We have previously discussed the bit error rate (BER) performance of M-QAM in AWGN. We now discuss the BER performance of M-QAM in Rayleigh fading. The one-tap Rayleigh fading channel is generated from two orthogonal Gaussian random variables with variance of 0.5 each. The complex random channel coefficient so generated has an amplitude which is Rayleigh distributed and a phase which is uniformly distributed. As usual the fading channel introduces a multiplicative effect whereas the AWGN is additive.
The function "QAM_fading" has three inputs, 'n_bits', 'M', 'EbNodB' and one output 'ber'. The inputs are the number of bits to be passed through the channel, the alphabet size and the Energy per Bit to Noise Power Spectral Density in dB respectively whereas the output is the bit error rate (BER).
The bit error rates of four modulation schemes 4-QAM, 16-QAM, 64-QAM and 256-QAM are shown in the figure above. All modulation schemes use Gray coding which gives a few dB of margin in the BER performance. As with the AWGN case each additional bit per symbol requires about 1.5-2 dB in signal to ratio to achieve the same BER.
Although not shown here similar behavior is observed for higher order modulation schemes such as 1024-QAM and 4096-QAM (the gap in the signal to noise ratio for the same BER is increased to about 5dB).
Lastly we explain some of the terms used above.
Rayleigh Fading is a commonly used term in simulation of Digital Communication Systems but it tends to differ in meaning in different contexts. The term Rayleigh Fading as used above means a single tap channel that varies from one symbol to the next. It has an amplitude which is Rayleigh distributed and a phase which is Uniformly distributed. A single tap channel means that it does not introduce any Inter Symbol Interference (ISI). Such a channel is also referred to as a Flat Fading Channel. The channel can also be referred to as a Fast Fading Channel since each symbol experiences a new channel state which is independent of its previous state (also termed as uncorrelated).
When using QAM modulation, each QAM symbol represents 2,3,4 or higher number of bits. That means that when a symbol error occurs a number of bits are reversed. Now a good way to do the bit-to-symbol assignment is to do it in a way such that no neighboring symbols differ by more than one bit e.g. in 16-QAM, a symbol that represents a binary word 1101 is surrounded by four symbols representing 0101, 1100, 1001 and 1111. So if a symbol error is made, only one bit would be in error. However, one must note that this is true only in good signal conditions. When the SNR is low (noise has a higher magnitude) the symbol might be displaced to a location that is not adjacent and we might get higher number of bits in error.
The concept of hard decision decoding is important when talking about channel coding, which we have not used in the above simulation. However, we will briefly explain it here. Hard decision is based on what is called "Hamming Distance" whereas soft decision is based on what it called "Euclidean Distance". Hamming Distance is the distance of a code word in binary form, such as 011 differs from 010 and 001 by 1. Whereas the Euclidean distance is the distance before a decision is made that a bit is zero or one. So if the received sequence is 0.1 0.6 0.7 we get a Euclidean distance of 0.8124 from 010 and 0.6782 from 001. So we cannot make a hard decision about which sequence was transmitted based on the received sequence of 011. But based on the soft metrics we can make a decision that 001 was the most likely sequence that was transmitted (assuming that 010 and 001 were the only possible transmitted sequences).
i found the formula ber in 16qam as (3/8)*erfc(sqrt(4/10)*snr)).
what is the formula of ber for 16qam in rayleigh channel?
I need full program to calculate BER QAM in rayleigh fading and gaussian noise..
This might be helpful (4-QAM = QPSK).
Thank you, this is very helpful. Do you have a tutorial to calculate the BER vs Eb/No in Rician channel?
Unfortunately at this moment I don't. I hope to post something on this soon.
Please help sir, whats the meaning of n in this program?
I need full program to calculate BER QAM in rayleigh fading. Thanks.
n is the number of points in the EbNodB vs BER plot. You need to first save the code given above in an m-file as QAM_fading.m and then run this wrapper program. The m-file QAM_fading.m will be called for each value of EbNodB and will return the value of BER for each value of EbNodB. Hope this helps.
I left that to you. So you can also learn to program yourself. But if you are having problems you can use the code below.
I have a question with your code at the receiver.
You should know h coefficient before dividing r=r./h. So could you please help to describe how can we obtain/estimate h by using pilot symbol?
That is a very good question. Indeed the channel coefficient 'h' must be known for it to be compensated at the receiver. If you transmitted pilot symbol 's_p' at the transmitter you would receive x = h*s_p+w at the receiver. Dividing 'x' by the known pilot symbol you can get an estimate of the channel i.e. h_est = x/h = s_p+(w/h). Hope this makes sense.
Sir, can you please give me the expression for error probability of M-QAM in Rayleigh fading channel by using Moment Generating method, and compare the result with the expression derived naturally… i.e., by direct integration by parts. And also a matlab program to simulate the results & check that they matches..!!
I am trying to simulate Symbol Error Rate for 16QAM (in Rayleigh fading channel).
I am having difficulties with the simulation. I tried using Simulink to simulate but failed.
Nice to know you. I'm interested in a research about performance of wimax downlink over haps channel in terms of capacity/ packet error rate (PER) vs HAP elevation angle. I'm still confused about that. Could you please provide some matlab scripts related to this topic. I look forward to your help.
I confused with BPSK performance….sorry.
Thanks for your plentiful and valuable information about digital communications. But I guess you may check the Rayleigh channel gain "h" because something strange with the plot. According to the theoretical calculation of 16-QAM BER over Rayleigh , BER = (1/2)*(1-sqrt(rm./(1+rm))) where rm is linear scale Eb/N0, the plot should be start from almost neat to 10^-1 at Eb/No 0dB. It means that the h=randn(n_bits/k,1)+1i*randn(n_bits/k,1). Could you check it again ?
I am student in communication engineering department in Egypt in Arab Academy for Science in Alexandria. I need change type of modulation from BPSK to PAM. Can you help me in change modulation from BPSK to PAM.
and what is the difference with this code?
It is the same code that I posted there.
What do you mean with this ? n_bits=k*1e5 ?? k is in the function. how can i call it with this value?
You are not changing the value of k which is equal to log2(M). You are just defining the number of bits to be passed through the channel, which must be divisible by k so that you can get a whole number for the number of symbols to be passed through the channel.
The answer will be a number, correct? the ber, how to make the graphic that you show here for the ber?
You need to create a "for loop" for different values of EbNodB and store the ber for each EbNodB. You can then plot these using "semilogy".
What are the values of n_bits, M, EbNodB that you used to get those results?
Number of elements in each channel of input X must be an integer multiple of log2(M).


Azerbaijan is a state in the South Caucasus. Russia is in the north, Georgia is in the north-west, Armenia is in the south-west, Iran is in the south, and Turkey in the south-west (Nakhichean border). The eastern border is drawn by the Caspian Sea. Some are in Europe and some in Asia. The capital is Baku. Azerbaijan is carried on a partial transportation. There are 2 shapes as our route. As Georgia we are carrying the other over Iran.
The deepest point of the Caspian Sea is at 1025 meters in Azerbaijan Azerbaijan is the fastest growing country in the world with a growth rate of 35 percent in 2006 and the most reformed country in 2009 according to the World Bank statistic. Azerbaijan is the world's oldest oil exporter. The first oil drilling in the world was made in Absheron in 1847. In Azerbaijan, the first country to open Caspian oil to the West, the world's best quality petroleum is being extracted.


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Brett Lee and Asha Bhosle
Albums by Brett Lee and Asha Bhosle
Including: Can You Tell A Girl
Tags on Brett Lee and Asha Bhosle
Popularity Brett Lee and Asha Bhosle
Review Brett Lee and Asha Bhosle
A total of 3 reviews for Brett Lee and Asha Bhosle:
Guest wrote on 15th May 2007, 20:01h:
excellent. Good to see east meeting west.
Login to vote!
Guest wrote on 1st Apr 2007, 21:03h:
please send me the lyrics for this song if you know it! i can't find it anywhere on the web...i like this song lol
Guest wrote on 27th Jan 2007, 21:08h:
i want lyrics for this song anyone can help me

redpajama_set_name: RedPajamaCommonCrawl


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Super Me – Netflix Movie Review
The Chinese Job
Super Me is an intriguing and fantastical Chinese movie, one that's as ambiguous as it is enjoyable. With some well shot action and a fast-paced story, this Chinese picture combines fantasy, romance and sci-fi nicely.
The story here plays out like a combination of Inception, Nightmare On Elm Street and The Matrix. At the heart of this lies our struggling screenwriter protagonist, Sang Yu. He's up to his eyeballs in debt, exhausted and desperately trying to stay awake. You see, when Sang Yu sleeps a strange demon chases and kills him. Unlike Elm Street however, Sang Yu's death is not permanent and each occurrence forces him awake again.
Torn over what to do, Sang Yu contemplates suicide before stumbling upon a revelation. He has the power to conjure forth treasures from his dreams and bring them into reality. And thus, Sang Yu sets out to become rich, stealing expensive antiques from his own dreams. However, this comes with its own set of problems, especially when a ruthless gangster called Qiang catches wind of what's happening.
Visually, Super Me looks fantastic and there's some really well-shot sequences throughout. Given the premise of the movie, the art department have done a wonderful job setting the scene for this moody story to take place. It's helped along by some solid camera work too, with lots of Dutch tilts, rotating cameras and obligatory slow motion money-shots during some of the more bombastic segments.
It's worth bearing in mind that this moving has a lot of ambiguity surrounding it. The open ending can be interpreted numerous ways. However, despite the wide-spanning ideas and cerebrally charged story, there's actually not a lot of depth to this one. In fact, most of the plot plays out with a really simple idea that's disappointingly predictable and never fleshed out as much as it could (or perhaps should) be. At least for the first two thirds of the run-time anyway.
The ambiguity surrounding the ending does challenge you to come up with your own interpretation of what's going on. There are a few Easter eggs along the way to help, but ultimately this will make or break your experience – especially if you're looking for a resolute ending.
Super Me flirts with some expansive and cerebrally charged ideas but never quite does enough with them. There's certainly room here for an Inception challenger but this Chinese picture never quite rises to the challenge.
If you can go in with an open mind and fancy something a bit different though, Super Me does have enough in the tank to recommend. However, it's also hard to ignore the flaws, making for a somewhat turbulent ride.
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Verdict - 6.5/10
Categories films, Sci-fi Tags films, movies, netflix, sci-fi


