer runs before the garage door will automatically shut.
We're going to pretend that I'm in the minicopter right now as I now clip away, just for the sake of demonstration. Now I leave it for a thirty second timer. For many people that doesn't seem like too much, for others it seems like way too long.
It depends on your ability with the minicopter, and you'll find the more you practice the easier it is to make a clean landing and get on inside there. It can automatically be forced shut by turning off the power supply, which makes it convenient. But as you'll see, thirty seconds has elapsed and the garage door will shut again.
This is great for cross maps, solo treks with the minicopter. As you return back in it's going to be very hard, especially if you have a limited platform to land, and get down and open the garage door. So this opens it up as you arrive with the HBHF. You would come on in here and park and then you can manually turn it off so you don't have to wait for this unnecessarily, and this will shut the garage doors.
Next up is that auto light setup. Very basic circuit. But as you'll see come nighttime, these lights will lose input. The power from the solar panels and all the lights will trigger. Now not many people know, but if you put your ceiling lights on the top floor of your base, the light will penetrate through the floors and actually provide light all throughout the base.
Now it depends on the angle that you come through as you see through the garage it provides more light, or in the angle that you're standing at. But it does penetrate all the way through to your base. A lot of convenience, a lot of save, it's very nice, especially if you have a top floor that's accessible to ceiling lights. So let's go ahead and change it back to daytime. You'll see that the auto lights turn off, and the battery will begin to refill again.
Lastly we have our boat dock. So we go back to the first floor. Now this one for me is an absolute must with a cargo ship. It uses the exact same circuit as the minicopter. But it is specifically for a boat. So you would have your boat here, you would turn this on, you can activate the timer, which I set to a much shorter timer of fifteen seconds.
You'd mount your boat, hop in, get going, come on out, and then your timer is set to fifteen seconds here. So as that timer duration elapses, that garage door shuts, and you're good to go. Upon re-approach, it does the exact same thing as the minicopter. It opens up the garage, gives you fifteen seconds to park, hop on out, and you can turn off the circuit yourself and the garage door will shut.
Now as far as opening the door, you could set up a door controller here with a little separate battery, but most of the time it's not necessary since you can open this door easily, exit, shut the door, and then come on re-approach when you're trying to quickly get into your base while the switch is on.
Plenty of things can be expanded upon, and much more can be added to your base. But the biggest thing that you're going to find in any vanilla or slightly modded server is the power that is required to have anything massive. Now with just these three devices, four hours on one large battery is plenty of time especially since they'll only be for the duration of time that you're not in your base.
But for anything more extensive, and for a constant power drain, you're going to want a bigger source of power, which many bases you'll see tend to have between five or six wind turbines that are linked together into one major power station, at which point is broken down into substations and smaller values of power.
A lot of people like to use counters to do this, but for this small solo/duo/trio, it is simply handled with two wind turbines and two solar panels. It's more than enough power to have the different devices that you need, and you can make some other funny, nifty kind of devices if you make a second large battery with more input.
As far as everybody else, I'd like to thank you guys for watching and continuously contributing. The schematics for all these builds will be found in the description as with any video, and you will have seen them in the beginning.
I'd like to thank everybody once again for joining the Discord, and I can't wait to show you guys more.
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{
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Michael Beer (19 August 1800, Berlin – 22 March 1833, Munich) was a German Jewish poet, author and playwright.
Early life
Beer was born to a wealthy Jewish family, the son of salonnière Amalie Beer. His elder brother was the composer Giacomo Meyerbeer; another brother was the astronomer Wilhelm Beer.
In the period 1817–1823 he frequently travelled with family members in Italy, where his brother Meyerbeer was studying.
In 1819 Beer was a founder member of the movement Verein für Cultur und Wissenschaft der Juden (Association for Culture and Science of the Jews), which attempted to provide an intellectual framework for considering the Jews as a people in their own right, and to validate their secular cultural traditions as being on an equal footing with those of the German people. Beer's co-founders included Eduard Gans, Moses Moser, Heinrich Heine and Leopold Zunz.
Works
The first of Beer's works to be performed was Klytemnestra (Clytemnestra), (1819), influenced by the classicism of Goethe. His second stage-work Die Bräute von Aragonien (The Brides of Aragon), was also suggested by Goethe's poetry.
Far superior to these early works was the one-act play Der Paria (The Pariah), premiered in Berlin in 1823, and admired by Goethe, which was soon played on stages across Germany. In the play, the pariah Gadhi and his wife Maja choose to die so as to enable their son to live freely. The work can be construed as a cry of pain about the pariah status of Judaism in early nineteenth-century Germany. This is a topic which constantly recurs in Beer's correspondence with Meyerbeer. Beer's 1827 drama Struensee (based on the life of the German-Danish reformer Johann Friedrich Struensee) was initially banned from production in Prussia, and was premiered in 1828 in Munich, where Beer had briefly settled and where he became a friend of Schelling. Not until 1846 (thirteen years after the author's death) did the relaxation of censorship enable a performance in Berlin; for this King Frederick William IV commissioned Meyerbeer to provide an overture and incidental music.
Beer's poetic output includes a series of 'Elegies' written in Italy, a protest at the injustice of criminal sentencing (Im Gerichtssaal), and a satirical poem on the paradoxes of extreme religiosity (Der fromme Rabbi).
Later life
Beer's personality is known mainly through his correspondence with his family and with the playwright Karl Leberecht Immermann. Beer spent many of his last years in Paris where he was acquainted with Heinrich Heine, Ferdinand Hiller and Felix Mendelssohn, who was an occasional chess-partner.
Beer's early death was attributed to neurasthenia. He is buried with his parents and siblings in the Jewish cemetery in Schönhauser Allee, Berlin.
Michael Beer Foundation
Beer was, in the tradition of his family, generous of his wealth and supported scholars and artists, including the orientalist Salomon Munk. He bequeathed a large fortune, which was turned into a foundation administered by the Berlin Academy of Arts. The annual income of the Michael Beer Foundation was awarded to two young artists, who had to be Jewish; this financed a one-year study period in Italy, of which they had to spend at least eight months in Rome.
References
Notes
Sources
Becker, Heinz & Gudrun, tr. Mark Violette (1989). Giacomo Meyerbeer: A Life in Letters. London: Christopher Helm.
Conway, David (2012). Jewry in Music - Entry to the Profession from the Enlightenment to Richard Wagner. Cambridge: Cambridge University Press. .
Espagne, Michel (1996). Les juif allemands de Paris à l'époque de Heine: la translation ashkénase. Paris: Presses Universitaires de France. .
Hiller, Ferdinand, tr. M.E. von Glehn (1874). Felix Mendelssohn: Letters and Recollections. London: Macmillan.
Jewish Encyclopedia (1906). 'Beer, Michael'
Kahn, Lothar (1976). 'Michael Beer (1800–1833)', in Leo Baeck Institute Yearbook 1976, pp. 149–160
Sachar, Howard M. (1990). The Course of Modern Jewish History. New York:Vintage. .
1800 births
1833 deaths
Writers from Berlin
19th-century German Jews
German poets
Jewish poets
Neurological disease deaths in Germany
Burials at Schönhauser Allee Cemetery, Berlin
German male poets
German male dramatists and playwrights
19th-century poets
19th-century German dramatists and playwrights
19th-century German male writers
19th-century German writers
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"What I disagree with is trying to force a new "system" on people. It is better to let things evolve driven by individual and collective choices. I am skeptical of "social engineering" because there are always unintended consequences."
Snap. But you support the Legal system?
Also, I believe we are 'socially engineered' from birth, by the Market.
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"redpajama_set_name": "RedPajamaC4"
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Huawei Honor Play 5X features a 5.5 inch full HD display with a resolution of 1080 x 1920 pixels. It runs on Android v5.1 (Lollipop) operating system and is backed by a 3 GB of RAM. The device houses a 13 MP rear camera and 5 MP front facing camera. It comes with 16 GB of built-in storage which can be expanded via microSD up to 128 GB. Huawei Honor Play 5X is equipped with a non-removable Li-Po 3000 mAh battery.
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"redpajama_set_name": "RedPajamaC4"
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Water authorities to ramp up releases from Menindee Lakes in response to demand
ABC Broken Hill
By Declan Gooch
Posted Wed 4 Jan 2017 at 12:44am Wednesday 4 Jan 2017 at 12:44am Wed 4 Jan 2017 at 12:44am , updated Wed 4 Jan 2017 at 1:49am Wednesday 4 Jan 2017 at 1:49am Wed 4 Jan 2017 at 1:49am
The MDBA will request flows of up to 6000 megalitres per day be released from the Menindee Lakes. (Cherie von Hörchner)
abc.net.au/news/water-authorities-to-ramp-up-releases-from-the-menindee-lakes/8160780
The Murray-Darling Basin Authority (MDBA) is set to increase releases of water from the Menindee Lakes, saying it is needed to meet demand downstream.
The shallow lake system sits on the Darling River in far-west NSW, about an hour away from Broken Hill, and filled up last year after a prolonged dry period.
Authorities are releasing 1.7 gigalitres per day downstream to improve fish habitats for native species in the Lower Darling, which ran dry in 2015 and 2016.
The MDBA said it would request NSW water authorities increase that rate to six gigalitres per day in the next week to meet demand from environmental licence holders, irrigators and other groups in the Lower Darling and Murray Rivers.
"[We are] not really sure how much water we'll be able to order from the lakes, because that depends on evaporation, further inflows and local usage," the MDBA's director of river management, David Dreverman, said.
"But whatever happens, we can't direct the release [to occur] once the volume in the lakes falls below 480 gigalitres," he said.
Once it falls below 480 gigalitres, only NSW authorities can order releases from the lakes.
Mr Dreverman said that if the lakes did not receive further inflows over summer, that trigger point could be reached in May.
Inflows into lake system slow
The filling of the lakes was met with jubilation in the region, but flows into the system are reducing and it is currently holding around 1,500 gigalitres, or 87 per cent of capacity.
Previous releases have caused concern in Broken Hill, which will continue to rely on the Menindee Lakes for drinking water until a pipeline from the Murray River is constructed.
The lakes are also a tourism drawcard for the town of Menindee, which suffered an economic slump when water levels were low.
Mr Dreverman said the releases from Menindee would allow authorities to keep more water in Victoria's Dartmouth Dam, which was deeper and had less evaporation than the lakes.
"The best thing to do is to keep as much water [in Dartmouth] as you can, where the evaporation is very low, and the reservoir is very deep, so it's a very efficient storage," Mr Dreverman said.
He said the Menindee Lakes would then be used to regulate summer flows down the Darling River.
Posted 4 Jan 2017 4 Jan 2017 Wed 4 Jan 2017 at 12:44am , updated 4 Jan 2017 4 Jan 2017 Wed 4 Jan 2017 at 1:49am
Menindee
Kyrgios goes from 'sleeping 17 hours a day' to booking a battle with men's world number two at Australian Open
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Neno Mirchev () (born 9 September 1909, date of death unknown) was a Bulgarian gymnast. He competed in eight events at the 1936 Summer Olympics.
References
External links
1909 births
Year of death missing
Bulgarian male artistic gymnasts
Olympic gymnasts of Bulgaria
Gymnasts at the 1936 Summer Olympics
Place of birth missing
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Mega-rich lifestyle of working-class Stokie who lives in luxury Dubai mansion and uses private jets
Even the lifebuoys by her pool come from exclusive designer brands.
Hannah Hiles
A woman born into a working-class Stoke-on-Trent family has told how her life in Dubai is one of decadent luxury including using private jets when she pops back to the UK.
Gaynor Scott, who splits her time between mansions in Dubai and Jersey, opened up about her experiences hob-nobbing with the mega-wealthy on BBC2's Inside Dubai: Playground of the Rich.
During the show, she shared the surprising detail about her Potteries background after viewers had already seen glimpses of her luxury lifestyle - including a team of live-in domestic staff from the Philippines.
More news here
Her glamorous villa boasts a pristine swimming pool - complete with Chanel and Hermès lifebuoys - and a room decorated in the red, green, white and black colours of the Emirates flag, right down to the hand-picked crystals on the chandeliers.
There's even a painting on the wall of former president Sheikh Zayed bin Sultan Al Nahyan, which Gaynor won in a charity auction in aid of needy children.
Have your say on this story in the comments section.
"My life hasn't always been like this," she said. "I grew up in Stoke-on-Trent in Staffordshire, in the Midlands, so just a very normal working class family. I'm sure there are people I went to school with who will probably go 'ugh, who does she think she is, living in Dubai with all these staff?' but I just feel like I have been lucky. A lot of us who live wealthy lives, we have been lucky, that's all.
"I don't think I'm anything special, you know, I just feel like I'm lucky."
Gaynor was interviewed as she was driven through the exclusive gated Emirates Hills estate, five miles inland, where she lives with her husband, adult son, young daughter - and her chef, nanny, driver and housekeeper.
Their neighbours include the family of the late Zimbabwean president Robert Mugabe, who apparently throw "amazing parties on New Year's Eve".
"One of the things I really like about living here is that Dubai is somewhere you can really enjoy your wealth," she said.
"When we came to look at our villa the estate agent told us that David and Victoria Beckham had rented [it] when David was on his way to the World Cup in Japan. So we have a bit of a laugh saying that David and Victoria have been in our shower and slept in our bedroom.
"Emirates Hills is an exclusive area. It's a gated community, ultra-private, because of the residents that live here. It's one of those destination places to live in in Dubai, so we came and had a look and we fell in love with it."
Gaynor, who is married to one of the richest businessmen in the Channel Islands, was shown enjoying a late afternoon dinner party to welcome her new neighbours - and boarding a private jet for an £80,000 round-trip to the other family home in Jersey.
The family received VIP treatment all the way - at the airport a concert pianist had been hired to serenade her with her favourite song and her coffee was served with an image of her face in the froth.
"There's good and bad things about seeing wealth all the time," said Gaynor. "We feel very lucky bringing up our young daughter here but when you live somewhere like this and you see a lot of wealth around you, I think it's really hard for the young age group.
"You know, she doesn't see me putting the washing in the washing machine or doing the ironing, so that's something we want her to be aware of - that we are lucky and not everybody's lives are like this."
How did Stokie Tara do on tonight's Language of Love?
Wakeboarding fan's Superman trick ends in Royal Stoke emergency
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\section{Introduction}
The time-dependent Hartree-Fock (TDHF) method was originally proposed as
early as 1930 by Dirac~\cite{Dirac-tdhf}. For a long time, it was merely a
formal tool to derive many-body approaches like, e.g., in \cite{Bro71aB} to
derive linear response theory. The enormous progress of computational
facilities has made TDHF a practical scheme for describing the dynamics
of many-body systems. By now it has found widespread applications in
various areas of physics. Under the label of time-dependent density-functional
theory it is used in electronic systems like atoms, molecules, clusters, and
solids, see e.g. \cite{Dreizler,Rei03a,tddft-notes}. The earliest practical
applications probably appeared in nuclear physics \cite{Bonche76}, where TDHF
is a powerful microscopic approach to simulate various dynamical scenarios in
the regime of large-amplitude collective motion, like fusion excitation
functions, fission, deep-inelastic scattering, and collective excitations; for
early reviews see, e.g., \cite{Svenne,Negele,Davies}.
These pioneering applications were still hampered by the
computational limitations of their time.
With the ongoing growth of computational power, fully three-dimensional TDHF
calculations without any symmetry restriction became feasible and so
renewed the interest in nuclear TDHF, for a few recent examples of
state-of-the-art TDHF calculations in many different processes see
\cite{Kim,Simenel,Nakatsukasa,Umar05a,Maruhn1,Guo08a}.
The TDHF approach allows the self-consistent
quantum-mechanical description of nuclear dynamics on a mean-field level.
Self-consistency means an unprejudiced description once a reliable energy
functional is given. This explains the versatility of TDHF. It
remains, however, an approximation
since it is a mean-field theory. TDHF misses dynamical
correlation effects stemming from nucleon-nucleon collisions, which contribute
to (two-body) dissipation and thermalization. Their inclusion in a fully
quantum mechanical treatment has so far only been achieved in homogeneous
systems like, e.g., \cite{Toe88a,Gre94a}. Including dynamical correlations
for finite nuclei is presently still restricted to a semiclassical
description \cite{Bertsch,Bonasera, Abe}. On the other hand, it was found that
nuclear TDHF calculations already include a great deal of (one-body) dissipation
if all terms of the functional, particularly the spin-orbit terms, are
properly accounted for \cite{Rei88d} and if all symmetry restrictions are
removed \cite{Mar06c}. This dissipation within TDHF does not result from
two-particle collisions but from collision of one particle with the boundaries of
the moving mean-field potential (``single-particle dissipation''
\cite{Swiatecki}) which randomizes the single-particle states. In a heavy-ion
collision, two pictures of single-particle dissipation can be
distinguished. The ``window'' picture describes dissipation of relative momentum
via nucleon exchange through a neck while the ``wall'' picture deals with the
dissipation of kinetic energy by reflection of the nucleons at a moving wall
\cite{Blocki,Randrup1,Randrup2}. The latter results in a net increase of the
nucleons' thermal energy provided there is no correlation between the nucleonic
and wall motions. However, these are idealized concepts which are not always
immediately applicable to realistic heavy-ion collisions
\cite{Maruhn2,Sierk,Koonin}. Until now it is not understood at a detailed
level how rapidly and how strongly equilibration works within the TDHF
approach.
A rough global measure of dissipation is given by comparing initial and final
kinetic energies of the fragments in a heavy-ion collision \cite{Mar06c}. More
detailed analysis should look at something like a local momentum
distribution. This naturally leads to the concept of a Wigner function which
provides a phase-space picture of a quantum state. Originally introduced in
\cite{Wigner}, it is often used for establishing the connection between quantum
and classical physics \cite{Bra97aB}. The result of such semiclassical
limits is a mean-field dynamics in classical phase-space called the Vlasov
equation \cite{Vlasov} which is widely used in simulating nuclear dynamics
\cite{Bertsch,Bonasera,Abe}. In this paper, we want to stay at the fully
quantum-mechanical level and employ the Wigner function as a useful
observable helping to analyze TDHF dynamics. An early analysis of that kind
is found in \cite{Maruhn3}. The Wigner function has the weakness that it is
not positive semidefinite, thus preventing a strict probabilistic
interpretation. This defect is cured by some phase-space smoothing leading to
the Husimi function \cite{Takahashi,Toscano}, which also turns out to be the
better starting point for the semiclassical expansion \cite{Eplattenier}. We
will also briefly address the Husimi function in connection with TDHF results.
As the Wigner function is six-dimensional and thus rather difficult to handle,
we deduce from it more
compact measures of dissipation and equilibration by considering local
quantities integrated with some weights over momentum space, e.g.,
the eccentricity of the momentum space distribution. These observables are
complemented by others computed without recurring to the
Wigner picture, e.g., the intrinsic excitation energy which is computed
from the local kinetic energy density. We will explore these different
analyzing tools for two realistic applications, collision of $^{16}$O+$^{16}$O
and $^{96}$Zr+$^{132}$Sn.
The paper is organized as follows: Section \ref{sec:numeric} describes briefly
the numerical handling of TDHF used in this work. In Section \ref{sec:Wigner}
we present the transformation from the TDHF wave function to the Wigner and
Husimi representations. Results for the ground states in static calculations
are presented to compare both pictures. Observables are defined in Section
\ref{sec:observ} to allow a quantitative discussion of equilibration. In
Section \ref{sec:results} we show results for dynamical calculations with
different nuclei, energies, and impact parameters.
For sake of generality the formal considerations of Section
\ref{sec:Wigner} are presented in $n$-dimensional coordinate
and $2n$-dimensional phase space. The results in this paper are
obtained in the reduced two-dimensional reaction plane
(assumed to be the $x$-$z$-plane). For clarity we will
label the number of coordinate dimensions $n$ of the applied
distribution function $f$ with $f^{(n)}$.
\section{Formal and numerical framework}
\label{sec:numeric}
The basis of the TDHF description is a set of occupied single-particle
wave functions $\psi_l(\mathbf{r},t)$ where $l$ labels the
states. These wave functions are two-component spinors.
The Skyrme mean-field Hamiltonian is computed for given densities and
currents in the standard manner \cite{Ben03aR}.
For all calculations reported here we
have used the Skyrme parametrization SkI3 \cite{Reinhard2}.
The TDHF equations are solved on a three-dimensional Cartesian
coordinate-space grid. Using the fast Fourier transformation (FFT) derivatives
can be evaluated very efficiently in Fourier-space. The mesh spacing is
${\rm d}x={\rm d}y={\rm d}z=1$\:fm.
The stationary ground states of the initial systems are computed via the
damped-gradient iteration algorithm \cite{Blum,Reinhard}. The initial state is
obtained by placing the ground states of the two fragments in a safe distance
and giving them a boost towards each other. These states are then propagated
in time by use of a Taylor-series expansion of the time-evolution operator
\cite{Flocard} where the expansion is taken up to sixth order. The actual
time step is $t=0.2$\:fm/c.
\section{Wigner and Husimi distributions}
\label{sec:Wigner}
The Wigner function is a transformation of the density matrix to a phase-space
function. There are various levels of density matrices in a many-body systems
and accordingly various Wigner functions. TDHF can be considered as
describing the dynamics of the one-body density matrix
$\rho(\mathbf{r},\mathbf{r}')$, neglecting all correlations
between the interacting nucleons above the mean-field level. This is related to
the one-body Wigner function which is obtained by a partial Fourier transform
acting on the relative coordinate $\mathbf{s}=\mathbf{r}-\mathbf{r}'$, i.e.
\begin{eqnarray}
f^{(n)}_\mathrm{W}(\mathbf{r},\mathbf{k},t)
&=&
\int\frac{{\rm d}^n s}{(2\pi)^n}\:
e^{-i\mathbf{k}\mathbf{s}}
\rho(\mathbf{r}\!-\!\frac{\mathbf{s}}{2},
\mathbf{r}\!+\!\frac{\mathbf{s}}{2},t)
\;,
\\
\rho(\mathbf{r},\mathbf{r}',t)
&=&
\sum_{l}\Psi^\dagger_{l}(\mathbf{r},t)\Psi_{l}(\mathbf{r}',t)
\:.
\end{eqnarray}
Note that these are, in fact, a spin-averaged density matrix and correspondingly a
spin-averaged Wigner function. The dimensionality of the transformation is a
very compact notation and needs some explanation. Of course, our TDHF
calculations are always 3D. The full Wigner function is then a six-dimensional
object, obviously a bit bulky. Therefore, we often take cuts and look at the
Wigner transformation in reduced dimensions. The notation $f^{(1)}_\mathrm{W}$ then
means that one coordinate, e.g. $x$, is transformed from the pair $(x,x')$ in
the density matrix to the pair $(x,k_x)$ in the Wigner function. The other
two coordinates, $y$ and $z$ in the example, are fixed at a certain value $y_0$
and $z_0$, usually at the center of the nucleus $y_0=0$ and $z_0=0$. In other
words, $f^{(1)}_\mathrm{W}(x,k_x)$ denotes $\rho(x,y_0,z_0;x',y_0,z_0;t)$
transformed in the $x$ dimension.
A direct interpretation of the Wigner function as a phase-space probability
distribution is not possible because $f_\mathrm{W}$ is not positive
semidefinite. There can arise situations where the quantum oscillations lead
to negative values. These problems are avoided by the Husimi distribution
\cite{Takahashi,Toscano}. The Husimi function
$f_\mathrm{H}(\mathbf{r},\mathbf{k},t)$ is obtained by a convolution of the
Wigner function with a Gaussian $\mathcal{G}(\mathbf{r},\mathbf{k})$
\begin{eqnarray}
&&
f^{(n)}_\mathrm{H}(\mathbf{r},\mathbf{k},t)
= \int {\rm d}^n r'd^n k '\:
\mathcal{G}(\mathbf{r}-\mathbf{r}',\mathbf{k}-\mathbf{k}')
\nonumber
\\
&&\hspace*{6em}\times\:f^{(n)}_{W}(\mathbf{r}',\mathbf{k}',t)
\:,
\\
&&
\mathcal{G}^{(n)}
(\mathbf{r}-\mathbf{r}',\mathbf{k}-\mathbf{k}')
=
\frac{1}{\pi^n}
e^{-\frac{\mathbf{r}^2}{2\Delta{r}^2}}
e^{-\frac{\mathbf{k}^2}{2\Delta{k}^2}}
\:,
\\
&&
\Delta r\Delta k
=
\frac{1}{2}
\:.
\end{eqnarray}
The Gaussian folding averages $f_\mathrm{W}$ over the minimal phase-space cell
of volume $(2\pi\hbar)^n$ and so successfully wipes out the negative
values. On the other hand, it induces some uncertainty which, however, is
physical because one cannot localize a particle in phase space better than
within a volume of $(2\pi\hbar)^n$. The Husimi folding has one free
parameter, the folding width. For best resolution in both directions it
should be chosen close to the width of the wave functions. As a basis for our
choice, we use here the nuclear harmonic oscillator model with frequency and
width parameter given as
\begin{equation*}
\hbar\omega
=
\frac{41\:\mathrm{MeV}}{A^{1/3}},
\quad
\lambda
=
\frac{m\omega}{\hbar}.
\end{equation*}
This yields the estimate
\begin{eqnarray}
\Delta r^2
&=&
\frac{1}{2\lambda}
=
\frac{\hbar^2}{2m}{\hbar\omega}
=
\frac{A^{1/3}\:\mathrm{fm}^2}{2}
\:,
\\
\Delta k^2
&=&
\frac{1}{4\Delta r^2}
=
\frac{\lambda}{2}
=
\frac{1}{2A^{1/3}\:\mathrm{fm}^2}
\:.
\end{eqnarray}
The choice is somewhat ambiguous for nuclear reactions because one could
insert the mass number $A$ for the compound system or the average $A$ of
projectiles, or fragments respectively. However, these are details which do
not hamper the analysis; a good order-of-magnitude guess suffices for the present
analysis.
\begin{figure}
\includegraphics*[width=8.3cm]{fig1a.eps}
\includegraphics*[width=8.3cm]{fig1b.eps}
\includegraphics*[width=8.3cm]{fig1c.eps}
\caption{\label{fig:static} (color online)
Comparison between slices through the one-dimensional
Wigner $f^{(1)}_\mathrm{W}(z,k_z=0)$ and Husimi $f^{(1)}_H(z,k_z=0)$
distribution functions for the static ground states of $^{16}O$ (top),
$^{96}Zr$ (middle), and $^{230}Th$ (bottom).}
\end{figure}
In a first round, we investigate the distributions for nuclear ground states,
to understand the basic pattern and to have a benchmark from a case certainly
free of excitation. Figure \ref{fig:static} shows slices through static
one-dimensional Wigner and Husimi distributions of the ground states for three
nuclei, a light, a medium heavy, and a heavy one. The Wigner distributions
show marked shell oscillations. The Husimi distributions have efficiently
removed these oscillations and represent a smooth curve averaged through the
Wigner distributions. The amplitude of the shell oscillations decreases with
increasing mass number, but very slowly such that smooth Wigner functions
(resembling classical phase-space distributions) are only reached at an order of
magnitude $A\approx 5000$ \cite{Bra97aB,Kri80a}. The Husimi distributions
look smooth already for the low mass numbers of really existing nuclei. This,
however, is achieved at the price of somewhat blurring the details due to the
folding procedure. This is acceptable for the analysis of the distributions
as such, i.e. in phase space. It may become misleading when reducing the
distributions to compact observables by integrating over phase space or parts
thereof, as will be done in Section \ref{sec:observ}). The Husimi folding may
add an offset to such averaged observables. In such a case, the integrations
suffice to average out the small-scale oscillations in the Wigner functions.
Therefore, we will in the following concentrate our investigations on the
use of the Wigner function only.
\section{More compact observables}
\label{sec:observ}
The Wigner and Husimi distributions are illustrative but difficult to handle,
being six-dimensional objects. They can be
looked at in some selected snapshots and by taking cuts through the 6D phase
space. Observables in lower dimensions down to single numbers are necessary
complements for the analysis of dynamical processes. In this section, we will
introduce local observables which are distributed in 3D coordinate space.
They are reduced to single-number observables by further spatial integration.
\subsection{Observables from the Wigner distribution}
It is a standard procedure in classical non-equilibrium statistical physics to
discuss dissipation dynamics in terms of the local momentum distribution, i.e.
the momentum distribution at a given space point \cite{Bal75}. The basic
features of the local momentum distribution can be characterized by
its moments. The first moment
\begin{equation}
\langle\mathbf{k}(\mathbf{r},t)\rangle_{(n)}
=
\frac{\int {\rm d}^n k\:\mathbf{k}\:f^{(n)}_\mathrm{W}(\mathbf{r},\mathbf{k},t)}
{\int {\rm d}^n k\:f^{(n)}_\mathrm{W}(\mathbf{r},\mathbf{k},t)}.
\end{equation}
plays a special role. It characterizes the center of the distribution and it
is associated with the average local flow. The higher moments are taken as
variances, i.e. relative to the first moment. For the $m$-th moment this reads
\begin{equation}
\langle\mathbf{k}^{(m)}(\mathbf{r},t)\rangle_{(n)}
=
\frac{\int {\rm d}^n k \:(\mathbf{k}-\langle\mathbf{k}(\mathbf{r},
t)\rangle)^{m}f^{(n)}_\mathrm{W}(\mathbf{r},\mathbf{k},t)}
{\int {\rm d}^n k\:f^{(n)}_\mathrm{W}(\mathbf{r},\mathbf{k},t)}
\:.
\end{equation}
These moments serve as raw material for further reduced observables.
Note that they depend on the dimensionality of the Wigner function
used in their definition. This is communicated by the index $(n)$ in
the moments.
The radial profile of the momentum distribution may be characterized
by the ratios of moments, in particular the $m=4$ to $m=2$ ratio
\begin{equation}
R^{(n)}(\mathbf{r},t)
=
\frac{\langle\mathbf{k}^{4}(\mathbf{r},t)\rangle_{(n)}}
{\langle\mathbf{k}^{2}(\mathbf{r},t)\rangle^2_{(n)}}
\:.
\label{eq:ratio}
\end{equation}
Reference value is the thermal equilibrium which corresponds in the high
temperature limit to a Maxwellian momentum distribution. These ``equilibrium''
values are given in Table \ref{tab:gauss} for various dimensions.
\begin{table}[htbp]
\caption{Analytic values of the ratio $R^{(n)}$ as defined in Eq.~(\ref{eq:ratio}) for
a Gaussian distribution function, depending on the spatial dimension
$n$ in which the ratio is evaluated.}
\begin{center}
\begin{tabular}{|c|c|}
\hline
dimension $n$ & $R^{(n)}_{\rm gauss}$ \\
\hline
$1$& $3$\\
$2$& $2$ \\
$3$& $5/3$ \\
\hline
\end{tabular}
\label{tab:gauss}
\end{center}
\end{table}
The ratios are plagued by the fact that cold equilibrium distributions are
Fermi functions rather than Gaussians and, more importantly, are significantly
smoothed by quantum effects. This hampers an analysis at a detailed level. A
more robust signature of equilibration is obtained by the deformation of the
momentum distribution. The leading term is the quadrupole deformation which
can be characterized by the eccentricity in the reaction plane, which reads
\begin{equation}
\varepsilon(\mathbf{r},t)
=
\frac{\langle k_x^{2}(\mathbf{r},t)\rangle-
\langle k_z^{2}(\mathbf{r},t)\rangle}
{\langle k_x^{2}(\mathbf{r},t)\rangle+
\langle k_z^{2}(\mathbf{r},t)\rangle}\:,
\end{equation}
where the dimensionality index has been skipped for simplicity.
The global eccentricity is obtained by spatial integration
\begin{eqnarray}
\varepsilon(t)
&=&
\int {\rm d}x\,dz\,\varepsilon(\mathbf{r},t)\rho(\mathbf{r},t)
\:,
\\
\rho(\mathbf{r})
&=&
\int {\rm d}k_x {\rm d}k_z\:f^{(2)}_\mathrm{W}(\mathbf{r},\mathbf{k},t)
\:,
\end{eqnarray}
with $\rho$ the local density.
\subsection{Intrinsic kinetic energy}
Another interesting observable is the intrinsic excitation energy. Ideally,
it is defined as the difference between the actual energy and a ``cold''
reference energy which is obtained from a stationary HF calculation
constrained to reproduce the density $\rho(\mathbf{r})$ and current
$\mathbf{j}(\mathbf{r})$ of the actual TDHF state \cite{Cus85a,Umar}. The
cumbersome density constrained calculations can be avoided when evaluating the
``cold'' reference state in Thomas-Fermi approximation. This shortcut was used
successfully in Cluster physics \cite{Calvayrac}. The so approximated
intrinsic kinetic energy reads
\begin{eqnarray}
E_{\rm int}(t)
&=&
E_{\rm kin}(t)-E_{\rm coll,kin}(t)-E_{\rm TFW}(t)
\:,
\\
E_{\rm kin}(t)
&=&
\frac{1}{2}\sum_i \int {\rm d}^3r\:|\nabla \varphi_i({\bf r},t)|^2
\:,
\\
E_{\rm coll,kin}(t)
&=&
\int {\rm d}^3 r \: \frac{{\bf j}^2({\bf r},t)}{2 \rho({\bf r},t)}
\:,
\\
E_{\rm TFW}(t)
&=&
\int {\rm d}^3r\: \tau_{\rm TFW}({\bf r},t)
\;,
\\
\tau_{\rm TFW}({\bf r},t)
&=&
\frac{3\hbar^2}{10m}(3\pi^2)^{2/3} \rho({\bf r},t)^{5/3}
\nonumber \\
&&
+\frac{\hbar^2}{18m}\frac{(\nabla\rho({\bf r},t))^2}{\rho({\bf r},t)}
\;.
\end{eqnarray}
It quantifies the non-adiabatic and non-collective component of the kinetic
energy, roughly corresponding to the intrinsic thermal energy. The first
ingredient for the calculation is the total kinetic energy, $E_{\rm kin}$, of
the system. The second term, $E_{\rm coll,kin}(t)$, subtracts the
hydrodynamic kinetic energy contained in the collective flow $\mathbf{j}$.
The third term, $E_{\rm TFW}(t)$, subtracts the instantaneous kinetic energy
of the zero-temperature ground state at the given density $\rho({\bf
r},t)$. The evaluation of this kinetic energy density $\tau({\bf r},t)$ is
done in the Thomas-Fermi-Weizs\"acker approximation \cite{Dreizler}.
\begin{figure*}[hbtp]
\includegraphics*[width=8.6cm]{fig2a.eps}
\includegraphics*[width=8.6cm]{fig2b.eps}
\includegraphics*[width=8.6cm]{fig2c.eps}
\includegraphics*[width=8.6cm]{fig2d.eps}
\caption{\label{fig:O16} (color online)
The one-dimensional Wigner distribution
$f^{(1)}_\mathrm{W}(z,k_z,t)$ for a central $^{16}O+$$^{16}O$ collision
is plotted at four different times $t$. Three contour lines are
plotted to highlight the levels of $f^{(1)}_\mathrm{W}(z,k_z,t)$ at
$2\cdot10^{-3},\:4\cdot10^{-3}$,\: and $6\cdot10^{-3}$.}
\end{figure*}
\subsection{An estimate for the fragment distance}
As a simple observable characterizing the geometry of a collisional stage, we
introduce the distance $d(t)$ between the fragments
\begin{equation}
d(t)
=
|\langle\mathbf{r}_1(t)\rangle-\langle\mathbf{r}_2(t)\rangle|
\:.
\end{equation}
The coordinates $\mathbf{r}_i $ of the right and left fragment were obtained
by splitting the density of the system symmetrically into two half spaces and
averaging over each half. This is an obvious definition for well separated
fragments. It becomes somewhat ambiguous in the overlap region, but still
remains a useful indicator of the overall geometry.
\section{Results}
\label{sec:results}
We present TDHF results for different reaction scenarios, $^{16}O+$$^{16}O$
collisions head-on and with finite impact parameter, and a
$^{96}Zr+$$^{132}Sn$ collision. The Skyrme parametrization SkI3
\cite{Reinhard2} is used for the calculations. We performed test calculations
with other Skyrme forces and found very similar results. Thus we report the
results only from this one force. The central collisions were computed on a
coordinate space mesh with $48\times24^2$ grid points and the non-central ones
with $36\times24^2$ points.
\subsection{$^{16}$O+$^{16}$O Collisions}
\subsubsection{$^{16}O+$$^{16}O$ Central}
First, we analyze a $^{16}O+$$^{16}O$ collision with a center-of-mass energy
of $E_{c.m.}=100$\:MeV and zero impact parameter $b=0$\:fm. Figure
\ref{fig:O16} shows the one-dimensional Wigner distribution
$f^{(1)}_\mathrm{W}(z,k_z,t)$ at four different stages of the collision.
Initially ($t=0.0$\:fm/c), there are two cold nuclei far apart from each
other. They are shifted in $k_z$-direction depending on their initial
boost. At the intermediate stage ($t=67.8$\:fm/c), the phase space volumes of
the two fragments seem to merge but are avoiding each other, i.e. they
maintain a division line. This is a consequence of the Pauli principle. After
a while ($t=155.8$\:fm/c), the phase space volumes start to separate, keeping
some contact still for some time.
The final stage ($t=273.8$\:fm/c) shows two separate fragments again. But
here the structure is quite different as compared to the initial state. Both
blobs become strongly asymmetric. The $k_z$ position of the maximal peaks
(red spots) are not lower than initially. But the asymmetry in the
distribution extends very much towards lower $k_z$ and also to values of
opposite sign. This indicates that the average slowdown in relative
c.m. motion in this case is not due to a global downshift of an otherwise
symmetric distribution, but to the strong asymmetry reducing significantly
the average $k_z$. A possible interpretation is that the
wave functions maintain their initial momentum structure to a large extent
but also components from the other fragment are mixed in.
\begin{figure}[htbp]
\includegraphics*[width=8.6cm]{fig3.eps}
\caption{\label{fig:eccent_eint} (color online)
The global eccentricity $\varepsilon(t)$ obtained from the
two-dimensional Wigner function $f^{(2)}_\mathrm{W}(x,z,k_x,k_z,t)$, the
distance between the fragments $d(t)$, and the internal kinetic
energy $E_{int}(t)$ for a central $^{16}O+$$^{16}O$
collision with a center-of-mass energy of $E_{c.m.}=100$\:MeV.}
\end{figure}
In a next step, we analyze the time evolution in terms of compact (single
number) observables. The time evolution of the intrinsic kinetic energy
$E_{int}(t)$ and of the global eccentricity $\varepsilon(t)$ are
shown, together with the fragment distance $d(t)$, in Figure
\ref{fig:eccent_eint}. The time of maximum overlap (compound stage) is
reached at 75 fm/c where $d(t)$ has a minimum. Both kinetic observables show
a pronounced maximum there. As the reaction continues the eccentricity is
strongly damped and keeps oscillating at a low level. This indicates some
thermalization. The internal energy reaches its maximum at
$E_{int}\approx86$\:MeV and saturates, again with some persisting
oscillations, at half of the maximal amount. As the potential energy plays a
huge role in the compound stage, the values for the kinetic energies have to
be taken with care here. The asymptotic values are more directly
interpretable. They show significant heating (from $E_{int}(t)$) and
great deal of equilibration already within the short time span of the
simulation.
\begin{figure}[htbp]
\includegraphics*[width=8.1cm]{fig4a.eps}
\includegraphics*[width=8.1cm]{fig4b.eps}
\includegraphics*[width=8.1cm]{fig4c.eps}
\caption{\label{fig:Osci} (color online)
A $k_x$-$k_z$ cut of the
two-dimensional momentum distribution $f^{(2)}_\mathrm{W}(k_x,k_z,t)$
plotted at the center of the fragment moving finally with
negative mean momentum in $k_z$-direction. The plots are taken at
three different times $t$ near the final stage of the calculation. Two
contour lines are plotted to highlight the levels of $f^{(2)}_\mathrm{W}$ at
$2\cdot10^{-4}$ and $4\cdot10^{-4}$.}
\end{figure}
To visualize the oscillations in $\varepsilon(t)$ and $E_{int}(t)$ we show in
Figure \ref{fig:Osci} the two-dimensional momentum distribution
$f^{(2)}_\mathrm{W}(k_x,k_z,t)$ in the exit channel for the fragment moving to
the left (with negative $\langle k_z\rangle$). The shape is
asymmetric and oscillates back and forth. This indicates that the largest
collective effect in the exit channel is residual octupole oscillations which
have their counterpart also in similar octupole oscillations of the fragments'
spatial shape.
We have checked the momentum ratio $R^{(n)}(\mathbf{r},t)$ as given in
Eq. (\ref{eq:ratio}) for $n=1,2$ at different times to probe the
closeness of the momentum distribution to a Maxwellian
distribution. The comparison with the analytic values $R_{\rm gauss}$ from
Tab. \ref{tab:gauss} is not conclusive, as quantum
effects blur the classical concept behind this ratio by making the distributions
too different from Gaussians. This casts some
doubts on the usefulness of the global ratio $R(t)$ in this still
predominantly quantum-mechanical domain. We will come back to this
observable in Section \ref{sec:ZrSn}.
\subsubsection{$^{16}$O+$^{16}$O Non-Central}
In this Section we analyze $^{16}$O+$^{16}$O fusion reactions with
non-zero impact parameter $b=2$\:fm and two different center-of-mass
energies $E_{c.m.}=20$\:MeV and $E_{c.m.}=50$\:MeV.
\begin{figure}[htbp]
\includegraphics*[width=8.6cm]{fig5a.eps}
\includegraphics*[width=8.6cm]{fig5b.eps}
\caption{\label{fig:fusion} (color online)
The global eccentricity $\varepsilon(t)$ obtained
from the two-dimensional Wigner function $f^{(2)}_\mathrm{W}(x,z,k_x,k_z,t)$,
the distance between the fragments $d(t)$, and the internal
kinetic energy $E_{int}(t)$ for two $^{16}$O+$^{16}$O
fusion reaction with a center-of-mass energy of $E_{c.m.}=20$\:MeV
(top) and $E_{c.m.}=50$\:MeV (bottom) and impact parameter $b=2$\:fm.}
\end{figure}
\begin{figure}[htbp]
\includegraphics*[width=8.6cm]{fig6.eps}
\caption{\label{fig:ZrSn} (color online)
The global eccentricity $\varepsilon(t)$ obtained from the
two-dimensional Wigner function $f^{(2)}_\mathrm{W}(x,z,k_x,k_z,t)$, the
quadrupole $Q_{20}$, and the internal kinetic energy $E_{int}(t)$
for a $^{96}$Zr+$^{132}$Sn fusion reaction with a center-of-mass
energy of $E_{c.m.}=250$\:MeV and impact parameter $b=2$\:fm.}
\end{figure}
\begin{figure}[htbp]
\includegraphics*[width=8.6cm]{fig7.eps}
\caption{\label{fig:ratio2d} (color online)
The local ratio $R^{(2)}(\mathbf{r})$ obtained from the
two-dimensional Wigner distribution $f^{(2)}_\mathrm{W}(x,z,k_x,k_z,t)$ for
a $^{96}Zr+$$^{132}Sn$ fusion reaction at
$t=1279.8$\:fm/c. Four points are marked in this plot to be analyzed
later more precisely (Figure \ref{fig:ZrSnMomentA} and Figure
\ref{fig:ZrSnMomentB}). The reference value from a Gaussian
distribution is $R^{(2)}_{\rm gauss}=2$. A Contour line is plotted to
highlight the level of $R^{(2)}(\mathbf{r})=1.5$.}
\end{figure}
\begin{figure*}[!htbp]
\includegraphics*[width=8.6cm]{fig8a.eps}
\includegraphics*[width=8.6cm]{fig8b.eps}
\includegraphics*[width=8.6cm]{fig8c.eps}
\includegraphics*[width=8.6cm]{fig8d.eps}
\caption{\label{fig:ZrSnMomentA} (color online)
The left column reviews the two-dimensional momentum
distribution $f^{(2)}_\mathrm{W}(k_x,k_z)$ in the selected (outer) points from
Figure \ref{fig:ratio2d}. Contour lines are plotted to highlight the
levels of $f^{(2)}_\mathrm{W}$ at $1\cdot10^{-4},2\cdot10^{-4}$, and
$3\cdot10^{-4}$. Slices through $f^{(2)}_\mathrm{W}(k_x,k_z)$ matching the
$k_x,k_z$-axis are shown in the right column.}
\end{figure*}
\begin{figure*}[!htbp]
\includegraphics*[width=8.6cm]{fig9a.eps}
\includegraphics*[width=8.6cm]{fig9b.eps}
\includegraphics*[width=8.6cm]{fig9c.eps}
\includegraphics*[width=8.6cm]{fig9d.eps}
\caption{\label{fig:ZrSnMomentB} (color online)
Same as Figure \ref{fig:ZrSnMomentA} but for the selected
(inner) points in Figure \ref{fig:ratio2d}.}
\end{figure*}
\begin{figure*}[!htbp]
\includegraphics*[width=8.6cm]{fig10a.eps}
\includegraphics*[width=8.6cm]{fig10b.eps}
\caption{\label{fig:Th230ZrSn} (color online)
Slices along $k_x$ (left) and $k_z$ (right) through
$f^{(2)}_\mathrm{W}(k_x,k_z)$ are shown for the merged $^{96}Zr+$$^{132}Sn$
system (red) at $t=1279.8$\:fm/c in the point $P_4$ from Figure
\ref{fig:ratio2d} in comparison with the ground state of $^{230}Th$
(blue).}
\end{figure*}
Figure \ref{fig:fusion} shows the global eccentricity $\varepsilon(t)$, the
internal kinetic energy $E_{int}(t)$, and the fragment distance $d(t)$ for both
reactions. The distance oscillates after the minimum reached on first impact. This
indicates that both cases describe a fused compound state. The eccentricity
$\varepsilon(t)$ follows the oscillations of the distance, reflecting
a continuing vivid interaction with the spatial deformation. This
indicates that we are far from equilibration. The intrinsic kinetic energy
grows initially and soon levels off, leaving small residual oscillations about
a constant mean value. This mean intrinsic energy is, of course, larger for
the higher-energy collision ($E_{c.m.}=50$\:MeV). The example demonstrates nicely
that one needs a couple of observables to conclude on equilibration. One may
be tempted to take the constant $E_{int}(t)$ as indicator of a thermalized
state. The still large values of eccentricity and the oscillations thereof
prove clearly that we are rather in a situation of substantial coherent
oscillations of the compound system. In this context, it is to be remembered
that the energy stored in the collective motion of the compound system is
subtracted in the evaluation of $E_{int}(t)$.
The demonstrated behavior of the observables was checked up to $t=4000$\:fm/c,
twice the time span shown in Figure \ref{fig:fusion}. The pattern carried on
unchanged also for these longer times.
\subsection{$^{96}$Zr+$^{132}$Sn}
\label{sec:ZrSn}
As an example for a much heavier nuclear system we present fusion of
$^{96}$Zr+$^{132}$Sn achieved with a center-of-mass energy of
$E_{c.m.}=250$\:MeV and impact parameter $b=2$\:fm. The reaction between the
neutron-rich $^{132}$Sn nucleus and $^{96}$Zr was already studied in TDHF with a
focus on barrier heights and widths of the heavy-ion potential as well as capture
cross sections \cite{Oberacker}.
It was not possible in the present analysis to calculate the distance $d(t)$
between the fragments as it was done in the $^{16}$O+$^{16}$O fusion
scenario. The numerical algorithm selecting the spatial expectation values for
a two-body system was not able to detect two distinct objects during the whole
calculation and in this asymmetric system a simple symmetric division of the
grid was not possible. In Figure \ref{fig:ZrSn} we therefore use the
expectation value $Q_{20}\equiv\langle \hat{Q}_{20}\rangle$ of the quadrupole
operator $\hat{Q}_{20}$ to visualize the global geometry of the reaction.
Again, large values indicate separated fragments and low values a compound
stage. It is obvious from the figure that the reaction ends in a compound
nucleus. The overall trends of intrinsic kinetic energy and eccentricity are to
some extent similar to the results in Figure \ref{fig:fusion}. However, the
final eccentricity is much smaller, still maintaining some small
oscillations. This indicates a better thermalization than seen for
$^{16}$O+$^{16}$O, which is no surprise because the single-particle phase space
is much larger for the heavier system. The trend of the intrinsic energy does
also differ in detail. There seem to be two stages of growth, a fast
initial rise on the way to the compound stage and a slower, but steady,
growth up to 1000 fm/c. This indicates that some thermalization processes
and energy transport from deformation energy to kinetic energy is still
going on. After 1000 fm/c we again see a rather constant $E_\mathrm{int}$
as seems to be typical for energetic compound nuclei.
Figure \ref{fig:ratio2d} shows the local ratio $R^{(2)}(\mathbf{r},t)$ as
defined in Eq. (\ref{eq:ratio}) in the reaction plane at
$t=1279.8$\:fm/c. The surface region is distinguished by large values coming
close to the Maxwellian reference values while much smaller ratios are seen
inside. In order to illuminate these results, we have a closer look at the more
detailed momentum distributions at four selected points
indicated in Figure \ref{fig:ratio2d}. Figure \ref{fig:ZrSnMomentA} shows
results for two the outer points at the surface. The first point $(P_1)$ is
taken at approximately half the maximum value of $f_\mathrm{W}$. The
distribution is very similar to a Gaussian overlayed by a slight
asymmetry. The next point $(P_2)$ shows a more pronounced asymmetric
shape. Moving further to the inner points $(P_3,P_4)$ reviewed in Figure
\ref{fig:ZrSnMomentB} the momentum distributions differ substantially from
Gaussians and come closer to the idea of a Fermi distribution, although
heavily overlayed by quantum shell oscillations.
An similar analysis of the momentum distribution at other points
near the nuclear center yields similar results. The inner region of
the merged systems seems to stay rather ``cold`` during the reaction.
The strong quantum mechanical shell oscillations hinder a fit of the
distribution functions shown in Figure \ref{fig:ZrSnMomentA} and
\ref{fig:ZrSnMomentB} to a Fermi function from which one eventually could read
off an estimate for the system's temperature distribution. Therefore we compare
the $^{96}$Zr+$^{132}$Sn system to be assumed "hot`` with the "cold" analogue
of this system. Figure \ref{fig:Th230ZrSn} shows the momentum
distribution of the $^{96}$Zr+$^{132}$Sn compound system at the point $P_4$
indicated in Figure \ref{fig:ZrSnMomentB}. This is compared with the result from
the prolate ground state of $^{230}$Th. The $^{96}$Zr+$^{132}$Sn system
consists of $p=90$ protons and $n=138$ neutrons. $^{230}$Th shares the same
proton number with two additional neutrons. The ground state nucleus shows
huge, fully developed shell oscillations. Compared to these, the remaining
quantum oscillations in the "hot" compound state become rather small. The
disappearance of quantum shell effects is a major thermalization effect
\cite{Bra97aB,Bra93}. The occupation of high momentum components, however,
which would also be expected for hot systems remains insignificant. This is due
to the fact that the nucleus is an open system from which high energy particles
escape, constantly depleting the high-momentum parts of the distribution.
This explains why a fit
to Fermi distributions failed. A measure of temperature may be deduced from the
suppression of the shell oscillations, but this analysis is blurred by the
large thermal fluctuations in the momentum-space density. For the time being,
the eccentricity remains the cleanest indicator of equilibration.
\section{Summary}
In this work we have analyzed from different perspectives the dynamics
of TDHF during various reactions including the nuclei $^{16}O$,
$^{96}Zr$, $^{132}Sn$, and $^{230}Th$ with various center-of-mass
energies and impact parameters. The key quantity of the analysis is the
Wigner distribution function which provides a detailed phase-space
picture of the quantum state. As complementing quantities, we also
considered three more compact observables in terms of local
distributions: the ratio $R(\mathbf{r},t)$ of the weighted moments
(weight four and two) of the local momentum distribution described by
the Wigner function (i.e. integrating the Wigner function over
momentum space for fixed local position), the eccentricity
$\epsilon(\mathbf{r},t)$ of the local momentum distribution, and the
intrinsic excitation energy $E_\mathrm{intr}(\mathbf{r},t)$ as deduced
from the kinetic energy density.
General properties of the Wigner distribution were discussed first for
stationary states. It shows oscillations which stem from the quantum
shell oscillations of the underlying single-particle states. We also
looked at the Husimi function derived from the Wigner function by
some phase-space smoothing. The latter indeed provides a cleaner and
more intuitive picture. We find, however, that the shell oscillations
are much reduced in the dynamical scenarios of heavy-ion collisions,
allowing us to continue the dynamical studies with the
Wigner function alone.
We have visualized the collision process through snapshots of a 2D cut
of the 6D Wigner function. This shows that the two initially separated
phase space blobs never fully merge, even at the compound stage.
The distributions of the emerging fragments acquire a strong asymmetry
in momentum space and nicely show the phase space rotations
associated with the remaining octupole oscillations of the final
fragments.
The moment ratio $R$ was intended as a means to
compare the shape of the TDHF-Wigner distribution with a Maxwellian
distribution corresponding to thermal equilibrium. We find for all reaction
parameters that the moment ratio remains below the Maxwellian
reference value, which means that thermalization could not be asserted
in this observable. The reason is that that the high-momentum tail of
the actual distribution is immediately depleted by particle emission.
This exemplifies the fact that a true equilibrium state is hard to
establish in an open system.
The eccentricity $\varepsilon$ turned out to be a more useful
indicator. It grows dramatically in the initial phase of the reaction
and relaxes to lower values quickly after the compound state and then
remains oscillating about some finite value. This means that the final
relaxation to a thermal state is probably underestimated in mere TDHF.
Similar patterns are shown by
the
intrinsic kinetic energy $E_{int}$.
The resulting ``asymptotic'' value of $E_{int}$ depends
strongly on the initial conditions, e.g., growing with the initial
collision energy.
We conclude that although TDHF includes dissipation owing to
single-particle viscosity, which acts strongly in the initial phase of
reactions, there is no evidence for complete equilibration.
\section*{Acknowledgment}
This work was supported by the Frankfurt Center for Scientific
Computing and by the BMBF under Contracts No. 06FY9086 and 06ER9063. We gratefully acknowledge
support by the Frankfurt Center for Scientific Computing.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Heinrich Lienhard (* 19. Januar 1822 in Bilten, Kanton Glarus; † 19. Dezember 1903 in Nauvoo, Illinois) war ein Schweizer Auswanderer, der 1843 erstmals in die Vereinigten Staaten reiste. Er verbrachte drei Jahre in Illinois und reiste 1846 nach Kalifornien, wo er bis 1850 blieb. Nach einem Aufenthalt von dreieinhalb Jahren in der alten Heimat kehrte er 1854 mit seiner Familie endgültig in die USA zurück, wo er bis zu seinem Tod in Nauvoo, Illinois, lebte. In den 1870er Jahren verfasste er seine Erinnerungen an die Jugend und ersten Jahre in Amerika, ein Manuskript, das eine wichtige historische Quelle für den California Trail, Johann August Sutter und den Goldrausch in Kalifornien darstellt.
Leben
Heinrich Lienhard wurde am 19. Januar 1822 auf dem Ussbühl in Bilten, Kanton Glarus, geboren. Er wuchs mit drei Geschwistern auf dem Bauernhof der Eltern in bescheidenen Verhältnissen auf. Seit seiner Kindheit träumte er davon, nach Amerika auszuwandern, wie es schon mehrere seiner Cousins getan hatten. Dieser Traum erfüllte sich, als der Vater nach langem Widerstand seinen Plänen endlich zustimmte. 1843 reiste Heinrich Lienhard zusammen mit einem Nachbarn nach Neu-Schweizerland, später Highland, in Illinois.
Die nächsten zweieinhalb Jahre waren eine Zeit des Fussfassens in der Neuen Welt. Lienhard hielt sich hauptsächlich in Illinois auf, wo er zuerst als Gehilfe bei verschiedenen Farmern in der Schweizer Siedlung arbeitete. Später fuhr er auch den Mississippi hinauf, machte Abstecher nach Iowa und Wisconsin und verrichtete unterwegs Gelegenheitsarbeiten, immer in der Hoffnung, bald ein besseres Auskommen zu finden. Als er im Frühjahr 1846 in einem Ladengeschäft in St. Louis arbeitete, traf er eines Tages zufällig einige Freunde aus Galena, mit denen er sich dort ein Jahr zuvor über eine mögliche Auswanderung nach Kalifornien unterhalten hatte. Sie waren nach St. Louis gekommen, um sich für ebendieses Abenteuer auszurüsten, und begeistert schloss Lienhard sich ihnen an.
Die Reise der «Five German Boys», wie die anderen Emigranten Heinrich Lienhard und seine vier Kameraden nannten, dauerte sechs Monate und führte von Independence, Missouri, nach New Helvetia, besser bekannt unter der Bezeichnung «Sutter's Fort», in Kalifornien. 1846 gab es für Emigranten mit Wagen noch keinen fest etablierten Trail in das von Mexiko beanspruchte Gebiet am Pazifik, weshalb vor allem die zweite Hälfte des Weges Menschen und Zugtieren oft das Äusserste an Kraft und Geschicklichkeit abverlangte. In seinen Erinnerungen beschreibt Heinrich Lienhard neben der genauen Route auch das vielseitige Alltagsleben auf dem Trail: die wechselhaften Beziehungen zwischen den Emigranten, die eindrücklichen, langsam sich verändernden Landschaftsformen, Begegnungen mit den einheimischen Indianern sowie Mühsal und Gefahren schwieriger Wegabschnitte wie beispielsweise die Überquerung der Grossen Salzwüste und der Sierra Nevada.
In Kalifornien erwartete die Immigranten noch vor ihrer Ankunft in Sutters Fort ein Werber der Armee der Vereinigten Staaten. Auf Drängen eines Kameraden, dem er ein paar Dollar schuldete, liess sich Lienhard wie andere mittellose Emigranten für einen dreimonatigen Freiwilligendienst im Krieg gegen Mexiko verpflichten. Die amerikanischen Truppen hatten den Auftrag, die Annexion aller von den Vereinigten Staaten beanspruchten Gebiete nördlich des Rio Grande durchzusetzen, ein Ziel, das die Regierung in Washington seit Jahrzehnten verfolgte. Bereits auf der Reise ins Hauptquartier von Monterey, der damaligen Hauptstadt Kaliforniens, erkrankte Lienhard jedoch schwer, verbrachte mehrere Wochen im Krankenhaus und wurde als Rekonvaleszenter anschliessend vom Dienst im Feld dispensiert.
Nach seiner Rückkehr von Monterey im Februar 1847 fand Lienhard Anstellung bei John A. Sutter. Das erste Halbjahr unterhielt er dessen Gemüsegarten am Yuba River rund fünfzig Meilen nördlich des Forts, ab September übernahm er für mehrere Monate die Aufseherstelle im Fort. Um die Jahreswende 1847/48 brachte er als Frachtbegleiter auf Sutters Schoner eine Ladung Weizen nach San Francisco, lehnte eine feste Anstellung in dieser Funktion jedoch ab. Sutter, der wusste, dass Lienhard sich gerne mit Gartenarbeit beschäftigte, bat ihn darauf, in Partnerschaft einen grossen Obst- und Gemüsegarten beim Fort anzupflanzen, ein Projekt, dem Lienhard sich in den folgenden Monaten mit Hingabe widmete.
Im Januar 1848 wurde am Südarm des American River, wo Sutter eine Sägemühle (Sutter's Mill) bauen liess, Gold entdeckt. Obwohl alle Arbeiter Sutters das Fort schon bald verliessen, um in den Flusstälern ihr Glück zu versuchen, blieb Lienhard bis im Sommer im Garten und begab sich erst im August in die Minen, als Sutter ihn dazu aufforderte. Dieser stellte ihm indianische Gehilfen, Arbeitsgeräte und Lebensmittel zur Verfügung und erhielt dafür von Lienhard die Hälfte des gewaschenen Goldes, eine Vereinbarung, die Sutter auch mit anderen Männern traf. Als im September Sutters ältester Sohn John August aus der Schweiz in Kalifornien eintraf, bat Sutter Lienhard, ihm leihweise auch seine eigene Hälfte des gewaschenen Goldes zu überlassen, damit er seinem Sohn eine möglichst grosse Ausbeute des Edelmetalls präsentieren könne. Als Lienhard aber später ins Fort zurückkehrte, war August Sutter, der inzwischen die Geschäfte seines tief verschuldeten Vaters übernommen hatte, nicht mehr in der Lage, Lienhard sein Gold wieder auszuhändigen. Nach Wochen vergeblichen Wartens willigte dieser schliesslich ein, an Zahlungs statt Sutters Schafherde zu übernehmen.
Den folgenden Winter 1848/49 verbrachte Lienhard mit einem Schweizer Landsmann namens Jakob Dürr auf der unweit des Forts gelegenen Schaffarm. Im Frühling kaufte Dürr Lienhard die Hälfte der Schafe ab, und im April zogen sie gemeinsam in die Minen, um Handel zu treiben. Nach mehreren Wochen verkaufte Lienhard Dürr auch seinen Teil der Herde und kehrte ins Fort zurück. Dort nahm er August Sutters Auftrag an, dessen Mutter und Geschwister aus der Schweiz nach Kalifornien zu bringen. Im Juni 1849 verliess er San Francisco, reiste über den Isthmus von Panama nach New York und von dort über England und Deutschland in die Schweiz. Im Spätherbst kehrte er auf gleichem Weg mit einer Gruppe von zehn Personen – es hatten sich noch Verwandte und Bekannte von Frau Sutter angeschlossen – nach San Francisco zurück, wo sie im Januar 1850 wohlbehalten eintrafen.
Ein halbes Jahr später beschloss Lienhard, Kalifornien endgültig zu verlassen. Der Abschied fiel ihm nicht leicht. Er liebte das Land mit seinem angenehmen Klima und seiner reichen Flora und Fauna; doch er konnte sich nicht mehr mit der Gesetzlosigkeit und der überhandnehmenden Gewalt abfinden, mit der den Einheimischen Land und Leben geraubt wurde und die ihr Schicksal auf brutale Art besiegelte. Nach einer abermals halbjährigen Reise und einem angenehmen Ausklang in Paris schritt er am 31. Dezember 1850 auf dem altvertrauten Fussweg wieder seinem Elternhaus auf dem Ussbühl entgegen.
Im Sommer 1851 heiratete Heinrich Lienhard Elsbeth Blumer von Bilten. Er kaufte die bäuerliche Liegenschaft «Auf Brunnen» in Kilchberg bei Zürich, wo die beiden Söhne Kaspar Arnold (1852) und Johann Heinrich (1853) geboren wurden. Doch dem Versuch, in der alten Heimat wieder sesshaft zu werden, war kein Erfolg beschieden: Nach zwei Jahren verkaufte Lienhard den Besitz in Kilchberg wieder und verliess mit seiner Familie im April 1854 die Schweiz für immer. Zuerst liessen sie sich in Madison, Wisconsin, nieder, wo der dritte Sohn, John Jacob, zur Welt kam. 1856 zogen sie nach Nauvoo, Illinois, einem malerisch gelegenen Ort am Mississippi, den eine grosse Mormonengemeinde zehn Jahre früher hatte verlassen müssen und der seither vor allem deutsch- und französischsprachige Einwanderer europäischer Herkunft anzog.
In Nauvoo verbrachte Heinrich Lienhard in einem prächtigen Haus mit Garten und grosszügigem Landbesitz 47 Jahre als erfolgreicher Farmer und geachteter Bürger. Hier gebar Elsbeth Lienhard sechs weitere Kinder, doch blieb die Familie nur wenige Jahre vollständig. 1878 verloren sie ihren ältesten Sohn Kaspar, Zahnarzt von Beruf, und 1884 ihre neunzehnjährige Tochter Dora, die an den Folgen eines unverschuldeten Zwischenfalls auf dem Schulhof innerlich verblutete. Wenige Monate später starb auch Lienhards Frau Elsbeth, und 1892 verlor er noch seine jüngste, erst sechzehnjährige Tochter Barbara Adela. Heinrich Lienhard starb am 19. Dezember 1903 nach kurzer Krankheit. Er wurde auf dem Familiengrab im presbyterianischen Friedhof von Nauvoo beigesetzt, wo sich auch die Gräber seiner Frau und sieben ihrer Kinder befinden.
Heinrich Lienhards Manuskript
Mitte der 1870er Jahre begann Heinrich Lienhard mit der Niederschrift seiner Erinnerungen an seine Kindheit und Jugend in der Schweiz bis zur Rückkehr aus Kalifornien Ende 1850, also an die ersten 29 Jahre seines Lebens. In regelmässigem, zügigem Duktus alter deutscher Schreibschrift füllte er nahezu eintausend Seiten, eine Arbeit, der er sich mehrere Jahre widmete und mit der er seinen Nachkommen ein Vermächtnis ganz besonderer Art hinterliess.
Wo immer Heinrich Lienhard sich in den Jahren seiner Wanderschaft aufhielt, galt seine ungebrochene Aufmerksamkeit der Natur in ihrer ganzen Vielfalt: der landschaftlichen Umgebung, den klimatischen Verhältnissen, der Bodenbeschaffenheit, geologischen Besonderheiten sowie ihm unbekannten Pflanzen und Tieren. Einen wichtigen Platz in seinen Erinnerungen nehmen auch die Menschen ein, die unterwegs seinen Weg kreuzten, Freundschaften, die Jahre dauerten, ebenso wie Begegnungen, die kurz und trotzdem unvergesslich waren. So setzte er mit seinen Porträts manchen Freunden und Bekannten, die heute längst vergessen wären, ein Denkmal, in dem sich immer auch seine eigene Persönlichkeit spiegelt. Dies zeigt sich besonders in seiner Beziehung zu John A. Sutter, dem Gründer Neu-Helvetiens, den er im Verlauf seiner Arbeit im Fort gut kennenlernte.
Lienhards Beobachtungsgabe beschränkte sich nicht auf Äusserlichkeiten, sie bedeutete vielmehr Wahrnehmen mit Augen, Herz und Verstand. Dies lässt sich besonders eindrücklich anhand seiner Begegnung mit den Indianern Kaliforniens verfolgen. Obwohl er diese als Einheimische des Landes respektierte, sind seine Bemerkungen nicht frei von der typisch ethnozentristischen Sehweise der Weissen. Angesichts des Goldrausches mit seinen dramatischen Folgen für die Indianer setzte jedoch ein Prozess des Umdenkens bei ihm ein, der dem damaligen Zeitgeist zuwiderlief. Ihre Wurzeln hatte seine zunehmend kritische Haltung in Mimal, wo er 1847 am Yuba River während sechs Monaten abgeschieden von anderen weissen Siedlern lebte und schon bald mit den Indianern der umliegenden Dörfer in Kontakt kam. Einige von ihnen trafen sich während seines Aufenthalts dort regelmässig bei seinem Haus, wo sie seine Tätigkeiten verfolgten, Tauschhandel trieben und ihm zwischendurch im Garten halfen. Sie bildeten ihn zu einem erstklassigen Bogenschützen aus, pflegten ihn, als er krank war, und nahmen ihn manchmal mit in ihr Dorf. So kam es, dass auch Lienhard begann, seine Nachbarn bei ihren alltäglichen Verrichtungen zu beobachten. Er staunte über ihre Fertigkeit im Herstellen von Gebrauchsgegenständen aller Art, über die Raffinesse ihrer Werkzeuge, über ihre Phantasie und ihren Schönheitssinn beim Verzieren ihrer meisterhaften Flechtarbeiten. Er begleitete sie bei der Jagd und beim Fischfang und beschreibt fasziniert die Geschicklichkeit, mit der sie dabei vorgingen, ebenso wie die Vielfalt ihrer Nahrungsbeschaffung und Zubereitung verschiedener Speisen.
Beobachtend begann Heinrich Lienhard zu verstehen, dass die Einheimischen ihre Lebensformen über viele Generationen äusserst sinnvoll an ihre Umgebung angepasst hatten und die reichen Ressourcen der kalifornischen Landschaften im Rhythmus der Jahreszeiten bestmöglich und nachhaltig zu nutzen wussten. Er begriff, dass das Fremde, obwohl anders, nicht zwangsläufig minderwertig war und dass die Verachtung, mit der die Weissen besonders den kalifornischen Indianern gegenübertraten, falsch und ungerecht war. Unvergesslich blieb ihm auch eine nächtliche Unterhaltung seiner Hüterjungen auf der Schaffarm, deren Zeuge er im Winter 1848/49 wurde. Sie sprachen von den Zeiten, bevor die weissen Siedler in ihre Täler eingedrungen waren, und von den grossen Veränderungen, die ihr Leben und das ihrer Eltern seither erfahren hatte. Auf Lienhard, der sich schlafend stellte, machten ihre Worte tiefen Eindruck: «Ich war durch das halblaut geführte Gespräch der Indianer recht Nachdenklich geworden. Ich suchte mich im Geiste in der Indianer Stelle zu versetzen und überlegte, ob ich wohl dann zufrieden sein würde, wenn man mich von meiner und meiner Voreltern Heimath derart verdrängen würde, wie es den armen Indianern wiederfuhr. Ich gestehe, dass mich dabei ein sehr rächegieriges Gefühl erfüllte, so dass ich Jedesmal zu dem Schluss kam, ich würde mich an den unverschämten, habgierigen Eindringlingen auf jede mögliche Weise rächen.» Doch wusste er aus eigener Erfahrung, dass für die Indianer, ob sie sich anpassten, wehrten oder flüchteten, der Kontakt mit den Weissen jederzeit mit dem Tod enden konnte.
Lienhards Text lässt somit verschiedene Betrachtungsweisen zu. Er fasziniert als detaillierte und spannende Beschreibung von Ereignissen und Menschen, Landschaften, Flora und Fauna. Weit mehr als ein Abenteuerbericht indessen sind seine Erinnerungen eine komplexe Reportage über rassische Eroberung. Der Raubbau an der Natur und den Tieren, die den Indianern auferlegte Zwangsarbeit, sexuelle Ausbeutung der Frauen, Vertreibung und Vernichtung der einheimischen Bevölkerung und Zerstörung ihrer jahrtausendealten Gemeinschaften durch die weissen Besetzer treten darin mit unerbittlicher Klarheit zutage. Heinrich Lienhards Text ist deshalb auch ein Tatsachenbericht über die angloamerikanische Eroberung der nördlichen Westhemisphäre mit ihrem Janus-Gesicht von Umweltzerstörung und rassischer Vernichtung einerseits und Aufbau einer kraftvoll pulsierenden angloamerikanischen Variante westlicher Kultur andererseits.
Publikationen
Heinrich Lienhards Manuskript blieb lange Zeit in Familienbesitz. Im Jahre 1949 verkaufte es eine Enkelin Lienhards an die Bancroft Library der University of California in Berkeley, Kalifornien, wo es heute im Original und auf Mikrofilm zugänglich ist. Es war aber bereits zu Lebzeiten Lienhards auch ausserhalb der Familie auf Interesse gestossen. Der erste, der sich mit dem Text befasste, war Kaspar Leemann, ein Freund aus der Zeit, als Lienhard in Kilchberg wohnte (1851–1854). Leemanns Bearbeitung erschien 1898, zwei Jahre später erfolgte ein Neudruck. Lienhards Text wurde für dieses Buch allerdings so stark gekürzt und verändert, dass vom Originaltext nicht viel übrig geblieben ist.
In den USA erschien die erste Teiledition 1941 in einer Bearbeitung von Marguerite E. Wilbur unter dem Titel A Pioneer at Sutter's Fort, 1846–1850. The Adventures of Heinrich Lienhard. Das Buch umfasst Lienhards Aufenthalt in Kalifornien, wobei die Herausgeberin sich vor allem für Sutter und andere bekannte Namen jener Zeit interessierte. Substanzielle Auslassungen von Lienhards persönlichen Erlebnissen und Interessen sowie unzutreffende Verbindungstexte verfälschen den Text an vielen Stellen. Transkriptions- und Übersetzungsfehler sowie die tendenziell kürzende Übersetzung berauben ihn zusätzlich seiner Authentizität. Lienhards Manuskript erfuhr aufgrund von Wilburs Buch in Kalifornien zu Unrecht harsche Kritik, was heute anhand der deutschsprachigen Edition (2010/2011) überprüft und richtiggestellt werden kann.
