How can we all help strengthen communities, those that are near and far? Students should invite that person to come and visit by describing the area, explaining why the person might want to settle there. They will have to run and change direction based on verbal commands and duck quickly to avoid being hit by the imaginary boom. Then have your students make posters that show ways to be a good citizen at home, in school, or in the community. American States and Regions, Grade 4 Students work in pairs to create museum exhibits and pamphlets after reading Have kids imagine they work for a museum that specializes in exhibits about your area and let them create a two-part exhibit. This serves as a brief introduction to the concept which will covered in a later unit.
Special Considerations: It is important to know your students. Using a simple story, the teacher guides students in identifying specific ways in which a community can keep people safe. Objective 1 Describe how producers and consumers work together in the making and using of goods and services. English Language Arts Key Concepts: basic needs, family Abstract: In this lesson students review what they learned about families in grade one. English Language Arts Key Concepts: community, human characteristics of place, physical characteristics of place Abstract : In this lesson students explore different kinds of communities. As a culminating activity, students create a short book illustrating in words and pictures why people live in communities. However, if you think of appropriate questions that are not on the list, you are free to ask those as well! Depending on what the helpers have planned, go with it.
Social Studies / Grade 2 MC3 Unit 1 What is a Community?
But first, I want everyone to think about your home. The other part is a written report about these features, as well as the climate, plants, and wildlife. Is it on flat land? Based on National Physical Education Standards, students should have been learning these skills for the last 4 years. Here you will find multiple lesson plans for second grade to help guide your standards-based instruction. What are their roles in the community? Many lessons are aligned to Common Core State Standards so you can easily integrate them into your existing curriculum.
But what if we flipped that idea around? After you pick one, draw a picture of them on the sheet and write a short description of what they do for the community. What are some of the communities that different people in the class are a part of? Encourage students to use color and pictures to communicate their ideas. Social Studies Vocabulary Students Should Know and Use: community, tradition, custom, immigrant, celebration, contribution, culture, group, state, nation, place, compare, contrast Standard 2 Citizenship : Students will recognize and practice civic responsibility in the community, state, and nation. Before long, more roads, bigger homes, apartment buildings, stores, and garages surround the Little House. Students begin to understand that people live in communities because communities help them meet these needs. Publish an Ancient Newspaper World History, Grade 6 First, have students. Then take them on a trip through time as they come face to face with Greek gods and American heroes alike through coloring pages, fascinating biographies, and paper doll activities.
Government Branches Neighborhoods and Community, Grade 2 Show students how to make a chart with three columns, one for each branch of the U. Students are then introduced to a Venn Diagram and how it is used to make comparisons with a teacher demonstration. Unit Abstract: In this foundational unit students explore characteristics of communities, the reasons people live in communities, and different kinds of communities. We appreciate their time and energy to come share with the class—so show them! Have the students make them whenever possible. Using photographs and illustrations from picture books, students identify human and physical characteristics of these three types of communities. In a categorization activity, the teacher guides students in grouping together the identified characteristics from both lists, using labels such as natural characteristics, buildings, transportation, etc. Finally, students review basic needs of people including food, clothing and shelter.
Create Good Citizen Posters The World Around Us, Kindergarten After reading , start a discussion with your class by asking questions like these: How can you help people at home? Objective 1 Examine and identify cultural differences within the community. Objective 1 Examine civic responsibility and demonstrate good citizenship. At each point in the story where the community changes, the teacher will stop to discuss what the students notice about the community around the house. Length of Time: About 45 This lesson is designed to help students understand Earth in relation to the Solar System. Pay close attention to how the area around the house changes.
What are some ways that you help out at your home? Teachers are facing time crunches, and dedicated social studies lessons are slowly being squeezed out of our elementary classrooms. Second Grade Social Studies Worksheets and Printables The world is filled with countries, stories, cultural traditions, and historical events to enrich your child's education. Objective 3 Investigate and show how communities, state, and nation are united by symbols that represent citizenship in our nation. Write down a brief description next to each name. This may be confusing for some students, so explain it well and try not to use the comparison more than you have to. Create a Museum Exhibit About Landforms.
They will decorate each slice and then exchange slices with classmates and then evaluate the fractions of slices that they have at the end. They use this knowledge to discover what type their community is as they explore a text to self connection, and then write a paragraph. Who else helps out at your home? What did we talk about yesterday? After the helpers have described their job and shared whatever they would like with the students, allow for a question and answer time. Specifically explain what you would like to share with the class and to have them emphasize their role and responsibilities in the community. First, a road appears, which brings horseless carriages and then trucks and steamrollers.
When distributing these materials, credit should be given to Utah State Board of Education. Abstract: This lesson connects to Lesson One and the discussion of how people meet the basic needs of food, clothing, shelter, and a sense of belonging with their family and community. The rosy-pink Little House, on a hill surrounded by apple trees, watches the days go by from the first apple blossoms in the spring through the winter snows. Throughout the rest of our unit about we community, we will talk more about them and how they help make our community great. What is the weather like? Objective 2 Identify individuals within the school community and how they contribute to the school's success. From Greek Mythology to historical figures to maps, your kids will have a better understanding of the world around them with our second grade social studies worksheets.


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redpajama_set_name: RedPajamaGithub


Sometimes we feel there is a gap – we cannot quite grasp – between who we are as individuals and how we lead as managers. A certain inadequacy between our true self and the way we think we should be and look like in our professional roles.
This often leads to pressure and great efforts in order to fit in a suit or mask we believe others expect from us. How can we bridge this gap? How to find the authenticity and feel comfortable with who we are as leaders and people at the same time? How to leave the burden behind and lead from who we are?
Everything starts not with what others necessarily expect from us, but what we believe these external expectations to be. So they do not always have a basis in reality, but they are rather connected to what we think others value, want or appreciate.
This is how we often strive to be these perfect leaders; to behave according to high standards and demands, to reach all the KPIs, to always be in the front line, to be inspirational and understand everything. You see how these implied or perceived expectations can actually hurt us and have the opposite effect of making us better: they can actually alienate us from who we are.
If this is what we face everyday, what should we do?
There is a subtle balance, a golden proportion between these two aspects. If you are always vulnerable and do not deliver results, the organisation might understandably start to say you no longer belong there. But if you are always rigid and do not care about the people in your team, if you are not preoccupied by their well-being and motivation, you also risk to be unfit.
The key is to speak your values and find a way to calibrate them with what your organisation needs.
Yes, you will need courage. Courage to position yourself, courage to take action, courage to speak from you heart and let's not forget about the courage to admit your mistakes. After all, we're all human; the stake is not how to be perfect as a leader, but to evolve, to strive to be better everyday.
In order to reach this balance between who you are and how you lead, the first step is to communicate and find out what the organisation needs as well. Talk about who you are, what are your values, find your strengths and see how you can build on them – in the benefit of others, of people in your team. So everything begins with an assessment, a thorough evaluation in order to find out how to reveal ourselves in our best possible version, but also in sync with our beliefs and character. And remember, this will not happen over night, we are talking about a process that takes time and patience.
Ten years ago, this sort of challenge – bridging the gap between who we are and how we lead – might not have been on our agenda or on our minds for that matter. Why? Because there was little room for leadership; being a manager was more important, structure, processes and rules were highly valued.
Only after big changes – new technologies, globalization, access to knowledge, the evolution of the Internet and so on –, leadership appeared as not only a need to keep people organised, but rather to involve them, to help them contribute, work together and see a bigger purpose in their work.
So it's a natural evolution of this paradigm that brought us questioning how to be and do all these things. Leaders have to be firm, yet mild. Authentic, yet professional. Human, yet capable. The mix between these very different and seemingly opposing traits exists and can be reached. It's a question of where we are in this bigger vision of the world today.
For eventually, it is not only about us. We are part of an organisational system. On the other hand, we cannot and should not hide who we are, as this will eventually rise to the surface anyway. Our most important abilities, talents and passions should just be put in the best light, in order for us and for the people from the organisation to move forward, make a change and bring bigger satisfaction to everyone they serve.


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Virginia class SSN. BAE Systems picture.