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/ Vs. / Spaceship vs Rocket: What Are the Main Differences?
Spaceship vs Rocket: What Are the Main Differences?
Updated: December 5, 2022 by Nate Williams
© Blan-k/Shutterstock.com
Spaceship vs Rocket: Five Must-Know Facts
Rockets and spaceships are two parts of the same launch process. In other words, rockets help the spaceship reach space.
The spaceship was invented nearly 30 years after the invention of the rocket.
Rockets tend to be much cheaper than spaceships. A rocket could cost around $50-60 million, while a spaceship can cost as much as $40-50 billion.
Spaceships are traditionally reused, while rockets don't always have to be.
Rockets are always unmanned, while spaceships can be crewed or uncrewed.
When you think about it, it's hard to wrap your head around the fact that man has achieved space travel. It's just an accepted part of our daily lives that mankind successfully managed to go from the surface of the earth to the outer reaches of space. Why pretend that's not incredible?
To truly appreciate this feat, it's worth understanding how all the moving parts work. Let's begin with spaceships and rockets.
Spaceship vs Rocket: Full Comparison
A spaceship is a spacecraft controlled by a crew.
©Evgeniyqw/Shutterstock.com
When hearing about the latest NASA launch, there are a lot of unfamiliar words that get tossed around by astronauts and experts alike. It can be difficult to know what it all means without an explanation for those of us who aren't NASA astronauts.
For instance, what are the similarities and differences between a spaceship and a rocket? What are the features that set them apart from one another?
This article will help outline those similarities and differences and attempt to make some sense of their distinguishing features. We'll begin with a side-by-side comparison, then look at the similarities and differences after that.
Spaceship vs Rocket: Side-by-Side Comparison
What it is: A vehicle used for space travel An engine used to create thrust
Main Use: Spaceflight missions Launching spacecraft into space
Invented: October 4th, 1957 March 16th, 1926
First Use: With its 1957 launch, Russia's Sputnik 1 became the first spaceship to reach orbit While rockets had been around since 1100 AD, the first liquid propellant rocket was invented in 1926
Created By: Energia Robert Goddard
Key Features: Thrusters, communication systems, aeroshell, parachutes Fuel load, fins, pumps, guidance system
Key Differences: Spaceships transport astronauts Space travel rockets are not manned
Spaceship vs Rocket: What's the Difference?
Now that we've seen the two compared side-by-side, let's take a look at the distinguishing features of both spaceships and rockets.
The key difference between Spaceship vs Rocket is rockets take the spaceship up to space and then fall back down to earth once releasing the ship. Spaceships are manned or unmanned but rockets are never manned. Also, rockets have much more power than spaceships.
The space shuttle is known to be very fast. It accelerates to 29,000 kmh. That's 18,000 miles per hour. Rockets have to reach that speed to be able to leave Earth's gravitational pull to get into space.
Spaceship: The Complete History
A spaceship — also known as spacecraft — is a vehicle made to transport astronauts to space and allow them to fly throughout the galaxy. Originally deriving their design and their use from satellites, spacecraft are now used for a variety of purposes including observation, meteorology, navigation, exploration, and more.
While there are countless differences between spaceships from different manufacturers, the average spaceship consists of a handful of key components: Thrusters (a guidance system), a communication system, fuel, and an aeroshell and parachutes.
The thrusters are used to move the spaceship this way or that, but they differ from rockets because they're much less powerful and don't fall away from the vehicle. The guidance system helps the spaceship land on a planet or dock at a space station.
The communication system and fuel are pretty self-explanatory; the former helps the astronauts talk back and forth with their mission control, and the latter helps the spaceships' thrusters. Lastly, the aeroshell and parachutes assist in the spaceship's re-entry into the earth's atmosphere so that the astronauts can return home safely.
Spaceships can be crewed, semi-crewed, or uncrewed, but their intent remains the same regardless – to fly in outer space, carry cargo, and make observations. Additionally, spaceships can be sub-orbital or orbital. Sub-orbital spaceships enter space and then come back to the earth's surface without fully orbiting the earth.
These spaceships are typically recoverable.
Orbital spaceships, on the other hand, enter a full orbit around the planet. These ships are traditionally unrecoverable.
Rocket: The Complete History
A rocket must obtain thrust from a rocket engine in order to launch.
©Everett Collection/Shutterstock.com
The word "rocket" can have several meanings. Beyond space travel rockets, which are used to help lift a launch vehicle or space shuttle out of the atmosphere and toward the stars and planets beyond, rockets can also refer to weapons, fireworks, and even a leafy vegetable. For all intents and purposes, we'll be discussing space travel rockets.
In essence, a rocket is a projectile that thrusts a spaceship into space. Its design is to expel propellant through the exhaust, propelling massive amounts of thrust toward the ground to push the spacecraft upward at the high speeds necessary to reach space.
Because of the science behind their design, rockets are capable of working in the vacuum of space. Some even work better in space than on earth.
The design of space travel rockets consists of several key components: fuel load, fins, pumps, and the guidance system. The fuel load is what the rocket uses to help the launch vehicle to the high speeds needed to travel to space.
The fins — located at the bottom of the rocket — give the launch vehicle stability, helping to keep the space shuttle on course.
Space shuttle in the space near Earth.
©Dima Zel/Shutterstock.com
The pumps are the part of the design that physically push the fuel where it needs to be. Lastly, the guidance system sets and maintains the course, allowing both the astronauts and the folks back at NASA to see the rocket's progress and assess any technical issues.
Once the rocket helps the launch vehicle reach space, it can officially clock out. In other words, the rocket is then programmed to fall away from the spaceship and return to earth.
While scientists haven't traditionally cared much about a rocket's landing as long as it is kept away from civilian harm, there have been several recent advances in landing technology.
These advances include safely landing the rocket back down to earth to be reused or repurposed again.
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Spaceship vs Rocket: What Are the Main Differences? FAQs (Frequently Asked Questions)
What are the differences between a rocket and a spaceship?
The primary differences between rockets and spaceships are as follows: Rockets take the spaceship to space, then fall back to earth. Spaceships can be manned or unmanned, but rockets are never manned. Additionally, rockets have more power than spaceships.
What are rockets and spaceships used for?
Rockets are used to help spaceships reach space. Spaceships are used for space travel.
What are the three main parts of a space shuttle?
The three main parts of a space shuttle are the solid rocket boosters, the external tank, and the orbiter.
What allows a rocket to be aerodynamic?
It all comes down to velocity. (Or, rather, the difference in velocity.) The motion of the rocket through the air creates the lift and drag that makes the rocket aerodynamic.
What were the six space shuttles?
The six NASA space shuttles were Enterprise, Columbia, Challenger, Discovery, Atlantis, and Endeavour.
How do rockets and spaceships land?
Rockets and spaceships use a combination of gravity, thrusters, and parachutes to land safely back to earth.
Where do rockets and spaceships launch from?
For the past 60 years or so, NASA rockets and spaceships have launched from Cape Canaveral, Florida.
Nate Williams
Nate Williams is a technical writer based in the Midwest. He frequently covers EVs, video games, space, science fiction, personal tech, cybersecurity, and the history of technology at large. His interests range from the simplest machines to the most complex creations (and everything in between). Outside of writing, Nate spends much of his free time hanging out with his wife and dog, watching movies, reading books, and trying to find the best dessert spots nearby.
grc NASA Available here: https://www.grc.nasa.gov/www/k-12/rocket/rktaero.html
grc NASA Available here: https://www.grc.nasa.gov/www/k-12/rocket/rockpart.html
NASA Available here: https://www.nasa.gov/audience/forstudents/k-4/stories/nasa-knows/what-is-a-rocket-k4.html
NASA Available here: https://www.nasa.gov/audience/forstudents/k-4/stories/nasa-knows/what-is-the-space-shuttle-k4.html
NASA Available here: https://www.nasa.gov/returntoflight/system/system_STS_prt.htm
The Shlayer Virus: How it Works and How to Protect Yourself