Im Jahre 1951 veröffentlichten J. Roderic Korns und Dale L. Morgan West from Fort Bridger, eine Untersuchung zum sogenannten «Hastings Cutoff», einem Abschnitt des California Trails. Die Autoren stützten sich dabei unter anderem auf Lienhards tägliche Aufzeichnungen, die sie textgetreu übersetzten und ausführlich kommentieren. Sie bezeichnen seine Beschreibung als «record of the highest importance» und waren damit die Ersten, die Lienhards genaue und verlässliche Art des Berichtens erkannten und ausserordentlich schätzten. 1961 übersetzten und edierten Erwin G. und Elisabeth K. Gudde unter dem Titel From St. Louis to Sutter's Fort den California Trail. Gudde hatte 1942 nach Erscheinen von Wilburs Buch in einer herabsetzenden Kritik Lienhards Glaubwürdigkeit in Zweifel gezogen, allerdings ohne selbst Einsicht in das Manuskript zu nehmen. Dies mag erklären, weshalb zwanzig Jahre später seine Übersetzung des Trails zwar Lienhards Text folgt, aber ziemlich spröde wirkt und der Glarner Humor unverstanden bleibt. Im Vorwort zu seinem Buch bezeichnet Gudde Lienhards Text als «einen der drei klassischen Berichte der grossen Westmigration von 1846».
John C. Abbott edierte im Jahr 2000 das Buch New Worlds to Seek. Es umfasst den ersten Teil des Manuskripts in englischer Übersetzung, nämlich Lienhards Jugend, seine erste Reise nach Amerika und seinen Aufenthalt in Illinois. Im Jahr 2010 edierte Christa Landert knapp die Hälfte von Lienhards Manuskript in deutscher Sprache unter dem Titel «Wenn Du absolut nach Amerika willst, so gehe in Gottesnamen!». Es handelt sich um die Jahre 1846 bis 1849, das heisst den California Trail und Lienhards Aufenthalt in Kalifornien bis zur ersten Reise in die Schweiz.
Zwei ausführliche Zeitungsartikel Heinrich Lienhards wurden unabhängig von seinem Manuskript publiziert. Der erste erschien nach Lienhards Aufenthalt in der Schweiz 1849. Er berichtet darin über Kalifornien, Sutters Fort, die Goldentdeckung und die Arbeit in den Minen, wie er sie aus eigener Erfahrung kannte. Zudem informiert er über die günstigste Reiseroute von der Schweiz nach Kalifornien, damals zweifellos für viele Leser von besonderem Interesse. Im zweiten Artikel, der 1885 im Daily Examiner in San Francisco veröffentlicht wurde, erinnert sich Lienhard unter anderem an seine Arbeit bei Sutter, die Ereignisse rund um die ersten Goldfunde und den darauf folgenden Goldrausch.
Werke
«Wenn Du absolut nach Amerika willst, so gehe in Gottesnamen!», Erinnerungen an den California Trail, John A. Sutter und den Goldrausch, 1846–1849. Herausgegeben von Christa Landert, mit einem Vorwort von Leo Schelbert. Zürich, Limmat Verlag 2010, 2011. ISBN 978-3-85791-504-8 pdf.
Memoirs of trip to California, life at Sutter's Fort and return to Switzerland: ms., 1846–1850. BANC MSS C-D 5024. Bancroft Library, Berkeley.
Schilderungen aus Kalifornien, die Entdeckung des Goldreichthums und dessen Folgen. In: Glarner Zeitung 95–99, 28. November, 1., 5., 8. und 12. Dezember 1849.
The Early Days: Reminiscences of a Pioneer Settler of '46. In: The Daily Examiner (San Francisco), 8 March 1885, p. 1, cols. 1–4.
Erwin G. Gudde and Elisabeth K. Gudde, eds. and transl.: From St. Louis to Sutter's Fort 1846, by Heinrich Lienhard. University of Oklahoma Press 1961.
J. Roderic Korns [and Dale L. Morgan], eds.: West from Fort Bridger: The Pioneering of the Immigrant Trails Across Utah, 1846–1850. Original Diaries and Journals Edited and with Introductions. Salt Lake City: Utah Historical Quarterly, vol. XIX (1951). Revised and Updated by Will Bagley and Harold Schindler, Logan: Utah State University Press, ISBN 0-87421-178-6, 1994.
John C. Abbott, ed.: New Worlds to Seek: Pioneer Heinrich Lienhard in Switzerland and America, 1824–1846. Foreword by John H. Lienhard IV. Carbondale and Edwardsville, Illinois: Southern Illinois University Press, 2000. ISBN 0-8093-2233-1.pdf.
Literatur
Benedikt Erenz: Karl May unplugged. In: Die Zeit, 16. Dezember 2010. (Rezension).
John Paul von Grüningen: An early migration to New Helvetia. In: The Swiss in the United States, Madison 1940, S. 71–87. Im Internet Archive.
Rachel Huber: "General Sutter", die obskure Seite einer Schweizer Heldenerzählung. In: Schweizerische Zeitschrift für Geschichte (SZG) 69 (2019), Nr. 3, S. 418–433. (R. Huber zitiert ausführlich aus den Erinnerungen von Heinrich Lienhard).
Uwe Kossack; Pascal Fischer: SWR2 Forum Buch vom 6. Februar 2011, 17.05 Uhr (Rezension). (Manuskript zur Sendung).
Christa Landert: Heinrich Lienhard von Bilten (1822–1903). Eine biographische Skizze. In: Jahrbuch des Historischen Vereins des Kantons Glarus, Heft 75 (Glarus: Kommissionsverlag Tschudi, 1995): 182–214. Digitalisat auf der Plattform E-Periodica. Englische Version dieses Artikels in: Yearbook of German-American Studies 25 (1990), S. 131–149.
Manfred Papst: Ein junger Glarner erlebt in Amerika den Goldrausch. In: NZZ am Sonntag, 26. Dezember 2010, S. 62. (Rezension).
Weblinks
Einzelnachweise
Geschichte der Vereinigten Staaten (1789–1849)
Schweizer
Geboren 1822
Gestorben 1903
Mann
Johann August Sutter
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Statoil/SDFI merger proposed
Strengthening Statoil with part of or all the government's directly-held petroleum assets (SDFI) and then partially privatising and listing Norway's state oil company has been recommended by its board of directors.
"The government now has a unique opportunity to enhance the value of its overall oil and gas assets, while laying a solid basis for a strong national oil and gas company which will be a world leader," says Statoil chairman Ole Lund.
At the request of the Ministry of Petroleum and Energy, the board has drawn up a report on future development of Statoil and the SDFI - the state's direct financial interest in Norwegian petroleum operations.
This document aims to identify solutions which maximise the state's overall asset value, and which can help to reduce its risk.
Its recommendations can be summarised as follows:
The value of the state's overall oil and gas asssets should be strengthened by merging all or a substantial part of the SDFI with Statoil to create a single commercial entity. This increase in value will find expression in the stock market after the enlarged group has been listed. A combined SDFI and Statoil meets the requirements for creating a strong Norwegian-based oil and gas company in international markets.
Statoil's ownership structure should be changed to provide the group with private shareholders and a stock exchange listing.
A new management ("caretaker") arrangement should be established for that part of the SDFI's resources which might not be reallocated to a commercial company. The board believes that the government should consider refraining from taking interests in new licences.
Commercialising the SDFI by transferring all or part of its portfolio to Statoil, combined with a partial privatisation of the group, would allow the state to realise substantial additional value by offering shares to private investors.
Calculations by Statoil and by Norwegian and international financial advisers show that a Statoil/SDFI would achieve a substantial value increase as a listed company.
Combining the two would provide a size and financial strength which provides opportunities to extend Statoil's leading role in Norway's offshore sector and its positions in international exploration and production as well as in the European gas market.
A commercialisation of the SDFI would thereby provide opportunities for increased value creation because its capital assets would be placed in the context of a commercial company.
This would boost the value of the state's oil and gas assets, secure and develop existing market positions and eliminate conflicts of roles between the government and the players.
"The board has concluded that the combination of transferring the SDFI to a company and partially privatising a strengthened Statoil would give the best financial gain for the state," says Mr Lund.
"Our recommendations identify an opportunity to increase the value of the state's oil and gas assets which also allows part of this wealth to be reallocated. In this way, they will also help to limit the state's exposure to changes in the oil and gas industry."
Both benefit
The board believes that both Statoil and its owner would benefit from a change in the ownership structure to give the group private shareholders and a stock exchange listing.
As a listed company, Statoil would gain the necessary continuous evaluation and supervision from the capital market. A new ownership structure would also normalise the group in relation to its competitors.
At the same time, the division of roles and responsibilities in Norwegian petroleum policy would become clearer since the government would no longer be Statoil's sole owners.
However, the state should retain a shareholding which makes it possible to maintain the group as a competitive Norwegian-based players in international energy markets.
Extensive changes
Statoil's directors note in their report that extensive changes have taken place in the market, which affect Norway as an oil nation and the state as a player in the petroleum sector. These have led to greater competition and higher risk.
The oil market is characterised by increased globalisation and more uncertainty. A prominent feature of today's oil market are the acquisitions and mergers which have made some of the international oil companies even bigger. Clear relationships can be seen between size and development in market value for oil and gas companies.
Deregulation in the European gas market has created a new position, which poses major new challenges for Norway as a gas nation and for the companies as players in a European market.
Further development of the market positions achieved for Norwegian gas operations will be crucial in securing the value of the Statoil and SDFI gas portfolios.
In developing Norwegian petroleum policy, the authorities have shown an ability to adapt instruments to changing conditions and competitive terms.
Today's policy instruments and structures were developed in the 1970s and 1980s, and have functioned as intended. But the board points in its report to the fundamental changes now being experienced by the oil and gas industry, both internationally and on the Norwegian continental shelf.
Oil and gas resources represent a major component in Norway's national wealth, and the state's share of these assets comprise several elements - anticipated tax revenues, the value of direct ownership through the SDFI and the value of companies owned wholly or partly by the state.
Managing these assets in the best possible way calls for policy instruments to be modified in order to maximise their overall value.
The state's position on the Norwegian continental shelf gives it substantial assets and great freedom of action, and the board notes that the government has several options for organising its participation in petroleum operations.
The board concludes that strengthening Statoil with all or a substantial part of the SDFI is the solution which gives the state's oil and gas assets their highest value.
Further information from:
John Ove Lindøe, senior vice president public affairs, tel: +47 51 99 68 81 (office), +47 90 57 20 89 (mobile)
Wenche Skorge, vice president public affairs, tel: +47 51 99 79 17 (office), +47 91 87 07 41 (mobile)
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{"url":"https:\/\/www.flapw.de\/MaX-5.0\/DFT-2019-tut\/densityOfStates\/","text":"1. Density of states\n\nBesides the band structure, the density of states (DOS) also provides a direct view on the electronic structure of a material. It is easy to construct it from a DFT calculation.\n\n1.1. Density of states for Si\n\nWe perform our first DOS calculations for Si. This starts by obtaining a self-consistent density with an 8x8x8 k point mesh\n\n<kPointMesh nx=\"8\" ny=\"8\" nz=\"8\" gamma=\"F\"\/>\n\n\nas it is typically generated with the default parameters (also use the other default parameters). We construct three different density of states on top of this calculation. Consider creating subfolders for each of them and copy the results (including the cdn.hdf file) of your self-consistent calculation in each of them.\n\nTo construct a DOS the inp.xml file has to be modified:\n\n1. output\/@dos has to be set tu \"T\".\n\n2. output\/densityOfStates\/@ndir has to be set to \"-1\".\n\nThe other XML attributes in output\/densityOfStates have the following use: With minEnergy and maxEnergy the energy window for the DOS is specified, with sigma one defines a broadening for each state to obtain a smooth result. By default sigma is 0.015. The three different DOS calculations will use different values for sigma: 0.015, 0.005, 0.0015\n\nPerform the three different DOS calculations on top of the already converged result. The calculations will generate a DOS.1 file, where the \"1\" relates to the spin. In spin-polarized calculations also a DOS.2 file will be generated (and for bandstructure calculations a bands.2 file). The DOS.1 file is a readable text file with several columns. The first colum defines an energy mesh. The second column is the total DOS and afterwards there are multiple columns for the projection of the DOS onto certain regions in the unit cell and onto certain orbital characters around each atom. A detailed description is available in the respective documentation page for the DOS.x file . We are interested in the energy mesh, the total DOS, and the s (column 7) and p (column 8) projections at the Si atoms. It may also be interesting to plot the DOS in the interstitial region (column 3).\n\nGenerate for each of the three calculations single plots for total DOS, s-, and p-projected DOS. The plots should feature on the y axis the energy and on the x axis the DOS. In combination with a bandstructure plot you can partially relate certain bands to the respective orbital character. Comparing the plots with each other you should see results strongly differing with respect to the local Gaussian averaging according to the specified sigma parameter. Obviously it is crucial to find a sigma that provides a good balance between a smoothing of the curves and a resolution of the features of the electronic structure.\n\nOf course, besides the sigma the smoothness of the curves also depends on the k point set. After all the density of states is obtained on the basis of the eigenstates at each k point. The Gaussian averaging is needed because the k point mesh is not infinitely dense. For the smallest sigma (0.0015) we will therefore also test other k point sets. It should be enough if this is performed on top of the already obtained self-consistent density for the coarser k point mesh. Test\n\n<kPointMesh nx=\"23\" ny=\"23\" nz=\"23\" gamma=\"F\"\/>\n\n\nand\n\n<kPointMesh nx=\"47\" ny=\"47\" nz=\"47\" gamma=\"F\"\/>\n\n\nfor the DOS calculation. How does it affect the result, how does it affect the computational demands?\n\n2. Exercises\n\n2.1. Band structure and DOS of a monatomic Cu wire (van Hove singularities)\n\nExperimentalists are capable of producing monatomic wires of certain chemical elements, either on some substrate or free standing wires obtained with break junctions or by pulling scanning tunneling microscope (STM) tips out of a sample. For each energy the conductivity along such a wire is limited by the conductance quantum $G_0 = \\frac{2e^2}{h}$ times the number of bands at the respective energy. Calculating the band structure of such a system therefore provides direct information on its ballistic electron transport properties.\n\nWe perform band structure and density of states calculations for a monatomic Cu wire. For this we set up a tetragonal unit cell with lattice parameters that provide a wide vacuum in two dimensions and the nearest neighbor distance between adjacent Cu atoms in the third dimension. Use $a=12.5~a_0$ and $c=4.82247~a_0$. For the self-consistent density calculation use a k point set of\n\n<kPointMesh nx=\"1\" ny=\"1\" nz=\"201\" gamma=\"F\"\/>\n\n\nFor the band structure calculation we explicitly provide the k point path by specifying \"special k points\" along the path:\n\n<kPointCount count=\"200\" gamma=\"F\">\n<specialPoint name=\"g\">0.0 0.0 0.0<\/specialPoint>\n<specialPoint name=\"X\">0.0 0.0 0.5<\/specialPoint>\n<\/kPointCount>\n\n\nFor the DOS calculation we use a very fine k point mesh such that the expected van Hove singularities at the band edges can easily be identified:\n\n<kPointMesh nx=\"1\" ny=\"1\" nz=\"7001\" gamma=\"F\"\/>\n\n\nThe number of energy mesh points for the DOS is fixed. As a consequence it is a good idea to refine the upper and lower limits of this mesh such that the mesh is not too coarse for our needs. Find out the Fermi energy obtained for the self-consistent density (grep -i fermi out) and adjust the energy window such that it only covers the interesting part of the band structure, e.g., from 0.1 Htr below the Fermi energy to 0.2 Htr above the Fermi energy. The sigma parameter should also be very small. Choose 0.0003.\n\nPlot the \"total DOS\" (column 2), the s- (column 7), and the d-projected (column 9) DOS. The van Hove singularities schould nicely be visible on every lower and upper band edge. Why is the s-projected DOS so much smaller than the d-projected DOS?","date":"2021-04-11 10:34:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 3, \"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\": 0.7259876728057861, \"perplexity\": 1416.6718358038343}, \"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-2021-17\/segments\/1618038061820.19\/warc\/CC-MAIN-20210411085610-20210411115610-00538.warc.gz\"}"}
| null | null |
{"url":"https:\/\/gap-packages.github.io\/permut\/","text":"# permut\n\nA package to deal with permutability in finite groups\n\nVersion 2.0.4\nReleased 2022-03-27\n\nThis project is maintained by Ram\u00f3n Esteban-Romero\n\n# GAP Package permut\n\nThis package provides functions for computing with permutability in finite groups.\n\nThe current version of this package is version 2.0.4, released on 2022-03-27. For more information, please refer to the package manual. There is also a README file.\n\n## Dependencies\n\nThis package requires GAP version 4.7.4\n\nThe following other GAP packages are needed:\n\n## Citing\n\nYou can get more info by typing Cite(\"permut\"); in the gap prompt.","date":"2022-10-03 22:18:26","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\": 0.21376273036003113, \"perplexity\": 8344.89744815625}, \"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\/1664030337432.78\/warc\/CC-MAIN-20221003200326-20221003230326-00552.warc.gz\"}"}
| null | null |
\section{Introduction}
The most known phenomenological model, accounting for many of the
properties of He II, given by {\it Tisza} \cite{Tisza} and {\it
Landau} \cite{Landau} is called the {two-fluid model}. The basic
assumption is that the liquid behaves as a mixture of two fluids:
the normal component with density $\rho_n$ and velocity ${\bf
v}_n$, and the superfluid component with density $\rho_s$ and
velocity ${\bf v}_s$. When the difference ${\bf V}:={\bf v}_n-{\bf
v}_s$ between the normal and superfluid velocities, known also as
counterflow velocity, exceed a certain critical velocity, a {\it
mutual friction} ${\bf F}_{sn}$ has to be included. This friction
force is attributed to an interaction of the normal component with
the vortices in the superfluid.
\par Quantized superfluid vortices play an important role in the
hydrodynamics of the fluid and they have been the object of many
studies. The state of the fluid in which vortices are present, is
referred to as the {\it superfluid turbulent state}. A review on
superfluid turbulence can be found in Tough's paper \cite{Tough}
and in chapter 7 of Donnelly's book \cite{Donnelly}. The quantized
vortices created by applying a thermal counterflow form an
irregular, spatially disordered tangle of lines. In this case, the
vortex line density $L$ (length of vortex line per unit volume) is
$L_H\approx\gamma^2V^2$, where ${\bf V}$ is the modulus of the
relative velocity between the two components of the mixture and
$\gamma$ a temperature-dependent coefficient \cite{Tough}. The
vortex system is almost isotropic, provided that one neglects a
small anisotropy induced by the imposed counterflow \cite{WSD}.
\par The creation of the vortices cannot be made only in this
way; in fact, the first studies of quantized vorticity involved a
sample of He II rotating at constant angular velocity $\Omega$
exceeding a certain small critical value. The results brought to
an ordered array of vortices aligned along the rotation axis,
whose number density per unit area is given by Feynman's rule
$L_R=2\Omega /\kappa$, where $\kappa=h/m=9.97\quad\!\!\! 10^{-4}$
cm$^2/$sec is the quantum of circulation, with $h$ Planck's
constant and $m$ mass of the helium atom.
\par Now, an important question naturally arises: what happens if
vortices are created by both rotation and counterflow? There has
been only one experiment of which we are aware \cite{SBD}, on the
formation of vortices in combined rotation and counterflow along
the rotational axis. This experiment suggests that there exists a
form of steady rotating turbulence, characterized by a vortex line
density at given counterflow velocity ${\bf V}$ and angular
velocity ${\bf \Omega}$. Swanson et al. \cite{SBD} found that at
slow rotation the critical counterflow velocity above which the
flow became turbulent was greatly reduced. The experimental
observations showed that the two effects (thermal counterflow and
rotation) are not merely additive, in fact for $V $ high the
measured values of $L$ are always less than $L_H+L_R $. However,
from our point of view the results of these experiments are purely
qualitative because the authors didn't take the anisotropy of the
vortex tangle in consideration which, as we will see in the last
section of this paper, is essential to know the spatial
distribution of the vortex tangle in liquid Helium II through
measurements of second sound attenuation.
The aim of this work is to study the propagation of longitudinal
density and temperature waves, and longitudinal and transversal
velocity waves and heat waves in the combined situation of a
rotating frame and of a cylindrical container in presence of
thermal counterflow. The studies of the two separate cases of pure
rotation and pure thermal counterflow are also considered in order
to give a more complete view of the wave propagation in these
three different situations. The influence of the parameters
characterizing the vorticity on the propagation of the waves is
shown explicitly. The practical interest of this research is to
obtain information on the vortex tangle from measurements on wave
propagation. This is an important issue, because under the
combined influence of rotation and counterflow the vortex tangle
cannot be assumed isotropic. Then, we must find not only the
vortex line density $L$ but also the geometrical characterization,
which requires, in principle, to consider wave propagation in
different directions, as well as a deeper full analysis of waves.
Note that here the tangle itself is not considered as a dynamical
quantity, because it is not modified by the second sound. For this
reason, evolution equations for the tangle are not needed here.
\par\noindent The plan of this paper is the following: Section 2
is concerned with the model for helium II, in which the use of a
pressure tensor associated to the vorticity has been considered;
in Section 3 and in Section 4 we study wave propagation in
rotating frame and in presence of thermal counterflow
respectively, pure rotation is analyzed in the general case in
which a component of the mutual friction force parallel to the
rotation axis is present; finally, in Section 5 we study wave
propagation in simultaneous rotation and counterflow, analyzing
two different situations about the relative direction of wave
propagation with respect to the rotation vector.
\section{Evolution equations}
Many observations have shown that both thermal conductivity
$\lambda_1$ and the relaxation time of the heat flux $\tau_1$ in
helium II are very high. As observed in \cite{M93} their ratio
$\frac{\lambda_1}{\tau_1}:=\zeta<\infty$ determines the velocity
of the second sound, which is a heat wave propagating in the
superfluid. As a consequence, it is natural to use a
thermodynamical theory where the heat flux $\bf q$ appears as a
further fundamental field. In this way, a linear macroscopic
one-fluid model of liquid helium II, based on Extended
Thermodynamics \cite{JCL, MR}, has been formulated \cite{M93}.
This model is able to describe the laminar flow of the superfluid
both in the presence and in absence of dissipative phenomena and
to predict the propagation of the two sounds in bulk liquid helium
II and of the fourth sound in liquid helium flowing in a porous
medium \cite{M93}, \cite{MPz}-\cite{MPJ}, in agreement with
microscopic and experimental data.
\par
In order to describe the presence of vortices in rotating helium
II, in superfluid turbulence or in combined rotation and thermal
counterflow, the use of a further additional pressure tensor ${\bf
P}_\omega$, associated to the vorticity, is necessary. The
simplified situations of a rotating frame and of pure thermal
counterflow have been considered in \cite{JLM}, where a
constitutive relation for ${\bf P}_\omega$ and its influence on
the dynamics of the heat flux has been studied.
In this work the more complex situation involving thermal
counterflow in a rotating cylinder, which is receiving much
attention recently \cite{BDV}-\cite{Tsubota2004}, is considered
too. We start from a linear macroscopic one-fluid model of liquid
helium II, whose fundamental fields are the density $\rho$, the
barycentric velocity ${\bf v}$, which is related to the two
velocities of the two-fluid model by the relation $\rho {\bf
v}=\rho_s {\bf v}_s+\rho_n {\bf v}_n$, the temperature $T$ and the
heat flux ${\bf q}$, related to the counterflow velocity ${\bf V}$
by the relation ${\bf q}=\rho_s T s {\bf V}$ (where $s$ is the
entropy of the Helium II). In the two-fluid model the natural
variables are ${\bf v}_s$ and ${\bf v}_n$, but in the experiments
it is ${\bf v}$ and ${\bf q}$ which are directly measured.
Therefore, the use of ${\bf v}$ and ${\bf q}$ appears suitable for
our analysis. Neglecting the bulk and shear viscosity and under
the hypothesis of small thermal dilatation (which in helium II are
indeed very small), the linearized system of field equations for
liquid helium II, in a non inertial frame, in absence of external
force, is \cite{JLM}:
\begin{equation}
\begin{cases}
\frac{\partial \rho}{\partial t}+\rho\frac{\partial v_j}{\partial
x_j }=0\cr \rho\frac{\partial v_i}{\partial t}+\frac{\partial
p}{\partial x_i}+{\bf i}^0_i+2\rho\left({\bf \Omega}\wedge {\bf
v}\right)_i=0\cr \frac{\partial T}{\partial t}+\frac{1}{\rho
c_V}\frac{\partial q_j}{\partial x_j}=0\cr \frac{\partial
q_i}{\partial t}+\zeta\frac{\partial T}{\partial x_i}+2\left({\bf
\Omega}\wedge{\bf q}\right)_i=\left({\vec \sigma}_\omega\right)_i
=-\left({\bf P}_\omega\cdot{\bf q}\right)_i.
\end{cases}\label{1}
\end{equation}
In this system, ${\bf i}^0+2\rho\left({\bf \Omega}\wedge {\bf v}\right)$ is the
inertial force, $\zeta$ is a positive coefficient linked to the
second sound velocity, and:
\begin{equation}
p=p_E(\rho,T)\quad \mbox{and}\quad c_V=\left(\frac{\partial \epsilon
(\rho ,T) }{\partial T}\right)_\rho
\end{equation}
are the thermostatic pressure and the specific heat respectively
($\epsilon$ is the specific internal energy). The effect of
vortices is described by incorporating the source term ${\bf
P}_\omega\cdot{\bf q}$ to the evolution equation of the heat flux.
As we will see, the expression of ${\bf P}_\omega$ will assume
different expressions in the different situations considered.
Now, a small comparison between the one-fluid model and the
two-fluid model could be useful. With the corresponding
transformations between the natural variables in the one fluid
model, ${\bf v}$ and ${\bf q}$, and those in the two-fluid model,
${\bf v}_s$ and ${\bf v}_n$, the evolution equations (\ref{1}) of
the one-fluid model are equivalent to those of the two-fluid model
in the linear approximation \cite{JLM}. A formal difference is
found in the form of the production term in the evolution equation
for the heat flux (\ref{1}d). When specified to pure rotation,
this production term, as given by (\ref{sigma2}), has the usual
Hall-Vinen form, whereas when specified to counterflow, the
production term, as given by (\ref{4.1}), yields the well-known
Gorter-Mellinck form. These two situations have been well explored
in the context of the two-fluid and one-fluid frameworks. In the
combined situation with simultaneous rotation and counterflow, the
general form of the production term in (\ref{1}d) is especially
useful, as expressed in (\ref{pesp2}) and (\ref{5.38}), because it
allows one to write in an explicit and appealing way the
anisotropy of the tangle, whose influence on the second sound is
one of our main concerns. Given the same geometrical conditions
for the tangle --- which here are given a priori, and whose form
is probed by means of second sound ---, the evolution equations of
the two-fluid model would coincide with those of the one-fluid
model. Thus, the dispersion relations obtained here should be
valid also in the context of the two-fluid model.
The one-fluid and the two-fluid models are not identical to each
other. However, their mutual differences arise in contexts which
are not relevant in the analysis presented here. For instance, one
difference arises in the fourth sound in helium through porous
media, in which some experimental results seem to support the
one-fluid model \cite{Mongiovi}. Anyway, the two-fluid model could
also cope with that situation provided the assumption that the
superfluid component carries no entropy is slightly relaxed by
assuming that it may carry a small but nonvanishing entropy. Other
differences arise concerning the interaction between second sound
and the vortex tangle. Here, we have assumed that second sound
does not modify the vortex line density nor the geometrical
structure of the tangle. If it is assumed that it may modify the
vortex tangle, more general evolution equations would be needed,
as for instance an evolution equation for the vortex line density
$L$ coupled with the rotation and counterflow, which have already
been explored in the literature \cite{MonJou07}. For instance,
the evolution equation for $L$ could be different --- a
generalized form of Vinen's equation with the mentioned couplings
has been proposed and studied \cite{JM04} --- but this is not
relevant here because an equation for $L$ is not necessary in this
paper, as $L$ is taken as fixed, and its value must be found from
wave experiments. Some other differences may appear, concerning,
for instance, the possibility of vortex density waves at high
frequencies in the one-fluid model \cite{JMS} that do not arise in
the Hall-Vinen-Bekarevich-Khalatnikov model \cite{Henderson}.
Since in this work we are focusing our attention to a situation in
which the interaction between the second sound and the tangle does
not distort the vortex lines nor the vortex density, the
dispersion relations obtained in this paper by using the
production terms ${\bf P}_\omega \left({\bf q}, {\bf
\Omega}\right) \cdot{\bf q}$ would be also valid in the two-fluid
context by using a production term of the form ${\bf P}_\omega
\left({\bf v}_n-{\bf v}_s, {\bf \Omega}\right) \cdot ({\bf
v}_n-{\bf v}_s)$ in an evolution equation for the relative
velocity ${\bf V}={\bf v}_n-{\bf v}_s$.
\section{Wave propagation in rotating frame}
\setcounter{equation}{0}
We generalize here the results of \cite{JLM} to the case in which
a small interaction between second sound and vortex line parallel
to the rotation axis is present. In \cite{HV56}, Hall and Vinen
described experiments of liquid Helium II in a rotating frame,
showing the main effects on the propagation and attenuation of the
second sound as a consequence of the interaction between
quasi-particles and vortex lines: these interactions are mainly
present in the planes orthogonal to the rotation axis. As
consequence of these experiments, in \cite{JLM}, Jou, Lebon and
Mongiov\`{\i} proposed an expression for the production term
${\vec{\sigma} }_{\omega}$ in (\ref{1}d), which takes into account
dissipative and non dissipative contributions of the interaction
between quasi-particles and vortex lines, but they did not
consider interactions parallel to the rotation axis.
In another experiment \cite{Snyder}, Snyder studied the component
of mutual friction along the rotational axis, and his result, in
agreement with \cite{BTMA2}, shows that this friction component
is very small compared with the orthogonal components but not
exactly zero. In this section, we consider the most general case
in which the axial component is included. In order to do that, the
following vorticity tensor ${\bf P_\omega}$ is used \cite{JM05}:
\bea \label{Pomega} {\bf P}_\omega^R=\frac{1}{2}\kappa L_R
\left[(B-B'')\left({\bf U}-{\bf \hat{\Omega} \hat{\Omega}}\right)+
B' {\bf W\cdot \bf \hat{\Omega}}+2B''\bf
\hat{\Omega}\hat{\Omega}\right],\eea where ${\bf U}$ is the unit
matrix, ${\bf W}$ the Ricci tensor, and $B$ and $B'$ are the
Hall-Vinen coefficients \cite{HV56} describing the orthogonal
dissipative and non dissipative contributions while $B''$ is the
friction coefficient along the rotational axis. Using the Eq.
(\ref{Pomega}), the production term in (\ref{1}d) can be expressed
as \cite{Donnelly,JM05}: \bea \label{sigma2} {
\vec{\sigma}}_\omega^R=\frac{1}{2}\kappa L_R \left[(B-B''){\bf
\hat{\Omega}}\wedge\left({\bf \hat{\Omega}}\wedge{\bf q}\right)+
B'{\bf \hat{\Omega}}\wedge {\bf q}-2B''\bf \hat{\Omega}\bf
\hat{\Omega}\cdot \bf q \right]. \label{2.4}\eea \\
The interest to consider spatial distribution of vortices and
anisotropy of mutual friction in rotating container has led
Mathieu et al. in \cite{MPS2} to analyze a more general case in
which a parallelepipedic cavity filled of helium II rotates around
an axis tilted an angle $\theta$ with respect to its wall. In the
following Subsection we will show that the results of the latter
experiments can be easily explained using the
general expression (\ref{sigma2}). \\
Substituting the expression (\ref{2.4}) into the system (\ref{1})
and choosing ${\bf \Omega}=\left(\Omega,0,0\right)$, the system
assumes the following form:
\begin{equation}
\begin{cases}
\frac{\partial \rho}{\partial t}+\rho\frac{\partial v_j}{\partial
x_j }=0\cr \rho\frac{\partial v_i}{\partial t}+\frac{\partial
p}{\partial x_i}+2\rho \Omega v_j\epsilon_{1ji}=0\cr \frac{\partial
T}{\partial t}+\frac{1}{\rho c_V}\frac{\partial q_j}{\partial
x_j}=0\cr \frac{\partial q_i}{\partial t}+\zeta\frac{\partial
T}{\partial x_i}+\left(2\Omega-\frac{1}{2}B' \kappa L_R \right)
q_j\epsilon_{1ji}=\frac{1}{2}\kappa
L_R[(B-B'')\left(-q_i+q_1\delta_{i1}\right)-2B'' q_1 \delta_{i1}],
\end{cases}\label{1.1}
\end{equation}
where $\epsilon_{kji}$ is the Ricci tensor.
\par
\noindent It is easily observed that a stationary solution of
this system is: \be \rho=\rho_0,\quad\!\!\! {\bf v}={\bf
0},\quad\!\!\! T=T_0,\quad\!\!\!{\bf q}={\bf 0}. \ee
\par
In order to study the propagation of plane harmonic waves of small
amplitude \cite{Whitham}, we put $\Gamma=(\rho, v_i, T, q_i)$, and
we look for solutions of the linearized system of field equations
(\ref{1}) of the form:
\begin{equation}
{\Gamma=\Gamma_0+\tilde \Gamma e^{i(Kn_jx_j-\omega t)}}, \label{1s1}
\end{equation}
where {$\Gamma_0=(\rho_0, 0, T_0, 0)$} denotes the unperturbed state,
{$\tilde \Gamma=\left(\tilde \rho, \tilde v_i, \tilde T, \tilde
q_i\right)$} small amplitudes whose products can be neglected,
{$K=k_r+ik_s$} is the wavenumber, {$\omega =\omega_r+i\omega_s$}
the frequency and {${\bf n}=(n_i)$} the unit vector orthogonal to
the wave front. Along this paper we will assume that the
propagating waves do not affect the vortex tangle, i.e. that they
do not contribute to the production nor the destruction of
vortices. In other terms, the waves are used to explore a given
vortex tangle, without modifying it. If the wave amplitude is high
enough, it could yield new contributions to the tangle.
\par
In the following we assume that $\Omega$ is small, so that the
term ${\bf i}_0$ in (\ref{1}b) can be neglected. For the sake of
simplicity, the subscript $0$, which denotes quantities referring
to the unperturbed state $\Gamma_0$, will be dropped out.
\par
\subsection{First case: ${\bf n}$ parallel to ${\bf \Omega}$}
In this subsection we analyze the case in which the unit vector
${\bf n}$ orthogonal to the wave front is parallel to the axis of
rotation, i.e. {${\bf n}=(1, 0, 0)$}. Substituting (\ref{1s1})
into the linearized system (\ref{1.1}) and letting {${\bf
t}_1=(0,0,1)$} and {${\bf t}_2=(0,1,0)$} as unit vectors tangent
to the wave front, the following homogeneous algebraic linear
system for the small amplitudes is obtained:
\begin{equation}\label{rotnpo}
\begin{cases}
-\omega \tilde \rho + \rho K \tilde v_1=0 \cr -\omega\tilde v_1
+K\frac{p_\rho}{\rho} \tilde \rho=0
\cr
-\omega \tilde T+\frac{K}{\rho c_V}\tilde q_1=0
\cr (-\omega-iB''\kappa L_R)\tilde q_1+\zeta K\tilde T =0
\cr \cr -\omega\tilde
v_3-2i \Omega\tilde v_2=0 \cr -\omega\tilde v_2+2i\Omega\tilde v_3=0
\cr\cr
\left(-\omega-\frac{i}{2}\kappa L_R (B-B'')\right)\tilde
q_3-i\left(2\Omega-\frac{1}{2}\kappa L_R B'\right)\tilde q_2=0 \cr
\left(-\omega-\frac{i}{2}\kappa L_R(B-B'')\right)\tilde
q_2+i\left(2\Omega-\frac{1}{2}\kappa L_R B'\right)\tilde q_3=0.
\end{cases}
\end{equation}
From the above system, it follows that longitudinal and
transversal modes evolve independently. The study of the
longitudinal modes furnishes the existence of two waves: the first
is known as {\it first sound} or {\it pressure wave} in which
density and velocity vibrate, and the second is known as {\it
second sound} or {\it temperature wave} in which temperature and
heat flux vibrate. Therefore, as observed in \cite{Snyder}, when
the wave is propagated parallel to the rotation axis, the
longitudinal modes are influenced by the rotation only through the
axial component of the mutual friction ($B''$ coefficient). In
fact, the first two equations of the system (\ref{rotnpo}), give
for the first sound $V_1:=\frac{\omega}{K}= \sqrt{p_\rho}$,
whereas the third and fourth equation, with the assumption
$K=k_r+ik_s$ and $\omega$ real, give second sound waves with the
following velocity and attenuation: \be \label{3.8}
w_2:=\frac{\omega}{k_r}= \sqrt{\frac{4 V_2^4 k_r^2}{4 V_2^2
k_r^2+B''^2 \kappa L_R^2}}\quad \textrm{and} \quad k_s=\frac{w_2
B'' \kappa L_R}{2 V_2^2}\ee where $V_2^2:=\frac{\zeta}{\rho c_V}$
is the velocity of the second sound in the absence of vortices.
Therefore, the following fields vibrate respectively:
\begin{center}
\begin{tabular}{|l|l|}
{$\omega_{1,2}=\pm k V_1$}&{$\omega_{3,4}=\pm \sqrt{\frac{4 V_2^4
k_r^4}{4
V_2^2 k_r^2+B''^2 \kappa L_R^2}}$} \\
\hline\hline & \\
{$\tilde \rho=\psi $}&{$\tilde \rho=0$}\\
{$\tilde v_1=\pm \frac{V_1}{\rho}\psi$}&{$\tilde v_1=0$} \\
{$\tilde T_0=0 $ } &{$\tilde T=T_0\psi $}\\
{$\tilde q_1=0 $ } &{$\tilde q_1=\pm \rho c_V T_0 \sqrt{\frac{4
V_2^4 k_r^4}{4
V_2^2 k_r^2+B''^2 \kappa L_R^2}} \ \psi$}\\
\end{tabular}
\end{center}
\par On the contrary, the transversal modes are influenced by the
rotation. In fact, by considering the fifth and the sixth equation
of (\ref{rotnpo}) they admit nontrivial solutions if and only if
its determinant vanishes; this yields $\omega_{5, 6}=\pm
2|\Omega|$.
\par
Now, we consider the equations seven and eight of the system
(\ref{rotnpo}) and, as above, we find the following dispersion
relation: \be \left(2\Omega-\frac{1}{2}\kappa L_R B'\right)^2-
\left(-\omega-\frac{i}{2}\kappa L_R (B-B'')\right)^2
=0,\label{rd0} \ee whose solutions are \be\label{3.16}
\omega_{7,8}=\pm (2\Omega-\frac{1}{2}\kappa L_R
B')-\frac{i}{2}\kappa L_R (B-B'').
\ee
These transversal modes are influenced from both
dissipative and nondissipative contributions $B$, $B'$ and $B''$ in the
interaction between quasi-particles and vortex lines.
\subsection{Second case: ${\bf n}$ orthogonal to ${\bf \Omega}$}
In this subsection we assume that the direction of propagation of
the waves is orthogonal to the rotation axis, i.e. for example,
${\bf n}=\left(0, 1, 0\right)$. The unit vectors tangent to the
wave front are ${\bf t}_1=\left(1,0,0\right)$ and ${\bf
t}_2=\left(0,0,1\right)$. Under these assumptions, substituting
(\ref{1s1}) into the linearized system (\ref{1.1}), the following
system is obtained:
\begin{equation}\label{rotnparo}
\begin{cases}
-\omega \tilde \rho + \rho K \tilde v_2=0 \cr -\omega\tilde v_2
+K\frac{p_\rho}{\rho} \tilde \rho+2i\Omega\tilde v_3=0
\cr
-\omega\tilde
v_3-2i\Omega\tilde v_2=0 \cr \cr
-\omega \tilde T+\frac{K}{\rho c_V}\tilde q_2=0
\cr \left(-\omega-\frac{i}{2}\kappa L_R (B-B'')\right)\tilde q_2+\zeta K\tilde
T+
i(2\Omega-\frac{1 }{2}\kappa L_R B')\tilde q_3=0
\cr\left(-\omega-\frac{i}{2}\kappa L_R (B-B'')\right)\tilde
q_3-i\left(2\Omega-\frac{1}{2}\kappa L_R B'\right)\tilde q_2=0\cr\cr
-\omega\tilde v_1=0\cr (-\omega-iB''\kappa L_R)\tilde
q_1=0.\end{cases}
\end{equation}
In this case, the longitudinal and transversal modes do not evolve
independently. The first sound is coupled with one of the two
transversal modes in which velocity vibrates; while the second
sound is coupled with a transversal mode in which heat flux
vibrates.
Studying the first three equations of the system (\ref{rotnparo}),
we obtain a dispersion relation whose solutions are: \bea \omega_1&=&0,\label{sol2b}\\
w_{2,3}&=& \pm
V_1\sqrt{\left(1-4\frac{\Omega^2}{\omega_{2,3}^2}\right)^{-1}}.\label{sol2}
\eea Summarizing:
\begin{center}
\begin{tabular}{|l|l|}
{$\omega_{1}=0$ }& {$\omega_{2,3}\simeq\pm K V_1+O(\Omega^2)$} \\
\hline\hline &
\\
{$\tilde \rho=\psi $} & {$\tilde \rho=\psi $}\\
{$\tilde v_2=0$} & {$\tilde v_2=\frac{\pm V_1}{\rho}\psi$}\\
{$\tilde v_3=i\frac{K V_1^2}{2\Omega\rho}\psi$} & {$\tilde v_3=-\frac{2i\Omega}{\rho K}\psi$}\\
\end{tabular}
\end{center}
The second three equations admit non trivial solutions if and only
if their determinant vanishes. Neglecting the second-order terms
in $\Omega$, the dispersion relation becomes: \be \label{relaz}
\left(-\omega-\frac{i}{2}\kappa L_R
(B-B'')\right)\left[-\omega\left(-\omega-\frac{i}{2}\kappa
L_R(B-B'')\right)-K^2V_2^2\right]=0 \ee For $\omega\in \Re$ and
$K=k_r+ik_s$ complex, one gets the solution $\omega_4=0$, which
represents a stationary mode; and two solutions which furnish the
following phase velocity and attenuation coefficient of the
temperature wave: \bea && w_2^2:=\frac{\omega^2}{k_r^2}=
V_2^2\frac{2}{1+\sqrt{1+\frac{(B-B'')^2\kappa^2 L_R^2}{4\omega^2}}},\\
&& k_s= \frac{(B-B'')\kappa L_R w_2}{4V_2^2}.\eea The approximated
solutions to second order in $\frac{(B-B'')\kappa L_R}{\omega}$
are: \bea &&
w_2\simeq V_2\left(1-\frac{(B-B'')^2\kappa^2 L_R^2}{32\omega^2}\right)+O\left(\frac{(B-B'')^4\kappa^4 L_R^4}{\omega^4}\right),\\
&& k_s\simeq \frac{(B-B'')\kappa
L_R}{4V_2}+O\left(\frac{(B-B'')^3\kappa^3 L_R^3}{\omega^2}\right)
\eea
Summarizing, when the direction of propagation of the waves is
orthogonal to the rotation axis, the temperature wave experiences
a strong attenuation, which grows with $\Omega$. The corresponding
modes are:
\begin{center}
\begin{tabular}{|l|l|}
$\omega_{4}=0$& {$\omega_{5,6}\simeq\pm k_r V_2\left(1-\frac{(B-B'')^2\kappa^2 L_R^2}{32\omega^2}\right)$} \\
\hline\hline
& \\
{$\tilde T=-\frac{i(2\Omega-\frac{1}{2}\kappa L_R B')}{\zeta K}\psi$}& {$\tilde T=T_0\psi $}\\
{$\tilde q_2=0$} & {$\tilde q_2=\frac{T_0\zeta}{V_2}\left(1-\frac{(B-B'')^2\kappa^2 L_R^2}{32\omega^2}\right)\psi$}\\
{$\tilde q_3=\psi$} & {$\tilde
q_3=\frac{i\left(2\Omega-\frac{1}{2}\kappa L_R B'\right)
T_0\zeta\left(1-\frac{(B-B'')^2\kappa^2 L_R^2}{32\omega^2}\right)}
{V_2\left[\pm k_r V_2\left(1-\frac{(B-B'')^2\kappa^2 L_R^2}{32\omega^2}\right)-\frac{i}{2}(B-B'')\kappa L_R\right]}\psi$}\\
\end{tabular}
\end{center}
We note that in the mode $\omega_4=0$, only
the transversal component of the heat flux is involved.\\
\noindent For $\omega=\omega_r+i\omega_s$ complex and $K\in \Re$,
the solutions of dispersion relation (\ref{relaz}) are:
\begin{eqnarray*} \omega_4
&=&-\frac{i}{2}(B-B'')\kappa L_R,\\
\omega_{5,6} &=& \pm
\sqrt{K^2V_2^2-\frac{1}{16}(B-B'')^2\kappa^2L_R^2}-i\frac{(B-B'')\kappa
L_R}{4}.
\end{eqnarray*}
The first mode, with $\omega_4=-\frac{i}{2}(B-B'')\kappa L_R$,
corresponds to an extremely slow relaxation phenomenon involving the
temperature wave and the transversal component of the heat flux:
\begin{center}
\begin{tabular}{|l|}
{$\omega_{4}=-\frac{i}{2}(B-B'')\kappa L_R$} \\
\hline\hline
\\
{$\tilde T=-\frac{i(2\Omega-\frac{1}{2}\kappa L_R B')}{\zeta K}\psi$}\\
{$\tilde q_2=0$}\\
{$\tilde q_3=\psi$}\\
\end{tabular}
\end{center}
which when $\Omega\rightarrow 0$, converges to a stationary mode.
The attenuation in $\omega_{5,6}$ (corresponding to $q_1$ and
$q_2$) is physically reasonable in view of (\ref{2.4}), where it
is seen that for ${\bf q}$ parallel to ${\bf \Omega}$ the only
component of the friction force is the axial one (related to the
coefficient $B''$), whereas for ${\bf q}$ orthogonal to the vortex
line (i.e. to ${\bf \Omega}$) there is an attenuation dependent on
the dissipative coefficient ($B-B''$).
\section{Wave propagation in presence of thermal counterflow}
\setcounter{equation}{0} In this section, we study wave
propagation in presence of pure thermal counterflow in liquid
Helium II to compare the results with those of \cite{JLM} and
those of Section 5. Let us consider a flow channel that connects
two He II reservoirs (as shown in fig. 1). When a steady heat is
applied to one end of the channel, there exists a temperature
difference $\Delta T$ between the two ends. From the microscopic
point of view using the two-fluid model, since only the normal
fluid component carries entropy and heat flow, it will move away
from the heat source (left reservoir) to the right reservoir and
then give up the heat. At the same time, the superfluid component
must counter-flow from right to left to conserve the mass. When it
arrives at the left reservoir, part of the superfluid component
will be converted to normal fluid by absorbing heat. Thus, a
relative counterflow between the normal fluid and superfluid
components is established, and this internal convection process is
termed thermal counterflow, which is associated to the heat flux
${\bf q}$ through the relation ${\bf q}=\rho_s T s \bf V$.
\par\noindent In this case, assuming that the vortex tangle caused
by the counterflow is isotropic, the vorticity tensor ${\bf
P}_\omega$, as indicated in \cite{JLM}, takes the following form:
\bea\label{4.1} {\bf
P}_\omega^H= \frac{1}{3} \kappa B L {\bf U} \ \ \ \ \Rightarrow \
\ \ \ {\vec{\sigma}}_\omega^H=-\frac{1}{3}\kappa B L{\bf q}, \eea
where $L=\gamma^2q^2$. Under this assumption, the linearized set
of field equations read as:
\begin{equation}\label{sist}
\begin{cases}
\frac{\partial \rho}{\partial t}+\rho\frac{\partial v_j}{\partial
x_j }=0\cr \rho\frac{\partial v_i}{\partial t}+\frac{\partial
p}{\partial x_i}=0\cr \frac{\partial T}{\partial t}+\frac{1}{\rho
c_V}\frac{\partial q_j}{\partial x_j}=0\cr \frac{\partial
q_i}{\partial t}+\zeta\frac{\partial T}{\partial
x_i}=-\frac{1}{3}\kappa B Lq_i
\end{cases}
\end{equation}
A stationary solution of the system (\ref{sist}) is \cite{JLM}:
\be \rho=\rho_0,\quad\!\!\! \dot{\bf v}={\bf 0},\quad\!\!\!
T=T(x)=T_0-\frac{\kappa B L}{3\zeta}q_0 x,\quad\!\!\!{\bf q}={\bf
q}_0 \ee where $x$ is the direction of the heat flux ${\bf q}={\bf
q}_0$. In order to study the propagation of harmonic plane waves
in the channel, we look for solutions of the system (\ref{sist})
of the form:
\begin{equation}
{\Gamma=\Gamma_0+\tilde \Gamma e^{i(Kn_jx_j-\omega t)}}, \label{1s}
\end{equation}
where {$\Gamma_0=(\rho_0, 0, T(x), {\bf q}_0)$}, and
the following homogeneous algebraic linear system for the small
amplitudes is obtained:
\begin{equation}
\begin{cases}
-\omega\tilde\rho+\rho K\tilde{v}_jn_j=0\cr
-\rho\omega\tilde{v}_i+p_\rho K\tilde\rho n_i=0\cr -\omega\tilde T
+\frac{K}{\rho c_V}\tilde q_jn_j=0\cr
\left(\omega+\frac{1}{3}i\kappa B L\right)\tilde q_i-\zeta K\tilde T
n_i=0.
\end{cases}\label{2b}
\end{equation}
The longitudinal modes are obtained projecting the vectorial
equations for the small amplitudes of velocity and heat flux on
the direction orthogonal to the wave front. It is observed that
the first sound is not influenced by the thermal counterflow,
while the velocity and the attenuation of the second sound are
influenced by the presence of the vortex tangle. The results are:
\[w_1 = \pm
\sqrt{p_\rho}
\]
with $p_\rho$ standing for $\pard p/\pard \rho $ and: \beas
w_2&=&V_2\sqrt{\left(1+\frac{k_s^2V_2^2}{\omega^2}\right)^{-1}}
\quad\Rightarrow\quad w_2\simeq
V_2\left(1-k_s^2\frac{V_2^2}{2\omega^2}\right),\\
k_s&=&\frac{1}{6}\kappa B L w_2.\eeas These results generalize
those of \cite{JLM} where the terms in $k_s^2$ have been
neglected.
\par
The transversal modes are obtained projecting the vectorial
equations for the small amplitudes of velocity and heat flux on the
wave front, obtaining:
\begin{equation}
\begin{cases}
-\omega\tilde{v}_\pi=0\cr \left(-\omega+\frac{i}{3}\kappa B
L\right)\tilde q_\pi=0
\end{cases}\label{2c}
\end{equation}
where $\pi$ denotes the tangential plane to the wave front. The
solutions of this equation are: $\omega_{1}=0$ and
$\omega_2=\frac{i}{3}\kappa B L$. The first mode ($\omega_{1}=0$)
is a stationary mode.
\section{Wave propagation with simultaneous rotation and counterflow}
\setcounter{equation}{0} The combined situation of rotation and
heat flux (as shown in fig. 2), is a relatively new area of
research \cite{JM04}-\cite{Tsubota2004}. The first motivation of
this great interest is that from the experimental observations one
deduces that the two effects are not merely additive; in
particular, for ${\bf q}$ or ${\bf \Omega}$ high, the measured
values of $L$ are always less than $L_H+L_R $.
Under the simultaneous influence of thermal counterflow {\bf $V$}
and rotation speed {\bf $\Omega$}, rotation produces an ordered
array of vortex lines parallel to rotation axis, whereas
counterflow velocity causes a disordered tangle. In this way the
total vortex line is given by the superposition of both
contributions so that the vortex tangle is anisotropic
\cite{JM05}, \cite{JMon}. Therefore, assuming that the rotation is
along the $x$ direction ${\bf \Omega}=\left(\Omega, 0, 0\right)$
and isotropy in the transversal $(y-z)$ plane, for the vorticity
tensor ${\bf P}_\omega$, in combined situation of counterflow and
rotation, the following explicit expression is taken:
\be \label{pesp2} {\bf P}_\omega=\gamma \kappa
L\left\{\frac{2}{3}(1-D){\bf U}+D
\left[\left(1-\frac{B''}{B}\right)\left({\bf U}-{\bf
\hat{\Omega}}{\bf \hat{\Omega}}
\right)+\frac{B'}{B}{\bf W}\cdot{\bf \hat{\Omega}}+2\frac{B''}{B}{\bf \hat{\Omega}}{\bf \hat{\Omega}}\right]\right\} \ee
where $\gamma$ is linked to the coefficient $B$ through the
relation $\gamma=B/2$ and $D$ is a parameter between $0$ and $1$
related to the anisotropy of vortex lines, describing the relative
weight of the array of vortex lines parallel to $\bf \Omega$ and
the disordered tangle of counterflow (when $D=0$ we recover an
isotropic tangle -- Eq. (\ref{4.1}), whereas when $D=1$ the
ordered array -- Eq. (\ref{Pomega})). Assuming
$b=\frac{1}{3}(1-D)+\frac{DB''}{B}$ and $c=\frac{B'D}{B}$, the
vorticity tensor (\ref{pesp2}) can be written as: \be\label{5.38}
{\bf P}_\omega=\gamma \kappa L\left\{\left(\begin{matrix}
2b&0&0\\
0&1-b&0\\
0&0&1-b
\end{matrix}\right)+\left(\begin{matrix}
0&0&0\\
0&0&c\\
0&-c&0
\end{matrix}\right)\right\}. \ee
Note that the isotropy in the $y-z$ plane may only be assumed when
both ${\bf \Omega}$ and ${\bf V}$ are directed along the $x$ axis. A
more general situations is yet an open topic.
\par\noindent Substituting the expression
(\ref{5.38}) into the linearized set of field equations (\ref{1}),
it assumes the following form:
\begin{equation}\label{siste1}
\begin{cases}
\frac{\partial \rho}{\partial t}+\rho\frac{\partial v_j}{\partial
x_j }=0\cr \rho\frac{\partial v_i}{\partial t}+\frac{\partial
p}{\partial x_i}+2\rho\Omega v_j\epsilon_{1ji}=0\cr \frac{\partial
T}{\partial t}+\frac{1}{\rho c_V}\frac{\partial q_j}{\partial
x_j}=0\cr \frac{\partial q_i}{\partial t}+\zeta\frac{\partial
T}{\partial x_i}+2\Omega q_j\epsilon_{1ji}=-\gamma\kappa L\left\{2b
q_1\delta_{1i}+\left[\left(1-b\right)q_2+cq_3\right]\delta_{2i}+
\left[\left(1-b\right)q_3-cq_2\right]\delta_{3i}\right\}
\end{cases}
\end{equation}
A stationary solution of this system is: \beas
&&\rho=\rho_0,\quad\!\!\! \dot{\bf v}={\bf 0},\quad\!\!\!{\bf
q}={\bf q}_0\equiv\left(q_{0}, 0, 0\right)\\
&&\!\!\!\!\!\!\!T=T(x_i)=T_0-2\frac{\gamma\kappa
L}{\zeta}bq_{0}\delta_{1i}x_i. \eeas
\par In order to study the propagation of harmonic plane waves,
we look for solutions of (\ref{siste1}) of the following form:
\begin{equation}
{\Gamma=\Gamma_0+\tilde \Gamma e^{i(Kn_jx_j-\omega t)}}, \label{1s}
\end{equation}
where {$\Gamma_0=(\rho_0,\quad\!\!\!\!\! 0,\quad\!\!\!\!\! T(x_i),\quad\!\!\!\!\! {\bf
q}_0)$} and $T(x_i)$ is a linear function of $x_i$.
Now, we investigate two different cases: ${\bf n}$ parallel to
${\bf \Omega}$ and ${\bf n}$ orthogonal to ${\bf \Omega}$; the
latter is the only case for which experimental data exist
\cite{SBD}.
\subsection{First case: ${\bf n}$ parallel to ${\bf
\Omega}$} In this subsection we analyze the case in which the unit
vector ${\bf n}$ orthogonal to the wave front is parallel to the
direction of the rotation, i.e. {${\bf n}=(1, 0, 0)$}. Letting
{${\bf t}_1=(0,1,0)$} and {${\bf t}_2=(0,0,1)$} as unit vectors
tangent to the wave front, the system (\ref{siste1}) for the small
amplitudes (\ref{1s}) is: \be
\begin{cases}
-\omega\tilde\rho+K\rho\tilde v_1=0\cr -\omega\tilde
v_1+K\frac{p_\rho}{\rho}\tilde\rho=0\cr -\omega\tilde
T+\frac{K}{\rho c_V}\tilde q_1=0\cr \left[-\omega -2i\gamma \kappa
Lb\right]\tilde q_1 +\zeta K\tilde T=0\cr \cr -\omega\tilde
v_2+2i\Omega\tilde v_3=0 \cr -\omega\tilde v_3-2i\Omega\tilde v_2=0
\cr\cr \left[-\omega-i\gamma \kappa L\left(1-b\right)\right]\tilde
q_2+\left(2i\Omega -i\gamma\kappa L c\right)\tilde q_3=0
\cr
\left[-\omega-i\gamma \kappa L\left(1-b\right)\right]\tilde
q_3-\left(2i\Omega -i\gamma\kappa L c\right)\tilde q_2=0\label{sc1}
\end{cases}
\ee In this case the longitudinal and transversal modes evolve
independently. In particular, we can observe that the first sound,
given by the study of the first two equations of the system
(\ref{sc1}), is not influenced by the presence of the vortex tangle:
\begin{center}
\begin{tabular}{|l|}
{$\omega_{1,2}=\pm k_r V_1$} \\
\hline\hline
\\
{$\tilde \rho=\psi$}\\
{$\tilde v_1=\frac{V_1}{\rho}\psi$}
\end{tabular}
\end{center}
whereas the second sound suffers extra attenuation due to the
vortex tangle. The third and fourth equation of the system
(\ref{sc1}) admit non trivial solutions if and only if their
determinant vanishes, obtaining in this way the following
dispersion relation: \be \omega^2+2i\gamma\kappa
Lb\omega-K^2V_2^2=0. \ee Supposing that $\omega$ is real and
$K=k_r+ik_s$ is complex, the dispersion relation admits the
solutions: \bea && w_2^2:=\frac{\omega^2}{k_r^2}=
V_2^2\frac{2}{1+\sqrt{1+\frac{4\gamma^2\kappa^2L^2b^2}{\omega^2}}},\label{5.45}\\
&& k_s=\frac{\gamma\kappa Lbw_2}{V_2^2}\label{5.46}.\eea When
$\Omega=0$ and $b=1/3$ the results of the Section 4 are obtained.
The approximate solutions to second order in $\frac{\gamma\kappa
Lb}{\omega}$ are: \bea &&
w_2\simeq V_2\left(1-\frac{\gamma^2\kappa^2L^2b^2}{2\omega^2}\right)+O\left(\frac{\gamma^4\kappa^4L^4b^4}{\omega^4}\right),\label{5.47}\\
&& k_s\simeq\frac{\gamma\kappa
Lb}{V_2}+O\left(\frac{\gamma^3\kappa^3L^3b^3}{\omega^2}\right)\label{5.48}.
\eea
Now, we study the transversal modes. The second subsystem (fifth
and sixth equation) of the system (\ref{sc1}) admits nontrivial
solutions if and only if its determinant vanishes; this yields:
\begin{equation}
\omega^2-4\Omega^2=0.
\end{equation}
The solutions of this equation are $\omega_{5, 6}=\pm 2|\Omega|$.
The respective modes are:
\begin{center}
\begin{tabular}{|l|}
{$\omega_{5,6}=\pm 2|\Omega|$} \\
\hline\hline
\\
{$\tilde v_3=\psi$}\\
{$\tilde v_2=\pm i\psi$}
\end{tabular}
\end{center}
and they correspond to extremely slow phenomena, which, when
$\Omega\rightarrow 0$, tend to stationary modes.
\par
Finally, we consider the last subsystem (equations seven and
eight), whose dispersion relation is: \be \omega^2+2i\gamma\kappa
L(1-b)\omega +\left[-\left(\gamma\kappa L(1-b) \right)^2
-4\Omega^2+4\gamma\Omega\kappa L c-(\gamma\kappa L c)^2\right]=0
\ee which admits the following exact solutions: \be
\omega_{7,8}=\pm \left(2\Omega-\gamma\kappa
Lc\right)-i\gamma\kappa L\left(1-b\right).\label{5.52} \ee The
corresponding modes are:
\begin{center}
\begin{tabular}{|l|}
{$\omega_{7,8}=\pm \left(2\Omega-\gamma\kappa
Lc\right)-i\gamma\kappa L\left(1-b\right)$} \\
\hline\hline
\\
{$\tilde q_3=\psi$}\\
{$\tilde q_2=\pm i\psi$}
\end{tabular}
\end{center}
From (\ref{5.45}), (\ref{5.46}) and (\ref{5.52}) one may obtain
the following quantities $L$, $b$ and $c$: \be \label{casoort}
L=\frac{-\omega_sw_2+V_2^2k_s}{\gamma\kappa w_2}, \quad\!\!\!
b=\frac{V_2^2k_s}{-\omega_sw_2+V_2^2k_s},\quad\!\!\!
c=\frac{-\omega_rw_2+2\Omega w_2}{-\omega_s w_2+V_2^2k_s}\ee where
we have put $\omega_7=\omega_r+i\omega_s$. \\ The results of this
section, from the physical point of view, imply that measurement
in a single direction are enough to give information on all the
variables describing the vortex tangle.
\subsection{Second case: ${\bf n}$ orthogonal to ${\bf \Omega}$}
Now we assume that the direction of propagation of the waves is
orthogonal to the rotation axis, i.e. for example, ${\bf n}=\left(0,
1, 0\right)$. The unit vectors tangent to the wave front are ${\bf
t}_1=\left(1,0,0\right)$ and ${\bf t}_2=\left(0,0,1\right)$. Under
these assumptions, the homogeneous algebraic linear system for the
small amplitudes is: \be
\begin{cases}
-\omega\tilde\rho+K\rho\tilde v_2=0\cr -\omega\tilde
v_2+K\frac{p_\rho}{\rho}\tilde \rho+2i\Omega\tilde v_3=0\cr -\omega
\tilde v_3-2i\Omega\tilde v_2=0\cr\cr -\omega\tilde T+\frac{K}{\rho
c_V}\tilde q_2=0\cr -\omega \tilde q_2+\zeta K\tilde
T+2i\Omega\tilde q_3=i\gamma \kappa L\left[(1-b)\tilde q_2+c\tilde
q_3\right] \cr
-\omega\tilde
q_3-2i\Omega\tilde q_2=i\gamma \kappa L\left[(1-b)\tilde q_3-c\tilde
q_2\right]\cr \cr -\omega\tilde v_1=0\cr \left[-\omega-i\gamma
\kappa 2Lb\right]\tilde q_1=0 \label{src2}
\end{cases}
\ee In this case the longitudinal and the transversal modes not
evolve independently. In particular, the first sound is coupled with
one of the two transversal modes in which velocity vibrates, while
the second sound is coupled with a transversal mode in which heat
flux vibrates.
\par \noindent As in the previous subsection,
the first subsystem (first three equations) of the system
(\ref{src2}), admits non trivial solutions if and only if its
determinant vanishes: \be
-\omega\left[\omega^2-4\Omega^2-K^2p_\rho \right]=0.\ee The
solutions of this equation, as also the corresponding modes, are
the same to the case of pure rotation (see equations
(\ref{sol2b})-(\ref{sol2})).
The second subsystem (fourth and fifth equations), has the
dispersion relation: \be \left(-\omega-i\gamma \kappa
L(1-b)\right)\left[\omega\left(-\omega-i\gamma \kappa
L(1-b)\right)+K^2 V_2^2\right]+\omega\left(2i\Omega-i\gamma \kappa
Lc\right)^2=0. \ee Assuming $\omega\in\Re$ and $K=k_r+ik_s$, one
obtains the following two equations: \bea
&&-\omega^3+\gamma^2\kappa^2
L^2(1-b)^2\omega+4\Omega^2\omega+\gamma^2\kappa^2L^2c^2\omega-4\gamma\kappa
L c \Omega\omega+\nonumber\\
&&\hspace{2.5cm} +k_r^2V_2^2\omega-k_s^2V_2^2\omega-2\gamma\kappa
L(1-b)k_rk_sV_2^2=0 ,\label{sol4b} \\
&& -2\gamma \kappa L(1-b)\omega^2+2k_rk_sV_2^2\omega+\gamma\kappa
L (1-b)(k_r^2-k_s^2)V_2^2=0.\label{sol3}\eea In the hypothesis of
small dissipation ($k_r^2\gg k_s^2$), from (\ref{sol3}) one
obtains: \be k_s=\gamma \kappa L
(1-b)\left(\frac{2w_2^2-V_2^2}{2w_2V_2^2}\right),\label{5.22} \ee
which substituting in (\ref{sol4b}), yields: \be
\omega^4-\left[\left(2\Omega-\gamma\kappa L c
\right)^2-\gamma^2\kappa^2L^2(1-b)^2\right]\omega^2-k_r^2V_2^2\omega^2-\gamma^2\kappa^2L^2(1-b)^2V_2^2k_r^2=0.\label{5.60}
\ee Putting $\tilde A=-\left[\left(2\Omega-\gamma\kappa L c
\right)^2-\gamma^2\kappa^2L^2(1-b)^2\right]$ and $\tilde
B=-\gamma^2\kappa^2L^2(1-b)^2$ and taking into account that
$w_2=\frac{\omega}{k_r}$, the Eq. (\ref{5.60}) becomes: \be
w_2^2\left[w_2^2\left(1+\frac{\tilde
A}{\omega^2}\right)-V_2^2\left(1-\frac{\tilde
B}{\omega^2}\right)\right]=0 \ee whose solutions are: \bea
&& w_2^2=0,\quad\mbox{and}\quad\nonumber\\
&& w_2^2=V_2^2\frac{\left(\omega^2-\tilde
B\right)}{\left(\omega^2+\tilde A\right)}=
V_2^2\frac{1}{1-\frac{\left(2\Omega-\gamma\kappa
Lc\right)^2}{\omega^2+\gamma^2\kappa^2L^2(1-b)^2}}.\label{5.26}\eea
We can remark that the coefficients $\tilde A$ and $\tilde B$ are
negative and that $w_2^2\geq V_2^2$ because $\omega^2+\tilde A
\leq \omega^2-\tilde B$ and, in particular, $w_2^2= V_2^2$ for
$\Omega=\frac{\gamma \kappa Lc}{2}$. Now, studying the transversal
modes, i.e. the third subsystem (equations seventh and eighth), we
obtain $ \omega_7=0$, which corresponds to a stationary mode,
and: \be \omega_8=-i\gamma \kappa 2Lb.\label{5.25} \ee
Summarizing, also in this case measurement in a single direction
are enough to given information on all the variables describing
the vortex tangle, namely $L$, $b$ and $c$, from equations
(\ref{5.22}), (\ref{5.26}) and (\ref{5.25}): \bea \label{casopara}
&&L=\frac{4k_sw_2V_2^2-\omega_s\left(2w_2-V_2^2\right)}{2\left(2w_2^2-V_2^2\right)\gamma\kappa},\nonumber \\
&&b=-\frac{\omega_s\left(2w_2^2-V_2^2\right)}{4k_sw_2V_2^2-\omega_s\left(2w_2-V_2^2\right)},\\
&&c=\frac{4\Omega(2w_2^2-V_2^2)-\sqrt{(1-V_2^2)(4k_r^2(2w_2^2-V_2^2)^2+16k_s^2V_2^4)}}
{4k_sw_2V_2^2-\omega_s(2w_2^2-V_2^2)}\nonumber \eea where we have
put $\omega_8=i\omega_s$ and $\omega_s=2\gamma \kappa L B$.
In this subsection we have analyzed wave propagation in the combined
situation of rotation and counterflow with the direction {\bf $n$}
orthogonal to {\bf $\Omega$}. In \cite{SBD} Swanson et al.
experimented the same situation, but they didn't represent the
attenuation neither the speed of the second sound but only the
vortex line density $L$ as function of {\bf
$\Omega$} and {\bf $V$}. Therefore, it is unknown how they plotted
these graphics, which the hypothesis were made and what was the
anisotropy considered. Instead, the results of these two
subsections allow to know the spatial distribution of the vortex
tangle simply by performing experiments on waves propagating
orthogonally to {\bf $\Omega$} (equations (\ref{casoort})) or
parallelly to {\bf $\Omega$} (equations (\ref{casopara})).
\section{Conclusions}
In this work we have studied the propagation of waves
(longitudinal density and temperature waves, longitudinal and
transversal velocity and heat waves) in turbulent superfluid
helium in the three situations: rotating frame, thermal
counterflow, and simultaneous
thermal counterflow and rotation.\\
From the physical point of view it is interesting to note that our
detailed analysis in Section 5 shows that, in contrast to which
one could intuitively expect, measurements in a single direction
are enough to give information on all the variables describing the
vortex tangle, namely $L$, $b$ and $c$, for instance, from one of
(\ref{5.45})-(\ref{5.46}) and (\ref{5.52}) or of
(\ref{5.22})-(\ref{5.26}) and (\ref{5.25}). This is not an
immediate intuitive result. Future analyses of work along this
direction could be, for instance, to consider that ${\bf \Omega}$
and ${\bf V}$ have arbitrary directions, i.e. that they are not
parallel to each other, in which case (\ref{5.38}) would not be
sufficient to describe the vortex tangle, because no isotropy in
the $y-z$ plane could be assumed.
\par
Another topic could be to assume that the external waves produce
vibrations in the vortex lines, without creating nor destroying
them. An example of that is the work of Barenghi et al.
\cite{BTMA2}. A more general possibility would be to consider that
nonlinear effects of the external waves create and destroy new
vortices. Yet another topic would be to consider what happens with
waves whose wavevector $\lambda$ become short enough to be
comparable with the average vortex separation, of the order
$L^{-1/2}$. In this case, one could study nonlocal effects in the
vortex \cite{MonJ05, MonJou05}. The first mentioned application
could be carried out within the existing physical model, at the
expenses of more cumbersome calculations. In contrast, the other
three applications need more progress in the basic physical
understanding of the problem.
\section*{Acknowledgments}
We acknowledge Prof. D. Jou (Departament de F\'{\i}sica, Universitat
Aut\`{o}noma de Barcelona) and Prof. M.S. Mongiov\`{\i}
(Dipartimento di Metodi e Modelli Matematici, Universit\`{a} di
Palermo) for the enlightening discussions useful to the deepening of
these arguments. This work is supported by MIUR of Italy under
"Azioni Integrate Italia-Spagna, anno 2005". R.A. P. is supported by
an "Assegno di ricerca dell'Istituto Nazionale di Alta Matematica
'{\it F. Severi}' (INdAM)" of Italy and M.S. is supported by an
"Assegno di ricerca MIUR" of Italy.