(other than the value 'special' which had its own route above).
When AsyncLoggerContextSelector is used to make all loggers asynchronous, make sure to use normal <root> and <logger> elements in the configuration. The AsyncLoggerContextSelector will ensure that all loggers are asynchronous, using a mechanism that is different from what happens when you configure <asyncRoot> or <asyncLogger>. The latter elements are intended for mixing async with sync loggers. If you use both mechanisms together you will end up with two background threads, where your application passes the log message to thread A, which passes the message to thread B, which then finally logs the message to disk. This works, but there will be an unnecessary step in the middle.
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Federal court strikes down ban on alcohol ads in Va. campus papers
April 1, 2008 December 4, 2018 Michael Beder
VIRGINIA — A federal judge on Monday struck down astate ban on alcohol advertisements in student media, ruling the ban violatedthe First Amendment rights of student publications.
The American Civil Liberties Union of Virginia filed a lawsuit in June 2006on behalf of two student papers: Virginia Tech University's CollegiateTimes and the University of Virginia's Cavalier Daily. In theircomplaint, filed in the U.S. District Court in Richmond, the papers estimatedthe ban cost each of them $30,000 a year in lost ad revenue.
"This is a long time coming, but I am very pleased to see it through," saidElizabeth Mills, who in January took over as the Cavalier Daily's editorin chief.
U.S. Magistrate Judge M. Hannah Lauck struck down two provisions of theVirginia Administrative Code. One regulation applied to alcohol ads in all printand electronic media, limiting advertisers to specific words and phrases todescribe their drinks and establishments. For example, it prohibits terms suchas "happy hour." The other provision applied specifically to studentpublications, banning them from running any alcohol ads except for limitedreferences in ads for restaurants.
Neither regulation was sufficiently effective or narrowly tailored enoughto justify infringing on the free speech of the papers and advertisers, Lauckruled.
Even assuming the government has a substantial interest in promotingtemperance, Lauck wrote, the more general regulation failed the court's analysisbecause the state showed "little evidence about this regulation at all, muchless evidence to explain why [allowed] generic phrases such as 'Mixed Drinks,"Exotic Drinks,' or even 'Polynesian Drinks' are more temperate than drink- orbrand-specific phrases."
The state does have a substantial interest in curbing underage drinking,Lauck wrote. But both sides agreed the majority of the college papers' readersare of legal drinking age. And banning alcohol ads in student media –while allowing them in other publications widely available on campus –does not materially advance the government's goal and thus is not a justifiedinfringement on the papers' First Amendment rights, Lauck ruled.
"Even presuming the Court could evaluate a 1970's regulation based solelyon its performance in the years after 2000, not a single witness testifies as tohow this regulation, which has been in effect for decades, has directly advancedthe admittedly substantial governmental interest of preventing underageconsumption of alcohol or abusive drinking," Lauck wrote.
A panel of the 3rd U.S. Circuit Court of Appeals in 2004 struck down asimilar law in Pennsylvania, a ruling Lauck cited in her decision.
Although Lauck ruled that "it appears an injunction should issue"permanently barring enforcement of the regulations, she allowed both sides fivedays to ask for a hearing on whether to grant the injunction.
Tucker Martin, a spokesman for the state attorney general's office, toldthe Associated Press the state was "disappointed" with the ruling and wouldconsider its options.
Tagged Cavalier Daily, Collegiate Times, University of Virginia, Virginia, Virginia Tech

redpajama_set_name: RedPajamaCommonCrawl


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HAZLETON, Pa. _ Salvadore DeFazio, the poet laureate of Hazleton, is on deadline. In the days leading to the Fourth of July, his hometown is celebrating its 150th birthday. The centerpiece of the commemoration is a history pageant DeFazio is furiously working to finish. He is searching for an ending, though he has settled on a musical theme in Aaron Copeland's "Fanfare for the Common Man."
It's a perfect fit, DeFazio says: "Hazleton is a fanfare for the common man."
Not everyone is thrilled with the choice of music.
"Fanfare for the Common Man burns me up," says Joseph Palaggi, 80, who has played clarinet with the Hazleton Philharmonic Symphonic Orchestra since its founding in 1954. Palaggi is the son of a shoemaker who came here from Italy at the age of 16. Having worked most of his days in shipping and receiving, his common man credentials are impeccable. But the Fanfare, he complains, was written for brass and percussion. At the sesquicentennial performance, he and the other woodwinds will be mostly idle with "all those measures rest."
For better or worse, these are the brass and percussion days in Hazleton.
A year ago this little city nestled high in the depleted coal country of northeastern Pennsylvania put itself on the map by enacting a law to fine landlords and employers who provide shelter or employment to people who are in the country illegally. It was July 14, Bastille Day, and Hazleton itself felt under siege by an influx of newcomers, some of them illegal immigrants. Mayor Lou Barletta complained of violent crime, drugs, gangs and graffiti, and a burden on the school system and hospital that the community could not bear.
Overnight, Hazleton literally and figuratively became a Lou Dobbs special, wildly cheered and just as passionately condemned, the ordinance emulated by small communities across the country even as its constitutionality is being weighed by the courts in a case that may eventually find its way to the Supreme Court.
But beneath the fanfare, Hazleton is also a small American town celebrating the Fourth of July. And more than most, it remains a timeless American place _ a throwback to an America before mobility and globalization exalted the new, the ersatz and the interchangeable over the settled, the staid and the distinct.
Parochial, perhaps, but, small as it is, Hazleton has its own poet laureate, symphony, and Liberty Band. The band _ like the philharmonic, a volunteer assemblage _ traces its history to the Civil War, when it reputedly provided the soundtrack of Lee's surrender to Grant outside Appomattox Courthouse, playing "Auld Lang Syne" (a number they will be reprising for the pageant).
As remarkable, Hazleton still has a drive-in with nightly first-run double features.
Hazletonians ought to know better than most that America has always been a melange. The town's multitude of church spires are monuments to its thick wickerwork of contending ethnic communities.
In the pageant script, DeFazio quotes a magazine from the 1890s, describing in lurid terms the fate that befell communities like Hazleton amid an onslaught of undesirable new immigrants from Eastern and Southern Europe: "One of the richest regions of the earth overrun with a horde of Hungarians, Slavs, Polanders, Italians, Sicilians, Russians and Tyrolese of the lowest class; a section almost denationalized by the scum of the Continent."
Or, as Mayor Barletta puts it, "Hazleton was built on diversity."
Like today's diversity, it was driven in part by demand for people willing to do work under conditions "Americans" would not _ in this case excavating the hard anthracite coal so prized because, though slow to ignite, it would burn longer and hotter.
The coal is gone. Hazleton rests on a hollowed core. By 2000 the Census described an aging city of only 23,000, about 93 percent white and about 5 percent Hispanic. Nearly nine out of 10 Hazletonians were native Pennsylvanians.
Then Hazleton was unexpectedly reborn in the ashes of Sept. 11. The only Hazeltonian to die in the attacks was a Puerto Rican man by the name of Efrain Franco Romero, who worked as a painter at the World Trade Center. Romero lived during the week in Jersey City, N.J., and on weekends joined his family in Hazleton, where they had moved to find a better life.
After Sept. 11, many Hispanics, including many immigrants and especially Dominicans, moved their families from New York and New Jersey to Hazleton. They were drawn by jobs, lower housing costs and Hazleton's quiet, small-town atmosphere, by word of mouth and family ties with the Latino community already budding there. They opened more businesses.
Hazleton's population swelled to more than 30,000, about a third Hispanic, according to the mayor, who estimates that among the immigrant population (which is not exclusively Hispanic), about 3,400 were not in the country legally (some of them left after the ordinance was enacted).
The immigrants were an answered prayer, says Amilcar Arroyo, a Peruvian immigrant who married a Hazleton woman he met working in a factory here, and who now runs the city's only Spanish-language newspaper, El Mensajero. "Downtown Hazleton five or six years ago was a ghost town," Arroyo said; now, property values have soared.
But it also meant that little Hazleton was contending in microcosm with the kind of demographic change the nation is encountering over a period of generations, all in less time than the series run of "The Sopranos."
"It happened too fast," says John T. Medashefski, 51, a native-born artist and proprietor of a cafe and gallery, who lives in the house where he grew up.
"That's my covenant. That's my blankie," he says of the family homestead, from which he flies an assortment of American flags (which also serve as a recurring theme in his art). "It's my heritage."
He explains, in the plain-speak of Hazleton: "I'm not a Polack, I'm an American."
This is something of mantra here. Antonio Rodriguez believes he was Hazleton's first Hispanic. He a married a local Irish girl and moved here nearly 40 years ago. But Rodriguez only wants to be known as an American.
He left Puerto Rico for the first time when he joined the Army, and when he arrived at basic training, one of nine Puerto Rican recruits, the sergeant separated them. "He was making us Americans," Rodriguez says. "I'm glad he did it. If you want to become American, you have to change."
That used to be the national consensus, DeFazio laments _ "Ozzie and Harriet weren't hyphenated Americans."
But even as he roams the hills and dales of the city assigning parts and delivering shards of script _ a school marm, a newsboy, a Lithuanian miner marching to his death in the Lattimer massacre of 1897 _ the poet laureate is still looking for Hispanic kids to embody the latest wave of immigrants.
Perhaps he will find them at the Correale Little League Stadium. A couple of years ago, some Hispanics moved to create a separate youth baseball league. A furious public reaction put the kibosh on it, and today the Little League games appear casually integrated.
The stadium and its fields, framed by a giant blue water tank on one side and a gaggle of radio towers on the other, are situated at what appears to be the highest point in the city. It's a blustery, heavenly remove from the turmoil below.
In the moonlit dusk of a recent evening, a young ballplayer with brown skin and a Latin lilt to his English dared his lighter-skinned friends to race. "The last one … " he began, then hesitated to find the right word to describe the stakes to be placed on this competition.
He thought, he beamed, and then, confident, reissued his challenge: "The last one stinks."


text:
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In fall of 2017 paris_monster began recording their first full length LP, set to be released in 2018. The album will solidify their dirty, glitchy, lyric-centered, groove-based sound, and will be a milepost in their search for a focused and flexible musical approach. The elements of funk and soul join with lyrical themes of bygone eras in rural settings, characters lost, and mortality confronted. Love achieved, and denied. Americana. Themes in stark contrast to the pounding grooves and electronic mayhem, creating depths and dimensions that could have been otherwise achieved only by dropping a modular synth in the corn fields of central New York or the woods of Connecticut. And maybe that's exactly what happened.