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Greater Boynton Beach City (North) PUMA, FL
10% growth
In 2017, Greater Boynton Beach City (North) PUMA, FL had a population of 160k people with a median age of 48.2 and a median household income of $55,842. Between 2016 and 2017 the population of Greater Boynton Beach City (North) PUMA, FL grew from 158,694 to 160,129, a 0.904% increase and its median household income grew from $53,703 to $55,842, a 3.98% increase.
The population of Greater Boynton Beach City (North) PUMA, FL is 61% White Alone, 21.4% Black or African American Alone, and 14.1% Hispanic or Latino. 26.6% of the people in Greater Boynton Beach City (North) PUMA, FL speak a non-English language, and 90.8% are U.S. citizens.
The largest universities in Greater Boynton Beach City (North) PUMA, FL are Florida Career College-Boynton Beach (214 degrees awarded in 2017), Hair Design Institute (34 degrees), and Saint Vincent de Paul Regional Seminary (22 degrees).
The median property value in Greater Boynton Beach City (North) PUMA, FL is $207,300, and the homeownership rate is 71.7%. Most people in Greater Boynton Beach City (North) PUMA, FL commute by Drove Alone, and the average commute time is 24.3 minutes. The average car ownership in Greater Boynton Beach City (North) PUMA, FL is 2 cars per household.
Greater Boynton Beach City (North) PUMA, FL borders Palm Beach Gardens & Riviera Beach Cities PUMA, FL, Lake Worth City, Lantana Town & Atlantis City PUMA, FL, Greater Delray Beach City PUMA, FL, and Wellington Village & Agricultural Reserve PUMA, FL.
Photo by Giuseppe Milo
United StatesFlorida
The economy of Greater Boynton Beach City (North) PUMA, FL employs 68.3k people. The largest industries in Greater Boynton Beach City (North) PUMA, FL are Restaurants & Food Services (5,115 people), Elementary & secondary schools (4,324 people), and Hospitals (3,268 people), and the highest paying industries are Scientific research & development services ($261,907), Specialized design services ($137,102), and Aircraft & parts manufacturing ($134,633).
Median household income in Greater Boynton Beach City (North) PUMA, FL is $55,842. Males in Greater Boynton Beach City (North) PUMA, FL have an average income that is 1.29 times higher than the average income of females, which is $47,212. The income inequality in Greater Boynton Beach City (North) PUMA, FL (measured using the Gini index) is 0.467, which is lower than than the national average.
Households in Greater Boynton Beach City (North) PUMA, FL have a median annual income of $55,842, which is less than the median annual income of $61,937 across the entire United States. This is in comparison to a median income of $53,703 in 2016, which represents a 3.98% annual growth.
Look at the chart to see how the median household income in Greater Boynton Beach City (North) PUMA, FL compares to that of it's neighboring and parent geographies.
In 2017, full-time male employees in Greater Boynton Beach City (North) PUMA, FL made 1.29 times more than female employees.
This chart shows the gender-based wage disparity in the 5 most common occupations in Greater Boynton Beach City (North) PUMA, FL by number of full-time employees.
In 2017 the highest paid race/ethnicity of Greater Boynton Beach City (North) PUMA, FL workers was White. These workers were paid 1.04 times more than Asian workers, who made the second highest salary of any race/ethnicity.
This chart shows the race- and ethnicity-based wage disparities in the 5 most common occupations in Greater Boynton Beach City (North) PUMA, FL by number of full-time employees.
In 2017, the income inequality in Greater Boynton Beach City (North) PUMA, FL was 0.467 according to the GINI calculation of the wage distribution. Income inequality had a 0.06% growth from 2016 to 2017, which means that wage distribution grew somewhat less even.
The 2017 the GINI for Greater Boynton Beach City (North) PUMA, FL was lower than than the national average of 0.479. In other words, wages are distributed more evenly in Greater Boynton Beach City (North) PUMA, FL in comparison to the national average.
This chart shows the number of workers in Greater Boynton Beach City (North) PUMA, FL across various wage buckets compared to the national average.
TotalWhiteBlackAsianPacific IslanderOtherTwo Or MoreWhite Non-HispanicHispanic
Weston, Pembroke Pines (Northwest) Cities & Southwest Ranches Town PUMA, FL
Coral Gables City, Pinecrest Village & Kendall (South) PUMA, FL
Wellington Village & Agricultural Reserve PUMA, FL
In 2018, the PUMA with the highest Median Household Income (Total) in Greater Boynton Beach City (North) PUMA, FL was Weston, Pembroke Pines (Northwest) Cities & Southwest Ranches Town PUMA, FL with a value of $109,903, followed by Coral Gables City, Pinecrest Village & Kendall (South) PUMA, FL and Wellington Village & Agricultural Reserve PUMA, FL, with respective values of $102,758 and $102,697.
The following map shows all of the PUMAs in Greater Boynton Beach City (North) PUMA, FL colored by their Median Household Income (Total).
11.9% of the population for whom poverty status is determined in Greater Boynton Beach City (North) PUMA, FL (18.9k out of 159k people) live below the poverty line, a number that is lower than the national average of 13.1%. The largest demographic living in poverty are Females 25 - 34, followed by Females 55 - 64 and then Females 75+.
10,273 ± 800
The most common racial or ethnic group living below the poverty line in Greater Boynton Beach City (North) PUMA, FL is White, followed by Black and Hispanic.
From 2016 to 2017, employment in Greater Boynton Beach City (North) PUMA, FL grew at a rate of 2.37%, from 66.7k employees to 68.3k employees.
The most common job groups, by number of people living in Greater Boynton Beach City (North) PUMA, FL, are Firstline supervisors of retail sales workers (2,251 people), Retail salespersons (2,217 people), and Elementary & middle school teachers (2,015 people). This chart illustrates the share breakdown of the primary jobs held by residents of Greater Boynton Beach City (North) PUMA, FL.
The most common jobs held by residents of Greater Boynton Beach City (North) PUMA, FL, by number of employees, are Firstline supervisors of retail sales workers (2,251 people), Retail salespersons (2,217 people), and Elementary & middle school teachers (2,015 people).
Compared to other pumas, Greater Boynton Beach City (North) PUMA, FL has an unusually high number of residents working as Environmental engineers (7.37 times higher than expected), Firstline supervisors of landscaping, lawn service, & groundskeeping workers (3.79 times), and Human resources assistants, except payroll & timekeeping (3.59 times).
The highest paid jobs held by residents of Greater Boynton Beach City (North) PUMA, FL, by average salary, are Miscellaneous life, physical, & social science technicians, including social science research assistants ($258,694), Physicians & surgeons ($229,342), and Chief executives & legislators ($164,599).
The most common employment sectors for those who live in Greater Boynton Beach City (North) PUMA, FL, are Restaurants & Food Services (5,115 people), Elementary & secondary schools (4,324 people), and Hospitals (3,268 people). This chart shows the share breakdown of the primary industries for residents of Greater Boynton Beach City (North) PUMA, FL, though some of these residents may live in Greater Boynton Beach City (North) PUMA, FL and work somewhere else. Census data is tagged to a residential address, not a work address.
The most common industries in Greater Boynton Beach City (North) PUMA, FL, by number of employees, are Restaurants & Food Services (5,115 people), Elementary & secondary schools (4,324 people), and Hospitals (3,268 people).
Compared to other pumas, Greater Boynton Beach City (North) PUMA, FL has an unusually high number of Textile product mills, except carpets & rugs (4.42 times higher than expected), Office supplies & stationery stores (3.76 times), and Soap, cleaning compound, & cosmetics manufacturing (3.65 times) industries.
The highest paying industries in Greater Boynton Beach City (North) PUMA, FL, by average salary, are Scientific research & development services ($261,907), Specialized design services ($137,102), and Aircraft & parts manufacturing ($134,633).
85.5% of the population of Greater Boynton Beach City (North) PUMA, FL has health coverage, with 36.2% on employee plans, 11.1% on Medicaid, 20.8% on Medicare, 15.8% on non-group plans, and 1.56% on military or VA plans.
Per capita personal health care spending in the puma of Greater Boynton Beach City (North) PUMA, FL was $8,076 in 2014. This is a 5.05% increase from the previous year ($7,688).
Primary care physicians in Greater Boynton Beach City (North) PUMA, FL see 1376 patients per year on average, which represents a 0.218% decrease from the previous year (1379 patients). Compare this to dentists who see 1735 patients per year, and mental health providers who see 703 patients per year.
Primary care physicians in Florida see an average of 1,376 patients per year. This represents a 0.218% decrease from the previous year (1,379 patients).
The following chart shows how the number of patients seen by primary care physicians has been changing over time in Florida in comparison to neighboring states.
Between N/A and 2016, the percent of uninsured citizens in Greater Boynton Beach City (North) PUMA, FL stayed by NaNk% from N/A% to 14.5%.
The following chart shows how the percent of uninsured individuals in Greater Boynton Beach City (North) PUMA, FL changed over time compared with the percent of individuals enrolled in various types of health insurance.
Greater Boynton Beach City (North) PUMA, FL is home to a population of 160k people, from which 90.8% are citizens. As of 2016, 20.9% of Greater Boynton Beach City (North) PUMA, FL residents were born outside of the country (33.2k people).
The ethnic composition of the population of Greater Boynton Beach City (North) PUMA, FL is composed of 97.7k White Alone residents (61%), 34.3k Black or African American Alone residents (21.4%), 22.5k Hispanic or Latino residents (14.1%), 3.06k Asian Alone residents (1.91%), 1.92k Two or More Races residents (1.2%), 458 Some Other Race Alone residents (0.286%), 130 American Indian & Alaska Native Alone residents (0.0812%), and 99 Native Hawaiian & Other Pacific Islander Alone residents (0.0618%).
The most common foreign languages spoken in Greater Boynton Beach City (North) PUMA, FL are Spanish (16,755 speakers), Haitian (14,516 speakers), and French (Incl. Cajun) (1,494 speakers).
In 2017, the median age of all people in Greater Boynton Beach City (North) PUMA, FL was 48.2. Native-born citizens, with a median age of 49, were generally older than than foreign-born citizens, with a median age of 46. But people in Greater Boynton Beach City (North) PUMA, FL are getting getting older. In 2016, the average age of all Greater Boynton Beach City (North) PUMA, FL residents was 48.
As of 2016, 20.9% of Greater Boynton Beach City (North) PUMA, FL residents (33.2k people) were born outside of the United States, which is higher than the national average of 13.7%. In 2015, the percentage of foreign-born citizens in Greater Boynton Beach City (North) PUMA, FL was 20.5%, meaning that the rate has been increasing.
The following chart shows the percentage of foreign-born residents in Greater Boynton Beach City (North) PUMA, FL compared to that of it's neighboring and parent geographies.
In 2017, the most common birthplace for the foreign-born residents of Greater Boynton Beach City (North) PUMA, FL was Haiti, the natal country of 12,080 Greater Boynton Beach City (North) PUMA, FL residents, followed by Colombia with 2,266 and Jamaica with 2,084.
As of 2017, 90.8% of Greater Boynton Beach City (North) PUMA, FL residents were US citizens, which is lower than the national average of 93.2%. In 2016, the percentage of US citizens in Greater Boynton Beach City (North) PUMA, FL was 90.7%, meaning that the rate of citizenship has been increasing.
The following chart shows US citizenship percentages in Greater Boynton Beach City (North) PUMA, FL compared to that of it's neighboring and parent geographies.
In 2017, there were 2.85 times more White Alone residents (97.7k people) in Greater Boynton Beach City (North) PUMA, FL than any other race or ethnicity. There were 34.3k Black or African American Alone and 22.5k Hispanic or Latino residents, the second and third most common racial or ethnic groups.
The following bar chart shows the 8 races and ethnicities represented in Greater Boynton Beach City (North) PUMA, FL as a share of the total population.
16,755 speakers (11%)
1,494 speakers (0.979%)
26.6% of Greater Boynton Beach City (North) PUMA, FL citizens are speakers of a non-English language, which is higher than the national average of 21.9%. In 2017, the most common non-English language spoken in Greater Boynton Beach City (North) PUMA, FL was Spanish. 11% of the overall population of Greater Boynton Beach City (North) PUMA, FL are native Spanish speakers. 9.51% speak Haitian and 0.979% speak Spanish, the next two most common languages.
Greater Boynton Beach City (North) PUMA, FL has a large population of military personnel who served in Vietnam, 1.02 times greater than any other conflict.
In 2017, universities in Greater Boynton Beach City (North) PUMA, FL awarded 270 degrees. The student population of Greater Boynton Beach City (North) PUMA, FL is skewed towards women, with 227 male students and 387 female students.
Most students graduating from Universities in Greater Boynton Beach City (North) PUMA, FL are Black or African American (125 and 46.5%), followed by Hispanic or Latino (74 and 27.5%), White (38 and 14.1%), and Unknown (24 and 8.92%).
The largest universities in Greater Boynton Beach City (North) PUMA, FL by number of degrees awarded are Florida Career College-Boynton Beach (214 and 79.3%), Hair Design Institute (34 and 12.6%), and Saint Vincent de Paul Regional Seminary (22 and 8.15%).
The most popular majors in Greater Boynton Beach City (North) PUMA, FL are Medical Assistant (113 and 41.9%), General Cosmetology (53 and 19.6%), and Medical Insurance Specialist (24 and 8.89%).
The median tuition costs in Greater Boynton Beach City (North) PUMA, FL are $N/A for private four year colleges, and $N/A and $N/A respectively, for public four year colleges for in-state students and out-of-state students.
1 to 2 Year Postsecondary CertificateAssociates DegreeMasters Degree
Medical Insurance Specialist
In 2015, the most common concentation for 1 to 2 Year Postsecondary Certificate recipients in Greater Boynton Beach City (North) PUMA, FL was General Cosmetology with 53 degrees awarded.
This visualization illustrates the percentage of students graduating with a 1 to 2 Year Postsecondary Certificate from schools in Greater Boynton Beach City (North) PUMA, FL according to their major.
Florida Career College-Boynton Beach
Hair Design Institute
Saint Vincent de Paul Regional Seminary
In 2017, the Greater Boynton Beach City (North) PUMA, FL institution with the largest number of graduating students was Florida Career College-Boynton Beach with 214 degrees awarded.
In 2017, 75 men were awarded degrees from institutions in Greater Boynton Beach City (North) PUMA, FL, which is 0.385 times less than the 195 female students who received degrees in the same year.
This chart displays the gender disparity between the institutions in Greater Boynton Beach City (North) PUMA, FL by degrees awarded.
In 2017 the majority of degrees awarded at institutions in Greater Boynton Beach City (North) PUMA, FL were to Black or African American students. These 125 degrees mean that there were 1.69 times more Black or African American students then the next closest race/ethnicity group, Hispanic or Latino, with 74 degrees awarded.
The median property value in Greater Boynton Beach City (North) PUMA, FL was $207,300 in 2017, which is 0.902 times smaller than the national average of $229,700. Between 2016 and 2017 the median property value increased from $188,400 to $207,300, a 10% increase. The homeownership rate in Greater Boynton Beach City (North) PUMA, FL is 71.7%, which is higher than the national average of 63.9%. People in Greater Boynton Beach City (North) PUMA, FL have an average commute time of 24.3 minutes, and they commute by Drove Alone. Car ownership in Greater Boynton Beach City (North) PUMA, FL is approximately the same as the national average, with an average of 2 cars per household.
In 2017, the median property value in Greater Boynton Beach City (North) PUMA, FL grew to to $207,300 from the previous year's value of $188,400.
The following charts display, first, the property values in Greater Boynton Beach City (North) PUMA, FL compared to it's parent and neighbor geographies and, second, owner-occupied housing units distributed between a series of property value buckets compared to the national averages for each bucket. In Greater Boynton Beach City (North) PUMA, FL the largest share of households have a property value in the $300k - $400k range.
In 2017, the median household income of the 65k households in Greater Boynton Beach City (North) PUMA, FL grew to $55,842 from the previous year's value of $53,703.
The following chart displays the households in Greater Boynton Beach City (North) PUMA, FL distributed between a series of income buckets compared to the national averages for each bucket. The largest share of households have an income in the $75k - $100k range.
This chart shows the households in Greater Boynton Beach City (North) PUMA, FL distributed between a series of property tax buckets compared to the national averages for each bucket. In Greater Boynton Beach City (North) PUMA, FL the largest share of households pay taxes in the $3k+ range.
In 2017, 71.7% of the housing units in Greater Boynton Beach City (North) PUMA, FL were occupied by their owner. This percentage declined from the previous year's rate of 72.1%.
This percentage of owner-occupation is higher than the national average of 63.9%. This chart shows the ownership percentage in Greater Boynton Beach City (North) PUMA, FL compared it's parent and neighboring geographies.
Using averages, employees in Greater Boynton Beach City (North) PUMA, FL have a shorter commute time (24.3 minutes) than the normal US worker (25.1 minutes). Additionally, 1.61% of the workforce in Greater Boynton Beach City (North) PUMA, FL have "super commutes" in excess of 90 minutes.
The chart below shows how the median household income in Greater Boynton Beach City (North) PUMA, FL compares to that of it's neighboring and parent geographies.
In 2017, the most common method of travel for workers in Greater Boynton Beach City (North) PUMA, FL was Drove Alone (83%), followed by those who Carpooled (6.97%) and those who Worked At Home (5.38%).
The following chart displays the households in Greater Boynton Beach City (North) PUMA, FL distributed between a series of car ownership buckets compared to the national averages for each bucket. The largest share of households in Greater Boynton Beach City (North) PUMA, FL have 2 cars, followed by 1 car.
Miami-Fort Lauderdale-West Palm Beach, FL
Lake Worth City, Lantana Town & Atlantis City PUMA, FL
Greater Delray Beach City PUMA, FL