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{"url":"https:\/\/encyclopediaofmath.org\/wiki\/Parameter,_method_of_variation_of_the","text":"# Parameter, method of variation of the\n\nA method for approximately solving non-linear (and linear) functional and operator equations $y = P( x)$, $x \\in X$, $y \\in Y$, in Banach spaces, and also for qualitatively investigating them. The method of variation of the parameter consists in the following: The equation $P( x) = 0$, where the operator $P( x)$ is continuously Fr\u00e9chet-differentiable up to the required order (cf. Fr\u00e9chet derivative), or a certain non-linear functional $\\Phi ( x)$ connected with the solution of this equation, is generalized by introducing an auxiliary numerical (or, in general, functional) parameter $\\lambda$ taking values in a finite or infinite interval $\\lambda _ {0} \\leq \\lambda \\leq \\lambda ^ \\star$ to $F( x, \\lambda ) = 0$, where $F( x, \\lambda ),$ $x \\in X$, $\\lambda _ {0} \\leq \\lambda \\leq \\lambda ^ \\star$, is an operator with values in $Y$ such that $P( x) = 0$ is obtained for $\\lambda = \\lambda ^ \\star$: $F( x, \\lambda ^ \\star ) = P( x)$, and the equation $F( x, \\lambda _ {0} ) = 0$ is either easy to solve or a solution $x _ {0}$ of it is known. Here it is assumed that $F( x, \\lambda )$ is continuously Fr\u00e9chet-differentiable in $x$ and $\\lambda$, that is, the partial derivatives $F _ {x} ( x, \\lambda )$ and $F _ \\lambda ( x, \\lambda )$ exist and are continuous, and that the operator $\\Gamma ( x, \\lambda ) = [ F _ {x} ( x, \\lambda )] ^ {-} 1$ from $Y$ into $X$ exists and is continuous. To construct a solution $x( \\lambda )$ of the equation $F( x, \\lambda ) = 0$ on the whole interval $\\lambda _ {0} \\leq \\lambda \\leq \\lambda ^ \\star$ one sets up the corresponding differential problem (Cauchy problem) under the assumption that $x( \\lambda )$ is a continuously-differentiable function with values in $X$, defined by the equation\n\n$$\\tag{1 } F _ {x} ( x, \\lambda ) \\frac{dx}{d \\lambda } + F _ \\lambda ( x,\\ \\lambda ) = 0,\\ \\ x( \\lambda _ {0} ) = x _ {0} ,$$\n\nor\n\n$$\\tag{2 } \\frac{dx}{d \\lambda } = - \\Gamma ( x, \\lambda ) F _ \\lambda ( x, \\lambda ) ,\\ \\ x( \\lambda _ {0} ) = x _ {0} .$$\n\nThe interval $[ \\lambda _ {0} , \\lambda ^ \\star ]$ is partitioned by points $\\lambda _ {0} < \\lambda _ {1} < \\dots < \\lambda _ {n} = \\lambda ^ \\star$ into finer subintervals of lengths $h _ {k} = \\lambda _ {k} - \\lambda _ {k-} 1$, $k = 1 \\dots n$, and to the Cauchy problem (2) or (1) one applies a method of numerical integration of ordinary differential equations with step $h _ {k}$( or several such methods). As a result, for constructing a solution $x( \\lambda )$ of $F( x, \\lambda ) = 0$ one obtains methods of variation of the parameter of corresponding types. The resulting value $x( \\lambda ^ \\star )$ is a solution of $P( x) = 0$.\n\nThe solution at every step of linear problems in $dx\/d \\lambda$ of the form (1), the inversion of the linear operators $F _ {x} ( x, \\lambda )$ in (2) or the successive approximation of the inverse operation $\\Gamma ( x, \\lambda )$ proceed by various methods or once again by variation of the parameter.\n\nThe steps $h _ {k}$ are chosen by various means, for example, from the condition of minimizing the norm of the discrepancy $\\| P( x _ {k+} 1 ) \\|$ as a function of, generally speaking, several variables. Here a joint choice of $h _ {k}$ and the free parameters of a method of numerical integration is also effective, for example, the Runge\u2013Kutta method of order of accuracy $s$, the use of the roots of Chebyshev and related polynomials, etc.\n\nThe Cauchy problem (2) is not only a means for determining an approximate solution of the equation in question, but also for proving the existence of the solution itself. A number of distinct ways of introducing the parameter $\\lambda$ have been studied. As the numerical parameter $\\lambda$ one may also use one of the natural parameters inherent in the problem.\n\nDepending on the way of introducing $\\lambda$, the method of variation of the parameter is a direct or an iterative method. A joint application of a direct and an iterative method is said to give a combined method. For example, the iterative method of improved Euler\u2013Cauchy type with step $h _ {k} = 1$( for $F( x, \\lambda ) = P( x) - ( 1- \\lambda ) P( x _ {0} )$, $\\lambda _ {0} = 0$ and $\\lambda ^ \\star = 1$) is of the third order of accuracy and has the following form:\n\n$$x _ {i+} 1 = x _ {i} - \\frac{1}{2} [ \\Gamma ( x _ {i} ) + \\Gamma ( x _ {i} - \\Gamma ( x _ {i} ) P( x _ {i} ))] P( x _ {i} ),$$\n\n$$i = 0, 1 , . . . ; \\ \\Gamma ( x) = [ P ^ \\prime ( x)] ^ {-} 1 .$$\n\nEvery method of numerical integration gives rise to an iterative method of variation of the parameter of a higher order of accuracy, and without the need to bring in the derivatives of $P( x)$ of an order higher than the first.\n\nThe use of methods of numerical integration in direct methods of variation of the parameter in conjunction with a correction of the results after each step by an iterative method of variation of the parameter (a combined method) is one of the most effective methods for solving non-linear equations.\n\nFor a broad class of problems the method of variation of the parameter has been worked-out sufficiently well. Originally it was proposed for systems of algebraic and transcendental equations, integral equations, ordinary and partial differential equations, and later for the solution of more general non-linear and operator equations. Conditions have been studied under which the solvability of the equation $P( x) = 0$ is guaranteed, as well as the possibility of constructing a solution of it by integrating the Cauchy problem (2) over $[ \\lambda _ {0} , \\lambda ^ \\star ]$ and establishing the domains of its location. Convergence conditions and error estimates have also been studied, as well as problems of applying a method of variation of the parameter for the inversion and pseudo-inversion of linear operators, for the construction of pseudo-solutions (and solutions) of linear functional equations with minimal deviation in norm (in a given space) from the initial value, for the summation of operator series and the construction of certain classes of projections, for the determination of initial approximations for iteration processes, for the solution of operator differential equations and problems in linear algebra, or for proving the solvability of non-linear systems connected with variational problems and the construction of solutions of them, for the minimization of functionals and in many others. Other research concerns extensive classes of effective modifications of the method of variation of the parameter, among them successive approximation of the inverse operator $\\Gamma ( x, \\lambda )$ or $\\Gamma ( x)$. Also, broad classes of branching problems and non-linear problems on eigen values have been studied.\n\n\n\n(However, the case of branching may be blocked by other ways of introducing a parameter $\\lambda$ or of an additional parameter $\\tau$.) The method of variation of the parameter has also been treated as a method of \"gradient\" type, and also without the assumption that $\\Gamma ( x)$ exists.\n\n#### References\n\n [1] D.F. Davidenko, \"On a new numerical solution method for systems of nonlinear equations\" Dokl. Akad. Nauk SSSR , 88\u00a0: 4 (1953) pp. 601\u2013602 (In Russian) [2] D.F. Davidenko, \"Application of the method of variation of parameters to the theory of nonlinear functional equations\" Ukrain Mat. Zh. , 7 (1955) pp. 18\u201328 (In Russian) [3] M.K. Gavurin, \"Nonlinear functional equations and continuous analogues of iteration methods\" Izv. Vuz. Mat. , 5 (1958) pp. 18\u201331 (In Russian) [4] B.T. Polyak, \"Gradient methods for solving equations and inequalities\" Zh. Vychisl. Mat. i Mat. Fiz. , 4\u00a0: 6 (1964) pp. 995\u20131005 (In Russian) [5] D.F. Davidenko, \"An application of the method of variation of parameters to the construction of iterative formulas of increased accuracy for numerical solutions of nonlinear integral equations\" Soviet Math. Dokl. , 6\u00a0: 3 (1965) pp. 702\u2013706 Dokl. Akad. Nauk SSSR , 162\u00a0: 3 (1965) pp. 499\u2013502 [6] S.G. Mikhlin, \"Numerical realization of variational methods\" , Moscow (1966) (In Russian) [7] H. Kleinmichel, \"Stetige Analoga und Iterationsverfahren f\u00fcr Gleichungen in Banachr\u00e4ume\" Math. Nachr. , 37\u00a0: 5\u20136 (1968) pp. 313\u2013343 [8] M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., \"Approximate solution of operator equations\" , Wolters-Noordhoff (1972) (Translated from Russian) [9] D.K. Lika, R.A. Shafiev, \"Continuous analogues of certain iteration methods for solving equations\" Izv. Akad. Nauk MoldavSSR Ser. Fiz-Tekhn. i Mat. Nauk , 2 (1970) pp. 13\u201318 (In Russian) [10] E.P., et al. Zhidkov, Fizika Element. Chast. i Atomn. Yadra , 4\u00a0: 1 (1973) pp. 127\u2013166 [11] D.F. Davidenko, \"The iteration method of parameter variation for the inversion of linear operations\" Zh. Vychisl. Mat. i Mat. Fiz. , 15\u00a0: 1 (1975) pp. 30\u201347, 274 (In Russian) [12] J.M. Ortega, W.C. Rheinboldt, \"Iteration methods for the solution of non-linear systems of equations in many unknowns\" , Acad. Press (1970) [13] H. Keller, , Methods of numerical and applied mathematics , 2 , Novosibirsk (1977) pp. 6\u201336 (In Russian) [14] D.F. Davidenko, , Mathematical programming and related problems. Computational methods , Moscow (1976) pp. 187\u2013212 (In Russian) [15] Yu.V. Kolyada, V.P. Sigorskii, Kibernetika\u00a0: 3 (1980) pp. 24\u201328\n\nThe general idea of the method of variation of the parameter as described above is to continuously transform (homotope) a (complicated) model problem $F$ into an easy model $H$ and then to go back taking a solution of $H$ along. Hence also the names homotopy method, continuation method and path following method for these and similar methods. The basic idea goes back to H. Poincar\u00e9. Bifurcation points may be met, so that this area of numerical analysis has much interrelation with numerical methods for the calculation of bifurcations. For a recent survey of the literature, available software and algorithms in this field cf. [a1].","date":"2020-12-02 09:34:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8725584149360657, \"perplexity\": 395.58452761163903}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141706569.64\/warc\/CC-MAIN-20201202083021-20201202113021-00454.warc.gz\"}"}
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At Hardins Country Store of Tupelo, MS, you'll get a variety of tasty dishes that will leave you asking for more. Pamper your taste buds with the finest food today!
Stop by our store to buy cold drinks, candy, and more. Enjoy personalized service from our owners, Bobby and Sarah Hardin at our full-range store.
You can shop freshly sourced groceries that will leave you delighted at our store. What's more - all our products are available at competitive prices.
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A quick view of the most influential metrics in Gallatin Farms.
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Conference slate released for Nebraska women's basketball
Andrew Ward
From: Nebraska Athletics
The Nebraska women's basketball team will face a loaded nine-game Big Ten home schedule this season, beginning with its league opener against defending conference tournament champion Iowa on Saturday, Dec. 28.
Nebraska's Big Ten tip-off against the Hawkeyes, who advanced to the NCAA Elite Eight a year ago, was the first of 18 conference dates for the Huskers announced by the Big Ten office and the Big Ten Network on Friday as part of the unveiling of the league-wide men's and women's basketball schedules for 2019-20.
No game times or television information were announced for any conference games. Those designations will come from the Big Ten at a later date.
Season tickets for Nebraska women's basketball are on sale now at Huskers.com/Tickets and by calling 800-8-BIG-RED during regular business hours Monday-Friday at the Nebraska Athletic Ticket Office. Reserved season tickets are just $180 for the Huskers' 18-game home schedule.
After opening Big Ten play at home against the Hawkeyes, the Huskers will hit the road for a New Year's Eve clash with another 2019 NCAA Tournament team when they face Michigan State in East Lansing. The Spartans advanced to the NCAA Second Round last season.
The Huskers open January at home with back-to-back games against Minnesota (Jan. 4) and Wisconsin (Jan. 9) before back-to-back road games at 2019 NCAA Tournament teams Rutgers (Jan. 12) and Maryland (Jan. 16). The Big Red then return home for games against 2019 NCAA qualifier Michigan (Jan. 19) and Purdue (Jan. 22).
Nebraska closes January much like it opens the month, with back-to-back games against Wisconsin (Jan. 25) and Minnesota (Jan. 30) – this time on the road – to complete season series with the Badgers and Golden Gophers.
The Huskers will play three of their first four games in February at Pinnacle Bank Arena, beginning with a battle against Ohio State (Feb. 2), before facing 2019 NCAA Tournament qualifier Indiana (Feb. 9) and Penn State (Feb. 13). Those three home games surround a road trip to Iowa City to complete the season series against the Hawkeyes (Feb. 6).
Nebraska closes the regular season with three of its final four games on the road, starting with trips to Northwestern (Feb. 16) and Ohio State (Feb. 19), before closing regular-season home action against Illinois on Saturday, Feb. 22. The game against the Fighting Illini will mark Senior Day for Huskers Nicea Eliely, Kristian Hudson, Grace Mitchell and Hannah Whitish.
The Big Red will conclude the regular season with a road trip to Bloomington to complete the regular-season series with Indiana (Feb. 27), before returning to the Hoosier State for the Big Ten Tournament at Bankers Life Fieldhouse in Indianapolis (March 4-8).
Coach Amy Williams, who enters her fourth season at the helm of the Huskers, returns five starters and each of the team's top seven scorers from a year ago. The 2018 Big Ten Conference Coach of the Year will lead a team that features three-year starters in seniors Nicea Eliely and Hannah Whitish along with returning starters in juniors Kate Cain and Taylor Kissinger. A strong group of four sophomores also returns for the Huskers, including Leigha Brown, Sam Haiby, Ashtyn Veerbee, and returning starter Kayla Mershon, who each played in all 30 games for Nebraska in 2018-19.
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The North Kensington Law Centre
On June 19, 2017 June 20, 2017 By Henry BrookeIn ACCESS TO JUSTICE
After I published my blog about law centres last night, Raji Hunjan[1], a former Director of the North Kensington Law Centre, sent me a message in which she told me that in the last five years the centre had fought to survive and to serve North Kensington despite huge cuts and little external support.
This led me to think it might be useful to publish a further blog, this time concentrating on this particular law centre, since it is in the eye of the firestorm which engulfed nearby Grenfell Tower last week.
Today it has published this notice on its website:
Following the recent tragic events at Grenfell Tower, we have been inundated with kind offers of help from members of the public.
We are open as usual, but are running a number of additional services to help the community here in North Kensington.
From Monday 19th June, we will be running daily legal clinics to help local residents affected by the disaster to get the legal support and access to justice they need. We will also be running sessions to help residents gather the documents and paperwork they will require for any future claims.
We will be holding a residents' meeting in our offices at 6.30pm on Monday 19th June. Please come along if you are a local resident who has been affected by the tragedy at Grenfell Tower, and what to learn more about your legal rights and how we can help.
We are committed to helping all residents take practical steps to start rebuilding their lives and to access justice. We look forward to seeing you all over the coming days.
As I have often said, if neighbourhood law centres did not exist, one would want to invent them. Never has the need been greater. Yet in what is said to be the richest borough in England this law centre, the first of its kind, – if it survives it will celebrate its half-century in 2020 – has been facing unprecedented difficulties in order to ensure that it continues to provide some kind of service in a hostile climate which wiped out so many other similar centres, as I described in my last blog.
If it had the means to do so, it would publish an annual report, like the Hackney and Southwark centres do, from which one could get a better idea of what it does, why it is so important, and how it obtains its funding. I have garnered the information in this blog from different sources on the web – and particularly the Centre's own website.
Between 2010 and 2015 its gross annual income was slashed from over £900,000 to just over £350,000. It provides legal help and representation in housing, immigration, employment, crime and welfare benefits law. It has a small staff, led by its director. Two of its immigration caseworkers possess Level 2 accreditation, which means that they can handle more complex cases. It also engages specialist casework volunteers for one or two days each week.
Its website describes how it has secured legal aid contracts in some of those areas of social welfare law which are still "in scope." It also describes a pilot scheme whereby the local council (RBKC) funded a project whereby it was able to give legal advice to the voluntary and community sector in topics like employment disputes, redundancy issues, the legal advice contained in staff handbooks, and staff contracts. Legal aid is no longer available for any kind of employment dispute, but Trust for London partly funded some of its work in that field. It says that it also has a small amount of funding from which it can help some people in welfare benefit cases free of charge.
It is one of the few law centres that provides services in fields of law that are "out of scope" for legal aid either on a low cost fixed fee basis (for immigration cases) or on a "no win no fee" basis for cases in which financial compensation is being sought. It describes an impressive win it achieved for a client in a discrimination case. A damages-based agreement was in place, and this enabled it to recover some of its costs out of the damages it recovered. It says:
We were instructed by our client, who had attended a job interview where she disclosed her disability. Her concern was, having raised the issue of her disability, the interviewer then proceeded to ask questions about her reliability and capability to do the job. This line of questioning made her feel uncomfortable and she wrote to the company to express her concerns. Rather than receive an apology, she was threatened with costs against her should she pursue her complaint any further.
The importance of this service is that I have heard again and again how the "telephone gateway" – the only facility funded by the Legal Aid Agency for discrimination cases – is wholly unsuitable for many discrimination sufferers who need the reassuring comfort of a face-to -face interview in order to bring out clearly the kind of treatment they have suffered from. But few services like the one at North Kensington have survived the brutal LASPO cuts.
It is not easy for an outsider who has not read deeply into the topic to appreciate the scale of the damage done to social cohesion by the LASPO cuts, and by the Treausry's insouciant belief that one can apply swingeing reductions to expenditure on "justice" without further disempowering those who are already severely disempowered.
This is why the current funding appeal for the North Kensington centre is so important, and why it is so important that with supportive and practical help from the Law Centres Network and, one would hope, the chastened local authority and other funding sources (including, perhaps, Her Majesty's Government), it achieves the £100,000 target of its current funding appeal [2]
Once it has done this, there must then surely be a determined effort to put its finances back on a really secure and sustainable footing, so that it does not have to turn so many needy people away in future.
I end by republishing a case study from their website about one of the many clients it has been able to help when they are faced with problems they could not possibly cope with on their own. [Emphasis added by me for two points that are particularly relevant today]
Housing Case Study
Mr D's case is very typical of the cases we handle at North Kensington Law Centre. He had been moved from London and housed in temporary accommodation in a coastal town for nearly a year, whilst his close networks remained in London. Mr D came to us because he was facing eviction from his temporary accommodation which would have left him homeless. We successfully fought this case and helped Mr D avoid eviction; we are now working to get him rehoused in London where he originally resided.
Like many of our housing clients, Mr D suffers severe mental health issues. He had previously fled from another country where he had been tortured by the Army. He was still suffering flashbacks about his traumatic past, which had a severe impact on his ability to interact with those in positions of authority. In particular he felt extremely vulnerable when dealing with the courts on his housing matter.
Mr D was living in temporary accommodation when the Landlord served notice to terminate the tenancy and issued possession proceedings in County Court. The Landlord served notice by arguing that Mr D did not live in the property. As evidence he used that fact that Mr D often visited his daughter in London, and that on inspection, he had not used the cooker in the temporary accommodation. In reality Mr D did not use the cooker because he had been told not to by the Landlord as it was unsafe to use.
We agreed to represent Mr D in the possession case, even though we were concerned about the strength of our defence because it was temporary accommodation and an unsecure tenancy. We defended that Mr D had no mental capacity to litigate and that this was a breach of Equality Act. At the first hearing the Judge reacted positively and commented that he would welcome more of these cases as proceedings of this type were rarely challenged. But even with this support, this case remained a very tough one to win.
With skill and patience we supported the client, as he was often afraid to even talk and our Housing Solicitor had to conduct all interviews in the presence of Mr D's daughter. His relatives would plead with him to attend court as the day before the hearings he would retreat, cover himself with quilts and refuse to talk to anyone because he feared he would be killed.
Funding Problems
Aside from Mr D's mental state, the other obstacle we faced was funding. Mr D's benefits had been stopped for not attending a medical examination. And so the only way we could secure Legal Aid was by proving his son and daughter's income, which was difficult because the son was unable to obtain a statement from the bank for an account that had been closed earlier in the year. We personally contacted the bank and demanded the evidence we required.
With no funding in place, we could not instruct an expert report on Mr K's mental capacity and without this we could not proceed. On this basis, the Landlord's solicitor made an application to strike out our defence and our Housing Solicitor produced a 50 page statement explaining actions taken to restore legal aid, exhibiting all the emails and telephone conversations with the Legal Aid Agency .
A satisfactory win
After much effort, legal aid was restored just in time for the second hearing and the judge accepted our Solicitor's statement and extended our time to comply with directions. We obtained an expert's report from an experienced Psychiatrist. The report gave a compelling case about Mr D's lack of mental capacity to litigate and why he should be placed near relatives in London. We used this report to negotiate with the other side that the possession order be dropped. Having achieved this, we then fought for our costs, and were awarded £20k. And the fight continues as we are now working with Mr D to get him housed in London.
This is not a world populated by the "fat cat lawyers", so often lazily stigmatised by unthinking politicians and some elements of the tabloid press. It is a world in which dedicated and skilled professional men and women – and others with much-needed ancillary skills – are determined to apply their skills to serve the poorest among us for quite a small financial return.
[1] She is now the Chief Executive of the Zacchaeus Trust, of which I am proud to be a patron, which does comparable work in the North Westminster area.
[2] In the last 30 hours nearly 150 people have contributed nearly £10,000, but there is still a long way to go.
Law Centres: Empowering the disempowered
2 thoughts on "The North Kensington Law Centre"
Pingback: The North Kensington Law Centre and Grenfell Tower – Henry Brooke
Catriona Jarvis
Thank you for this Henry. It is extremely important that the word be spread in relation to all the issues you raise. After asking questions, in particular about those who may be deterred from coming forward because of difficulties with their immigration status, I had been assured that there were few immigration problems. However, this article appears to show otherwise and will help to show where and how further contributions may be usefully made.
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\section{Introduction}
Since about 25 years, it is known that Galactic GCs host a fraction of stars characterized by chemical patterns which appear to be peculiar when compared with that of the bulk of the cluster stellar population. However, in this last decade the possibility of performing accurate multi-objects spectroscopy has clearly shown that what seemed to be an anomaly, was indeed a normal behaviour among GC stars. More in detail, extensive spectroscopical surveys of several GCs have shown the existence of well-defined chemical patterns among stars within individual clusters, as the existence of light-elements (anti-) correlations - the most famous being Na-O anti-correlation (Gratton, Carretta \& Bragaglia~2012). These empirical findings have challenged the idea that GCs are the prototype of Simple Stellar Populations (SSPs), i.e. they host coeval stars formed with the same initial chemical composition.
This notwithstanding, the damning evidence supporting the presence of multiple stellar populations in a GC has been provided by high-precision HST photometry. In fact, accurate photometric investigations have revealed the existence of multiple
Main Sequence (MS) and/or Sub-Giant Branch (SGB) and/or Red Giant Branch (RGB) sequences in the CMD of various GCs. Actually the features observed in the CMD change significantly from one cluster to another, and their properties do strongly depend on the adopted photometric systems (see Milone et al.~(2010), and G. Piotto in this volume).
The commonly accepted scenario is that, in any GC, a second (and in some cases also more) generation(s) of stars can form from the ejecta of intermediate-mass and/or massive stars belonging to the first stellar population formed
during the early phase of the cluster evolution, whose chemical composition is modified by high-temperature proton captures.
Although some amount of dilution between pristine (unpolluted) matter and nuclearly processed matter seems to be unavoidable in order to explain many observational evidence, the second generation stars would be formed via matter characterized by light-element (anti-)correlations and He enhancement.
The ability to trace both spectroscopically and photometrically the various sub-populations hosted by each individual GC, allows
now the identification of both the primordial stellar component (the first generation, FG) and the second generation (SG) stars.
The existence of a strong correlation between the spectroscopic signatures of the distinct sub-populations and their
distribution along the multiple CMD sequences, has suggested that the peculiar chemical patterns of SG stars have to affect both the
evolutionary properties of these stars as well as their spectral energy distribution.
To date, investigations have provided clear evidence that the multiple CMD sequences can be interpreted as due to \lq{quantized}\rq\ He abundances, as in the case of $\omega$~Cen and NGC~2808, distinct CNO abundance pattens as in the case of M~22 and (the still-debated case of) NGC1851, and the presence of light-element (anti-)correlations.
To trace correctly the various sub-populations from CMD analyses, stellar models must properly account
for the observed chemical patterns in both FG and SG stars.
In the last decade, a substantial effort has been devoted to investigate the effect
of the chemical patterns characteristic of the multiple population
phenomenon on both stellar models and model atmospheres, and the corresponding evolutionary tracks, isochrones and
colour - ${\rm T_{eff}}$ transformations.
In this contribution we discuss how these chemical peculiarities affect the evolution of SG stars, as well as
their photometric properties.
\section{Theoretical framework}
\subsection{He-enhancement}
Helium is one of the most important chemical species in the context of stellar evolution, because any change of its abundance hugely affects the structural and evolutionary properties of stars. In more detail, a change of the He abundance affects low-temperature, radiative opacities, for an increase of He causes a reduction of the opacity. This is due to the fact that when increasing the helium abundance at fixed metallicity, the H abundance has to decrease, and given that H is a major opacity source via the ${\rm ^-H}$ ion, this causes a global reduction of the opacity. This effect explains why He-enhanced stellar models have hotter ${\rm T_{eff}}$ values in comparison with normal He abundance stars. The opacity reduction due to He-enhancement contributes also to make brighter the stars during the core H-burning stage, although the larger contribution to the change in the stellar surface luminosity is due to the change in the mean molecular weight associated to the He abundance increase. In fact, the H-burning efficiency is strongly dependent on the value of the mean molecular weight $\mu$: ${\rm L_H\propto\mu^7}$. When He increases the mean molecular weight (at fixed metallicity) has to increase and this translates in a larger H-burning efficiency. Given that
${\rm \Delta{L_H}/L_H=7\Delta\mu/\mu}$, a change $\Delta{Y}=0.10$ -- lower than the maximum He enhancements expected for the bluest MS stars in GCs $\omega$~Cen and NGC~2808 -- causes a $\sim 50$\% variation of the H-burning efficiency compared to normal-He stars. The combined effect of radiative opacity decrease and H-burning efficiency increase causes He-rich stellar models to be brighter and hotter during the MS stage. As a consequence, their core H-burning lifetime ($t_H$) is significantly reduced: for a $0.8M_\odot$, $t_H$ decreases by $\sim48$\% when increasing the He content from the primordial value Y=0.245 to 0.40.
Fig.~\ref{fig_elio} shows different isochrones computed with the same [Fe/H] and age, but various values for the initial He content. There are some interesting features disclosed by this plot: {\sl i)} the He-rich MS runs parallel in the luminosity interval from the MS Turn-off (TO) and the location of stars with mass $\sim0.5M_\odot$; {\sl ii)} the MS effective temperature is sensitive to the He increase (${\Delta{T_{eff}}}/{\Delta{Y}}\approx 2.3\times10^3 K$); {\sl iii)} at fixed age, the mass at the MS TO significantly decreases when increasing the initial He content. For a 12Gyr old isochrone it is equal to $0.806M_\odot$ for Y=0.245 and $0.610M_\odot$ for Y=0.40; {\sl iv)} for any given age, the SGB is not affected by a He change; {\sl v)} the ${\rm T_{eff}}$ values of RGB stellar models are also affected by an He increase, although to a smaller extent than the MS locus.
The predicted behaviour of the MS as a function of the initial He abundance justifies the interpretation of the MS splitting observed in some GCs, like $\omega$~Cen and NGC~2808.
When considering the RGB evolution, there are two important features that are affected by a change of the initial He abundance: the RGB bump and the brightness of the RGB tip (TRGB). Model computations predict: {\sl a)} an increase of the bump brightness (see Fig.~\ref{fig_elio}), and {\sl b)} a smaller luminosity excursion during the RGB bump stage. The first effect is due to the lower envelope opacity of He-rich stars, that causes the discontinuity of the H abundance left over by the {\sl first dredge-up} to be located in more external layers. As a consequence, the H-burning shell encounters the discontinuity at later times, hence at a brighter luminosity. The second effect is caused by the fact that in He-rich stars, the jump of the H abundance at the discontinuity is smaller than in normal He stars. As a consequence the surface stellar luminosity is less affected when the H-burning shell crosses the discontinuity. From an observational point of view, the impact of an He enhancement on the RGB bump brightness in GCs has been discussed by Bragaglia et al.~(2010) . On the other hand, the effect on the RGB bump luminosity excursion, hence evolutionary lifetime, has been used by Nataf et al.~(2011) to interpret the anomalously
small number of RGB bump stars in the Galactic bulge. A number smaller than predictions by He-normal stellar models has been considered a proof that bulge stellar populations are He-enhanced.
\begin{figure}[]
\resizebox{\hsize}{!}{\includegraphics[clip=true]{fig1_cassisi.eps}}
\caption{\footnotesize
Comparison between 12~Gyr-old, Z=0.002, isochrones computed for various assumptions about the initial He abundance. The two insets
shows the location of the ZAHB, and the trend of the average RGB bump brightness as a
function of the initial He abundance for a ${\rm 0.8M_\odot}$.}
\label{fig_elio}
\end{figure}
The TRGB brightness is significantly affected by an He-enhancement. For a given total stellar mass increase of the initial He content
decreases the TRGB brightness. This is due to the fact that the interiors of He-rich stars are hotter at the end of the core H-burning stage, and when they reach the RGB stage they develop a significantly lower level of electron degeneracy in their He core. At the same time, as a consequence of the larger H-burning efficiency, the He core mass grows at a faster rate. Both effects make \lq{easier}\rq\ to achieve the thermal conditions required by the $3\alpha$ reaction ignition, and He ignition is attained with
a smaller He core mass. Due to the existence of a {\sl He core mass - luminosity} relation for RGB stars, the TRGB brightness decreases in He-rich, low-mass giants. Unfortunately, due to the low number of stars populating the brighter portion of the RGB and the still unsettled issue of the GC distance scale, it is almost impossible to verify observationally this prediction.
We also note that, as a consequence of the reduction of the value of $t_H$ for He-rich stars,
the mass of stars at the TRGB is expected to be significantly smaller in He-enhanced stellar populations, when age is kept constant
(and if RGB mass loss does not have a strong dependence on He).
This occurrence explains why He-enhanced stellar populations are characterized by a bluer Horizontal Branch (HB) morphology with respect to a FG stellar population.
When moving to the core He-burning stage, one notices that Zero Age HB (ZAHB) brightness is a strong function of the initial He content. When Y increases, the ZAHB becomes brighter for ${\rm T_{eff}}$ lower that $\sim20000$K, and fainter at higher ${\rm T_{eff}}$
(see Fig.~\ref{fig_elio}). This behaviour is the consequence of both the decrease of the He-core mass at the TRGB, and increased efficiency of the shell H-burning in He-rich stars. In ZAHB objects cooler than $\sim20000$~K, the second effect dominates over the first one and the stars appear brighter, whereas in the hottest portion of the ZAHB -- due to the tiny envelope mass -- the H-burning shell is not efficient enough, and the decrease of the He core mass is the dominating effect. This has the important consequence that the slope of the ZAHB in the H-R diagram (as well as in the various observational planes) is strongly dependent on the (spread in the) initial He abundances of the various sub-populations hosted within a given GC. This theoretical prediction can provide a direct explanation for the existence of a tilted HB in GCs like NGC6388, NGC6441 and NGC1851. He-enhancement has an additional important implication for the global HB morphology. Due to the combined effect of the lower RGB evolving mass and more extended blue loops that characterize the off-ZAHB evolution of He-rich stars, for a given average efficiency of the mass loss along the RGB the predicted HB location of He-enhanced models will be on average hotter, i.e. bluer, than that of He-normal stars. This helps explaining the existence the very blue HB morphology -- as well as the presence of extended HB blue tail -- in GCs hosting He-enhanced sub-populations.
\begin{figure*}[t!]
\resizebox{\hsize}{!}{\includegraphics[clip=true]{fig2_cassisi.eps}}
\vskip -1.0truecm
\caption{\footnotesize
Theoretical isochrones for the same age and metallicity (see label) but for various assumptions about the light-element distribution and
He-enhancement. More in detail, {\sl reference}: canonical $\alpha-$enhanced mixture, Y=0.248; {\sl CNONa}: light-element (anti-)correlation but the same CNO sum and He abundance of the $\alpha-$enhanced composition; $(CNO)_{enh}Na$: as before but the CNO sum is now enhanced by a factor $\sim2$; $(CNO)_{enh}Na-He$: as before but now the initial He abundance is equal to Y=0.40.
}
\label{fig_anti}
\end{figure*}
\subsection{Light-element (anti-)correlations}
Light element abundance changes can affect the stellar properties via the effects induced on both the radiative opacity evaluations and -- at least in the case of C, N and O, i.e. the CNO-cycle catalysts -- the H-burning efficiency. Many investigations have been devoted to this topic (Salaris et al.~2006, Cassisi et al.~2008, Pietrinferni et al.~2009, Ventura et al.~2009, Vandenberg et al.~2012) and the main results can be summarized as follows:
\begin{itemize}
\item even in case of extreme light-element anti-correlations, as long as the sum of CNO elements is kept constant, the evolution in the H-R diagram is unchanged compared to standard $\alpha-$enhanced stellar models;
\item if the CNO sum is enhanced, the morphology of the evolutionary tracks in the H-R diagram is modified.
For a given iron content and stellar mass, the SGB appears fainter in comparison with standard $\alpha-$enhanced models.
This behaviour is mainly due to the fact that the efficiency of the CNO-cycle increases when the CNO sum is enhanced.
The ${\rm T_{eff}}$ scale for both MS and RGB CNO-enhanced stellar models is marginally affected (by less than 20~K);
\item when computing isochrones, unless the CNO sum is increased with respect the reference $\alpha-$enhanced mixture,
the effect of the light-element (anti-)correlations is negligible. When the C+N+O sum is increased, a separation appears along the SGB, and a CNO-enhanced
isochrone is almost perfectly mimicked by a \lq{canonical}\rq\ $\alpha-$enhanced, $\sim(1.5-2)$~Gyr older, isochrone.
\end{itemize}
Isochrones for various assumptions about the heavy element distribution and/or the initial
He abundance, but for the same age and [Fe/H], are shown in Fig.~\ref{fig_anti};
\section{On the photometric appearance of multiple populations}
The isochrones shown in Fig.~\ref{fig_anti} reveal that in the theoretical H-R plane, the only possibility to observe a separation between isochrones with a standard $\alpha-$enhanced isochrone and with the composition of SG stars, is to assume
a huge He enhancement (that affect MS and RGB) and/or a significant CNO-enhancement (that affect the SGB sequence).
Needless to say that this theoretical evidence is in stark contrast with observations, that reveal the presence of splitting/broadening of the photometric sequences (in particular, the RGB one) also in those GCs that do not show (spectroscopical) evidence of a significant CNO enhancement and/or huge He enhancement. The chemical peculiarities of second generation stars must therefore have
an effect also on the spectral energy distribution.
Sbordone et al.~(2011) have demonstrated the crucial role played by CN and NH molecules (whose abundances in SG stars are very different from those in FG stars) in modifying the stellar spectrum at wavelengths shorter than $\sim400$~nm. This has the important implication that only magnitudes corresponding to photometric filters bluer than the standard Johnson B filter are affected by the peculiar chemical patterns of multiple populations; as a consequence, the photometric appearance of these sub-populations depends on the adopted photometric systems:
\noindent
{\underline{\it $BVI$ CMDs}}:
a splitting (or a spread) of sequences along the MS up
to the Turn-off (TO), and to a lesser degree of the RGB can only be achieved by varying
the helium content. The CNONa (anti-)correlations do not influence significantly
stellar models and spectral energy distribution, when the C+N+O abundance is unchanged.
On the other hand, a variation of the CNO sum with respect the \lq{canonical}\rq\ value leads to a split of the SGB; this is a
purely evolutionary effect.
\noindent
{\underline{\it $UBV$- and $uy$ Str{\"o}mgren CMDs}}:
(anti-)correlations in CNONa abundances as well as He differences may
lead to multiple sequences from the MS to the RGB, where the effect
tends to be larger, and may reach 0.2--0.3~mag. This multiplicity is
independent of the CNO sum. However, the individual element variations are decisive.
We note that He enhancement works in the opposite direction than CNONa (anti-)correlations.
\noindent
{\underline{\it $vy$ CMDs}}: as in the case of the $BVI$-colours, a splitting of the MS up to
the TO can be achieved only by varying the He abundance. A split of the SGB is the result of a
change in the C+N+O abundance. Additionally, a split along the RGB may result both from both helium
and C+N+O variations; this is different from the $BVI$-case.
\noindent
{\underline {\it $m_1$uy CMDs}}: light element (anti-)correlations lead to splits along the MS, the SG and RGB; also helium variations lead to colour differences. However, the sign of the colour change is different for the lower and upper part of the RGB.
\noindent
{\underline{\it $c_y$V CMDs}}:
here, all the evolutionary sequences in the CMD show the influence of both element
anticorrelations and helium variations, and a large separation between the various sequences can be easily achieved. It is worth
noticing that recently a new photometric index ${\rm C_{UBI}=(U-B)-(B-I)}$ has been defined (M. Monelli, this volume) as a very powerful tool for tracing various sub-populations along the RGB. This photometric index is based on broadband photometric filters and, hence, is quite more efficient than ${\rm c_y}$, based on narrowband filters.
Model predictions about the ${\rm c_{UBI}}$ index qualitatively agree very well with the observational findings.
\begin{acknowledgements}
SC warmly thanks the organizers for inviting him to this interesting and pleasant conference, and wishes to express his gratitude to Franca for her friendship and for her painstaking effort as {\sl "Lens Maker..."}. Financial support from PRIN INAF 2012 (PI: E. Carretta) and PRIN MIUR 2010-2011, project ``The Chemical and Dynamical Evolution of the Milky Way and Local Group Galaxies'', prot. 2010LY5N2T " is acknowledged.
\end{acknowledgements}
\bibliographystyle{aa}
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{"url":"https:\/\/astronomy.stackexchange.com\/tags\/planetary-science\/hot","text":"# Tag Info\n\nAccepted\n\n### Why is there so little nitrogen in the Martian and Venusian atmospheres?\n\nNitrogen, with a molecular mass of 28 atomic mass units, is too light to have remained in Mars's atmosphere. Carbon dioxide, with a molecular mass of 44 amu, could (and does) exist on Mars, but it is ...\n\u2022 27.1k\n\n### Why is there so little nitrogen in the Martian and Venusian atmospheres?\n\n3.5% of all atmosphere in Venus still accounts for more partial pressure of nitrogen than on Earth. Venus has ~90bar pressure at the surface, 3.5% of them are ~3.2 bar nitrogen. Earth has only 0.8 bar ...\n\u2022 1,770\nAccepted\n\n### Do some comets spin? If so, how fast?\n\nYes, comets spin although measuring it can be tricky due to the coma and outgassing from the nucleus. It's easiest to measure the rotation period when the comet is inactive near aphelion although this ...\n\u2022 7,135\n\n### Is there a way to tell the difference between earth andesite from Mars\n\nIf this is something that you have found (rather than purchased as a meteorite) the chances are very small that it is a meteorite. Even if it is a meteorite, the chances it's a Martian one are even ...\n\u2022 7,135\nAccepted\n\n### How fast is Neptune getting brighter? When was was this first noticed and reported?\n\nTL;DR: There was apparent 11% increase of Neptune brightness during 1980 and 2000. This could be due to multiple reasons. Recent observation suggested the reason to be change in the amount and ...\n\u2022 3,226\n\n### Which JWST instrument modes are compatible with observations of the bright trans-Earth planets Mars, Jupiter, and Saturn? Which aren't?\n\nThere are approved proposals for Cycle 1 to point the JWST at The Jovian system Jupiter's great red spot Mars Saturn and its moons and rings In those PDFs, they describe exactly what instruments ...\n\n### Do gas giants have a core?\n\nGas giants are believed to have a solid core. They first formed as icy planets, and were heavy enough to accrete hydrogen and helium from the protoplanetary cloud they were in. Saturn, for instance, ...\n\u2022 16.7k\n\n### What is k2, how does it relate to Io's volcanism and how can Juno constrain its value?\n\nIt is called Tidal love number. The definition is as follows: In Newtonian gravitational theory, a tidal Love number relates the mass multipole moment created by tidal forces on a spherical body to ...\n\u2022 3,226\nAccepted\n\n### How can it be known that Venus does not have plate tectonics?\n\nThere's much less data available from Venus. Some data exists. As mentioned in HDE 226868's answer, maps of Venus's surface exist. Like Earth's atmosphere, Venus's atmosphere is transparent to some ...\n\u2022 27.1k\n\n### Caves traced in nine planets of solar system\n\nA key requirement for caves is solid substance: while there may be rock cores inside the giant planets, the rest is liquid or gas. Metallic hydrogen may be solid or liquid in Jupiter, but presumably ...\n\u2022 10.2k\nAccepted\n\n### Why is Saturn invisible in this radar image of its rings?\n\nThe main issue is that there is relatively little material in Saturn's atmosphere that can efficiently scatter radar waves, so the radar basically just gets absorbed. The key point is that it's much ...\n\u2022 14.7k\nAccepted\n\n### Is the boulder on the peak of Tycho Crater the core of the impactor, or is it a random rock?\n\nIt is just a rock. A complex crater like Tycho is formed in several stages as the rock behaves like a fluid. The initial impact completely destroys the impactor and excavates a large cavity in the ...\n\u2022 87.7k\n\n### Which JWST instrument modes are compatible with observations of the bright trans-Earth planets Mars, Jupiter, and Saturn? Which aren't?\n\nAll modes can be used. But for bright targets, observations are limited to specific filters, subarrays, regions of the target planet, or spectral intervals. James Norwood and colleagues wrote a paper ...\n\u2022 1,764\n\n### How many things are wrong in this \"artist view\" of the TRAPPIST-1 system?\n\nThe 2 big planets are probably f & g and they don't look up to scale to me. While f is the size of Earth, or almost 4 times bigger than the Moon, but it's also further away, ~ 1.3 million ...\n\u2022 201\nAccepted\n\n### Could rocks from Earth have reached the Kuiper belt, or Neptune at least? If so, how?\n\n(I have tracked down the reference that I made in my comment on the question.) Presumably earth rocks once blasted into space (by volcanism or meteorite impact) gain or lose energy primarily by ...\n\u2022 2,661\n\n### How long does lunar opposition surge last? Are there measurements of the full Moon getting suddenly brighter?\n\nThe lunar opposition surge has been well studied, likely because we can study it in detail, we have surface samples, and so it serves as a baseline for other bodies in the solar system (as it does for ...\n\u2022 1,843\nAccepted\n\n### What is k2, how does it relate to Io's volcanism and how can Juno constrain its value?\n\n$k_2$ is one of three tidal Love-Shida numbers related to how gravitation of another body (Jupiter in this case) changes a planet-like body's second degree spherical harmonics (Io in this case). Three ...\n\u2022 27.1k\n\n\u2022 31.4k\nAccepted\n\n### ...but where did Mars' atmosphere actually GO?\n\nAssuming most of the escaping Martian atmosphere is entrained in the solar wind, it will flow outward until it reaches the termination shock, and then slow down in the heliosheath until it reaches the ...\n\u2022 14.7k\n\n### Could a magnetosphere be created for Venus by recreated by spinning-up the planet to a 24 hour day?\n\nPlanetary magnetic fields are not produced by their rotation, they are produced by convection in the core. The rotation has an effect on the patterns of convection through Coriolis forces, but is not ...","date":"2022-05-27 12:53:50","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\": 0.46038320660591125, \"perplexity\": 2444.069438823001}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662647086.91\/warc\/CC-MAIN-20220527112418-20220527142418-00210.warc.gz\"}"}
| null | null |
Q: My re.sub statement gets hung up I'm a Python (and regex) rookie with relatively little programming experience outside of statistical packages (SAS & Stata). So far, I've gotten by using Python tutorials and answers to other questions on stackoverflow, but I'm stuck. I'm running Python 3.4 on Mac OS X.
I've written a script which downloads and parses SEC filings. The script has four main steps:
*
*Open the URL and load the contents to a string variable
*remove HTML encoding using BeautifulSoup
*remove other encoding with regex statements (like jpg definitions, embedded zip files, etc.)
*save the resulting text file.
My goal is to remove as much of the "non-text" information as possible from each filing before saving to my local drive. I have another script written where I do the actual analysis on the residual text.
I'm running into a problem with step 3 on at least one filing. The line that is causing the hangup is:
_content1 = re.sub(r'(?i).*\.+(xls|xlsx|pdf|zip|jpg|gif|xml)+?[\d\D]+?(end)',r'',_content1)
where _content is a string variable containing contents of the SEC filing. The regex statement is supposed to capture blocks beginning with a line ending in a file extension (xls, pdf, etc.) and ending on the word "end."
The above code has worked fine for entire years' worth of filings (i.e., I've analyzed all of 2001 and 2002 without issue), but my script is getting hung up on one particular filing in 2013 (http://www.sec.gov/Archives/edgar/data/918160/0000918160-13-000024.txt). I'm unsure how to debug as I'm not getting any error message. The script just hangs up on that one line of code (I've verified this with print statements before and after). Interestingly, if I replace the above line of code with this:
_content1 = re.sub(r'(?i)begin*.*(xls|xlsx|pdf|zip|jpg|gif|xml)+?[\d\D]+?(end)',r'',_content1)
Then everything works fine. Unfortunately, certain kinds of embedded files in the filings don't start with "begin" (like zip files), so it won't work for me.
I'm hoping one of the resident experts can identify something in my regex substitution statement that would cause a problem, as going match-by-match through the linked SEC filing probably isn't feasible (at least I wouldn't know where to begin). Any help is greatly appreciated.
Thanks,
JRM
EDIT:
I was able to get my script working by using the following REGEX:
_content1 = re.sub(r'(?i)begin|\n+?.+?(xls|xlsx|pdf|zip|jpg|gif|xml)+?[\d\D]+?(end)',r'\n',_content1)
This seems to be accomplishing what I want, but I am still curious as to why the original didn't work if anyone has a solution.
A: I think your biggest problem is the lack of anchors. Your original regex begins with .*, which can start matching anywhere and won't stop matching until it reaches a newline or the end of the text. Then it starts backtracking, giving back one character at a time, trying to match the first falsifiable component of the pattern: the dot and the letters of the file extension.
So it starts at the beginning of the file and consumes potentially thousands of characters, only to backtrack all the way to the beginning before giving up. Then it bumps ahead and does the same thing starting at the second character. And again from the third character, from the fourth, and so on. I know it seems incredibly dense, but that's the tradeoff we make for the power and compactness of regexes.
Try this regex:
r"(?im)^[^<>\n]+\.(?:xlsx?|pdf|zip|jpg|gif|xml)\n(?:(?!end$)\S+\n)+end\n"
The start anchor (^) in multiline mode makes sure the match can only start at the beginning of a line. I used [^<>\n]+ for the first part of the line because I'm working with the file you linked to; if you've removed all the HTML and XML markup, you might be able to use .+ instead.
Then I used (?:(?!end$).+\n)+ to match one or more complete lines that don't consist entirely of end. It's probably more efficient than your [\d\D]+?, but the most important difference is that, when I do match end, I know it's at the beginning of the line (and the $ ensures it's at the end of the line).
A: Try using the following REGEX
_content1 = re.sub(r'(?i).*?\.+(xls|xlsx|pdf|zip|jpg|gif|xml)+?[\d\D]+?(end)',r'',_content1)
I've converted your * operation to *? which is non-greedy which is most likely what you want.
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 3,383
|
\section{Introduction}
Differential renormalization
\cite{FJL,LMV} was invented as an alternative renormalization scheme
useful for calculations strictly in four dimensions \cite{appl,O}.
The basic idea of this renormalization\footnote{In \cite{P}
a renormalization
prescription of differential style was much earlier formulated at the
level of primitively divergent diagrams, using the language of the
$\alpha$-representation.}
is to
represent products of propagators in coordinate
space\footnote{Euclidean and Minkowski spaces can be treated
on the same footing.
For simplicity in what follows Feynman amplitudes will be considered
in four-dimensional Euclidean space-time.}
\begin{equation}
\Pi_{\Gamma} (x_1, \ldots , x_N) =
\prod_{l} G(x_{\pi_+(l)} - x_{\pi_-(l)}),
\label{F0b}
\end{equation}
through derivatives of sufficient order acting on locally integrable
functions.
Here the product is over the lines of a given graph $\Gamma$,
$\pi_{\pm}(l)$ are respectively beginning and the end of a line
$l$,
\begin{equation}
G(x) = P(\partial / \partial x, m) \:
\frac{m}{4\pi^2 \sqrt{x^2}} K_1 (m \sqrt{x^2})
\label{PROP}
\end{equation}
is a propagator, with $P$ polynomial and $K_1$ a modified Bessel
function.
This procedure explicitly characterizes the $R$-operation (i.e.
renormalization at diagrammatic level)
as an extension of the functional $\Pi_{\Gamma}(x_1, \ldots , x_n)$
from the subspace of test functions which vanish
in a vicinity of points where the
coordinates $x_i$ coincide to the whole space
${\cal D}({\cal R}^{4n})$.
The first step within initial version of differential
renormalization \cite{FJL,LMV} is to reduce the problem to the case of
diagrams depending on one coordinate difference.
To do this at low orders of perturbation theory, it suffices to exploit
certain
manipulations based on the Leibniz rule. At higher orders, the only way
of performing such a reduction is to integrate over all coordinate
differences
except one.
However it is then possible to run into infrared problems
since this `naive' integration generally induces infrared divergences.
In \cite{S}
the original version of differential renormalization was supplied
with simple prescriptions
which enabled infrared troubles to be avoided so that
differentially renormalized expressions could be found
with no more difficulty than determining the corresponding
counterterms
in dimensional renormalization. It was also shown that
in writing down
differentially renormalized quantities it is very useful to apply
calculational experience based on dimensional regularization.
The second step
\cite{FJL,LMV}
is performed with prescriptions of the following type:
\begin{eqnarray}
\frac{1}{x^4} & \to &
- \frac{1}{4} \Box \frac{\ln \mu^2 x^2}{x^2},
\label{F1a}\\
\frac{\ln \mu^2 x^2}{x^4} & \to &
- \frac{1}{8} \Box
\frac{\ln^2 \mu^2 x^2 + 2 \ln \mu'^2 x^2}{x^2},
\label{F1b}\\
\frac{1}{x^6} & \to &
- \frac{1}{32} \Box^2 \; \frac{\ln \mu^2 x^2}{x^2},
\label{F1c}
\end{eqnarray}
etc., where $\Box = \partial_{\alpha} \partial_{\alpha}$ is the usual Laplacian,
$x^4 = (x^2)^2, x^6 = (x^2)^3$, and
$\mu,\mu'$ are massive parameters which play the role of
subtraction points.
For $x \neq 0$, the expressions in the left-hand side and
the right-hand side of (\ref{F1a}--\ref{F1c}) are identical. By definition,
the extension of
functionals in the left-hand side from the subspace of test functions which
vanish near $x=0$ to the whole space is determined by the right-hand side.
Note that all the derivatives involved are understood in
the distributional sense, i.e. a derivative $D^{\alpha}f$ of
a distribution $f$ acts on a test function $\phi$ as
\begin{equation}
(D^{\alpha}f,\phi) = (-1)^{| \alpha |}(f,D^{\alpha} \phi),
\label{DD} \end{equation}
$| \alpha |$ being the order of the derivative.
In \cite{SZ} a second version of differential renormalization
was presented in the case of scalar massless logarithmically
divergent diagrams.
It was
based on `pulling out' another differential operator instead of the
Laplacian. In particular, (\ref{F1a}) is replaced by
\begin{equation}
\frac{1}{x^4} \to
\hat{S} \: \frac{\ln \mu^2 x^2}{x^4},
\label{F1aa}
\end{equation}
where
\begin{equation}
\hat{S} = \frac{1}{2} \frac{\partial}{\partial x_{\alpha}} x_{\alpha}.
\label{SH0}
\end{equation}
Within this version, there is no necessity of reducing the problem of
renormalization to propagator-type diagrams.
(This reduction is as usual important in
renormalization group calculations --- see below.) Thus there is no
asymmetry of treating vertices of the given graph.
The purpose of this paper is to present a general prescription of
this version of differential
renormalization which is applicable for arbitrary diagrams
including massive ones.
The status
of the renormalized action principle within differential
renormalization will be also discussed. Another task is to show
that some constants that naturally arise within this approach \cite{SZ}
are straightforwardly related to the renormalization group coefficients.
It will be proved that the beta function and anomalous dimensions
are expressed through these constants
by the same formulae that,
in the case of the MS scheme, the RG coefficients are expressed
through counterterms.
Note that the differential renormalization happens to be a
mass-independent scheme.
The plan of the paper is as follows.
In the next section necessary differential operators similar to
(\ref{SH0})
will be presented and standard formulae for the $R$-operation
will be listed. Then in Section 3 renormalization of massless lower
order diagrams is characterized. In Section 4 an auxiliary technique
necessary for renormalization in the massive case is introduced
through examples of
lower order graphs.
In Section 5 the general prescriptions are formulated and justified.
Section 6 is devoted to discussion of the action principle within
differential renormalization. In Section 7 explicit formulae for RG
coefficients will be derived. Finally, Section 8 contains discussion of
the results obtained.
\section{Notation}
\subsection{Differential operators}
Let us define the following differential operator:
\begin{equation}
\hat{S}_{\underline{x} } = \frac{1}{2} \sum_{i=1}^{n}
\frac{\partial}{\partial x_{i \alpha}}
(x_{i \alpha} - \overline{x_{\alpha}}),
\label{F1}
\end{equation}
where $\underline{x} \equiv x_1, \ldots, x_n$ is a set of $n$
four-dimensional variables, and
$\overline{x} = \frac{1}{n} \sum x_i$.
If $F(\underline{x} )$ is a translationally invariant function, i.e.
$F(\underline{x} +a)=F(\underline{x} )$, then
$F(\underline{x} )=f(\underline{u} )$ for
$u_i=x_i-x_{i_0}, \; i \neq i_0$ and
\begin{equation}
\hat{S}_{\underline{x} }F(\underline{x} )
=\hat{S}_{\underline{u} }f(\underline{u} ),
\label{F1ab}
\end{equation}
where
\begin{equation}\hat{S}_{\underline{u} } = \frac{1}{2} \sum_{i \neq i_0}
\frac{\partial}{\partial u_{i \alpha}} u_{i \alpha}.
\label{F2}
\end{equation}
Since the Feynman amplitudes are translationally invariant we will use
subsequently this form for the operator $\hat{S}$.
In fact, the homogeneity properties of Feynman amplitudes play an essential
role. If $f(u_1, \ldots, u_{n-1})$ is a homogeneous function or
distribution of degree $\lambda$ then
\[
\hat{S}f = \frac{1}{2} (\lambda + 4(n-1)) f .
\]
Note that the operator $\hat{S}$ involves a preliminary multiplication
by variables $u_i$ which vanish at points where initial
amplitudes are singular.
Correspondingly, these singularities are reduced.
In case the ultraviolet
divergence is logarithmic, it disappears if the subsequent
differentiation is understood in the distributional sense --- see
(\ref{DD}).
In the case of linear divergences multiplication by a monomial of the
first degree in coordinates is not sufficient. A second order monomial
is necessary so that it is natural to apply the following operator
\begin{equation}\hat{S}^{(1)} = \frac{1}{4} \sum_{i,j,\alpha , \beta}
\frac{\partial}{\partial x_{i \alpha}} \frac{\partial}{\partial x_{j \beta}}
(x_{i \alpha} - \overline{x}_{\alpha}) (x_{j \beta} - \overline{x}_{\beta}) .
\label{F25}
\end{equation}
For massless graphs, it is sufficient to
apply (\ref{F2}), (\ref{F25}) and their generalizations. If
massive lines are present, we may use homogeneity of Feynman amplitudes with
respect to coordinates and inverse masses. Then a natural analog
of (\ref{F2}) is given by
\begin{equation}\hat{S} = \frac{1}{2} \sum_{i} \frac{\partial}{\partial u_{i}} u_{i}
- \frac{1}{2} \sum_{l}
m_{l} \frac{\partial}{\partial m_{l}} ,
\label{SXM}
\end{equation}
since differentiation in masses also improves the ultraviolet
behaviour.
In the general case of degree of divergence $\omega$ let us apply
the following differential operator:
\begin{equation}\hat{S}^{(\omega)} =
N \{ \underbrace{ \hat{S}^0 \ldots \hat{S}^0 }_{ \omega } \} ,
\label{F29}
\end{equation}
where $\omega$ is the degree of divergence,
$\hat{S}^0 \equiv \hat{S}$ is defined by (\ref{SXM}),
and the symbol of the $N$-product implies that all the derivatives
$\frac{\partial}{\partial u_{i}}$ are to the
left of all $u_{i'}$ while all the derivatives
$\frac{\partial}{\partial m_{j}}$ are to the right of $m_{j'}$.
It is not difficult to show that
\begin{equation}\hat{S}^{(\omega)} = \hat{S} (\hat{S} + 1/2)
\ldots (\hat{S} + \omega/2 ) .
\label{F31}
\end{equation}
In particular
\begin{eqnarray}
\hat{S}^{(1)} = \hat{S} (\hat{S} + 1/2),
\label{S1} \\
\hat{S}^{(2)} = \hat{S} (\hat{S} + 1/2 )
(\hat{S} + 1) .
\label{S2}
\end{eqnarray}
The following commutation relation will be also of use:
\begin{equation}
\ln^{k} \mu^2 x^2 = \frac{1}{k+1} \left(
\hat{S} \ln^{k+1} \mu^2 x^2 -
\ln^{k+1} \mu^2 x^2 \hat{S} \right) .
\label{CR1}
\end{equation}
Its generalization for $k=0$ in the case of the operator
$\hat{S}^{(\omega)}$ looks like
\begin{equation}
1 = a_{\omega} \hat{S}^{(\omega)} \ln \mu^2 x^2
- ( \ln \mu^2 x^2 - 4 b_{\omega} ) (\hat{S} + \omega/2)
\label{CR1a}
\end{equation}
and is understood in the sense that it acts on a quantity that
vanishes after the action of the square of the
operator $(\hat{S} + \omega/2)$ (in the
second term of the right-hand side the second and higher powers of this
operator are omitted). Here
\[ a_{\omega} = (-2)^{\omega}/\omega!, \;
b_{\omega} = 1 + 1/2 + \ldots + 1/\omega. \]
Generalizations of (\ref{CR1a}) for arbitrary $k$ can be also derived
but we shall not write them explicitly.
Furthermore, we shall need the following commutation relation:
\begin{equation}
\left(\hat{S}_{\Gamma} + \omega_{\Gamma}/2 \right) \Pi_{\Gamma}
= \Pi_{\Gamma \setminus \gamma}
\left(\hat{S}_{\gamma} + \omega_{\gamma}/2 \right) \Pi_{\gamma},
\label{CR2}
\end{equation}
where $\Gamma \setminus \gamma$ denotes the subgraph which consists of lines
that do not belong to the subgraph $\gamma$.
\subsection{$R$-operation}
Unrenormalized Feynman amplitudes are obtained from the
products $\Pi_{\Gamma}$
by integrating over coordinates associated with internal
vertices:
\begin{equation}
F_{\Gamma}(x_1, \ldots, x_n) =
\int \mbox{d} x_{n+1} \ldots x_N \Pi_{\Gamma} (x_1, \ldots, x_N) .
\label{FPI} \end{equation}
The ultraviolet divergences manifest themselves through local
non-integrability of the function $\Pi_{\Gamma}$. The $R$-operation
transforms this function into a locally integrable function
$R\Pi_{\Gamma}$ which therefore can be naturally regarded as a
distribution.
The integration at large $x_i$ does not influence
the ultraviolet divergences so that when defining
$R\Pi_{\Gamma}$ all the vertices can be treated as external.
As is well-known, the renormalization can be based either on subtractions
for each complete subgraph (a subgraph is called complete if,
in case it contains the endpoints of some line,
it necessarily contains the line itself, i.e. from $\pi_{\pm}(l) \in \gamma$
it follows that $l \in \gamma$),
or for all 1PI subgraphs.
The first type of renormalization was used in many early works on
renormalization --- see, e.g. \cite{BP,H} and is designed for
theories with Lagrangians and composite operators with normal
ordering.
The corresponding $R$-operation
acts on the Feynman amplitude
$\Pi_{\Gamma}$ for the graph $\Gamma$ as
\begin{eqnarray}
R\Pi_{\Gamma} = \sum_{{\cal V} =
{\cal V}_1 \cup \ldots \cup {\cal V}_j}
\Lambda({\cal V}_1) \ldots \Lambda({\cal V}_j) \Pi_{\Gamma} \nonumber \\
\equiv R'\Pi_{\Gamma} + \Lambda(\Gamma) \Pi_{\Gamma} .
\label{ROP}
\end{eqnarray}
The sum is over all decompositions of the set of vertices
$\cal V$ of the graph $\Gamma$ into non-empty non-intersecting subsets
${\cal V}_1, \ldots, {\cal V}_j$. Moreover,
$\Lambda({\cal V}_i)$ is the counterterm operation for the subgraph
$\gamma({\cal V}_i)$
composed of vertices ${\cal V}_i$ and all lines that are internal to
these vertices.
Remember that $\Lambda ({\cal V}_i)=1$ if $\gamma({\cal V}_i)$ is an
isolated
vertex, and $\Lambda ({\cal V}_i)=0$, if $\gamma({\cal V}_i)$
is not an 1PI divergent subgraph.
The operation $R'$
is called incomplete $R$-operation. This operation removes all
subdivergences of the diagram but does not include the overall
counterterm $\Lambda(\Gamma)$.
This implies that the function
$R'\Pi_{\Gamma}$ is locally integrable in the space of coordinates
except at the point where all the coordinates coincide.
Thus, the problem reduces to the extension of this function
to a distribution defined on the whole space.
For many reasons, a second type of renormalization is commonly
used.\footnote{For example, the scheme based on subtractions at zero
momenta is in the first case the BPH renormalization \cite{BP,H} while
its analog of the second type is the BPHZ renormalization \cite{Z}.
For dimensional renormalization, only the second type is
used in practice. In contrast to the first type, it provides a
mass independent renormalization scheme.}
The corresponding $R$-operation looks like
\begin{eqnarray}
R\Pi_{\Gamma} = \sum_{\gamma_1, \ldots, \gamma_j}
\Delta(\gamma_1) \ldots \Delta(\gamma_j) \Pi_{\Gamma} \nonumber \\
\equiv R'\Pi_{\Gamma} + \Delta(\Gamma) \Pi_{\Gamma} .
\label{R1PI}
\end{eqnarray}
where $\Delta(\gamma)$ is the corresponding counterterm operation, and
the sum is over all sets $\{\gamma_1, \ldots, \gamma_j\}$
of disjoint divergent 1PI subgraphs,
with $\Delta(\emptyset)=1$.
Note that these two types of renormalization coincide in the massless
case, due to zero values of massless vacuum diagrams.
\section{Lower order examples in the massless case}
In the case of graph of Fig.~1a
for $u \neq 0$ we have
\begin{equation}\Pi_{1a}(u) \equiv
\frac{1}{16\pi^4} \frac{1}{u^4} =
\frac{1}{16\pi^4} \hat{S}\, \frac{\ln \mu^2 u^2}{u^4} .
\label{F3}
\end{equation}
The left-hand side of (\ref{F3}) is ill-defined as a
distribution because this function is non-integrable in the vicinity
of the point $u = 0$. However the right-hand side is correctly defined
as a distribution everywhere in ${\cal R}^4$, since the operator $\hat{S}$
involves preliminary multiplication by $u_{\alpha}$, and the function
$u_{\alpha} / u^4$ is already locally integrable.
By definition, an extension of the functional in the
left-hand side to the whole space ${\cal R}^4$ is determined
with the help of the
right-hand side. The arbitrariness
of this extension is explicitly
contained in the parameter $\mu$ which plays the same role as the
corresponding parameters in the frameworks of dimensional and analytic
renormalizations.
Let us thus define the `differentially
renormalized' Feynman amplitude for the graph 1a by
\begin{equation}
R \, \Pi_{1a} = \frac{1}{16\pi^4} \hat{S} \frac{\ln \mu^2 u^2}{u^4}
\label{F9}
\end{equation}
so that, in accordance with (\ref{DD}),
the action of this distribution on a test function
$\phi(u) \in {\cal D} ({\cal R}^4)$ is defined by
\begin{equation}
(R \Pi_{1a}, \phi) = -
\frac{1}{16\pi^4}
\int {\rm d} u \frac{\ln \mu^2 u^2}{u^4}
u_{\alpha} \frac{\partial}{\partial u_{\alpha}} \phi(u).
\label{F9z}
\end{equation}
The counterterm operation $\Delta (\Gamma)$ for the graph 1a can be
`formally' represented as
\begin{equation}
\Delta \Pi_{1a} = \frac{1}{16\pi^4} \left( \hat{S}\, \frac{\ln \mu^2 u^2}{u^4}
- \frac{1}{u^4} \right) .
\label{F9a}
\end{equation}
This quantity alone (as well as other counterterms and unrenormalized
or partially renormalized Feynman amplitudes) does not make sense as a
functional on the whole space of test functions.
However one can combine the sum of contributions of counterterm
operations into renormalized quantities in such a way that all the
obtained combinations will be sensible under integration.
It should be noted that the counterterm
(\ref{F9a}) vanishes for $u \neq 0$.
The functional $\hat{S} R \Pi_{1a}$
equals zero for $u \neq 0$ and therefore its support coincides with the
point $u = 0$. It is easy to verify that its action
on test functions that are zero at this point is zero.
Hence
\begin{equation}
\hat{S}R \Pi_{1a} = c_{1a} \delta^{(4)} (u) .
\label{F5}
\end{equation}
To calculate the constant $c_{1a}$ one may introduce analytic
regularization, to write (\ref{F9}) through
$\frac{\mbox{d}}{\mbox{d} \lambda} \hat{S} (x^2)^{\lambda-2}$ at $\lambda=0$ and apply the
expansion
\begin{equation}
(x^2)^{\lambda-2} = \frac{\pi^2}{\lambda} \delta^{(4)} (x) + O(\lambda^0),
\label{L}
\end{equation}
with $\lambda$ in the neighbourhood of the origin of the complex plane.
The result is
\begin{equation} c_{1a} = \frac{1}{16 \pi^2}. \label{C1} \end{equation}
The next example is the graph of Fig.~1b.
The subdivergence is removed according to prescription for the graph 1a.
The `incomplete' $R$-operation (i.e. without the last subtraction), when
applied to the Feynman amplitude under consideration, gives
\begin{equation}
R' \, \Pi_{1b} \equiv (1+\Delta(\gamma_1)) \Pi_{1b}
= \frac{1}{(4 \pi^2)^4}
\frac{1}{v^2 (u-v)^2} \hat{S}\, \frac{\ln \mu^2 u^2}{u^4} ,
\label{F6}
\end{equation}
where $\gamma_1$ is graph 1a as a subgraph in 1b.
Using (\ref{CR1}) at $k=0$ one observes that if not all the coordinates
$u,v,0$ of the graph 1b coincide the following
equation is valid:
\begin{equation}
R' \, \Pi_{1b} = \frac{1}{(4 \pi^2)^4} \left\{
\hat{S}_{u,v}\, \frac{\ln \mu'^2 v^2}{v^2 (u-v)^2} \hat{S}_{u}\, \frac{\ln \mu^2
u^2}{u^4}
- \frac{\ln \mu'^2 v^2}{v^2 (u-v)^2} \,\hat{S}_u R \frac{\ln \mu^2
u^2}{u^4} \right\} .
\label{F7}
\end{equation}
Let us now use (\ref{F5}) and the equation
\begin{equation}
\frac{\ln \mu' v^2}{v^4} = \frac{1}{2}
\hat{S}\, \frac{\ln^2 \mu'^2 v^2}{v^4}, \; \; v \neq 0 ,
\label{F7a}
\end{equation}
which enables us to differentially renormalize graph 1a with an additional
logarithm:
\begin{equation}
R \frac{\ln \mu' v^2}{v^4} \equiv R \ln \mu' v^2 \Pi_{\Gamma/\gamma_1}
=\frac{1}{2} \hat{S}\, \frac{\ln^2 \mu'^2 v^2}{v^4}.
\label{F77a}
\end{equation}
After that, as for Fig.~1a, the right-hand side of (\ref{F7})
turns out to be defined as a functional on the whole space
${\cal D}({\cal R}^8)$. The differentially renormalized Feynman amplitude for
the graph 1b is defined as the corresponding extension of functional
(\ref{F7}) from the subspace of test functions vanishing in a
vicinity of the point $u=v=0$.
As a result we obtain
\begin{eqnarray}
R \, \Pi_{1b} & = &
\hat{S} \ln \mu'^2 v^2 \, R' \Pi_{\Gamma}
- c_{\gamma_1} R \ln \mu'^2 v^2 \, \Pi_{\Gamma / \gamma_1}
\nonumber \\
& \equiv &
\frac{1}{(4 \pi^2)^4} \left\{
\hat{S}_{u,v}\, \frac{\ln \mu'^2 v^2}{v^2 (u-v)^2} \hat{S}_u \, \frac{\ln \mu^2
u^2}{u^4}
- \frac{1}{2} c_1 \hat{S}_v \,
\frac{\ln^2 \mu'^2 v^2}{v^4} \delta (u) \right\},
\label{F8}
\end{eqnarray}
with $\Gamma =$ 1b, $\gamma =$ 1a.
The arbitrariness of the subtraction operation for the graph 1b itself
appears explicitly in the parameter $\mu'$.
It is possible, for example, to introduce a unique mass scale
$\mu$ that determines an energy scale for perturbation
theory and fix all $\mu$-parameters which may arise as
$\mu_{\Gamma} = \zeta_{\Gamma} \mu$, with some constants $\zeta_{\Gamma}$.
With this prescription $\mu$ determines a one-parametrical subgroup
of RG transformations
and is quite analogous, in its character, to
the 't~Hooft mass $\mu$ in dimensional renormalization.
The action of the counterterm operation for the graph 1b is formally
written as
\begin{equation}
\Delta \Pi_{1b} = R \Pi_{1b} - R' \Pi_{1b} .
\label{F10}
\end{equation}
The counterterm (\ref{F10}) vanishes everywhere outside the point
$u = v = 0$.
Note that for all other points
\[
\hat{S} \, R \, \Pi_{1b} =
c_{1a} \hat{S} \Pi_{\Gamma/\gamma_1} .
\]
Therefore the left-hand and right-hand sides of this equation differ
by a functional with support localized at the point $u=v=0$.
Using the same arguments as in the first example we obtain
\begin{eqnarray}
\hat{S} \, R \, \Pi_{1b}
& = &
c_{\Gamma} \delta^{(8)}(u,v) +
c_1 \hat{S}\, \frac{\ln \mu'^2 v^2}{v^4} \delta^{(4)} (u)
\nonumber \\
& \equiv &
c_{\Gamma} \delta^{(8)}(u,v) + c_{\gamma_1} R \Pi_{\Gamma / \gamma_1},
\label{F10a}
\end{eqnarray}
with some constant $c_{\Gamma} \equiv c_{1b}$.
To calculate this constant let us integrate (\ref{F10a}) over $u$:
\begin{equation}
c_{\Gamma} \delta^{(4)}(v) = \frac{1}{(4 \pi^2)^4}
\left\{ \hat{S}_{v} \frac{1}{v^2}
\int \mbox{d} u \frac{1}{(u-v)^2} \hat{S}_{u} \frac{\ln \mu^2 u^2}{u^4}
- \pi^2 \hat{S}_{v} \frac{\ln \mu^2 v^2}{v^4} \right\} .
\label{2l1}
\end{equation}
The integral in the braces can be evaluated by introducing analytic
regularization
\begin{equation}
\int \mbox{d} u \frac{1}{(u-v)^2} \hat{S}_{u} \frac{\ln \mu^2 u^2}{u^4}
= \frac{\mbox{d}}{\mbox{d} \lambda} \left. \left[ (\mu^2)^{\lambda} \lambda
\int \mbox{d} u \frac{1}{(u-v)^2(u^2)^{2-\lambda}}
\right] \right|_{\lambda=0}
\label{2l2}
\end{equation}
and applying one-loop massless formula in four dimensions
\begin{equation}
\int \mbox{d} u \frac{1}{(u^2)^{\lambda_1}((u-v)^2)^{\lambda_2}}
=\pi^2 G(\lambda_1,\lambda_2)
\frac{1}{(v^2)^{\lambda_1+\lambda_2-2}},
\label{2l3}
\end{equation}
with four-dimensional $G$-function given by
\begin{equation}
G(\lambda_1,\lambda_2) =
\frac{\Gamma(\lambda_1+\lambda_2-2) \Gamma(2-\lambda_1) \Gamma(2-
\lambda_2)}{\Gamma(\lambda_1)\Gamma(\lambda_2)\Gamma(4-\lambda_1-\lambda_2)} .