You've been charged with driving under the influence of marijuana, and now you don't know what to expect. You don't want to go to jail, but you know that's a possibility. If you've been arrested for DWI marijuana in Fort Worth TX, what penalties are you facing? And can you avoid or reduce the charges against you?
At Hoeller McLaughlin PLLC, we have experience fighting DWI marijuana cases in Tarrant County. Contact us today to learn more about how a DWI lawyer can help.
After a second DWI charge, the offense is a third-degree felony. This is a serious offense, and will stay on your permanent legal record. It can be difficult to find a job and housing if convicted.
By this point, the judge knows you will likely not stop driving under the influence of marijuana. This is the reason for such severe consequences.
How Can a DWI Marijuana Attorney Help You?
The only way to avoid jail time is to have the charges dismissed or reduced. This is not easy, as the law takes driving under the influence of marijuana very seriously. The good news is that an experienced DWI attorney can help. An experienced attorney who specializes in DWI cases may be able to reduce or dismiss the charges against you, especially if this is your first offense.
Many lawyers will first look for the reasonable suspicion for the traffic stop. The police officer must have reasonable suspicion the driver committed a crime to be pulled over. Possible reasonable suspicions may include speeding, risky driving, or failing to obey traffic signs. In other words, a police officer cannot pull you over just because he suspected you were under the influence.
A judge can dismiss the charges if the police officer did not document traffic violations. The Fourth Amendment expunges legal issues found during traffic stops not conducted under reasonable suspicion. In other words, since the police officer did not see any crime committed before stopping you, the DWI charge cannot stand. DWI lawyers know how to use this to your advantage in court.
If you have prior DWIs, this doesn't mean the penalties cannot be reduced. It's just much harder to convince the court because repeat offenses indicate that the previous penalties weren't enough to prevent the offense from happening again.
You have a choice when it comes to representation in court. You can have a public defender appointed to you who has a lot of experience with DUI/DWI law, or you can hire a private attorney.
Although hiring a private attorney comes with a cost, it's worth it if you want to convince the court not to sentence you to jail time, pay thousands of dollars, and/or lose your license. An attorney can present your case in court in the best way possible to give you the best possible outcome.
At Hoeller & McLAughlin PLLC, we may be able to get your DWI charges dismissed or reduced. Contact us today for a consultation of your case.


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Rascal flatts bless the broken road music
- 27.09.2020 27.09.2020 - Daijin
Oct 25, · Mix up the verse strum patterns, e.g. play pattern 1, 2, 1, 1 for the first two lines, then pattern 1, 2, 2, 2 and end with 1, 2, 1, 1. Can vary as you wish or play verse 2 slightly differen/5(). Download BLESS THE BROKEN ROAD by RASCAL FLATTS free. #1 rated music site. Million songs. Get lyrics ♫ music videos for your iPhone®. Download Rascal Flatts Bless The Broken Road sheet music notes and chords for Easy Piano. This Country composition includes 6 page(s). SKU: Free preview. Transpose, save as PDF.
Karaoke Bless The Broken Road YouTube, time: 4:01
Rascal Flatts - Bless The Broken Road (Live) "Watch a live performance of ""Bless the Broken Road"" performed by Rascal Flatts Best of Rascal Flatts: out-n-about.de Subscribe here: out-n-about.de Music video by Rascal Flatts performing Bless The Broken Road. Mar 05, · Country music trio Rascal Flatts weren't the first to record "Bless The Broken Road," but they did make the song extremely popular! Country music group Rascal Flatts' "Bless the Broken Road" was co-written by Marcus Hummon, Bobby Boyd and Jeff . Print and download Bless the Broken Road sheet music by Rascal Flatts. Sheet music arranged for Piano/Vocal/Guitar in C Major (transposable). SKU: MN/5(28). Download BLESS THE BROKEN ROAD by RASCAL FLATTS free. #1 rated music site. Million songs. Get lyrics ♫ music videos for your iPhone®. Oct 25, · Mix up the verse strum patterns, e.g. play pattern 1, 2, 1, 1 for the first two lines, then pattern 1, 2, 2, 2 and end with 1, 2, 1, 1. Can vary as you wish or play verse 2 slightly differen/5(). Download Rascal Flatts Bless The Broken Road sheet music notes and chords for Easy Piano. This Country composition includes 6 page(s). SKU: Free preview. Transpose, save as PDF. Watch the video for Bless the Broken Road from Rascal Flatts's Feels Like Today for free, and see the artwork, lyrics and similar artists. Nov 01, · Check out Bless The Broken Road (Remastered Version) by Rascal Flatts on Amazon Music. Stream ad-free or purchase CD's and MP3s now on out-n-about.de(47).The highest-charting rendition is by the country music group Rascal Flatts, who cut the song for the Feels Like Today album. God Bless the Broken Road () SoundTracks on IMDb: Memorable quotes and exchanges from movies, TV series and more Performed by Rascal Flatts. Lyrics to "Bless The Broken Road" song by Rascal Flatts: I set out on a narrow way (originally by Nitty Gritty Dirt Band) That God blessed the broken road. Bless The Broken Road. By Rascal Flatts. • 1 song More by Rascal Flatts Listen to Rascal Flatts in full in the Spotify app Music; ℗ Maranatha!. "God blessed the broken road that led me straight to you." the Broken Road" had a complicated journey before it made its way to Rascal Flatts in MH: The first was on the (Nitty Gritty Dirt Band's) album "Acoustic.". Now I'm just rollin' home into my lover's arms. This much I know is true. That God blessed the broken road that led me straight to you. Back to Music. Bless The Broken Road by Rascal Flatts song meaning, lyric interpretation, video This song was originally recorded by the Nitty Gritty Dirt Band on their -
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The rumors about Ariana Grande dating Saturday Night Live star Pete Davidson are seemingly true. Sources are confirming that the couple have only recently started seeing each other, but that things are moving along nicely in the relationship.
According to a May 21 report by E! Online, Ariana Grande and Pete Davidson both announced that they had split with their longtime significant others this month. Grande confirmed her split from Mac Miller, while Davidson announced that he had ended his two-year relationship with Cazzie David.
Ariana and Pete spent time together back in 2016 when she hosted Saturday Night Live, but sources have not yet revealed how the two ended up together. While insiders claim that Grande and Davidson's relationship is still in the early stages, and very "casual," the "Into You" singer is said to be "very happy" with the budding romance.
When the split was announced, Grande took to her Instagram page to release a statement about her love and respect for Miller, saying that she still considers him one of her best friends and that she will continue to "adore" him no matter where their lives take them.
However, Mac may not be coping with the break up well. Miller was arrested earlier this month for a DUI and a hit and run. The L.A. Police Department confirmed that Mac was driving his 2016 G-Wagon when he hit a pole. He and his two passengers then fled the scene on foot. Following the shocking arrest, Ariana took to Twitter to post a cryptic tweet that many believe to be in regards to her ex-boyfriend.
"Pls take care of yourself," she wrote.
Meanwhile, Pete Davidson's split with Cazzie David was not as dramatic. The SNL star revealed that he and his longtime love were no longer together, calling her a "very talented girl" and claiming that she would "be fine" following the breakup. He and Ariana Grande have yet to go public with their romance.


text:
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text:
\section{Introduction}

The x-ray phase problem, and inverse problems more generally, are often characterized as overdetermined or underdetermined \cite{Millane, Hauptman}. A successful, i.e. unique, phase reconstruction belongs to the former class, where the number of data exceed the number of free variables (atomic coordinates, contrast pixels, etc.). This way of formulating the feasibility criterion, however, may not be appropriate for any of several reasons. Intensity data is subject to noise, and in the extreme shot-noise limit can hardly be treated as a collection of continuous constraints. On the other side of the equation, the variables to be reconstructed may be quite different from simple real numbers. Examples are the binary-valued contrast in magnetic scattering from an Ising magnet, or the contrast of a set of identical atoms at low resolution. These examples highlight the fact that in many applications the actual information we wish to extract from the data is a small fraction of the information in a general image, and as a result the reconstruction should be able to tolerate significant noise in the data.

The question of uniqueness takes on a different character when we depart from the Maxwell constraint counting principle as applied to continuous contrast values and constraints. We will use the example of magnetic scattering to frame this question in precise terms. Suppose we are trying to reconstruct a 2D Ising domain pattern from its circular dichroism contrast in an x-ray scattering experiment \cite{experiment}. Given the x-ray wavelength  and the maximum scattering angle, the photons collected at the detector provide information about the domain pattern at a resolution of finitely many pixels, say $N$. Each pixel has one of two contrast values that we wish to determine. Further suppose that the total number of collected photons is a certain multiple of the number of pixels, say $\mu N$. Strong limits on the number of photons may arise in the case of stroboscopic single-shot experiments with x-rays, or simply due to weakness of the contrast. A basic question given these circumstances is whether some fraction of the pixels will always be uncertain for any $\mu$, or whether there is a critical $\mu_c$ above which this fraction is so small that a unique reconstruction, in a practical sense, is possible in principle.

\section{Communication channel analogy}\label{analogy}

The above question has a close analogy with problems studied in communication theory \cite{Shannon}. Consider a scheme where information, in the form of binary sequences, is transmitted on a noisy communication channel. The first stage in the process is to encode blocks of size $N$ by their Fourier intensities. Whereas in a diffraction experiment this is accomplished by the quantum mechanics of scattering, we will adhere to our information processing scenario where the intensities are obtained as the squared magnitudes of the discrete Fourier transform applied to the sequence of bits. Noise in our channel arises from the fact that it is not the continuous intensity values that are ``received" by the detector, but discrete photons. The final stage of our communication channel is thus the Poisson sampling of the Fourier intensities to give a sequence of $N$ non-negative photon counts. We specify the noise in the channel in terms of the mean number $\mu$ of photons per intensity sample. Examples of received signals (photon counts) for a particular binary input sequence, at various values of $\mu$, are shown in Table 1.