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text:
\section{introduction}
\label{sec:intro}

Perhaps the simplest example of finite-time singularities in surface motion is that which occurs when the two
loops supporting a catenoid-shaped soap film are pulled apart beyond a critical separation, rendering the film
unstable.  Extensive experimentation \cite{RobinsonSteen} has shown that the collapsing axisymmetric surface eventually breaks up
through self-contact at multiple points, producing a series of satellite soap bubbles (Fig. \ref{fig1}). Theoretical work on this
dynamical process ranges from Maxwell's original stability analysis \cite{Maxwell_catenoid} to
much more recent computational studies of fluid dynamical models \cite{ListerHinch,ChenSteen} focused on
the nature of the singularities.

In a broader sense, there have been two schools of thought in the study of singularity formation by moving surfaces.  On the one hand,
there is a very substantial body of rigorous work on the simplest geometrical law, namely
motion by mean curvature \cite{mean_curvature_flow}.  However, while this law is a physically realistic description
of interface motion arising from surface diffusion, it cannot be applied to the motion of soap films because
it can not account for the dynamics of the surrounding fluid and the conservation laws that follow.
There are modifications of mean curvature flow that conserve volume enclosed by the surface, but they are not
faithful representations of the dynamics of the surrounding fluid.
On the other hand, there is the large body of more phenomenological work on singularity formation in fluid mechanics
\cite{eggers_review}, where simplified PDE models have been developed to address
self-similar dynamics near singularities.  These models are physically realistic but have generally
lacked rigor.

The two examples where the gap between these two approaches has been bridged is in the context of interface
motion in two dimensions, where a systematic procedure to derive PDEs for the evolution of asymptotically thin fluid layers
has been developed from the exact boundary-integral formulation
\cite{GPS_PRL,PughShelley,GPS_PRL2,GPS_PF}.  These analyses have put on a solid foundation empirical models
\cite{Constantin,Dupont,EggPont}
derived within lubrication theory.
Given these results, the question then arises of whether there is a comparable physical setup in three dimensions in which
a PDE can be derived for the motion of a surface surrounded by an incompressible fluid. The simplest example of this
would naturally be an axisymmetric surface surrounded by an inviscid fluid.
If there is an affirmative answer to this question, such a PDE should have an explicit flux form.

In the inviscid limit, the moving interface can be represented by a vortex sheet with surface tension.
The predominant approach to this problem has been through a Lagrangian formulation \cite{Moore,CaflischLi}, which is the most
appropriate for computational studies \cite{BakerOrzag,NieBaker,Nie,MikeJohn}. However, such an approach does not readily lend
itself to the development of asymptotic models appropriate to thin necks, as would be relevant in the neighborhood
of singularities.  In contrast, an Eulerian formulation would not only be amenable to asymptotic analysis, but would also
be subject more easily to rigorous studies.

\begin{figure}[t]
\centering
\includegraphics[width=0.95\textwidth]{Pesci_etal_figure1.pdf}
\caption{Collapse of a catenoid.}
\label{fig1}
\end{figure}

Here, as a first step, and starting from the Lagrangian formulation, we derive an exact Eulerian dynamics for
axisymmetric vortex sheets with surface tension, and show that it has an explicit flux form.  Naturally,
because the problem is deeply nonlocal due to the Biot-Savart interactions between distant elements of the vortex sheet,
the flux is an integral over the entire sheet whose kernel is precisely the mutual induction between two coaxial loops as
derived by Helmholtz for fluids \cite{Helmholtz_loops} and by Maxwell for electrical currents \cite{Maxwell_loops}.  As
the mutual induction is, by definition, the flux through one loop due to the circulation in the other, this is a
very intuitive result.  We show that known stability results for cylindrical vortex sheets can be recovered by direct
calculation.
While nonlocal, the mutual induction between two coaxial loops of the same radius
is sharply peaked as the distance between the loops vanishes, and this feature suggests a natural asymptotic analysis to
reduce systematically the dynamics to a local PDE.  In this limit we recover at first order
the inviscid version \cite{EggSIAM} of the empirical model first proposed by Eggers and Dupont \cite{EggPont}
for this problem.

\section{Flux Equation}
\label{sec:flux}

We consider an infinite three-dimensional inviscid fluid of density $\rho$ within which is a vortex sheet that is axisymmetric
about the $z$ axis, and whose time-dependent radius is $r(z,t)$.    In this
inviscid limit there is in general a discontinuity in the tangential fluid velocity across the sheet, and this
jump defines the vortex sheet strength $\gamma$.
In the same way that we use a minimal surface as an idealization of a static soap film, here the vortex sheet is
the idealized representation of a moving film endowed with
surface tension $\sigma$.  As is well-known \cite{CaflischLi}, the problem of the self-induced motion of the sheet
reduces to the coupled dynamics of $r(z,t)$ and $\gamma(z,t)$, for which the vortex sheet strength evolves as
\begin{equation}\label{strength}
{\partial \gamma \over \partial t} = -{\sigma \over \rho} {\partial \kappa \over \partial \alpha},
\end{equation}
where $\kappa $ is the mean curvature of
the surface,
\begin{equation}
\kappa=\frac{r_{zz}}{\left(1+r_z^2\right)^{3/2}}-\frac{1}{r\left(1+r_z^2\right)^{1/2}}~,
\end{equation}
and
the radial and axial Lagrangian
velocities are the principal value integrals
\begin{subequations}
 \begin{align}
z_t\equiv {dz\over dt} &= -{1 \over 4 \pi}  \Xint= \gamma^{\prime} \left[ {r^{\prime}}^2  J_2 - r r^{\prime} J_1 \right] d\alpha^{\prime},
\label{CL1a} \\
r_t \equiv {dr\over dt} &= -{1 \over 4 \pi}  \Xint= \gamma^{\prime} r^{\prime} (z - z^{\prime}) J_1 d\alpha^{\prime},
\label{CL1b}
\end{align}
\label{CL1}
\end{subequations}
where
$\alpha'$ is a Lagrangian label, $r'\equiv r(\alpha',t)$, $\gamma'\equiv \gamma(\alpha',t)$, and
\begin{subequations}
\begin{align}
J_1 &=  \int_0^{2\pi} \cos \theta^{\prime} \left[ (z - z^{\prime})^2 + r^2 +
{r^{\prime}}^2  - 2 r r^{\prime}\cos \theta^{\prime} \right]^{-3/2} d\theta^{\prime},\label{J1define} \\
J_2 &= \int_0^{2\pi} \left[ (z - z^{\prime})^2 + r^2 +
{r^{\prime}}^2  - 2 r r^{\prime}\cos \theta^{\prime} \right]^{-3/2} d\theta^{\prime}.
\label{J2define}
\end{align}
\label{Jdefine}
\end{subequations}
Performing the integrations in $\theta'$ yields the following expressions for the velocities in terms of
$K$ and $E$, the complete elliptic integrals of the first and second kind, respectively,
with argument $k^2 =  4 r r^{\prime}/\left[(z - z^{\prime})^2 + (r +r^{\prime})^2\right]$,
\begin{subequations}
 \begin{align}
z_t &= {1 \over \pi}  \Xint= d\alpha' \gamma' r' \left( {k^2 \over 4 r r'}\right)^{3/2}\left[ {r' - r \over 1-k^2}
E(k) -{2r\over k^2} \left[ E(k) - K(k) \right] \right] \equiv \Xint= d\alpha' \gamma' P(\alpha,\alpha') \label{CL2a} \\
r_t &= {1 \over \pi}  \Xint= d\alpha' \gamma' r' (z - z') \left( {k^2 \over 4 r r'}\right)^{3/2}\left[ {2\over k^2}
 \left[ E(k) - K(k) \right]  + {1 \over 1-k^2} E(k) \right] \equiv \Xint= d\alpha' \gamma' Q(\alpha,\alpha').
 \label{CL2b}
 \end{align}
 \label{CL2}
\end{subequations}

In order to obtain an Eulerian equation of motion
from these Lagrangian velocities we make the change of variables from $\alpha$ to $z=z(\alpha,t)$.
This yields \cite{PughShelley}
\begin{subequations}
 \begin{align}
  {\partial r \over \partial t} &= r_t- z_t {\partial r \over \partial z}\label{strength1a} \\
  {\partial \gamma \over \partial t}+z_t {\partial \gamma \over \partial z} &=
  -{\sigma \over \rho}z_{\alpha} {\partial \kappa \over \partial z}.
  \label{strength1b}
 \end{align}
 \label{strength1}
\end{subequations}
The quantity
$\tilde\gamma=\gamma/s_{\alpha}$ represents the actual jump in tangential velocity \cite{Peng,Herrmann}, where $s$ is the arclength, whereas it is the unnormalized
vortex sheet strength
$\gamma$ that appears in  Eqs. \ref{CL2} and \ref{strength1}.  For an Eulerian description
in which we make the change of variables from $\alpha$ to $z=z(\alpha,t)$, it is
natural to consider instead the quantity $\eta=\gamma/z_{\alpha}$ , so
$d\alpha' \gamma'=dz'\eta'$.