\label{2l4}
\end{equation}
Therefore the first term in the braces in (\ref{2l1}) is rewritten as
\begin{equation}
\pi^2 \hat{S}_{v} \frac{1}{v^2}
\frac{\mbox{d}}{\mbox{d} \lambda} \left. \left[ (1+\lambda)
(\mu^2)^{\lambda} (v^2)^{\lambda-1} \right] \right|_{\lambda=0} =
\pi^2 \hat{S}_{v} \left[ \frac{1}{v^4} + \frac{\ln \mu^2 v^2}{v^4}
\right] .
\label{2l5}
\end{equation}
As a result we obtain the value $c_{1b}= 1/(16\pi^2)^2$.
For Fig.~1c
using relation (\ref{F9a}) we have
\begin{eqnarray}
R'\, \Pi_{1c} & = & [1 + \Delta(\gamma_1)
+ \Delta(\gamma_2) ] \Pi_{1c} \nonumber \\
& = & \frac{1}{(4\pi^2)^4} \left\{
\frac{1}{(u-v)^4} \hat{S}\, \frac{\ln \mu^2 u^2}{u^4}
+ \frac{1}{u^4} \hat{S}\, \frac{\ln \mu^2 (u-v)^2}{(u-v)^4}
- \frac{1}{u^4 (u-v)^4} \right\},
\label{F11}
\end{eqnarray}
where $\gamma_1$ and $\gamma_2$ are respectively left and right simple
loops 1a in 1c.
As in the case of Fig.~1b, using equations (\ref{F5}),
(\ref{F7a}), (\ref{CR1})
and extending
functional (\ref{F11}) from the space of test functions determined in
${\cal R}^8$ with `deleted' point $u=v=0$ to the whole space
${\cal D}({\cal R}^8)$
we come to the following result:
\begin{equation}
R \, \Pi_{1c} =
\hat{S} \ln \mu'^2 v^2 \, R' \Pi_{\Gamma}
- c_{\gamma_1} (R \ln \mu'^2 v^2 \Pi_{\Gamma / \gamma_1}
+ R \ln \mu'^2 v^2 \Pi_{\Gamma / \gamma_2} ) .
\label{F12}
\end{equation}
In the case of the `catseye' diagram Fig.~1d we have
\begin{equation}
R'\, \Pi_{1d} = [1 + \Delta(\gamma_1) + \Delta(\gamma_{21})
+ \Delta(\gamma_{22}) ] \Pi_{1d} ,
\label{F13}
\end{equation}
where
\begin{equation}
(1 + \Delta(\gamma_1)) \Pi_{1d} =
\frac{1}{(4\pi^2)^6} \frac{1}{u^2 v^2 (u-w)^2 (v-w)^2}
\hat{S}\, \frac{\ln \mu_1^2 (u-v)^2}{(u-v)^4} ,
\label{F14}
\end{equation}
\begin{eqnarray}
\Delta(\gamma_{21}) \Pi_{1d} =
\frac{1}{(4\pi^2)^6} \frac{1}{(u-w)^2 (v-w)^2} \left\{
\hat{S}\, \frac{\ln \mu_2^2 u^2}{u^2 v^2}
\hat{S}\, \frac{\ln \mu_1^2 (u-v)^2}{(u-v)^4} \right. \nonumber \\
\left. - \frac{1}{2} c_1 \hat{S}\,
\frac{\ln^2 \mu_2^2 u^2}{u^4} \delta (u-v)
- \frac{1}{u^2 v^2} \hat{S}\, \frac{\ln \mu_1^2 (u-v)^2}{(u-v)^4}
\right\} ,
\label{F15}
\end{eqnarray}
and $\Delta(\gamma_{22}) \Pi_{1d}$ is obtained by replacing
$u$ by $w-u$ and $v$ by $w-v$. Here $\gamma_1$ is the central simple
loop; $\gamma_{21}$ and $\gamma_{22}$ are respectively left and right
graphs 1b as subgraphs of 1d.
Let us consider the space ${\cal R}^{12}$ with deleted origin $u=v=w=0$,
For each point in the vicinity of the origin we have at least one of the
following two possibilities:
({\em i}) $u \neq 0$ or/and $v \neq 0$,
({\em ii}) $u \neq w$ or/and $v \neq w$. We consider first case ({\em ii}).
Then the contribution from the counterterm of the subgraph $\gamma_{22}$
disappears and $R'$ takes the form
\begin{equation}
R'\Pi_{1d} =
\frac{1}{(4\pi^2)^6} \frac{1}{(u-w)^2 (v-w)^2}
R \Pi_{\gamma_{22}} .
\label{R1d}
\end{equation}
Using the
procedure described above, in particular (\ref{CR1}) at $k=0$,
(\ref{F5}) and (\ref{F10a}),
$R' \Pi_{1d}$ can be represented in the form
\begin{equation}
R' \, \Pi_{\Gamma} = \hat{S} \ln \mu^2 w^2 R' \, \Pi_{\Gamma}
- c_{\gamma_1} \ln \mu^2 w^2 R' \, \Pi_{\Gamma / \gamma_1}
- c_{\gamma_{22}} \ln \mu^2 w^2 R' \Pi_{\Gamma / \gamma_{22}} .
\label{F15a}
\end{equation}
The functional $\hat{S} \ln \mu^2 w^2 R \, \Pi_{\Gamma}$ is naturally
extended to the whole ${\cal R}^{12}$; the extension of simple loops
(with `additional' logarithms)
$\ln \mu^2 w^2 R' \, \Pi_{\Gamma / \gamma_{2i}}, \; i=1,2$
was described in the case of Fig.~1b. For the graph of Fig.~1c with
an additional
logarithm $\ln \mu^2 w^2 R' \, \Pi_{\Gamma / \gamma_1}$,
the procedure that was used for
Fig.~1c itself and applies (\ref{CR1}) at $k=2$ can be straightforwardly
generalized. A similar expression is obtained for the case ({\em i}).
As a result we obtain the following expression for the differentially
renormalized diagram 1d which is valid in the whole space:
\begin{equation}
R \, \Pi_{\Gamma} = \hat{S} \ln \mu^2 w^2 R' \, \Pi_{\Gamma}
- c_{\gamma_{1a}} R \ln \mu^2 w^2 \Pi_{\Gamma / \gamma_1}
- c_{\gamma_{1b}} R \ln \mu^2 w^2 \Pi_{\Gamma / \gamma_{21}}
- c_{\gamma_{1b}} R \ln \mu^2 w^2 \Pi_{\Gamma / \gamma_{22}}.
\label{F16}
\end{equation}
\section{Lower order examples in the massive case}
Renormalization of the diagrams of Fig.~1 in the massive case is performed
by formulae similar to the previous section, with
$\hat{S}$ given by (\ref{SXM}), e.g. for
graph 1a
\begin{equation}
R \, \Pi_{1a} \equiv R G(x)^2
= \hat{S} \ln \mu^2 x^2 G(x)^2.
\label{F99}
\end{equation}
The constants $c_{\gamma}$ involved here
have the same values as in the massless case.
To see the additional problems that can arise
let us consider other simple examples.
New problems
can appear for vacuum graphs or those with one external vertex,
e.g. for tadpoles.
Let us distinguish contributions (see Fig.~2a) which are formally given
by $G(0)$
--- the value of propagator at the origin where it is singular.
They can
exist separately or belong to other graphs. In the latter
case, such contributions appear as independent factors. All the other tadpoles
are products of propagators integrated over all coordinates but one.
The methods of differential renormalization can be applied for these
tadpoles without problem.
Note that even within dimensional regularization
$G(0)$ is not defined without some further prescription.
A standard way of handling this quantity
is write it down as momentum space
$d$-dimensional integral
$\int \mbox{d}^d k/(k^2+m^2)$, then calculate
and renormalize it. In the limit $d\to 4$,
one obtains $G(0)=\frac{1}{16\pi^2}m^2\ln m^2/\mu^2$
(up to finite renormalization).
At $d=4$ it is possible to say that $G(0)$ is understood as the value at
$x=0$ of $G(x)$
from which the singular terms of the small $x$ expansion
\begin{equation}
G(x) = \frac{1}{4\pi^2} \left\{ \frac{1}{x^2}
+ \frac{m^2}{4}\left( \ln m^2 x^2 + 2 \gamma_{\rm E} -1
\right) \right\} + o(x^2),
\end{equation}
are subtracted
($\gamma_{\rm E}$ the Euler constant). This procedure
leads to the reasonable result
$G(0)=\frac{1}{16\pi^2}m^2 \left( \ln m^2/\mu^2 + 2 \gamma_{\rm E} -1\right)$.
At first sight the strategy of extending functionals from some space of test
functions
to the whole space does not have anything to do with defining $G(0)$.
Therefore the general recipes of differential renormalization seem to be
of no use here. One possibility is to introduce
additional prescriptions, and there can be at least
two variants: to set $G(0)=0$ or to take
$G(0)=\frac{1}{16\pi^2}m^2\ln m^2/\mu^2$ (or some similar value for non-scalar
propagators) --- the former variant of course reduces to the latter one at
$\mu=m$.
There is also a new problem of another type in the massive case. Consider
the `setting sun' diagram shown in Fig.~2b.
In the case of
renormalization (\ref{ROP}) based on complete subgraphs
the only subtraction involved is for the overall graph.
However the operator (\ref{S2})
is not sufficient to remove all the ultraviolet divergences because
differentiation in the mass removes only the most singular part of
$G(x)^3$.
If one tries to perform the second type of renormalization
(\ref{R1PI}) in the
differential style
one will observe that the insertion of counterterms for three (overlapping)
simple loops does not change the diagram at $x \neq 0$. Furthermore,
in the whole space, insertion of these counterterms does not make
sense. It seems that we do not have `enough space' to perform consecutive
extensions of the initial functional in two steps that correspond
respectively to renormalization of above three subgraphs and the graph
itself.
To overcome these problems let us exploit the following trick that was
used in \cite{L}. Instead of propagators in coordinate space
(\ref{PROP}), let us consider its Fourier transform in respect to two
additional components, $m_1$ and $m_2$ by considering the square of
mass $m^2$ in the propagator as the square of this two-dimensional
vector $m=(m_1,m_2)$.
Denoting the corresponding coordinate-space variable by
$y=(y_1,y_2)$ we obtain the massless propagator in six-dimensional
space
\begin{equation}
G(x,y) = \frac{1}{4\pi^3} \frac{1}{(x^2+y^2)^2}
\label{Lu}
\end{equation}
which satisfies
\begin{equation}
(\Box_x + \Box_y) G(x,y) = - \delta^{(4)}(x)\delta^{(2)}(y).
\end{equation}
For general Feynman diagram let us introduce Fourier transformation
for each massive line. As a result $\Pi_{\Gamma}$ is expressed
as a product of propagators $G(x,y)$ depending on $4N$ usual
coordinate-space variables and $2N_m$ additional ones, $N_m$ being the
number of massive lines.
Remember that
Feynman amplitudes must be well-defined distributions in order that
Fourier transforms to momentum space
(where physical quantities are calculated)
are possible.
Therefore it is necessary
to perform integration over coordinates; at this step locally
non-integrable singularities manifest themselves as the source of the
ultraviolet divergences. Now, for the same reason, it is natural to
consider the
product of propagators $\Pi_{\Gamma}(\underline{x}, \underline{y})$, with
$\underline{y} = (y_1, \ldots, y_{N_m})$,
as a distribution not only in $x_i$ but also in $y_i$. Indeed, in
the end it is necessary to perform inverse Fourier transformation and
come back to masses (and put them equal to each other if there was
initially only one mass).
In this approach the ultraviolet divergences manifest
themselves in integrations over small coordinate differences {\em and}
also the variables $y_i$.
Note that it is possible to introduce Fourier
transformation in masses even for massless lines. In this case, it is
necessary, in the end of calculation, to integrate over the
corresponding $y$-variables.
Therefore, as products of propagators, we now have
\begin{equation}
\Pi_{\Gamma} (\underline{x},\underline{y}) =
\prod_{l} \frac{1}{4\pi^3}
\frac{1}{((x_{\pi_+(l)} - x_{\pi_-(l)})^2+y_l^2)^2} .
\label{PXY}
\end{equation}
One can show that usual power counting turns out to be
the same and is governed by the same degree of divergence $\omega$.
It is now possible to apply above formulae of differential
renormalization (for the massless case) using the operator
\begin{equation}
\hat{S}_{x,y} = \frac{1}{2}
\left( \sum_{i} \frac{\partial}{\partial x_{i }} x_{i }
+ \sum_{l} \frac{\partial}{\partial y_{l}} y_{l} \right) .
\label{SXY}
\end{equation}
We can now write the value of the propagator at $x=0$
\begin{equation}
G(x,y)\big |_{x=0} = \frac{1}{4\pi^3} \frac{1}{(y^2)^2} ,
\label{G0}
\end{equation}
without running into division by zero. To make sense of it for
all $y$,
i.e. as distribution in $y$,
let us apply the operator (since the divergence is quadratic)
$\hat{S}^{(2)}_{y}$ given by (\ref{S2}):
\begin{equation}
R G(0,y) = \frac{1}{4\pi^3} 2\hat{S}^{(2)}_y \frac{\ln \mu^2 y^2}{(y^2)^2} .
\label{G00}
\end{equation}
Its inverse Fourier transformation in $y$ gives
\begin{equation}
R G(0;m) = \frac{1}{16\pi^2}m^2
\left( \ln m^2/4\mu^2 + 1 + 2\gamma_{\rm E} \right) .
\label{G0m}
\end{equation}
The renormalized simple loop (\ref{F99}) is rewritten in the
language of $y$-variables as
\begin{equation}
R \, \Pi_{1a} = \frac{1}{(4\pi^3)^2} \hat{S}_{x,y_1,y_2}
\frac{\ln \mu^2 V}{(x^2+y_1^2)^2 (x^2+y_2^2)^2} ,
\label{F90}
\end{equation}
with the corresponding counterterm represented as
\begin{equation}
\Delta \Pi_{1a} = \frac{1}{(4\pi^3)^2}
\left[ \hat{S}_{x,y_1,y_2}
\frac{\ln \mu^2 V}{(x^2+y_1^2)^2 (x^2+y_2^2)^2}
- \frac{1}{(x^2+y_1^2)^2 (x^2+y_2^2)^2} \right] .
\label{F90a}
\end{equation}
One can use different arguments of the logarithm involved, for
instance, (a) $V=x^2$; (b) $V = y_1^2$ or $V=y_2^2$,
(c) $V = x^2+y_1^2$ or $V = x^2+y_2^2$; (d) $V = x^2+y_1^2+y_2^2$.
It is possible to show that in cases (a) and (b) the limit $m\to 0$ exactly
reproduces the `pure massless' prescription (\ref{F9}).
Let us now return to the `setting-sun' diagram.
In the language of $y$-variables the product of
corresponding propagators is
\begin{equation}
\Pi_{2b} = \prod_{i=1,2,3} \left(
\frac{1}{4\pi^3} \frac{1}{(x^2+y_i^2)^2} \right) .
\label{2a}
\end{equation}
Inserting counterterms for three subgraphs 1a gives
\begin{equation}
R'\Pi_{2b} = \Pi_{2b}
+ G(0,y_3)
\left[ \hat{S}_{x,y_1,y_2}
\ln \mu^2 x^2 G(x,y_1) G(x,y_2) - G(x,y_1) G(x,y_2)
\right] + \ldots ,
\label{R'2a}
\end{equation}
where the dots stand for permutations.
This quantity is meaningful as a functional everywhere except the
origin with respect to the variables $(x,y_1,y_2,y_3)$.
Let us now apply (\ref{CR1a}) at $\omega=2$ and use
the values of one-loop counterterms to write down the incompletely
renormalized diagram as
\begin{equation}
R'\Pi_{2b} = 2\hat{S}^{(2)} \ln \mu^2 V \; R' \Pi_{2b}
- \frac{1}{4\pi^3} (\ln \mu^2 y_3^2 -6) \frac{1}{(y_3^2)^2}
\frac{1}{16\pi^2} \delta^{(4)}(x) \delta^{(2)}(y_1) \delta^{(2)}(y_2) -\ldots ,
\label{R'2a1}
\end{equation}
with $V$ chosen as $V = x^2 + y_1^2 +y_2^2 +y_3^2$.
Now we extend this functional to the whole space of variables
$(x,y_1,y_2,y_3)$ with the help of the renormalization of the $G(0)$
(see (\ref{G00})) and its analog with an additional logarithm:
\begin{equation}
R \frac{\ln \mu^2 y^2}{y^4} =
\hat{S}^{(2)} \; \frac{\ln^2 \mu^2 y^2 + 6 \ln \mu^2 y^2}{y^4}.
\label{PL}
\end{equation}
We obtain
\begin{eqnarray}
R\Pi_{2b} & = &
2\hat{S}^{(2)} \ln \mu^2 V \; R' \Pi_{2b} \nonumber \\
&&{} - \frac{1}{4\pi^3} \hat{S}^{(2)}_{y_1}
\frac{\ln^2 \mu^2 y_1^2 -6\ln \mu^2 y_1^2}{y_1^4}
\frac{1}{16\pi^2} \delta^{(4)}(x) \delta^{(2)}(y_2) \delta^{(2)}(y_3) - \ldots ,
\label{R2a}
\end{eqnarray}
where we did not use a freedom to introduce a new parameter $\mu$
associated with the overall graph.
Let us now calculate the constants $c_{\gamma}$ which enter the following
formula:
\begin{eqnarray}
(\hat{S}+1) R\Pi_{2b} & = &
(c_x \Box_x + \sum_i c_{y_i} \Box_{y_i}) \delta^{(4)}(x)
\prod_{i} \delta^{(2)}(y_i) \nonumber \\
&& {}+ R \ln \mu^2 y_1^2
G(0,y_3) c_{1a} \delta^{(4)}(x) \delta^{(2)}(y_2) \delta^{(2)}(y_3)
+ \ldots .
\label{S+1}
\end{eqnarray}
To calculate $c_x$ we multiply (\ref{S+1}) by $x^2$ and integrate it
over $x$ and $y_i$.
To calculate $c_y$ we integrate (\ref{S+1})
over $x,y_2,y_3$ and perform inverse Fourier transformation in $y_1$.
The results are
\begin{equation}
c_x = \frac{1}{2} \frac{1}{(16 \pi^2)^2} , \;
c_{y_i} = \frac{2}{(16 \pi^2)^2}.
\label{cuy}
\end{equation}
\section{General prescriptions}
By generalizing procedure described in the above examples let us
define a renormalization procedure which will be naturally called as
differential renormalization.
To present quite general prescription let us apply the language of
$y$-variables described in the previous section even in the case when
some of the lines are massless. Let us consider renormalization
of products (\ref{PXY}) multiplied by powers of logarithms:
$ \ln^k (\mu^2_{\Gamma} V_{\Gamma}(\underline{x},\underline{y})) \, \Pi_{\Gamma}$.
Here we imply two possible variants:
$V_{\Gamma}= u^2_{\Gamma} + \sum_{l \in \Gamma} y_l^2$
and
$V_{\Gamma}= \sum_{l \in \Gamma} y_l^2$, where
$u_{\Gamma}=x_i-x_i'$ is any difference variable of the considered
Feynman amplitude.
{\bf Definition.}
Let a renormalization $R$ of the graph $\Gamma$ be given by the following
recursive formulae:
\begin{equation}
R\, \ln^k (\mu^2_{\Gamma} V_{\Gamma}) \Pi_{\Gamma} =
\frac{1}{k+1}
\hat{S} \ln^{k+1} (\mu^2_{\Gamma} V_{\Gamma})
\; R'\, \Pi_{\Gamma} -
\frac{1}{k+1} \sum_{\gamma \subset \Gamma}
R \ln^{k+1} (\mu^2_{\Gamma} V_{\Gamma}) \, {\cal C}_{\gamma}
\Pi_{\Gamma},
\label{F17}
\end{equation}
for $\omega=0$ and integer $k \geq 0$;
\begin{equation}
R\,\Pi_{\Gamma} =
a_{\omega} \hat{S}^{(\omega)} \ln (\mu^2_{\Gamma} V_{\Gamma})
\; R'\, \Pi_{\Gamma} -
\frac{1}{k+1} \sum_{\gamma \subset \Gamma}
R ( \ln (\mu^2 V_{\Gamma}) - 4 b_{\omega} ) (\hat{S} + \omega/2)
{\cal C}_{\gamma}
\Pi_{\Gamma},
\label{F17a}
\end{equation}
for $\omega>0$ and integer $k = 0$, as well by other relations for arbitrary
$\omega$ and $k$ which follow from the corresponding generalizations of
(\ref{CR1a}).
Here $R'$ is incomplete $R$-operation (\ref{ROP}).
The sum in the second term of the right-hand side of (\ref{F17}) and
(\ref{F17a}) runs over all 1PI
proper subgraphs $\gamma$ of $\Gamma$.
Furthermore,
\begin{equation}
\Delta(\Gamma) \Pi_{\Gamma} = R \Pi_{\Gamma} - R'\, \Pi_{\Gamma}
\label{Dl}.
\end{equation}
Finally, the operations ${\cal C}_{\gamma}$
are determined from equations
\begin{eqnarray}
\left( \hat{S} + \omega /2 \right) R\, \Pi_{\Gamma} =
\sum_{\gamma \subseteq \Gamma}
{\cal C}_{\gamma} R \Pi_{\Gamma}
\equiv \sum_{\gamma \subset \Gamma}
R {\cal C}_{\gamma} \Pi_{\Gamma }
+ {\cal C}_{\Gamma} \Pi_{\Gamma },
\label{F18}
\end{eqnarray}
where the operation
${\cal C}_{\gamma}$ inserts a polynomial ${\cal P}_{\gamma}$
of degree $\omega(\gamma)$ in masses
of $\gamma$ and its external momenta into the reduced vertex of the graph
$\Gamma /\gamma$. Symbolically we write
\begin{equation}
{\cal C}_{\gamma} \Pi_{\Gamma } = \Pi_{\Gamma /\gamma} \circ {\cal P}_{\gamma}
\label{CG}
\end{equation}
where $\circ$ denotes the insertion operation. In the language of
coordinate space,
\begin{equation}
{\cal C}_{\Gamma} \Pi_{\Gamma } (\underline{x},\underline{y}) =
{\cal P}_{\Gamma} (\partial /\partial x_i, \partial /\partial y_l) \prod_{i \in \Gamma}
\delta^{(4)}(x_i-x_1) \prod_{l \in \Gamma} \delta^{(2)}(y_l) .
\label{CGC}
\end{equation}
Indeed relations (\ref{F17})--(\ref{F18}) and their
generalizations for arbitrary $\omega$ and $k$ enable us to obtain
$R \ln^k (\mu^2_{\Gamma} V_{\Gamma})\;
\Pi_{\Gamma}$, $\Delta(\Gamma)$ and ${\cal C}_{\Gamma}$ provided we know the corresponding
quantities for all proper subgraphs of $\Gamma$ and its reduced graphs:
in particular, ${\cal P}_{\Gamma}$ is expressed from (\ref{F18}).
The following proposition is valid.
{\bf Proposition.} The procedure $R$ defined by relations
(\ref{F17})--(\ref{F18}) is a correct $R$-operation.
{\em Proof.}
To prove this proposition we should show that ({\em i}) the expression
(\ref{F17}) is finite, ({\em ii}) (\ref{F17}) is obtained from
$R'\Pi_{\Gamma}$ as an extension from the space from deleted origin
of the whole space, i.e. the corresponding counterterm
(\ref{Dl}) is local,
({\em iii}) the quantity $ {\cal C}_{\Gamma} \Pi_{\Gamma }$ found from
(\ref{F18}) is local.
Let us consider, in the space of coordinates, an arbitrary point
$\underline{x^0} \equiv \{ x^0_1, \ldots, x^0_n\}$ in which at least
one of the difference variables (e.g. $u^0_1 = x^0_1-x^0_n$) is
non-zero: $u^0_1 \neq 0$.
In respect to the point
$\underline{x^0}$ all the set of vertices $\cal V$ of the graph
$\Gamma$ is naturally decomposed over non-intersecting subsets
${\cal V}_{r}$, with ${\cal V} = \bigcup_{r} {\cal V}_{r}$,
$x^0_i - x^0_{i'} =0, \; \forall i,i' \in {\cal V}_{r}$, and
$x^0_i - x^0_{i'} \neq 0, \; \forall i \in {\cal V}_{r}, \;
i' \in {\cal V}_{r '}$ for $r \neq r'$. In accordance with the
assumption about the point $\underline{x^0}$, the number of subsets
${\cal V}_{r}$ is not less than two.
Let $\Gamma_{r}$ be subgraphs constructed with help of vertex sets
${\cal V}_{r}$: by definition, each $\Gamma_{r}$ contains any line
that connects a pair of vertices from ${\cal V}_{r}$. Note that the
subgraphs $\Gamma_{r}$ can be one-particle-reducible or disconnected.
Let us denote by $\cal W$ the set of maximal 1PI subgraphs of the
graph $\Gamma^0 = \bigcup_{r} \Gamma _{r}$. Furthermore, let
$\overline{\cal W}$ be the set of all divergent subgraphs of the
graph $\Gamma^0$.
Let $U^0$ be a sufficiently small
vicinity of this point so that for all $\underline{x} \in U^0$
the same properties hold,
$x^0_i - x^0_{i'} =0, \; \forall i,i' \in {\cal V}_{r}$, and
$x^0_i - x^0_{i'} \neq 0, \; \forall i \in {\cal V}_{r}, \;
i' \in {\cal V}_{r '}$ for $r \neq r'$.
In the domain $U^0$, the incomplete $R$-operation $R'$ does not include
counterterms contributed by subgraphs containing vertices from
different subsets ${\cal V}_{r}$. Therefore, in $U^0$, we have the
equation
\begin{equation}
\hat{S} R' \, \Pi_{\Gamma} =
\hat{S} \{ \sum
\Delta(\gamma_1) \ldots \Delta(\gamma_j) \}
\Pi_{\Gamma},
\label{FA2}
\end{equation}
where the sum is over decompositions
${\cal V} = {\cal V}_1 \cup \ldots \cup {\cal V}_j$ such that any
$\gamma_i$ belongs to $\overline{\cal W}$. In other words, the sum in
(\ref{FA2}) can be represented as analogous sum over decompositions in
which any $\gamma_i$ happens to be an element from the set $\cal W$.
After that each of the factors $\Delta(\gamma_i)$ involved transforms
into the $R$-operation $R(\gamma_i)$ that acts on the Feynman
amplitude for the subgraph $\gamma_i$. Thus,
(commutation relations (\ref{CR2}) are used)
\begin{equation}
(\hat{S}+\omega_{\Gamma}/2) R' \, \Pi_{\Gamma} =
\hat{S}
\left\{ \prod_{\gamma \in {\cal W}} (R \Pi_{\gamma}) \right\}
\Pi_{\Gamma \setminus \Gamma^0}
= \sum_{\gamma \in {\cal W}}
((\hat{S}+\omega_{\gamma}/2) R \Pi_{\gamma})
) \prod_{\gamma' \neq \gamma} ( R \Pi_{\gamma '}) \; \Pi_{\Gamma \setminus \Gamma^0}.
\label{FA3}
\end{equation}
Since in $U^0$ the propagators of lines which connect different
elements of ${\cal W}$ are not singular we may apply the Leibniz
rule encoded in (\ref{CR1}), (\ref{CR1a}) and their generalizations. For
example, in the case $\omega=0$ we have
\begin{equation}
\ln^k (\mu^2_{\Gamma} V_{\Gamma}) \; R' \Pi_{\Gamma} =
\frac{1}{k+1}
\hat{S} \ln^{k+1} (\mu^2_{\Gamma} V_{\Gamma})
\; R'\, \Pi_{\Gamma} -
\frac{1}{k+1}
\ln^{k+1} (\mu^2_{\Gamma} V_{\Gamma})\, \hat{S} R' \Pi_{\Gamma}.
\label{LR}
\end{equation}
Let us now apply relation (\ref{F18}) for subgraphs and turn
from summation over elements $\gamma \in {\cal W}$ and subgraphs of
each $\gamma$ to summation of subgraphs of $\Gamma^0$. After that we
obtain the following expression for the second term in the
right-hand side of (\ref{LR}):
\begin{equation}
\frac{1}{k+1}
\ln^{k+1} (\mu^2_{\Gamma} V_{\Gamma}) \; \sum_{\gamma \subseteq {\cal W}}
R' {\cal C}_{\gamma} \Pi_{\Gamma}.
\label{LR1}
\end{equation}
Here only subgraphs and reduced graphs with a smaller number of
loops are involved. Therefore we know how to renormalize these
quantities
$\ln^{k+1} (\mu^2_{\Gamma} V_{\Gamma}) \, R' {\cal C}_{\gamma} \Pi_{\Gamma}$
by extending them as functionals to the whole space and arriving
at
$R \ln^{k+1} (\mu^2_{\Gamma} V_{\Gamma}) \; {\cal C}_{\gamma} \Pi_{\Gamma}$.
As to the first term in (\ref{LR}) it does not have divergences
because all subdivergences are removed by $R'$ and the overall
divergence is removed by the operator
$\hat{S}^{(\omega)}$.
Thus we arrive at a finite differentially renormalized quantity
(\ref{F17}) which is obtained by extension of the functional
$R'\Pi_{\Gamma}$ to the whole space (i.e. by adding a local
counterterm).
Note that in (\ref{F17}) the summation is all divergent 1PI
subgraphs of $\Gamma$; in each vicinity $\Gamma^0$ this summation
reduces to the corresponding set ${\cal W}$.
To prove ({\em iii}) it is sufficient to repeat the same
manipulations as for ({\em ii}) starting from
$(\hat{S}+\omega/2) R'\, \Pi_{\Gamma}$
instead of
$\hat{S} \ln^{k+1} (\mu^2_{\Gamma} V_{\Gamma})\; R'\, \Pi_{\Gamma}$.
{\em Comments.} ({\em a}) For renormalizable theories in
the pure massless case
there is no need to Fourier transform to $y$-variables. One can choose
$V_{\Gamma}=u^2_{\Gamma}$ where
$u_{\Gamma}=x_i-x_i'$ is any difference variable of the considered
Feynman amplitude
such that the vertices $i$ and $j$ do not belong to
the same divergent subgraph --- see examples in Section 3.
({\em b}) The renormalization prescriptions for Feynman amplitudes
(\ref{FPI}) are obtained form the above prescriptions for the products
$\Pi_{\Gamma}(\underline{x},\underline{y})$ by 1) integrating over coordinates associated with
internal vertices
2) Fourier transforming in $y_l$ and putting the corresponding $m^2_l$
equal to squares of masses, e.g. $m_l=0$.
It is then natural to consider the insertion polynomials
${\cal P}_{\Gamma}$
dependent only on derivatives in coordinates that correspond to
{\em external} vertices of the given graph.
\section{Commutativity of $R$-operation with \newline
differentiation and the action principle}
It is natural to check whether differential renormalization is in
agreement with the action principle which expresses basic properties of
quantum field theories such as equations of motion and the gauge
invariance.
Within dimensional renormalization the strategy for proving the
renormalized action principle \cite{BM} is
as follows: equations of motion, Ward identities etc.
are proved for unrenormalized
quantities, then for regularized quantities
and finally (by more or less obvious commutativity of differentiation in
coordinates with renormalization) for renormalized quantities.
The most non-trivial point is to justify the relevant symmetries for
regularized quantities.
In the context of differential renormalization
there will be no such intermediate steps following this program
because this is essentially
a renormalization without regularization.
To see what problems arise in reducing the problem to the case of
unrenormalized quantities (via commutation of differentiation with
respect to $R$-operation) let us consider the simplest example
of Fig.~1a:
\begin{equation} R G(x)^2 = \hat{S} \ln \mu^2 x^2 G(x)^2 .\end{equation}
Let us try to see what is
\[ \partial_{\alpha} R G(x)^2 ,\]
with $\partial_{\alpha} = \partial/\partial x_{\alpha}$.
First, remember that
both $\partial_{\alpha}$ and $\hat{S}$ are
understood in the distributional sense.
We have
\begin{equation} \partial_{\alpha} \hat{S} = (\hat{S}+1/2) \partial_{\alpha} ,
\end{equation}
and hence
\[ (\hat{S}+ 1/2) \partial_{\alpha}
\ln \mu^2 x^2 G(x)^2 . \]
Proceeding naively, for the moment, we continue to apply $\partial_{\alpha}$ using the
Leibniz rule. After this operation we obtain expressions that are
ultraviolet divergent.
However in a distributional sense the
derivative is not defined because it acts on
{\em unrenormalized} expression that is defined only at $x\neq 0$.
This is a manifestation of an important difference with respect to
dimensional renormalization: we do not
have a regularization associated with differential
renormalization.
A different prescription is necessary in order to apply $\partial_{\alpha}$.
The point is that our
differential $R$-operation should take account of the degree of
divergence. For $G(x)^2$ it is equal to zero while formally it is
$\omega=1$ for $\partial_{\alpha} G(x)^2$.
According to our prescriptions when the degree of
divergence $\omega$ is equal to one
it is possible to use the operator (\ref{S1}) and
the corresponding differentially renormalized expression is
\begin{equation} R \partial_{\alpha} G(x)^2
= -2 (\hat{S}+1/2) \hat{S} \ln \mu^2 x^2
\partial_{\alpha} G(x)^2.
\label{RDG}
\end{equation}
Let us use the following equation:
\begin{equation}
\hat{S} = - 2\hat{S} (\hat{S}-1/2) + 2 \hat{S}^2, \end{equation}
with (\ref{F5}), (\ref{C1}) to show that
\begin{equation}
(\hat{S}+1/2) \hat{S} \partial_{\alpha}
G(x)^2 = c_1 \partial_{\alpha} \delta(x). \end{equation}
With the help of the auxiliary analytic regularization it is easy to
find
$-2c_1 = c_0 \equiv c_{1a}= 1/16\pi^2$.
We have
\[\partial_{\alpha} \hat{S} (\hat{S}-1/2) =
(\hat{S}+1/2) \hat{S} \partial_{\alpha} \equiv\hat{S}^{(1)} \partial_{\alpha}.\]
Hence we may now apply $\partial_{\alpha}$ to $\ln \mu^2 x^2 /{x^4}$
naively, in the sense of ordinary functions rather than in a
distributional sense,
because the operator $\hat{S}^{(1)}$ ensures that the overall expression is
well defined.
As a result we see that commutativity breaks down:
\begin{equation} \partial_{\alpha} \hat{S} \ln \mu^2 x^2 G(x)^2
= -2 (\hat{S}+1/2) \hat{S} \ln \mu^2 x^2
\partial_{\alpha} G(x)^2
+ \frac{3}{2} c_0 \partial_{\alpha} \delta(x).
\label{NC}
\end{equation}
One can however define the renormalization of
$\partial_{\alpha} G(x)^2$ with another $\mu$-parameter, $\mu'$. This results in
\begin{equation} \partial_{\alpha} \hat{S} \ln \mu^2 x^2 G(x)^2
= -2 (\hat{S}+1/2) \hat{S} \ln \mu'^2 x^2
\partial_{\alpha} G(x)^2
+ (3/2 -\ln(\mu'^2 / \mu^2) ) c_0 \partial_{\alpha} \delta(x).
\label{Co}
\end{equation}
and so we may recover commutativity when $\ln(\mu'^2 / \mu^2) =3/2$.
This simple example shows that the commutativity of differentiation in
coordinates with the $R$-operation is not satisfied automatically in
differential renormalization.
It is necessary to adjust renormalization parameters to provide it.
Another possibility is to use desired commutation relations
as definitions for renormalization of diagrams that are obtained as
derivatives of some other diagrams \cite{O}. In the above example,
this amounts to applying
$\partial_{\alpha} R G(x)^2$ (rather than the right-hand side of (\ref{RDG}))
as a definition of
$R \partial_{\alpha} G(x)^2$.
Bearing in mind this conclusion let us
consider, for example, the following equations in the $\phi^4$
theory:
\begin{eqnarray}
m^2 \frac{\partial}{\partial m^2} RG^{(n)} = - R D_m G^{(n)}, \label{AP1} \\
g \frac{\partial}{\partial g} RG^{(n)} = - R D_4 G^{(n)}, \label{AP2} \\
R \left( D_m - D_2 + 2 D_4 + \frac{n}{2} \right) G^{(n)} = 0,
\label{AP3}
\end{eqnarray}
where $D_m G^{(n)}, D_2 G^{(n)}, D_4 G^{(n)}$
denote, respectively, $n$-point Green functions with insertions of
the following operators:
\begin{equation}
\int \mbox{d} x \, m^2 \phi^2 (x)/2 , \; \int \mbox{d} x \, (\partial \phi)^2 (x) /2 , \;
\int \mbox{d} x \, g\phi^4 (x) /4!,
\label{VDO}
\end{equation}
In dimensional renormalization these equations
directly follow from the
renormalized action principle \cite{BM} (in particular,
(\ref{AP3}) is the
equation of motion).
However
the renormalized action principle is not automatically guaranteed for
differential
renormalization. Nevertheless, it is possible to adjust finite
arbitrariness (at a diagrammatic level)
in renormalization of diagrams that contribute to the first two
operators in (\ref{VDO}) in such a
way that equations (\ref{AP1}--\ref{AP3}) hold.
In fact, it is sufficient to define renormalization of diagrams with one
insertion of operators $m^2\phi^2/2$
or $(\partial \phi)^2 /2$ to satisfy
\begin{eqnarray}
R \int \mbox{d} x \, m^2 G(x-x_1) G(x-x_2) \overline{\Pi}(x_1,x_2, \ldots)
\nonumber \\
= - m^2 \frac{\partial}{\partial m'^2}
R G(x_1-x_2;m') \overline{\Pi}(x_1,x_2, \ldots) \big |_{m'=m} \; ,
\label{AP4} \\
R \int \mbox{d} x \left[ (m^2-\Box_x) G(x-x_1) \right] G(x-x_2)
\overline{\Pi}(x_1,x_2,\ldots)
\nonumber \\
= R G(x_1-x_2) \overline{\Pi}(x_1,x_2, \ldots),
\label{AP5}
\end{eqnarray}
where $\overline{\Pi}$ is the rest of the product of the propagators,
and in (\ref{AP4})
the mass derivative acts only on the first
propagator $G(x_1-x_2)$.
Since we certainly have possibility to adjust
coefficients of proportionality $\zeta_{\Gamma}$ of the `overall'
$\mu$-parameter
$\mu_{\Gamma} = \zeta_{\Gamma} \mu$ to satisfy (\ref{AP4},\ref{AP5}), it is
sensible to use Eqs. (\ref{AP4},\ref{AP5}) as definitions of the
left-hand side.
\section{Renormalization group coefficients}
In dimensional renormalization the renormalization group
equation
\begin{equation}
\left( \mu^2 \frac{\partial}{\partial \mu^2} +
\beta(g) \frac{\partial}{\partial g} - \gamma_m(g) m^2 \frac{\partial}{\partial m^2}
+\frac{n}{2} \right) RG^{(n)} =0
\label{RGE}
\end{equation}
can be derived \cite{C} from the so-called
diagrammatic RG equation
\begin{equation}
-\mu^2 \frac{\partial}{\partial \mu^2} R F_{\Gamma} + \epsilon h_{\Gamma} R F_{\Gamma} =
\sum_{\gamma \subseteq \Gamma}
h_{\gamma} R \left( \Delta^{(1)} (\gamma) F_{\Gamma} \right) ,
\label{DRG}
\end{equation}
where $h_{\gamma}$ is the loop number,
the operation $\Delta^{(1)} (\gamma) \equiv \epsilon
\hat{K}_{\epsilon}^{(1)} \Delta(\gamma)$ is obtained from the
counterterm operation $\Delta^{\rm MS}(\gamma)$ of the MS-scheme as the
residue of the simple pole in $\epsilon$.
Then one uses relations (\ref{AP1}--\ref{AP3}) and
obtains the well-known formulae for the RG coefficients:
\begin{eqnarray}
\beta(g) & = & g \frac{\partial Z_g^{(1)}}{\partial g} =
g(2\gamma_2(g) - \gamma_4(g)) ,
\label{RGC1} \\
\gamma_{m^2}(g) & = & g \frac{\partial Z_{m^2}^{(1)}}{\partial g}
= g_2 (g) - \gamma_{\phi^2}(g) ,
\label{RGC2} \\
\gamma_{i}(g)& = & - g \frac{\partial Z_{i}^{(1)}}{\partial g} , \; i=2,\phi^2,4 ,
\label{RGC3}
\end{eqnarray}
where $Z_i^{(1)}$ are contributions of simple poles to the
counterterms
\begin{eqnarray}
Z_2 & = & 1+
\frac{\partial}{\partial p^2} \hat{K}_{\epsilon} R' G^{(2)}(p^2,m^2;\epsilon), \\
Z_{\phi^2} & = & 1+ \frac{\partial}{\partial m^2}
\hat{K}_{\epsilon} R' G^{(2)}(p^2,m^2;\epsilon), \\
Z_{4} & = & 1+\hat{K}_{\epsilon} R' G^{(4)}(p_1,\ldots,p_4,m^2;\epsilon).
\end{eqnarray}
In differential renormalization we have an equation similar to
(\ref{DRG}):
\begin{equation}
-\mu^2 \frac{\partial}{\partial \mu^2} R F_{\Gamma}
= \sum_{\gamma \subseteq \Gamma}
R \left( {\cal C}_\gamma F_{\Gamma} \right) ,
\label{DiRG}
\end{equation}
where ${\cal C}_{\gamma}$ is the operation (given by (\ref{CG}) and
(\ref{CGC}))
that inserts a finite polynomial of degree $\omega$ in external momenta into the
reduced graph $\Gamma / \gamma$.
To prove (\ref{DiRG}) it is sufficient to repeat arguments
applied in proving (\ref{F18}). Moreover, (\ref{DiRG}) is in
agreement with homogeneity of renormalized Feynman amplitudes in
coordinates, inverse masses and $1/\mu$.
Under these conditions,
for evaluation of RG coefficients in differential renormalization,
we can apply the following formulae that are quite similar to
(\ref{RGC1}--\ref{RGC3}):
\begin{eqnarray}
\beta(g) & = & c_g = g(2\gamma_2(g) - \gamma_4(g)) ,
\label{RGCD1} \\
\gamma_{m^2}(g) & = & c_{m^2}
= g_2 (g) - \gamma_{\phi^2}(g) ,
\label{RGCD2} \\
\gamma_{i}(g) & = & - c_i, \; i=2,\phi^2,4 ,
\label{RGCD3}
\end{eqnarray}
with
\begin{eqnarray}
(c_{m^2} m^2-c_2\Box_x) \delta(x) =
{\cal C} RG^{(2)}(x), \label{FC1} \\
c_4 \prod_{i=2,3,4} \delta(x_i-x_1) = {\cal C} RG^{(4)}(x_1,\ldots,x_4).
\label{FC2}
\end{eqnarray}
The constants $c_i$ are calculated as sums of diagrammatic
contributions ${\cal C}_{\Gamma} F_{\Gamma}$.
Note that there is no factor $g \frac{\partial}{\partial g}$ in
(\ref{RGCD1}--\ref{RGCD3}) because, in contrast to (\ref{DRG}), Eq.
(\ref{DiRG}) does not involve the loop number $h_{\gamma}$.
Thus, within differential renormalization, the constants
$c_i$ play the same role as the residues of the simple poles
in dimensional renormalization counterterms.
\section{Conclusion}
We have seen that the action principle in differential renormalization is
not satisfied automatically.
It is necessary to adjust renormalization parameters to satisfy
equations of motion etc. Moreover it is clearly natural and very
useful to employ equations of motion etc.
as definitions for renormalization of derivatives of the quantities
involved
(e.g. Green functions with insertions of composite operators) whenever
possible \cite{O}.
Based on subtractions on all divergent 1PI subgraphs differential
renormalization turns out to be a mass-independent scheme --- this
property corresponds to locality of counterterms in the auxiliary
parameters $y_l$ as described in Sections 4 and 5. Renormalization group
calculations in this version of differential renormalization are rather
simple.
Since the problem reduces to
calculations of constants $c_{\gamma}$ then one can use
the method of infrared rearrangement \cite{V} which is based on
possibility to put to zero masses and external momenta ($\equiv$
integration over some coordinates, from coordinate-space point of
view) --- see examples of such calculations in Sections 3 and 4.
An alternative approach to renormalization group calculations within
another version of differential renormalization \cite{O} is based, in
the massive case, on the short distance expansion of propagator in
coordinate space. This also results in a mass-independence property of
the renormalization. Up to two-loop order, such scheme is successfully
applied in various situations.
However, for higher orders, the relevant short-distance expansion of the
propagator should involve many terms which can essentially complicate
the situation. It seems that the language of the auxiliary
$y$-variables is a necessary price that must be paid
in order to have a practical
strictly four-dimensional scheme for multiloop calculations.
\vspace{3mm}
{\em Acknowledgements.}
I am very much grateful to K.G.~Chetyrkin and H.~Osborn
for valuable discussions.
|
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"redpajama_set_name": "RedPajamaArXiv"
}
| 7,104
|
It virtually never fails: about 6 weeks into a health and fitness program, most people begin to lose motivation, become less focused, and ultimately revert to old habits. In many cases this means they stop exercising and begin to fall back into the trap of eating an unhealthy diet. Needless to say, this lull in motivation has to be overcome if you're to achieve your long-term fitness goals.
The good news is that the longer you are able to stick with your program, the more likely you are to make the type of permanent lifestyle changes that lead to lasting success.
So if you're stuck in a bit of a rut, if you feel your motivation slipping, here are a couple quick ways to refocus your efforts and get back on track.
1. Get your calendar out. Write down on the calendar which days you are going to exercise and block out the target time to do it. It is extremely important that you take the time to actually writing this down, especially if you're prone to procrastination. As I've said before, we make all kinds of appointments in our daily lives and we keep them. Make an appointment with yourself for your own health and fitness—and stick to it!
bother "looking" for the time to train and plan your nutrition and supplement programs—it's not there. You need to "make" time for these priorities in your life. Enough said.
3. Plan your nutrition. The key to eating a healthy diet is planning. When you're hungry and there's no healthy food nearby, you'll eat what you can get . . . and usually what you can get is fast food that isn't very good for you. Have a well-focused plan for keeping yourself nourished throughout the day. Bring healthy snacks wherever you go, be it work, school, or wherever. Use Meal Replacements to make eating wisely even easier.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,097
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Vitals offers CNA Exam Prep classes taught by Vital's Nurse Owners to students who want to quickly challenge the Florida Certified Nursing Assistant Examination.
Our CNA Exam Prep Course is designed to provide you with the knowledge and skills required for the Florida CNA Licensure Exam. You will have the opportunity to complete this CNA Prep course via weekly 3-day classes Mon-Wed or weekly 2-day classes Sat-Sun with Nurse Instructors AND/OR through our Learning Management System (LMS) online. You can access training courses and videos from your cell phone or your home computer. It is not mandatory that you come into the campus but for best academic learning, it would be essential to do both, in-class and on-line class and review. These online courses offer content that is essential to both clinical and theory and include videos for all 22 clinical skills and also hundreds of written theory practice questions based on State of Florida CNA Exam criteria. In-class and On-line will give you the needed advantage for excelling on your State Boards.
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Vital Medical Training ensures that equal educational opportunities are offered to students, regardless of race, color, national origin, age, gender, gender expression, sexual orientation, religion, veteran status, ancestry, or disability. Questions in reference to educational opportunities may be directed to the Administrator of Vitals Medical Training, who is responsible for gender equity (Title IX), minorities, (Title VI), and the Americans with Disabilities Act (ADA) and also serves as the director of disability services, for issues regarding students with disabilities.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 971
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\section{Introduction}
Predicting the future progression of patients with neurodegenerative diseases (NDD) is a key challenge to treat patients at an earlier stage than today or to better evaluate drug efficacy in clinical trials. Longitudinal data sets, consisting of repeated observations of the same patients over time, play a central role to describe and predict disease progression. Machine Learning techniques, trained on sequence data (multiple observations per patient), have seize this challenge. However, the databases often lack patients with sufficient follow-up visits, a future visit to predict and a sufficiently large delay in between, leading to poor generalization on test data. More importantly, the different experimental settings - time to prediction or patients/visits presenting the considered feature(s) - are difficult to compare as they involve a subset of the initial cohort that has specific characteristics in terms of number of patients, number of follow-up visits and duration. It is often impossible to balance the training and test set for all these characteristics. This problem prevents from a reliable comparison of the algorithms, some being better due to the size or characteristics of the training set, rather than the intrinsic performance of the algorithm or choice of the features
To increase the size of real data sets, data augmentation techniques have been developed : virtual data are drawn with the intention to reproduce the characteristics of the data in the initial cohort. Most of the literature focuses on techniques for independent and identically distributed observations such as image classification \cite{perez2017effectiveness}, text categorization \cite{lu2006enhancing} or speech recognition \cite{cui2015data}. Due to the unrealistic hypothesis for sequence data, some techniques have been proposed for uni-dimensional time-series. They rely on a continuous transformation of the time domain by warping, slicing or sliding the time window \cite{le2016data}. Such techniques do not apply to NDD as the temporal pattern is key in the disease progression. Recently, Generative Adversarial Networks (GAN) have received interest due to the characteristic of the generative part of the model: it can sample virtual realistic data. It is however non-trivial to generate sequence data as there is no straightforward way to propagate the gradient updates from the discriminator to the generator \cite{yu2017seqgan}. Furthermore, these models rely on large training data sets, which are typically inaccessible in the targeted medical applications.
In this paper, we propose to use a simulation framework to perform data augmentation for sequence data in the presence of small training samples, and to evaluate to which extend it increases the performance of a predictive algorithm. The model introduced in \cite{schiratti2017bayesian} recombines short-term individual observations at different disease stages to estimate a long-term scenario of disease progression. The model estimates also parameters that change the pattern of progression, the pace of progression and the age at onset, so that it can reconstruct a continuous trajectory by fitting individual data, or simulate entirely synthetic trajectories by sampling the empirical distribution of the parameters. This model, once trained on a small and unbalanced training data set, may be used therefore to re-sample the trajectories of the training subjects by adding new visits and covering larger time span, and even increase the number of subjects by adding data sampled from simulated trajectories. This augmented virtual cohort may then be used in lieu of the training samples to train a predictive algorithm. We propose to evaluate the performance of the algorithm trained with this augmented and virtualized data, as compared to the original training set, for the prediction of the cognitive decline in subjects with mild cognitive impairments.
\begin{figure}[bt!]
\includegraphics[width=\textwidth]{figures/prediction_setting.pdf}
\caption{The top row describes the standard prediction setting where the data set is split in a train and test set. A fixed time-delay $\Delta T$ is set between the input visits (blue dots) and the target prediction (red dot). It leads to discarding visits in between (grey dots) or entire subjects that do not present sufficient follow up visits (discarded set). The bottom row corresponds to the procedure with simulated data : the training set is composed of virtual patients that are simulated thanks to the estimation set.} \label{fig:setting}
\end{figure}
\section{Sequence-based prediction}
In the following, we consider a longitudinal data set $\mathbf{y} = (\mathbf{y}_{ij}, t_{ij})_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n_i}}$ where the $i-$th subject has been observed $n_i$ times at ages $ t_{1i} < \dots < t_{i n_i}$ with $\mathbf{y}_{ij} \in R^d$ a set of biomarkers. Each observation corresponds to a snapshot of the individual spatiotemporal trajectory. We aim to predict the value of one biomarker in $\Delta T$ years after a given visit, knowing the values of the biomarkers at the previous visits.
\subsection{Standard prediction setting}
In a standard setting, one needs to discard patients that do not have sufficient follow-up visits to cover the required temporal span, i.e. $t_{i n_{i}} - t_{i1} < \Delta T$, as shown on the top row of Figure \ref{fig:setting}. For the remaining patients, the input biomarkers $(\mathbf{y}_{ij})_{1 \leq j \leq k_i}$ at some early visits $(t_{ij})_{1 \leq j \leq k_i}$ (blue dots) are used to predict the biomarker $\mathbf{y}_{i p_{i}^*}$ at age $t_{i p_{i}^*}$ (red dot) such that $t_{i p_{i}^*} = t_{i k_i} + \Delta T$. This task may be achieved by any machine learning algorithm, for instance a neural network. In this setting, possible intermediate visits $t_{ij}$ such that $t_{ik_i} < t_{ij} < t_{i p_{i}^* }$ are discarded (grey dots). If multiple splits of input/output visits $((\mathbf{y}_{ij}, t_{ij})_{1 \leq j \leq k_i}, (\mathbf{y}_{i p_{i}^*}, t_{i p_{i}^*}))$ are possible, one is selected at random. Once the input and target visits have been selected for each patient, they are split into the train and test set.
The longer $\Delta T$, the fewer patients remain in the train and test set and the fewer visits per remaining patients there are. Therefore, when $\Delta T$ is varied, the size and composition of the train and test set may vary dramatically, and thus the performance of the predictive algorithm. This problem is even more critical if several biomarkers that are not observed at every visit are used.
\subsection{Sequence data simulation}
We consider a generative mixed-effect model such that the individual observations at time-point $t_{ij}$ is $\mathbf{y}_{ij} = f(\theta, \mathbf{z}_i, t_{ij}) + \epsilon_{ij} \quad \text{,} \, \, \epsilon_{ij} \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\mathbf{0}, \sigma^2 \text{Id}_{d})$ where $\theta$ corresponds to the fixed-effects and $\mathbf{z}_i$ the random effects associated to the subject $i$. Assuming that it is possible to estimate $\theta$ and $(\mathbf{z}_i)_{1 \leq i \leq n}$ given a training data set, we can draw a new sample $\mathbf{z}_{i'}$ from the empirical distribution of the $(\mathbf{z}_i)_{1 \leq i \leq n}$. It corresponds to a new individual for which is it possible to simulate new observations at arbitrary time-point $(t_{i'j})_{1 \leq j \leq n_{i'}}$.
An example of such model is introduced in \cite{schiratti2017bayesian}, where the authors consider that the fixed-effects $\theta$ define the group-average disease progression and its variability in the population. This fixed-effects, of small dimension relatively to the feature space, can be estimated, thanks to the MCMC-SAEM \cite{delyon1999, allassonniere2015}, with short-term observations for a relatively small number of subjects. As for the $i$-th individual trajectory, the authors consider that it derives from the group-average trajectory according to the random-effects $\mathbf{z}_i = (\alpha_i, \tau_i, (s_{ij})_{1 \leq j \leq N_s})$ where $\alpha_i$ corresponds to the pace of progression and $\tau_i$ relates about the time delay between the group-average and the individual scenario. On top of these temporal parameters that impact the observation coordinates similarly, the space-shifts $(s_{ij})_{1 \leq j \leq N_s}$ characterize the inter-coordinates variations ($N_s \leq N$). These random-effects are learnt by optimizing the individual complete likelihood $p((\mathbf{y}_{ij})_{1 \leq j \leq n_i}, \mathbf{z}_i ; \theta) = p((\mathbf{y}_{ij})_{1 \leq j \leq n_i} | \mathbf{z}_i ; \theta) p(\mathbf{z}_i ; \theta)$ thanks to the L-BFGS-B method \cite{zhu1997algorithm}.
It is possible to draw a set of individual parameters $\mathbf{z}_{i'}$ by, first, simulating the temporal parameters $(\alpha_{i'}, \tau_{i'})$ with a kernel density estimation on the empirical distribution $(\alpha_{i}, \tau_{i})_{1 \leq i \leq n}$. Then, considering the multivariate Gaussian distribution $\mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$ estimated on the whole learnt random effects $(\mathbf{z}_i)_{1 \leq i \leq n}$, it is possible to draw $((s_{i'j})_{1 \leq j \leq N_s} | \alpha_{i'}, \tau_{i'}) \sim \mathcal{N}(\mathbf{\tilde{\mu}}, \mathbf{\tilde{\Sigma}})$ where $\mathbf{\tilde{\mu}}$ and $\mathbf{\tilde{\Sigma}}$ are functions of $\mathbf{\mu}$ and $\mathbf{\Sigma}$ \cite{petersen2008matrix}.
\subsection{Prediction setting with simulated patients}
From the whole data set, we first select an estimation subset that includes the subjects that were discarded in the standard procedure along with some subjects that have more follow-up duration, as shown on the bottom row of Figure \ref{fig:setting}. The remaining subjects form the test set, with same split constraints on the input and target visits as in the standard prediction setting.
Once $\theta$ and $(\mathbf{z}_i)_{1 \leq i \leq n}$ are learnt on the estimation set, we can simulate an arbitrary number of patients and visits per individual that will be used as the training set. To have similar characteristics in the training and test set, the time between two simulated visits is kept similar to the one of the real patient (e.g. one year), and, the input and target visits are split as before. It is important to mention that the estimation procedure now uses the visits $t_{ij}$ such that $t_{ik_i} < t_{ij} < t_{ij^*}$ that were previously discarded (grey dots).
In the following, to assess the quality of the simulated data only, we prevent ourselves from using patients from the estimation set that are eligible in term of number of visits to be used in the training set (upper part of the estimation set of Figure \ref{fig:setting}) : that the training procedure relies on the simulated patients only. In other cases, it is indeed possible - and recommended - to add them to the training set.
\begin{figure}[h]
\begin{center}
\subfloat[Normalized ADAS 11]{
\includegraphics[width=0.32\textwidth]{figures/ADAS11.pdf}
\label{fig:sub-ADAS}
}
\subfloat[Normalized MMSE]{
\includegraphics[width=0.32\textwidth]{figures/MMSE.pdf}
\label{fig:sub-MMSE}
}
\subfloat[Normalized MOCA]{
\includegraphics[width=0.32\textwidth]{figures/MOCA.pdf}
\label{fig:sub-MOCA}
}
\caption{Empirical histograms and cumulative distributions of original cohort (blue) and of the simulated data (red) based on a model that estimate the evolution of the normalized ADAS Cog with 11 and 13 items and the normalized MOCA.}
\label{fig:data-simulation}
\end{center}
\end{figure}
\section{Experimental results}
\subsection{Data and experimental setting}
The experiments focus on the prediction of the mini-mental state examination (MMSE) for subjects with mild cognitive impairment (MCI) from the ADNI database. Therefore, our cohort includes early and late MCI, but also MCI who converted to Alzheimer's Disease and stable MCI. We considered predictions at 1, 2, 3 and 4 years, based on a set of features among the MMSE, Alzheimer's disease assessment scale - cognitive subscale (ADAS-Cog) with 11 or 13 items, clinical dementia rating sum of boxes (CDRSB), the Montreal cognitive assessment (MOCA) and the functional assessment questionnaire (FAQ). One estimation set was defined per subset of features used in the predictive model, e.g. an estimation set that simulate patients with MMSE, ADAS-11 and ADAS-13 was defined for the predictions that are based on these features.
To assess the performance of the simulation framework, we create a virtual cohort with the same characteristics (number of patients and time-points) as the estimation set. The empirical data distribution for the real and virtual cohort are nearly identical as shown on Figure~\ref{fig:data-simulation} for three different features.
We choose a long short-term memory (LSTM) neural network, with 10 hidden dimensions, stacked with a linear layer, as our algorithm to predict future feature values. The Mean Squared Error (L2-norm) loss is optimized thanks to the ADAM optimizer (learning rate of $10^{-3}$ and weighted decay of $10^{-5}$). To prevent the model from overfitting, a subset of the real patients, namely the validation set, is used to apply the early stopping criterion procedure : it stops the training if no loss improvement is detected from a given number of epoch on the validation set. The code is available at [shown after paper acceptance].
The results report the Mean Absolute Error (MAE). To estimate the variance of the estimation procedure, the results are presented with error-bars corresponding to the mean and standard deviation of 10 independent runs with different test splits. The results are compared to, first, the constant prediction, i.e. the hypothesis that there is no change of MMSE within the time interval, and, on the other side, the noise in the data. For the MMSE, \cite{clark1999variability} reports two noise values : a standard deviation of 1.3 and 2.8 (out of 30) for respectively cognitively normal and MCI patients. Once normalized and converted to absolute values, it corresponds to MAE errors of 0.035 and 0.074, represented by a pale orange interval on Figures \ref{fig:algorithm-comparison} and \ref{fig:prediction-accuracy}.
\begin{figure}[h]
\begin{center}
\subfloat[Standard prediction]{
\includegraphics[width=0.5\textwidth]{figures/initial.pdf}
\label{fig:standard-setting}
}
\subfloat[Augmented data]{
\includegraphics[width=0.5\textwidth]{figures/augmented.pdf}
\label{fig:augmented-setting}
}
\caption{Prediction of the MMSE in 1 (blue), 2 (orange), 3 (green) and 4 (red) years with different sets of variables (upper part of each column). The colored dashed lines corresponds to the error for the corresponding constant prediction. The pale orange area corresponds to the noise interval in the data. The number at the bottom presents the training and test set sizes. }
\label{fig:algorithm-comparison}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\subfloat[MMSE prediction in 3 years]{
\includegraphics[width=0.5\textwidth]{figures/3_years.pdf}
\label{fig:three-years-prediction}
}
\subfloat[MMSE prediction in 4 years]{
\includegraphics[width=0.5\textwidth]{figures/4_years.pdf}
\label{fig:four-years-prediction}
}
\caption{MMSE prediction based on MMSE, ADAS-11, ADAS-13, MOCA, FAQ and CRDSB. The red value on the left corresponds to the MAE without simulated data. Then, each column corresponds to a different size of the estimation set. Within each column, we simulate, from left to right, 50, 100, 250, 500 and 1000 virtual patients.}
\label{fig:prediction-accuracy}
\end{center}
\end{figure}
\subsection{Prediction of MMSE}
The prediction accuracy in the standard prediction setting, without simulated patients, are presented on Figure \ref{fig:standard-setting}. The different columns correspond to different sets of markers used as input. It is possible to reach noise level prediction up to 2 years in advance with the MMSE, ADAS-11, ADAS-13, MOCA, FAQ and CDRSB. At 3 and 4 years, the prediction, although often better than the constant prediction, is still larger than noise level.
Replacing the training set by a simulated cohort leads to a significant improvement of the prediction, as first shown on Figure \ref{fig:prediction-accuracy}. It corresponds to a decrease of the MAE of 20\% (resp. 37\%) for prediction 3 years (resp. 4 years) in advance. As the part of the patients used in the estimation set may vary, we tested different scenarios that lead to better results when more patients were used. On the contrary, the number of simulated patients does not seem to have a great impact on the quality of the prediction. A possible but preliminary explanation lies in the fact that even though there are not a lot of simulated patients, they already incorporate more (simulated) visits than real patients.
\subsection{Fair comparison of algorithms}
To better exhibit the problem of comparison between different prediction settings, we refer to Figure \ref{fig:standard-setting} where the lower figures represent the number of training and test patients in the related model. These numbers decrease for longer time to prediction but also for different sets of features, as not all the examinations have been assessed at each patients' visit.
To solve this issue, we simulated 500 virtual patients for each scenario and estimated the MAE on real patients, as shown on Figure \ref{fig:augmented-setting}. The prediction at 3 and 4 years on the left column corresponds to the values of Figure \ref{fig:prediction-accuracy}. The first result to notice, comparatively to \ref{fig:standard-setting}, is that the MAE variance over the 10 runs is reduced, probably due to the increased test set. More interestingly, the predictive power of the ADAS-11, ADAS-13 and MMSE is not better than with the MMSE alone, a result that could not have been stated from the standard prediction. It essentially means that the MMSE alone is a predictor as good as the three variables but needs more patients to train the model on. In the same spirit, FAQ, MOCA and/or CDRSB provide substantial information that allow to reach noise level prediction up to 4 years in advance.
\section{Discussion}
We proposed a data augmentation technique for small data sets that allow to increase the accuracy of the MMSE prediction for MCI subjects. We believe this technique to be a milestone in the ability to accurately compare various algorithms and features for different time to prediction, as it helps simulating training cohort that are comparable in terms of number of subjects, number of visits per subject and overall follow-up duration.
In this regard, we need to further evaluate the simulation procedure by measuring, for instance, the impact of the number of visits simulated, the time-interval between them, or the selection of the first visit. This could benefit other studies by providing a more accurate comparison of the predictive quality of models or new biomarkers. Overall, it gives a better idea of the generalisation errors of such predictive algorithms in a real clinical setting.
\vspace{5mm}
\textbf{This work has been partly funded by ERC grant N\textsuperscript{o}678304, H2020 EU grant N\textsuperscript{o}666992, and ANR grant ANR-10-IAIHU-06.}
\bibliographystyle{splncs04}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 35
|
require 'image_searcher/version'
require 'httparty'
require 'image_searcher/api'
require 'image_searcher/client'
module ImageSearcher; end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,249
|
This Liquid Liner Define is a must-have in your makeup bag . Available in black, this Modern liquid liner draws a high-pigment, flawless line with is precision brush applicator. Features a super-fine felt tip for precise application. Perfect for any kind of look to make your eyes look brighter.
Draw liner across upper and lower lash lines, starting from the outer corner of eye. For high impact, apply slightly heavier at the outer corners.
Aqua, Glycerin, Mica, PEG-33, Acrylates/Beheneth-25 Methacrylate Copolymer, PEG-8 Dimethicone, Silica, Triethanolamine, PEG-14, Caprylyl Glycol, Phenoxyethanol, Sorbic Acid. May contain (+/-): CI 77499, CI 77266, CI 77491, CI 77492, CI 77007.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,903
|
Кръсточовките (Loxia) са род врабчоподобни птици със специфичен външен вид и с интересно поведение и биология. Отличителен белег при тях е човката ѝ на която двете половини са кръстосани, откъдето са получили и името си. С помощта на кръстосания си клюн могат да изваждат семената от шишарките на иглолистните дървета. За разлика от другите птици те гнездят не само през пролетта, а и през останалите сезони – дори и през зимата.
Биолозите обясняват формата на клюна с еволюционна адаптация към стриктната диета от семената в шишарките. Според легендата, когато Исус Христос бил разпънат на кръста, кръсточовката се опитала да извади гвоздеите, с които той бил прикован. Тогава клюнът ѝ се изкривил, а кръвта му обагрила перата ѝ в червено.