\renewcommand\arraystretch{1.5}
\begin{table}[!t]
\begin{center}
\begin{tabular}{c|*{13}{r}}
$\mu$&$+1$&$-1$&$+1$&$-1$&$-1$&$+1$&$-1$&$+1$&$+1$&$+1$&$+1$&$+1$&$-1$\\
\hline
5&4& 8& 2& 0& 2& 10& 6& 8& 10& 3& 0& 1& 9\\
10&8& 16& 3& 4& 0& 16& 9& 16& 31& 1& 3& 2& 18\\
15&5& 28& 2& 1& 4& 38& 21& 26& 38& 5& 3& 2& 21\\
\end{tabular}
\end{center}
\caption{Example of a binary-valued $\pm 1$ sequence (top row) and its encoding (lower rows) by Poisson samples (photon counts) of its Fourier transform intensity for three values of the mean photon number per bit, $\mu$. Our analysis shows that the original binary sequence can be reliably decoded from the Poisson sampled intensities, up to symmetries, when $\mu$ exceeds a value near 10.}
\end{table}

A central question addressed by communication theory is the extent to which the received signal in the noisy channel can be decoded to recover the original message. This hinges upon two things: (1) a capacity intrinsic to the characteristics of the channel, and (2) the encoding/decoding protocol. Whereas the first point also applies to the problem of uniqueness in imaging, the analogy breaks down somewhat when it comes to the second point. In the context of communications, Shannon \cite{Shannon} showed that information transmission approaching arbitrarily close to the rate given by the channel capacity is achievable by protocols that use carefully constructed codes. This flexibility in extracting the maximum information in a diffraction experiment is usually not available. For example, in the case of magnetic imaging the ``code" is already set by the form of the contrast: binary-valued pixels with some degree of correlation. The communications channel analogy thus applies less in the construction of capacity-achieving codes and more in the performance of given codes at various levels of noise.

There are trivial and exotic forms of non-uniqueness that qualify the binary-contrast decoding problem. These all apply even in the case of zero noise ($\mu\to \infty$). Trivial non-uniqueness arises as a result of symmetry. Cyclically shifting a binary sequence, reflecting it, or reversing the contrast values (when these are $\pm 1$), all do not change the Fourier intensities. This follows from the fact that these operations preserve the sequence-autocorrelation, which by a Fourier transform is equivalent to the intensities. Exotic non-uniqueness can be understood by the same device. Suppose a binary sequence $b$ factors as the cyclic convolution product of two binary sequences: $b=b_1\ast b_2$. The convolution $b'=b_1\ast R(b_2)$, of $b_1$ with the reversal of $b_2$, will then have the same cyclic autocorrelation, and therefore intensity, as $b$. Consequently if $b'$ is binary and not related by a symmetry to $b$, then we have at least two decodings. The two sequences of length $N=13$ below are an example of this phenomenon:
\begin{equation*}
\begin{array}{rrrrrrrrrrrrr}
+1& +1 &+1& +1& +1 &-1& +1& -1 &-1& +1 &+1 &+1 &-1,\\
+1& +1& +1& +1& +1& +1& -1& +1& -1& +1& +1& -1& -1.
\end{array}
\end{equation*}
We could dispense with the above cases of non-uniqueness in the absence of noise by agreeing that the actual content of the ``messages" is the sequence-autocorrelation. For the most part this is the same as treating the magnetic contrasts as symmetry classes, the exotic cases being relatively rare\footnote{We believe the fraction of sequences which have the same autocorrelation as a sequence not in the same symmetry class vanishes exponentially with the sequence length $N$.}.

Error-correcting codes in communications exploit block structure to realize information rates approaching Shannon's channel capacity. With suitable encoding/decoding the probability of even one bit being flipped can be made to vanish exponentially in the size of the block when the information rate is below the channel capacity. We show below that a similar phenomenon appears to apply to diffractive imaging, where the ``encoding" is always the same (and not under the control of the imaging scientist). In concrete terms this means that the fidelity of magnetic contrast reconstruction is typically much better than naive estimates would predict.

\section{Capacity of the Poisson channel}

In communications theory the discrete-time Poisson channel is defined by the conditional probability (or transition matrix)
\begin{equation}\label{Poisson}
p(k|w)=\frac{w^{k}}{k!}\exp{(-w)},
\end{equation}
for receiving $k$ photons when the input intensity\footnote{By ``intensity" we actually mean photon fluence integrated over the area of a detector pixel, a dimensionless quantity.} signal is $w$. The capacity of the channel \cite{Shannon} is found by maximizing the mutual information associated with the joint probability
\begin{equation}
p(k,w) = p(w) p(k|w)
\end{equation}
with respect to the prior distribution $p(w)$. Mutual information is an information-theoretic measure of the degree of correlation of two random variables\footnote{We use capitalized variable names for all random variables.} and in this case is written $I(K,W)$.  The mutual information is the difference of two entropies
\begin{equation}
I(K,W)=H(K)-H(K|W)
\end{equation}
that, in our case, quantify the information provided by the photon counts $k\in K$ about the intensity $w\in W$, with due regard to the loss of information (second term) due to entropy in the counts even when $w$ is known precisely. The entropy and conditional entropy have the following forms when expressed in terms of the probability distributions:
\begin{eqnarray}\label{H(K|W)}
H(K|W)&=&-\sum_{k\in K}\sum_{w\in W}p(k,w)\,\log_2{\,p(k|w)}\\ \label{H(K)}
H(K)&=&-\sum_{k\in K}\sum_{w\in W}p(k,w)\,\log_2{ \sum_{w'\in W}p(k,w')}.
\end{eqnarray}

The maximization of $I(K,W)$ with respect to $p(w)$, to determine the channel capacity, is usually performed with some constraints on $p(w)$ that fix the mean or maximum value of $W$. Since one has very little control over the distribution $p(w)$ in diffractive imaging, other than its mean value, it makes sense to define the capacity of the Poisson channel for the particular form of prior $p(w)$ that applies in most cases. In diffraction theory, the form
\begin{equation}\label{Wilson}
p_\mu(w)=\exp{(-w/\mu)}/\mu
\end{equation}
is known as Wilson statistics \cite{Wilson} and arises when the complex-valued radiation amplitude has a Gaussian distribution as it would when the contrast pixels are modeled as independent random variables. Here and below, $\mu$ is the mean number of photons associated with one intensity measurement.

The information capacity of the Poisson channel, for the prior distribution (\ref{Wilson}), can be calculated using the expressions (\ref{H(K|W)},\ref{H(K)}) and takes the following form \cite{PCC},
\begin{equation}\label{IP}
I_P(\mu)=(\mu+1)\log_2{(\mu+1)}-\frac{\gamma \mu}{\log{2}} -\sum_{k=2}^\infty \frac{ \log_2{k}}{(1+\frac{1}{\mu})^k},
\end{equation}
where $\gamma$ is Euler's constant. We are not aware of any work that considers the construction of block codes that achieve this information capacity. A block code of length $N$ in this context would be a collection of blocks of intensities $(w_0,\ldots,w_{N-1})$ with average value $\mu$. We show below that block codes arise naturally in the context of diffractive imaging and it is their error correcting capacity that reconstruction algorithms can take advantage of.

\section{The Fourier-Poisson channel}

For diffractive imaging it makes sense to define the ``communication channel" in a way that takes into account the block structure of the signal. We will continue our discussion for the case of signals in one dimension, as introduced in section \ref{analogy}, but the same construction applies to imaging in two and three dimensions.

For our signals we will take vectors of contrast values $x=(x_0,\ldots,x_{N-1})$ subject to some prior distribution $p(x)$. In many applications the contrast is real-valued and that will be the case we consider here. The first stage of our channel encodes the contrast into a vector of Fourier intensities $w=(w_0,\ldots,w_{N-1})$, where
\begin{equation}\label{intens}
w_n(x)=\frac{\mu}{N}\left|\sum_{m=0}^{N-1}e^{i \frac{2\pi}{N} m n} \,x_m\right|^2.
\end{equation}
The normalization provided by $\mu$ is such that if the contrast prior $p(x)$ satisfies the constraint $\langle x\cdot x\rangle=N$ on the average power, then the average intensity satisfies $\sum_{n=0}^{N-1} \langle w_n\rangle = N\mu$. The final stage of our channel is the Poisson sampling of the intensities with (\ref{Poisson}) to give the vector of photon counts (nonnegative integers) $k=(k_0,\ldots,k_{N-1})$.

We will refer to the combined operations of Fourier intensity encoding followed by Poisson sampling as the Fourier-Poisson channel, or FPC. The pair of channel variables are $X$, representing the contrast in the imaging experiment and its probability distribution, and $K$,
the photon counts recorded by the detector. The associated mutual information, $I(X,K)$, is the information acquired in a typical experiment about the contrast in a typical sample. Our choice of constructing the ``block code" by the discrete Fourier transform in one dimension corresponds to imaging a one dimensional crystal. The $N$ intensities $w_0(x),\ldots,w_{N-1}(x)$ are the set of Bragg intensities extending to a resolution where the contrast $x=(x_0,\ldots,x_{N-1})$ is sampled at $N$ equally spaced points within the one dimensional unit cell. Because $x$ is real, the intensities have the Friedel symmetry property\footnote{In all our expressions indices are given up to an (irrelevant) multiple of $N$.}: $w_n(x)=w_{-n}(x)$.