For this system, conservation of fluid volume should be expressible as a flux form involving the cross-sectional
area $\pi r^2$, that is,
\begin{equation}
{\partial r^2 \over \partial t} = -{\partial F \over \partial z} ,
\label{flux}
\end{equation}
for some function $F$ which we now seek.  The fact that $2r\partial r/\partial t=\partial r^2/\partial t$, implies that
$2r(r_t-z_t\partial r/\partial z)=-\partial F/\partial z$.  However, to infer $F$ by direct substitution of \eqref{CL2a}
and \eqref{CL2b} into this expression
is cumbersome and non-trivial.  A much better way to find $F$ is to make use of Eqs. \eqref{CL1} and \eqref{Jdefine}.
After some algebra and integrations by
parts, we find
\begin{equation}\label{flux2}
F =  {1\over 2 \pi }\Xint= \eta^{\prime} r r^{\prime} dz^{\prime}
\int_0^{2\pi} { \cos \theta^{\prime} \over \left[ (z - z^{\prime})^2 + r^2 +
{r^{\prime}}^2  - 2 r r^{\prime}\cos \theta^{\prime} \right]^{1/2}} d\theta^{\prime}.
\end{equation}
The integral over the variable $\theta^{\prime}$ can be calculated in terms of elliptic integrals, with the result
\begin{equation}\label{fluxMaxwell}
{\partial r^2 \over \partial t} =  -{1\over  \pi } {\partial \over \partial z}\int dz^{\prime}
\eta^{\prime} {\sqrt {r r^{\prime}} }
M(k)~,
\end{equation}
where $M(k)$ is Maxwell's function first derived for the mutual induction of a pair of circular current loops of radii $r$ and $r'$
at locations $z$ and $z'$,
\begin{equation}\label{Mfunction}
M = {2 \over k} \left(K - E  - {k^2 \over 2} K\right).
\end{equation}
That this remarkably compact result has not been previously obtained in this context may be a consequence of the
natural emphasis on the Lagrangian formulation and its computational applications.  The results in \eqref{fluxMaxwell} and \eqref{Mfunction}
are intuitive, in that the mutual induction $M$ is, by definition, the flux passing through one loop due to another, and so
$F$ is simply the sum of all the individual fluxes through the loop at position $z$.  Moreover, in the case of a vortex loop, the
integrand in $F$ was identified by Helmoltz \cite{Helmholtz_loops} as the stream function of the flow.

In closing this section, we note that if we rewrite the pair \eqref{CL2} as
$z_t = W$ and $r_t=V$, then $\eta_t+W_z\eta=-(\sigma/\rho) \kappa_z$.  Considering $\eta(\alpha,t)=\tilde\eta(z(\alpha,t),t)$, and similarly
$r(\alpha,t)=R(z(\alpha,t),t)$, the Eulerian evolution equations for $\eta$ and $R$ that parallel \eqref{strength1} are
\begin{subequations}
 \begin{align}
  {\partial R \over \partial t} &= V- W {\partial R \over \partial z}\label{Mikea} \\
  {\partial \tilde\eta \over \partial t}+{\partial \over \partial z}\left(W\tilde\eta\right) &= -{\sigma \over \rho} {\partial \kappa \over \partial z}.
  \label{Mikeb}
 \end{align}
 \label{Mike}
\end{subequations}
Multiplying \eqref{Mikeb} by $R$ and integrating the incompressibility relationship $(1/r)(rV)_r+W_z=0$, one can easily show
\begin{equation}
 \frac{\partial}{\partial t}\left(\frac{1}{2}R^2\right)=-\frac{\partial}{\partial z}\int_0^{R(z)}\!\! dr' r' W(z,r').
 \label{tada}
\end{equation}
Introducing the Stokes stream function $\Psi$, which we identify as the integral on the rhs of \eqref{tada},
with $rV=-\partial \Psi/\partial z$ and $rW=\partial \Psi/\partial r$, we
obtain
\begin{equation}
 \frac{\partial}{\partial t}\left(\frac{1}{2}R^2\right)=-\frac{\partial \Psi}{\partial z},
 \label{tada1}
\end{equation}
thus confirming the connection between the functions $F$ and $\Psi$.

\section{Hamiltonian Structure}
\label{Hamiltonian}

In this section we discuss some issues regarding a possible Hamiltonian formulation of the present system, for which
there is a vortex sheet strength with nontrivial dynamical evolution.  It is useful to contrast this case with that
of a system of discrete vortex rings and its continuous limit \cite{Meleshko}, for despite some fundamental distinctions involving
conservations laws, there are common mathematical structures involved.
As first shown by Dyson \cite{Dyson}, the Lagrangian dynamics of a discrete set of coaxial vortex rings with centers $Z_i$ on
the $z$-axis, with radii $R_i$ and circulation $\Gamma_i$, can be written as
\begin{subequations}
 \begin{align}
  R_i{\dot Z}_i &=\frac{\Gamma_i}{4\pi}\left(\log\frac{8R_i}{a_i}-\frac{1}{4}\right)+\sum_{j=1, j\neq i}^{N}
  \frac{\Gamma_j}{2\pi}\frac{\partial I_{ij}}{\partial R_i}\label{ring1}\\
  R_i{\dot R}_i &=-\sum_{j=1, j\neq i}^{N}
  \frac{\Gamma_j}{2\pi}\frac{\partial I_{ij}}{\partial Z_i}\label{ring2},
 \end{align}
\label{ring}
\end{subequations}
where $I_{ij}=\sqrt{R_i R_j}M(k_{ij})$ is the mutual inductance between the two loops $i$ and $j$. The first
term on the rhs of \eqref{ring1} is the approximate self-induced velocity of ring $i$, where $a_i$ is the core radius,
which serves as a cutoff for the localized induction approximation.

The evolution equation \eqref{fluxMaxwell} can now be seen as the continuum limit of \eqref{ring2}, expressed in an
Eulerian form.  A direct calculation shows a less obvious result, namely that the axial velocity $z_t$ in \eqref{CL2a} is
the equivalent of \eqref{ring1},
without the self-induction term.  In fact,
\begin{equation}
r z_t = {1\over  \pi }{\partial \over \partial r}\int_{-\infty}^{\infty} d\alpha^{\prime}\gamma' {\sqrt {r r^{\prime}} M(k)}
={\partial F \over \partial r}.
\label{fluxMaxwell2}
\end{equation}
By analogy to the discrete case, in which
one can introduce an energy that is quadratic in the circulations,
$\Gamma_i$, namely
\begin{equation}
 U=\sum_{i=1}^{N}\sum_{j=1, j\neq i}^{N}
  \frac{\Gamma_i\Gamma_j}{2\pi}I_{ij},
\end{equation}
and from which one obtains the equations of motion
\begin{subequations}
 \begin{align}
  \Gamma_i R_i{\dot Z}_i &=\frac{\Gamma_i^2}{4\pi}\left(\log\frac{8R_i}{a_i}-\frac{1}{4}\right)+\frac{\partial U}{\partial R_i}
  \label{ring1a}\\
  \Gamma_iR_i{\dot R}_i &=-\frac{\partial U}{\partial Z_i}\label{ring2a},
 \end{align}
\label{ringa}
\end{subequations}
one can take the continuum limit to obtain \cite{Novikov}
\begin{equation}
 {\cal U}=\frac{1}{2\pi}\int\! d\alpha \int d\alpha' \gamma(\alpha)\gamma(\alpha')
  \sqrt{r r'}M(k),
\end{equation}
which, generalizing to functional derivatives, yields
\begin{equation}
\gamma r z_t =  {\delta {\cal U} \over \delta r} \ \ \ \ {\rm and} \ \ \ \
\gamma r r_t = -{\delta  {\cal U} \over \delta z}.
\label{pseudoHam}
\end{equation}
This is of the same form as the discrete dynamics \eqref{ringa}, but with the important distinction that the vortex
sheet strength $\gamma$ itself depends on time, whereas the individual circulations $\Gamma_i$ in the discrete case
do not.  Moreover, in the discrete case it is possible to rewrite the dynamics so the left-hand-sides are total time
derivatives, rendering them truly Hamiltonian.  In contrast, the dynamics \eqref{pseudoHam} do not obviously have
this feature.  The fact that \eqref{strength} can be throught of as a nonholonomic constraint may offer a path to
obtain an ``almost-Hamiltonian'' dynamics \cite{Fernandez}.

\section{Stability Analysis}
\label{stability}

In this section we show how the vortex sheet dynamics in the Eulerian form reproduces known stability results
\cite{Alterman,Sterling} for
capillary jets, both with and without a streaming velocity within the fluid enclosed by the sheet.  Note that our
assumption that the fluids inside and outside the sheet have the same density precludes recovering the original
stability result of Rayleigh \cite{Rayleigh}, which assumed vacuum outside.

We first consider the case with a quiescent fluid on both sides of a vortex sheet of radius $R$, and linearize
the equations of motion for small perturbations in $\gamma$ and $r$ of the form
\begin{equation}
\gamma = {\hat \gamma} e^{ iqz + \beta t} \ \ \ \ {\rm and} \ \ \ \
r  = R +  {\hat \zeta} e^{ iqz + \beta t}.
\label{perts}
\end{equation}
At this order the mean curvature has the
simplified form $\kappa \simeq r_{zz}-r^{-1}$.  The resulting vortex sheet evolution equation is
\begin{equation}\label{strength_noflow}
 \beta {\hat \gamma} =  i P (1 - P^2){\sigma \over  \rho R^3}~{\hat \zeta},
\end{equation}
where $P=qR$.  Since, in the absence of background fluid motion, $\gamma$ is first order in the perturbation, the
remaining factors on the rhs of \eqref{fluxMaxwell} are those corresponding to a cylinder.  Let
\begin{equation}
k_R^2 = {4 R^2 \over  (z - z^{\prime})^2 + 4R^2}, \ \ \ \ {\rm and} \ \ \ \ x = {(z^{\prime}-z) \over 2R }.
\label{kRdefine}
\end{equation}
Then, the linearization of \eqref{fluxMaxwell} is
\begin{equation}
 \beta {\hat \zeta} =  -{4\over  \pi }i P {\hat \gamma} \int_{0}^{\infty}  M(k_R) \cos(2Px) ~dx
 \label{hatzeta}
\end{equation}
After calculating the integral (see Appendix), we substitute $\hat\zeta$ from \eqref{hatzeta} into
\eqref{strength_noflow} to obtain, in agreement with previous results \cite{Alterman,Sterling},  the growth rate
\begin{equation}
\beta^2 = {\sigma \over  \rho R^3}~ P^2\left(1 - P^2\right)I_1(P)  K_1(P)
\label{linstab1}
\end{equation}
where  $I_n$ and $K_n$ are the $n$-th order modified Bessel functions of the first and second kind, respectively.

Demonstrating that the flux-form PDE reproduces the stability results in the presence of streaming flows, obtained by Alterman \cite{Alterman} and by Sterling and Sleicher \cite{Sterling}, follows the same procedure as the calculation above,
but requires a more delicate analysis.
This complexity is related to the limiting procedure of vanishing sheet thickness $\delta$ and
vanishing viscosity
used to arrive at the evolution equation for the vortex sheet strength \eqref{strength1}.  In particular, these limits
preclude the determination of a a unique value of the tangential sheet velocity $z_t$.   For concreteness, consider the situation in which the streaming
velocities inside and outside of the cylindrical vortex sheet are $U$ and $0$, respectively.  In the
Lagrangian formalism, this ambiguity is eliminated by setting  $z_t=U/2$ \cite{CaflischLi}.
However, this choice cannot satisfy
the boundary conditions on either side of the sheet in the Eulerian frame. It is therefor necessary to reinstate the
appropriate boundary
conditions, which is equivalent to taking the limit $\delta \to 0$ at the end of the calculation.