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Фосилна летопис
По костни останки отпреди около 2,25 млн. г. от находище край гр. Вършец палеоорнитологът проф. Златозар Боев е описал първият изкопаем вид от кръсточовките – патева кръсточовка (Loxia patevi), наречена на името на бележития български орнитолог Павел Патев
Източници
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,480
|
package org.apache.jsp.include_005fjsp.myMessage;
import javax.servlet.*;
import javax.servlet.http.*;
import javax.servlet.jsp.*;
import net.violet.platform.util.StaticTools;
import net.violet.platform.util.SessionTools;
import net.violet.platform.datamodel.Lang;
import net.violet.platform.util.DicoTools;
import net.violet.platform.util.MyConstantes;
public final class inc_005fmyMessagesReceived_jsp extends org.apache.jasper.runtime.HttpJspBase
implements org.apache.jasper.runtime.JspSourceDependent {
private static java.util.List _jspx_dependants;
static {
_jspx_dependants = new java.util.ArrayList(1);
_jspx_dependants.add("/include_jsp/utils/inc_dico.jsp");
}
private org.apache.jasper.runtime.TagHandlerPool _jspx_tagPool_bean_define_type_property_name_id_nobody;
private org.apache.jasper.runtime.TagHandlerPool _jspx_tagPool_logic_notEmpty_property_name;
private org.apache.jasper.runtime.TagHandlerPool _jspx_tagPool_logic_iterate_property_name_id;
private org.apache.jasper.runtime.TagHandlerPool _jspx_tagPool_bean_define_property_name_id_nobody;
private org.apache.jasper.runtime.TagHandlerPool _jspx_tagPool_bean_write_property_name_nobody;
private org.apache.jasper.runtime.TagHandlerPool _jspx_tagPool_logic_empty_property_name;
private org.apache.jasper.runtime.TagHandlerPool _jspx_tagPool_logic_equal_value_property_name;
private org.apache.jasper.runtime.TagHandlerPool _jspx_tagPool_logic_greaterThan_value_property_name;
private org.apache.jasper.runtime.TagHandlerPool _jspx_tagPool_logic_notEqual_value_property_name;
public Object getDependants() {
return _jspx_dependants;
}
public void _jspInit() {
_jspx_tagPool_bean_define_type_property_name_id_nobody = org.apache.jasper.runtime.TagHandlerPool.getTagHandlerPool(getServletConfig());
_jspx_tagPool_logic_notEmpty_property_name = org.apache.jasper.runtime.TagHandlerPool.getTagHandlerPool(getServletConfig());
_jspx_tagPool_logic_iterate_property_name_id = org.apache.jasper.runtime.TagHandlerPool.getTagHandlerPool(getServletConfig());
_jspx_tagPool_bean_define_property_name_id_nobody = org.apache.jasper.runtime.TagHandlerPool.getTagHandlerPool(getServletConfig());
_jspx_tagPool_bean_write_property_name_nobody = org.apache.jasper.runtime.TagHandlerPool.getTagHandlerPool(getServletConfig());
_jspx_tagPool_logic_empty_property_name = org.apache.jasper.runtime.TagHandlerPool.getTagHandlerPool(getServletConfig());
_jspx_tagPool_logic_equal_value_property_name = org.apache.jasper.runtime.TagHandlerPool.getTagHandlerPool(getServletConfig());
_jspx_tagPool_logic_greaterThan_value_property_name = org.apache.jasper.runtime.TagHandlerPool.getTagHandlerPool(getServletConfig());
_jspx_tagPool_logic_notEqual_value_property_name = org.apache.jasper.runtime.TagHandlerPool.getTagHandlerPool(getServletConfig());
}
public void _jspDestroy() {
_jspx_tagPool_bean_define_type_property_name_id_nobody.release();
_jspx_tagPool_logic_notEmpty_property_name.release();
_jspx_tagPool_logic_iterate_property_name_id.release();
_jspx_tagPool_bean_define_property_name_id_nobody.release();
_jspx_tagPool_bean_write_property_name_nobody.release();
_jspx_tagPool_logic_empty_property_name.release();
_jspx_tagPool_logic_equal_value_property_name.release();
_jspx_tagPool_logic_greaterThan_value_property_name.release();
_jspx_tagPool_logic_notEqual_value_property_name.release();
}
public void _jspService(HttpServletRequest request, HttpServletResponse response)
throws java.io.IOException, ServletException {
JspFactory _jspxFactory = null;
PageContext pageContext = null;
HttpSession session = null;
ServletContext application = null;
ServletConfig config = null;
JspWriter out = null;
Object page = this;
JspWriter _jspx_out = null;
PageContext _jspx_page_context = null;
try {
_jspxFactory = JspFactory.getDefaultFactory();
response.setContentType("text/html;charset=UTF-8");
pageContext = _jspxFactory.getPageContext(this, request, response,
null, true, 8192, true);
_jspx_page_context = pageContext;
application = pageContext.getServletContext();
config = pageContext.getServletConfig();
session = pageContext.getSession();
out = pageContext.getOut();
_jspx_out = out;
out.write("\n\r\n\r\n");
response.setContentType("text/html;charset=UTF-8");
out.write("\r\n\n\r\n\r\n\r\n\r\n");
out.write('\n');
out.write('\r');
out.write('\n');
out.write("\r\n\r\n\n");
Lang dico_lang = SessionTools.getLangFromSession(session, request);
out.write("\r\n\r\n");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_0 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_type_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_0.setPageContext(_jspx_page_context);
_jspx_th_bean_define_0.setParent(null);
_jspx_th_bean_define_0.setName("myMessagesListForm");
_jspx_th_bean_define_0.setProperty("index");
_jspx_th_bean_define_0.setId("index");
_jspx_th_bean_define_0.setType("Integer");
int _jspx_eval_bean_define_0 = _jspx_th_bean_define_0.doStartTag();
if (_jspx_th_bean_define_0.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_type_property_name_id_nobody.reuse(_jspx_th_bean_define_0);
return;
}
_jspx_tagPool_bean_define_type_property_name_id_nobody.reuse(_jspx_th_bean_define_0);
Integer index = null;
index = (Integer) _jspx_page_context.findAttribute("index");
out.write('\r');
out.write('\n');
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_1 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_type_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_1.setPageContext(_jspx_page_context);
_jspx_th_bean_define_1.setParent(null);
_jspx_th_bean_define_1.setName("myMessagesListForm");
_jspx_th_bean_define_1.setProperty("nabcast");
_jspx_th_bean_define_1.setId("nabcast");
_jspx_th_bean_define_1.setType("Integer");
int _jspx_eval_bean_define_1 = _jspx_th_bean_define_1.doStartTag();
if (_jspx_th_bean_define_1.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_type_property_name_id_nobody.reuse(_jspx_th_bean_define_1);
return;
}
_jspx_tagPool_bean_define_type_property_name_id_nobody.reuse(_jspx_th_bean_define_1);
Integer nabcast = null;
nabcast = (Integer) _jspx_page_context.findAttribute("nabcast");
out.write("\r\n\r\n\n<div id=\"myPlayerMp3_r\" class=\"messagesPlayer\" ></div>\n\r\n<form action=\"../action/myMessagesList.do\" method=\"post\" id=\"update_list\">\r\n\t<input type=\"hidden\" name=\"action\" value=\"received\">\n\r\n<table width=\"100%\" cellspacing=\"0\" cellpadding=\"0\" class=\"messagesListeTable\" id=\"messagesListeReceived\">\n\t\t<thead>\n\t\t\t<tr>\t\t\n\t\t\t\t<th class=\"date\">");
out.print(DicoTools.dico(dico_lang, "myMessages/date"));
out.write("</th>\n\t\t\t\t<th class=\"sender\">");
out.print(DicoTools.dico(dico_lang, "myMessages/sentby"));
out.write("</th>\n\t\t\t\t<th class=\"title\">");
out.print(DicoTools.dico(dico_lang, "myMessages/message_title"));
out.write("</th>\n\t\t\t\t<th class=\"played\">");
out.print(DicoTools.dico(dico_lang, "myMessages/played"));
out.write("</th>\n\t\t\t\t<th class=\"select\"> </th>\n\t\t\t\t<th class=\"select\"> </th>\t\t\t\n\t\t\t</tr>\n\t\t</thead>\n\t\t<tbody>\n\t\t");
// logic:notEmpty
org.apache.struts.taglib.logic.NotEmptyTag _jspx_th_logic_notEmpty_0 = (org.apache.struts.taglib.logic.NotEmptyTag) _jspx_tagPool_logic_notEmpty_property_name.get(org.apache.struts.taglib.logic.NotEmptyTag.class);
_jspx_th_logic_notEmpty_0.setPageContext(_jspx_page_context);
_jspx_th_logic_notEmpty_0.setParent(null);
_jspx_th_logic_notEmpty_0.setName("myMessagesListForm");
_jspx_th_logic_notEmpty_0.setProperty("listeMessages");
int _jspx_eval_logic_notEmpty_0 = _jspx_th_logic_notEmpty_0.doStartTag();
if (_jspx_eval_logic_notEmpty_0 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\n\t\t\t");
// logic:iterate
org.apache.struts.taglib.logic.IterateTag _jspx_th_logic_iterate_0 = (org.apache.struts.taglib.logic.IterateTag) _jspx_tagPool_logic_iterate_property_name_id.get(org.apache.struts.taglib.logic.IterateTag.class);
_jspx_th_logic_iterate_0.setPageContext(_jspx_page_context);
_jspx_th_logic_iterate_0.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEmpty_0);
_jspx_th_logic_iterate_0.setName("myMessagesListForm");
_jspx_th_logic_iterate_0.setProperty("listeMessages");
_jspx_th_logic_iterate_0.setId("messageReceivedData");
int _jspx_eval_logic_iterate_0 = _jspx_th_logic_iterate_0.doStartTag();
if (_jspx_eval_logic_iterate_0 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
java.lang.Object messageReceivedData = null;
if (_jspx_eval_logic_iterate_0 != javax.servlet.jsp.tagext.Tag.EVAL_BODY_INCLUDE) {
out = _jspx_page_context.pushBody();
_jspx_th_logic_iterate_0.setBodyContent((javax.servlet.jsp.tagext.BodyContent) out);
_jspx_th_logic_iterate_0.doInitBody();
}
messageReceivedData = (java.lang.Object) _jspx_page_context.findAttribute("messageReceivedData");
do {
out.write("\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_2 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_2.setPageContext(_jspx_page_context);
_jspx_th_bean_define_2.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_iterate_0);
_jspx_th_bean_define_2.setName("messageReceivedData");
_jspx_th_bean_define_2.setProperty("id");
_jspx_th_bean_define_2.setId("message_id");
int _jspx_eval_bean_define_2 = _jspx_th_bean_define_2.doStartTag();
if (_jspx_th_bean_define_2.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_2);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_2);
java.lang.Object message_id = null;
message_id = (java.lang.Object) _jspx_page_context.findAttribute("message_id");
out.write("\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_3 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_3.setPageContext(_jspx_page_context);
_jspx_th_bean_define_3.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_iterate_0);
_jspx_th_bean_define_3.setName("messageReceivedData");
_jspx_th_bean_define_3.setProperty("userSenderId");
_jspx_th_bean_define_3.setId("userSenderId");
int _jspx_eval_bean_define_3 = _jspx_th_bean_define_3.doStartTag();
if (_jspx_th_bean_define_3.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_3);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_3);
java.lang.Object userSenderId = null;
userSenderId = (java.lang.Object) _jspx_page_context.findAttribute("userSenderId");
out.write("\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_4 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_4.setPageContext(_jspx_page_context);
_jspx_th_bean_define_4.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_iterate_0);
_jspx_th_bean_define_4.setName("messageReceivedData");
_jspx_th_bean_define_4.setProperty("url");
_jspx_th_bean_define_4.setId("event_url");
int _jspx_eval_bean_define_4 = _jspx_th_bean_define_4.doStartTag();
if (_jspx_th_bean_define_4.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_4);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_4);
java.lang.Object event_url = null;
event_url = (java.lang.Object) _jspx_page_context.findAttribute("event_url");
out.write("\n\t\t\t\t\t<tr>\t\t\n\t\t\t\t\t\t<td>");
if (_jspx_meth_bean_write_0(_jspx_th_logic_iterate_0, _jspx_page_context))
return;
out.write("</td>\n\t\t\t\t\t\t<td><a onclick=\"TB_show('', 'myProfil.do?height=515&width=800&user_id=");
out.print(userSenderId);
out.write("')\" href=\"javascript:;\">");
if (_jspx_meth_bean_write_1(_jspx_th_logic_iterate_0, _jspx_page_context))
return;
out.write("</a></td>\n\t\t\t\t\t\t<td>\n\t\t\t\t\t\t<a onclick=\"loadPersoPlayer('");
out.print(event_url);
out.write("', '100%', true, 'myPlayerMp3_r')\" title=\"Ecouter\" href=\"javascript:;\">\n\t\t\t\t\t\t");
if (_jspx_meth_bean_write_2(_jspx_th_logic_iterate_0, _jspx_page_context))
return;
out.write("\n\t\t\t\t\t\t</a>\n\t\t\t\t\t\t</td>\n\t\t\t\t\t\t<td>");
if (_jspx_meth_bean_write_3(_jspx_th_logic_iterate_0, _jspx_page_context))
return;
out.write("x</td>\n\t\t\t\t\t\t<td width=\"1%\"><input class=\"genericBt\" type=\"button\" value=\"");
out.print(DicoTools.dico(dico_lang, "myMessages/reply"));
out.write("\" onclick=\"sendMsgTo('");
if (_jspx_meth_bean_write_4(_jspx_th_logic_iterate_0, _jspx_page_context))
return;
out.write("');\" /></td>\n\t\t\t\t\t\t<td><input type=\"checkbox\" name=\"checkListMsg\" value=\"");
out.print(message_id);
out.write("\" onclick=\"messages_selectMsg(this);\"></td>\n\t\t\t\t\t</tr>\n\t\t\t");
int evalDoAfterBody = _jspx_th_logic_iterate_0.doAfterBody();
messageReceivedData = (java.lang.Object) _jspx_page_context.findAttribute("messageReceivedData");
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
if (_jspx_eval_logic_iterate_0 != javax.servlet.jsp.tagext.Tag.EVAL_BODY_INCLUDE) {
out = _jspx_page_context.popBody();
}
}
if (_jspx_th_logic_iterate_0.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_iterate_property_name_id.reuse(_jspx_th_logic_iterate_0);
return;
}
_jspx_tagPool_logic_iterate_property_name_id.reuse(_jspx_th_logic_iterate_0);
out.write('\n');
out.write(' ');
out.write(' ');
int evalDoAfterBody = _jspx_th_logic_notEmpty_0.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_notEmpty_0.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_notEmpty_property_name.reuse(_jspx_th_logic_notEmpty_0);
return;
}
_jspx_tagPool_logic_notEmpty_property_name.reuse(_jspx_th_logic_notEmpty_0);
out.write('\n');
out.write(' ');
out.write(' ');
// logic:empty
org.apache.struts.taglib.logic.EmptyTag _jspx_th_logic_empty_0 = (org.apache.struts.taglib.logic.EmptyTag) _jspx_tagPool_logic_empty_property_name.get(org.apache.struts.taglib.logic.EmptyTag.class);
_jspx_th_logic_empty_0.setPageContext(_jspx_page_context);
_jspx_th_logic_empty_0.setParent(null);
_jspx_th_logic_empty_0.setName("myMessagesListForm");
_jspx_th_logic_empty_0.setProperty("listeMessages");
int _jspx_eval_logic_empty_0 = _jspx_th_logic_empty_0.doStartTag();
if (_jspx_eval_logic_empty_0 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\n\t\t\t<tr><td colspan=\"10\">");
out.print(DicoTools.dico(dico_lang, "main/no_message"));
out.write("</td></tr>\n\t\t");
int evalDoAfterBody = _jspx_th_logic_empty_0.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_empty_0.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_empty_property_name.reuse(_jspx_th_logic_empty_0);
return;
}
_jspx_tagPool_logic_empty_property_name.reuse(_jspx_th_logic_empty_0);
out.write("\t\t\n\t\t</tbody>\n\t</table>\r\n\r\n\t<ul class=\"messages-list-func\">\r\n\t\t<li><a href=\"javascript:;\" onclick=\"messages_select_all('messagesListeReceived')\">");
out.print(DicoTools.dico(dico_lang, "myMessages/select_all"));
out.write("</a></li>\r\n\t\t<li>\r\n\t\t\t");
// logic:equal
org.apache.struts.taglib.logic.EqualTag _jspx_th_logic_equal_0 = (org.apache.struts.taglib.logic.EqualTag) _jspx_tagPool_logic_equal_value_property_name.get(org.apache.struts.taglib.logic.EqualTag.class);
_jspx_th_logic_equal_0.setPageContext(_jspx_page_context);
_jspx_th_logic_equal_0.setParent(null);
_jspx_th_logic_equal_0.setName("myMessagesListForm");
_jspx_th_logic_equal_0.setProperty("nabcast");
_jspx_th_logic_equal_0.setValue("0");
int _jspx_eval_logic_equal_0 = _jspx_th_logic_equal_0.doStartTag();
if (_jspx_eval_logic_equal_0 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\t\t\t<a href=\"javascript:void(0);\" onclick=\"divChangeUrl('contentRecu' , '../action/myMessagesList.do?action=received&index=");
out.print(index);
out.write("&nabcast=1');\">");
out.print(DicoTools.dico(dico_lang, "myMessages/display_nabcast"));
out.write("</a>\r\n\t\t\t");
int evalDoAfterBody = _jspx_th_logic_equal_0.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_equal_0.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_equal_value_property_name.reuse(_jspx_th_logic_equal_0);
return;
}
_jspx_tagPool_logic_equal_value_property_name.reuse(_jspx_th_logic_equal_0);
out.write("\r\n\t\t\t\r\n\t\t\t");
// logic:equal
org.apache.struts.taglib.logic.EqualTag _jspx_th_logic_equal_1 = (org.apache.struts.taglib.logic.EqualTag) _jspx_tagPool_logic_equal_value_property_name.get(org.apache.struts.taglib.logic.EqualTag.class);
_jspx_th_logic_equal_1.setPageContext(_jspx_page_context);
_jspx_th_logic_equal_1.setParent(null);
_jspx_th_logic_equal_1.setName("myMessagesListForm");
_jspx_th_logic_equal_1.setProperty("nabcast");
_jspx_th_logic_equal_1.setValue("1");
int _jspx_eval_logic_equal_1 = _jspx_th_logic_equal_1.doStartTag();
if (_jspx_eval_logic_equal_1 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\t\t\t<a href=\"javascript:void(0);\" onclick=\"divChangeUrl('contentRecu' , '../action/myMessagesList.do?action=received&index=");
out.print(index);
out.write("&nabcast=0');\">");
out.print(DicoTools.dico(dico_lang, "myMessages/no_display_nabcasts"));
out.write("</a>\r\n\t\t\t");
int evalDoAfterBody = _jspx_th_logic_equal_1.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_equal_1.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_equal_value_property_name.reuse(_jspx_th_logic_equal_1);
return;
}
_jspx_tagPool_logic_equal_value_property_name.reuse(_jspx_th_logic_equal_1);
out.write("\r\n\t\t</li>\r\n\t\t<li><a href=\"myTerrier.do?onglet=blackList\" >");
out.print(DicoTools.dico(dico_lang, "myMessages/display_blacklist"));
out.write("</a></li>\r\n\t</ul>\r\n\t\r\n\t<div class=\"messages-select-action\">\r\n\t\t<select name=\"selectChoice\" style=\"width:120px;\">\r\n\t\t\t<!-- <option value=\"\">");
out.print(DicoTools.dico(dico_lang, "myMessages/action"));
out.write("</option> -->\r\n\t\t\t<option value=\"archive_msg\" selected=\"selected\">");
out.print(DicoTools.dico(dico_lang, "myMessages/archive"));
out.write("</option>\r\n\t\t\t<option value=\"delete_msg\">");
out.print(DicoTools.dico(dico_lang, "myMessages/delete"));
out.write("</option>\r\n\t\t\t<option value=\"replay_msg\">");
out.print(DicoTools.dico(dico_lang, "myMessages/replay"));
out.write("</option>\r\n\t\t\t<option value=\"blacklist\">");
out.print(DicoTools.dico(dico_lang, "myMessages/blacklist"));
out.write("</option>\r\n\t\t</select>\r\n\t\t<input type=\"button\" value=\"");
out.print(DicoTools.dico(dico_lang, "myMessages/button_ok"));
out.write("\" class=\"genericBt\" onclick=\"submitAjaxForm('update_list','contentRecu');\"/> \t\r\n\t</div>\r\n\r\n</form>\r\n\r\n\r\n\r\n<div class=\"paginate\">\r\n\t<ul>\r\n\t\t<form action=\"../action/myMessagesList.do\" name=\"pageSelector\" method=\"post\" >\r\n\t\t\t");
// logic:greaterThan
org.apache.struts.taglib.logic.GreaterThanTag _jspx_th_logic_greaterThan_0 = (org.apache.struts.taglib.logic.GreaterThanTag) _jspx_tagPool_logic_greaterThan_value_property_name.get(org.apache.struts.taglib.logic.GreaterThanTag.class);
_jspx_th_logic_greaterThan_0.setPageContext(_jspx_page_context);
_jspx_th_logic_greaterThan_0.setParent(null);
_jspx_th_logic_greaterThan_0.setName("myMessagesListForm");
_jspx_th_logic_greaterThan_0.setProperty("nombre_pages");
_jspx_th_logic_greaterThan_0.setValue("1");
int _jspx_eval_logic_greaterThan_0 = _jspx_th_logic_greaterThan_0.doStartTag();
if (_jspx_eval_logic_greaterThan_0 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_5 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_5.setPageContext(_jspx_page_context);
_jspx_th_bean_define_5.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_define_5.setId("page_indexD");
_jspx_th_bean_define_5.setName("myMessagesListForm");
_jspx_th_bean_define_5.setProperty("page_indexD");
int _jspx_eval_bean_define_5 = _jspx_th_bean_define_5.doStartTag();
if (_jspx_th_bean_define_5.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_5);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_5);
java.lang.Object page_indexD = null;
page_indexD = (java.lang.Object) _jspx_page_context.findAttribute("page_indexD");
out.write("\r\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_6 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_6.setPageContext(_jspx_page_context);
_jspx_th_bean_define_6.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_define_6.setId("page_indexMM");
_jspx_th_bean_define_6.setName("myMessagesListForm");
_jspx_th_bean_define_6.setProperty("page_indexMM");
int _jspx_eval_bean_define_6 = _jspx_th_bean_define_6.doStartTag();
if (_jspx_th_bean_define_6.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_6);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_6);
java.lang.Object page_indexMM = null;
page_indexMM = (java.lang.Object) _jspx_page_context.findAttribute("page_indexMM");
out.write("\r\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_7 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_7.setPageContext(_jspx_page_context);
_jspx_th_bean_define_7.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_define_7.setId("page_indexM");
_jspx_th_bean_define_7.setName("myMessagesListForm");
_jspx_th_bean_define_7.setProperty("page_indexM");
int _jspx_eval_bean_define_7 = _jspx_th_bean_define_7.doStartTag();
if (_jspx_th_bean_define_7.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_7);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_7);
java.lang.Object page_indexM = null;
page_indexM = (java.lang.Object) _jspx_page_context.findAttribute("page_indexM");
out.write("\r\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_8 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_8.setPageContext(_jspx_page_context);
_jspx_th_bean_define_8.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_define_8.setId("page_index");
_jspx_th_bean_define_8.setName("myMessagesListForm");
_jspx_th_bean_define_8.setProperty("page_index");
int _jspx_eval_bean_define_8 = _jspx_th_bean_define_8.doStartTag();
if (_jspx_th_bean_define_8.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_8);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_8);
java.lang.Object page_index = null;
page_index = (java.lang.Object) _jspx_page_context.findAttribute("page_index");
out.write("\r\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_9 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_9.setPageContext(_jspx_page_context);
_jspx_th_bean_define_9.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_define_9.setId("page_indexP");
_jspx_th_bean_define_9.setName("myMessagesListForm");
_jspx_th_bean_define_9.setProperty("page_indexP");
int _jspx_eval_bean_define_9 = _jspx_th_bean_define_9.doStartTag();
if (_jspx_th_bean_define_9.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_9);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_9);
java.lang.Object page_indexP = null;
page_indexP = (java.lang.Object) _jspx_page_context.findAttribute("page_indexP");
out.write("\r\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_10 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_10.setPageContext(_jspx_page_context);
_jspx_th_bean_define_10.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_define_10.setId("page_indexPP");
_jspx_th_bean_define_10.setName("myMessagesListForm");
_jspx_th_bean_define_10.setProperty("page_indexPP");
int _jspx_eval_bean_define_10 = _jspx_th_bean_define_10.doStartTag();
if (_jspx_th_bean_define_10.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_10);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_10);
java.lang.Object page_indexPP = null;
page_indexPP = (java.lang.Object) _jspx_page_context.findAttribute("page_indexPP");
out.write("\r\n\t\t\t\t");
// bean:define
org.apache.struts.taglib.bean.DefineTag _jspx_th_bean_define_11 = (org.apache.struts.taglib.bean.DefineTag) _jspx_tagPool_bean_define_property_name_id_nobody.get(org.apache.struts.taglib.bean.DefineTag.class);
_jspx_th_bean_define_11.setPageContext(_jspx_page_context);
_jspx_th_bean_define_11.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_define_11.setId("page_indexF");
_jspx_th_bean_define_11.setName("myMessagesListForm");
_jspx_th_bean_define_11.setProperty("page_indexF");
int _jspx_eval_bean_define_11 = _jspx_th_bean_define_11.doStartTag();
if (_jspx_th_bean_define_11.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_11);
return;
}
_jspx_tagPool_bean_define_property_name_id_nobody.reuse(_jspx_th_bean_define_11);
java.lang.Object page_indexF = null;
page_indexF = (java.lang.Object) _jspx_page_context.findAttribute("page_indexF");
out.write("\r\n\r\n\t\t\t<li>\r\n\t\t\t\t<a href=\"javascript:;\" onclick=\"messagesChangePage('contentRecu', ");
if (_jspx_meth_bean_write_5(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write(")\"> << </a>\r\n\t\t\t</li>\r\n\t\t\t\r\n\t\t\t<li>\r\n\t\t\t\t<a href=\"javascript:;\" onclick=\"messagesChangePage('contentRecu', ");
if (_jspx_meth_bean_write_6(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write(")\"> < </a>\r\n\t\t\t</li>\r\n\r\n\t\t\t");
if (_jspx_meth_logic_notEqual_0(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write("\r\n\t\t\t\r\n\t\t\t");
if (_jspx_meth_logic_notEqual_1(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write("\r\n\r\n\t\t\t<li class=\"current\">\r\n\t\t\t\t<a href=\"javascript:;\">");
if (_jspx_meth_bean_write_11(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write("</a>\r\n\t\t\t</li>\r\n\r\n\t\t\t");
if (_jspx_meth_logic_notEqual_2(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write("\r\n\t\t\t\r\n\t\t\t");
if (_jspx_meth_logic_notEqual_3(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write("\r\n\r\n\t\t\t<li>\r\n\t\t\t\t");
if (_jspx_meth_logic_equal_2(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write("\r\n\t\t\t\t");
if (_jspx_meth_logic_notEqual_4(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write("\t\t\t\t\t\t\r\n\t\t\t</li>\r\n\t\t\t\r\n\t\t\t<li>\r\n\t\t\t\t<a href=\"javascript:;\" onclick=\"messagesChangePage('contentRecu', ");
if (_jspx_meth_bean_write_18(_jspx_th_logic_greaterThan_0, _jspx_page_context))
return;
out.write(")\"> >> </a>\r\n\t\t\t</li>\r\n\t\t\t\r\n\t\t\t");
int evalDoAfterBody = _jspx_th_logic_greaterThan_0.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_greaterThan_0.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_greaterThan_value_property_name.reuse(_jspx_th_logic_greaterThan_0);
return;
}
_jspx_tagPool_logic_greaterThan_value_property_name.reuse(_jspx_th_logic_greaterThan_0);
out.write("\r\n\t\t</form>\r\n\t</ul>\r\n\t\r\n</div>\r\n\r\n\r\n\r\n<script type=\"text/javascript\">\r\n\t\r\n\tgetNewMessages();\r\n\r\n\tmessageListColorization(\"messagesListeReceived\");\r\n\r\n\t// affiche le lecteur dés le départ\r\n\t//loadPersoPlayer('', '100%', true, 'myPlayerMp3_r' );\r\n\r\n\r\n\tvar msg = \"\";\r\n\tvar col = gErrorColor;\r\n\r\n\t");
if (_jspx_meth_logic_equal_3(_jspx_page_context))
return;
out.write("\r\n\t\t\r\n\t");
if (_jspx_meth_logic_equal_4(_jspx_page_context))
return;
out.write("\r\n\t\r\n\t");
if (_jspx_meth_logic_equal_5(_jspx_page_context))
return;
out.write("\r\n\r\n</script>\t\r\n");
} catch (Throwable t) {
if (!(t instanceof SkipPageException)){
out = _jspx_out;
if (out != null && out.getBufferSize() != 0)
out.clearBuffer();
if (_jspx_page_context != null) _jspx_page_context.handlePageException(t);
}
} finally {
if (_jspxFactory != null) _jspxFactory.releasePageContext(_jspx_page_context);
}
}
private boolean _jspx_meth_bean_write_0(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_iterate_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_0 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_0.setPageContext(_jspx_page_context);
_jspx_th_bean_write_0.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_iterate_0);
_jspx_th_bean_write_0.setName("messageReceivedData");
_jspx_th_bean_write_0.setProperty("dateOfDelivery");
int _jspx_eval_bean_write_0 = _jspx_th_bean_write_0.doStartTag();
if (_jspx_th_bean_write_0.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_0);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_0);
return false;
}
private boolean _jspx_meth_bean_write_1(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_iterate_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_1 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_1.setPageContext(_jspx_page_context);
_jspx_th_bean_write_1.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_iterate_0);
_jspx_th_bean_write_1.setName("messageReceivedData");
_jspx_th_bean_write_1.setProperty("sender_name");
int _jspx_eval_bean_write_1 = _jspx_th_bean_write_1.doStartTag();
if (_jspx_th_bean_write_1.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_1);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_1);
return false;
}
private boolean _jspx_meth_bean_write_2(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_iterate_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_2 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_2.setPageContext(_jspx_page_context);
_jspx_th_bean_write_2.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_iterate_0);
_jspx_th_bean_write_2.setName("messageReceivedData");
_jspx_th_bean_write_2.setProperty("title");
int _jspx_eval_bean_write_2 = _jspx_th_bean_write_2.doStartTag();
if (_jspx_th_bean_write_2.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_2);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_2);
return false;
}
private boolean _jspx_meth_bean_write_3(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_iterate_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_3 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_3.setPageContext(_jspx_page_context);
_jspx_th_bean_write_3.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_iterate_0);
_jspx_th_bean_write_3.setName("messageReceivedData");
_jspx_th_bean_write_3.setProperty("nbPlayed");
int _jspx_eval_bean_write_3 = _jspx_th_bean_write_3.doStartTag();
if (_jspx_th_bean_write_3.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_3);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_3);
return false;
}
private boolean _jspx_meth_bean_write_4(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_iterate_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_4 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_4.setPageContext(_jspx_page_context);
_jspx_th_bean_write_4.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_iterate_0);
_jspx_th_bean_write_4.setName("messageReceivedData");
_jspx_th_bean_write_4.setProperty("sender_name");
int _jspx_eval_bean_write_4 = _jspx_th_bean_write_4.doStartTag();
if (_jspx_th_bean_write_4.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_4);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_4);
return false;
}
private boolean _jspx_meth_bean_write_5(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_5 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_5.setPageContext(_jspx_page_context);
_jspx_th_bean_write_5.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_write_5.setName("myMessagesListForm");
_jspx_th_bean_write_5.setProperty("page_indexD");
int _jspx_eval_bean_write_5 = _jspx_th_bean_write_5.doStartTag();
if (_jspx_th_bean_write_5.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_5);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_5);
return false;
}
private boolean _jspx_meth_bean_write_6(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_6 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_6.setPageContext(_jspx_page_context);
_jspx_th_bean_write_6.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_write_6.setName("myMessagesListForm");
_jspx_th_bean_write_6.setProperty("page_indexM");
int _jspx_eval_bean_write_6 = _jspx_th_bean_write_6.doStartTag();
if (_jspx_th_bean_write_6.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_6);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_6);
return false;
}
private boolean _jspx_meth_logic_notEqual_0(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// logic:notEqual
org.apache.struts.taglib.logic.NotEqualTag _jspx_th_logic_notEqual_0 = (org.apache.struts.taglib.logic.NotEqualTag) _jspx_tagPool_logic_notEqual_value_property_name.get(org.apache.struts.taglib.logic.NotEqualTag.class);
_jspx_th_logic_notEqual_0.setPageContext(_jspx_page_context);
_jspx_th_logic_notEqual_0.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_logic_notEqual_0.setName("myMessagesListForm");
_jspx_th_logic_notEqual_0.setProperty("page_indexMM");
_jspx_th_logic_notEqual_0.setValue("0");
int _jspx_eval_logic_notEqual_0 = _jspx_th_logic_notEqual_0.doStartTag();
if (_jspx_eval_logic_notEqual_0 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\t\t\t<li>\r\n\t\t\t\t\t<a href=\"javascript:;\" onclick=\"messagesChangePage('contentRecu', ");
if (_jspx_meth_bean_write_7(_jspx_th_logic_notEqual_0, _jspx_page_context))
return true;
out.write(")\"> ");
if (_jspx_meth_bean_write_8(_jspx_th_logic_notEqual_0, _jspx_page_context))
return true;
out.write(" </a>\r\n\t\t\t\t</li>\r\n\t\t\t");
int evalDoAfterBody = _jspx_th_logic_notEqual_0.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_notEqual_0.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_0);
return true;
}
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_0);
return false;
}
private boolean _jspx_meth_bean_write_7(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_notEqual_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_7 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_7.setPageContext(_jspx_page_context);
_jspx_th_bean_write_7.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEqual_0);
_jspx_th_bean_write_7.setName("myMessagesListForm");
_jspx_th_bean_write_7.setProperty("page_indexMM");
int _jspx_eval_bean_write_7 = _jspx_th_bean_write_7.doStartTag();
if (_jspx_th_bean_write_7.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_7);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_7);
return false;
}
private boolean _jspx_meth_bean_write_8(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_notEqual_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_8 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_8.setPageContext(_jspx_page_context);
_jspx_th_bean_write_8.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEqual_0);
_jspx_th_bean_write_8.setName("myMessagesListForm");
_jspx_th_bean_write_8.setProperty("page_AffIndexMM");
int _jspx_eval_bean_write_8 = _jspx_th_bean_write_8.doStartTag();
if (_jspx_th_bean_write_8.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_8);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_8);
return false;
}
private boolean _jspx_meth_logic_notEqual_1(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// logic:notEqual
org.apache.struts.taglib.logic.NotEqualTag _jspx_th_logic_notEqual_1 = (org.apache.struts.taglib.logic.NotEqualTag) _jspx_tagPool_logic_notEqual_value_property_name.get(org.apache.struts.taglib.logic.NotEqualTag.class);
_jspx_th_logic_notEqual_1.setPageContext(_jspx_page_context);
_jspx_th_logic_notEqual_1.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_logic_notEqual_1.setName("myMessagesListForm");
_jspx_th_logic_notEqual_1.setProperty("page_indexM");
_jspx_th_logic_notEqual_1.setValue("0");
int _jspx_eval_logic_notEqual_1 = _jspx_th_logic_notEqual_1.doStartTag();
if (_jspx_eval_logic_notEqual_1 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\t\t\t<li>\r\n\t\t\t\t\t<a href=\"javascript:;\" onclick=\"messagesChangePage('contentRecu', ");
if (_jspx_meth_bean_write_9(_jspx_th_logic_notEqual_1, _jspx_page_context))
return true;
out.write(")\"> ");
if (_jspx_meth_bean_write_10(_jspx_th_logic_notEqual_1, _jspx_page_context))
return true;
out.write(" </a>\r\n\t\t\t\t</li>\r\n\t\t\t");
int evalDoAfterBody = _jspx_th_logic_notEqual_1.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_notEqual_1.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_1);
return true;
}
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_1);
return false;
}
private boolean _jspx_meth_bean_write_9(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_notEqual_1, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_9 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_9.setPageContext(_jspx_page_context);
_jspx_th_bean_write_9.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEqual_1);
_jspx_th_bean_write_9.setName("myMessagesListForm");
_jspx_th_bean_write_9.setProperty("page_indexM");
int _jspx_eval_bean_write_9 = _jspx_th_bean_write_9.doStartTag();
if (_jspx_th_bean_write_9.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_9);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_9);
return false;
}
private boolean _jspx_meth_bean_write_10(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_notEqual_1, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_10 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_10.setPageContext(_jspx_page_context);
_jspx_th_bean_write_10.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEqual_1);
_jspx_th_bean_write_10.setName("myMessagesListForm");
_jspx_th_bean_write_10.setProperty("page_AffIndexM");
int _jspx_eval_bean_write_10 = _jspx_th_bean_write_10.doStartTag();
if (_jspx_th_bean_write_10.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_10);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_10);
return false;
}
private boolean _jspx_meth_bean_write_11(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_11 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_11.setPageContext(_jspx_page_context);
_jspx_th_bean_write_11.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_write_11.setName("myMessagesListForm");
_jspx_th_bean_write_11.setProperty("page_AffIndex");
int _jspx_eval_bean_write_11 = _jspx_th_bean_write_11.doStartTag();
if (_jspx_th_bean_write_11.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_11);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_11);
return false;
}
private boolean _jspx_meth_logic_notEqual_2(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// logic:notEqual
org.apache.struts.taglib.logic.NotEqualTag _jspx_th_logic_notEqual_2 = (org.apache.struts.taglib.logic.NotEqualTag) _jspx_tagPool_logic_notEqual_value_property_name.get(org.apache.struts.taglib.logic.NotEqualTag.class);
_jspx_th_logic_notEqual_2.setPageContext(_jspx_page_context);
_jspx_th_logic_notEqual_2.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_logic_notEqual_2.setName("myMessagesListForm");
_jspx_th_logic_notEqual_2.setProperty("page_indexP");
_jspx_th_logic_notEqual_2.setValue("0");
int _jspx_eval_logic_notEqual_2 = _jspx_th_logic_notEqual_2.doStartTag();
if (_jspx_eval_logic_notEqual_2 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\t\t\t<li>\r\n\t\t\t\t\t<a href=\"javascript:;\" onclick=\"messagesChangePage('contentRecu', ");
if (_jspx_meth_bean_write_12(_jspx_th_logic_notEqual_2, _jspx_page_context))
return true;
out.write(")\"> ");
if (_jspx_meth_bean_write_13(_jspx_th_logic_notEqual_2, _jspx_page_context))
return true;
out.write("</a>\r\n\t\t\t\t</li>\r\n\t\t\t");
int evalDoAfterBody = _jspx_th_logic_notEqual_2.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_notEqual_2.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_2);
return true;
}
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_2);
return false;
}
private boolean _jspx_meth_bean_write_12(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_notEqual_2, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_12 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_12.setPageContext(_jspx_page_context);
_jspx_th_bean_write_12.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEqual_2);
_jspx_th_bean_write_12.setName("myMessagesListForm");
_jspx_th_bean_write_12.setProperty("page_indexP");
int _jspx_eval_bean_write_12 = _jspx_th_bean_write_12.doStartTag();
if (_jspx_th_bean_write_12.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_12);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_12);
return false;
}
private boolean _jspx_meth_bean_write_13(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_notEqual_2, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_13 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_13.setPageContext(_jspx_page_context);
_jspx_th_bean_write_13.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEqual_2);
_jspx_th_bean_write_13.setName("myMessagesListForm");
_jspx_th_bean_write_13.setProperty("page_AffIndexP");
int _jspx_eval_bean_write_13 = _jspx_th_bean_write_13.doStartTag();
if (_jspx_th_bean_write_13.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_13);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_13);
return false;
}
private boolean _jspx_meth_logic_notEqual_3(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// logic:notEqual
org.apache.struts.taglib.logic.NotEqualTag _jspx_th_logic_notEqual_3 = (org.apache.struts.taglib.logic.NotEqualTag) _jspx_tagPool_logic_notEqual_value_property_name.get(org.apache.struts.taglib.logic.NotEqualTag.class);
_jspx_th_logic_notEqual_3.setPageContext(_jspx_page_context);
_jspx_th_logic_notEqual_3.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_logic_notEqual_3.setName("myMessagesListForm");
_jspx_th_logic_notEqual_3.setProperty("page_indexPP");
_jspx_th_logic_notEqual_3.setValue("0");
int _jspx_eval_logic_notEqual_3 = _jspx_th_logic_notEqual_3.doStartTag();
if (_jspx_eval_logic_notEqual_3 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\t\t\t<li>\r\n\t\t\t\t\t<a href=\"javascript:;\" onclick=\"messagesChangePage('contentRecu', ");
if (_jspx_meth_bean_write_14(_jspx_th_logic_notEqual_3, _jspx_page_context))
return true;
out.write(")\"> ");
if (_jspx_meth_bean_write_15(_jspx_th_logic_notEqual_3, _jspx_page_context))
return true;
out.write("</a>\r\n\t\t\t\t</li>\r\n\t\t\t");
int evalDoAfterBody = _jspx_th_logic_notEqual_3.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_notEqual_3.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_3);
return true;
}
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_3);
return false;
}
private boolean _jspx_meth_bean_write_14(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_notEqual_3, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_14 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_14.setPageContext(_jspx_page_context);
_jspx_th_bean_write_14.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEqual_3);
_jspx_th_bean_write_14.setName("myMessagesListForm");
_jspx_th_bean_write_14.setProperty("page_indexPP");
int _jspx_eval_bean_write_14 = _jspx_th_bean_write_14.doStartTag();
if (_jspx_th_bean_write_14.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_14);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_14);
return false;
}
private boolean _jspx_meth_bean_write_15(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_notEqual_3, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_15 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_15.setPageContext(_jspx_page_context);
_jspx_th_bean_write_15.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEqual_3);
_jspx_th_bean_write_15.setName("myMessagesListForm");
_jspx_th_bean_write_15.setProperty("page_AffIndexPP");
int _jspx_eval_bean_write_15 = _jspx_th_bean_write_15.doStartTag();
if (_jspx_th_bean_write_15.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_15);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_15);
return false;
}
private boolean _jspx_meth_logic_equal_2(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// logic:equal
org.apache.struts.taglib.logic.EqualTag _jspx_th_logic_equal_2 = (org.apache.struts.taglib.logic.EqualTag) _jspx_tagPool_logic_equal_value_property_name.get(org.apache.struts.taglib.logic.EqualTag.class);
_jspx_th_logic_equal_2.setPageContext(_jspx_page_context);
_jspx_th_logic_equal_2.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_logic_equal_2.setName("myMessagesListForm");
_jspx_th_logic_equal_2.setProperty("page_indexP");
_jspx_th_logic_equal_2.setValue("0");
int _jspx_eval_logic_equal_2 = _jspx_th_logic_equal_2.doStartTag();
if (_jspx_eval_logic_equal_2 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\t\t\t\t<a href=\"javascript:;\" onclick=\"messagesChangePage('contentRecu', ");
if (_jspx_meth_bean_write_16(_jspx_th_logic_equal_2, _jspx_page_context))
return true;
out.write(")\"> > </a>\r\n\t\t\t\t");
int evalDoAfterBody = _jspx_th_logic_equal_2.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_equal_2.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_equal_value_property_name.reuse(_jspx_th_logic_equal_2);
return true;
}
_jspx_tagPool_logic_equal_value_property_name.reuse(_jspx_th_logic_equal_2);
return false;
}
private boolean _jspx_meth_bean_write_16(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_equal_2, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_16 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_16.setPageContext(_jspx_page_context);
_jspx_th_bean_write_16.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_equal_2);
_jspx_th_bean_write_16.setName("myMessagesListForm");
_jspx_th_bean_write_16.setProperty("page_indexF");
int _jspx_eval_bean_write_16 = _jspx_th_bean_write_16.doStartTag();
if (_jspx_th_bean_write_16.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_16);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_16);
return false;
}
private boolean _jspx_meth_logic_notEqual_4(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// logic:notEqual
org.apache.struts.taglib.logic.NotEqualTag _jspx_th_logic_notEqual_4 = (org.apache.struts.taglib.logic.NotEqualTag) _jspx_tagPool_logic_notEqual_value_property_name.get(org.apache.struts.taglib.logic.NotEqualTag.class);
_jspx_th_logic_notEqual_4.setPageContext(_jspx_page_context);
_jspx_th_logic_notEqual_4.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_logic_notEqual_4.setName("myMessagesListForm");
_jspx_th_logic_notEqual_4.setProperty("page_indexP");
_jspx_th_logic_notEqual_4.setValue("0");
int _jspx_eval_logic_notEqual_4 = _jspx_th_logic_notEqual_4.doStartTag();
if (_jspx_eval_logic_notEqual_4 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\t\t\t\t<a href=\"javascript:;\" onclick=\"messagesChangePage('contentRecu', ");
if (_jspx_meth_bean_write_17(_jspx_th_logic_notEqual_4, _jspx_page_context))
return true;
out.write(")\"> > </a>\r\n\t\t\t\t");
int evalDoAfterBody = _jspx_th_logic_notEqual_4.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_notEqual_4.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_4);
return true;
}
_jspx_tagPool_logic_notEqual_value_property_name.reuse(_jspx_th_logic_notEqual_4);
return false;
}
private boolean _jspx_meth_bean_write_17(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_notEqual_4, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_17 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_17.setPageContext(_jspx_page_context);
_jspx_th_bean_write_17.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_notEqual_4);
_jspx_th_bean_write_17.setName("myMessagesListForm");
_jspx_th_bean_write_17.setProperty("page_indexP");
int _jspx_eval_bean_write_17 = _jspx_th_bean_write_17.doStartTag();
if (_jspx_th_bean_write_17.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_17);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_17);
return false;
}
private boolean _jspx_meth_bean_write_18(javax.servlet.jsp.tagext.JspTag _jspx_th_logic_greaterThan_0, PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// bean:write
org.apache.struts.taglib.bean.WriteTag _jspx_th_bean_write_18 = (org.apache.struts.taglib.bean.WriteTag) _jspx_tagPool_bean_write_property_name_nobody.get(org.apache.struts.taglib.bean.WriteTag.class);
_jspx_th_bean_write_18.setPageContext(_jspx_page_context);
_jspx_th_bean_write_18.setParent((javax.servlet.jsp.tagext.Tag) _jspx_th_logic_greaterThan_0);
_jspx_th_bean_write_18.setName("myMessagesListForm");
_jspx_th_bean_write_18.setProperty("page_indexF");
int _jspx_eval_bean_write_18 = _jspx_th_bean_write_18.doStartTag();
if (_jspx_th_bean_write_18.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_18);
return true;
}
_jspx_tagPool_bean_write_property_name_nobody.reuse(_jspx_th_bean_write_18);
return false;
}
private boolean _jspx_meth_logic_equal_3(PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// logic:equal
org.apache.struts.taglib.logic.EqualTag _jspx_th_logic_equal_3 = (org.apache.struts.taglib.logic.EqualTag) _jspx_tagPool_logic_equal_value_property_name.get(org.apache.struts.taglib.logic.EqualTag.class);
_jspx_th_logic_equal_3.setPageContext(_jspx_page_context);
_jspx_th_logic_equal_3.setParent(null);
_jspx_th_logic_equal_3.setName("myMessagesListForm");
_jspx_th_logic_equal_3.setProperty("errorMsg");
_jspx_th_logic_equal_3.setValue("now_in_blackList");
int _jspx_eval_logic_equal_3 = _jspx_th_logic_equal_3.doStartTag();
if (_jspx_eval_logic_equal_3 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
out.write("\r\n\t\tmsg = msg_txt['now_in_blackList'];\t\r\n\t\t$(\"div.mainTabBody\").msgPopup(msg, col, 4000);\r\n\t");
int evalDoAfterBody = _jspx_th_logic_equal_3.doAfterBody();
if (evalDoAfterBody != javax.servlet.jsp.tagext.BodyTag.EVAL_BODY_AGAIN)
break;
} while (true);
}
if (_jspx_th_logic_equal_3.doEndTag() == javax.servlet.jsp.tagext.Tag.SKIP_PAGE) {
_jspx_tagPool_logic_equal_value_property_name.reuse(_jspx_th_logic_equal_3);
return true;
}
_jspx_tagPool_logic_equal_value_property_name.reuse(_jspx_th_logic_equal_3);
return false;
}
private boolean _jspx_meth_logic_equal_4(PageContext _jspx_page_context)
throws Throwable {
PageContext pageContext = _jspx_page_context;
JspWriter out = _jspx_page_context.getOut();
// logic:equal
org.apache.struts.taglib.logic.EqualTag _jspx_th_logic_equal_4 = (org.apache.struts.taglib.logic.EqualTag) _jspx_tagPool_logic_equal_value_property_name.get(org.apache.struts.taglib.logic.EqualTag.class);
_jspx_th_logic_equal_4.setPageContext(_jspx_page_context);
_jspx_th_logic_equal_4.setParent(null);
_jspx_th_logic_equal_4.setName("myMessagesListForm");
_jspx_th_logic_equal_4.setProperty("errorMsg");
_jspx_th_logic_equal_4.setValue("already_in_blackList");
int _jspx_eval_logic_equal_4 = _jspx_th_logic_equal_4.doStartTag();
if (_jspx_eval_logic_equal_4 != javax.servlet.jsp.tagext.Tag.SKIP_BODY) {
do {
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{
"redpajama_set_name": "RedPajamaGithub"
}
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Günter Klein ist der Name folgender Personen:
* Günter Klein (Politiker, 1900) (1900–1963), deutscher Jurist, Verwaltungsbeamter und Politiker (SPD)
Günter Klein (Komponist) (1921–2010), deutscher Komponist
Günter Klein (Theologe) (1928–2015), deutscher Theologe
Günter Klein (Politiker, 1930) (1930–1998), deutscher Jurist und Politiker (CDU), MdB
Günter Klein (Maler) (* 1943), deutscher Maler
Günter Klein (Sportjournalist) (* 1962), deutscher Sportjournalist
Günter Klein (Veterinärmediziner) (1964–2016), deutscher Veterinärmediziner und Mikrobiologe
siehe auch:
Günther Klein
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
Random $n$-mappings, or simply random functions from the set of integers $[n] \colonequals\left\lbrace 1, 2, \ldots, n \right\rbrace$ into itself, appear in various applications: for example in the birthday paradox, the coupon collector problem and occupancy problems, just to name a few.
Structural properties of the functional digraphs of random mappings have widely been studied, see e.g.\ the work of Arney and Bender~\cite{arney1982random}, Kolchin~\cite{kolchin1986random}, Flajolet and Odlyzko~\cite{FlaOdl1989}.
For instance, it is well known that the expected number of connected components in a random $n$-mapping is asymptotically $1/2 \cdot \log(n)$, the expected number of cyclic nodes is $\sqrt{\pi n/2}$ and the expected number of terminal nodes, i.e., nodes with no preimages, is $e^{-1}n$.
In functional digraphs corresponding to random mappings the nodes' labels play an important role and thus it is somewhat surprising that occurrences of label patterns have, so far, not received much attention.
The simplest of all label patterns in mappings are ascents, i.e., nodes with label $i$ for which it holds that $i < f(i)$ where $f(i)$ denotes the image of $i$.
In a random $n$-mapping, it is clear that the probability of $i$ being an ascent is simply equal to $(n-i)/n$.
Ascents in random mappings are thus rather easily dealt with.
E\u{g}ecio\u{g}lu and Remmel~\cite{egecioglu1986bijections} showed how results on ascents in mappings can be translated to results for ascents in Cayley trees by providing a family of weight preserving bijections.
As shown by Clark~\cite{clark2008ascents}, these results can be used to provide central limit theorems for the number of ascents in random mappings and in random Cayley trees.
In this context we also mention the recent paper of Seo and Shin \cite{SeoShi2012}, where the size of the maximal subtree without ascents containing the root node has been studied for Cayley trees.
Some recent research concerned with patterns formed by the labels in a functional digraph has been performed in \cite{panholzer2013alternating}, where \emph{alternating mappings} have been studied.
These are a generalization of the concept of alternating permutations to mappings.
They can be defined as those mappings for which every iteration orbit forms an alternating sequence.
Alternating mappings can thus also be seen as mappings that do not contain two consecutive ascents or descents.
Also, we would like to mention the PhD thesis of Okoth~\cite{Okoth2015} who has very recently studied local extrema in trees (called sources and sinks there).
His studies also led to results for the corresponding quantities in mappings.
In this paper we perform an analysis of a fundamental \emph{label pattern} in random mappings.
Namely, we generalize the notion of ascending runs from permutations to mappings.
In the corresponding functional digraphs, ascending runs are maximal ascending paths.
Our goal is to enumerate mappings by their size and by their number of ascending runs and to characterize the typical behaviour of this label pattern in random mappings.
This paper is organized as follows:
We start by briefly recalling the basic definitions that will be used.
In Section~\ref{sec:Cayley_formula}, we present a bijective proof of Cayley's formula that will turn out to be useful for the study of ascending runs in Cayley trees and mappings.
Section~\ref{Mappings/sec:runs} is devoted to the main topic of this paper, namely to the exact and asymptotic study of ascending runs in mappings.
This leads to results for Cayley trees as well.
We use a combinatorial decomposition of Cayley trees and mappings according to their smallest node.
This leads to a recurrence relation for the studied numbers that can be translated into a partial differential equation for the corresponding generating functions.
Solving this PDE allows for extraction of coefficients and for a characterization of the limiting distribution of the number of ascending runs in a random mapping or tree.
The bijection presented in Section~\ref{sec:Cayley_formula} will allow to explain a simple connection between the results for trees and for mappings.
Finally, we close this paper by setting our results in relation with the corresponding results that have been obtained for permutations and pointing out directions for future research.
\section{Preliminaries}
The combinatorial objects of interest in this paper are Cayley trees and mappings.
These are defined as follows:
\begin{definition}
A \emph{Cayley tree} of size $n$ is a unordered rooted labelled tree with $n$ nodes.
In a labelled tree of size $n$ every node carries a distinct integer from the set $[n]$ as a label.
\end{definition}
We always consider the edges in a Cayley tree to be oriented towards the root node, as depicted in Figure~\ref{cayley/fig:example} on page~\pageref{cayley/fig:example}; thus, for an edge $(w,v)$, $v$ is closer to the root node and therefore the parent of $w$.
The number of Cayley trees of size $n$ is $T_n=n^{n-1}$.
This formula is attributed to Arthur Cayley and thus also referred to as ``Cayley's formula''.
The exponential generating function $T(z) \colonequals \sum_{n \geq 0} \frac{T_n}{n!}z^n$ of Cayley trees, the so-called \emph{tree function}, is defined by the functional equation $T(z)=z e^{T(z)}$.
The tree function is closely related to the so-called (Lambert) $W$-function \cite{corless1996lambertw} that is defined by the functional equation $z= W(z) e^{W(z)}$.
The simple connection between the Cayley tree function and the Lambert $W$-function is: $T(z)=-W(-z)$.
\begin{definition}
A function $f$ from the set $[n]$ into itself is called an \emph{$n$-mapping}.
Given an $n$-mapping $f$, its \emph{functional digraph} $G_f$ is defined to be the directed graph on the vertex set $[n]$ where a directed edge $(i,j)$ is drawn whenever $f(i)=j$.
\end{definition}
\begin{figure}
\centering
\input{functional_graph}
\caption{Functional digraph of a $19$-mapping.}
\label{Mappings/fig:mapping_graph}
\end{figure}
For an example of the functional digraph of a $19$-mapping, see Figure~\ref{Mappings/fig:mapping_graph}.
The structure of mappings is simple and is well described in \cite{flajolet2009analytic}: the weakly connected components of their digraphs are simply cycles of Cayley trees.
That is, each connected component consists of rooted labelled trees whose root nodes are connected by directed edges such that they form a cycle.
In this context, we will call a node $j$ that lies on a cycle in a mapping $f$, i.e., for which there exists a $k \ge 1$ such that $f^{k}(j) = j$, a \emph{cyclic node}.
Using the symbolic method (see~\cite{flajolet2009analytic} for an introduction), this structural connection between Cayley trees and mappings can also easily be taken to the level of generating functions.
Indeed, if $\mathcal{T}$, $\mathcal{C}$ and $\mathcal{M}$ denote the combinatorial classes of Cayley trees, connected mappings and mappings and if $T(z)$, $C(z)$ and $M(z)$ denote their exponential generating functions we have the following connection:
\begin{align*}
& \mathcal{C} = \CYC{\mathcal{T}} && \Longrightarrow &&C(z) = \log \left( \frac{1}{1-T(z)}\right), \\
& \mathcal{M} = \SET{\mathcal{C}} && \Longrightarrow &&M(z) = \exp(C(z))= \frac{1}{1-T(z)}.
\end{align*}
However, for the problem considered here the above relation between $\mathcal{T}$ and $\mathcal{C}$ cannot be applied directly. Instead we will use a decomposition of the objects with respect to the node with smallest label.
Note that whenever we speak about a random $n$-mapping $f$, this means that $f$ has been chosen uniformly at random among all $n$-mappings.
Similarly, a random Cayley tree of size $n$ is chosen according to the uniform distribution on all Cayley trees of the same size.
In this paper, we shall generalize the concept of ascending runs from permutations to Cayley trees and to mappings.
Let us start by defining ascents for permutations:
An \emph{ascent} of a permutation $\pi$ is a position $i$ for which it holds that $\pi(i)< \pi(i+1)$.
Similarly, a descent of a permutation $\pi$ is a position $i$ for which it holds that $\pi(i)> \pi(i+1)$.
The descents of a permutation $\pi$ partition it into so-called \emph{ascending runs}.
These are increasing contiguous subsequences of maximal length.
The following holds: if $\pi$ has $k$ descents, it is the union of $k+1$ ascending runs.
Throughout this paper, we will write $n^{\underline{m}}$ for the falling factorials $n \cdot (n-1) \cdots (n-m+1)$.
\section{A bijective proof of Cayley's formula}
\label{sec:Cayley_formula}
In~\cite{BruPan2015Parking}, the present authors generalized the concept of parking functions to Cayley trees and to mappings.
In this context, they presented a bijection between parking functions on trees and parking functions on mappings.
The precise statement of this bijection can be found in Theorem 3.6 in~\cite{BruPan2015Parking}.
When restricting this bijection to trees and mappings only, we obtain (to the best of our knowledge) a new bijective proof of Cayley's formula.
We will be able to use this bijection in the following section that concerns ascending runs in trees and mappings.
In the following, we will denote by $T(v)$ the parent of node $v$ in the tree $T$.
That is, for $v \neq \textsf{root}(T)$, $T(v)$ is the unique node such that $(v, T(v))$ is an edge in $T$.
For the sake of convenience, let us define $T \left( \textsf{root}(T)\right) =\textsf{root}(T)$.
\begin{theorem}[Corollary of Theorem 3.6 in \cite{BruPan2015Parking}]\label{cayley/thm:bijection}
For each $n \ge 1$, there exists a bijection $\varphi$ from the set of pairs $(T,w)$, with $T \in \mathcal{T}_{n}$ a tree of size $n$ and $w \in T$ a node of $T$, to the set of $n$-mappings. Thus
\begin{equation*}
n \cdot T_{n} = M_{n}, \quad \text{for $n \ge 1$}.
\end{equation*}
\end{theorem}
\begin{proof}
Given a pair $(T,w)$, we consider the unique path $w \rightsquigarrow \textsf{root}(T)$ from the node $w$ to the root of $T$.
It consists of the nodes $v_{1}=w$, $v_2=T(v_1), \ldots, v_{i+1}=T(v_i), \ldots, v_{r} = \textsf{root}(T)$ for some $r \geq 1$.
We denote by $I = (i_{1}, \dots, i_{t})$, with $i_{1} < i_{2} < \cdots < i_{t}$ for some $t \ge 1$, the indices of the right-to-left maxima in the sequence $v_1, v_2, \ldots, v_r$, i.e.,
\begin{equation*}
i \in I \Longleftrightarrow v_{i} > v_{j}, \quad \text{for all } j > i.
\end{equation*}
The corresponding set of nodes in the path $w \rightsquigarrow \textsf{root}(T)$ will be denoted by $V_{I} \colonequals \{v_{i} : i \in I\}$. Of course, if follows from the definition that the root node is always contained in $V_{I}$, i.e., $v_{r} \in V_{I}$.
We can now describe the function $\varphi$ by constructing an $n$-mapping $f$.
The $t$ right-to-left maxima in the sequence $v_1, v_2, \ldots, v_r$ will give rise to $t$ connected components in the functional digraph $G_{f}$.
Moreover, the nodes on the path $w \rightsquigarrow \textsf{root}(T)$ in $T$ will correspond to the cyclic nodes in $G_{f}$. We describe $f$ by defining $f(v)$ for all $v \in [n]$, where we distinguish whether $v \in V_{I}$ or not.
\begin{itemize}
\item[$(a)$] Case $v \notin V_{I}$: We set $f(v) \colonequals T(v)$.
\item[$(b)$] Case $v \in V_{I}$: We set $f(v_{i_1}) \colonequals v_1$ and
$f(v_{i_{\ell}}) \colonequals T\left( v_{i_{\ell-1}}\right)$ for $l >1$.
This means that the nodes on the path $w \rightsquigarrow \textsf{root}(T)$ in $T$ form $t$ cycles $C_{1} \colonequals (v_{1}, \dots, v_{i_{1}})$, \dots, $C_{t} \colonequals (T(v_{i_{t-1}}), \dots, v_{r}=v_{i_t})$ in $G_{f}$.
\end{itemize}
It is now easy to describe the inverse function $\varphi^{-1}$.
Given a mapping $f$, we sort the connected components of $G_f$ in decreasing order of their largest cyclic elements.
That is, if $G_f$ consists of $t$ connected components and $c_i$ denotes the largest cyclic element in the $i$-th component, we have $c_1 > c_2 > \ldots > c_t$.
Then, for every $1 \leq i \leq t$, we remove the edge $(c_i, d_i)$ where $d_i= f(c_i)$.
Next we reattach the components to each other by establishing the edges $(c_i, d_{i+1})$ for every $1 \leq i \leq t-1$.
This leads to the tree $T$.
Note that the node $c_t$ is attached nowhere since it constitutes the root of $T$.
Setting $w=d_1$, we obtain the preimage $(T,w)$ of $f$.
\end{proof}
\begin{example}
Taking the image of the pair $(T,1)$ where the tree $T$ is depicted in Figure~\ref{cayley/fig:example} leads to the mapping described in Figure~\ref{Mappings/fig:mapping_graph}.
We consider the unique path from the node labelled $1$ to the root of $T$.
It consists of the following nodes: $1,7,11,17,4,14$ and $10$.
Within this sequence, the right-to-left maxima are $17, 14$ and $10$ which are marked by gray nodes in the figure.
When creating the image of $(T,1)$ under the map $\varphi$, the edges $(17,4)$ and $(14,10)$ are removed and the edges $(17,1)$, $(14,4)$ and $(10,10)$ are created.
\demo\end{example}
\begin{figure}
\centering
\input{bijection_cayley_trees}
\caption{Taking the image of the pair $(T,1)$ where $T$ is depicted above leads to the mapping described in Figure~\ref{Mappings/fig:mapping_graph}. The unique path from $1$ to the root is marked by dashed edges.
Right-to-left-maxima on this path are marked by gray nodes.}
\label{cayley/fig:example}
\end{figure}
The bijection described above has the property of preserving ascents as well as elements that do not lie at the beginning of an ascending run:
\begin{proposition}
Consider the pair $(T,w)$ where $T$ is a Cayley tree and $w$ is a node in $T$.
Let $f$ be the image of $(T,w)$ under the bijection $\varphi$ described in Theorem~\ref{cayley/thm:bijection}.
Then the following holds for every node $v$ in $T$:
\begin{enumerate}
\item $T(v)>v \Longleftrightarrow f(v) > v,$
\item There exists a node $x < v$ in $T$ with $T(x)=v$
$\Longleftrightarrow$ There exists an element $x < v$ in $f$ with $f(x)=v.$
\end{enumerate}
\label{cayley/prop:properties_bijection}
\end{proposition}
\begin{remark}
These properties are fulfilled because of the following observation: When creating the image of some $(T,w)$ under the map $\varphi$, the only edges that are removed are decreasing ones. The edges that are created instead in $G_f$ are also decreasing ones.
\end{remark}
\begin{proof}[Proof of Proposition~\ref{cayley/prop:properties_bijection}] \mbox{}
\begin{enumerate}
\item As in the proof of Theorem~\ref{cayley/thm:bijection} we distinguish whether the node $v$ in $T$ is contained in $V_I$ or not:
If $v$ is not contained in $V_I$, $f(v)=T(v)$ and thus the statement is clearly true.
If $v \in V_I$, $v$ is a right-to-left maximum in the unique path from $w$ to the root of $T$.
It thus holds $v=v_{i_{\ell}}$ for some $\ell$ and clearly $T(v) \leq v$.
We need to show that this implies that $f(v) \leq v$ holds as well.
By definition, we have $f(v_{i_1}) \colonequals v_1$ and
$f(v_{i_{\ell}}) \colonequals T\left( v_{i_{\ell-1}}\right)$ for $l >1$
Thus, $f(v_{i_{\ell}})$ is either equal to $v_{i_{\ell}}$ or $f(v_{i_{\ell}})$ is not a right-to left maximum in which case it is clearly strictly smaller than $v_{i_{\ell}}$.
This proves the first statement of this proposition.
\item $(\Longrightarrow):$ If the node $v$ has a child $x$ with $x < v$, then $x$ cannot belong to the set of nodes $V_I$.
Thus $f(x)=T(x)=v$ and in the mapping $f$ the node $v$ has a preimage $x$ with $x < v$.
$(\Longleftarrow):$ Let $x$ be an element with $x< v$ and $f(x)=v$.
Suppose $x=v_{i_{\ell}}$ for some $\ell$.
Then $v=f(x)=T(v_{i_{\ell -1}}) \leq x$, a contradiction.
Thus $x \notin V_I$ and we have $f(x)=T(x)$.
Therefore $x$ is a child of $v$ in $T$ that fulfils $x < v$.
\end{enumerate}
\end{proof}
\section{Main results}
\label{Mappings/sec:runs}
We now consider the notion of ascending runs in a mapping.
Generalizing the definition of ascending runs for permutations, an ascending run in a mapping $f$ is a maximal ascending sequence $i < f(i) < f^{2}(i) < \cdots < f^{k}(i)$, i.e., an ascending sequence which is not contained in a longer such sequence.
\begin{figure}[h]
\centering
\input{label_patterns}
\caption{The label pattern studied in this paper: Counting ascending runs is equivalent to counting the occurrences of the label pattern depicted above.}
\label{Mapping/fig:label_patterns}
\end{figure}
Given a specific mapping $f$ and its functional digraph $G_{f}$, the number of runs in $f$ can be counted easily by considering the first element of a run:
Indeed, an element $j$ is the first element of a run iff each label $i$ of a preimage of $j$ is not smaller than $j$, i.e., $(f(i)=j) \; \Rightarrow \; (i \ge j)$.
We thus need to count nodes in the mapping that form the label pattern depicted in Figure~\ref{Mapping/fig:label_patterns}.
Note that elements with no preimages at all, i.e., terminal nodes in the graph, are of course always at the beginning of a run.
\begin{example}
Consider the $19$-mapping depicted in Figure~\ref{Mappings/fig:runs}.
In its functional digraph, the nodes that correspond to first elements of an ascending run are marked by gray nodes.
There are $13$ such nodes and thus our mapping has $13$ ascending runs.
Moreover, edges $(i,j)$ where $i<j$ are marked with dashed lines and edges with $j>i$ are marked with full lines.
An ascending run is thus a path of maximal length consisting of dashed edges only (or no edges at all).
An element that is the start of an ascending run can be characterized as a node where no incoming edges are dashed.
\demo\end{example}
\begin{figure}
\centering
\input{example_runs}
\caption{A $19$-mapping with $13$ ascending runs. If an element is the
first element in a run, its node is gray; otherwise it is white. Increasing edges, i.e., edges $(i,j)$ with $i <j$, are marked by dashed lines.}
\label{Mappings/fig:runs}
\end{figure}
Recall that for permutations the following simple connection between descents and ascending runs holds: if a permutation $\pi$ has $k$ descents it can be decomposed into $(k+1)$ ascending runs.
Moreover, by taking the complement $\pi^c$ of $\pi$, i.e. the permutation mapping $i$ to $n-\pi(i)+1$, one obtains a permutation with $k$ ascents.
Thus, studying the distribution of the following parameters in permutations is equivalent: ascents, descents, ascending runs and descending runs.
This is clearly not the case for mappings: The presence of $k$ descents does not yield a decomposition into $(k+1)$ ascending runs.
For example, in the mapping depicted in Figure~\ref{Mappings/fig:runs} there are $10$ ascents and thus $9$ descents but the number of ascending runs is equal to $13$.
Thus, the study of ascending runs has to be performed separately and requires, as we will see, more involved techniques than the one of ascents.
Concerning the study of ascents, recall that the probability of $i$ being an ascent in a random $n$-mapping is simply equal to $(n-i)/n$.
See the work of E\u{g}ecio\u{g}lu and Remmel~\cite{egecioglu1986bijections} as well as Clark~\cite{clark2008ascents} for a treatment of ascents in mappings and Cayley trees.
In the following, $R_{n}$ will denote the random variable that counts the number of ascending runs in a random $n$-mapping.
Because of the combinatorial structure of mappings, we will first analyse the corresponding random variable for trees before we study $R_{n}$.
\subsection{Ascending runs in Cayley trees}
To study $R_{n}$ we first consider the corresponding quantity in Cayley trees.
Thus, we introduce the random variable $F_{n}$ which counts the number of runs in a random Cayley tree of size $n$.
Let us further introduce the generating function
\begin{align*}
F(z,v) & = \sum_{n \ge 1} \sum_{m \ge 0} \mathbb{P}\{F_{n}=m\} T_{n} \frac{z^{n}}{n!} v^{m}
= \sum_{n \ge 1} \sum_{m \ge 0} F_{n,m} \frac{z^{n}}{n!} v^{m},
\end{align*}
where $T_{n} = n^{n-1}$ is the number of trees of size $n$ and $F_{n,m}$ is the number of trees of size $n$ with exactly $m$ ascending runs.
Clearly, $F_{1,m}=\delta_{m,1}$.
In order to establish a recurrence relation for the numbers $F_{n,m}$, we decompose a tree of size $n \geq 2$ with respect to the node labelled $1$. Two different cases, as shown in Figure~\ref{Mappings/fig:decomp_runs_tree}, have to be considered:
\begin{figure}
\includegraphics[width=\columnwidth]{runs_tree.pdf}
\caption{Decomposition of a tree with respect to its smallest node.}
\label{Mappings/fig:decomp_runs_tree}
\end{figure}
\medskip
\noindent \underline{Case (1):} The node $1$ is the root node.
In this case a set of $r \geq 1$ subtrees $T_1, T_2, \ldots, T_r$ is attached to the root $1$.
Since the root has the smallest of all labels, it will always constitute a run of its own.
Thus, if the subtrees $T_i$ have respective sizes $n_i$ and $m_i$ runs each, the entire tree will consist of $n$ nodes and have $m$ runs iff $\sum_{i=1}^r n_i=n-1$ and $\sum_{i=1}^r m_i=m-1$.
The contribution to $ F_{n,m}$ is therefore:
\begin{align*}
F_{n,m}^{(1)} = \sum_{r \geq 1} \frac{1}{r!} \sum_{\substack{ \sum n_i =n-1, \\ \sum m_i=m-1}} F_{n_1,m_1} \cdots F_{n_r,m_r} \cdot \binom{n-1}{n_1, \ldots, n_r},
\end{align*}
where the multinomial coefficient reflects the number of possibilities of redistributing the labels in $[n-1]$ order-isomorphically to the subtrees.
\medskip
\noindent \underline{Case (2):} The node $1$ is not the root node.
In this case the node $1$ is attached to some node labelled $i$ of a tree $T_0$ of size $n_0$ and with $m_0$ runs and has itself a set of $r \geq 0$ subtrees $T_1, T_2, \ldots, T_r$ attached to it.
Depending on where in $T_0$ the node $1$ is attached, a new run is created or not. If $1$ is attached at the beginning of a run in $T_0$ no new run is created -- there are $m_0$ possibilities of attaching $1$ in this way.
If $1$ is however attached somewhere in the middle of a run, a new run will be created -- for this case there are $(n_0-m_0)$ possibilities.
The contribution to $ F_{n,m}$ is therefore:
\begin{align*}
F_{n,m}^{(2a )} & = \hspace{-0.5cm} \sum_{\substack{1 \leq n_0 \leq n-1, \\ 1\leq m_0 \leq m}} m_0 \cdot \sum_{r \geq 0} \frac{1}{r!} \sum_{\substack{ \sum n_i =n-n_0-1, \\ \sum m_i=m-m_0}} \hspace{-0.5cm} F_{n_0,m_0} \cdots F_{n_r,m_r} \binom{n-1}{n_0, \ldots, n_r}
\end{align*}
for the case that the node $1$ is attached to the beginning of a run in $T_0$ and
\begin{align*}
F_{n,m}^{(2b )} & = \hspace{-0.5cm} \sum_{\substack{1 \leq n_0 \leq n-1, \\ 1\leq m_0 \leq m}} \hspace{-0.5cm} (n_0- m_0) \cdot \sum_{r \geq 0} \frac{1}{r!} \sum_{\substack{ \sum n_i =n-n_0-1, \\ \sum m_i=m-m_0-1}} \hspace{-0.5cm} F_{n_0,m_0} \cdots F_{n_r,m_r} \binom{n-1}{n_0, \ldots, n_r}
\end{align*}
in the other cases.
The goal is now to translate this recurrence relation for $F_{n,m}= F_{n,m}^{(1)} + F_{n,m}^{(2a)}+ F_{n,m}^{(2b)}$ into the language of generating functions.
In order to alleviate notation we write $F$ for $F(z,v)$, $F_z$ and $F_v$ for the partial derivatives of $F$ with respect to $z$ and $v$.
First, let us note that for $F_{n,m}^{(1)}$ we have the following:
\begin{align*}
F_{n,m}^{(1)} & = \sum_{r \geq 1} \frac{(n-1)!}{r!} \sum_{\substack{ \sum n_i =n-1, \\ \sum m_i=m-1}} [z^{n_1}v^{m_1}]F \cdots [z^{n_r}v^{m_r}]F \\
& = (n-1)! [z^{n-1}v^{m-1}]\exp(F) = (n-1)! [z^{n}v^{m}]zv \exp(F).
\end{align*}
Recall that $m_0 \cdot F_{n_0,m_0}/(n_0)! = [z^{n_0}v^{m_0}] v F_v$.
We then obtain for $F_{n,m}^{(2a)}$:
\begin{align*}
F_{n,m}^{(2a)} & = (n-1)! \sum_{n_0, m_0} \left([z^{n_0}v^{m_0}]v F_v \right)\cdot \left([z^{n-n_0-1}v^{m-m_0}]\exp (F) \right) \\
& = (n-1)! [z^{n}v^{m}]zv F_v \exp(F).
\end{align*}
Similarly, for $F_{n,m}^{(2b)}$:
\begin{align*}
F_{n,m}^{(2b)} & = (n-1)! [z^{n}v^{m}]zv \left( zF_z - vF_v\right)\exp(F).
\end{align*}
Thus, we can write $F_{n,m}$ as follows:
\begin{align*}
F_{n,m} & = n! [z^{n}v^{m}] F = (n-1)! [z^{n}v^{m}]zv \exp(F) \left( 1+ z F_z + (1-v)F_v\right),
\end{align*}
or equivalently,
\[
\sum_{n \geq 1} \sum_{m \geq 1} n [z^{n}v^{m}] F = z F_z = zv \exp(F) \left( 1+ z F_z + (1-v)F_v\right).
\]
Thus the bivariate generating function $F(z,v)$ is defined by the following first-order quasi-linear
partial differential equation:
\begin{align*}
\left ( 1-zv e^{F(z,v)} \right) F_{z}(z,v) = v(1-v) e^{F(z,v)} F_{v}(z,v) + v e^{F(z,v)},
\end{align*}
with the initial condition $F(0,v)=0$.
This PDE can be solved using the method of characteristics (see, e.g., \cite{evans1998partial}).
Let us regard $z$, $v$ and $F$ as variables depending on $t$. We then obtain the following system of ordinary differential equations, the so-called system of characteristic differential equations:
\begin{subequations}
\label{Mappings/eqn:char_runs_trees}
\begin{align}
\frac{dz}{dt} &= 1-zv \cdot e^{F(z,v)}, \label{Mappings/eqn:char_runs_trees_z}\\
\frac{dv}{dt} &= e^{F(z,v)} \cdot v(v-1), \label{Mappings/eqn:char_runs_trees_v}\\
\frac{dF}{dt} &= v \cdot e^{F(z,v)}. \label{Mappings/eqn:char_runs_trees_R}
\end{align}
\end{subequations}
We are now looking for first integrals of the system~\eqref{Mappings/eqn:char_runs_trees}, i.e., for functions $\zeta(z,v,F)$ that are constant along any solution curve -- a so-called characteristic curve -- of~\eqref{Mappings/eqn:char_runs_trees}.
From~\eqref{Mappings/eqn:char_runs_trees_v} and~\eqref{Mappings/eqn:char_runs_trees_R} one obtains the following differential equation
\[
\frac{dv}{dF}= v-1
\]
which has the general solution
\[
v=c_1 \cdot e^{F}+1.
\]
Thus a first integral of~\eqref{Mappings/eqn:char_runs_trees} is given by:
\begin{equation}
\zeta_1(z,v,F)=c_1=e^{-F} \cdot (v-1).
\label{Mappings/eqn:first_int_runs_trees1}
\end{equation}
From~\eqref{Mappings/eqn:char_runs_trees_z} and~\eqref{Mappings/eqn:char_runs_trees_R} one obtains, after substituting $v=c_1 \cdot e^{F}+1$, the following differential equation
\[
\frac{dz}{dF}= \frac{1}{(c_1e^{F}+1)\cdot e^{F}}-z.