In the case of the Gaussian prior
\begin{equation}\label{Gaussian}
p(x) = (2\pi)^{-N/2} \exp{(-{\textstyle \frac{1}{2}}\, x\cdot x)}
\end{equation}
the mutual information of the FPC can be directly related to the capacity of the simple Poisson channel (\ref{IP}). The discrete Fourier transform of $x$, written $\hat{x}$, is a linear transformation of $x$ with the property $\hat{x}\cdot\hat{x}^\ast=x\cdot x$. We may therefore work instead with the Gaussian distribution on the complex variables $\hat{x}_0,\ldots,\hat{x}_{N-1}$. These variables are an equivalent representation provided one takes into consideration the symmetry property $\hat{x}_{-n}=\hat{x}_{n}^\ast$. Since the intensities (\ref{intens}) are given by $w_n=\mu |\hat{x}_n|^2$, the FPC mutual information depends on only $\lfloor N/2+1\rfloor\approx N/2$ independent intensity distributions. With the exception of $w_0$ and $w_{N/2}$ (when $N$ is even), these are identical Wilson distributions (\ref{Wilson}). The exceptions are not of the form (\ref{Wilson}) because the corresponding $\hat{x}$ is real; however, their contribution to the mutual information in the large $N$ limit can be neglected.

\begin{figure}[t]
\begin{center}
\includegraphics{fig1.eps}
\end{center}
\caption{Mutual information of the Fourier-Poisson channel as a function of the mean photon number $\mu$ when the contrast of the crystal has a Gaussian prior. The function $I_G(\mu)$ gives the number of bits of information available per Bragg peak when the average peak has $\mu$ photons.}
\end{figure}

In the preceding we showed that the mutual information for the Gaussian prior is, for large $N$, the mutual information associated with $N/2$ independent and identical joint distributions, a particular one involving $k_n$, $k_{-n}$ and $w_n$ (which always equals $w_{-n}$). The value of $I(X,K)$ for large $N$ is therefore given by $N/2$ times the mutual information associated with the joint distribution
\[
p(k_n,k_{-n}, w_n)=p(k_n|w_n)\, p(k_{-n}|w_n)\, p_\mu(w_n).
\]
Expressing this in terms of the variable $k_n^+=k_n+ k_{-n}$ (instead of $k_{-n}$), the distribution takes the form
\[
p(k_n^+,k_n,w_n)=\frac{w_n^{k_n^+}}{k_n! (k_n^+-k_n)!}\exp{(-2 w_n)}\,p_\mu(w_n)
\]
which is the product of a function of $k_n$ and a function of $w_n$. The mutual information for this pair of variables therefore vanishes and thus the mutual information is the same as that for the marginal distribution
\[
p(k_n^+, w_n) = \sum_{k_n=0}^{k_n^+}p(k_n^+,k_n,w_n)=\frac{(2 w_n)^{k_n^+}}{k_n^+!}\exp{(-2 w_n)}\,p_\mu(w_n).
\]
Expressing this in terms of the variable $w_n^+=2 w_n$, we obtain
\[
p(k_n^+, w_n^+) = \frac{(w_n^+)^{k_n^+}}{k_n^+!}\exp{(-w_n^+)}\,p_{2 \mu}(w_n^+),
\]
which is the same as the probability distribution that defines the simple Poisson channel but with $\mu$ replaced by $2\mu$. We therefore obtain the following result for the mutual information per contrast element of the FPC with Gaussian contrast prior:
\begin{eqnarray}
I_G(\mu) &=& \lim_{N\to\infty}I(X,K)/N\\
&=&\frac{1}{2}I_P(2\mu).
\end{eqnarray}

The function $I_G(\mu)$ is plotted in Figure 1. Although it was derived for the case of a one dimensional crystal, the same result holds for periodic contrast in any number of dimensions.  In concrete terms, this mutual information measures the maximum number of bits of information per Bragg peak we can obtain about the contrast in a crystal as a function of the mean number $\mu$ of photons detected in a typical Bragg peak. For example, suppose we have a weakly scattering crystal and collect only 100 photons per Bragg peak on average. In this case the information in $M$ Bragg peaks will provide about 2 bits of information about the contrast (e.g. electron density) at $M$ sample points in the unit cell. In the remainder of this paper we will explore the implications of this function for a type of contrast where even 1 bit of information is sufficient to reconstruct the essential structure of the sample.

\section{Fidelity of binary contrast reconstruction}\label{Fidelity}

We now turn to the problem of reconstructing a signal $x$ known to be binary, that is, where each component takes only the values $\pm 1$. In this case the prior probability $p(x)$ is the uniform distribution on the hypercube $B=\{-1,1\}^N$. Alternatively, we can continue to work with the FPC defined by the Gaussian prior and view the set $B$ as a code.
From the capacity $I_G(\mu)$ per contrast element of the FPC we derived in the previous section we obtain the threshold value $\mu_c\approx 9.543$ where the capacity exceeds 1 bit. The most optimistic result for binary contrast reconstruction would therefore be that the code $B$ achieves exactly this capacity for $\mu>\mu_c$. We will see that although $B$ fails to achieve the channel capacity in a strict sense, something close to this does indeed hold.

To investigate reconstruction fidelity, that is, the error correction properties of our chosen binary code of signals, we consider the behavior of the optimum decoder. By definition, the latter is given by the maximum likelihood principle, where the decoded $x\in B$ maximizes the conditional probability
\begin{equation}\label{p(k|x)}
p(k|x)=\prod_{n=0}^{N-1}\frac{w_n(x)^{k_n}}{k_n!}\exp{(-w_n(x))}
\end{equation}
for any given received vector of photon counts $k$. We can simplify this decoding function by taking its logarithm and summing pairs of photon counts $k_n^+=k_n+ k_{-n}$ that arise from equal intensities:
\begin{equation}
d(k^+|x)=\sum_{n=1}^{\lceil N/2\rceil-1}k^+_n \log{w_n(x)}.
\end{equation}
For simplicity we have omitted from the sum the terms $n=0$ and $n=N/2$ (for even $N$) which are unpaired and can be neglected in the limit of large $N$, and henceforth abbreviate the index of the final term as $N/2$. We have also omitted the factorials of the photon counts since they are irrelevant when comparing the codewords $x$. Similarly, since
\[
\sum_{n=0}^{N-1} w_n(x) = \mu N
\]
for all $x\in B$, it too can be omitted in the definition of the decoding function.

\begin{figure}[t]
\begin{center}
\includegraphics{fig2.eps}
\end{center}
\caption{Error rate (ER) in binary contrast decoding as a function of the mean photon number $\mu$ for block codes up to size $N=1024$. Each curve gives the probability that the contrast sampled at $N$ points within the unit cell of a crystal will be reconstructed with one flipped value when the best possible (maximum likelihood) reconstruction criterion is used. The dashed line marks the value $\mu=\mu_c$ where the Fourier-Poisson channel has capacity 1 bit.}
\end{figure}

The severest challenge for the decoding function is its ability to detect a single flipped bit. This follows from the fact that the Fourier transform, as a unitary transformation, preserves the Euclidean distance between codewords. A pair of codewords $x$ and $x'$ differing in just one bit (one sign reversal) have $2=\|x-x'\|=\|\hat{x}-\hat{x}'\|$, that is, Fourier transforms with the smallest possible separation. This is a sufficient, though not necessary condition for the corresponding intensities $w(x)$ and $w(x')$ to be close and difficult to discriminate. A pair of Fourier transforms might have a larger separation and yet have nearly the same intensities if all the complex amplitudes were mostly phase rotations. But apart from symmetry related codewords this scenario is statistically unlikely.

Our fidelity tests have therefore focused on the ability to detect a single flipped bit. We performed numerical simulations where a random codeword $x\in B$ is selected and the corresponding intensity vector $w(x)$ is Poisson sampled, and pairwise summed, to give a vector of photon counts $k^+$. The decoding function $d(k^+|x)$ is then evaluated and compared with $d(k^+|x')$, where $x'$ differs from $x$ by a single flipped bit. A decoding error is declared if $d(k^+|x')>d(k^+|x)$ for any of the $N$ bits that can be flipped. Finally, the decoding error rate, or ER, is obtained from the frequency of decoding errors when this procedure is applied to a million codewords $x$ selected uniformly from $B$.

Results of our fidelity tests are shown in Figure 2 and appear consistent with there being a  transition to unique reconstructions at a critical mean photon number $\mu_c$ near 10. For example, the small ER at $\mu=10$ for size $N=1024$ shows that with as few as ten photons per Bragg peak it is very unlikely that an optimal reconstruction will make an error in even one flipped contrast element out of all $1024$ in the unit cell. The Figure also shows that the ER tail extends beyond the analytically determined value $\mu_c\approx 9.543$ where the capacity of the FPC is 1 bit. To properly decide whether the binary block code achieves this capacity it is necessary to investigate the behavior of the ER in this tail, in particular, whether the ER vanishes in the limit of large $N$ for $\mu>\mu_c$.

\begin{figure}[t]
\begin{center}
\includegraphics{fig3.eps}
\end{center}
\caption{Bit error rate (BER) corresponding to the data shown in Figure 2. For fixed mean photon number $\mu$, even in the region $\mu>\mu_c$, the BER does not vanish with the block size $N$ and instead approaches a limiting value given by the error function (dashed curve).}
\end{figure}

To study the more rigorous decoding criterion we consider a quantity related to the ER that has a nicer limiting behavior with the block size $N$. This is the bit error rate, or BER, defined by
\begin{equation}
\mathrm{BER}=1-(1-\mathrm{ER})^{1/N}.
\end{equation}
Solving for ER in terms of BER we see that the latter corresponds to an interpretation of the ER as arising from independent bit-wise errors throughout the block. Figure 3 shows the BER obtained from the same data of Figure 2 on a logarithmic scale emphasizing the tail region. The convergence of the tail to a limiting form shows that the binary code does not achieve the full capacity of the FPC in the limit of large block size --- the rate of decoding errors remains finite for $\mu>\mu_c$.