The calculation proceeds by systematic perturbation of the terms within the flux integral.  We expand theseparately
the two, obtaining
\begin{equation}
M(k) = M(k_R ) + {dM \over dk}\bigg|_{k_R} (k - k_R) + \cdots \ \ \ \ {\rm and} \ \ \ \
\gamma^{\prime} {\sqrt {r r^{\prime}} } \simeq  R ( \gamma_0 + {\tilde \gamma}) \left( 1 + { \zeta^{\prime}+\zeta \over 2 R} \right) + \cdots,
\label{22}
\end{equation}
where $k_R$ is defined in \eqref{kRdefine}.
Using
\begin{equation}
k - k_R \simeq k_R (1 -k_R^2) \left( { \zeta^{\prime}+\zeta \over 2 R} \right),
\label{23}
\end{equation}
and collecting terms, we obtain (see Appendix)
\begin{equation}
\gamma^{\prime} {\sqrt {r r^{\prime}} } M(k) \simeq \gamma_0  R M(k_R)
+ R {\tilde \gamma^{\prime}}M(k_R)-
  \gamma_0 R k_R [E(k_R) - K(k_R)] \left( { \zeta^{\prime}+\zeta \over 2 R} \right).
\end{equation}
Substituting into the equations of motion and retaining only first order terms, we find
\begin{equation}
\beta{\hat \zeta}=  - {1\over  \pi }\left[ 2 i P {\hat \gamma} \int_{0}^{\infty}  M(k_R) \cos(2Px) ~dx -
i {P \gamma_0
{\hat \zeta}\over R}
\int_{0}^{\infty} \left[ 1 + \cos(2Px)\right] k_R \left[E(k_R) - K(k_R)\right] ~dx \right]
\end{equation}
After some laborious calculation of these non-trivial integrals we obtain
\begin{equation}
\beta{\hat \zeta} =  - i P  I_1(P) K_1(P){\hat \gamma} - i {\gamma_0
\over 2 R}  P\left[ 1 + P \left(I_0(P)K_1(P)  - I_1(P) K_0(P)\right) \right]~ {\hat \zeta}.
\end{equation}
Using the Bessel function identity $P(I_0 K_1 + I_1 K_0) = 1$, we
further reduce this expression to
\begin{equation}\label{stability1}
\left( \beta + i {P \gamma_0 \over  R} K_1(P) I_0(P) \right) {\hat \zeta}=  -2 i P  I_1(P)K_1(P) ~ {\hat \gamma}
\end{equation}

This equation is the evolution for perturbations to the mean radius, i.e. the mean location of the vortex sheet:
$ r = (r^+ + r^-)/2)$, where $r^+$ and $r^-$ are
the outer and inner radius of the infinitesimaly thin sheet, respectively ($r^+ - r^- = \delta \to 0$).
To reinstate the proper boundary conditions we note that \eqref{stability1}
should be split into two parts, the contribution from $r^+$ and the one from $r^-$. To do so we rewrite \eqref{stability1} as
\begin{equation}\label{stabilityboth}
\left( P \left( \beta + i {P \gamma_0 \over  R}\right) K_1(P) I_0(P)+ P( \beta -0) K_0(P) I_1(P)\right){\hat \zeta} =  -{ i P \over 2}{\hat \gamma} K_1(P) I_1(P)
-{ i P \over 2}{\hat \gamma} K_1(P) I_1(P)
\end{equation}
In this equation all terms involving $I_0$ are connected to the inner region, that is $r^-$, while those involving $I_1$ correspond to the
outer one ($r^+$) yielding two equations, one for each region:
\begin{equation}\label{stability-}
P \left( \beta + i {P \gamma_0 \over  R}\right) K_1(P) I_0(P){\hat \zeta} =  -{ i P \over 2}{\hat \gamma} K_1(P) I_1(P)
\end{equation}
and
\begin{equation}\label{stability+}
P  \beta {\hat \zeta} K_0(P) I_1(P) =  -{ i P \over 2}{\hat \gamma} K_1(P) I_1(P)
\end{equation}
A similar expansion of the vortex sheet evolution equation \eqref{strength} yields for the inner region
\begin{equation}\label{strength-}
\left( \beta + i {P \gamma_0 \over  R}\right){\hat \gamma} =  {\sigma \over  \rho} { i P (1 - P^2)\over R^3}{\hat \zeta},
\end{equation}
and for the outer one
\begin{equation}\label{strength+}
 \beta {\hat \gamma} =  {\sigma \over  \rho} { i P (1 - P^2)\over R^3}{\hat \zeta} .
\end{equation}

Substituting \eqref{strength-} and \eqref{strength+} into \eqref{stability-} and \eqref{stability+}, respectively, and adding the
contributions to obtain the total expansion for $r = (r^-/2)+(r^+/2)$ yields
\begin{equation}\label{strength2}
\beta^2  + 2 i \beta \frac{\gamma_0}{R} P^2 I_0(P)K_1(P) =
\frac{\sigma}{\rho R^3} P^2\left(1-P^2\right)I_1(P) K_1(P)+\frac{\gamma_0^2}{R^2}P^3 I_0(P)K_1(P)~,
\end{equation}
and which coincides with previous results \cite{Sterling,Alterman} for the case where the inner fluid velocity is  $U = \gamma_0 $ and
the outer fluid is stationary.

\section{Derivation of a local PDE}
\label{sec:pde}
To obtain an approximate PDE that describes the dynamics of slender necks,
it is necessary to find a suitable approximation to $M(k)$ that
would make it possible to calculate the integral in the flux equation in a controlled manner. With this purpose in mind,
it is very useful to
note \cite{GradRyz} that $M(k) = Q_{1/2} (\chi)$, with $Q_{1/2} (\chi)$ the associated Legendre
function of index $1/2$ with variable $\chi = (2/k^2) -1$,
which obeys the differential equation
\begin{equation}
(1 -\chi^2) {d^2 Q \over d \chi^2} - 2 \chi {d^2 Q \over d \chi^2} + {3 \over 4}Q = 0.
\label{Legendre}
\end{equation}
This identity makes it possible to obtain a uniform approximation for $M(k)$ by matching the inner
and outer solutions, ($\chi \to 1$ and $\chi \to \infty$, respectively).  In the inner region the limiting behavior is
\begin{equation}
Q_{1/2} \simeq A \ln \left(\sqrt{{2\over \chi-1}}\right) \ \ \ {\rm for } \ \ \ \chi \to 1,
\end{equation}
while in the outer one is
\begin{equation}
Q_{1/2} \simeq B \sqrt{ { (\chi+1)^3 (\chi-1) \over 16}}  \ \ \ {\rm for } \ \ \ \chi \to \infty ,
\end{equation}
where $A$ and $B$ are constants to be adjusted by matching the inner and outer solutions, with the final result in terms of $k$ given by
\begin{equation}
M(k) \simeq M_u(k)= {1\over 4} \left[ 3 \ln \left( {1\over (1-k^2)}\right)-
\pi k^{3/2} {\sqrt {1-k^2}}  \right].
\label{Muniform}
\end{equation}
The fit of $M(k)$ provided by the uniform approximation $M_u(k)$ is remarkable; $M(k)- M_u(k) \gtrapprox 0$ only in a
small neighbourhood of $ k^2 = 1/2 $; everywhere else it
is almost impossible to distinguish $M_u(k)$ from the exact solution (see Fig. \ref{fig2}).

\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{Pesci_etal_figure2.pdf}
\caption{Uniform approximation to the Maxwell function.  Solid black line is the exact expression
$M(k(x))$.  Dash-dot and dashed blue lines are the inner and outer approximations, respectively.
Dashred red line is the uniform approximation $M_u(k)$.}
\label{fig2}
\end{figure}

We can now approximate $M(k)$ by $M_u(k)$ in the evolution equation for $r^2(t)$ \eqref{fluxMaxwell}.
In order to expression $M_u(k)$ in a form that allows calculation of the relevant integral, we Further,
using the expression for $k$ in terms of $z$, $z^{\prime}$, $r$, and $r^{\prime}$ it is possible to rewrite  in a way that
allows to calculate the integrals. In particular, using
\begin{equation}
k^2 = {4 r r^{\prime} \over {(z^{\prime}-z)^2 + (r + r^{\prime})^2}},
\end{equation}
the denominator inside the logarithm in \eqref{Muniform} is
expandable in Taylor series in the limit $ z \to z^{\prime}$ because
the ratio $ (r^{\prime}- r)/(z^{\prime}-z) \to r_z$.
In this limit, $ r \to r^{\prime}$, and rescaling $r = \epsilon R$ we find
\begin{subequations}
\begin{align}
\ln \left( {1\over (1-k^2)}\right) &\simeq \ln \left[
{(z^{\prime}-z)^2 + \epsilon^2(2R)^2 \over
 {(z^{\prime}-z)^2 \left(1 +  \epsilon^2 R_z^2\right)}} \right]  = \ln \left[
{x^2 + \epsilon^2\over
 x^2}\right] - \ln \left( 1 +  \epsilon^2 R_z^2\right), \\
k^2 {\sqrt{1-k^2}} &\simeq 8\epsilon^3 R^3
{|z^{\prime}-z|\left(1 +  \epsilon^2 R_z^2\right)^{1/2}\over {\left[(z^{\prime}-z)^2 + \epsilon^2 (2R)^2\right]^2}}
={\epsilon^3 |x|\over (x^2+\epsilon^2)^2} \left(1 +  \epsilon^2 R_z^2\right)^{1/2},
\end{align}
\end{subequations}
where $x$ is the same as defined in \eqref{kRdefine}.  The contribution to the flux $F$ arising from the
two terms in the uniform approximation \eqref{Muniform} involves $I_1$ and $I_2$, where, by the
symmetry of the integrand,
\begin{subequations}
\begin{align}
I_1 &= {3 \over 2} \epsilon \gamma R^2\int_0^{\infty}
\ln \left[ {x^2 + \epsilon^2\over x^2}\right] dx + O(\epsilon^3 R R_z^2)\\
I_2 &=  \pi \epsilon^4 \gamma R^2 \int_0^{\infty} {x \over \left[ x^2 + \epsilon^2\right]^2} dx
+ O(\epsilon^3 R R_z^2)
\end{align}
\end{subequations}
Performing the integrations we obtain
\begin{equation}
I_1= {3 \pi \over 2} \epsilon^2 \gamma R^2 \ \ \ \ {\rm and} \ \ \ \
I_2 = {\pi \over 2} \epsilon^2 \gamma R^2 .
\end{equation}

Collecting terms and substituting $r = \epsilon R$ in the LHS of the flux equation \eqref{fluxMaxwell} we obtain
\begin{equation}
{\partial R^2 \over \partial t}= -{\partial \over \partial z} \left(\gamma R^2 \right).
\label{lastone}
\end{equation}

While this result was obtained in a systematic way that would be useful in cases where it is necessary,
or desirable to obtain the next terms of the expansion, the procedure is not very intuitive. In fact, the
meaning and procedence of \eqref{lastone} can be more easily understood by recognising that the first order approximation corrresponds to a cylindrical
vortex sheet, where $r = r'$ and $\gamma' = \gamma$. Then, starting as before from \eqref{fluxMaxwell} we may write
\begin{equation}
{\partial r^2 \over \partial t} =  -{2\over  \pi } {\partial \over \partial z} \left( \gamma r^2 \int_{-\infty}^{+\infty} dx M(k(x))\right)~,
\label{cylinder}
\end{equation}
where we have used a variant of the usual variable $x$, namely $x= (z' -z)/2r$. Because the value of the integral
is  $\pi/2$, \eqref{cylinder}
yields the same result \eqref{lastone} that was obtained for the first order term by employing the expansion method
making clear its geometric interpretation.