\]
The general solution of this first order linear differential equation is
\[
z=\frac{ F-\ln\left( c_1 \cdot e^{F} +1 \right) +c_2}{e^{F}}.
\]
After backsubstituting $c_1=e^{-F} \cdot (v-1)$ one obtains the following first integral of~\eqref{Mappings/eqn:char_runs_trees} which is independent of~\eqref{Mappings/eqn:first_int_runs_trees1}:
\begin{equation}
\zeta_2(z,v,F)=c_2=z \cdot e^{F}- F +\ln(v).
\label{Mappings/eqn:first_int_runs_trees2}
\end{equation}
The general solution of~\eqref{Mappings/eqn:char_runs_trees} is thus given by:
\[
G(\zeta_1(z,v,F),\zeta_2(z,v,F))=\text{const.},
\]
where $G$ is an arbitrary differentiable function in two variables.
One can also solve this equation with respect to the variable $z$ and obtains:
\[
z= e^{-F} \cdot \left( F -\ln(v) + g\left( \frac{v-1}{e^{F}}\right) \right),
\]
where $g$ is an arbitrary differentiable function in one variable.
In order to specify this function $g$, we plug the initial condition $F(0,v)=0$ into this equation.
This leads to:
\[
g(v-1) =\ln(v),
\]
and finally we obtain that the solution of~\eqref{Mappings/eqn:char_runs_trees} is given by the following functional equation:
\begin{align}
z = \frac{\ln\left(\frac{e^{F(z,v)}-1+v}{v}\right)}{e^{F(z,v)}}.
\label{Mappings/eqn:func_run_tree}
\end{align}
\subsection{Ascending runs in (connected) mappings}
Having computed the generating function $F(z,v)$ of ascending runs in trees in the previous section, we now turn to our actual problem, namely the enumeration of mappings by size and number of ascending runs.
We introduce the following bivariate generating function
\begin{equation}
R(z,v) = \sum_{n \ge 0} \sum_{m \ge 0} \mathbb{P}\{R_{n}=m\} M_{n} \frac{z^{n}}{n!} v^{m},
\label{Mappings/eqn:GF_runs_mappings}
\end{equation}
with $M_{n} = n^{n}$ the number of $n$-mappings.
Actually, it is easier to study the quantity for connected mappings first.
We thus introduce the corresponding generating function $C(z,v)$.
Due to the combinatorial construction of mappings as sets of connected mappings, it holds that
\begin{equation*}
R(z,v) = e^{C(z,v)}.
\end{equation*}
Note however that such a simple relation does not hold when passing from Cayley trees to cycles of Cayley trees.
Indeed, the number of ascending runs in a connected mapping is not equal to the sum of the numbers of ascending runs in the Cayley trees constituting this connected mapping.
Thus, it does not hold that $C(z,v)$ is equal to $\log \left( 1/(1-F(z,v))\right)$ as one might be tempted to think.
In order to establish a functional equation for $C(z,v)$, we will therefore decompose a connected mapping graph according to the node labelled $1$ as done before for Cayley trees.
Here, we have to consider three different cases, two of which are depicted in Figure~\ref{Mappings/fig:decomp_runs_mapping}.
\begin{figure}
\includegraphics[width=\columnwidth]{runs_connectedMappings.pdf}
\caption{Decomposition of a connected mapping with respect to its smallest node.}
\label{Mappings/fig:decomp_runs_mapping}
\end{figure}
\noindent \underline{Case (1):} The node $1$ is a cyclic node in a cycle of length one.
In this case the mapping graph is simply a tree with root node $1$ and an additional loop-edge $(1,1)$.
We thus have exactly the same situation as in the first case for trees. See the left hand side of Figure~\ref{Mappings/fig:decomp_runs_tree}.
The contribution to $ C_{n,m}$ is therefore:
\begin{align*}
C_{n,m}^{(1)} & = \sum_{r \geq 1} \frac{1}{r!} \sum_{\substack{ \sum n_i =n-1, \\ \sum m_i=m-1}} F_{n_1,m_1} \cdots F_{n_r,m_r} \cdot \binom{n-1}{n_1, \ldots, n_r}\\
& = (n-1)! [z^{n} v^{m}] zv \exp(F).
\end{align*}
\noindent \underline{Case (2):} The node $1$ is a cyclic node in a cycle containing more than one element; this case is depicted on the left hand side of Figure~\ref{Mappings/fig:decomp_runs_mapping}.
The structure of such a mapping can be understood as follows:
There is a (non-empty) tree $T_0$ in which the node $1$ has been attached to some node $i$.
Moreover, the root of $T_0$ is attached to the node $1$, thus forming a cycle consisting of the path from $i$ to the root of $T_0$ and $i$.
Since $1$ can have arbitrarily many children, we pick a set of $r \geq 0$ subtrees and attach them to $1$.
Whether the node $1$ contributes to a new run or not depends on whether it is attached at the beginning or somewhere in the middle of a run (as in the second case for trees).
Indeed, if $T_0$ is of size $n_0$ and has $m_0$ runs itself, there are $m_0$ possibilities of attaching $1$ at the beginning of a run in $T_0$ thus not creating a new run.
Moreover, there are $(n_0-m_0)$ possibilities of attaching $1$ somewhere in the middle of a run and of creating a new run.
The contribution to $ C_{n,m}$ is therefore:
\begin{align*}
C_{n,m}^{(2 )} & = \hspace{-0.5cm} \sum_{\substack{1 \leq n_0 \leq n-1, \\ 1\leq m_0 \leq m}} m_0 \sum_{r \geq 0} \frac{1}{r!} \sum_{\substack{ \sum n_i =n-n_0-1, \\ \sum m_i=m-m_0}} \hspace{-0.5cm} F_{n_0,m_0} \cdots F_{n_r,m_r} \binom{n-1}{n_0, \ldots, n_r}\\
& + \hspace{-0.5cm} \sum_{\substack{1 \leq n_0 \leq n-1, \\ 1\leq m_0 \leq m}} \hspace{-0.5cm} (n_0- m_0) \cdot \sum_{r \geq 0} \frac{1}{r!} \sum_{\substack{ \sum n_i =n-n_0-1, \\ \sum m_i=m-m_0-1}} \hspace{-0.5cm} F_{n_0,m_0} \cdots F_{n_r,m_r} \binom{n-1}{n_0, \ldots, n_r}\\
& (n-1)! [z^{n} v^{m}] zv \exp(F) \left(z F_{z} + (1-v) F_{v}\right).
\end{align*}
\noindent \underline{Case (3):} The node $1$ is not a cyclic node; this case is depicted on the right hand side of Figure~\ref{Mappings/fig:decomp_runs_mapping}.
Here, the node $1$ is attached to some node $i$ of a connected mapping $C_0$ and a set of $r \geq 0$ trees are attached to $1$.
Again, the node $1$ contributes to a new run or not depending on where it is attached.
Similarly as in the second case, we obtain that the contribution to $ C_{n,m}$ is:
\begin{align*}
C_{n,m}^{(3)} & = \hspace{-0.5cm} \sum_{\substack{1 \leq n_0 \leq n-1, \\ 1\leq m_0 \leq m}} m_0 \sum_{r \geq 0} \frac{C_{n_0,m_0}}{r!} \sum_{\substack{ \sum n_i =n-n_0-1, \\ \sum m_i=m-m_0}} \hspace{-0.5cm} F_{n_1,m_1} \cdots F_{n_r,m_r} \binom{n-1}{n_0, \ldots, n_r} \\
& + \hspace{-0.5cm} \sum_{\substack{1 \leq n_0 \leq n-1, \\ 1\leq m_0 \leq m}} \hspace{-0.5cm} (n_0- m_0) \cdot \sum_{r \geq 0} \frac{C_{n_0,m_0}}{r!} \sum_{\substack{ \sum n_i =n-n_0-1, \\ \sum m_i=m-m_0-1}} \hspace{-0.5cm} F_{n_1,m_1} \cdots F_{n_r,m_r} \binom{n-1}{n_0, \ldots, n_r} \\
& = (n-1)![z^nv^m] zv \exp(F) \left( z C_z + (1-v)C_v\right).
\end{align*}
Adding up the contributions of these three cases and summing over $n \geq 1$ and $m \geq 1$ we obtain the following:
\[
C_z = v \exp(F) \cdot \left( 1+ (1-v)F_v + z F_z + (1-v)C_v + zC_z \right).
\]
Thus, the bivariate generating function $C(z,v)$ is defined by the
following first-order linear partial differential equation:
\begin{align*}
C_{z}\left(1-zv\cdot \exp(F)\right) - v (1-v) \exp(F) C_{v} = F_{z},
\end{align*}
where $F(z,v)$ is defined by equation~\eqref{Mappings/eqn:func_run_tree}.
Using the auxiliary function $H(z,v)$ that is defined by
\begin{align}
\exp(H) = \frac{\exp(F)-1+v}{v}
\label{Mappings/eqn:runs_connection_H_F}
\end{align}
and that satisfies the following functional equation
\begin{equation}
z = \frac{H}{v e^{H}+1-v},
\label{Mappings/eqn:runs_func_eqnH}
\end{equation}
we can solve this PDE.
We find that the solution is given as follows:
\begin{equation*}
C(z,v) = \ln\left(\frac{v e^{H(z,v)} + 1 - v}{v e^{H(z,v)} \big(1-H(z,v)\big)+1-v}\right),
\end{equation*}
with $H(z,v)$ as given above.
Checking that this function indeed is a solution to the partial differential equation can be done easily by using implicit differentiation in order to compute the partial derivatives.
We thus also obtain the solution for ascending runs in arbitrary mappings:
\begin{align}
R(z,v) & = e^{C(z,v)} = \frac{v e^{H(z,v)} + 1 - v}{v e^{H(z,v)} \big(1-H(z,v)\big)+1-v} \notag \\
& = \frac{H/z}{H/z-Hv \exp(H)} = \frac{1}{1-zv \cdot \exp(H(z,v))},
\label{Mappings/eqn:runs_func_eqnR}
\end{align}
with $H(z,v)$ satisfying \eqref{Mappings/eqn:runs_func_eqnH}.
\subsection{Exact enumeration formul{\ae}}
Having found the generating functions $F(z,v)$ and $R(z,v)$ for trees and mappings, respectively, we can prove the following interesting connection between ascending runs in trees and in mappings:
\begin{theorem}
Let $F_{n,m}$ denote the number of Cayley trees of size $n$ with exactly $m$ ascending runs and $R_{n,m}$ the number of $n$-mappings with exactly $m$ ascending runs. Then for all $n \geq 1$ and $m \geq 1$ the following identity holds:
\[
R_{n,m} = n \cdot F_{n,m}.
\]\label{Mappings/thm:connection_trees_mappings}
\end{theorem}
\begin{proof}
We will prove the assertion above at the level of generating functions.
Combining equation~\eqref{Mappings/eqn:runs_connection_H_F} and \eqref{Mappings/eqn:runs_func_eqnR} we have:
\[
R(z,v) = \frac{1}{1-z \cdot (\exp(F(z,v)-1+v))}.
\]
Moreover, implicit differentiation of~\eqref{Mappings/eqn:func_run_tree} with respect to $z$ leads to:
\[
F_z(z,v)=\frac{\exp(F(z,v))-1+v}{1-z \cdot (\exp(F(z,v)-1+v))}.
\]
Thus one easily sees that
\[
R(z,v) = 1 + z \cdot F_z(z,v).
\]
Extracting coefficients for $n \geq 1$ and $m \geq 1$ leads to
\[
R_{n,m} = n! [z^n v^m] R(z,v) = n! [z^{n-1} v^m] F_z(z,v) = n \cdot F_{n,m}.
\]
\end{proof}
A combinatorial proof of this result can be found in the following section.
Combining this result with the functional equation for the generating function $F(z,v)$ we obtain the following enumeration formul{\ae}:
\begin{theorem}
The number $F_{n,m}$ of Cayley trees of size $n$ with exactly $m$ runs is given as:
\begin{align*}
F_{n,m} = (n-1)^{\underline{m-1}} \Stir{n}{m}
\end{align*}
and the number $R_{n,m}$ of $n$-mappings with exactly $m$ runs is given as:
\begin{align}
R_{n,m} = n^{\underline{m}} \Stir{n}{m} = \frac{n!}{(n-m)!} \Stir{n}{m},
\label{Mappings/eqn:mapping_stirling}
\end{align}
where $\Stir{n}{m}$ denotes the Stirling numbers of the second kind that count partitions of the set $[n]$ into $m$ non-empty subsets.
\end{theorem}
\begin{proof}
Extracting coefficients from the functional equation \eqref{Mappings/eqn:func_run_tree} for $F(z,v)$, the generating function of ascending runs in Cayley trees, can be done by applying Lagrange's inversion formula (see, e.g., \cite{flajolet2009analytic}) with $\varphi(F)=\exp(F)F/\ln\left(\frac{e^{F}-1+v}{v}\right)$ and twice Cauchy's integral formula:
\begin{align*}
[z^n] F(z,v) & = \frac{1}{n} [F^{n-1}] \frac{e^{n F}\cdot F^n}{\ln\left(\frac{e^{F}-1+v}{v}\right)^n} \\
& =\frac{1}{n\cdot 2 \pi i} \oint \frac{e^{n F}}{\ln\left(\frac{e^{F}-1+v}{v}\right)^n} dF \\
& =\frac{1}{n\cdot 2 \pi i} \oint \frac{(e^S\cdot v -1 +v)^{n-1}\cdot e^Sv}{S^n} dS \\
& = \frac{1}{n} [S^{n-1}] (e^S\cdot v -1 +v)^{n-1}\cdot e^Sv\\
& =\frac{v}{n} \sum_{k=0}^{n-1} \frac{1}{(n-1-k)!} \sum_{\ell=0}^{n-1}\binom{n-1}{\ell} \frac{\ell^{k}v^{\ell}}{k !} (v-1)^{n-1-\ell} \\
& =\frac{v}{n} \sum_{\ell=0}^{n-1} \binom{n-1}{\ell} v^{\ell} (v-1)^{n-1-\ell} \frac{(l+1)^{n-1}}{(n-1)!}.
\end{align*}
When passing from the second to the third line above we use the substitution $S=\ln\left(\frac{e^{R}-1+v}{v}\right)$ for which it holds that
$dR = \frac{e^R-1+v}{e^R} dS = \frac{ve^S}{ve^S-1+v} dS$.
What remains to do now is to extract the coefficients in $v$:
\begin{align*}
F_{n,m} & = n! [v^mz^n] F(z,v)= \sum_{\ell=0}^{n-1} \binom{n-1}{\ell} (\ell+1)^{n-1} [v^{m-\ell-1}] (1-v)^{n-1-\ell} \\
& = \sum_{\ell=0}^{n-1} \binom{n-1}{\ell} (\ell+1)^{n-1} \binom{n-\ell-1}{m-\ell-1} (-1)^{m- \ell -1}. \end{align*}
Finally, we obtain the following explicit solution for the numbers $F_{n,m} = T_{n} \mathbb{P}\{F_{n}=m\} = n! [z^{n} v^{m}] F(z,v)$ of size-$n$ trees with exactly $m$ runs:
\begin{align*}
F_{n,m} = \binom{n-1}{m-1} \sum_{\ell=0}^{m-1} (\ell+1)^{n-1} (-1)^{m-1-\ell} \binom{m-1}{\ell}.
\end{align*}
These numbers can also be described with the help of the Stirling numbers of the second kind.
Since the Stirling numbers of second kind $\Stir{n}{m}$ satisfy
\begin{equation}
\Stir{n}{m} = \frac{1}{m!} \sum_{\ell=0}^{m} \binom{m}{\ell} (-1)^{m-\ell} \ell^{n},
\end{equation}
we obtain the following for $n \geq 1$:
\begin{align}
F_{n,m} & = \notag \binom{n-1}{m-1} \sum_{\ell=1}^{m} \ell^{n-1} (-1)^{m-\ell} \binom{m-1}{\ell-1} \\\notag
& = \binom{n-1}{m-1} \sum_{\ell=1}^{m} \ell^{n-1} (-1)^{m-\ell} \cdot \left( \binom{m}{\ell} - \binom{m-1}{\ell} \right) \\ \notag
& = \binom{n-1}{m-1} \left( m! \Stir{n-1}{m} + (m-1)! \Stir{n-1}{m-1} \right) \\ \notag
& = \binom{n-1}{m-1} \cdot (m-1)! \left( m \Stir{n-1}{m} + \Stir{n-1}{m-1} \right) \\ \notag
& = \binom{n-1}{m-1} \cdot (m-1)! \Stir{n}{m} \\
& = (n-1)^{\underline{m-1}} \Stir{n}{m}.
\label{Mappings/eqn:tree_stirling}
\end{align}
Using the connection between the numbers $F_{n,m}$ and $R_{n,m}$ as obtained in Theorem~\ref{Mappings/thm:connection_trees_mappings}, we obtain the corresponding result for runs in mappings.
\end{proof}
\subsection{Bijective proofs}
\label{Mappings/sec:runs_bijective_proofs}
In the previous section, we proved two results about ascending runs in trees and mappings where a combinatorial, i.e., bijective explanation is desirable.
First, the fact that $R_{n,m} = n \cdot F_{n,m}$ and second, the fact that the numbers $R_{n,m}$ can very easily be expressed with the help of the Stirling numbers of the second kind.
In this section we will thus provide bijective proofs for these statements.
\begin{theorem}
There is a bijection between pairs $(T,w)$, where $T$ is a tree of size $n$ with exactly $m$ ascending runs and $w$ is a node in $T$ and $n$-mappings $f$ with exactly $m$ ascending runs.
\end{theorem}
\begin{proof}
This statement follows quite directly from Proposition~\ref{cayley/prop:properties_bijection}.
For a pair $(T,w)$, let $f$ be its image under the bijection described in Theorem~\ref{cayley/thm:bijection}.
We need to show the following: A node $v$ is at the start of a run in $T$ if and only if $v$ is at the start of a run in $f$.
As remarked earlier, a node $v$ is at the start of a run in a tree or mapping if and only if all children or preimages $x$ of $v$ have labels larger than $x$.
Then the statement of this theorem follows by negating the second property in Proposition~\ref{cayley/prop:properties_bijection}.
\end{proof}
\begin{example}
As described in Section~\ref{sec:Cayley_formula}, the pair $(T,1)$ where $T$ is depicted in Figure~\ref{cayley/fig:example} is identified with the mapping $f$ depicted in Figure~\ref{Mappings/fig:runs}.
As can be seen very easily both $T$ and $f$ have $13$ ascending runs and the nodes that are at the start of a run are the following:
$1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 15, 18$ and $19$.
\demo\end{example}
Now let us turn to the second interesting fact that asks for a bijective proof, namely that the number $R_{n,m}$ of $n$-mappings with $m$ ascending runs can be expressed with the help of the Stirling numbers of the second kind as in Theorem~\ref{Mappings/eqn:mapping_stirling}.
We shall prove the following:
\begin{theorem}
There is a bijection between the set of $n$-mappings with exactly $m$ runs and the set of pairs $(S,x)$, where $S$ is a set-partition of $[n]$ into $m$ parts and $x=(n_{1},\dots,n_{m})$ is an integer sequence of length $m$.
The set partition is given as $S=(S_{1},S_{2}, \dots, S_{m})$ where the parts are ordered decreasingly according to the largest element in each part, i.e., it holds $\max(S_{1}) > \max(S_{2}) > \cdots > \max(S_{m})$.
The sequence $x$ then has to fulfil the following restriction: $n_{j} \in [n] \setminus \left(\bigcup_{i=1}^{j-1} \min\{\ell \in S_{i} : \ell > \max(S_{j})\}\right)$.
\label{Mappings/thm:bij_runs_partitions}
\end{theorem}
\begin{remark}
The idea of the bijection is to successively decompose the mapping into ascending runs.
This is done by starting with a run ending at the largest element of the mapping, then one ending at the next-largest element that has not been involved yet, and so on.
The runs then correspond to blocks of the partition.
In order to keep track of how these runs where ``glued'' together and to be able to reconstruct the mapping, we additionally store the image of the last element of each run in the sequence $x$.
\end{remark}
\begin{figure}
\begin{minipage}[h]{0.45\linewidth}
\centering
\begin{tabular}{|c|c|} \hline
$S_1=\left\lbrace 19\right\rbrace$ & $n_1=13$ \\ \hline
$S_2=\left\lbrace 18\right\rbrace$ & $n_2=13$ \\ \hline
$S_3=\left\lbrace 17, 13\right\rbrace$ & $n_3=1$ \\ \hline
$S_4=\left\lbrace 16, 9\right\rbrace$ & $n_4=4$ \\ \hline
$S_5=\left\lbrace 15\right\rbrace$ & $n_5=7$ \\ \hline
$S_6=\left\lbrace 14, 4 \right\rbrace$ & $n_6=4$ \\ \hline
$S_7=\left\lbrace 12\right\rbrace$ & $n_7=7$ \\ \hline
$S_8=\left\lbrace 11,7,6\right\rbrace$ & $n_8=17$ \\ \hline
$S_9=\left\lbrace 10, 8 \right\rbrace$ & $n_9=10$ \\ \hline
$S_{10}=\left\lbrace 5\right\rbrace$ & $n_{10}=17$ \\ \hline
$S_{11}=\left\lbrace 3\right\rbrace$ & $n_{11}=10$ \\ \hline
$S_{12}=\left\lbrace 2\right\rbrace$ & $n_{12}=1$ \\ \hline
$S_{13}=\left\lbrace 1\right\rbrace$ & $n_{13}=7$ \\ \hline
\end{tabular}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[h]{0.45\linewidth}
\centering
\input{ex_bijection_runs}
\end{minipage}
\caption{Example of the bijection described in the proof of Theorem~\ref{Mappings/thm:bij_runs_partitions} for the mapping depicted in Figure~\ref{Mappings/fig:mapping_graph}.}
\label{Mappings/tab:ex_bij}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{Mappings/thm:bij_runs_partitions}]
First we remark that this indeed will prove that
\[R_{n,m} = n^{\underline{m}} \Stir{n}{m}, \]
since the number of set-partitions of $[n]$ into $m$ parts is given by $\Stir{n}{m}$ and the number of sequences $x$ satisfying the restrictions is given by $n \cdot (n-1) \cdots (n-m+1) = n^{\underline{m}}$.
To prove the theorem we consider an $n$-mapping with exactly $m$ runs and iterate the following procedure, where we colour the elements of the mapping until all elements are coloured.
\begin{itemize}
\item In the $j$-th step we consider the largest element in the mapping which has not been coloured so far; let us denote it by $s_{1}^{(j)}$. Consider all preimages of $s_{1}^{(j)}$ with a label smaller than $s_{1}^{(j)}$ and, if there are such ones, take the one with largest label amongst them; let us denote this element by $s_{2}^{(j)}$. Then iterate this step with $s_{2}^{(j)}$, i.e., amongst all preimages of $s_{2}^{(j)}$ with a label smaller than $s_{2}^{(j)}$ take the one with largest label, which is denoted by $s_{3}^{(j)}$. After finitely many steps we arrive at an element $s_{k_{j}}^{(j)}$, which does not have preimages with a smaller label. We then define the set $S_{j} := \{s_{1}^{(j)}, \dots, s_{k_{j}}^{(j)}\}$.
Note that in the mapping graph this corresponds to a path $s_{k_{j}}^{(j)} \to \dots \to s_{2}^{(j)} \to s_{1}^{(j)}$ with increasing labels on it.
\item Additionally we store in $n_{j}$ the image of $s_{1}^{(j)}$.
Clearly $s_{1}^{(j)}$ is in $[n]$.
Due to the construction further restrictions hold:
Indeed, if $i < j$, $n_{j}$ cannot be the smallest element in $S_{i}$ larger than $s_{1}^{(j)}$, since otherwise $s_{1}^{(j)}$ would have been chosen during the construction of the set $S_{i}$.
\item
Finally colour all elements of the mapping contained in $S_{j}$.
\end{itemize}
Since the mapping contains exactly $m$ runs and the smallest element in each set $S_{j}$ corresponds to the minimal element of a run the procedure stops after exactly $m$ steps.
It thus defines a pair of a set partition $S=(S_{1}, \dots, S_{m})$ and a sequence $x=(n_{1}, \dots, n_{m})$ with the given restrictions.
If the pair $(S,x)$ is given, the corresponding mapping can easily be reconstructed.
Indeed, the partition $S$ gives us a decomposition of the mapping into ascending runs and the sequence $x$ tells us how these runs have to be linked to each other.
The inverse of this bijection can therefore be defined in a straightforward way.
\end{proof}
\begin{example}
The construction of the partition $S$ and the sequence $x$ for the mapping described in Figure~\ref{Mappings/fig:mapping_graph} can be found in Figure~\ref{Mappings/tab:ex_bij}.
Let us exemplarily explain how the set $S_8$ is constructed.
At this point, the elements in $\cup_{i=1}^7 S_i$, i.e., $19, 18, 17, 13, 16, 9, 15,14, 4$ and $12$, have already been coloured.
Thus, the largest element that has not been coloured so far is $11= s_1^{(8)}$.
For $s_2^{(8)}$, we consider the preimages of $11$ that have a label smaller than $11$. The only such element is $7$ and thus $s_2^{(8)}=7$.
Next, the preimages of $7$ are $6, 12$ and $15$ and thus $s_3^{(8)}=6$.
Since $6$ does not have any preimages, we stop here and $\mathcal{S}_8= \left\lbrace 11,7,6 \right\rbrace$.
Since the image of $11$ is $17$, we set $n_8 =17$.
\demo\end{example}
\subsection{Limiting distribution}
In order to show that the random variable $R_n$ has linear mean and variance and that it is asymptotically normally distributed when suitably standardized, we apply H.-K.\ Hwang's quasi-power theorem:
\begin{theorem}[Hwang's quasi-power theorem \cite{hwang1998convergence}]\label{Prelimin/thm:Hwang_quasi}
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of discrete random variables supported by $\mathbb{N}_0$.
Assume that the moment generating functions $g_n (s)= \mathbb{E}\left( e^{sX_n}\right) = \sum_{k \geq 0} \mathbb{P}(X_n=k)e^{sk}$ are analytic in
a fixed complex neighbourhood of $s = 0$ in which they have an expansion of the form
\[
g_n (s) = \exp\big( {\phi_n U(s) +V(s)}\big) \cdot \left( 1 + \mathcal{O}\left( \frac{1}{\kappa_n}\right) \right) ,
\]
where $\phi_n , \kappa_n \rightarrow \infty$ and $U(s)$, $V(s)$ do not depend of $n$ and are analytic at $s=0$.
Assume finally that
$U(s)$ satisfies the so-called ``variability condition'' $U''(0) \neq 0$.
Under these conditions, the mean and variance of $X_n$ satisfy
\begin{align*}
\mathbb{E}(X_n) & = \phi_n U'(0) + V'(0) + \mathcal{O}(\kappa_n^{-1}), \\
\mathbb{V}(X_n) & = \phi_n U''(0) + V''(0) + \mathcal{O}(\kappa_n^{-1}).
\end{align*}
Furthermore, $X_n$ is after standardization asymptotically normally distributed, with speed of convergence $\mathcal{O}(\kappa_n^{-1}+ \phi_n^{-1/2})$:
\[
\mathbb{P}\left\lbrace \frac{X_n-\mathbb{E}(X_n)}{\sqrt{\mathbb{V}(X_n)}} \leq x\right\rbrace = \Phi(x) + \mathcal{O}\left( \frac{1}{\kappa_n} + \frac{1}{\sqrt{\phi_n}}\right),
\]
where $\Phi(x)$ is the distribution function of the standard normal distribution.
\end{theorem}
We then obtain the following:
\begin{theorem}
Let $R_n$ denote the random variable counting the number of ascending runs in a random $n$-mapping.
Then the expectation and variance satisfy
\begin{align*}
\mathbb{E}(R_n) & =(1-e^{-1})n +\mathcal{O}(1) \text{ and } \\
\mathbb{V}(R_n) & = (e^{-1}-2e^{-2})n +\mathcal{O}(1).
\end{align*}
Furthermore, the standardized random variable
\[
\frac{R_n - \mathbb{E}(R_n)}{\sqrt{\mathbb{V}(R_n)}}
\]
is asymptotically Gaussian distributed with a rate of convergence of order $\mathcal{O}(n^{-1/2})$.
\label{thm:distribution_runs}
\end{theorem}
\begin{remark}
Note that Theorem~\ref{Mappings/thm:connection_trees_mappings} implies that the random variable $R_n$ counting the number of runs in $n$-mappings and the random variable $F_n$ counting the number of runs in trees of size $n$ follow exactly the same distribution.
The theorem above thus holds for $F_{n}$ as well.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{thm:distribution_runs}]
Before we get our hands on the bivariate generating function $R(z,v)$ defined in equation~\eqref{Mappings/eqn:GF_runs_mappings}, we will need to take a closer look at the function $H(z,v)$ from equation~\eqref{Mappings/eqn:runs_func_eqnH} that is involved in the functional equation~\eqref{Mappings/eqn:runs_func_eqnR} for $R(z,v)$.
Once we have a singular expansion of $H(z,v)$, we can obtain an expansion of $R(z,v)$.
Extracting coefficients $[z^n]$ in $R(z,v)$ will then allow us to obtain an asymptotic expansion of the probability generating function $p_n(v)=\sum_{k \geq 0} \mathbb{P}(X_n=k)v^{k}$ and the moment generating function $g_n(s)=p_n(e^s)$.
This will finally allow us to apply the quasi-power theorem and to obtain the statement of this Theorem.
\emph{Asymptotic expansion of $H$}.
The function $H=H(z,v)$ is defined by a functional equation of the form $H=z \varphi(H)$ where the function $\varphi$ is defined as $\varphi(u)=v \cdot e^u + 1- v$.
Singularities of $H$ can be found with the help of \emph{singular inversion} (see Theorem VI.6 in \cite{FlajOdl1990}).
For this, we need the \emph{characteristic equation} of $\varphi$:
\begin{align}
& \varphi(\tau) - \tau \cdot \varphi'(\tau)=0 \notag\\
\Leftrightarrow \, \, & v e^{\tau} +1-v-\tau\cdot v e^{\tau} =0\notag \\
\Leftrightarrow \, \, & \tau = 1 + \frac{1-v}{v e^{\tau}}. \label{Mappings/eqn:runs_tau}
\end{align}
For $v=1$ we have $\tau=1$ and whenever $v$ is close to $1$ a unique solution to equation~\eqref{Mappings/eqn:runs_tau} exists as can be seen by applying the Implicit Function Theorem.
Note that $\tau$ is a function of $v$. This will however be omitted in the notation and in the following, $\tau$ will always denote the unique solution of equation~\eqref{Mappings/eqn:runs_tau}.
The unique dominant singularity of $H$, considered as a function in $z$, is at $z=\rho$, with:
\begin{align}
\rho & = \frac{1}{\varphi'(\tau)}=\frac{1}{v e^{\tau}} \label{Mappings/eqn:runs_rho} \\ & \stackrel{\eqref{Mappings/eqn:runs_tau}}{=} \begin{cases} \frac{1}{e}, &\mbox{if } v=1, \\
\frac{\tau -1}{1-v}, & \mbox{if } v \neq 1. \end{cases}
\label{Mappings/eqn:runs_rho_v_neq1}
\end{align}
Before we continue by giving the asymptotic expansion of $H$, we want to characterize its unique dominant singularity $\rho$ with the help of a functional equation.
From \eqref{Mappings/eqn:runs_tau} and $\rho= \tau/\varphi(\tau)$ it follows that:
\begin{align*}
\tau & = 1 + \rho(1-v) \text{ and} \\
\tau & = \rho \cdot \left( v e^\tau +1 -v\right) = \rho \cdot \left( v e^{1 + \rho(1-v)} +1 -v\right).
\end{align*}
Dividing by $\rho$ then leads to
\[
\frac{1}{\rho}+1-v = v e^{1 + \rho(1-v)} +1 -v,
\]
or, equivalently,
\[
\rho =
\frac{1}{v \cdot e \cdot e^{\rho(1-v)}}.
\]
Thus
\begin{align}
(1-v)\rho e^{\rho(1-v)} =
\frac{1-v}{v \cdot e }
\label{Mappings/eqn:func_eq_rho}
\end{align}
and $\rho$ can be expressed with the help of the Lambert $W$-function for $v \neq 1$ but in a neighbourhood of $1$:
\[
\rho = \frac{1}{1-v}W\left( \frac{1-v}{ev}\right).
\]
Note that the above expression is not defined for $v=1$.
However, taking limits as $v$ tends to $1$, the right-hand side tends to $\exp(-1)$ as expected from equation~\eqref{Mappings/eqn:runs_rho_v_neq1}.
The function $H$ admits the following asymptotic expansion around its unique dominant singularity $\rho$, i.e, for $z \to \rho$:
\begin{align*}
H(z,v) & = \tau - \sqrt{\frac{2 \varphi(\tau)}{\varphi''(\tau)}} \cdot \sqrt{1- \frac{z}{\rho}} + K \cdot \left( 1- \frac{z}{\rho}\right) + \mathcal{O}\left( 1- \frac{z}{\rho}\right)^{3/2}, \\
& = \tau - \sqrt{2 \tau}\sqrt{1- \frac{z}{\rho}} + K \cdot \left( 1- \frac{z}{\rho}\right) + \mathcal{O}\left( 1- \frac{z}{\rho}\right)^{3/2},
\end{align*}
where $K$ is a computable constant that will not be needed explicitly in the following.
\emph{Asymptotic expansion of $R$.}
We can now use this result in order to describe the analytic behaviour of the function $R(z,v)$.
Indeed, $R$ inherits a unique dominant singularity at $z=\rho$ from $H$.
This can be seen as follows:
Let us first recall that $(R(z,v))^{-1}=1- zv\cdot \exp(H(z,v))$.
Thus, $R$ has a singularity whenever $zv\cdot \exp(H)=1$ or equivalently, whenever $H=-\ln(zv)$.
Using the functional equation for $H$ given in equation~\eqref{Mappings/eqn:runs_func_eqnH}, this is equivalent to:
\begin{align*}
& z = \frac{-\ln(zv)}{v\cdot \exp(-\ln(zv))+1 -v}
\Leftrightarrow 1 = \frac{-\ln(zv)}{1+z - zv}
\Leftrightarrow z \exp(z(1-v)) = \frac{1}{e v}, \\
\end{align*}
which is exactly the same as equation~\eqref{Mappings/eqn:func_eq_rho} for $\rho$.
Thus $z=\rho$ is the unique dominant singularity of $R$.
In order to provide the asymptotic expansion of $R$ for $z \to \rho$, let us write $z$ as $(z - \rho) + \rho$ and $H$ as $(H - \tau) +\tau$:
\begin{align*}
\frac{1}{R} & = 1 - v \rho e^{\tau} \cdot e^{H-\tau} + v \rho \left( 1 - \frac{z}{\rho}\right)e^{\tau} \cdot e^{H-\tau} = 1 - e^{H-\tau} + \left( 1 - \frac{z}{\rho}\right) \cdot e^{H-\tau},
\end{align*}
since $v \rho e^{\tau}=1$ by definition of $\rho$ (see equation~\eqref{Mappings/eqn:runs_rho}).
Using the power series expansion of the exponential function and the asymptotic expansion for $H(z,v)$ we obtain the following (for $z \to \rho$):
\begin{align*}
e^{H-\tau} & = 1 + (H- \tau) + \mathcal{O}((H - \tau)^2) = 1 - \sqrt{2 \tau} \sqrt{1-\frac{z}{\rho}} + \mathcal{O}\Big(1-\frac{z}{\rho}\Big)
\end{align*}
and thus
\begin{align*}
\frac{1}{R} & = \sqrt{2 \tau} \sqrt{1-\frac{z}{\rho}} + \mathcal{O}\Big(1-\frac{z}{\rho}\Big).
\end{align*}
Finally, with $\tau = 1 +\rho(1-v)$, we obtain
\begin{align}
R(z,v) & = \frac{1}{\sqrt{2 \tau}\sqrt{1-\frac{z}{\rho}}+ \mathcal{O}\left( 1-\frac{z}{\rho}\right)} \notag \\
& = \frac{1}{\sqrt{2}\sqrt{1+\rho(1-v)}\sqrt{1-\frac{z}{\rho}}} \cdot \left( 1 + \mathcal{O}\left( \sqrt{1-\frac{z}{\rho}}\right) \right).
\label{Mappings/eqn:runs_sing_expansionR}
\end{align}
\emph{Asymptotic expansion of $p_n(v)$.}
By a standard application of singularity analysis (see \cite{flajolet2009analytic} for an introduction) we obtain asymptotics of the coefficients of $R(z,v)$:
\begin{align*}
[z^n]R(z,v) & = \frac{1}{\rho^n}[z^n]R(z \rho, v) \\
& = \frac{1}{\rho^n} \cdot \left[ \frac{1}{\sqrt{2}\sqrt{1+\rho(1-v)}} \cdot \binom{n +\frac{1}{2}-1}{n} + \mathcal{O}\left(\frac{1}{n} \right)\right] \\
& = \frac{1}{\rho^n} \cdot \frac{1}{\sqrt{2}\sqrt{1+\rho(1-v)}} \cdot \frac{1}{\sqrt{\pi n}} \cdot \left( 1 + \mathcal{O}\left( \frac{1}{\sqrt{n}} \right) \right).
\end{align*}
Since $p_n(v)=n!/n^n \cdot [z^n]R(z,v)$, we obtain the following expansion of the probability generating function by applying Stirling's formula:
\begin{align}
p_n(v) & = \frac{n!}{n^n}\frac{1}{\rho^n} \cdot \frac{1}{\sqrt{2}\sqrt{1+\rho(1-v)}} \cdot \frac{1}{\sqrt{\pi n}} \cdot \left( 1 + \mathcal{O}\left( \frac{1}{\sqrt{n}} \right) \right) \notag \\
& = \frac{n^n\sqrt{2 \pi n}}{e^nn^n} \cdot \left( 1 + \mathcal{O}\left( \frac{1}{n} \right) \right)\frac{1}{\rho^n} \notag \\
& \quad \cdot \frac{1}{\sqrt{2}\sqrt{1+\rho(1-v)}} \cdot \frac{1}{\sqrt{\pi n}} \cdot \left( 1 + \mathcal{O}\left( \frac{1}{\sqrt{n}} \right) \right) \notag \\
& = \frac{1}{(e \cdot \rho)^n \cdot \sqrt{1+\rho(1-v)}} \cdot \left( 1 + \mathcal{O}\left( \frac{1}{\sqrt{n}} \right) \right)
\label{Mappings/eqn:runs_sing_expansionPn(v)}
\end{align}
\emph{Applying the quasi-power theorem.}
Expansion~\eqref{Mappings/eqn:runs_sing_expansionPn(v)} now immediately gives an asymptotic expansion for the moment generating function:
\begin{align*}
g_n(s) & = \frac{1}{(e \cdot \rho)^n \cdot \sqrt{1+\rho(1-e^s)}} \cdot \left( 1 + \mathcal{O}\left( \frac{1}{\sqrt{n}} \right) \right) \\
& = \exp\left( - n \cdot \ln(e \cdot \rho) - \ln \left(\sqrt{1+\rho(1-e^s)}\right)\right) \cdot \left( 1 + \mathcal{O}\left( \frac{1}{\sqrt{n}} \right) \right).
\end{align*}
This is precisely the situation where the quasi-power theorem, Theorem~\ref{Prelimin/thm:Hwang_quasi}, can be applied.
The involved functions are here defined as follows:
\[
\begin{array}{ll}
\phi_n = n, & U(s)= - \ln(e \cdot \rho), \\
\kappa_n = \sqrt{n} \quad \quad \text{ and } & V(s)= - \ln \left(\sqrt{1+\rho(1-e^s)}\right). \\
\end{array}
\]
Note once again that $\rho$ is not a constant but depends on $e^s$.
We only need to check the variability condition $U''(0) \neq 0 $; the other conditions are clearly fulfilled.
For this purpose, let us use equation~\eqref{Mappings/eqn:runs_rho} and write $U(s)$ as follows:
\[
U(s) = - \ln \left( \frac{e}{e^s e^{\tau}}\right) = -1 + s + \tau.
\]
We thus obtain that the first and second derivative of $U(s)$ are given as follows:
\begin{align*}
U'(s) & = 1+ \tau'(e^s)\cdot e^s, \\
U''(s) & = e^s \cdot \left( \tau''(e^s) + \tau'(e^s)\right),
\end{align*}
where $\tau$ is considered as a function in $v=e^s$.
In order to determine $\tau'$ and $\tau''$, we will make use of implicit differentiation and the functional equation for $\tau$ given in \eqref{Mappings/eqn:runs_tau}:
\[
\tau' = - \frac{1}{v^2 e^{\tau}} - \frac{1-v}{v e^{\tau}} \cdot \tau', \text{ implying } \tau' = \frac{1}{v \cdot ( v-1 -ve^{\tau})}.
\]
Furthermore,
\begin{align*}
\tau'' & = - \frac{(v-1 -ve^{\tau})+v \cdot (1- e^{\tau}-v e^{\tau} \cdot \tau')}{v^2 (v-1 -ve^{\tau})^2}\\
& = \frac{e^{\tau}}{v (v-1 -ve^{\tau})^3} + \frac{2ve^{\tau}+1-2v}{v^2 (v-1 -ve^{\tau})^2}.
\end{align*}
Recalling that $\tau=1$ for $e^s=v=1$, we obtain:
\begin{align*}
U'(0) & = 1+ \tau'(1) = 1 + \frac{1}{-e^{\tau(1)}} = 1-e^{-1} \approx 0.632\ldots , \\
U''(0) & = \tau''(1)+ \tau'(1)= \frac{e}{-e^3} + \frac{2e-1}{e^2}-e^{-1} = -2e^{-2}+ e^{-1} \approx 0.0972\ldots.
\end{align*}
In a similar way, we could also determine $V'(0)$ and $V''(0)$.
Since the calculations are rather tedious and the additive constants in $\mathbb{E}(R_n)$ and $\mathbb{V}(R_n)$ are not of particular interest, we omit this here.
It suffices to note that $V'(0)$ and $V''(0)$ are both real constants.
This finally gives us the desired result that
\[
\mathbb{E}(R_n) =(1-e^{-1})n +\mathcal{O}(1) \text{ and } \mathbb{V}(R_n) = (e^{-1}-2e^{-2})n +\mathcal{O}(1),
\]
and that the standardized random variable $(R_n - \mathbb{E}(R_n))/\sqrt{\mathbb{V}(R_n)}$ converges to a standard normal distribution.
Moreover, since $\kappa_n^{-1}=n^{-1/2}=\phi_n^{-1/2}$, the speed of convergence is of order $\mathcal{O}(n^{-1/2})$.
\end{proof}
\section{Summary of the results and future work}
Let us briefly discuss the results of this paper.
For the number of ascending runs in Cayley trees and mappings, we obtained exact enumeration formul{\ae} and a connection to the Stirling numbers of the second kind that we could also explain with the help of bijective proofs.
We also obtained limiting distribution results and showed convergence to a normal distribution.
Let us compare this to the known results for the Eulerian numbers that enumerate permutations by their length and their number of ascents.
The random variable $X_n$ counting the number of ascents in a random permutation of length $n$ satisfies a central limit theorem~\cite{Carlitz1972eulerian}.
Indeed, with $\mathbb{E}(X_n)=\frac{n-1}{2}$ and $\mathbb{V}(X_n)=\frac{n+1}{12}$ it holds that the standardized random variable $(X_n - \mathbb{E}(X_n))/\sqrt{\mathbb{V}(X_n)}$ converges in distribution to a standard normal distribution.
Equivalently, the random variable $Y_n=X_n+1$ counting the number of ascending runs in a random permutation of length $n$ satisfies a central limit theorem with $\mathbb{E}(Y_n)=\frac{n+1}{2}$ and $\mathbb{V}(Y_n)=\mathbb{V}(X_n)$.
Thus, both for ascents and for ascending runs, it holds that $\mathbb{E} \sim n/2$ and $\mathbb{V}\sim n/12=0.08 \dot{3} \cdot n$.
Here, we could show similar results for ascending runs in trees and mappings.
Indeed, we also obtained a linear mean and variance and convergence in distribution to a standard normal distribution of the standardized random variable.
However, the involved coefficients are not the same:
\begin{align*}
\mathbb{E}(R_n) & \sim (1-e^{-1}) n \text{ with } 1-e^{-1} \approx 0.632 \ldots \text{ and } \\
\mathbb{V}(R_n)& \sim (e^{-1}-2e^{-2}) n \text{ with } e^{-1}-2e^{-2} \approx 0.0972 \ldots .
\end{align*}
The combinatorial decomposition of Cayley trees and mappings employed in this paper can also be used to study other problems in connection with label patterns. First, we remark that with the generating functions approach it is indeed possible to carry out a joint study of ascending and descending runs, although (as might be expected) the occurring functions are more involved.
Furthermore, our method can also be applied to count the number of local minima or maxima, thus generalizing the notion of valleys and peaks in permutations.
However, the results contained in Okoth's thesis~\cite{Okoth2015} are more general and thus the study of local minima and maxima was not included here.
Moreover, preliminary studies show that our method can also be applied to what we call \emph{up-ups}.
For an $n$-mapping $f$, these are elements $i$ in $[n]$ for which it holds that $i < f(i) < f^2(i)$.
The notion of up-ups thus generalizes consecutive $123$-patterns in permutations.
Permutations avoiding this pattern have been studied, e.g., in \cite{elizalde2003consecutive}.
Finally, we remark that with the same techniques one could also attack corresponding label pattern problems for other combinatorial tree families, such as labelled ordered trees and labelled binary trees.
All label patterns mentioned so far in this paper were consecutive patterns, i.e., they concern the images or preimages of nodes in a mapping.
A more involved task would be to study non-conse\-cutive label patterns in mappings.
For instance, the simplest non-trivial non-consecutive pattern $21$ corresponds to an inversion in the mapping.
The number of inversions in trees has been studied in several papers, see, e.g., \cite{GesSagYeh1995,PanSei2012}, and it seems that limiting distribution results carry over from Cayley trees to mappings.
Of course, it would be interesting to obtain results also for longer non-consecutive patterns in mappings or Cayley trees.
\bibliographystyle{abbrv}
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{
"redpajama_set_name": "RedPajamaArXiv"
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President Obama: Orlando Families' Grief Is 'Beyond Description'
By Maya Rhodan
June 16, 2016 5:19 PM EDT
Clad in dark suits with their brows furrowed, President Barack Obama and Vice President Joe Biden laid 49 roses — one for each of the Pulse nightclub shooting victims — at a makeshift memorial on Thursday afternoon at the Phillips Center, adjacent to Orlando's City Hall.
President Obama has been here before, addressing the families of victims and survivors in the aftermath of a deadly mass shooting. But here he was again, traveling to Orlando almost a week after 49 were shot dead and 53 others were wounded at the Pulse nightclub. The president's trip to Orlando was his 10th visit to the scene of a mass shooting.
During the president's remarks, after meeting with families of victims and survivors, the poignancy of the visit rung true. He described the families' grief as "beyond description." Obama again called the shooting an act of "terror" and an act of "hate." The shooting, he said, was an attack on the LGBT community.
"Today, once again, as has been true too many times before, I held and hugged grieving family members and friends and they asked why does this keep happening. They pleaded that we do more to stop the carnage," Obama said Thursday. "They don't care about the politics. Neither do I. Niether does [Vice President] Joe [Biden]."
Obama added, "the notion that the answer to this tragedy would be to make sure that more people in a nightclub are similarly armed to the killer, defies common sense," recalling some arguments by pro-gun activists.
President Obama said though the city was "shaken by an evil, hateful act." "The worst of humanity reared it's evil but the best of humanity came roaring back," he said.
President Obama traveled to Orlando with the Vice President, Congresswoman Corrine Brown and Sens. Bill Nelson and Marco Rubio of Florida. Rubio and Brown traveled via Air Force One with the President, while Sen. Nelson flew with the Vice President. Before meeting with families at Orlando's Amway Center, President Obama had an opportunity to thank Orlando Mayor Buddy Dyer and first responders for their response to the shooting. During a press conference on Monday, authorities said the actions of officials who took down the shooter "saved many, many lives." The president also met with staff members from Pulse nightclub, which lost two staff members during the attack.
Though the president visited Orlando to meet with families, he waded into what has become a contentious debate about who and what is to blame for the attack. He called on all levels of government to do more to prevent terrorists from attacking Americans, saying his administration would continue working to destroy ISIS. Shortly before the president spoke, the Arizona Sen. John McCain blamed the administration's policies for the rise of ISIS and, in turn, the attack itself. Earlier this week, likely Republican presidential nominee Donald Trump said the president "claims to know our enemy, and yet he continues to prioritize our enemy over our allies, and for that matter, the American people."
Obama blamed U.S. politics for Omar Mateen's ability, despite having been interviewed by the FBI on three separate occasions due to suspected ties to terrorism, to purchase a firearm. He said though the motives of shooters in Aurora, Newtown, and San Bernardino, and now Orlando may have differed, "the instruments of death were so similar." Now, he said, another 49 are dead and some 53 will have scars that will last a lifetime.
"Unfortunately, our politics have conspired to make it as easy as possible for a terrorist or just a disturbed indivual like those in Aurora or Newtown to buy extraordinary powerful weapons and they can do so legally," Obama said. "This debate needs to change."
He said the families he met with in Orlando "don't care about politics" and urged the Senate, where a 15 hour filibuster urging gun action wrapped early Thursday, to "do the right thing" and "save some lives."
"Those who defend the easy accessibility of assault weapons should meet these families," Obama said.
GOP Agrees to Gun Control Vote
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 2,754
|
\section{Introduction}\label{sec:introduction}
An insulating bulk energy gap along with gapless edge states is a hallmark of a two-dimensional (2D) topological insulator (TI).\cite{Kane-PRL-2005a,Kane-PRL-2005b,Qi-RMP-2011,Ando-JPSJ-2013} At each boundary, two counterpropagating edge states with opposite spin-polarization and wave numbers form Kramers pairs, i.e.~two distinct degenerate states connected by time-reversal symmetry. These states are denoted helical edge states due to their connection between spin and propagation direction. Due to time-reversal symmetry, elastic backscattering of a single electron from a helical edge state (HES) to its Kramers partner is not possible by any time-reversal invariant potential, e.g. disorder.\cite{Xu-PRB-2006} Thereby an important mechanism to hinder ballistic transport is absent and quantized conductance of $2e^2/h$ per pair of HESs is within reach. The first experimentally realized 2D TI in a HgTe quantum well (QW) indeed found quantized conductance in micrometer-sized samples,\cite{Konig-Science-2007,Roth-Science-2009,Brune-nature-phys-2012,Konig-PRX-2013} along with evidence of edge state transport in both two-terminal \cite{Konig-Science-2007} and multiterminal \cite{Roth-Science-2009} configurations. Prior to the experiments, HgTe QWs were in fact predicted to be 2D TIs beyond a certain QW thickness.\cite{Bernevig-Science-2006} These efforts also resulted in a rather generic Dirac-like model describing the essential physics of some 2D TIs, which is now know as the Bernevig-Hughes-Zhang (BHZ) model. Recently, also InAs/GaSb double QWs were suggested theoretically to be 2D TIs described using the BHZ model,\cite{Liu-PRL-2008} which afterwards have been tested experimentally.\cite{Knez-PRL-2011,Suzuki-PRB-2013,Knez-PRL-2014,Du-PRL-2015,Nichele-et-al-2015}
\begin{figure}
\includegraphics[width=0.96\linewidth]{fig1a.png}
\caption{\label{fig:GHES_sketch} Illustration of the dispersion relation and spin orientation for (a) helical edge states with constant spin orientation and (b) generic helical edge states with energy-dependent spin orientation. In this paper, we analyse the generic helical edge states and their spin orientation variation due to the Rashba spin-orbit coupling within the BHZ model.}
\end{figure}
Deviations from the quantized conductance have also been found experimentally for longer edges in both HgTe\cite{Konig-Science-2007,Roth-Science-2009,Gusev-PRB-2011,Grabecki-PRB-2013,Konig-PRX-2013,Gusev-et-al-PRB-2014} and InAs/GaSb\cite{Spanton-PRL-2014} QW TIs. Conduction reduction due to \emph{inelastic} backscattering has been studied theoretically,\cite{Schmidt-PRL-2012,Strom-PRL-2010,Budich-PRL-2012,Crepin-PRB-2012,Lezmy-PRB-2012,Geissler-PRB-2014,Kainaris-PRB-2014,Vayrynen-PRL-2013} since it is not a priorly ruled out by time-reversal invariance. Most studies of inelastic backscattering combine some energy-exchange mechanism (e.g. phonons \cite{Crepin-PRB-2012} or electron-electron interactions \cite{Schmidt-PRL-2012,Strom-PRL-2010,Crepin-PRB-2012,Kainaris-PRB-2014}) with a way to manipulate the spin (often some form of spin-orbit coupling \cite{Crepin-PRB-2012,Geissler-PRB-2014,Kainaris-PRB-2014}). Scattering of localized spins \cite{Tanaka-PRL-2011,Lunde-PRB-2012,Eriksson-PRB-2012,Eriksson-PRB-2013,Probst-arxiv-2014} such as magnetic impurities or nuclear spins \cite{Lunde-PRB-2013} has also been analyzed.
In a particularly interesting proposal for inelastic backscattering, Schmidt et al. \cite{Schmidt-PRL-2012} considered HESs without axial spin symmetry. The Rashba spin-orbit coupling (RSOC)\cite{Rothe-NJP-2010,Virtanen-PRB-2012,Rothe-PRB-2014} and bulk inversion asymmetry (BIA)\cite{Konig-JPSJ-2008,Liu-PRL-2008,Ostrovsky-PRB-2012} can break the axial spin symmetry of the HESs. In this case, a pair of HESs acquire a more generic and intriguing spin-structure than merely having opposite and constant spin-orientations independently of energy. Time-reversal symmetry still dictates that the two counterpropagating Kramers partners have orthogonal spinors, but it does not require equal spinors at different energies as illustrated in Fig. 1. These states were named \emph{generic} helical edge states (GHESs).\cite{Schmidt-PRL-2012} Recently, Kainaris et al. \cite{Kainaris-PRB-2014} extended the original work\cite{Schmidt-PRL-2012} on transport in short GHESs with electronic interaction and disorder to longer ones. Furthermore, the spin-structure of the GHESs was also shown to change the noninteracting transport properties of a point contact and of disordered 2D TI strips. \cite{Orth-PRB-2013} Moreover, the spin-structure of GHESs plays a role in the umklapp-scattering-induced energy gap suggested to host parafermions, a generalisation of Majorana Fermions.\cite{Orth-PRB-2015}
These studies show that it is worthwhile to analyse the GHESs and their microscopic origin further, which is the purpose of this paper. Very recently, Rod \emph{et al.}\cite{Rod-PRB-2015} studied the spin texture of GHESs due to BIA within the BHZ model and also numerically for the Kane-Mele model.\cite{Kane-PRL-2005a,Kane-PRL-2005b} In contrast, we consider how RSOC\cite{Rothe-NJP-2010} can produce GHESs within the BHZ model. We develop analytical models for the GHESs appearing at an isolated boundary and in the case of a finite width ribbon, where the overlap of the edge states on different boundaries plays an important role. For an isolated edge, we are able to give an analytical formula for the so-called spin-structure parameter, which describes how much the spin-orientation of the GHESs change. This parameter was originally introduced phenomenologically.\cite{Schmidt-PRL-2012} Using realistic numbers for a HgTe QW, we find that an isolated edge is in fact rather robust against spin rotation produced by the RSOC. In contrast, we discover that the combination of RSOC and finite width enhanced significantly the spin rotation versus energy of the GHESs. Throughout the paper, spin rotation refers to the spin orientation variations of the GHES. Furthermore, we show that our analytical models compare well to full numerical tight-binding calculations.
We organise the paper as follows: First, we outline the phenomenology of the GHESs (Sec.~\ref{sec:helical_edge}) and the BHZ model including the RSOC that breaks the axial spin symmetry (Sec.~\ref{sec:BHZ}). Then, we consider an isolated pair of GHESs at a single boundary in Sec.~\ref{subsec:Semi} and finally analyze the case of a finite width ribbon both analytically and numerically (Sec.~\ref{subsec:twoedges}). Sec.~\ref{conclusions} summarizes the paper and the Appendices give various technical details.
\section{Phenomenology of\\ the generic helical edge states}\label{sec:helical_edge}
In this section, we discuss the GHESs phenomenologically. GHESs can be modelled as two counterpropagating one-dimensional (1D) states with linear dispersion relations $\e_{k,\pm}=\pm \hbar v k$, i.e.
\begin{equation}
H_0=\sum_{k,\eta=\pm1} \eta \hbar v k c^{\dagger}_{k \eta} c^{\phantom{\dagger}}_{k \eta}
\end{equation}
as in Refs.~\onlinecite{Schmidt-PRL-2012,Orth-PRB-2013,Kainaris-PRB-2014,Orth-PRB-2015}. Here $c^{\dagger}_{k \eta}$ ($c^{\phantom{\dagger}}_{k \eta}$) creates (annihilates) a state $\ket{k,\eta}$ with momentum $k$ and propagating direction $\eta$. The states of opposite $k$ and $\eta$ are Kramers partners such that elastic scattering due to e.g. impurities is still absent. The spin $s_j$ ($j=x,y,z$) expectation value of the Kramers partners are also opposite, i.e.
\begin{equation}
\bra{k,+} s_j \ket{ k,+} = - \bra{-k,-} s_j \ket{ -k,-},\ \ j=x,y,z.
\end{equation}
The counterpropagating states at each $k$ can be related to the spin states $\sigma=\uparrow,\downarrow$ along a definite direction by a momentum dependent (and thereby energy dependent) SU(2) matrix $B_k$ as\cite{Schmidt-PRL-2012}
\begin{equation}\label{eq:Bk-full}
\left(\begin{array}{c} c_{k\uparrow} \\
c_{k\downarrow} \end{array}\right) = B_k \left(\begin{array}{c} c_{k +} \\ c_{k-} \end{array} \right).
\end{equation}
Time-reversal symmetry and $B_k \in$ SU(2) lead to $B_k=B_{-k}$. Consistent with these facts, Schmidt {\em et al.} \cite{Schmidt-PRL-2012} introduced the following expansion for small $|k| \ll k_0$:
\begin{equation}
\label{eq:bk}
B_k=\left(
\begin{array}{cc}
1-k^4/(2k_0^4) & -k^2/k_0^2 \\
k^2/k_0^2 & 1-k^4/(2k_0^4)
\end{array} \right),
\end{equation}
where the spin-quantization axis is chosen such that at the band crossing point $k=0$, we have $c_{k=0,+}=c_{k=0\uparrow}$ and $c_{k=0,-}=c_{k=0\downarrow}$ as in Fig.~\ref{fig:GHES_sketch}(b). In other words, a constant rotation of all the spins regardlessly of $k$ has been removed from $B_k$ in Eq.(\ref{eq:bk}) following Ref.~\onlinecite{Schmidt-PRL-2012}. Such a $k$-independent rotation corresponds to a constant rotation matrix and can be removed by choosing a rotated basis for the spin. Importantly, a phenomenological spin-structure parameter $k_0$ has been introduced in the expansion (\ref{eq:bk}), which measures the velocity of spin rotation in momentum space. Schmidt {\em et al.}\cite{Schmidt-PRL-2012} showed using perturbation theory that the correction to the quantized conductance due to backscattering processes possible within a pair of GHESs scales as temperature to the forth power with a prefactor depending on $k_0$. In this paper, we find $k_0$ analytically within the BHZ model including the RSOC for an isolated edge.
To gain more insights into the spin structure of the GHESs, we also evaluate the total spin rotation of the edge states, which we define as
\begin{equation}
T_s=\int
dk \left(\left|\left< k_1,\uparrow | k_1, + \right> \right|^2-\left|\left<k, \uparrow | k, + \right> \right|^2 \right).
\label{eq:ts}
\end{equation}
Here $k_1$ is a fixed reference momentum and the integration is over the range of $k$-space, where the edge states exist. The idea behind $T_s$ is to quantify the total variation of the spin orientation of the edge state $\ket{k,+}$ over all relevant $k$. We have constructed $T_s$ such that if $\ket{k,+}$ is a HES (i.e.~$\ket{k,+}=\ket{k,\uparrow}$), then $T_s=0$. Likewise, if the spin of $\ket{k,+}$ is rotated by the same amount for all $k$, then we still get $T_s=0$. This is due to the reference term $|\langle k_1,\uparrow | k_1, + \rangle|^2$ with an arbitrary, but fixed, momentum $k_1$. In our calculations, we choose $k_1$ to be the momentum where the edge state dispersion cross the upper bulk band gap edge. In the cases we have analyzed, the reference term $|\langle k_1,\uparrow\! | k_1, + \rangle|^2$ is very close to one and quite unaffected by small changes in $k_1$. However, generally the choice of $k_1$ does affect the numerical value of $T_s$, but not its variation versus some physical parameter. The behaviour of the spin rotation is more complex for a ribbon than for a single edge, especially for narrower ribbons, as the edge state wave function can have components on both edges. The quantity $T_s$ is useful in that case as this kind of behaviour is difficult to capture with the parameter $k_0$, which quantifies the rotation close to $k=0$. The unit of both $T_s$ and $k_0$ is inverse length.
Before proceeding, we consider a simple 1D model Hamiltonian for a pair of HESs with a generic linear spin-orbit coupling, i.e.
\begin{equation} \label{eq:simplest-case}
H=\hbar v k\sigma_z+(a_x\sigma_x+a_y\sigma_y)k.
\end{equation}
Here $\sigma_i, i=x,y,z$, are the Pauli matrices and $a_x,a_y$ are the spin-orbit coupling strengths. By diagonalization, we see that this often used \cite{Eriksson-PRB-2012,Eriksson-PRB-2013} simple model does not introduce a $k$-dependent $B_k$, since all matrix elements are linear in momentum $k$. Thereby, it does not give rise to energy-dependent spin-orientation and to GHESs, i.e.~$T_s=0$. This is consistent with the lack of the lowest order inelastic backscattering due to a linear spin-orbit coupling combined with a phonon exchange.\cite{Budich-PRL-2012} In order to get non-trivial GHESs, we resort to calculations for the realistic BHZ model with RSOC.
\section{The BHZ model with Rashba spin-orbit coupling}
\label{sec:BHZ}
The BHZ model is an effective four band model describing the basic physics of a 2D TI.\cite{Bernevig-Science-2006} It was derived using $\mathbf{k \cdot p}$ theory for the band structure of a HgTe QW and therefore valid for small wavevectors $\mathbf{k}=(k_x,k_y)$, i.e.~close to the $\Gamma$ point. It accounts correctly for the physics of HgTe QWs close to the critical well thickness, which marks the transition between a normal semiconductor band structure and an inverted band structure with topologically protected edge states.\cite{Bernevig-Science-2006} The BHZ Hamiltonian consists of two disconnected blocks connected by time reversal symmetry. Each block has the form of a massive Dirac model in 2D in addition to quadratic terms crucial for the band inversion and thereby the topological properties of the material. In fact, the Dirac-like nature makes the BHZ model rather generic for 2D TIs --- even though it grew out of a specific material choice. The BHZ basis states consist of two Kramer pairs of electron-like, $\ket{E\pm}$, and hole-like, $\ket{H\pm}$, states, respectively. The states labeled with $+$ ($-$) are often referred to as the spin-up (spin-down), since they have positive (negative) total angular momentum projection.\cite{Lunde-PRB-2013} In this sense, the time-reversed blocks of the BHZ model have opposite spin. In the basis $\{\ket{E+},\ket{H+},\ket{E-},\ket{H-}\}$, the BHZ Hamiltonian is
\begin{equation}\label{eq:BHZ-H}
H_0=
\left(\begin{array}{cccc}
\e_k+M_k & Ak_+ & 0 & 0 \\
Ak_- & \e_k-M_k & 0 & 0 \\
0 & 0 & \e_k+M_k & -Ak_- \\
0 & 0 & -Ak_+ & \e_k-M_k
\end{array}\right),
\end{equation}
where $k_\pm=k_x\pm ik_y$, $\e_k=-Dk^2$, $M_k=M_0-Bk^2$ and $k^2=k_x^2+k_y^2$. The sign of $M_0/B$ determines the existence of the HESs\cite{Zhou-PRL-2008} and $D\neq0$ induces particle-hole asymmetry in $H_0$. Table \ref{table:parameters} gives the parameters for two different systems modelled by the BHZ model, namely HgTe QWs\cite{Bernevig-Science-2006} and InAs/GaSb double QWs.\cite{Liu-PRL-2008}
In this paper, we utilize an extension of the BHZ model derived in Ref.~\onlinecite{Rothe-NJP-2010} for the inclusion of structural inversion asymmetry (SIA) terms including the RSOC. Importantly, the RSOC couples the two blocks of $H_0$ such that the axial spin symmetry is broken. Here, we include only the most important RSOC linear in momentum, i.e.
\begin{equation}\label{Rashba_hamil}
H_R =
\left(\begin{array}{cccc}
0 & 0 & -iR_0k_- & 0 \\
0 & 0 & 0 & 0 \\
iR_0k_+ & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right),
\end{equation}
and therefore our full Hamiltonian is $H=H_0 + H_R$. Interestingly, the Rashba term in $H_R$ only couples the electron-like bands, which makes our model more complex than the simple $2\times2$ model Hamiltonian in Eq.(\ref{eq:simplest-case}). Moreover, GHES are now possible as we shall see below. The strength of the RSOC, $R_0$, depends of the amount of SIA, which is often related to an internal or external electric field. For a HgTe QW one can control the RSOC with an external field,\cite{Rothe-NJP-2010} whereas it is an internal field for InAs/GaSb double QWs.\cite{Liu-PRL-2008} Rothe \emph{et al.}\cite{Rothe-NJP-2010} also derives higher order RSOC terms in momentum as we briefly discuss in Appendix \ref{app:higher-order-Rashba}.
The HgTe has a zincblende crystal structure such that inversion symmetry of the crystal is lacking. Therefore, bulk-inversion-asymmetric terms can in principle be included,\cite{Konig-JPSJ-2008} but are often disregarded due to their small size.\cite{Konig-JPSJ-2008,Qi-RMP-2011} However, in InAs/GaSb double QWs BIA terms are in fact significant,\cite{Liu-PRL-2008} so our calculations for this system without BIA terms are not an attempt to model the real system in detail.
\begin {table}[H]
\begin{center}
\begin{tabular}{| l | l | l |}
\hline
Material & HgTe & InAs/GaSb \\ \hline
$A$ (meV nm) & 365.0 & 37.0 \\ \hline
$B$ (meV nm$^2$) & -686.0 & -66.0 \\ \hline
$D$ (meV nm$^2$) & -512.0 & -58.0 \\ \hline
$M_0$ (meV) & -10.0 & -7.8 \\ \hline
$R_0$ (meV nm) & 15.6 $\epsilon_z$ & -7.0 \\ \hline
\end{tabular}
\end{center}
\caption{The BHZ model parameters for two QW systems in the topological regime.\cite{ZHANG-BOOK-2013} The parameters for the HgTe QW correspond to a well width of 7 nm, while the values for the InAs/GaSb double QWs are for equal widths of 10nm for both wells. The RSOC constant $R_0$ includes a value for the external electric field $\epsilon_z$ (in meV) in the case of HgTe.}
\label{table:parameters}
\end{table}
Before we proceed to the GHESs, we briefly comment on the bulk bands including the RSOC. By diagonalizing $H=H_0 + H_R$, we find
\begin{subequations}\label{eq:bulk-bands}
\begin{align}
E^{n}_{1,2}&=
-D k^2
\pm\frac{R_0 k}{2}-\frac{\sqrt{J_k+K^{\pm}_k}}{2},
\\
E^{p}_{3,4}&=
-D k^2
\pm\frac{R_0 k}{2}+\frac{\sqrt{J_k+K^{\pm}_k}}{2},
\end{align}
\end{subequations}
where $p$ ($n$) is the bulk band with positive (negative) energy and we define $J_k=4 A^2 k^2+4B^2 k^4+R_0^2 k^2+4M_0^2$ and $K^{\pm}_k=-4 B k^2 (2 M_0 \pm R_0k)\pm4 M_0 R_0 k$ and $k=\sqrt{k_x^2+k_y^2}>0$. Fig.~\ref{fig:dispersions} shows the bulk energy bands with RSOC (together with the edge state dispersions that we consider below). The bulk bands shift due to the RSOC such that the band gap for HgTe becomes indirect. Moreover, the size of the bulk band gap is changed slightly, but not enough to change the topology of the system.
\begin{figure}
\includegraphics[width=\linewidth]{fig2b2.png}
\includegraphics[width=\linewidth]{Fig2d2.png}
\caption{\label{fig:dispersions} Bulk and edge state dispersion with the RSOC for a single edge (top panel) and a ribbon (bottom panel) in a HgTe QW. The parameters are given in Table \ref{table:parameters} and the electric field $\epsilon_z$ is such that $R_0=A$. The bulk gap is marked by dashed horizontal lines and the bulk bands Eq.(\ref{eq:bulk-bands}) are for $H_0+H_R$ with periodic boundary conditions in both directions. Analytical results are not shown in the entire bulk energy gap, because our method requires the existence of both edge states at the same time. The insets show the edge state dispersions close to $k=0$.}
\end{figure}
\section{Generic helical edge states}\label{sec:GHES}
The RSOC breaks the spin degeneracy of the BHZ model in such a way that GHESs with energy-dependent spin orientation now becomes feasible. We treat the GHESs below in two cases: (i) A single isolated edge and (ii) a finite width ribbon with two parallel edges. The isolated edge case offers more analytical insights and we are able to extract the spin-structure parameter $k_0$ defined in Eq.(\ref{eq:bk}).
Without the RSOC, it is possible to obtain the HESs analytically for both an isolated edge and a ribbon with two edges.\cite{Zhou-PRL-2008} In the appendix \ref{appendix:HES}, we give the details of the analytical wave functions and dispersion relations of the HESs in both cases. Including the RSOC, it becomes much more challenging to obtain exact analytical forms by the same method, since it is now a problem with four coupled differential equations, see Appendix \ref{subsec:HES-derivation}. Nevertheless, we are able to obtain analytical results by assuming that the GHESs with RSOC are combinations of the HESs without RSOC, neglecting the possible contribution of the bulk bands. This is a good approximation, since the edge states naturally have a small spatial overlap with the bulk states as long as the edge states are well-localized at the boundary. This is well satisfied especially for momenta close to zero and well into the bulk gap. We find that the bulk gap is reduced as $R_0$ increases and so does the range of applicability of the analytical results. Moreover, we also compare our analytical results with the solution via exact diagonalization of a tight-binding regularization of the BHZ Hamiltonian for a ribbon of width $W$ with periodic boundary conditions in the $x$ direction and edges at $y=-W/2$ and $y=W/2$. The details of the tight-binding formulation is discussed in Appendix \ref{app:tight-binding} and follows Ref.~\onlinecite{book_Shen}. This calculation allows us to unambiguously check the validity of our analytical model.
In the next subsection, we find the GHESs in the presence of RSOC for an isolated edge. We obtain explicit expressions for the spin orientation versus energy and find good agreement with the large-width limit of the numerics. We show that for a single edge the spin orientation is only weakly dependent on energy for a real HgTe sample, i.e. the spin orientation is actually quite robust against RSOC. The following subsection is devoted to a ribbon. Now the expressions become much more complicated but the results as a function of the width of the sample show more interesting patterns, where spin rotation versus energy cannot be neglected.