The slight failure of the binary contrast code to achieve the 1 bit capacity of the FPC for $\mu>\mu_c$ may be more of interest to coding theory than the reconstruction problem in diffractive imaging. In any case, it is interesting to see that the BER vanishes apparently exponentially\footnote{A naive estimate based on the signal-to-noise ratio $\sqrt{\mu}$ in each of multiple independent intensity measurements would predict a much higher error rate.} in $\mu$ and it is this behavior that we now turn to.

Consider the random variable defined by
\begin{eqnarray}
\Delta(k^+,x)&=&d(k^+|x)-d(k^+|x')\\
&=&\sum_{n=1}^{N/2}k_n^+\log{\left(w_n(x)/w_n(x')\right)},\label{Delta}
\end{eqnarray}
where as before $x'$ is a single bit-flip applied to $x$. The vectors $x$ and $k^+$ are themselves random variables: $x$ has a uniform distribution on the set of binary codewords $B$ and $k_n^+$ is sampled from the Poisson distribution with mean $2 w_n(x)$. By symmetry the position of the flipped bit does not affect the distribution of $\Delta$ and we may take it to be the first bit. An event where $\Delta$ is negative corresponds to a decoding error since then the flipped vector $x'$ has higher likelihood than the vector $x$ from which the photon counts where sampled. If we can evaluate the probability of a negative $\Delta$ we will know the BER under the assumption that this quantity is dominated by single flip errors.

It is a straightforward exercise to determine the probability of a negative $\Delta$ if we can assume the distribution of this variable is Gaussian in the large $N$ limit. Both the mean and variance are found to have finite limits as $N\to\infty$:
\begin{equation}
\langle\Delta\rangle=4\mu\qquad\qquad
\langle\Delta^2\rangle-\langle\Delta\rangle^2=8\mu.
\end{equation}
Since these are the only parameters that determine a Gaussian, the BER is given as
\begin{eqnarray}
\mathrm{BER}&=&\frac{1}{\sqrt{2\pi}}\int_{\sqrt{2\mu}}^\infty \exp{(-t^2/2)}\, dt\\
&=&\frac{1}{2}\mathrm{erfc}(\sqrt{\mu})\\
&\sim&\exp{(-\mu)}/\sqrt{4\pi \mu}.
\end{eqnarray}
The exponential decay of the BER with $\mu$ is well supported by the simulations (Fig. 3). Although suboptimal in the sense of a code, for not vanishing for $\mu>\mu_c$, the rapid decay of the BER with photon counts has practical significance for diffractive imaging where the block size $N$ is fixed by the resolution and $\mu$ can be increased through the incident flux of radiation. The exponential behavior implies that substantial fidelity enhancement can be realized through rather modest increases of flux.

We conclude this section by speculating why Fourier intensity encoding of binary sequences apparently gives a very good, if not optimal, block code for the Poisson channel. Shannon \cite{Shannon} showed that a capacity achieving code, in the limit of large blocks, is realized by the construction where codewords are selected as random independent samples of the source distribution. In the case of the FPC with block size $N$ and $\mu>\mu_c$, where the capacity is 1 bit, this corresponds to drawing $2^N$ samples $x$ from the Gaussian (\ref{Gaussian}) and computing their intensities $w(x)$. Although this set of block-intensities is a capacity achieving code, it is not practical because the decoder needs the very long list of random $x$ used to generate the intensities. We believe the good behavior of the intensity code formed by just the binary $x$ is explained by the fact this set of block intensities is ``close" to being a random code as in Shannon's construction. Consider an arbitrary (not necessarily consecutive) selection of Fourier components $\hat{x}_1, \hat{x}_2, \ldots,\hat{x}_n$ with $n<N$ fixed. From the central limit theorem we know that these have independent complex-Gaussian distributions in the $N\to\infty$ limit even when $x$ is restricted to the binary contrasts $B$. The corresponding intensities $|\hat{x}_1|^2, |\hat{x}_2|^2, \ldots,|\hat{x}_n|^2$, in this fixed set of components, thus looks like a random code. The correlations that spoil the optimality of the binary construction are therefore rather global in scale, involving a number of Fourier components that grows with the size of the block. This reasoning also applies to our analysis above of the decoder variable $\Delta$ as it is a sum over Fourier components (\ref{Delta}). The independence properties of finite sets of these terms (as $N\to\infty$) lends support to the Gaussian statistics we assumed for their sum.

\section{Spin glass interpretation of the uniqueness transition}

The transition from a regime of image reconstructions with high rates of error to a regime of unique, practically error-free reconstructions, has an analog in statistical mechanics: the spin glass \cite{SpinGlass}. We explore this analogy for the case of binary contrast reconstruction but expect it to apply to phase retrieval problems more generally.

A spin glass is characterized by a paramagnetic phase at high temperature and multiple ``ordered" equilibrium phases at low temperature. In our reconstruction problem the role of temperature is played by shot noise and the control parameter is the mean photon number $\mu$. The thermodynamic phase behavior is exhibited not by any one particular reconstruction problem, but by the ensemble\footnote{It is in this respect that our treatment differs from other applications of spin glass ideas to the noisy channel coding problem \cite{Sourlas, MM}.} that comprises \textit{all} reconstruction problems of a particular size $N$. Either the algorithm (assumed to be optimal or near optimal) gives the correct reconstruction for almost all problems, or it almost always gets it wrong. This same ensemble appears in the definition of the mutual information $I_B(X,K)$ between binary codewords $X$ and their diffraction intensities as transmitted by the photon counts $K$. The distribution of the vector of photon counts
\begin{equation}\label{p(k)}
p(k)=\frac{1}{2^N}\sum_{x\in B}p(k|x).
\end{equation}
has the form of a sum of $2^N$ sub-distributions (\ref{p(k|x)}), grouped into $M$ symmetry classes (with respect to cyclic shifts, reflection, reversal) of binary codewords that have unique Fourier intensities. When the sub-distributions are well separated, as is required for a reconstruction algorithm (decoder) to have a low error rate, then the entropy $H(K)$ of the distribution $p(k)$ is simply the entropy $H(K|X)$ of a typical sub-distribution $p(k|x)$ augmented by the logarithm of the number of sub-distributions, or $\log_2{M}\sim N$. In this case we therefore have
\[
I_B(\mu)=\lim_{N\to\infty} I_B(X,K)/N=\lim_{N\to\infty} \log_2{M}/N=1
\]
for the mutual information per bit. On the other hand, at small $\mu$ or high noise, when the sub-distributions are not well separated, the entropy of $p(k)$ is close to the entropy of $p(k|x)$ for typical $x$ and the mutual information is small. The noisy reconstruction problem would exhibit a true thermodynamic phase transition in the large $N$ limit at some noise threshold $\mu_c$ if the sub-distributions became increasingly well separated with increasing $N$ for $\mu>\mu_c$.

In the previous section, however, it was shown that the decoding error rate for binary contrast remains finite (although very small) in the limit of large $N$. The sub-distributions $p(k|x)$ therefore do not become perfectly separated in the thermodynamic limit, and there is no phase transition in the strict sense. On the other hand, because the error rate vanishes so rapidly --- exponentially in $\mu$ --- the spin glass is still a useful analogy. The marginal distribution of photon counts (\ref{p(k)}) for the FPC is interesting as a definition of a (quasi) spin glass since the effective Hamiltonian\footnote{The effective Hamiltonian given by the logarithm of (\ref{p(k)}) is more complex in structure than standard spin glass Hamiltonians.}, when $p(k)$ is interpreted as a Boltzmann distribution, has no quenched disorder. Had we used $2^N$ random Gaussian samples (rather than simply all the binary samples) in the construction of our code for the FPC, then by Shannon's argument the error rate would vanish in the thermodynamic limit and the resulting statistical model would have a true spin glass transition. The effective Hamiltonian in this case would, however, have quenched disorder.

To support the spin glass analogy for the case of binary contrast, where the corresponding Hamiltonian has no quenched disorder,
we performed numerical simulations on systems up to block size $N=23$. The Metropolis algorithm was used to sample the distribution $p(k)$ defined by (\ref{p(k)}), (\ref{p(k|x)}) and (\ref{intens}). In every update the change in each component was limited to $\Delta k_n\in\{-1,0,1\}$. From the average $\langle -\log_2{p(k)}\rangle$ we obtained the entropy $H(K)$. The conditional entropy $H(K|X)$ involved less effort because the sum over counts $k$ can be evaluated explicitly, leaving a single sum over the approximately $M=2^{N-2}/N$ symmetry classes of binary sequences to be performed numerically. By contrast, such a sum needs to be performed for every update in the sampling of $p(k)$. The mutual information is obtained from the difference of entropies, which we now normalize as $I_B(\mu)=I_B(X,K)/\log_2{M}$ to take into account finite size effects arising from the symmetry classes.