\section{Discussion} The present work has shown hwo the familiar axisymmetric vortex sheet dynamics can be recast exactly in
an Eulerian form which clearly displays the inherent flux form of the motion.  Previously established linear stability
results are easily recovered with this formulation.  Most signficantly, the Eulerian dynamics is well-suited to the development
of a systematic reduction to a local approximation when the aspect ratio of the system is suitably small.  This controlled
approximation, which at leading order recovers well-known empirical results \cite{EggPont},
should lend itself to more rigorous studies both of the form and validity of the governing PDEs as well as
the singularities they produce.  As the underlying model includes the dynamics of an incompressible fluid surrounding
the vortex sheet, it consitutes a more physically relevant starting point for understanding singularity formation by
moving soap films than that provided by mean curvature flows.

In light of the large body of existing work on axisymmetric vortex sheets, there are many possible extensions of the present work,
including the introduction of swirl \cite{CaflischLiShelley} or helical symmetry \cite{CaflischLi}.  Greater challenges
would be to incorporate weak viscous effects \cite{Lundgren,Ceniceros}, and to complete the Hamiltonization of the inviscid Eulerian
case in the presence of surface tension.

\section{acknowledgments} We are indebted to Keith Moffatt for stimulating discussions
at an early stage of this project, and to John Hinch for discussions on the asymptotic analysis and the significance
of the flux form.
This work was supported in part by Established Career Fellowship EP/M017982/1 from the EPSRC (REG \& AIP).
REG and AIP are grateful to the I.H.E.S., and especially Patrick Gourdon, for hospitality during an extended visit supported
by the Schlumberger Visiting Professorship (REG).

\section{Appendix}

Here we show some intermediate steps in the stability analysis.   We begin from the first equation of
\eqref{22},
\begin{equation}
M(k) = M(k_R ) + {dM \over dk}\bigg|_{k_R} (k - k_R)
\end{equation}
where $k_R$ is as defined in \eqref{kRdefine}, and using the relationships between the elliptic functions
and its derivatives we find
\begin{equation}
{dM \over dk} = {2 \over k^2} (E - K)  - {2\over k} \left[{dE \over dk} -{dK \over dk}\right] - K - k  {dK \over dk}.
\end{equation}
The corresponding first-order expansions of the relevant quantities are
\begin{equation}
k^2 = {4 (R + \zeta^{\prime}) (R +\zeta ) \over  (z - z^{\prime})^2 + (2R +\zeta^{\prime}+ \zeta )) ^2 }
\simeq k_R^2 \left[ 1 + (1 -k_R^2) \left( { \zeta^{\prime}+\zeta \over R} \right)\right],
\end{equation}
which yields \eqref{23},
and
\begin{equation}
{dM \over dk}\bigg|_{k_R} = {2 - k_R^2 \over k_R^2 (1-k_R^2)} E(k_R) - {2\over k_R^2}  K(k_R),
\end{equation}
where we have used the relationships $dE/dk = (E-K)/k$ and  $k dK/dk = E/(1-k^2) - K$ \cite{GradRyz},  and have substituted
 $k= k_R$. Introducing the convenient notation
\begin{equation}
M_0 \equiv M(k_R) \ \ {\rm and} \ \ M_1 \equiv  {dM \over dk}\bigg|_{k_R} k_R(1-k_R^2) ,
\end{equation}
we note that, after straightforward algebra, the quantity $ M_0 + M_1$, which will be needed later, reduces to $ -k_R (E-K)$.

Utilizing the second of the equations in \eqref{22} and collecting all the contributions we obtain
\begin{equation}
\gamma^{\prime} {\sqrt {r r^{\prime}} } M(k) \simeq \gamma_0  R M_0 + \gamma_0 R (M_0 + M_1) \left( { \zeta^{\prime}+\zeta \over 2 R} \right)
+ {\tilde \gamma} R M_0.
\label{62}
\end{equation}
Using again the identity $M(k_R) = Q_{1/2} (\chi)$, with argument $\chi = 2/k_R -1$,
and substituting into \eqref{62} we obtain the first order result
\begin{equation}
\gamma^{\prime} {\sqrt {r r^{\prime}} } M(k) \simeq \gamma_0  R M(k_R) + R {\tilde \gamma^{\prime}} Q_{1/2}( {\chi}_R) -
  \gamma_0 R k_R [E(k_R) - K(k_R)] \left( { \zeta^{\prime}+\zeta \over 2 R} \right).
  \label{63}
\end{equation}
Substituting \eqref{63} into the equations of motion and retaining only first order terms we find
\begin{equation}
2 R {\partial \zeta \over \partial t} =  - {1\over  \pi } {\partial \over \partial z}\left[\int_{-\infty}^{\infty} d\alpha^{\prime}\gamma_0  M_0 R
 + R \int_{-\infty}^{\infty} d\alpha^{\prime} {\tilde \gamma} Q_{1/2} - \gamma_0
\int_{-\infty}^{\infty}  k_R [E(k_R) - K(k_R)]\left( { \zeta^{\prime}+\zeta \over 2 R} \right) ~d\alpha^{\prime} \right].
\label{perturbedr}
\end{equation}
The first integral on the rhs does not contribute to the equation of motion, for $\gamma_0=0$
identically when the
fluids on both sides of the
vortex sheet are stationary, and the integral is a constant when there
is a streaming velocity in either one of the two regions.
The other integrals involve the Lagrangian parameter $\alpha$, and to be able to obtain the result in the Eulerian frame as a function of
the coordinate $z$ we note that $ds = {\sqrt {r_{\alpha}^2 + z_{\alpha}^2}}~ d{\alpha}$ and $ds = {\sqrt {1 + r_z^2}}~ dz$.
Using these identities allows us to rewrite
\begin{equation}
dz = {\sqrt {1 + r_z^2 \over r_{\alpha}^2 + z_{\alpha}^2} }~ d{\alpha} \simeq (1 + {\cal O}(\zeta^2)) d{\alpha}
\end{equation}
making it possible to approximate $d{\alpha'} \simeq dz'$ to the required order. Replacing the expressions for the perturbations \eqref{perts} into\eqref{perturbedr}, rewriting
all exponentials in terms of $z' -z$ and expressing them in terms of sines and cosines makes it clear that the only non-zero contributions to
the integrals come from the cosine terms which are the only ones that produce even integrands. Then,
defining $P = qR$, and performing the usual change of variables $x = (z'-z) /2R $ we find $k_R(x) = 1/(1+x^2)$,
and $\chi(x) = 1+2 x^2 $, yielding the first one of the two stability equations
\begin{equation}
 \beta {\hat \zeta} =  - {1\over  \pi }\left[ 2 i P {\hat \gamma} \int_{0}^{\infty}  ~ Q_{1/2}(\chi(x)) \cos(2Px) ~dx - i {P \gamma_0
{\hat \zeta}\over R}
\int_{0}^{\infty} \left[ 1 + \cos(2Px)\right] k_R [E(k_R) - K(k_R)] ~dx \right].
\label{horrible}
\end{equation}

The integrals have been calculated with the help of \cite{Okui} and \cite{GradRyz}. In particular, after a further change
of variable, $ y = {\sqrt 2} \ x$, the first integral acquires its canonical form 7.162.6 \cite{GradRyz} with $\nu = 1/2$, and $a = {\sqrt 2} \ P$, thus
\begin{equation}
\int_{0}^{\infty}  ~ Q_{1/2}(1+y^2) \cos({\sqrt 2} \  P y) ~dy = {\pi \over 2} K_1(P) I_1(P).
\end{equation}
The second integral can be divided in two contributions
\begin{subequations}
\begin{align}
 {\cal I}_1 &=  \int_{0}^{\infty} k_R [E(k_R) - K(k_R)] ~dx,  \\
 {\cal I}_2 &= \int_{0}^{\infty} \cos(2Px)~ k_R [E(k_R) - K(k_R)] ~dx.
 \end{align}
\end{subequations}

To calculate ${\cal I}_1$ we use the identity 3.2 (2) in \cite{Okui} for $p = 1$ and $a = x$
\begin{equation}
{ k_R \over 2 \pi p^2} [E(k_R) - K(k_R)]= \int_{0}^{\infty}  ~ e^{-2 p t}  N_0(a t) J_0(at) ~dt,
\end{equation}
where $N_0$ and $J_0$ are the Bessel functions. With this replacemente we obtain ${\cal I}_1 = - \pi /2$.
The second integral can be solved using identity 6.3(5) also in \cite{Okui}, for $a = x$ and $b =1$
\begin{equation}
{ k_R \over b^2} [E(k_R) - K(k_R)]= \int_{0}^{\infty}  ~ K_0(a t) J_1(at) ~dt ,
\end{equation}
to obtain
\begin{equation}
{\cal I}_2 = {\pi \over 2} \int_{0}^{\infty} {t J_1(t) \over {\sqrt{ 4P^2 + t^2}} } ~dt.
\end{equation}
This can be reduded to the standard form 6.565.1 with $a = 2P$  and $b = \alpha$ found in \cite{GradRyz} by noting that ${\cal I}_2$
can also be written as
\begin{equation}
{\cal I}_2 ={\pi \over 2} \lim_{\alpha \to 1} {\partial \over \partial {\alpha}} \left[
\int_{0}^{\infty} { J_0(\alpha t) \over {\sqrt{ 4P^2 + t^2}} } ~dt \right] = {\pi \over 2} P \bigg[ I_1(P) K_0(P) - I_0(P) K_1(P) \bigg]
\end{equation}

Finally, after collecting all the contributions and replacing them into \eqref{horrible} we find the first order equation
\begin{equation}
\beta {\hat \zeta}=  - i P {\hat \gamma} K_1(P) I_1(P) - i {P \gamma_0
{\hat \zeta}\over 2 R}  \bigg[ 1 + P \left[ K_1(P) I_0(P) - K_0(P) I_1(P)\right] \bigg] .
\end{equation}
Making use of standard Bessel
functions identities, it is possible to further reduce this expression to
\begin{equation}\label{stability1a}
\left( \beta + i {P \gamma_0 \over  R} K_1(P) I_0(P) \right){\hat \zeta} =  - i P  K_1(P) I_1(P) {\hat \gamma}.
\end{equation}