\subsection{The case of a single isolated edge}
\label{subsec:Semi}
Now we find the pair of GHESs appearing at an isolated boundary of a 2D TI described by the BHZ model including the RSOC. As mentioned above, the starting point is the exact HESs without the RSOC. The HES dispersions are linear,\cite{Wada-PRB-2011} i.e. $E_{\sigma k_y}=E_0+\mathfrak{s}\h vk_y$, where $\mathfrak{s}=+(-)$ for $\sigma=\uparrow (\downarrow)$, $v$ is the constant velocity and $E_0$ an energy shift. The HESs located at the boundary of the half-plane $x>0$ are given by
\begin{align}\label{eq:HES-R0-zero}
\psi_{k_y\sigma}(x,y)=\frac{1}{\sqrt{L}}e^{ik_yy}g_{\mathfrak{s}k_y}(x)\hat{\phi}_\sigma,
\end{align}
i.e.~a plane-wave running along the $y$-axis combined with a transverse wave function $g_{\mathfrak{s}k_y}(x)$ determining the width of the HES and a $k_y$-independent four-component spinor $\hat{\phi}_\sigma$. There is one spinor from each time-reversed block of the BHZ model, i.e. $\hat{\phi}_{\uparrow}$ ($\hat{\phi}_{\downarrow}$) only has non-zero components on the two first (last) entries with positive (negative) total angular momentum projection. Periodic boundary conditions are used along the edge of length $L$. The HES wave functions and dispersions are given explicitly using the BHZ parameters in Appendix \ref{subsec-Appendix:HES-isolated-boundary}.
To include the RSOC analytically, we write the full Hamiltonian $H=H_0+H_R$ in a basis of the HESs for $R_0=0$ given in Eq.(\ref{eq:HES-R0-zero}), i.e.
\begin{align}\label{eq:2_quantization}
\mathcal{H}_0 + \mathcal{H}_R=&\sum_{k_y,\sigma\in\{\uparrow,\downarrow \}}
E_{\sigma k_y} c_{\sigma k_y}^\dag c^{}_{\sigma k_y}
\nonumber\\
&+
\sum_{k_y,k_y'}
\sum_{\sigma\sigma'}
\langle\psi_{k_y\sigma}|H_{R}|\psi_{k'_y\sigma'}\rangle
c_{\sigma k_y}^\dag c_{\sigma'k_y'}^{},
\end{align}
where $c_{\sigma k_y}^\dag$ ($c_{\sigma k_y}^{}$) creates (annihilates) a particle in the HES $\psi_{k_y\sigma}$. In this approach, we neglect the matrix elements between the edge and bulk states. These are presumably very small, since bulk and edge states to a very large extend are spatially separated. This is an excellent assumption well within the bulk gap close to the $\Gamma$ point, whereas the bulk states begin to play a role close to the bulk band gap edge as our numerics show. The full Hamiltonian (\ref{eq:2_quantization}) simplifies by noting that the matrix elements of $H_R$ are diagonal in $k_y$ due to the plane-wave part of the HESs (\ref{eq:HES-R0-zero}). Moreover, $H_R$ only couples opposite spins, so we find
\begin{align}
\mathcal{H}=
\sum_{k_y}\!
\big(
c_{\uparrow k_y}^\dag, c_{\downarrow k_y}^\dag
\big)\!
\left(
\begin{array}{cc}
E_0+\h vk_y & k_y\alpha_{k_y}\\
k_y\alpha_{k_y} & E_0-\h vk_y\\
\end{array}
\right)\!
\left(
\begin{array}{c}
c_{\uparrow k_y}\\
c_{\downarrow k_y}\\
\end{array}
\right)\!,
\label{eq:full-non-diagonal-H}
\end{align}
where an effective RSOC $\alpha_{k_y}\equiv\langle\psi_{k_y\uparrow}|H_{R}|\psi_{k_y\downarrow}\rangle /k_y$ is introduced. In terms of the BHZ parameters, we find
\begin{align} \label{eq:alpha-ky}
\alpha_{k_y}&=
R_0\frac{B-D}{2Bk_y}
\int_{0}^{\infty}\!\! dx
g_{k_y}(x)\big[\p_x+k_y\big] g_{-k_y}(x)
\nonumber\\
&=
R_0\frac{(B-D)^2}{2B^2}\Big(1-ak_y^2\Big)+\mathcal{O}\Big[k_y^4\Big].
\end{align}
where $a=\frac{D^2 [A^2 B+2 (B^2-D^2) M_0]}{2 B (B^2-D^2) M_0^2}$ and we expanded in $k_y$ in the last step. The exact result for $\alpha_{k_y}$ and details of the calculation are found in Appendix \ref{subsec:details-RSOC-isolated}.
The effective RSOC (\ref{eq:alpha-ky}) only includes the first order RSOC in the BHZ basis given in Eq.(\ref{Rashba_hamil}). In Appendix \ref{app:higher-order-Rashba}, we discuss the effects of higher order RSOC terms. We show that the second order term does not contribute to $\alpha_{k_y}$, while the third order term in principle could contribute even though we face technical difficulties in this case due to the hard wall boundary condition used to find the HESs analytically.
However, the third order RSOC term cannot introduce terms of a different order in $k_y$ in $\alpha_{k_y}$ than the ones found here. Therefore it cannot change the physics of the GHESs discussed below. Moreover, the magnitude of the effects of the third order term can partly be incorporated into the prefactor $R_0$.
The form of $\mathcal{H}$ in Eq.(\ref{eq:full-non-diagonal-H}) is clearly very similar to the simple 1D Hamiltonian for a pair of HESs with a generic spin-orbit coupling Eq.(\ref{eq:simplest-case}), since the effective spin-orbit term $\alpha_{k_y}\sigma_xk_y$ resembles $a_{x}\sigma_xk$. The important difference is that our effective RSOC $\alpha_{k_y}$ depends on $k_y$ and therefore gives rise to GHESs with $k_y$-\emph{dependent} (or equivalently energy-dependent) spin orientation as we shall see shortly. In contrast, the spin-orbit coupling in Eq.(\ref{eq:simplest-case}) only leads to a constant wavevector-independent spin rotation. In other words, the effective spin-orbit term $\alpha_{k_y}\sigma_xk_y$ has to be nonlinear in $k_y$ for GHESs to arise.
By diagonalizing $\mathcal{H}$ in Eq.(\ref{eq:full-non-diagonal-H}), we get the dispersion relations including the RSOC
\begin{subequations}
\begin{align}
E^{\textsc{rsoc}}_{k_y,\pm}
&=E_0\pm \h v_{\alpha_{k_y}} k_y ,
\label{eq:dispersions_single_edge_including_R0}
\end{align}
and the eigenstates in $k_y$-space
\begin{align}\label{eq:eigenstate-single-edge-with_R0}
\Psi_{k_y,\pm}&= \frac{1}{\sqrt{2}}
\left(
\begin{array}{c}
\pm\sqrt{1\pm\frac{v}{v_{\alpha_{k_y}}}}\\
\sqrt{1\mp\frac{v}{v_{\alpha_{k_y}}}}
\end{array}
\right),
\end{align}
\end{subequations}
where $\pm$ corresponds to two different edge states with the renormalized velocity $v_{\alpha_{k_y}}=\sqrt{v^2+(\alpha_{k_y}/\h)^2}$. For $R_0=0$, the states are simply $\psi_{k_y\uparrow}$ and $\psi_{k_y\downarrow}$, whereas for $R_0\neq0$ they become a superposition of both spins. Moreover, they are GHESs due to their $k_y$-dependent spin orientation. The case described by the model Hamiltonian in Eq.(\ref{eq:simplest-case}) is included here: if $\alpha_{k_y}$ is \emph{in}dependent of $k_y$, then so are $\Psi_{k_y,\pm}$ and no GHESs appear. Due to time-reversal symmetry, the eigenstates constitute a Kramers pair with opposite spin orientations (i.e.~orthogonal spinors). This is seen by applying the time-reversal operator $\Theta$ to $\Psi_{k_y,\pm}$ and using $\alpha_{-k_y}{=}\alpha_{k_y}$, i.e.~$\Theta\Psi_{k_y,\pm}= \mp\Psi_{-k_y,\mp}$ (see Appendix \ref{app:TR}). Finally, we observe that the RSOC does not open a gap in the spectrum in accordance with time-reversal symmetry, but merely renormalizes the velocity close to $k_y=0$ and creates a slight nonlinearity for larger $k_y$.
The GHES dispersions (\ref{eq:dispersions_single_edge_including_R0}) for a HgTe QW with $R_0=A$ are shown in the top panel of Fig.~\ref{fig:dispersions} along with a comparison to our numerical results using the tight-binding regularization for $W=1000$nm. We find that the effect of the RSOC on the dispersions is rather weak for a HgTe sample. We also present similar calculations for a InAs/GaSb double QW in Fig.~\ref{fig:InAs}. Our analytical method only works if both HESs without RSOC exist simultaneously, hence the dispersions do not cover the entire bulk gap as seen in Figs.~\ref{fig:dispersions} and \ref{fig:InAs}. Although the bulk bands are quite different for the InAs/GaSb and HgTe QWs, we find that the behavior of the GHESs is very similar for similar values of $R_0$ --- both in the single edge case and for the ribbon discussed in the next section. Therefore, we do not show more figures for InAS/GaSb with the understanding that the results for the latter are similar to our results for HgTe in the presence of an electric field such that $R_0 \approx 0.2A$.
\begin{figure}
\includegraphics[width=0.95\linewidth]{InAs_new.png}
\caption{\label{fig:InAs} Dispersion relation for InAs/GaSb QWs using the BHZ Hamiltonian with the parameters given by table \ref{table:parameters} without taking the BIA terms into account. Analytical results for an isolated edge and numerical results for $W=1000$ nm for the edge states almost coincide.}
\end{figure}
Next, we consider the $k_y$-dependence of the spin orientation of the GHESs in the case of an HgTe TI. In Fig.~\ref{fig:proj_single_edge}, we show the amount of spin $\downarrow$ in the state $\Psi_{k_y,+}$, which is a spin $\uparrow$ state for $R_0=0$, i.e. the projection $P=|\langle k_y,\downarrow\! | k_y,+\rangle|^2=|\langle\psi_{k_y\downarrow} |\Psi_{k_y,+}^{}\rangle|^2$. We find a reasonably good comparison between the analytical results for the isolated edge and the numerical results for a large width of $W=1000$nm. The small discrepancy between the analytical and numerical projections could be due to the truncation of the Hilbert space in the analytical calculation. As seen in the figure, the spin rotation is rather small in a realistic HgTe QW even for relatively large values of $R_0$, i.e.~the spin orientation of the edge states is rather robust against large external electric field. The analytical projection is found from the GHESs in Eq.(\ref{eq:eigenstate-single-edge-with_R0}) using the exact RSOC $\alpha_{k_y}$ in Eq.(\ref{eq:alpha-single-edge-exact}) in Appendix \ref{subsec:details-RSOC-isolated}. The analytical theory requires simultaneous existence of both HESs without RSOC. The analytical projection in Fig.~\ref{fig:proj_single_edge} is shown in both the bulk band gap region (full curve) and in the region of coexistence between edge and bulk states (dotted curve). In the coexistence regime, the HESs gradually widen and finally the penetration length divergences well within the bulk states as seen in Fig.~\ref{fig:penetration-length} in Appendix \ref{subsec-Appendix:HES-isolated-boundary}. By using the projection, we obtain the total spin rotation $T_s$ Eq.(\ref{eq:ts}). From the numerical results for the entire $k$-space, we find that $T_s$ is proportional to $R_0^2$ to a good approximation.
\begin{figure}
\includegraphics[width=0.9\linewidth]{projr0eq05cont-new.png}
\caption{\label{fig:proj_single_edge} The projection $P=|\langle\psi_{k_y\downarrow}|\Psi_{k_y,+}^{}\rangle|^2$ of the GHES $\Psi_{k_y,+}$ with $R_0=0.5A$ into the $R_0=0$ spin-$\downarrow$ state as a function of $k_y$ using the parameters of a HgTe QW in table \ref{table:parameters}. The figure shows a comparison of the analytical results with the numerics with $W=1000$ nm. The analytical projection is seen in both the bulk energy band gap (full black curve) and in the regime of coexistence of bulk and edge states (dotted black curve). Moreover, we have manually removed the numerical results close $k=0$ where a very narrow peak appears due to finite size effect, see Fig.~\ref{fig:projW} and the discussion in Sec.~\ref{subsec:twoedges}.}
\end{figure}
Now, we find the analytical form of the spin structure parameter\cite{Schmidt-PRL-2012} $k_0$ in Eq.(\ref{eq:bk}) for the BHZ model including the RSOC. We do this by introducing two unitary transformations, which together diagonalize $\mathcal{H}$ in Eq.(\ref{eq:full-non-diagonal-H}). The first transformation is $k_y$-independent and rotate the spin basis such that it removes the $k_y$-independent part of $\alpha_{k_y}$. This part does not lead to GHESs as discussed above. This rotation is convenient such that we use the same choice of spin-quantization axis as in Ref.~\onlinecite{Schmidt-PRL-2012}, i.e. the spins point along the new rotated spin-quantization axis at $k=0$. The second unitary transformation is $k_y$-dependent and transforms between the eigenstates and the new rotated spin basis. In other words, it is the matrix $B_k$ in Eq.(\ref{eq:Bk-full}). Now we perform the steps explicitly. First, we define $\delta\alpha_{k_y} \equiv \alpha_{k_y}-\alpha_0$, where $\alpha_0=\alpha_{k_y=0}^{}$ is $k_y$-independent. Thereby, we can diagonalize the $\alpha_0$ part of $\mathcal{H}$, i.e.
\begin{align}
\mathcal{H}
&=
\sum_{k_y}
C^{\dagger}_{k_y}
\left(
\begin{array}{cc}
E_0+\h vk_y & k_y(\alpha_0+\delta\alpha_{k_y})\\
k_y(\alpha_0+\delta\alpha_{k_y}) & E_0-\h vk_y\\
\end{array}
\right)
C_{k_y}
\nonumber\\
&=
\sum_{k_y}
C^{\dagger}_{k_y}
U
\left(
\begin{array}{cc}
E_0+ \h v_{\alpha_0}k_y & 0\\
0 & E_0-\h v_{\alpha_0}k_y\\
\end{array}
\right)U^{\dag}C^{}_{k_y}
\nonumber\\
&\ +
\sum_{k_y}
C^{\dagger}_{k_y}U
k_y\delta\alpha_{k_y}
\left(
\begin{array}{cc}
\sin(\theta) & \cos(\theta)\\
\cos(\theta) & -\sin(\theta)\\
\end{array}
\right)
U^{\dag}
C^{}_{k_y}.
\label{eq:trivial-rotation-performed}
\end{align}
Here $C^{\dagger}_{k_y}=\big(c_{\uparrow k_y}^\dag, c_{\downarrow k_y}^\dag\big)$, $\h v_{\alpha_0}=\sqrt{(\h v)^2+\alpha_0^2}$ and the first $k_y$-independent unitary transformation $U$ is
\begin{align}
U=\left(
\begin{array}{cc}
\cos(\theta/2) & -\sin(\theta/2)\\
\sin(\theta/2) & \cos(\theta/2)\\
\end{array}
\right)
\end{align}
where $\cos(\theta)\equiv v/v_{\alpha_0}$ and $\sin(\theta)\equiv \alpha_0/(\h v_{\alpha_0})$. This transformation is simply a $k_y$-independent rotation to a new spin basis,
\begin{align}
\left(
\begin{array}{c}
c_{\uparrow' k_y}^{}\\
c_{\downarrow' k_y}^{}\\
\end{array}
\right)
=
U^{\dagger}
\left(
\begin{array}{c}
c_{\uparrow k_y}^{}\\
c_{\downarrow k_y}^{}\\
\end{array}
\right),
\end{align}
where $\uparrow'$ and $\downarrow'$ are the eigenstates of $\mathcal{H}$ at $k_y=0$. Finally, we diagonalize the Hamiltonian completely by a second unitary transformation and obtain
\begin{align}
\mathcal{H}
&{=}
\sum_{k_y}\!
C_{k_y}^{'\dag}
\mathcal{V}_{k_y}^{}
\!\left(\!
\begin{array}{cc}
E_0\!+\!k_y \h v_{\alpha_{k_y}} & 0\\
0 & E_0\!-\!k_y\h v_{\alpha_{k_y}} \\
\end{array}
\!\right)\!
\mathcal{V}_{k_y}^{\dag}
C_{k_y}^{'},
\nonumber
\end{align}
where $C_{k_y}^{'\dagger}=\big(c_{\uparrow'k_y}^\dag, c_{\downarrow' k_y}^\dag\big)$ and $\h v_{\alpha_{k_y}}=\sqrt{(\h v)^2+\alpha_{k_y}^2}$. As expected, we find the same dispersions as in Eq.(\ref{eq:dispersions_single_edge_including_R0}). More importantly, we acquire an analytical form of the unitary transformation $\mathcal{V}_{k_y}$, which by construction is exactly $B_{k_y}$ from Eq.(\ref{eq:Bk-full}), i.e.
\begin{align}
B_{k_y}=
\mathcal{V}_{k_y}=
\left(
\begin{array}{cc}
\cos(\phi/2) & -\sin(\phi/2)\\
\sin(\phi/2) & \cos(\phi/2)\\
\end{array}
\right),
\end{align}
where $\cos(\phi)\equiv[(\h v)^2+\alpha_{k_y}\alpha_0]/(\h^2 v_{\alpha_0}v_{\alpha_{k_y}})$ and $\sin(\phi)\equiv \delta\alpha_{k_y} v /(v_{\alpha_0}\h v_{\alpha_{k_y}})$. Therefore, we have now found the $k_y$-dependent matrix $B_{k_y}$ relating the GHESs to the HESs with a fixed spin-axis for a specific model, namely the BHZ model including the RSOC. We remark that $\delta\alpha_{k_y}=0$ at $k_y=0$ by definition, so $\mathcal{V}_{k_y=0}$ is the unity matrix and therefore $\uparrow'$ and $\downarrow'$ become eigenstates at $k_y=0$.
We can now find the spin structure parameter $k_0$ in Eq.(\ref{eq:bk}) controlling the amount of spin-rotation for small $k_y$. By expanding $B_{k_y}=\mathcal{V}_{k_y}$ around $k_y=0$, we obtain
\begin{align}\label{eq:k0-general}
\!\frac{1}{k_0^2}
&=
\frac{D^2 |R_0 A (B{-}D)| \left|A^2 B+2 M_0(B^2-D^2)\right|}{2 \sqrt{B^2{-}D^2} M_0^2 \left| 4 A^2 B^2 (B{+}D) {+}(B{-}D)^3 R_0^2\right|}.
\end{align}
Thereby, we have an explicit expression for $k_0$ --- a parameter originally introduced based on symmetry arguments.\cite{Schmidt-PRL-2012} Such an expression in terms of the BHZ parameters is valuable beyond the case of HgTe QWs due to the generic Dirac-like nature of the BHZ model. Interestingly, we observe that the particle-hole asymmetry parameter $D$ plays an essential role for $k_0$, i.e. for $D=0$ no spin rotation appears and therefore no GHESs come out in the case studied here. This is valid beyond the expansion of $B_{k_y}$ in $k_y$, since the effective RSOC in Eq.(\ref{eq:alpha-ky}) is $k_y$-independent to all orders, $\alpha_{k_y}^{(D=0)}=R_0/2$, for $D=0$ such that $B_{k_y}$ is the unity matrix (see Eq.(\ref{eq:alpha-D_equal_0}) in Appendix \ref{subsec:details-RSOC-isolated}). Curiously, the parameter $D$ is often removed in many theoretical discussions of topology\cite{book_BH} and thereby the interesting physics of GHESs might be missed. Furthermore, Eq.(\ref{eq:k0-general}) also reveals that $k_0$ depends on the Dirac mass $M_0$ and the RSOC strength $R_0$ in rather non-trivial ways.
Before proceeding, we briefly discuss the effect of the lowest order BIA terms given by\cite{Konig-JPSJ-2008,Qi-RMP-2011}
\begin{equation}\label{eq:BIA}
H_{BIA}=
\left(\begin{array}{cccc}
0 & 0 & 0 & -\Delta \\
0 & 0 & \Delta & 0 \\
0 & \Delta & 0 & 0 \\
-\Delta & 0 & 0 & 0
\end{array}\right),
\end{equation}
where $\Delta$ is a constant. Including the $H_{BIA}$ in the basis of the HESs for an isolated edge Eq.(\ref{eq:HES-R0-zero}) as we did for $H_R$ in Eq.(\ref{eq:2_quantization}), we find $\langle\psi_{k_y\sigma}|H_{BIA}|\psi_{k'_y\sigma'}\rangle=0$ for all $k_y\sigma$ and $k'_y\sigma'$. Hence, within our analytic approach, the lowest order BIA terms does not affect the HESs nor their spin orientation for an isolated edge. The second order RSOC terms has the same structure in the anti-diagonal as $H_{BIA}$ and therefore also has zero matrix elements, see Appendix \ref{app:higher-order-Rashba}. Including the small overlaps between the bulk and edge states, a modest effect on the energy dispersions is found due to $H_{BIA}$ close to the bulk gap edge, where these overlaps matter the most.\cite{Michetti-Semicond-Sci-Tech-2012} For a ribbon, the $H_{BIA}$ was found to couple opposite edges.\cite{Krueckl-PRL-2011} Very recently, Rod \emph{et al.}\cite{Rod-PRB-2015} found GHESs for both ribbon and disk geometries due to $H_{BIA}$ in the BHZ model. For both geometries, a finite $k_0^{-2}$ was extracted numerically in the limit of a particle-hole symmetric BHZ model (i.e.~$D=0$), where both edge and bulk states were included in their calculations.
\subsection{The case of a ribbon with two parallel edges} \label{subsec:twoedges}
In this section, we consider the GHESs for a ribbon with two parallel edges using the BHZ model including the RSOC. Thereby, four edge states come into play, since a pair of GHESs exist on each edge for well-separated edges. We pay special attention to how the finite size effects can enhance spin orientation variation of the GHESs as the width of the ribbon $W$ gets smaller and the edge states on opposite sides begin to overlap.
Before including the RSOC, we briefly summarize the HESs and their dispersions without RSOC for a ribbon.\cite{Zhou-PRL-2008} We refer to Appendix \ref{subsec:HES-ribbon} for details. An important difference between the ribbon and the single-edge case is that we do not have the energy dispersions in closed analytical forms for the ribbon, but instead as the solutions to a cumbersome equation (see Eq.(\ref{eq:implicit-eq-for-energies-for-finite-width})). Nevertheless, the physical consequence of the finite width is clear: A gap opens at the crossing point of the dispersions found for the isolated edge, see Fig.~\ref{fig:dispersion_noR}.\cite{Zhou-PRL-2008} The dispersions for a ribbon have a limiting cusp form for a wide ribbon, i.e.
\begin{align}\label{eq:cusp-form}
E^{e=\pm}_{k_x}\rightarrow E_0\pm\h v |k_x|
\quad \textrm{for}\quad
\ W \rightarrow \infty,
\end{align}
where $E^{+}_{k_x}$ ($E^{-}_{k_x}$) is the energy dispersion above (below) the gap for $W\leq\infty$. Therefore, the label $e=\pm$ should not be confused with the single-edge case, where $\pm$ often refers to the sign of the velocity. The velocity $v$ and energy shift $E_0$ are identical to the single-edge case. Noticeably, $E^{e=\pm}_{k_x}$ are independent of the spin $\sigma$, since equal spins travel in opposite directions on the two edges.
A ribbon with edges at $y=\pm W/2$ has four HESs without RSOC\cite{Zhou-PRL-2008} $\psi_{k_x\sigma}^{e}(x,y)$, where $e=\pm$ labels the energy $E^{e}_{k_x}$ to which the state belongs. As for an isolated edge, the states have a plane-wave part running along the edges, i.e. $\psi_{k_x\sigma}^{e}\propto e^{ik_xx}$. Only the first (last) two components of the states $\psi_{k_x\uparrow}^{e}$ ($\psi_{k_x\downarrow}^{e}$) are non-zero, corresponding to the spin-up (spin-down) block of $H_0$. However, in contrast to the single-edge case, the spinors are not constant, but the relative weight of the two components vary with both $k_x$ and $y$. A particular state $\psi_{k_x\sigma}^{e}$ is not always localized on the same edge. Instead, the localization changes gradually from one edge to the other when crossing $k_x=0$. For $k_x>0$, the states
\begin{subequations}\label{eq:localization-of-HESs}
\begin{align}
\psi_{k_x\uparrow}^{+},\
\psi_{k_x\downarrow}^{-} \ & \textrm{are localized close to}\ y=W/2\ \textrm{and} \\
\psi_{k_x\uparrow}^{-},\
\psi_{k_x\downarrow}^{+}\ & \textrm{are localized close to}\ y=-W/2,
\end{align}
\end{subequations}
and vice versa for $k_x<0$.
As for the isolated edge, we build an analytical model using only the HESs without RSOC. Since this approach leaves out the overlaps between bulk and edge states, it becomes less good for a narrow ribbon, where bulk and edge states become comparable in spatial extend. Therefore, our analytical results are most reliable for small momenta well within the bulk gap as we shall see.
By including the RSOC in the subspace of the HESs without RSOC, $\psi_{k_x\sigma}^{e}$, the Hamiltonian becomes
\begin{align}
\mathcal{H}=&\mathcal{H}_0+\mathcal{H}_R
=\sum_{\sigma,k_x,e}{E_{k_x}^{e}(c^{e}_{k_x\sigma})^\dagger c_{k_x\sigma}^e}
\nonumber\\
&+
\sum_{k_x,k'_x}\sum_{\sigma,\sigma'}\sum_{e,e'}
{\langle\psi_{k_x\sigma}^{e}|H_R| \psi_{k'_x\sigma'}^{e'}\rangle
(c^{e}_{k_x\sigma})^\dagger c_{k'_x\sigma'}^{e'}},
\end{align}
where $(c^{e}_{k_x\sigma})^\dagger$ [$c_{k_x\sigma}^{e}$] creates [annihilates] a particle in the HES $\psi_{k_x\sigma}^{e}$ of energy $E_{k_x}^{e}$. The RSOC Eq.(\ref{Rashba_hamil}) only couples opposite spins and the Hamiltonian is diagonal in $k_x$, since $\langle\psi_{k_x\sigma}^{e}|H_R| \psi_{k'_x\sigma'}^{e'}\rangle\propto\delta_{k^{}_x,k'_x}$. We order the basis as $\{ | \psi_{k_x\uparrow}^{+}\rangle, \, | \psi_{k_x\downarrow}^{-}\rangle,\, | \psi_{k_x\uparrow}^{-}\rangle,\,| \psi_{k_x\downarrow}^{+}\rangle\}$ such that the first two entries are localized on the opposite edge of the last two as seen in Eq.(\ref{eq:localization-of-HESs}), i.e.
\begin{align}\label{full_hamiltonian}
&\mathcal{H}=\mathcal{H}_0+\mathcal{H}_R
\nonumber\\
&=
\sum_{k_x}
\mathbf{C}^{\dagger}_{k_x}\!
\left(\!
\begin{array}{cccc}
E^{+}_{k_x}&ib&0&id_{+}\\
-ib&E^{-}_{k_x}&id_{-}&0 \\
0&-id_{-}&E^{-}_{k_x}&-ib \\
-id_{+}&0&ib& E^{+}_{k_x}
\end{array}
\right)
\!\mathbf{C}_{k_x},
\end{align}
where $\mathbf{C}^{\dagger}_{k_x}=\big[(c_{k_x\uparrow}^+)^\dagger,(c_{k_x\downarrow}^-)^\dagger,(c_{k_x\uparrow}^-)^\dagger,(c_{k_x\downarrow}^+)^\dagger\big]$ and we introduced the \emph{inter}-edge matrix elements
\begin{subequations}\label{eq:inter-matrix-elements}
\begin{align}
id_{+}=\langle\psi_{k_x\downarrow}^{+} | H_R| \psi_{k_x\uparrow}^{+}\rangle,\\
id_{-}=\langle\psi_{k_x\uparrow}^{-} | H_R| \psi_{k_x\downarrow}^{-}\rangle,
\end{align}
\end{subequations}
and the \emph{intra}-edge matrix element
\begin{align}\label{eq:intra-matrix-element}
ib_{}=\langle\psi_{k_x\downarrow}^{-} | H_R| \psi_{k_x\uparrow}^{+}\rangle,
\end{align}
which all depend on $k_x$. In Eq.(\ref{full_hamiltonian}), we used that the intra-edge matrix elements on opposite edges are related as $ib=\langle\psi_{k_x\downarrow}^{-} | H_R| \psi_{k_x\uparrow}^{+}\rangle=-\langle\psi_{k_x\downarrow}^{+} | H_R| \psi_{k_x\uparrow}^{-}\rangle$, as discussed in Appendix \ref{subsec:ribbon-details}. Thereby, we are left with three matrix elements only, which depend on the implicitly known dispersions relations $E^{\pm}_{k_x}$. The detailed formulas are given in Appendix \ref{subsec:ribbon-details}.
Due to the ordering of the basis, the Hamiltonian (\ref{full_hamiltonian}) has two $2\times2$ blocks in the diagonal, one for each edge. Each $2\times2$ block resembles the Hamiltonian (\ref{eq:full-non-diagonal-H}) found for an isolated edge and an effective \emph{intra}-edge RSOC could be introduced as $b/k_x$ as in Sec.~\ref{subsec:Semi}. However, due to the limiting cusp form of the energy dispersions Eq.(\ref{eq:cusp-form}), one should instead define the effective intra-edge RSOC as $\alpha_{k_x}^{\textrm{intra}}=-b/|k_x|$. With this definition, $\alpha_{k_x}^{\textrm{intra}}$ corresponds to the effective RSOC for the isolated edge in Eq.(\ref{eq:alpha-ky}) in the wide ribbon limit. However, as the width gets smaller, we find an increased $k_x$-dependence of $\alpha_{k_x}^{\textrm{intra}}$ for small $k_x$. This indicates an increased spin-orientation change for small $k_x$ as $W$ decreases, which we also find below by direct calculation.
The opposite edges of the ribbon are coupled by the \emph{inter}-edge elements $d_{\pm}$ in the anti-diagonal of $\mathcal{H}$, which vanish for $W\rightarrow \infty$. Finally, we mention that performing unitary transformations of $\mathcal{H}$ to find $k_0$ as in Sec.~\ref{subsec:Semi} is difficult, since we do not have closed formulas for $E_{k_x}^e$.
By diagonalization of the Hamiltonian (\ref{full_hamiltonian}), the dispersion relations including the RSOC become
\begin{align}\label{eq:energy-rsoc-ribbon}
E^{\textsc{rsoc}}_{k_x,s \tau}
=&
\tau
\frac{1}{2}\sqrt{[s(d_{-}-d_{+})+E_{k_x}^--E_{k_x}^+]^2+4 b^2}
\nonumber\\
&+\frac{1}{2} \left[s(d_{+}+d_{-})+E_{k_x}^++E_{k_x}^-\right]
\end{align}
where $ s=\pm1$ and $\tau=\pm1$. In the wide ribbon limit, where the inter-edge matrix elements $d_\pm$ are insignificant, these dispersions resemble the isolated-edge dispersions Eq.(\ref{eq:dispersions_single_edge_including_R0}) (disregarding the cusp limit of $E^{\pm}_{k_x}$). For a finite width $W$, however, the inter-edge matrix elements $d_{\pm}$ come into play and create four different dispersions. As shown in the bottom panel of Fig.~\ref{fig:dispersions} and in Fig.~\ref{fig:ribbon-dispersions-anticross}, two gaps arise symmetrically with respect to $k_x=0$. Therefore, we have found that the spin degeneration present for $R_0=0$ between $\psi_{k_x\uparrow}^{e}$ and $\psi_{k_x\downarrow}^{e}$ is broken by the interplay of RSOC and a finite width, where both ingredients are necessary. A similar effect of SIA combined with finite size have also been found in one dimension higher, namely for the 2D Dirac surface states on a 3D TI.\cite{Shan-NJP-2010}
\begin{figure}
\includegraphics[width=0.95\linewidth]{gap5.png}
\includegraphics[width=0.95\linewidth]{kgap3.png}
\caption{\label{fig:gapW}
Comparison of analytical and numerical results for the value of the gap $\Delta_{\textrm{edge}}$ [$meV$] vs.~width of the ribbon (top panel) and position of the gap in momentum $k_{\textrm{gap}}$ [$nm^{-1}$] vs.~width of the ribbon (bottom panel). We use the HgTe parameters in Table \ref{table:parameters}.}
\end{figure}
In Fig.~\ref{fig:gapW}, we show the position $k_{\textrm{gap}}$ and value $\Delta_{\textrm{edge}}$ of the gaps as a function of $W$ for different $R_0$. We compare numerical and analytical calculations on a logarithmic scale. As the width is increased, the position of the gap goes rapidly towards $k_x=0$ and the value of the gap goes to zero, such that we recover the result for an isolated edge. Interestingly enough, there are several values of the width, depending on the value of $R_0$, where the gap is particularly reduced. The reason is essentially that the actual transverse wave function including RSOC for an isolated edge has a form similar to an exponential times a sine. This means that the solution for an isolated edge state (including the gapless dispersions) also becomes the solution for a finite ribbon, when the zeros of the transverse wave function match the width. This destructive interference has been studied before both with\cite{Takagaki-PRB-2014} and without\cite{Mehdi-JPCM-2012} RSOC. Similar physics have also been discussed for thin films of 3D TIs.\cite{Linder-PRB-2009,Liu-PRB-2010-3DTI,Lu-PRB-2010} In Ref.~\onlinecite{Takagaki-PRB-2014}, Takagaki showed that the gap vanishes periodically with a period almost inversely proportional to the strength of the RSOC. In these particular values of the width, the coupling between the edges is cancelled without reaching the large width limit. As shown in Fig. \ref{fig:gapW}, this effect is not captured by the analytical theory, although it correctly gives the essential decaying trends of both the gap size $\Delta_{\textrm{edge}}$ and position $k_{\textrm{gap}}$.
The eigenvectors in $k_x$-space in the basis presented above, i.e. $\{ | \psi_{k_x\uparrow}^{+}\rangle, | \psi_{k_x\downarrow}^{-}\rangle, | \psi_{k_x\uparrow}^{-}\rangle, | \psi_{k_x\downarrow}^{+}\rangle\}$ are:
\begin{align}\label{two_rashba}
\Psi_{k_x,s\tau}=
\frac{1}{\sqrt{8b^2+2\zeta^2_{s\tau}}}
\left(
\begin{array}{c}
is 2b\\
-s\zeta_{s\tau}\\
i\zeta_{s\tau}\\
2b
\end{array}
\right)
\end{align}
where $s=\pm$ and $\tau=\pm$ and we defined
\begin{align}
\zeta_{s\tau}=
&s(d_{+}-d_{-})+E_{k_x}^+-E_{k_x}^-
\nonumber\\
&-\tau\sqrt{\big[s(d_{-}-d_{+})+E_{k_x}^- -E_{k_x}^+\big]^2+4 b^2}.
\end{align}
Here the two first and the two last components of $\Psi_{k_x,s\tau}$ represent spinors localized on opposite edges. The four states $\Psi_{k_x,s\tau}$ are clearly present on both edges, but in a very particular way: The spinor localized on one edge, $\varphi_a\propto (is 2b, -s\zeta_{s\tau})^T$, is always orthogonal to the spinor $\varphi_b\propto (i\zeta_{s\tau},2b)^T$ localized on the opposite edge, since $\varphi_a^\dag\varphi_b=0$. In other words, the squared projections on the basis states on opposite edges are pairwise identical, i.e. $|\langle\psi_{k_x\uparrow}^{+}|\Psi_{k_x,s\tau}\rangle|^2=|\langle\psi_{k_x\downarrow}^{+}|\Psi_{k_x,s\tau}\rangle|^2$ and $|\langle\psi_{k_x\downarrow}^{-}|\Psi_{k_x,s\tau}\rangle|^2=|\langle\psi_{k_x\uparrow}^{-}|\Psi_{k_x,s\tau}\rangle|^2$. Thus, the states always have half of the weight on each edge, i.e. $|\langle\psi_{k_x\uparrow}^{+}|\Psi_{k_x,s\tau}\rangle|^2+|\langle\psi_{k_x\downarrow}^{-}|\Psi_{k_x,s\tau}\rangle|^2=1/2$ independently of $k_x$. Moreover, the Kramers partner of $\Psi_{k_x,\pm\pm}$ is $\Psi_{-k_x,\mp\pm}$, which can be seen by using that $E_{k_x}^\pm$ and $b$ are even in $k_x$ and $d_\pm$ is odd such that $\zeta_{s\tau}(-k_x)=\zeta_{-s\tau}(k_x)$, see Appendix \ref{subsec:ribbon-details}. Furthermore, the dispersions $E^{\textsc{rsoc}}_{k_x,s \tau}$ (\ref{eq:energy-rsoc-ribbon}) and eigenstates (\ref{two_rashba}) depend on both $E_{k_x}^+$ and $E_{k_x}^-$, and therefore only well-defined for momenta $k_x$, where both $E_{k_x}^-$ and $E_{k_x}^+$ are well-defined, see Figs.~\ref{fig:dispersions} and \ref{fig:dispersion_noR}.
\begin{figure}
\includegraphics[width=0.95\linewidth]{projw-new.png}
\caption{\label{fig:projW} The analytical projection of a ribbon edge state with primarily spin up character (on the lower edge $y=-W/2$ and $k>0$) into the spin down subspace, i.e $P=|\langle\psi_{k_x\downarrow}^{+}|\Psi_{k_x,+-}\rangle|^2$. In other words, the last component squared of $\Psi_{k_x,+-}$ in Eq.(\ref{two_rashba}). The parameters for HgTe in table \ref{table:parameters} and $R_0=A$ are used.}
\end{figure}
Now, we will argue that the eigenstates $\Psi_{k_x,s\tau}$ are GHESs, since their spin orientation on a single edge depends on $k_x$. Due to the structure of $\Psi_{k_x,s\tau}$ discussed above, we observe that the two edges of the ribbon suffer the same --- but opposite --- spin rotation. Due to the coupling between the two edges and the RSOC, the dispersions have two avoided crossings. These avoided crossings induce some particular characteristics of the spin rotation. Fig.~\ref{fig:projW} shows the analytical results for the projection onto the spin down subspace, $P=|\langle\psi_{k_x\downarrow}^{+}|\Psi_{k_x,s=1\tau=-1}\rangle|^2$, as a function of $k_{x}$ for one of the edge states which asymptotically is more than 99$\%$ spin up for different values of $W$ on lower edge ($y=-W/2$) and $k_x>0$. We can see that the projection reaches a relatively high value, higher for larger widths, but in a very narrow range of $k_x$, smaller for larger widths. The peaks of the projections are located close to the position of the gap $k_{\textrm{gap}}$. For clarity, we only show a range of $W$ from $140$nm to $300$nm, but the trend goes on indefinitely.
The avoided crossings of the ribbon dispersions and the associated spin-structure of the GHESs are illustrated on Fig.~\ref{fig:ribbon-dispersions-anticross}. As discussed above, the GHESs $\Psi_{k_x,s\tau}$ Eq.(\ref{two_rashba}) are always equally present on the lower ($y=-W/2$) and the upper ($y=W/2$) edge. Fig.~\ref{fig:ribbon-dispersions-anticross} only shows the spin-structure of the lower edge. We illustrate how to understand this by using the state $\Psi_{k_x,+-}$ as an example. Away from the avoided crossing (the green region), we find
\begin{align} \label{eq:pure-spin-GHES+-}
\Psi_{k_x,+-}=\frac{1}{\sqrt{2}}(-\psi_{k_x\downarrow}^-+i\psi_{k_x\uparrow}^-)
\end{align}
with more than $99\%$ accuracy. For $k_x>0$, $\psi_{k_x\uparrow}^-$ ($\psi_{k_x\downarrow}^-$) is localized near the lower (upper) edge and vice versa for $k_x<0$, see Eq.(\ref{eq:localization-of-HESs}). Thus, in this sense, $\Psi_{k_x,+-}$ is spin $\uparrow$ for $k_x>0$ (blue region) and spin $\downarrow$ for $k_x<0$ (yellow region) \emph{on the lower edge}, while the upper edge has the opposite spin-structure. In between these regions of almost pure spin $\uparrow$ or $\downarrow$, the states become genuine GHESs with sizable amounts of both spin $\uparrow$ and $\downarrow$ present on each edge (the green region). These regions are quantified by the peaks in the projections shown in Fig.~\ref{fig:projW}. Noticeably, the weight of each spin-component in the almost pure spin regions (blue/yellow) of the GHESs is only $1/2$, see e.g.~Eq.(\ref{eq:pure-spin-GHES+-}). Thus, the entire weight of one spin-component on one edge is carried by two different dispersion curves. This is vastly different from the two simple linear dispersions found for an isolated edge. Therefore, it is now clear that our states $\Psi_{k_x,s\tau}$ indeed are GHESs with $k_x$-dependent spin orientation. A remarkable difference to the isolated edge case is that the spin-orientation change is enhanced a great deal by the finite size.
\begin{figure}
\includegraphics[width=0.98\linewidth]{spin-texture-ribbon-dispersion-figur.pdf}
\caption{\label{fig:ribbon-dispersions-anticross} The ribbon energy dispersions $E^{\textsc{rsoc}}_{k_x,s \tau}$ Eq.(\ref{eq:energy-rsoc-ribbon}) and the spin-structure of the GHESs close to the lower edge at $y=-W/2$. The combination of finite width and RSOC produce splitting in $k_x$-space and energy gaps at $k_{\textrm{gap}}\neq0$. The spin-structure associated to the two avoided crossings is illustrated by the colors: The states are more than 99$\%$ pure spin $\uparrow$ ($\downarrow$) on the lower edge in the blue (yellow) part of the dispersions, whereas the spin-orientation rotates when all states come close together (green regions). The upper edge at $y=W/2$ has the opposite spin-structure (see the main text). Therefore, the states become true GHESs in the green regions, which coincide with the peaks in the projection seen in Fig.~\ref{fig:projW}. The parameters for HgTe in table \ref{table:parameters} are used together with $R_0=A$ and $W=200$nm.}
\end{figure}
In Fig.~\ref{fig:integral}, we show that the total spin rotation $T_s$ of $\Psi_{k_x,+-}$ scales with $R_0^2$ for not too large values of $R_0$. The total spin rotation is essentially the integral of the projections in Fig.~\ref{fig:projW} due to our choice of $k_1$ in Eq.(\ref{eq:ts}). We only show the numerical results as the analytical states (\ref{two_rashba}) are only available in the small $k_x$ range, where both $E_{k_x}^+$ and $E_{k_x}^-$ are well-defined. Nevertheless, using the analytical states to find $T_s$ very similar results are obtained for analytically feasible values of $R_0$ and $W$. Although the maximum of the spin projection in Fig.~\ref{fig:projW} increases, the total value of the integral is reduced, when the ribbon widens to the single edge limit. The scaling with $R_0^2$ works perfectly well for $R_0\lesssim0.5A$ except for very small values of the width. (Note that we only show $W>100$nm in Fig.~\ref{fig:integral}.) For larger values of $R_0$, the scaled total spin rotation $T_s/R_0^2$ increases compared to the values of $R_0\leq0.5A$. However, for very large spin-orbit couplings (like $R_0=2A$ in Fig. 8) and large widths, we obtain smaller $T_s/R_0^2$ probably due to the reduced bulk gap of the system.
\begin{figure}
\includegraphics[width=0.95\linewidth]{integral4b.png}
\caption{\label{fig:integral} Numerical value of the total spin rotation $T_s$ of the GHES $\Psi_{k_x,+-}$ rescaled with $R_0^2$ as a function of the ribbon width for different values of $R_0$.}
\end{figure}
\section{Summary}
\label{conclusions}
We have analyzed the spin-structure of the generic helical edge states appearing at the boundary of 2D TIs without axial spin symmetry. For the usual helical edge states in a 2D TI, the spin and propagation direction are locked in such a way that the spin-orientation is energy independent. However, for the GHESs the spin-orientation varies with energy or equivalently momentum $k$. This is possible in systems without axial spin symmetry, broken for instance by spin-orbit coupling. Importantly, time reversal symmetry still ensures counterpropagating states to be Kramers partners with orthogonal spins, but the spin-orientation of neighbouring states with different energies are not identical. This opens the possibility of inelastic scattering and thereby deviations from quantized conductance.\cite{Schmidt-PRL-2012}
Our study is focused on the GHESs produced by Rashba spin-orbit coupling within the BHZ model. We use HgTe QWs and InAs/GaSb double QWs as concrete examples. We analyze two situations: (i) a pair of GHESs at an isolated edge and (ii) the two pairs of GHESs in a ribbon with two parallel boundaries. In both cases, we employ an analytical approach, where the GHESs \emph{with} RSOC are found within a reduced basis consisting of the HESs \emph{without} RSOC. This is a good approximation, since the bulk and edge states are usually well separated spatially --- especially for small $k$ within the bulk energy gap. We also use a numerical tight-binding regularization of the BHZ model including RSOC to verify the analytical approach and, moreover, obtain independent valuable information.
For an isolated boundary, our analytical approach gives rise to a $2\times2$ Hamiltonian Eq.(\ref{eq:full-non-diagonal-H}), which is formally equivalent to a simple 1D model of a pair of HESs with a phenomenological spin-orbit coupling. From this analogy, we discover that GHESs are produced, when the effective spin-orbit coupling term is nonlinear in the momentum. In contrast, no GHESs appear for a linear effective spin-orbit coupling term within our framework. Moreover, we find the effective RSOC $\alpha_{k_y}$ in terms of the BHZ parameters. We also obtain the pair of GHESs in Eq.(\ref{eq:eigenstate-single-edge-with_R0}), where the velocity has been renormalized. Using our insights into linear versus nonlinear effective RSOC terms, we are able to provide an explicit expression for the so-called spin-structure parameter $k_0$, which measures the amount of spin-orientation variation for small $k$. The spin-structure parameter $k_0$ was originally deduced by symmetry arguments\cite{Schmidt-PRL-2012} and it is interesting to have an expression in a concrete case. For instance, it shows that $k_0$ depends on the RSOC strength $R_0$ and the Dirac mass $M_0$ in non-trivial ways. Moreover, $1/k_0^2$ vanishes when the particle-hole symmetry parameter $D$ of the BHZ model is zero. This statement is in fact more general: the effective RSOC term becomes exactly linear for $D=0$ such that only ordinary HESs appear in this case. For realistic HgTe and InAs/GaSb TIs, we observe that the spin-orientation of the edge states are quite robust against even large RSOC strengths $R_0$ for the single-edge case. Nevertheless, the spin-orientation does change slightly with energy. Moreover, we find good agreement between the numerical and analytical approaches.
Now we turn to the case of a ribbon, where the change in the spin-orientation of the GHESs is enhanced substantially for realistic HgTe TIs. The new physical element of the ribbon compared to the isolated edge, is the coupling of the GHESs on opposite edges. This finite size effect --- even without RSOC --- produce a gap in the HES spectrum.\cite{Zhou-PRL-2008} Now combining the finite width and the RSOC, two gaps and two associated avoided crossings arise in the GHESs spectrum symmetrically around $k=0$ as shown in Fig~\ref{fig:ribbon-dispersions-anticross}. Our analytical approach shows that the inter-edge RSOC is responsible for the avoided crossings to take place at finite momenta, which is evident from the dispersions in Eq.(\ref{eq:energy-rsoc-ribbon}). Moreover, we find the position in momentum of these gaps and their size $\Delta_{\textrm{edge}}$ versus the ribbon width. The analytical and numerical results for these quantities compare well, except at certain widths where the full numerical calculation reveals an interesting destructive interference effect. From our analytical approach, we find the GHESs including the RSOC in Eq.(\ref{two_rashba}). Remarkably, they consist of two orthogonal spinors, one on each side of the ribbon. Thus, the states are equally distributed on the two parallel edges. The states become true GHESs with a sizable variation in the spin-orientation close to the two avoided crossings in the GHES spectrum, where all the states are close in energy. We show in Fig.~\ref{fig:projW} that the region in $k$-space of sizable spin-orientation variation becomes wider, if the ribbon becomes narrower. On the other hand, widening the ribbon increases the maximal value of the projection, which measures the change in spin-orientation. To quantify this further, we find the total spin rotation $T_s$ Eq.(\ref{eq:ts}), which is related to the integral of the spin-orientation variation over the entire region of $k$-space. The numerical calculations show that the total spin rotation decreases with the ribbon width and, moreover, that $T_s\propto R_0^2$ for values of $R_0 \lesssim 0.5A$.
Our analytical GHESs for both the isolated edge and the ribbon open the possibility to study other effects in the presence of RSOC. For instance, scattering of magnetic impurities or the nuclear spins in the crystal could be studied. Furthermore, it would be interesting to explore the transport properties of a ribbon, since we found a significant spin-orientation change.
\acknowledgements
This work has been funded by Spanish Government projects: FIS2012-33152, FIS2012-34479, MAT2014-58241-P and CAM research consortium QUITEMAD+ S2013/ICE-2801. AML acknowledges A.~Fernandez Romero and financial support from the Carlsberg Foundation.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Search Saint Michaels, Arizona Foreclosure homes. See information on particular Foreclosure home or get Saint Michaels, Arizona real estate trend as you search. Besides our Foreclosure homes, you can find other great Saint Michaels, Arizona real estate deals on RealtyStore.com.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 4,720
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Lemonade Makin' Mama: Lighting food on fire for the fun of it.
I have no real back story for these photos.
Other than the fact that they make food look fun.
And I was there with some of my favorite women on earth... my book club girls.
And things were getting lit on fire.
All meals should include food on fire.
In college I worked at a sushi/teppan place and food was lit on fire nightly.
funny but people would always touch the grill. Yes, it is hot.. which is why we were just able to cook your food on it.
I'll be right back with some ice for your fingers.
OOooh I love teppanyaki bars! So much fun!
Reminds me of Kyoto Restaurant; loved that place. We once had a waiter that accidentally put too much oil in the onion... nearly singed the eyebrows off of everyone at our table! Too much fun.
Three cheers to that girl !! Here here !!!
I clicked over to your beach weekend post. Adore stuff like that. My favorite photo of all? The one of your arm with the bracelet. Why? I immediately saw your sweet freckles and thought it was MY arm! lol! Yep, I'm a freckly girl, too.
I've never been to a restaurant like that, but it looks to be so much fun! Girls' nights always are.
We love to take the kids to the Japanese restaurant, always so fun. Food on fire is a good thing.
...and now I'm thinking about how the last time we had hibachi, the chef lost his grip on his knife and it slid up on the table, right in front of my ribs. Like, the tip of the knife was touching my sweater. Close call!
I hope it tasted good!!! Celebrations are fun!
i love tokyo japanese steakhouse in federal way. so good.
just wondering if you have ever done a post about how and where you organize all of your homeschool books/supplies and where you do school. i would love to read it if you have and see a new one if you haven't!!
btw, your friend erin looks so familiar to me. i'm sure i know her from somewhere!!
Traditional English christmas pudding is covered in brandy and then set on fire! I think im going to have to come up with a less, um, fruit cake type pudding and post the recipe! Not a fan of traditional fruit cake! Maybe a Lemon sponge covered in Limoncello would look cool on fire?
Wowza...great shots girlie. Love the crazy flamin' food. Have no idea what in the world it was, but it looked like a great night!
hope you had a good day and that you have a wonderful evening!!
I LOVE your site! It is so fun to stop by and see what you are doing and your fabulous pictures!!
The pictures of your kiddos are precious!!
just found our blog and I LOVE it!
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{
"redpajama_set_name": "RedPajamaC4"
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The stack grows downward in memory. That is, the top of the stack has the lowest memory address. When a program is executed, the top of the process stack contains 1) the argument count (conventionally referred to as argc in source code), followed by 2) pointers to the argument strings, 3) zero, 4) pointers to the environment strings, 5) zero, and 6) additional data (including the data that the argument/environment pointers reference).
echo and printenv can be implemented in assembly language by traversing the stack and printing out the relevant strings.
Both assembly programs have a helper macro, print, for writing to standard output. They also share a helper function, strlen, which returns the length of a string.
The assembly code for echo iterates over the argument pointers on the stack, printing the string corresponding to each argument. The iteration starts at the second element on the stack (past the first element, argument count), and stops when reaching a zero.
The full source code for echo, including the print macro and strlen function, is available at https://gist.github.com/dstein64/890e02e8e277f17d931c8a250ceaaf44.
The assembly code for printenv is similar to the code above for echo, but starts iteration a few elements deeper into the stack, at the first environment variable pointer. It uses the argument count on the stack to jump past the argument pointers.
The full source code for printenv, including the print macro and strlen function, is available at https://gist.github.com/dstein64/a52146a3c6a12c8c0b84cfd4e084bb15.
This entry was tagged assembly, echo, linux, printenv, unix. Bookmark the permalink.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Q: Error connecting to 127.0.0.1:27017 :: caused by :: Connection refused mongodb I have 2 terminal windows, first to run the server, the second one to use as a client to connect to the db server.
I have successfully connected to my local mongoDB server.
Terminal window 1
bogdanmac:~ iliebogdanbarbulescu$
bogdanmac:~ iliebogdanbarbulescu$ brew services start mongodb-community@4.2
Service `mongodb-community` already started, use `brew services restart mongodb-community` to restart.
Unfortunately, I can not access the database from the client.
Terminal window 2
ilies-mbp:mongodb iliebogdanbarbulescu$ pwd
/usr/local/var/mongodb
ilies-mbp:mongodb iliebogdanbarbulescu$ ls
WiredTiger diagnostic.data
WiredTiger.lock index-1-454351138292104502.wt
WiredTiger.turtle index-3-454351138292104502.wt
WiredTiger.wt index-5-454351138292104502.wt
WiredTigerLAS.wt index-6-454351138292104502.wt
_mdb_catalog.wt journal
collection-0-454351138292104502.wt mongod.lock
collection-2-454351138292104502.wt sizeStorer.wt
collection-4-454351138292104502.wt storage.bson
ilies-mbp:mongodb iliebogdanbarbulescu$ mongo
MongoDB shell version v4.2.5
connecting to: mongodb://127.0.0.1:27017/?compressors=disabled&gssapiServiceName=mongodb
2020-04-16T17:13:02.269+0100 E QUERY [js] Error: couldn't connect to server 127.0.0.1:27017, connection attempt failed: SocketException: Error connecting to 127.0.0.1:27017 :: caused by :: Connection refused :
connect@src/mongo/shell/mongo.js:341:17
@(connect):2:6
2020-04-16T17:13:02.271+0100 F - [main] exception: connect failed
2020-04-16T17:13:02.271+0100 E - [main] exiting with code 1
ilies-mbp:mongodb iliebogdanbarbulescu$
Logs at /usr/local/var/log/mongodb/mongo.log
2020-04-16T16:54:03.370+0100 I CONTROL [main] ***** SERVER RESTARTED *****
2020-04-16T16:54:03.373+0100 I CONTROL [main] Automatically disabling TLS 1.0, to force-enable TLS 1.0 specify --sslDisabledProtocols 'none'
2020-04-16T16:54:03.385+0100 W ASIO [main] No TransportLayer configured during NetworkInterface startup
2020-04-16T16:54:03.387+0100 I CONTROL [initandlisten] MongoDB starting : pid=76453 port=27017 dbpath=/usr/local/var/mongodb 64-bit host=bogdanmac.mynet
2020-04-16T16:54:03.387+0100 I CONTROL [initandlisten] db version v4.2.5
2020-04-16T16:54:03.387+0100 I CONTROL [initandlisten] git version: 2261279b51ea13df08ae708ff278f0679c59dc32
2020-04-16T16:54:03.387+0100 I CONTROL [initandlisten] allocator: system
2020-04-16T16:54:03.387+0100 I CONTROL [initandlisten] modules: none
2020-04-16T16:54:03.387+0100 I CONTROL [initandlisten] build environment:
2020-04-16T16:54:03.387+0100 I CONTROL [initandlisten] distarch: x86_64
2020-04-16T16:54:03.387+0100 I CONTROL [initandlisten] target_arch: x86_64
2020-04-16T16:54:03.387+0100 I CONTROL [initandlisten] options: { config: "/usr/local/etc/mongod.conf", net: { bindIp: "127.0.0.1" }, storage: { dbPath: "/usr/local/var/mongodb" }, systemLog: { destination: "file", logAppend: true, path: "/usr/local/var/log/mongodb/mongo.log" } }
2020-04-16T16:54:03.387+0100 E NETWORK [initandlisten] Failed to unlink socket file /tmp/mongodb-27017.sock Permission denied
2020-04-16T16:54:03.387+0100 F - [initandlisten] Fatal Assertion 40486 at src/mongo/transport/transport_layer_asio.cpp 684
2020-04-16T16:54:03.387+0100 F - [initandlisten]
***aborting after fassert() failure
A: Thanks to @Abhishek kumar , after changing the permission on the .sockfile, I could start the client with the command mongo
*
*set the permissions of the .sock file to the current user:
sudo chown whoami /tmp/mongodb-27017.sock
A: So, we can conclude this post with the solution to check the .sock file permission and change the permission of the .sock file.
Change the permission of .sock file.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Roger Select is a wireless microphone that helps you to understand better in noise and over distance.
In what situations can Roger Select be used?
How do I use Roger Select?
If you are sitting at a table, simply place the Roger Select in the middle of the table. If you are in a group but you have no table, you can also hold Roger Select in your open palm. To hear a single conversation partner, attach the clip or the lanyard and ask your partner to wear Roger Select on the chest.
What can I do to improve understanding using Roger Select?
Always bring Roger Select as close to the speakers as possible.
When Roger Select is worn on the chest, then it should be worn in the center of the chest.
When Roger Select is on the table, you may tap the device to activate it only into the directions where a talker sits.
Avoid hiding Roger Select behind glasses, computer screens or other objects.
How far can I move away from the Roger Select and still hear the signal?
Typically, you can be up to 10 meters / 30 feet away from the Roger Select. In line of sight, which means you can see the Roger Select, you can even be up to 20 meters / 60 feet away.
How long is the operating time?
After a full charge, the Roger Select can transmit up to 8 hours.
What options do I have to hear phone calls better?
Roger Select can be connected to any device that is capable of doing phone calls such as cell phone, landline phones, computers, etc. The easiest way is to pair your Roger Select with the phone / computer using Bluetooth wireless technology. When you use your computer, you can make phone calls using VoIP services such as Skype.
When your phone does not have Bluetooth, there might be a solution using adaptor cables, please ask your hearing healthcare professional.
Does Roger Select work with my hearing aids?
Roger Select works with most hearing aids, cochlear implants and Bahas through a Roger receiver that is attached to your hearing aids or worn on the body. Your hearing care professional will help you finding the correct Roger receiver for you.
Is it possible to listen to an audio device and the microphone in parallel?
Yes, after plugging in the audio cable you can activate the microphone by tapping the center of the device.
Is Roger Select compatible to my phone / computer?
Most likely yes. Roger Select supports Bluetooth version 4.0. Ask the dealer of your phone or computer to clarify compatibility questions.
Can I turn off Roger Select while it still remains connected to my phone / computer?
Yes, a short press on the on/off button will put Roger Select into standby mode, this means the microphones are turned off but you are still connected with your phone / computer. A long press on the on/off button will switch off Roger Select completely, this means you are no longer connected with your phone / computer.
It is recommended to switch Roger Select off completely if no phone call is expected to reduce battery consumption.
Can I listen to music via Bluetooth?
Most phones / computers stream music using the A2DP profile which is not supported by Roger Select. Some phone settings or apps allow you to convert music into the headset or hands-free profile and in those cases you can listen to music via Roger Select. Please ask your phone dealer for options.
Another option is to use the short audio cable which can be connected to the headphone output of your audio device and Roger Select.
Why can Roger Select not be muted when worn?
The mute function is disabled while Roger Select is worn to prevent inadvertent muting during handling. In such situations it is advised to simply turn off Roger Select.
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{
"redpajama_set_name": "RedPajamaC4"
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| 7,322
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With more than 15,000 flight hours as captain on the Boeing 737 and the new Boeing 787, Captain Pat BOONE is launching a brand new system quiz app for these airplanes. The Question Bank mobile app comes with 1,000 technical questions covering all FCOM systems, as well as FCTM, QRH, Performance and the Exterior Inspection. Simply set-up your quiz profile in accordance with your company fleet (e.g. our fleet are B737-700 with winglets and carbon brakes) and the app will only display questions that meet the profile.
The Question Bank app is available for Boeing 737 (Classic, NG and Max), Boeing 787 and the Embraer E-170/190 series.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,528
|
Tarkio és una població dels Estats Units a l'estat de Missouri. Segons el cens del 2000 tenia una població de 1.935 habitants.
Demografia
Segons el cens del 2000, Tarkio tenia 1.935 habitants, 749 habitatges, i 468 famílies. La densitat de població era de 541,4 habitants per km².
Dels 749 habitatges en un 27% hi vivien nens de menys de 18 anys, en un 48,1% hi vivien parelles casades, en un 10% dones solteres, i en un 37,4% no eren unitats familiars. En el 33,6% dels habitatges hi vivien persones soles el 17,9% de les quals corresponia a persones de 65 anys o més que vivien soles. El nombre mitjà de persones vivint en cada habitatge era de 2,26 i el nombre mitjà de persones que vivien en cada família era de 2,87.
Per edats la població es repartia de la següent manera: un 29,1% tenia menys de 18 anys, un 7,5% entre 18 i 24, un 22,9% entre 25 i 44, un 20,8% de 45 a 60 i un 19,6% 65 anys o més.
L'edat mediana era de 38 anys. Per cada 100 dones de 18 o més anys hi havia 86,4 homes.
La renda mediana per habitatge era de 28.144 $ i la renda mediana per família de 34.625 $. Els homes tenien una renda mediana de 26.900 $ mentre que les dones 18.681 $. La renda per capita de la població era de 14.160 $. Entorn del 12,4% de les famílies i el 16,3% de la població estaven per davall del llindar de pobresa.
Poblacions més properes
El següent diagrama mostra les poblacions més properes.
Referències
Entitats de població del comtat d'Atchison
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,945
|
\section*{This is an unnumbered first-level section head}
\section{Introduction and main results}
The Floquet (Bloch) theory indicates that the spectrum of the Schr\"{o}dinger operator
\begin{equation}
\label{E:Schrodinger operator}
Lf:=-f''(x)+\nu(x)f(x), \ \ \ x\in\mathbb{R}
\end{equation}
with a smooth real-valued periodic potential $\nu(x)$ has a band gap structure. If we further assume that $\nu(x)$ is of periodic $\pi$ and set
\begin{equation*}
\nu(x)=-\sum_{n=1}^{\infty}\theta_{n}\cos(2nx)-\sum_{m=1}^{\infty}\phi_{m}\sin (2mx),
\end{equation*}
where $\theta_{n}$ and $\phi_{m}$ are real, then (\ref{E:Schrodinger operator}) can be written as:
\begin{equation}
Lf:=-f''(x)-\left[\sum_{n=1}^{\infty}\theta_{n}\cos(2nx)+\sum_{m=1}^{\infty}\phi_{m}\sin (2mx)\right]f(x).
\end{equation}
Moreover, there are \cite{Eastham} two monotonically increasing infinite sequence of real numbers
\begin{equation*}
\lambda_{0}^{+}, \ \lambda_{1}^{+}, \ \lambda_{2}^{+}, \cdots
\end{equation*}
and
\begin{equation*}
\lambda_{1}^{-}, \ \lambda_{2}^{-}, \ \lambda_{3}^{-}, \cdots
\end{equation*}
such that the Hill equation
\begin{equation}
Lf=\lambda f
\end{equation}
has a solution of period $\pi$ if and only if $\lambda=\lambda_{n}^{+}$, $n=0, 1, 2, \cdots$, and a solution of semi-period $\pi$ (i.e., $f(x+\pi)=-f(x)$) if and only if $\lambda=\lambda_{n}^{-}$, $n=1, 2, 3, \cdots$. The $\lambda_{n}^{+}$ and $\lambda_{n}^{-}$ satisfy the inequalities
\begin{equation*}
\lambda_{0}^{+}<\lambda_{1}^{-}\leq \lambda_{2}^{-}\leq \lambda_{1}^{+}\leq \lambda_{2}^{+}<\lambda_{3}^{-}\leq \lambda_{4}^{-}<\lambda_{3}^{+}\leq \lambda_{4}^{+}<\cdots
\end{equation*}
and the relations
\begin{equation*}
\lim_{n\rightarrow\infty}\lambda_{n}^{+}=\infty, \ \ \ \lim_{n\rightarrow\infty}\lambda_{n}^{-}=\infty.
\end{equation*}
Besides, $\gamma_{n}:=(\lambda_{n+1}^{-}-\lambda_{n}^{-})$ for odd $n$ and $\gamma_{n}:=(\lambda_{n}^{+}-\lambda_{n-1}^{+})$ for even $n$ are referred to as band gaps or instability intervals,
where $n\geq 1$.
It is well-known that there is an extensive theory for the Mathieu operator, where the potential $\nu(x)$ is a single trigonometric function, i.e.,
\begin{equation}
\nu(x)=-B \cos 2x.
\end{equation}
Ince \cite{Ince 3} proved that all instability intervals of the Mathieu operator are open, i.e., no closed gaps for the Mathieu operator. In 1963, Levy and Keller \cite{Levy} gave the asymptotics of $\gamma_{n}=\gamma_{n}(B)$, i.e., for fixed $n$ and real nonzero number $B$, when $B \to 0$,
\begin{equation}
\gamma_{n}=\frac{8}{[(n-1)!]^{2}}\cdot \left(\frac{B}{8}\right)^{n} (1+O(B)).
\end{equation}
18 years later, Harrell \cite{Harrell} gave the asymptotics of the band gaps of the Mathieu operator for fixed $B$ and $n\rightarrow\infty$, i.e.,
\begin{equation}
\gamma_{n}=\lambda_{n}^{+}-\lambda_{n}^{-}=\frac{8}{[(n-1)!]^{2}}\cdot \left(\frac{|B|}{8}\right)^{n}\left(1+O\left(\frac{1}{n^{2}}\right)\right).
\end{equation}
Compared with the Mathieu potential, the band gaps for the Whittaker-Hill potential
\begin{equation}
\label{E:Whittaker-Hill potential}
\nu(x)=-(B\cos 2x+C\cos 4x)
\end{equation}
may be open or closed.
Specifically, if $B=4\alpha t$ and $C=2\alpha^{2}$, for any real $\alpha$ and natural number $t$, it is already known that for odd $t=2m+1$, all the even gaps are closed except the first $m$, but no odd gap disappears; similarly, for even $t=2m$, except for the first $m$, all the odd gaps are closed, but even gaps remain open (see Theorem 11, \cite{Djakov 1} and Theorem 7.9, \cite{Maguns}).
In 2007, P. Djakov and B. Mityagin (see \cite{Djakov 2}) gave the asymptotics of the instability intervals for the above special Whittaker-Hill potential, namely, for real $B, C\neq 0$, $B=4 \alpha t$ and $C=2 \alpha^{2}$, they have the following results, where either both $\alpha$ and $t$ are real numbers if $C>0$ or both $\alpha$ and $t$ are pure imaginary numbers if $C<0$.
\begin{theorem}[\cite{Djakov 2}]
\label{T:Djakov 1}
Let $\gamma_{n}$ be the $n-$th band gap of the Whittaker-Hill operator
\begin{equation}
Lf=-f''-[4\alpha t \cos 2x+ 2\alpha^{2} \cos 4x]f,
\end{equation}
where either both $\alpha$ and $t$ are real, or both are pure imaginary numbers. If $t$ is fixed and $\alpha\rightarrow 0$, then for even $n$
\begin{equation}
\gamma_{n}=\left|\frac{8\alpha^{n}}{2^{n}[(n-1)!]^{2}}\prod_{k=1}^{n/2}(t^{2}-(2k-1)^{2})\right|(1+O(\alpha)),
\end{equation}
and for odd $n$
\begin{equation}
\gamma_{n}=\left|\frac{8\alpha^{n}t}{2^{n}[(n-1)!]^{2}}\prod_{k=1}^{(n-1)/2}(t^{2}-(2k)^{2})\right|(1+O(\alpha)).
\end{equation}
\end{theorem}
\begin{theorem}[\cite{Djakov 2}]
\label{T:Djakov 2}
Let $\gamma_{n}$ be the $n-$th band gap of the Whittaker-Hill operator
\begin{equation}
Lf=-f''-[4\alpha t \cos 2x+ 2\alpha^{2} \cos 4x]f,
\end{equation}
where either both $\alpha$ and $t\neq 0$ are real, or both are pure imaginary numbers. Then the following asymptotic formulae hold for fixed $\alpha$, $t$ and $n\rightarrow\infty$:
for even $n$
\begin{equation}
\gamma_{n}=\frac{8|\alpha|^{n}}{2^{n}[(n-2)!!]^{2}}\left|\cos\left(\frac{\pi}{2}t\right)\right|\left[1+O\left(\frac{\log n}{n}\right)\right],
\end{equation}
and for odd $n$
\begin{equation}
\gamma_{n}=\frac{8|\alpha|^{n}}{2^{n}[(n-2)!!]^{2}}\frac{2}{\pi}\left|\sin\left(\frac{\pi}{2}t\right)\right|\left[1+O\left(\frac{\log n}{n}\right)\right],
\end{equation}
where $(2m-1)!!=1\cdot3\cdots(2m-1)$, \ \ \ $(2m)!!=2\cdot4\cdots(2m)$.
\end{theorem}
In this paper, a more general Whittaker-Hill operator
\begin{equation}
L=-D^{2}+(bq^{m_{1}}\cos 2x+cq^{m_{2}}\cos 4x)
\end{equation}
is considered and the asymptotics of the instability intervals are derived, where $b$, $c$, $q$, $m_{1}$ and $m_{2}$ are real. Our theorems are stated as follows, in particular, we can deduce P. Djakov and B. Mityagin's results by choosing $m_{1}=1$, $m_{2}=2$, $b=-4\alpha t$ and $c=-2\alpha^{2}$.
\begin{theorem}
\label{T:1}
Let the Whittaker-Hill operator be
\begin{equation}
Ly=-y''+(bq^{m_{1}}\cos 2x+cq^{m_{2}}\cos 4x)y,
\end{equation}
where $b$, $c$ and $q$ are real. If $q\rightarrow 0$ and $m_{1}$, $m_{2}$ are positive real parameters, then one of the following results holds:
\begin{enumerate}
\item
when $m_{1}> \frac{m_{2}}{2}$,
\\
(i)
\begin{equation}
\gamma_{2m}=\left|\frac{32\cdot(\frac{c}{2})^{m}\cdot q^{m_{2}m}}{2^{4m}[(m-1)!]^{2}}\right|+O\Big(q^{m_{2}(m-\frac{1}{2})+m_{1}}\Big),
\end{equation}
\\
(ii)
\begin{equation}
\gamma_{1}= |bq^{m_{1}}|+O(q^{2m_{1}-\frac{m_{2}}{2}}), \ \ \ \gamma_{3}=\frac{|bcq^{m_{1}+m_{2}}|}{8}+O(q^{2m_{1}+\frac{m_{2}}{2}}),
\end{equation}
\begin{equation}
\begin{split}
&\gamma_{2m-1}= \Big|\left(\frac{c}{2}\right)^{m-1}\cdot b\cdot q^{m_{2}(m-1)+m_{1}}\cdot \frac{8}{2^{3m}}\cdot
\Big\{ \frac{1}{[(2m-3)!!]^{2}}\\
&\cdot \sum_{i=1}^{m-2} \frac{(2m-2i-3)!!\cdot (2i-1)!!}{i!\cdot (m-1-i)!} + \frac{2}{(2m-3)!!\cdot(m-1)!} \Big\}\Big|+O\Big(q^{m_{2}(m-\frac{3}{2})+2m_{1}}\Big)
\ \ \ \text{for} \ \ \ m\geq 3;
\end{split}
\end{equation}
\item
when $m_{1}< \frac{m_{2}}{2}$,
\begin{equation}
\gamma_{1}=\left|bq^{m_{1}}\right|+O(q^{m_{2}-m_{1}}), \ \ \ \gamma_{2}=\left|cq^{m_{2}}+\frac{b^{2}q^{2m_{1}}}{8}\right|+O(q^{m_{2}}),
\end{equation}
\begin{equation}
\gamma_{n}=\left|\frac{8\cdot b^{n}\cdot q^{m_{1}n}}{2^{3n}\cdot [(n-1)!]^{2}}\right|+O(q^{m_{1}n+m_{2}-2m_{1}})
\ \ \ \ \ \ \ \text{for} \ \ \ n\geq 3;
\end{equation}
\item
when $m_{1}= \frac{m_{2}}{2}$ and $c<0$,
\\
(i)
\begin{equation}
\gamma_{1}=\left|bq^{m_{1}}\right|+O(q^{3m_{1}}), \ \ \ \gamma_{2}=\left|cq^{m_{2}}+\frac{b^{2}q^{2m_{1}}}{8}\right|+O(q^{4m_{1}}),
\end{equation}
\\
(ii)
\begin{equation}
\gamma_{2m}=8\left|\frac{\prod_{k=1}^{m}\Big(\Big(\frac{b}{2}\Big)^{2}+8c\Big(k-\frac{1}{2}\Big)^{2}\Big)\cdot q^{2m_{1}\cdot m}}{2^{4m}\cdot[(2m-1)!]^{2}}\right|+O(q^{2m_{1}(m+1)}) \ \ \ \ \ \ \ \text{for} \ \ \ m\geq 2,
\end{equation}
\\
(iii)
\begin{equation}
\gamma_{2m-1}=32\left|\frac{\frac{b}{2}\prod_{k=1}^{m-1}\Big(\Big(\frac{b}{2}\Big)^{2}+8ck^{2}\Big)\cdot q^{m_{1}\cdot (2m-1)}}{2^{4m}\cdot[(2m-2)!]^{2}}\right|+O(q^{m_{1}(2m+1)}) \ \ \ \ \ \ \ \text{for} \ \ \ m\geq 2.