The counterpart of the Edwards-Anderson spin glass order parameter \cite{EdwardsAnderson}, for our model, is given by
\begin{equation}
q(\mu)=\frac{1}{N-1}\sum_{n=1}^{N-1}\left(\langle k_n(t)\rangle_t/\mu-1\right)^2,
\end{equation}
where a time average is taken over the photon counts $k(t)$ generated by local $k$-step dynamics. In the ergodic phase, where $\langle k_n(t)\rangle_t=\mu$ is independent of $n$ and $q$ vanishes, the time series origin need not be specified. But because $k(0)$ does matter in the spin glass phase, we additionally average $q(\mu)$ with respect to $k(0)$ in all the simulations. In this ``basin average", we sample $k(0)$ by uniformly sampling the binary sequence intensities $w(x)$, $x\in B$, and setting $k_n(0)=\lfloor w_n\rfloor$ (\textit{i.e.} the most likely photon counts for that sequence). Assuming, in the spin glass phase, that the dynamics is restricted to the basin specified by a single intensity distribution $w$, we have $\langle k_n(t)\rangle_t=w_n$ and the basin average, indicated by the overbar, takes the form
\begin{equation}\label{qave}
\bar{q}(\mu)=\frac{1}{N-1}\sum_{n=1}^{N-1}\left( \overline{w_n^2}/\mu^2-2\,\overline{w_n}/\mu+1\right).
\end{equation}
In the limit of large $N$ the intensities (for $n\ne 0, N/2$) have Wilson statistics with $\overline{w_n}=\mu$ and $\overline{w_n^2}=2\mu^2$, giving the basin average $\bar{q}(\mu)=1$ in the spin glass phase. The basin averages in our simulations used 200 samples.

\begin{figure}[t]
\begin{center}
\includegraphics{fig4.eps}
\end{center}
\caption{Normalized mutual information, computed numerically for binary codewords of length $N=23$, as a function of the mean photon number per bit, $\mu$. The function $I_B(\mu)$ would equal unity for $\mu>\mu_\mathrm{c}\approx 9.543$ (dashed line) in the case of a random Gaussian code. For binary codewords, however, the saturation is imperfect by an exponentially decaying function of $\mu$ that is not resolved by the computation.}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics{fig5.eps}
\end{center}
\caption{Counterpart of the Edwards-Anderson order parameter $\bar{q}(\mu)$, computed numerically for the binary codeword spin glass model with $N=23$ as a function of $\mu$. The averaging time $T$ (number of Metropolis updates) for ergodic behavior grows dramatically as $\mu$ approaches the transition region at $\mu_c\approx 9.5$.}
\end{figure}

Simulation results for the mutual information $I_B(\mu)$ and the order parameter $\bar{q}(\mu)$ are shown in Figures 4 and 5. We see that $I_B(\mu)$ saturates quickly to 1 for $\mu$ near the value $\mu_c\approx 10$ where the FPC has 1 bit capacity. From our analysis of decoding error rates in the previous section, however, we expect that $I_B(\mu)$ fails to saturate by an amount that is exponentially decreasing in $\mu$ even at infinite $N$. Our simulation data show that these corrections are actually quite negligible. The spin glass quasi-transition, in the case of $\bar{q}(\mu)$, is most evident in the behavior with respect to the averaging time period. We see in Figure 5 that as $\mu_c$ is approached from below, the vanishing of the order parameter, and the restoration of ergodicity, requires increasingly long averaging times. For $\mu>\mu_c$ the kinetics of basin-hoping is likely dominated by pairs of sub-distributions $p(k|x)$ with sequences $x$ differing by a single flipped bit. Since hopping to a new sub-distribution is similar to making a decoding error, we expect the time scale for ergodic behavior to grow exponentially with $\mu$ in this (quasi) spin glass ordered phase.

\section{Performance of iterative phase retrieval algorithms}

Practical phase retrieval algorithms, for reconstructing contrast from intensity data, are generally not designed to deal with noise. Maximum likelihood decoding (section \ref{Fidelity}), although optimal from the perspective of noise, is not feasible as it requires an exhaustive search. For the binary contrast problem we have seen that unique (zero flipped bit) reconstructions are impossible for $\mu<\mu_c\approx 9.543$ and that above $\mu_c$ the error rate decays so rapidly that essentially perfect reconstructions are possible in principle. The behavior of practical phase retrieval algorithms in this transition regime is therefore of interest. We have performed extensive simulations with the difference map algorithm \cite{diffmapPNAS} that show its ability to deal with noise is close to that of the best possible algorithm.
Because the simulations in the present study used exactly the same constraint projections and parameters given previously \cite{diffmapPNAS}, we only need to specify how the algorithm was adapted to work with noisy data.

\begin{figure}[t!]
\begin{center}
\includegraphics{fig6.eps}
\end{center}
\caption{Time series of the difference map error $\epsilon$ for two values of the noise parameter and $N=50$; top: $\mu=15$, bottom: $\mu=20$. Incompatibility of constraints as a result of noise prevents $\epsilon$ from vanishing. As the uniqueness threshold $\mu\approx 10$ is approached, the two-state distribution seen in the lower panel tends to the single steady-state of the upper panel. The reconstructions obtained at the lowest $\epsilon$ in both of these experiments yielded valid solutions.}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics{fig7.eps}
\end{center}
\caption{Phase retrieval success rate of the difference map algorithm as a function of the mean photon number per bit, $\mu$, for four different binary sequence lengths $N$.}
\end{figure}

The input for each phase retrieval experiment was a single Poisson sampling, $k_n$, of the true intensities $w_n$ of a binary sequence selected at random. Given a photon count $k_n$ the most-probable intensity estimate, based only on Poisson statistics, is $\tilde{w}_n=k_n/(1+1/\mu)$. Although this is simply a uniform rescaling, it needs to be applied in order to preserve the $\pm 1$ scale of the binary contrast projection used by the algorithm. After symmetrization of Friedel pairs, these intensity values were used as hard constraints in the algorithm's Fourier magnitude constraint projection. Because of the errors in the magnitude estimates, the constraint satisfaction problem no longer has a true solution, and a protocol must be established to recover binary sequence candidates in the absence of fixed points. Our procedure was to let the algorithm run for a fixed number of iterations and keep a record of the smallest error metric $\epsilon$ over the course of the run. The binary sequence returned by the binary-value projection, at the iteration with smallest $\epsilon$, was then output as the solution candidate. By repeating such experiments, each derived from a different, randomly selected sequence, we compiled reconstruction success rates for various values of $\mu$. A reconstruction was considered a success if the Fourier intensities of the solution candidate exactly matched those of the sequence that produced the noisy data. Samples of the $\epsilon$ time series for successful reconstructions at two levels of noise are compared in Figure 6.

Plots of the reconstruction success rates are shown in Figure 7 for a range of sequence lengths up to $N=60$. For each $N$, the same number of iterations was used for each value of $\mu$, and this number was increased until there was no change in the success rates, as measured by the mean of $10^3$ trials. At the largest size, $N=60$, each solution attempt used $10^7$ iterations.
These simulations show that the success rate of the difference map algorithm is close to the best possible. The behavior of the success rates with respect to $N$, specifically the narrowing of the transition region, is consistent with the existence of a thermodynamic transition.

\section{Summary and conclusions}

We have shown that in the case of binary contrast reconstruction, where the free variables are not continuous, and in the limit of strong photon shot noise, such that the intensities do not provide precisely known data, an information theoretic framework is useful and should replace constraint counting as the means for assessing the feasibility of phase retrieval. The usual criterion of a reconstruction problem being under- or over-constrained is replaced by a criterion based on the mutual information between noisy data and contrast variables. When the mutual information exceeds the target entropy of the contrast, a unique reconstruction is possible in principle. More generally, the mutual information provides a measure of the information that can be extracted from the intensity measurements regardless of noise.

The mutual information exhibits a transition much like that of a spin glass, from an equilibrium ``paramagnetic" phase at high noise, where very many contrasts are equally compatible with the data, to an ``ordered" phase  at low noise comprising multiple equilibrium states, each associated with a unique pairing of data with contrast. We believe this correspondence extends beyond the minimalist model studied here, and reflects favorably on the robustness and prospects of diffractive imaging in general.

This study has focused on a model with periodic contrast. A result of general interest is the information content $I_G(\mu)$ of an average Bragg peak as a function of the mean photon count $\mu$ when the contrast has only a weak (Gaussian) prior. This information measure was then used to obtain the noise threshold for obtaining just one bit of information per contrast pixel, as would be the case for an image of magnetic domains.

\begin{figure}[t!]
\begin{center}
\includegraphics{fig8.eps}
\end{center}
\caption{Simulations of realistic magnetic domain reconstruction \cite{realistic}. Top row: simulated Ising domain pattern in a circular aperture and one quadrant of its continuous diffraction pattern. Bottom row: domains reconstructed using the difference map algorithm from the noisy data on the right (6047 photons from all four quadrants, or about 0.3 photons per aperture pixel).}
\end{figure}

A natural extension of this work would be the study of non-periodic contrast with a known support. The mutual information would then depend not just on the number of photons detected per pixel of the support, but presumably also on the support shape. Encouraged by the results of this study we have performed simulations of magnetic domain reconstruction within a circular support at very low photon counts. Preliminary results shown in Figure 8 indicate that high fidelity reconstructions in this more realistic setting can be achieved at very high levels of noise.

\ack
V. E. thanks Jim Sethna for his comments on the manuscript and the Kavli Institute for Theoretical Physics where this work was completed; KITP is supported by the
National Science Foundation under Grant No. NSF PHY05-51164. Support for this work was provided by the Department of Energy grant DE-FG02-05ER46198.