\bibliographystyle{aipauth4-1}

redpajama_set_name: RedPajamaArXiv


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Boys varsity basketball team beats Ross on the road
By Ambrose Clancy
The Shelter Island varsity boys basketball team took a closely contested road game from league rival Ross on Monday.
The Indians beat the Cosmos 59-55, with Eric Thilberg draining 3 out of 4 clutch free throws in the fourth quarter to secure the win.
It shouldn't have been as close as the score indicated, said Coach Jay Card Jr. after the game. "We had some self-inflicted pain," the coach said, with too many turnovers and "poor passing decisions."
Shelter Island has secured a Suffolk County Class D playoff berth, Coach Card said. It's a formality that the school will ask the league to be included, and the request will be accepted. Class D is a designation for schools with small enrollments.
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Today, Thursday, the varsity, along with the junior varsity, hosts Southold. The JV game tips off at 5:45 p.m. and the varsity game is scheduled for 7:30 p.m.
Ambrose Clancy has been the editor of the Shelter Island Reporter since 2012. He's worked as a staff reporter for The North Shore Sun, the Southampton Press and was associate editor of the Riverhead News-Review and an editor at Long Island Business News.
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OFFSPRING TONIGHT I DO LYRICS"
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REVIEW: The Taylors – Salvation's Song
March 1, 2020 James Hales Reviews 0
Producer: Wayne Haun & Chris Krunk
Record Label: StowTown Records
Website: www.thetaylorsmusic.com
About 10 years ago, the Taylor siblings began making themselves known across the industry. I wouldn't say they "burst" on the scene, but over time they quietly made great strides and have consistently put out great music. This latest release is probably the best collection of tunes they've ever released. Albeit, it's shorter than their previous efforts with only 7 songs, it's a stronger collective and just about any of these tunes would do well on the charts for them.
The recording starts with the uptempo title song, "SALVATION'S SONG". Penned by Wayne Haun, Joseph Habedank and Michael Farren, the song reminds us of the joy that is found in our salvation…"It says we're forgiven, it says we're redeemed, it says that our sins are gone forever, all because of Calvary…it's the joyful, grateful sound of salvation's song".
"YOU RESTORE" keeps things light and happy before the tempo slows down a bit for the anthem, "I'LL NEVER KNOW". Probably the strongest song on the recording, the song is a grateful proclamation of the things we'll never know because Christ took our place on the cross…"I'll never know the cold steel of justice, driven like nails into flesh, I'll never know the sound of the gavel declaring the sentence of death, I'll never know the cost of redemption that Jesus paid for my soul, why He'd take the cross and count everything loss, there are some things that I'll never know".
The remainder of the recording features uptempo tunes with varying styles such as the horn/organ infused "THE SAME", the bluesy and fun "HOUSE OF GOD", the country feel of "I CHOOSE JOY" and the happy-go-lucky feel of the closing tune, "LOVE REALLY CHANGED THE WORLD".
This is a very upbeat and uplifting recording. There isn't a power ballad or true slow song found here. If you need a nice pick-me-up, this is the one to pull to lift your spirits. It's an excellent piece of work and continues their string of consistently strong recordings.
(stars denote personal favorites)
Salvation's Song**
You Restore
I'll Never Know**
House of God**
I Choose Joy**
Love Really Changed the World
READER RATING:
James Hales
Head Reviewer at Absolutely Gospel
James Hales, from Durham, North Carolina, has been a writer for AbsolutelyGospel.com since 2000. James is our featured reviewer and also contributes to monthly features periodically.
Chris Krunk
StowTown Records
The Taylors
Wayne Haun
StowTown Records Announces Upcoming Digital Release From Paid In Full With New Single "Too Much"
REVIEW: The Taylors – Hope & Healing
The Erwins – Watch & See


Mila 23 Pension is located on the right bank of the Old Danube in Mila 23 village, one of the most famous fishing villages of the Danube Delta.
Our pension has 32 places in 14 rooms with bathroom, air conditioning and satellite TV. In the backyard we have patio, pool, barbecue, terrace. The dining room with 40 seats, the open kitchen and the open bar are some of the ingredients of a pleasant stay at Mila 23.
We organize banquets, team building, business meetings, events and social gatherings.
We can provide boat transfers to and from Tulcea, organize trips, boat rental with or without motor, with or without a guide.
Whether you are a fisherman or not you can spend great holidays here, the possibilities of recreational activities in the middle of the nature, almost untouched by man, being multiple.


text:
Leonardo Pinta
leonardo.pinta@shearman.com
Nachrichten und Events
Leonardo Pinta, an Italian lawyer in the Corporate and Capital Markets Groups of the Rome office, practices in the areas of mergers and acquisitions, corporate law, corporate governance, finance and capital markets.
Before joining Shearman & Sterling in 2014, Leonardo worked for the corporate department of an International law firm in Rome.
He is admitted to practice in Italy.
Mitsubishi Chemical Carbon Fiber and Composite GmbH, a German affiliate of Mitsubishi Chemical Corporation, in the acquisition of interests in the share capital of the Italian CPC S.r.l., a leading technology company in the carbon fiber business, supplying, among the others, leading worldwide manufacturers in the luxury automotive sector
Dover Italy Holdings S.r.l., an Italian subsidiary of the US Dover Corporation, in the acquisition of the entire share capital of the Italian Ravaglioli S.p.A., one of the world's top-ranking companies in the garage equipment industry with offices in Italy, France, Spain, Germany, UK and USA
McCormick Italy Holdings S.r.l., an Italian subsidiary of the US Mc Cormick Corporation, in the acquisition of the entire share capital of the Italian Enrico Giotti S.p.A., an Italian leading company in the flavor industry and manufacturer of world-renowned natural and organic aromas, plant extracts, fruit juice concentrates and aromatic bases
Pometon S.p.A., in the formation of a Korean joint venture entity with Seojincam Co. Ltd. for the manufacturing in South Korea of metal powders (ferrous/non-ferrous powders, and advanced steel granules)
Digitouch S.p.A. an AIM listed entity, leader in the digital advertising market, in the acquisition of Optimized Group S.r.l., a web agency specialized in the SEO (Search Engine Optimization) and SEM (Search Engine Marketing), with a consolidated client portfolio
Digitouch S.p.A. an AIM listed entity, leader in the digital advertising market, in the formation of Mediamob, a joint venture entity with Mobyt S.p.A., operating as independent agency on new media and digital marketing solutions (connected tv, Internet of things, wearable devices, etc.)
Dover Italy Holdings S.r.l., an Italian subsidiary of the US Dover Corporation, in the acquisition of the entire share capital of the Italian JK Group S.p.A. leading manufacturer of inks for digital print with offices in Italy, China and USA
Italian Sugar Holding S.r.l., an Italian subsidiary of the ASR Group, in the restructuring and recapitalization of its interests in SFIR Raffineria di Brindisi S.p.A.
Digitouch S.p.A. an AIM listed entity, leader in the digital advertising market, in the acquisition of E3 S.r.l., a distinguished entity with strong expertise in social networks communications and Facebook Preferred Marketing Developer.
Blackstone, a world's leading investment and advisory firm in the acquisition of the quotas of Marangi Immobiliare S.r.l., a company owning the "Palmanova Outlet Village" shopping center
Munich Re, in the sale of 100% of its Italian subsidiary Newmed to RBHold., a subsidiary of Previmedical, the Italian market leader on health related insurance services
Minority shareholders and managers of Visiant Galyleo S.p.A., active in the Italian IT business, in connection with the sale of a majority stake to INDRA Sistemas, a leader in the IT systems sector
MEAG, the asset management company of the Munich Re group, in the acquisition of a portfolio of four photovoltaic solar parks in Italy with an overall capacity of 29.3 MWp
NGK Insulators, Ltd., a market leader in large-scale battery storage, in negotiation of a framework agreement for supply of NAS (sodium-sulfur) battery storage systems in favor of Terna
RWE Innogy Italia in the sale of its 80 per cent share in the 18.7 MW Enna biomass plant in Sicily to FriEl Green Power
LUISS University
Ph.D. in Corporate Law
LUISS - Libera Università Internazionale degli Studi Sociali Guido Carli
Laurea Magistrale in Diritto d'Impresa, magna cum laude
Laurea in Scienze Giuridiche, magna cum laude


As the US begins to integrate biomass crops and residues into its mix of energy feedstocks, tools are needed to measure the long-term sustainability of these feedstocks. Two aspects of sustainability are long-term potential for profitably producing energy and protection of ecosystems influenced by energy-related activities. The Soil and Water Assessment Tool (SWAT) is an important model used in our efforts to quantify both aspects. To quantify potential feedstock production, we used SWAT to estimate switchgrass yields at a national scale. The results from this analysis produced a map of the potential switchgrass yield along its natural eastern range. To quantify ecological protection, we are using the SWAT model to forecast changes in water quality and fish richness as result of landscape alterations due to incorporating bioenergy crops. We have implemented the SWAT model in the Arkansas-Red-White region, which drains into the Mississippi River, and we present our methods here. We identified two sub-watershed for sensitivity analysis and calibration of the water quality results, and then, explored ways to apply the calibration results to the whole region and validate the model setup. We also present an overview of our research in which results from the calibrated regional SWAT model were used to analyze potential changes in fish biodiversity. Only by evaluating the energy and environmental implications of landscape changes can we make informed decisions about bioenergy at the national scale, and the SWAT model will enable us to reach that goal.


text:
Country Hill Casa
our life living at country hill
Lovely Ladytron, Damn Good Datarock!
OK, so this might be the last of our late nights out in Boston to go to concerts for…well…a LOOOONGG time (like when our kid turns 18). But we thought we'd do it up right with Ladytron at Paradise Rock Club.
Opening for them was Datarock, which might just be the most genius "nerd rock" band since Devo. The group (from Bergen, Norway!) came on stage wearing matching red jogging suits and ray ban glasses.
They totally got the crowd involved, doing silly dances in unison (very Devo-esque!), and having people sing along and clap. They rocked the scene with cheesy casio keyboards and mini guitars. At one point, one even pulled out a saxophone and played it rather well. How can you not love a band who brings such cheesy fun, and who has song titles like "Molly Ringwald" and "Computer Camp Love".
Apparently, they have been compared to DEVO so many times that even DEVO themselves remixed a song of theirs and have become fans of them!
myspace page for Datarock: http://www.myspace.com/datarock
They definitely got us pumped up for LADYTRON…
At this point, the not-so-good sound qualities of Paradise Rock Club started to show. Any time there are more than a few instruments playing at a time, its as if the speakers can't handle it and the sound becomes drowned out.
That said, although we couldn't hear very well, Ladytron put on a very good show. They played most of their songs off the new "Velocifero" album, and a few goodies from "Witching Hour". They created a good atmosphere with flashing strobe lights that enhanced their badass/dramatic dancehall vibe. By the time they returned for an encore with "Destroy Everything You Touch", the crowd went nuts!
We were protected by a guardian angel bodyguard, who, after finding out I was pregnant and didn't want to get too overheated in the big crowd, found a spot for us to sit and enjoy the rest of the show in relative peace. When the show ended, he even snuck us into a "secret" exit so I wouldn't have to push my way through the big crowd. It was so nice of him, we kinda felt like VIPS! (very important pregnant people). We even wrote to the club to thank him.
It was a Monday night and we got home at 1:30 (relatively early compared to some of our other Boston jaunts!), well worth it. But we know we can't pull nights like this too much longer!
This weekend we're going to New Jersey to spend forth of July with my sister and the kids. Should be fun we hope…more later!
July 1, 2008 carnini31
Listing of Printable Coupon Sites
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End of the Year Totals
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Price Chopper Triplers
$2.00 off Lightlife products
littlemaccc on End of the Year Totals
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Stop and Shop Shoppi… on Rare Flatout Coupon
Crafty Coin on Staples Gift Card Promotion at…
hannafords
harman xxv