\end{equation}
\end{enumerate}
Here, $m$ is a positive integer and $\gamma_{n}$ is the n-th instability interval.
\end{theorem}
\begin{theorem}
\label{T:2}
Let the Whittaker-Hill operator be
\begin{equation}
Ly=-y''+(bq^{m_{1}}\cos 2x+cq^{2m_{1}}\cos 4x)y,
\end{equation}
where $b$, $q$ are real, $c<0$ and $m_{1}>0$. Then the following asymptotic formula holds for fixed $b$, $c$, $q$ and $n\rightarrow\infty$.
\begin{equation}
\gamma_{2m}=\frac{q^{2m_{1}\cdot m}\cdot |c|^{m}}{2^{3m-3}\cdot[(2m-2)!!]^{2}}\cdot \left|\cos\left(\frac{b\pi}{4\sqrt{-2c}}\right)\right|\cdot\left[1+O\left(\frac{\log m}{m}\right)\right],
\end{equation}
\begin{equation}
\gamma_{2m-1}=\frac{q^{m_{1}(2m-1)}\cdot |c|^{m-1}\cdot\sqrt{-2c}}{2^{3m-5}\cdot[(2m-3)!!]^{2}\cdot \pi}
\cdot \left|\sin\left(\frac{b\pi}{4\sqrt{-2c}}\right)\right|\cdot\left[1+O\left(\frac{\log m}{m}\right)\right].
\end{equation}
Here, $m$ is a positive integer and $\gamma_{n}$ is the n-th instability interval.
\end{theorem}
\section{Some lemmas}
\begin{lemma}[\cite{Djakov 2}]
\label{L:1}
Let the Schr\"{o}dinger operator
\begin{equation}
Ly=-y''+v(x)y
\end{equation}
be defined on $\mathbb{R}$, with a real-valued periodic $L^{2}([0, \pi])$-potential $v(x)$, where $v(x)=\sum_{m\in \mathbb{Z}} V(m)\exp(imx)$, $V(m)=0$ for odd $m$, then $\|v\|^{2}=\sum|V(m)|^{2}$.
(a) If $\|v\|<\frac{1}{9}$, then for each $n=1,2,\cdots$, there exists $z=z_{n}$ such that
~\\$|z|\leq 4\|v\|$, and
\begin{equation}
\label{E:length estimation}
2|\beta_{n}(z)|(1-3\|v\|^{2}/n^{2})\leq \gamma_{n}\leq 2|\beta_{n}(z)|(1+3\|v\|^{2}/n^{2}),
\end{equation}
where
\begin{equation}
\beta_{n}(z)=V(2n)+\sum_{k=1}^{\infty}\sum_{j_{1},\cdots,j_{k}\neq \pm n}\frac{V(n-j_{1})V(j_{1}-j_{2})\cdots V(j_{k-1}-j_{k})V(j_{k}+n)}
{(n^{2}-j_{1}^{2}+z)\cdots(n^{2}-j_{k}^{2}+z)}
\end{equation}
converges absolutely and uniformly for $|z|\leq 1$, and $\gamma_{n}$ is the n-th instability interval.
(b) If $V(0)=\frac{1}{\pi}\int_{0}^{\pi}v(x)dx=0$, then there is $N_{0}=N_{0}(v)$ such that (\ref{E:length estimation}) holds for $n\geq N_{0}$ with $z=z_{n}$, $|z_{n}|<1$.
\end{lemma}
\begin{lemma}[\cite{Volkmer}]
\label{L:Volkmer}
The Ince equation
\begin{equation}
\label{E:general Ince}
(1+a\cos 2t)y''(t)+b(\sin 2t)y'(t)+(c+d\cos 2t)y(t)=0
\end{equation}
can be transformed into the Whittaker-Hill equation by assuming
\begin{equation}
a=0,\ \ b=-4q,\ \ c=\lambda+2q^{2}, \ \ d=4(m-1)q.
\end{equation}
Moreover,
\begin{equation}
\label{E:semifinite band gap 1}
\mathrm{sign} (\alpha_{2n}-\beta_{2n})=\mathrm{sign} \ q^{2}\cdot \mathrm{sign} \prod_{p=-n}^{n-1}(2p-m+1)
\end{equation}
and
\begin{equation}
\label{E:semifinite band gap 2}
\mathrm{sign} (\alpha_{2n+1}-\beta_{2n+1})=\mathrm{sign} \ q \cdot \mathrm{sign} \prod_{p=-n}^{n-1}(2p-m),
\end{equation}
where $a$, $b$ and $d$ are real; $\alpha_{2n}$ and $\beta_{2n+2}$ are defined by the eigenvalues corresponding to non-trivial even and odd solutions with period $\pi$, respectively; and $\alpha_{2n+1}$ and $\beta_{2n+1}$ are defined by the eigenvalues corresponding to non-trivial even and odd solutions with semi-period $\pi$, respectively.
\end{lemma}
\begin{lemma}[\cite{Maguns}]
\label{L:Maguns}
The Whittaker-Hill equation
\begin{equation}
f''+[\lambda+4mq\cos 2x+2q^{2}\cos 4x]f=0
\end{equation}
can have two linearly independent solutions of period $\pi$ or $2\pi$ if and only if $m$ is an integer. If $m=2l$ is even, then the odd intervals of instability on the $\lambda$ axis disappear, with at most $|l|+1$ exceptions, but no even interval of instability disappears. If $m=2l+1$ is odd, then at most $|l|+1$ even intervals of instability remain but no interval of instability disappears.
\end{lemma}
\begin{lemma}
\label{L:2}
The Whittaker-Hill operator
\begin{equation}
L=-D^{2}-(B\cos 2x+C\cos 4x)
\end{equation}
admits all even gaps closed except the first $n$ when $\pm\frac{B}{4\sqrt{2C}}=-n-\frac{1}{2}$, $n\in\mathbb{Z}_{\geq 0}$; and all odd gaps closed except the first $n+1$ when $\pm\frac{B}{4\sqrt{2C}}=-n-1$, $n\in\mathbb{Z}_{\geq 0}$.
\end{lemma}
\begin{proof}
By Lemma \ref{L:Maguns}, we obtain that the Whittaker-Hill equation
\begin{equation}
\label{E:W-H-real}
f''(x)+(A+B\cos 2x+C\cos 4x)f(x)=0
\end{equation}
have two linearly independent solutions of period or semi-periodic $\pi$ if and only if $\frac{B}{2\sqrt{2C}}\in \mathbb{Z}$. Moreover, we transform (\ref{E:W-H-real})
into the Ince equation
\begin{equation}
g''(x)+4\sqrt{\frac{C}{2}}\sin 2x\cdot g'(x)+\left[(A+C)+\left(B+4\sqrt{\frac{C}{2}}\right)\cos 2x\right]g(x)=0.
\end{equation}
via $f(x)=e^{-\sqrt{\frac{C}{2}}\cos 2x}\cdot g(x)$. From Lemma \ref{L:Volkmer}, we can write the parameters $q$, $\lambda$ and $m$ of equation (\ref{E:general Ince}) in terms of $A$, $B$ and $C$, i.e.,
\begin{equation*}
q=-\sqrt{\frac{C}{2}}, \ \ \ \lambda=A, \ \ \ m=-\frac{B}{2\sqrt{2C}}.
\end{equation*}
(1) If $m=2n+1$, $n\in \mathbb{N}^{+}\cup\{0\}$, i.e., $\frac{B}{2\sqrt{2C}}=-2n-1$, and the solutions satisfy the periodic boundary conditions, then we deduce from Lemma \ref{L:Volkmer} that the first $2n+1$ eigenvalues are simple, and others are double.
(2) If $m=2n+2$, $n\in \mathbb{N}^{+}\cup\{0\}$, i.e., $\frac{B}{2\sqrt{2C}}=-2n-2$, and the solutions satisfy the semi-periodic boundary conditions, then we also deduce from Lemma \ref{L:Volkmer} that the first $2n+2$ eigenvalues are simple, and others are double.
Besides, we can also transform (\ref{E:W-H-real}) into the Ince equation
\begin{equation}
g''(x)-4\sqrt{\frac{C}{2}}\sin 2x\cdot g'(x)+\left[(A+C)+\left(B-4\sqrt{\frac{C}{2}}\right)\cos 2x\right]g(x)=0.
\end{equation}
via $f(x)=e^{\sqrt{\frac{C}{2}}\cos 2x}\cdot g(x)$. Thus, we have similar conclusions.
\end{proof}
In order to prove our results, let us consider all possible walks from $-n$ to $n$. Each such walk is determined by the sequence of its steps
\begin{equation}
x=(x_{1}, \cdots, x_{\nu+1}),
\end{equation}
or by its vertices
\begin{equation}
j_{s}=-n+\sum_{k=1}^{s}x_{k}, \ \ \ s=1, \cdots, \nu.
\end{equation}
The relationship between steps and vertices are given by the formula
\begin{equation}
x_{1}=n+j_{1};\ \ \ x_{k}=j_{k}-j_{k-1}, \ k=2, \cdots, \nu; \ \ \ x_{\nu+1}=n-j_{\nu}.
\end{equation}
\begin{definition}
\label{D:1}
Let $X$ denote the set of all walks from $-n$ to $n$ with steps $\pm 2$ or $\pm 4$. For each $x=(x_{s})_{s=1}^{\nu+1}\in X$ and each $z\in \mathbb{R}$, we define
\begin{equation}
B_{n}(x,z)=\frac{V(x_{1})\cdots V(x_{\nu+1})}{(n^{2}-j_{1}^{2}+z)\cdots(n^{2}-j_{\nu}^{2}+z)}.
\end{equation}
\end{definition}
\begin{definition}
\label{D:2}
Let $X^{+}$ denote the set of all walks from $-n$ to $n$ with positive steps equal to $2$ or $4$. For each $\xi\in X^{+}$, let $X_{\xi}$ denote the set of all walks $x\in X\backslash X^{+}$ such that each vertex of $\xi$ is a vertex of $x$ also. For each $\xi\in X^{+}$ and $\mu\in\mathbb{N}$, let $X_{\xi,\mu}$ be the set of all $x\in X_{\xi}$ such that $x$ has $\mu$ more vertices than $\xi$. Moreover, for each $\mu-$tuple $(i_{1}, \cdots, i_{\mu})$ of integers in $I_{n}=(n+2\mathbb{Z})\setminus \{\pm n\}$, we define $X_{\xi}(i_{1}, \cdots, i_{\mu})$ as the set of all walks $x$ with $\nu+1+\mu$ steps such that $(i_{1}, \cdots, i_{\mu})$ and the sequence of the vertices of $\xi$ are complementary subsequences of the sequence of the vertices of $x$.
\end{definition}
From Definition \ref{D:2}, we deduce
\begin{equation}
X_{\xi,\mu}=\bigcup_{(i_{1}, \cdots, i_{\mu})\in (I_{n})^{\mu}}X_{\xi}(i_{1}, \cdots, i_{\mu}).
\end{equation}
\begin{lemma}[\cite{Djakov 2}]
\label{L:3}
If $\xi\in X^{+}$ and $n\geq 3$, then for $z\in[0,1)$
\begin{equation}
1-z\frac{\log n}{n}\leq \frac{B_{n}(\xi,z)}{B_{n}(\xi,0)}\leq 1-z\frac{\log n}{4n},
\end{equation}
and for $z\in(-1,0]$
\begin{equation}
1+|z|\frac{\log n}{2n}\leq \frac{B_{n}(\xi,z)}{B_{n}(\xi,0)}\leq 1+|z|\frac{2\log n}{n}.
\end{equation}
\end{lemma}
\begin{lemma}[\cite{Djakov 2}]
\label{L:4}
For each walk $\xi\in X^{+}$ and each $\mu-$tuple $(i_{1}, \cdots, i_{\mu})\in (I_{n})^{\mu}$,
\begin{equation}
\sharp X_{\xi}(i_{1}, \cdots, i_{\mu})\leq 5^{\mu}.
\end{equation}
\end{lemma}
\begin{lemma}
\label{L:5}
If $\xi\in X^{+}$ and $|z|\leq 1$, then there exists $n_{1}$ such that for $n\geq n_{1}$,
\begin{equation}
\sum_{x\in X_{\xi}}|B_{n}(x,z)|\leq |B_{n}(\xi,z)|\cdot \frac{K \log n}{n},
\end{equation}
where $K=40 \left(|\frac{b}{2}q^{m_{1}}|+|\frac{c}{2}q^{m_{2}}|+|\frac{b^{2}}{2c}q^{2m_{1}-m_{2}}|\right)$.
\end{lemma}
\begin{proof}
By Definition \ref{D:2}, we have
\begin{equation}
\sum_{x\in X_{\xi}}|B_{n}(x,z)|=\sum_{\mu=1}^{\infty}\sum_{x\in X_{\xi,\mu}}|B_{n}(x,z)|.
\end{equation}
Moreover,
\begin{equation}
\sum_{x\in X_{\xi,\mu}}|B_{n}(x,z)|\leq \sum_{(i_{1}, \cdots, i_{\mu})}\sum_{X_{\xi}(i_{1}, \cdots, i_{\mu})}|B_{n}(x,z)|,
\end{equation}
where the first sum on the right is taken over all $\mu-$tuples $(i_{1}, \cdots, i_{\mu})$ of integers $i_{s}\in n+2\mathbb{Z}$ such that $i_{s}\neq \pm n$.
Fix $(i_{1}, \cdots, i_{\mu})$, if $x\in X_{\xi}(i_{1}, \cdots, i_{\mu})$, then
\begin{equation}
\frac{B_{n}(x,z)}{B_{n}(\xi,z)}=\frac{\prod_{k}V(x_{k})}{\prod_{s}V(\xi_{s})}\cdot \frac{1}{(n^{2}-i_{1}^{2}+z)\cdots(n^{2}-i_{\mu}^{2}+z)}.
\end{equation}
Note that $V(\pm 2)=\frac{b}{2}q^{m_{1}}$ and $V(\pm4)=\frac{c}{2}q^{m_{2}}$.
If each step of $\xi$ is a step of $x$, then
\begin{equation}
\frac{\prod_{k}\left|V(x_{k})\right|}{\prod_{s}\left|V(\xi_{s})\right|}\leq C^{\mu},
\end{equation}
where $C:=|\frac{b}{2}q^{m_{1}}|+|\frac{c}{2}q^{m_{2}}|+|\frac{b^{2}}{2c}q^{2m_{1}-m_{2}}|$. For the general case, let $(j_{s})_{s=1}^{\nu}$ be the vertices of $\xi$, and let us put for convenience $j_{0}=-n$ and $j_{\nu+1}=n$. Since each vertex of $\xi$ is a vertex of $x$, for each $s$, $1\leq s\leq \nu+1$,
\begin{equation}
\xi_{s}=j_{s}-j_{s-1}=\sum_{k\in J_{s}}x_{k},
\end{equation}
where $x_{k}$, $k\in J_{s}$, are the steps of $x$ between the vertices $j_{s-1}$ and $j_{s}$. Fix an $s$, $1\leq s \leq \nu+1$. If $\xi_{s}=2$, then there is a step $x_{k^{*}}$, $k^{*}\in J_{s}$ such that $|x_{k^{*}}|=2$, otherwise, $\xi_{s}$ would be a multiple of $4$. Hence, $|V(\xi_{s})|=|V(x_{k}^{*})|$, which implies that
\begin{equation}
\frac{\prod_{J_{s}}|V(x_{k})|}{|V(\xi_{s})|}\leq C^{b_{s}-1},
\end{equation}
where $b_{s}$ is the cardinality of $J_{s}$.
If $\xi_{s}=4$, there are two possibilities. (1) If there is $k_{*}\in J_{s}$ with $|x_{k_{*}}|=4$, then $|V(\xi_{s})|=|V(x_{k_{*}})|$, so the above inequality holds. (2) There are $k', k''\in J_{s}$ such that $|x_{k'}|=|x_{k''}|=2$, hence,
\begin{equation}
\frac{|V(x_{k'})V(x_{k''})|}{|V(\xi_{s})|}=\left|\frac{b^{2}}{2c}q^{2m_{1}-m_{2}}\right|\leq C,
\end{equation}
so the above inequality also holds. Note that
\begin{equation}
\sum_{s}(b_{s}-1)=\mu,
\end{equation}
we get
\begin{equation}
\frac{\prod_{k}V(x_{k})}{\prod_{s}V(\xi_{s})}\leq C^{\mu}
\end{equation}
holds for the general case.
By
\begin{equation}
\frac{1}{|n^{2}-i^{2}+z|}\leq \frac{2}{|n^{2}-i^{2}|},
\end{equation}
where $i\neq \pm n$, $|z|\leq 1$, we have
\begin{equation}
\frac{|B_{n}(x,z)|}{|B_{n}(\xi,z)|}\leq \frac{(2C)^{\mu}}{|n^{2}-i_{1}^{2}|\cdots |n^{2}-i_{\mu}^{2}|}, \ \ \ x\in X_{\xi}(i_{1}, \cdots, i_{\mu}).
\end{equation}
By Lemma \ref{L:4}, we derive
\begin{equation}
\sum_{x\in X_{\xi}(i_{1}, \cdots, i_{\mu})}\frac{|B_{n}(x,z)|}{|B_{n}(\xi,z)|}\leq \frac{(10C)^{\mu}}{|n^{2}-i_{1}^{2}|\cdots |n^{2}-i_{\mu}^{2}|}.
\end{equation}
Combining Lemma \ref{L:3}, it yields
\begin{equation}
\begin{split}
\sum_{x\in X_{\xi,\mu}}\frac{|B_{n}(x,z)|}{|B_{n}(\xi,z)|}&\leq \sum_{(i_{1}, \cdots, i_{\mu})}\frac{(10C)^{\mu}}{|n^{2}-i_{1}^{2}|\cdots |n^{2}-i_{\mu}^{2}|}\leq\left(\sum_{i\in(n+2\mathbb{Z})\setminus \{\pm n\}} \frac{10C}{|n^{2}-i^{2}|}\right)^{\mu}\\
&\leq (10C)^{\mu}\left(\frac{1+\log n}{n}\right)^{\mu}\leq \left(\frac{20C \log n}{n}\right)^{\mu}.
\end{split}
\end{equation}
Thus,
\begin{equation}
\sum_{x\in X_{\xi,\mu}} |B_{n}(x,z)|\leq |B_{n}(\xi,z)|\cdot \left(\frac{20C \log n}{n}\right)^{\mu}.
\end{equation}
Hence,
\begin{equation}
\sum_{x\in X_{\xi}}\frac{|B_{n}(x,z)|}{|B_{n}(\xi,z)|}\leq \sum_{\mu=1}^{\infty}\left(\frac{20C \log n}{n}\right)^{\mu}.
\end{equation}
We can choose $n_{1}\in \mathbb{N}^{+}$ such that $\frac{20C \log n}{n}\leq \frac{1}{2}$ for $n\geq n_{1}$. Then
\begin{equation}
\sum_{x\in X_{\xi}}\frac{|B_{n}(x,z)|}{|B_{n}(\xi,z)|}\leq \frac{40C \log n}{n}.
\end{equation}
Therefore, there exists $n_{1}$ such that for $n\geq n_{1}$,
\begin{equation}
\sum_{x\in X_{\xi}}|B_{n}(x,z)|\leq |B_{n}(\xi,z)|\cdot \frac{K \log n}{n},
\end{equation}
where $K:=40C=40 \left(|\frac{b}{2}q^{m_{1}}|+|\frac{c}{2}q^{m_{2}}|+|\frac{b^{2}}{2c}q^{2m_{1}-m_{2}}|\right)$.
\end{proof}
\section{Proof of Theorem \ref{T:1}}
Note that
\begin{equation}
V(\pm 2)=\frac{bq^{m_{1}}}{2}, \ \ \ V(\pm 4)=\frac{cq^{m_{2}}}{2}
\end{equation}
and
\begin{equation}
\|v\|^{2}=\frac{1}{2}\Big(b^{2}q^{2m_{1}}+c^{2}q^{2m_{2}}\Big),
\end{equation}
by Lemma \ref{L:1}, we have
\begin{equation}
\gamma_{n}=\pm2\Big(V(2n)+\sum_{k=1}^{\infty} \beta_{k}(n,z)\Big)\Big(1+O(q^{2\cdot\min\{m_{1},m_{2}\}})\Big),
\end{equation}
where
\begin{equation}
\label{E:belta}
\begin{split}
\beta_{k}(n,z)&=\sum_{j_{1},\cdots,j_{k}\neq \pm n}\frac{V(n-j_{1})V(j_{1}-j_{2})\cdots V(j_{k-1}-j_{k})V(j_{k}+n)}
{(n^{2}-j_{1}^{2}+z)\cdots(n^{2}-j_{k}^{2}+z)}\\
&=\sum_{j_{1},\cdots,j_{k}\neq \pm n}\frac{V(n+j_{1})V(j_{2}-j_{1})\cdots V(j_{k}-j_{k-1})V(n-j_{k})}
{(n^{2}-j_{1}^{2}+z)\cdots(n^{2}-j_{k}^{2}+z)}
\end{split}
\end{equation}
and $z=O(q)$. Moreover, all series converge absolutely and uniformly for sufficiently small $q$.
Note that
\begin{equation}
(n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})=2n,
\end{equation}
and
\begin{equation}
\frac{V(n+j_{1})V(j_{2}-j_{1})\cdots V(j_{k}-j_{k-1})V(n-j_{k})}
{(n^{2}-j_{1}^{2}+z)\cdots(n^{2}-j_{k}^{2}+z)}\neq 0
\end{equation}
when
\begin{equation}
(n+j_{1}),(j_{2}-j_{1}),\cdots,(j_{k}-j_{k-1}),(n-j_{k})\in \{\pm2, \pm4\}.
\end{equation}
We distinguish three cases to discuss.
{\noindent\bf Case 1.} If $m_{1}> \frac{m_{2}}{2}$, then
\begin{equation}
V(n+j_{1})V(j_{2}-j_{1})\cdots V(j_{k}-j_{k-1})V(n-j_{k})
\end{equation}
is a monomial in $q$ of degree at least
\begin{equation}
\frac{m_{2}}{4}\cdot\Big[|n+j_{1}|+|j_{2}-j_{1}|+\cdots+|j_{k}-j_{k-1}|+|n-j_{k}|\Big].
\end{equation}
The minimum case occurs when
\begin{equation}
(n+j_{1}),(j_{2}-j_{1}),\cdots,(j_{k}-j_{k-1}),(n-j_{k})\in \{\pm4\},
\end{equation}
then
\begin{equation}
(n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})\in 4\mathbb{Z},
\end{equation}
while
\begin{equation}
(n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})=2n.
\end{equation}
If $n$ is even, i.e., $n=2m$, $m\in\mathbb{Z}_{>0}$, since
\begin{equation}
\begin{split}
&|n+j_{1}|+|j_{2}-j_{1}|+\cdots+|j_{k}-j_{k-1}|+|n-j_{k}|\\
&\geq (n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})\\
&=2n=4m,
\end{split}
\end{equation}
we obtain
\begin{equation}
V(n+j_{1})V(j_{2}-j_{1})\cdots V(j_{k}-j_{k-1})V(n-j_{k})
\end{equation}
is a monomial in $q$ of degree at least $m_{2}\cdot m$. Such monomial in $q$ of degree $m_{2}\cdot m$ corresponds to a walk from $-n$ to $n$ with vertices $j_{1}, j_{2}, \cdots, j_{k}\neq \pm n$ and positive steps of length $4$. Thus,
\begin{equation}
\gamma_{2m}=\pm P_{2m}(t)q^{m_{2}m}+O\Big(q^{m_{2}(m-\frac{1}{2})+m_{1}}\Big),
\end{equation}
where
\begin{equation}
P_{2m}(t)q^{m_{2}m}=2\Big(V(4m)+\sum_{k=1}^{\infty} \beta_{k}(2m,z)\Big).
\end{equation}
We have
\begin{equation}
P_{2}(t)q^{m_{2}}=2 V(4)=c q^{m_{2}}
\end{equation}
and
\begin{equation}
\begin{split}
&P_{2m}(t)q^{m_{2}m}=2 \sum_{k=1}^{\infty} \beta_{k}(2m,z)\\
&=2 \cdot \Big(\frac{c}{2}\Big)^{m} \cdot q^{m_{2}m}\cdot \prod_{j=1}^{m-1}\Big((4m^{2}-(-2m+4j)^{2})\Big)^{-1}\\
&=\frac{32\cdot(\frac{c}{2})^{m}\cdot q^{m_{2}m}}{2^{4m}[(m-1)!]^{2}}
\end{split}
\end{equation}
for $m\geq 2$. Therefore,
\begin{equation}
\gamma_{2m}=\left|\frac{32\cdot(\frac{c}{2})^{m}\cdot q^{m_{2}m}}{2^{4m}[(m-1)!]^{2}}\right|+O\Big(q^{m_{2}(m-\frac{1}{2})+m_{1}}\Big).
\end{equation}
If $n$ is odd, i.e., $n=2m-1$, $m\in\mathbb{Z}_{>0}$, since
\begin{equation}
(n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})=2n=4m-2,
\end{equation}
then
\begin{equation}
V(n+j_{1})V(j_{2}-j_{1})\cdots V(j_{k}-j_{k-1})V(n-j_{k})
\end{equation}
is a monomial in $q$ of degree at least
\begin{equation}
\begin{split}
&\frac{m_{2}}{4}\cdot\Big[|n+j_{1}|+|j_{2}-j_{1}|+\cdots+|j_{k}-j_{k-1}|+|n-j_{k}|\Big]+m_{1}-\frac{m_{2}}{2}\\
&\geq \frac{m_{2}}{4}\cdot(4m-2)+m_{1}-\frac{m_{2}}{2}\\
&=m_{2}(m-1)+m_{1}.
\end{split}
\end{equation}
Such monomial in $q$ of degree $m_{2}(m-1)+m_{1}$ corresponds to a walk from $-n$ to $n$ with vertices $j_{1}, j_{2}, \cdots, j_{k}\neq \pm n$ and positive steps. Specifically, except for one step with length $2$, the others are of length $4$. Thus,
\begin{equation}
\gamma_{2m-1}=\pm P_{2m-1}(t)q^{m_{2}(m-1)+m_{1}}+O\Big(q^{m_{2}(m-\frac{3}{2})+2m_{1}}\Big),
\end{equation}
where
\begin{equation}
P_{2m-1}(t)q^{m_{2}(m-1)+m_{1}}=2\Big(V(4m-2)+\sum_{k=1}^{\infty} \beta_{k}(2m-1,z)\Big).
\end{equation}
We obtain
\begin{equation}
P_{1}(t) q^{m_{1}}=2 V(2)=bq^{m_{1}},
\end{equation}
\begin{equation}
P_{3}(t) q^{m_{1}+m_{2}}=2 \sum_{k=1}^{\infty} \beta_{k}(3,z)=2 \left(\frac{bq^{m_{1}}}{2}\right)\left(\frac{cq^{m_{2}}}{2}\right)\left(\frac{1}{3^{2}-1^{2}}+\frac{1}{3^{2}-1^{2}}\right)
=\frac{bcq^{m_{1}+m_{2}}}{8},
\end{equation}
\begin{equation}
\begin{split}
&P_{5}(t) q^{m_{1}+2m_{2}}=2 \sum_{k=1}^{\infty} \beta_{k}(5,z)\\
&=2 \left(\frac{bq^{m_{1}}}{2}\right)\left(\frac{cq^{m_{2}}}{2}\right)^{2}
\left[\frac{1}{(5^{2}-3^{2})(5^{2}-1^{2})}+\frac{1}{(5^{2}-1^{2})(5^{2}-1^{2})}+\frac{1}{(5^{2}-3^{2})(5^{2}-1^{2})}\right]\\
&=\frac{bc^{2}q^{m_{1}+2m_{2}}}{3^{2}\cdot 2^{6}},
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&P_{2m-1}(t)q^{m_{2}(m-1)+m_{1}}=2 \sum_{k=1}^{\infty} \beta_{k}(2m-1,z)\\
&=2 \left(\frac{c}{2}\right)^{m-1}\cdot \left(\frac{b}{2}\right)\cdot q^{m_{2}(m-1)+m_{1}}\cdot
\Big\{\sum_{i=1}^{m-2} \prod_{j=1}^{i}\Big[(2m-1)^{2}-(-2m+1+4j)^{2}\Big]^{-1}\\
&\cdot \prod_{j=i}^{m-2}\Big[(2m-1)^{2}-(-2m+3+4j)^{2}\Big]^{-1}+\prod_{j=0}^{m-2}\Big[(2m-1)^{2}-(-2m+3+4j)^{2}\Big]^{-1}\\
&+\prod_{j=1}^{m-1}\Big[(2m-1)^{2}-(-2m+1+4j)^{2}\Big]^{-1}\Big\}\\
&=2 \left(\frac{c}{2}\right)^{m-1}\cdot \left(\frac{b}{2}\right)\cdot q^{m_{2}(m-1)+m_{1}}\cdot \frac{8}{2^{3m}}\cdot
\Big\{ \frac{1}{[(2m-3)!!]^{2}}\cdot \sum_{i=1}^{m-2} \frac{(2m-2i-3)!!\cdot (2i-1)!!}{i!\cdot (m-1-i)!} \\
&+ \frac{2}{(2m-3)!!\cdot(m-1)!} \Big\}.
\end{split}
\end{equation}
Hence,
\begin{equation}
\gamma_{1}= |bq^{m_{1}}|+O(q^{2m_{1}-\frac{m_{2}}{2}}), \ \ \ \gamma_{3}=\frac{|bcq^{m_{1}+m_{2}}|}{8}+O(q^{2m_{1}+\frac{m_{2}}{2}}),
\end{equation}
\begin{equation}
\begin{split}
&\gamma_{2m-1}= \Big|\left(\frac{c}{2}\right)^{m-1}\cdot b\cdot q^{m_{2}(m-1)+m_{1}}\cdot \frac{8}{2^{3m}}\cdot
\Big\{ \frac{1}{[(2m-3)!!]^{2}}\\
& \cdot \sum_{i=1}^{m-2} \frac{(2m-2i-3)!!\cdot (2i-1)!!}{i!\cdot (m-1-i)!}
+ \frac{2}{(2m-3)!!\cdot(m-1)!} \Big\}\Big|+O\Big(q^{m_{2}(m-\frac{3}{2})+2m_{1}}\Big)
\end{split}
\end{equation}
for $m\geq 3$.
{\noindent\bf Case 2.} If $m_{1}< \frac{m_{2}}{2}$, then
\begin{equation}
V(n+j_{1})V(j_{2}-j_{1})\cdots V(j_{k}-j_{k-1})V(n-j_{k})
\end{equation}
is a monomial in $q$ of degree at least
\begin{equation}
\frac{m_{1}}{2}\cdot\Big[|n+j_{1}|+|j_{2}-j_{1}|+\cdots+|j_{k}-j_{k-1}|+|n-j_{k}|\Big].
\end{equation}
The minimum case occurs when
\begin{equation}
(n+j_{1}),(j_{2}-j_{1}),\cdots,(j_{k}-j_{k-1}),(n-j_{k})\in \{\pm2\},
\end{equation}
then
\begin{equation}
(n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})\in 2\mathbb{Z},
\end{equation}
while
\begin{equation}
(n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})=2n.
\end{equation}
Since
\begin{equation}
\begin{split}
&|n+j_{1}|+|j_{2}-j_{1}|+\cdots+|j_{k}-j_{k-1}|+|n-j_{k}|\\
&\geq (n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})\\
&=2n,
\end{split}
\end{equation}
we have
\begin{equation}
V(n+j_{1})V(j_{2}-j_{1})\cdots V(j_{k}-j_{k-1})V(n-j_{k})
\end{equation}
is a monomial in $q$ of degree at least $m_{1}\cdot n$. Such monomial in $q$ of degree $m_{1}\cdot n$ corresponds to a walk from $-n$ to $n$ with vertices $j_{1}, j_{2}, \cdots, j_{k}\neq \pm n$ and positive steps of length $2$. Thus,
\begin{equation}
\gamma_{n}=\pm P_{n}(t) q^{m_{1}n}+O(q^{m_{1}n+m_{2}-2m_{1}}),
\end{equation}
where
\begin{equation}
P_{n}(t) q^{m_{1}n}=2\Big(V(2n)+\sum_{k=1}^{\infty} \beta_{k}(n,z)\Big).
\end{equation}
We deduce
\begin{equation}
P_{1}(t)q^{m_{1}}=2V(2)=bq^{m_{1}}, \ \ \ P_{2}(t)q^{2m_{1}}=2 \left(V(4)+\frac{\Big(\frac{bq^{m_{1}}}{2}\Big)^{2}}{2^{2}}\right)=cq^{m_{2}}+\frac{b^{2}q^{2m_{1}}}{8},
\end{equation}
\begin{equation}
\begin{split}
&P_{n}(t) q^{m_{1}n}=2\sum_{k=1}^{\infty} \beta_{k}(n,z)\\
&=2\cdot\Big(\frac{b}{2}\Big)^{n}\cdot q^{m_{1}\cdot n}\cdot \prod_{j=1}^{n-1}(n^{2}-(-n+2j)^{2})^{-1}\\
&=2\cdot \Big(\frac{b}{2}\Big)^{n}\cdot q^{m_{1}\cdot n}\cdot \prod_{j=1}^{n-1}(n^{2}-(-n+2j)^{2})^{-1}\\
&=2\cdot \frac{(\frac{b}{2})^{n}\cdot q^{m_{1}\cdot n}}{4^{n-1}\cdot[(n-1)!]^{2}}\\
&=\frac{8\cdot b^{n}\cdot q^{m_{1}n}}{2^{3n}\cdot [(n-1)!]^{2}}
\end{split}
\end{equation}
for $n\geq 3$. Therefore,
\begin{equation}
\gamma_{1}=\left|bq^{m_{1}}\right|+O(q^{m_{2}-m_{1}}), \ \ \ \gamma_{2}=\left|cq^{m_{2}}+\frac{b^{2}q^{2m_{1}}}{8}\right|+O(q^{m_{2}}),
\end{equation}
\begin{equation}
\gamma_{n}=\left|\frac{8\cdot b^{n}\cdot q^{m_{1}n}}{2^{3n}\cdot [(n-1)!]^{2}}\right|+O(q^{m_{1}n+m_{2}-2m_{1}})
\end{equation}
for $n\geq 3$.
{\noindent\bf Case 3.} If $m_{1}= \frac{m_{2}}{2}$, then
\begin{equation}
V(n+j_{1})V(j_{2}-j_{1})\cdots V(j_{k}-j_{k-1})V(n-j_{k})
\end{equation}
is a monomial in $q$ of degree
\begin{equation}
\frac{m_{1}}{2}\cdot\Big[|n+j_{1}|+|j_{2}-j_{1}|+\cdots+|j_{k}-j_{k-1}|+|n-j_{k}|\Big].
\end{equation}
Since
\begin{equation}
\begin{split}
&|n+j_{1}|+|j_{2}-j_{1}|+\cdots+|j_{k}-j_{k-1}|+|n-j_{k}|\\
&\geq (n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})\\
&=2n,
\end{split}
\end{equation}
we have
\begin{equation}
V(n+j_{1})V(j_{2}-j_{1})\cdots V(j_{k}-j_{k-1})V(n-j_{k})
\end{equation}
is a monomial in $q$ of degree at least $m_{1}\cdot n$, and each such monomial of degree $m_{1}\cdot n$ corresponds to a walk from $-n$ to $n$ with vertices $j_{1}, j_{2}, \cdots, j_{k}\neq \pm n$ and positive steps of length $2$ or $4$. The minimum case occurs when $n+j_{1}$, $j_{2}-j_{1}$, $\cdots$, $j_{k}-j_{k-1}$ and $n-j_{k}$ are of the same sign,
while the second smallest degree is for one step of length $2$ with opposite sign. Thus,
\begin{equation}
\gamma_{n}=\pm P_{n}(t) q^{m_{1}n}+O(q^{m_{1}(n+2)}),
\end{equation}
where
\begin{equation}
P_{n}(t) q^{m_{1}n}=2\Big(V(2n)+\sum_{k=1}^{\infty} \beta_{k}(n,z)\Big).
\end{equation}
We obtain
\begin{equation}
P_{1}(t)q^{m_{1}}=2V(2)=bq^{m_{1}}, \ \ \ P_{2}(t)q^{2m_{1}}=2 \left(V(4)+\frac{\Big(\frac{bq^{m_{1}}}{2}\Big)^{2}}{2^{2}}\right)=cq^{m_{2}}+\frac{b^{2}q^{2m_{1}}}{8},
\end{equation}
\begin{equation}
\begin{split}
&P_{n}(t) q^{m_{1}n}=2\sum_{k=1}^{\infty} \beta_{k}(n,z)\\
&=2\cdot P_{n}\Big(\frac{b}{2}\Big)\cdot q^{m_{1}\cdot n}\cdot \prod_{j=1}^{n-1}(n^{2}-(-n+2j)^{2})^{-1}\\
&=2\cdot P_{n}\Big(\frac{b}{2}\Big)\cdot q^{m_{1}\cdot n}\cdot \prod_{j=1}^{n-1}(n^{2}-(-n+2j)^{2})^{-1}\\
&=8\cdot \frac{P_{n}\big(\frac{b}{2}\big)\cdot q^{m_{1}\cdot n}}{2^{2n}\cdot[(n-1)!]^{2}}
\end{split}
\end{equation}
for $n\geq 3$. Therefore,
\begin{equation}
\gamma_{1}=\left|bq^{m_{1}}\right|+O(q^{3m_{1}}), \ \ \ \gamma_{2}=\left|cq^{m_{2}}+\frac{b^{2}q^{2m_{1}}}{8}\right|+O(q^{4m_{1}}),
\end{equation}
\begin{equation}
\gamma_{n}=\left|8\cdot \frac{P_{n}\big(\frac{b}{2}\big)\cdot q^{m_{1}\cdot n}}{2^{2n}\cdot[(n-1)!]^{2}}\right|+O(q^{m_{1}(n+2)})
\end{equation}
for $n\geq 3$, where $P_{n}\big(\frac{b}{2}\big)$ is a polynomial of $\frac{b}{2}$ with degree $n$ and leading coefficient $1$.
Specifically, if $n$ is even, i.e., $n=2m$, $m\in\mathbb{Z}_{>0}$, then
\begin{equation}
(n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})=4m,
\end{equation}
which implies that each walk from $-2m$ to $2m$ has even number of steps with length $2$. We have
\begin{equation}
P_{2m}\Big(\frac{b}{2}\Big)=\prod_{k=1}^{m}\Big(\Big(\frac{b}{2}\Big)^{2}-x_{k}\Big),
\end{equation}
where $x_{k}$, $k=1,\cdots, m$, depend on $m$. By Lemma \ref{L:2}, we obtain all even gaps closed except the first $k$ if $\big(\frac{b}{2}\big)=-8c\big(k+\frac{1}{2}\big)^{2}$, which yields
\begin{equation}
P_{2m}\Big(\frac{b}{2}\Big)=\prod_{k=1}^{m}\Big(\Big(\frac{b}{2}\Big)^{2}+8c\Big(k-\frac{1}{2}\Big)^{2}\Big).
\end{equation}
Hence,
\begin{equation}
\gamma_{2m}=8\left|\frac{\prod_{k=1}^{m}\Big(\Big(\frac{b}{2}\Big)^{2}+8c\Big(k-\frac{1}{2}\Big)^{2}\Big)\cdot q^{2m_{1}\cdot m}}{2^{4m}\cdot[(2m-1)!]^{2}}\right|+O(q^{2m_{1}(m+1)})
\end{equation}
for $m\geq 2$. If $n$ is odd, i.e., $n=2m-1$, $m\in\mathbb{Z}_{>0}$, then
\begin{equation}
(n+j_{1})+(j_{2}-j_{1})+\cdots+(j_{k}-j_{k-1})+(n-j_{k})=2n=4m-2,
\end{equation}
which implies that each walk from $-2m$ to $2m$ has odd number of steps with length $2$. We have
\begin{equation}
P_{2m-1}\Big(\frac{b}{2}\Big)=\frac{b}{2}\prod_{k=1}^{m-1}\Big(\Big(\frac{b}{2}\Big)^{2}-y_{k}\Big),
\end{equation}
where $y_{k}$, $k=1,\cdots, m-1$, depend on $m$. By Lemma \ref{L:2}, we deduce
\begin{equation}
P_{2m-1}\Big(\frac{b}{2}\Big)=\frac{b}{2}\prod_{k=1}^{m-1}\Big(\Big(\frac{b}{2}\Big)^{2}+8ck^{2}\Big).
\end{equation}
Hence,
\begin{equation}
\gamma_{2m-1}=32\left|\frac{\frac{b}{2}\prod_{k=1}^{m-1}\Big(\Big(\frac{b}{2}\Big)^{2}+8ck^{2}\Big)\cdot q^{m_{1}\cdot (2m-1)}}{2^{4m}\cdot[(2m-2)!]^{2}}\right|+O(q^{m_{1}(2m+1)})
\end{equation}
for $m\geq 2$.
\section{Proof of Theorem \ref{T:2}}
Since $V(\pm 2)=\frac{b}{2}q^{m_{1}}$ and $V(\pm 4)=\frac{c}{2}q^{2m_{1}}$, thus,
\begin{equation}
\|v\|^{2}=\frac{1}{2}\Big(b^{2}q^{2m_{1}}+c^{2}q^{4m_{1}}\Big).
\end{equation}
By Lemma \ref{L:1}, we get
\begin{equation}
\gamma_{n}=2\left|\sum_{x\in X}B_{n}(x,z)\right|\left(1+O\left(\frac{1}{n^{2}}\right)\right),
\end{equation}
where $z=z_{n}$ depends on $n$, but $|z|<1$.
Set $\sigma_{n}=\sum_{\xi\in X^{+}}B_{n}(\xi,0):=\sigma_{n}^{+}+\sigma_{n}^{-}$, where
$\sigma_{n}^{\pm}:=\sum_{\xi:B_{n}(\xi,0)\gtrless 0}B_{n}(\xi,0)$.
When $\xi\in X^{+}$,
\begin{equation}
B_{n}(\xi,0)=\frac{V(x_{1})\cdots V(x_{\nu+1})}{(n^{2}-j_{1}^{2})\cdots(n^{2}-j_{\nu}^{2})},
\end{equation}
where $x_{i}=2$ or $4$ for $i=1, \cdots, \nu+1$.
Note that $X\setminus X^{+}= \bigcup_{\xi\in X^{+}} X_{\xi}$, we choose disjoint sets $X_{\xi}'\subset X_{\xi}$ so that
\begin{equation}
X\setminus X^{+}=\bigcup_{\xi\in X^{+}}X_{\xi}'.
\end{equation}
Then
\begin{equation}
\sum_{x\in X\setminus X^{+}}B_{n}(x,z)=\sum_{\xi\in X^{+}}\left(\sum_{x\in X_{\xi}'}B_{n}(x,z)\right),
\end{equation}
therefore, we have
\begin{equation}
\begin{split}
&\sum_{x\in X}B_{n}(x,z)=\sum_{\xi\in X^{+}}\left(B_{n}(\xi,z)+\sum_{x\in X_{\xi}'}B_{n}(x,z)\right)\\
&=\sum_{\xi:B_{n}(\xi,0)>0}\left(B_{n}(\xi,z)+\sum_{x\in X_{\xi}'}B_{n}(x,z)\right)
+\sum_{\xi:B_{n}(\xi,0)<0}\left(B_{n}(\xi,z)+\sum_{x\in X_{\xi}'}B_{n}(x,z)\right)\\
&:\Sigma=\Sigma^{+}+\Sigma^{-},
\end{split}
\end{equation}
where $\Sigma^{\pm}:=\sum_{\xi:B_{n}(\xi,0)\gtrless 0}\left(B_{n}(\xi,z)+\sum_{x\in X_{\xi}'}B_{n}(x,z)\right)$.
By Lemma \ref{L:3} and Lemma \ref{L:5}, we get there exists a constant $C_{1}>0$ such that
\begin{equation}
\left[1\mp C_{1}\frac{\log n}{n}\right]\sigma_{n}^{\pm}\leq\Sigma^{\pm}\leq \left[1\pm C_{1}\frac{\log n}{n}\right]\sigma_{n}^{\pm},
\end{equation}
which is followed by
\begin{equation}
\label{E:estimation}
\left|\frac{\Sigma}{\sigma_{n}}-1\right|\leq C_{1} \frac{|\sigma_{n}^{-}|+\sigma_{n}^{+}}{|\sigma_{n}|} \cdot \frac{\log n}{n}.
\end{equation}
If $\xi\in X^{+}$, then $V(x_{1})\cdots V(x_{\nu+1})$ is a monomial in $q$ of degree $\frac{m_{1}}{2}\cdot (x_{1}+\cdots+x_{\nu+1})=m_{1}\cdot n$. From Case 3 of Theorem \ref{T:1}, we have
\begin{equation}
\sigma_{2m}=\sum_{\xi\in X^{+}}B_{2m}(\xi,0)=\frac{q^{2m_{1}\cdot m}}{4^{2m-1}\cdot[(2m-1)!]^{2}}\cdot\prod_{k=1}^{m}\left(\left(\frac{b}{2}\right)^{2}+8c\left(k-\frac{1}{2}\right)^{2}\right)
\end{equation}
and
\begin{equation}
\sigma_{2m-1}=\sum_{\xi\in X^{+}}B_{2m-1}(\xi,0)=\frac{q^{m_{1}(2m-1)}}{4^{2m-2}\cdot[(2m-2)!]^{2}}\cdot\frac{b}{2}
\cdot\prod_{k=1}^{m-1}\left(\left(\frac{b}{2}\right)^{2}+8ck^{2}\right).
\end{equation}
Moreover, $\sigma_{2m}\neq 0$ when $\frac{b}{2}\neq 2\sqrt{-2c}\cdot (k-\frac{1}{2})$ and $\sigma_{2m-1}\neq 0$
when $\frac{b}{2}\neq 2\sqrt{-2c} \cdot k$, where $c<0$. So
\begin{equation}
\label{E:upper bound 1}
\frac{|\sigma_{2m}^{-}|+\sigma_{2m}^{+}}{|\sigma_{2m}|}
=\frac{\prod_{k=1}^{m}\left(1-\frac{b^{2}}{8c(2k-1)^{2}}\right)}{\prod_{k=1}^{m}\left|1+\frac{b^{2}}{8c(2k-1)^{2}}\right|}
\leq \frac{\prod_{k=1}^{\infty}\left(1-\frac{b^{2}}{8c(2k-1)^{2}}\right)}{\prod_{k=1}^{\infty}\left|1+\frac{b^{2}}{8c(2k-1)^{2}}\right|}
=\left|\frac{\cosh \left(\frac{b\pi}{4\sqrt{-2c}}\right)}{\cos \left(\frac{b\pi}{4\sqrt{-2c}}\right)}\right|.
\end{equation}
Similarly, we have
\begin{equation}
\label{E:upper bound 2}
\frac{|\sigma_{2m-1}^{-}|+\sigma_{2m-1}^{+}}{|\sigma_{2m-1}|}\leq \left|\frac{\sinh \left(\frac{b\pi}{4\sqrt{-2c}}\right)}{\sin \left(\frac{b\pi}{4\sqrt{-2c}}\right)}\right|.
\end{equation}
By (\ref{E:estimation}), we obtain
\begin{equation}
\sum_{x\in X}B_{2m}(x,z)=\sigma_{2m}\left[1+O\left(\frac{\log m}{m}\right)\right]=\left(\sum_{\xi\in X^{+}}B_{2m}(\xi,0)\right)\left[1+O\left(\frac{\log m}{m}\right)\right]
\end{equation}
and
\begin{equation}
\sum_{x\in X}B_{2m-1}(x,z)=\sigma_{2m-1}\left[1+O\left(\frac{\log m}{m}\right)\right]=\left(\sum_{\xi\in X^{+}}B_{2m-1}(\xi,0)\right)\left[1+O\left(\frac{\log m}{m}\right)\right].
\end{equation}
Notice that
\begin{equation}
\cos \left(\frac{b\pi}{4\sqrt{-2c}}\right)=\prod_{k=1}^{\infty}\left(1+\frac{b^{2}}{8c(2k-1)^{2}}\right)
\end{equation}
and
\begin{equation}
\sin \left(\frac{b\pi}{4\sqrt{-2c}}\right)=\frac{b\pi}{4\sqrt{-2c}}\prod_{k=1}^{\infty}\left(1+\frac{b^{2}}{8c(2k)^{2}}\right),
\end{equation}
then
\begin{equation}
\cos \left(\frac{b\pi}{4\sqrt{-2c}}\right)=\prod_{k=1}^{m}\left(1+\frac{b^{2}}{8c(2k-1)^{2}}\right)\left[1+O\left(\frac{1}{m}\right)\right]
\end{equation}
and
\begin{equation}
\sin \left(\frac{b\pi}{4\sqrt{-2c}}\right)=\frac{b\pi}{4\sqrt{-2c}}\prod_{k=1}^{m-1}\left(1+\frac{b^{2}}{8c(2k)^{2}}\right)\left[1+O\left(\frac{1}{m}\right)\right].
\end{equation}
Hence,
\begin{equation}
\sum_{\xi\in X^{+}}B_{2m}(\xi,0)=\frac{q^{2m_{1}\cdot m}\cdot(-1)^{m}\cdot c^{m}}{2^{3m-2}\cdot[(2m-2)!!]^{2}}\cdot \cos\left(\frac{b\pi}{4\sqrt{-2c}}\right)\cdot \left[1+O\left(\frac{1}{m}\right)\right]
\end{equation}
and
\begin{equation}
\sum_{\xi\in X^{+}}B_{2m-1}(\xi,0)=\frac{q^{m_{1}(2m-1)}\cdot(-1)^{m-1}\cdot c^{m-1}\cdot\sqrt{-2c}}{2^{3m-4}\cdot[(2m-3)!!]^{2}\cdot \pi}
\cdot \sin\left(\frac{b\pi}{4\sqrt{-2c}}\right)\cdot \left[1+O\left(\frac{1}{m}\right)\right].
\end{equation}
Combining (\ref{E:estimation}), (\ref{E:upper bound 1}) and (\ref{E:upper bound 2}), we deduce
\begin{equation}
\begin{split}
&\sum_{x\in X}B_{2m}(x,z)=\left(\sum_{\xi\in X^{+}}B_{2m}(\xi,0)\right)\left[1+O\left(\frac{\log m}{m}\right)\right]\\
&=\frac{q^{2m_{1}\cdot m}\cdot(-1)^{m}\cdot c^{m}}{2^{3m-2}\cdot[(2m-2)!!]^{2}}\cdot \cos\left(\frac{b\pi}{4\sqrt{-2c}}\right)\cdot\left[1+O\left(\frac{\log m}{m}\right)\right]
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
&\sum_{x\in X}B_{2m-1}(x,z)=\left(\sum_{\xi\in X^{+}}B_{2m-1}(\xi,0)\right)\left[1+O\left(\frac{\log m}{m}\right)\right]\\
&=\frac{q^{m_{1}(2m-1)}\cdot(-1)^{m-1}\cdot c^{m-1}\cdot\sqrt{-2c}}{2^{3m-4}\cdot[(2m-3)!!]^{2}\cdot \pi}
\cdot \sin\left(\frac{b\pi}{4\sqrt{-2c}}\right)\cdot\left[1+O\left(\frac{\log m}{m}\right)\right].
\end{split}
\end{equation}
Therefore,
\begin{equation}
\gamma_{2m}=\frac{q^{2m_{1}\cdot m}\cdot |c|^{m}}{2^{3m-3}\cdot[(2m-2)!!]^{2}}\cdot \left|\cos\left(\frac{b\pi}{4\sqrt{-2c}}\right)\right|\cdot\left[1+O\left(\frac{\log m}{m}\right)\right]
\end{equation}
and
\begin{equation}
\gamma_{2m-1}=\frac{q^{m_{1}(2m-1)}\cdot |c|^{m-1}\cdot\sqrt{-2c}}{2^{3m-5}\cdot[(2m-3)!!]^{2}\cdot \pi}
\cdot \left|\sin\left(\frac{b\pi}{4\sqrt{-2c}}\right)\right|\cdot\left[1+O\left(\frac{\log m}{m}\right)\right].
\end{equation}
\bibliographystyle{amsplain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,523
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Tourism Destinations by Interest
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Jammu and Kashmir Information
Brief Introduction of Jammu & Kashmir
Jammu & Kashmir is a newly created Union Territory in India consisting of two divisions: Jammu Division & Kashmir Division, both of which are administered by the Central Government of India. It is located to the north of Himachal Pradesh & Punjab and to the west of Ladakh. Jammu is known as the City of Temples & offers plentiful sightseeing opportunities with its gardens, palaces, forts & religious attractions, the most famous of which is Mata Vaishno Devi in Katra. Kashmir Valley is famous for its meadows, lakes, high altitude passes, hill stations, Mughal Gardens, Dal Lake, Shikara Ride & ancient religious sites.
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History of Jammu & Kashmir
Among the many interesting facts about Jammu and Kashmir, one is that it was a princely state during the rule of the British East India Company & the British Raj from 1846 to 1947. The princely state was formed after the 1st Anglo Sikh War. During the Partition of India & its political integration, Hari Singh, the Maharaja of Jammu & Kashmir, delayed his decision regarding integration with India.
However, on 26th October, 1947, the Maharaja acceded to India in return for military aid during the Indo-Pakistan War of 1947-48 by signing the Instrument of Accession. That was how Jammu and Kashmir came to be a part of India. Article 370, which gave special status to Jammu & Kashmir, a separate constitution, state flag & autonomy over its internal administration, was incorporated into the Constitution.
A new chapter in the history of Jammu & Kashmir was added on 6th August, 2019, when the Government of India removed Article 370 and consequently, the special status of Jammu & Kashmir. It also passed the Jammu & Kashmir Reorganisation Act, which created 2 Union Territories - Jammu & Kashmir in the west & Ladakh in the East. There are now 3 administrative divisions: Jammu Division, Kashmir Division & Ladakh.
Population of Jammu & Kashmir
According to the latest census conducted in 2011, the population in Jammu is 5.04 lakhs, while the population in Kashmir Valley is 69.1 lakhs.
Climate of Jammu & Kashmir
The climate of Jammu Region is different from Kashmir Valley, even though they receive three seasons: summer, monsoon & winter.
Summer Season: Summer in Jammu starts from March & continues till May, with the temperature ranging between 35°C going as high as 45°C. Summer in Kashmir Valley starts from May and lasts till August, with the temperature ranging between 14°C to 30°C.
Monsoon Season: Monsoon in Jammu starts from June & lasts till September, with the temperature ranging between 32°C to 35°C. The rainfall causes a sharp increase in the humidity levels. In Kashmir Valley, monsoon starts from July & lasts till August, with the temperature ranging from 13°C to 17°C.
Winter Season: Winter in Jammu & Kashmir starts from October & lasts till February. While it gets quite cold in both Jammu Region & Kashmir Valley during winters, it gets much more colder in Kashmir Valley, with the temperature dropping as low as -2°C.
The temperature in Kashmir Valley ranges between -2°C to 10°C. In Jammu, the temperature in winter ranges between 4°C to 12°C, and is known as the "Winter Capital of Jammu & Kashmir", since it offers an escape from the freezing temperatures of Kashmir Valley.
Religion of Jammu & Kashmir
The major religion of Jammu Region is Hinduism, and it is home to important Hindu pilgrimage sites, including the famous Mata Vaishno Devi Temple in Katra.
In Kashmir Valley, Islam is practised by the majority of people. Amarnath Cave, a major Hindu Pilgrimage Site, is located about 141 kilometers from Srinagar.
District of Jammu & Kashmir
Districts in Jammu Region: There are 10 districts in Jammu Region. These are Kathua, Jammu, Samba, Udhampur, Reasi, Rajouri, Poonch, Doda, Ramban & Kishtwar.
Districts in Kashmir Region: Kashmir Valley Region consists of 10 districts, which are Anantnag, Kulgam, Pulwama, Shopian, Budgam, Srinagar, Ganderbal, Bandipora, Baramulla & Kupwara.
Literacy Rate in Jammu and Kashmir
As per the Census 2011 (the last census conducted in India), Jammu & Kashmir has a literacy rate of 67.16%. Male literacy is 76.75%, while female literacy rate is 56.43%.
Lifestyle of Jammu & Kashmir People
The lifestyle of the Jammu people is not very different from the rest of the people of India. They are deeply religious, and value their culture heritage a lot. Religion plays a central role in their lives, and regularly visit the numerous temples scattered around the city.
Both men & women can be seen in modern as well as traditional attire. The majority of the Jammu people speak Dogri, Gojri, Pahadi, Kashmiri, Hindi, Punjabi & Urdu.
In the Kashmir Valley, people follow a traditional lifestyle, while accommodating the modern influences to a certain extent. Historically, Central Asian & Persian influences on Kashmir have been quite strong.
The traditional dress of the people here is Pheran & Poots, which is worn by both men and women. Along with this, Mughal style turbans, headgear, taranga belt of pashmina & coloured scarf are also worn by the people. The principal languages spoken are Kashmiri & Urdu.
Source of Economy
One of the sources of economy in Jammu are a number of small industries in Jammu that produce a variety of items like electronic goods and carpets. Tourism also contributes in a big way to the economy of Jammu, since it is dotted with cultural, historical & spiritual sites.
Some of the most famous are Bahu Fort, Raghunath Temple, Mubarak Mandi Palace, Bagh-e-Bahu Garden & one of the most visited Hindu pilgrimage places in India, the Mata Vaishno Devi Temple in Katra.
The primary source of revenue for the people of Kashmir Valley is agriculture & related activities along with tourism. Sericulture and cold water fisheries are other industries that provide livelihood to people in Kashmir Valley. High quality bats known as Kashmir Willow is made from the wood found in this region.
A variety of agricultural exports are also made in Kashmir including those of barley, cherries, corn, millet, orange, rice, peaches, pears, saffron & vegetables. Apples grown in Kashmir are exported throughout India and the world.
Tourism also contributes a large part to the economy of Kashmir Valley, and attracts tourists both from India and around the world, not just for its spectacular landscape but its rich culture & heritage.
Music & Dance of Jammu & Kashmir
The music of Jammu & Kashmir has been influenced by a variety of musical influences, including that of Central, Eastern & Southern Asia. Some of the most famous musical forms practiced in Kashmir are Chakri, Henzae, Ladishah, Rouf, Hindustani Classical & Sufiana Kalam.
The rich culture of Jammu & Kashmir also includes several dances that are performed during birthdays, festivals & other special occasions. Some of the traditional dance forms are Dumhal, Kud, Bhand Pather, Rouf, Hafiza & Bacha Nagma.
Cuisine of Jammu & Kashmir
Both the Jammu Region & the Kashmir Valley boast a rich cuisines, with a variety of vegetarian & non vegetarian dishes.
The cuisine of Jammu incorporates the use of several items like pulses, lentils, rice and potatoes. The Dogri dishes are a major part of its cuisine and includes a variety of dishes like Ambal, Kulthi ki Dal, Khatta Meat, Dal Patt, Maa da Madra, and Auraiya.
Pickles are also an important part of the Jammu cuisine, and greatly enjoyed by the people. Kasrod, Jimikand, Girgle, Tyaoo & Seyoo are some of the pickles served along with the main dish. Desserts like chocolate barfi, patista & sund panjeeri are the major sweet dishes in Jammu.
The cuisine of Kashmir Valley reflects Central Asian, Persian & Afghan influences. Spices like cardamom, cinnamon, fennel & cloves are widely used. One of the most famous dishes in Kashmir is Wazwan, a collection of 32 vegetarian & non vegetarian dishes.
The most popular dishes greatly enjoyed by the people are Tabakhmaaz, Shab Deg, Dum Olav/Dum Aloo, Aab Gosh, Lyader Tschaman, Runwagan Tschaman, Riste, Nader ti Gaad, Herath, Novroze, Yakhni, and the widely acclaimed Rogan Josh.
Bread is also greatly relished by the local people, with various kinds of breads like Tsot & Tsochvor, Sheermal, Lavas & Kulcha. Kehwa, a type of tea mixed with Kashmiri green tea leaves, nuts, saffron & whole spices, is served to guests and greatly enjoyed by the people too.
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}
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San Antonio Rock 'n' Roll Marathon
The 2022 San Antonio Rock 'n' Roll Marathon will come through Joint Base San Antonio-Fort Sam Houston between 7-11 a.m. Dec. 4, 2022. Residents should expect traffic delays and drivers need to watch for runners.
Rock 'N' Roll Marathon comes through JBSA-Fort Sam Houston
502nd Air Base Wing Public Affairs
The 2022 San Antonio Rock 'n' Roll Marathon will come through Joint Base San Antonio-Fort Sam Houston between 7-11 a.m. Dec. 4. Residents should expect traffic delays and drivers need to watch for runners.
When entering the Rock N' Roll Marathon area, drivers should proceed with caution and plan ahead to avoid delays. The marathon will run from Harry Wurzbach West Gate (Pershing gate) onto Stanley Road by the Old BAMC/building 1000 and continue from Stanley Road onto Staff Post Road between Quarters #10 - #15 and on through the Quadrangle.
Cole High School will perform across from Old BAMC/building 1000, the Band of The West will be performing on the corner of Stanley Road and New Braunfels, and "Fort Sam Houston's Own" Army Band will perform inside the Quadrangle.
Vendors will be available during the event. Parking for spectators will be available at buildings 40, 247, 128, 2250, 2270, 367, 325, 2373, 2791, 2841 and 368.
Security Forces will facilitate traffic flow through the marathon course. Traffic control points will be at these locations: Gorgas Circle, Stanley Road and Harry Wurzbach Road, Schofield Road and Stanley Road, N. New Braunfels Avenue and Stanley Road, and Soapsuds Row and Stanley Road.
2022 San Antonio Rock 'n' Roll Marathon
Rock 'n' Roll San Antonio Marathon
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{
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| 3,015
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Musique et Sciences Sociales
AccueilNumérosHors-série 2Interview and CommentariesGoing Underground: The Politics o...
Hors-série 2 | 2020
Sound, Music and Violence
Interview and Commentaries
Going Underground: The Politics of Free Music around 1968
Going underground : politiques de la musique libre autour de 1968
Timothy Scott Brown
https://doi.org/10.4000/transposition.4863
Résumé | Index | Notes de la rédaction | Texte | Citation | Auteur
Going Underground situates the demand for "free music" as part of a broader contestation of the terms of cultural consumption in the radical milieu of the long 1960s. At stake in the mobilizations recounted in the reflections of Action Directe-member Jean-Marc Rouillan was not just access to popular music, but the validity of the subversive meanings ascribed to cultural production under capitalism. Struggling with the system's ability to co-opt challenges to its hegemony by putting them up for sale, activists insisted that it was they, and not promoters or other financially-interested middle men, who had the right to determine the conditions under which liberatory cultural expression such as rock'n'roll would be consumed. The insistence that music be "free" embodied a characteristic demand of the radical moment around 1968: that culture actually matter.
Going underground resitue la revendication d'une "musique libre" (free music) au sein d'une contestation plus large, dans le milieu radical des années 1960, des termes de la consommation culturelle. Dans les mobilisations dont Jean-Marc Rouillan, membre d'Action directe, rend compte dans ses réflexions, ce qui est en jeu n'est pas simplement l'accès à la musique populaire, mais aussi la validité de la charge subversive accordée à la production culturelle en régime capitaliste. Luttant contre la capacité qu'a le système à intégrer – en le commercialisant – ce qui défie son hégémonie, les activistes insistent : c'est bien eux, et non des promoteurs et autres intermédiaires intéressés par l'argent, qui ont le droit de déterminer les conditions de consommation d'expressions culturelles libératoires telles que le rock'n'roll. L'insistance sur le fait que la musique doit être "libre" (free) rejoint une revendication caractéristique du moment radical entourant 1968 : que la culture compte bel et bien.
DIY, récupération, rock, subculture, underground
DIY, recuperation, rock, subculture, underground
The following is a commentary on: Velasco-Pufleau Luis, "From Music to Armed Struggle, from 1968 to Action Directe: An Interview with Jean-Marc Rouillan", Transposition, no. Hors-série 2, 2020, https://journals.openedition.org/transposition/3780. DOI : https://doi.org/10.4000/transposition.3780
1When Jean-Marc Rouillan observes that his "political commitment in 1968 was preceded by musical commitment", he touches on an oft-emphasized point in the recent scholarship on the "global 1960s". It was not just that culture and politics were intimately bound together around 1968, but that innovations in the former often precipitated developments in the latter. The point is particularly relevant where popular music is concerned. In no other area of cultural production was the social impact so explosive nor the politics of consumption so fraught. And as the scholarship of the last dozen or so years has shown and Rouillan's recollections again confirm, popular music—above all rock'n'roll—was the sound of revolt par excellence. Rouillan touches on this fact directly when he remarks that "[p]laying music was like going underground…in anticipation of a confrontation". That this statement could be uttered with reference to the moment of 1968 was in part a function of rock'n'roll's newness, and the ease with which that newness allowed it to articulate with other cultural and political innovations emerging around the same time. In this respect, rock'n'roll gained its cultural-political import through a process of synergy in which it came to stand in for a broader rebellion.
2Rouillan's comment simultaneously suggests a spatial relationship—one in which the socially-valuable in music, as in politics, can take root only in the subterranean spaces where society's demands for conformity fail fully to penetrate. Here, the "underground" as a term/concept has a double valence—as the space simultaneously in which the left-wing political desperadoes of the post-'68 moment marshaled their forces for all out war with the forces of the state, and the place where cultural militants sought to create authentic artistic expression free from the deforming pressures of mass commerce. The most culturally militant sounds, like the most politically militant agitation, were to be found not at the surface level of society, again, but underneath it, in the realm of subculture. To "go underground" was the act not just of the political militant who rejected all accommodation with bourgeois-parliamentary forms, but of the cultural militant who, similarly, rejected the forms dictated by the market.
3The liberatory qualities of music, in Rouillan's account especially the music of Jimi Hendrix, foreshadowed other possibilities. With Hendrix, remembers Rouillan, "we felt that the door was open. What was not possible four years before, Jimi Hendrix said, 'It's possible now. We have decided that it's possible.'" This voluntaristic act was, as Rouillan puts it, a "decision", one that could lead to others. "From that point [the appearance on the scene of Hendrix] we could do whatever we wanted. Musically, he did what he wanted because he was capable of doing it. In the same way, as a political movement, we were fed up with political parties and unions. So we said to ourselves, 'We're going to do things differently.'"
4Almost as soon as it was enunciated, however, that emancipatory decision faced the danger of being rendered meaningless. "Jimi Hendrix would say 'I'm free' in a seven- or eight-minute solo;" says Rouillan; "Then, right away, people showed up to sell this freedom—major labels and concert managers. They played with this feeling of freedom, in order to sell it to our entire generation. They turned it into commercial products. We spent two or three years buying it, and then we thought, 'No, we're getting fucked.'" The demand for free music was a response to this realization. A key threat perceived by radicals in the long 1960s was posed by what the Situationists dubbed recuperation, a term meant to invoke capitalism's ability to heal itself from challenges to its hegemony. Across the whole range of cultural production, from mainstream publishing programs calculated to cash in on the demand for the writings of Che and Mao to the commerce-safe recapitulation of countercultural values represented by the musical Hair, capitalism proved adept at placing the revolution up for sale almost as quickly as it could be created. Radicals worried about recuperation because capitalism's ability to assimilate the cultural gave it the ability to disable the political. The "underground" in which subcultures flourished was never a pure sphere of rebellion, but constantly under threat of having its content siphoned away. Capitalism was all too adept, in the words of a band that figures centrally in Rouillan's account—the Clash—of "turning rebellion into money".
5The effort to "free the music from the merchants" was a transnational one. It is no surprise, for example, that Action Directe's West German counterparts, the Movement 2nd June, were first politicized in precisely such a refusal of the logic of the "big concert". This was in the infamous 1965 riot at the Rolling Stones concert in West Berlin, which saw militants-to-be crash the show and participate in a riot that destroyed the venue. The same group of militants later carried out a smoke bomb attack on the West Berlin debut of Hair, denouncing it as an attempt to "gratify capitalist demands" at the expense of the "real subculture". The attack against Hair took place against the backdrop of ongoing police pressure against the Zodiak Arts Lab, a key site of subcultural experimentation in West Berlin. Such attempts to mobilize against the twin threat of police pressure and capitalist recuperation could be multiplied across a range of European and North American cases. Meanwhile the conceptual basis of these attempts—the assumption of a unity between political and cultural forms of struggle—was symbolized in an ubiquitous image of 1968 on both sides of the Atlantic: the juxtaposition of the machine gun and the electric guitar.
6The pervasiveness of the association between political and cultural militancy and the linked struggle against recuperation offers more evidence of the importance of the transnational around 1968. It was not just a matter of ideas moving across national borders, however; rather, activists everywhere were responding to the issues imbedded in industrial society in general and capitalism in particular. That militants expected a political message from their music or ascribed political significance to music whether it was openly political or not was symptomatic of the long-1960s moment. But it also suggests for us a key link between the activism of the moment around 1968 and that of the decade that followed. It is no accident that Rouillan references both Hendrix and the Clash, the former a musical avatar of the 1960s rebellion, the latter the most-explicitly political of the bands associated with the punk explosion of the late-1970s. In both cases, music held a political valence; more importantly, punk—along with the proto-punk exemplified by bands such as West Germany's Ton Steine Scherben—directly embodied an approach to cultural production in which the act of making and distributing the music became as important as the messages embedded in the music itself.
7This DIY ("Do it yourself") approach is a key feature linking the rebellion of 1968 and the rebellion of punk, calling into question too-easy assumptions about the extent to which punk actually broke with the cultural politics of a hippie rebellion that, on the surface, it violently rejected. More important is that DIY, as both mode of cultural production and political ethos, exists at the very heart of the particular understanding of music put forward by Jean-Marc Rouillan. The demand that music be free was simultaneously an anti-capitalist act and an act of cultural rebellion that sought to protect the integrity of the musical-political gesture. "Free music was a political struggle", as Rouillan puts it; "we understood it as such".
Timothy Scott Brown, « Going Underground: The Politics of Free Music around 1968 », Transposition [En ligne], Hors-série 2 | 2020, mis en ligne le 15 mars 2020, consulté le 16 janvier 2021. URL : http://journals.openedition.org/transposition/4863 ; DOI : https://doi.org/10.4000/transposition.4863
Timothy Scott Brown is Professor and Chair of History at Northeastern University, and Senior Fellow at the Institute for European Studies at the University of California, Berkeley. He is a recent Fellow of the American Council of Learned Societies and Berlin Prize Fellow of the American Academy in Berlin. His books include West Germany in the Global Sixties: The Anti-Authoritarian Revolt, 1962-1978 (Cambridge 2013); The Global Sixties in Sound and Vision: Media, Counterculture, Revolt (Palgrave 2014); and Between the Avantgarde and the Everyday: Subversive Politics in Europe, 1957 to the Present (Berghahn 2011). His latest book, Sixties Europe, is forthcoming with Cambridge University Press in 2020.
La revue Transposition est mise à disposition selon les termes de la Licence Creative Commons Attribution - Partage dans les Mêmes Conditions 4.0 International.
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