\section*{References}

redpajama_set_name: RedPajamaArXiv


text:
A New Voice in the Book World: The Author
By Porter Anderson, March 21st 2014
http://tcat.tc/1en5mxu
I want all authors to stand up for each other, ask for better pay, fairer terms, saner contracts. This only works if writers of all stripes get together and do what the screenwriters were able to do when Hollywood digitized. That should be our model. — Hugh Howey, author, Wool, Shift, Dust, Sand, Peace in Amber
Are you a reader of books? Are you a writer of them?
Are you a publisher of books? Are you an agent, who finds good authors and sells their manuscripts to that publisher? Or are you an editor who takes that material at the publishing house and mines it for the author's best intents? Maybe you're the cover designer who creates a visual metaphor for the story.
Where do you stand? It will affect how the digital disruption sounds to you.
Because of all things, the digital dynamic in publishing means that for the first time in history we're starting to hear the collective voice of our authors unfiltered by publishing houses.
One Divine Disruption
I stand for the ability of those who choose to write for a living to have the best opportunities possible. It's a narrow focus, but it's one I'm passionate about. I've been passionate about this for longer than I've been writing. It goes back to my book review and bookstore employee days. As a reader who loved stories, I cared for those who created them. Now that I'm on the other side and have become friends with storytellers, this cause is strengthened. And the more I learn about the abuses authors suffer, the more I want to speak out.
That's more from author Hugh Howey, in an essay, Bread and Roses.
Depending on your position in our culture's most extravagant treasure—our writings, our use of language, literature—those comments may sound inflammatory or nurturing; bold or cowardly; wise or naïve.
The contradictory nature of their reception darkly mirrors the whiplash confusions of the digital disruption in publishing today.
To a traditional publisher, digital can appear to be giving away the farm. It's making it possible for authors to publish and sell all by themselves; to reach right out and contact readers; to build community around their work; and to keep most of the money their books bring in, instead of handing over the cash to the publishers.
As crazy as this sounds, many publishers don't know their own books' readers. Until very recently, the "customer" to a publisher was the buyer of books at a chain of bookstores, or the distributor who would broker those books to the shops.
And digital for authors? Self-publishing is hard work and comes without the distribution leverage and prestige of the big houses. Many authors would prefer having a traditional contract. But those who take their manuscripts into their own hands talk of the control they have over their output. And access: after they publish their own books and track their own sales and collect their own revenues, they then can basically pick up the phone of social media and call their own readers. Howey speaks of self-publishing writers being "maniacally focused" on readers. They are close to the customer in ways never imagined by old publishing regimes.
No wonder publishers feel suddenly sidelined and frequently blamed for what many self-publishing authors complain are standard industry practices: paying authors too little; providing them with next to no details about their sales; locking down their rights for punishingly long periods of time; treating authors, in some instances, as talented airheads.
Overexposure: In the Eye of Which Beholder?
Many in publishing today like to call the author Hugh Howey "overexposed." And, as is usually the case when you hear this about a newsmaker, it means his messages are hitting home. For all the fine writers we have working today, we have not, until now, had one who had the time, the temperament, the visibility, and the stamina to articulate with real force the author's experience of the digital dynamic. This is a firebrand, a smart one.
To his detractors' frustration, Howey's own case is as complex as the digital dynamic's many effects on publishing itself.
For someone usually seen as a self-publishing icon, Howey has a lot of traditional contracts. More than 30. He just signed a big one for his new novel with the biggest of all publishers, Random House, its UK division, for publication of his novel Sand in both print and ebook editions. The message?—today's author need not be one thing or another. You do what's best for each book in each country and each readership.
Howey says that standard industry surveys are scaring off authors from considering self-publishing. And so he has mounted his own data-scraping analysis of online ebook sales and rankings—AuthorEarnings.com—in an effort to demonstrate that financial success may be as much in reach for self-publishers as it is for traditionally published authors. In the initial AuthorEarnings report, he writes:
Our fear is that authors are selling themselves short and making poor decisions based on poor data. That is the main purpose for fighting for earnings transparency: helping aspiring writers choose the path that's best for them.
Actual sales data? Not available. The great retailers (not just Amazon) keep it private. That's why the term data suffers a little overexposure of its own these days in publishing. They can't stop talking about how badly they need it.
The people of the publishing world actually do not know how many books there are on the market today.
They do not know how many people are publishing their own books.
They do not know how many sales are being made in the book market.
Anytime you hear "so-and-so million books were published last year," you're hearing a guess, an estimate. No one has those numbers.
This is an industry that has lost control of its own market. And it turns out, that was exactly the chance the authors needed to start talking.
Finding Their Voices
Far from being alone in all this, Howey is simply the most visibly consistent and eloquent of a widening class of entrepreneurial authors.
He's joined, for example, by Bella Andre, Barbara Freethy, Stephanie Bond, Candice Hern, Liliana Hart, and Jasinda Wilder next month at the London Book Fair. They'll be sharing an "Indies Bestseller" booth at Earls Court.
Together, those seven entrepreneurial authors have sold more than 15 million ebooks.
In New York in May, Howey, Andre and Freethy will be joined by CJ Lyons and HM Ward in headlining BookExpo America's first-ever Author Hub on the floor of the trade show at the Javits Center.
By then, Howey will have spoken in conference settings from PubSmart in Charleston to Klopotek's Publishers Forum in Berlin.
Some representatives (not all) of the industry! the industry!—that part of the publishing world that still worries more about the business than the books—will continue to chant that self-publishing is an insignificant part of the marketplace. (Not that they have the actual data to be sure of that.)
Some of them (not all) will commend Howey on his fiction and condemn his efforts at nonfiction each time he tries to address a corporate world they claim he doesn't understand.
More of them (not all) will listen a bit more carefully.
Because one of the most interesting effects of the rising voice of the author, amid the digital dynamic's techno-twists and -turns, is that everyone, top to bottom, side to side, must reassess why he or she got into publishing and books in the first place.
When the digital dynamic rolled over the news industry—and flattened what once was the richness of American journalism into infotainment—each journalist had to ask him- or herself whether, if starting out all over again, the new media would be something they'd want to go into?
Now, it's publishing's turn. What once were "citizen journalists" in newsrooms are self-publishing authors in bookstores. Digital has unlocked the door and amateur authors, aspirational writers, are stampeding the marketplace. We're awash in the early-stage celebration of your mother's memoir and your dad's lawnmower-repair book.
Each time a publishing worker gets quiet and thinks it through—is this what I want to be doing now?—the answer is going to arrive in a voice that sounds a lot like an author. The volume of the creative corps is rising, with typos, awful covers and occasional masterpieces as syncopation.
Just as we might hope it would, that authorial voice, as Howey is diligently, obsessively, painstakingly delivering it, arrives in tones of diplomacy and grace frequently missed by those who feel threatened. In More Pie, Please, he writes:
There's no war here. There's nothing to fight over…When we in the publishing business come at each other with trust, love, and respect, I believe we will find there's plenty of pie to go around. Our goal should not be to point fingers or humiliate, but to lower barriers, to work for contracts that treat people like people, and to allow the great folks in publishing to do what's right instead of what's handed down from on high.
Those aren't fighting words. They're much more promising.
And the authors are waiting for us to answer them.
A version of this post originally appeared at JaneFriedman.com
featured image – Shutterstock
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Our Star Authors: Special Victims Unit
New Howey Report: Self-Published eBook Authors May Out-Earn The Rest By 27%
In Self-Publishing, The Gatekeepers Are Dead. Long Live The Gatekeepers!
Getting Past The 'Bookstore Barrier': Freethy And Ingram Go Into Print
http://porteranderson.com/new-voice-book-world-author/	A New Voice in the Book World: The Author - Porter Anderson
[…] Writing on the Ether: A New Voice in the Book World: The Author […]
http://storyarcs.wordpress.com/2014/04/04/a-belated-sharing-the-wealth-442013/	A Belated Sharing the Wealth: 4/4/2013 | Story Arcs
[…] found both of these on the Passive Voice. Porter Anderson discusses the new voice in the literary world, authors. It's a quaint idea, I know, but the "standard industry practices" of publishing […]


Mandalas have long been used to help people meditate by moving past the "Monkey Mind" that distracts from true focus. From a human's viewpoint, mandalas represent a microcosm of the universe. A small piece of the big picture, a mandala (especially its center) gives a focal point where the mind can expand.
"Mandala" translates to "circle" in Sanskrit and is a powerful symbol for balance, wholeness, and harmony. As seen in atoms, tree rings, galaxies, and crystals, mandalas are unmistakably a pART of nature and have become a sacred symbol in every culture. Despite how far our minds may travel, the geometric (and often symmetrical) shapes tend to draw attention back towards the center, both literally and metaphorically. As a form of meditation, mandalas are healing tools. Participating in drawing, coloring and looking at mandalas can boost the immune system, reduce stress, combat depression, reduce pain, lower blood pressure, and stimulate the release of melatonin – a hormone that can slow aging and promote good sleep. Recent studies have shown mandalas are especially useful in helping deal with coping in regards to illness, enhancing physical functioning, and helping quit harmful habits (like smoking) by relieving stress and occupying the mind (and hands). Mandalas have the potential of changing how people see themselves, the planet, and even their life purpose. Based on Hindu philosophy, the Sri Yantra (the triangular symbol in the center of this mandala page) is an object of devotion. What a perfect symbol to focus on while relating to this piece!