text:
Speedway Moves on Hess Conversions
California Pilot Flying J on the Market for $17 Million
EG Group to Sell Off U.S. Retail Assets?
Insider's View: 2023 M&A Outlook
Analysis: 2022 M&A Year in Review
Nearly 90 sites rebranded since October acquisition, GasBuddy says By Samantha Oller on Jan. 21, 2015
ENON, Ohio -- Speedway LLC is steadily making progress on rebranding the 1,200 gas station and convenience store sites included in its October 2014 acquisition of Hess' retail portfolio.
According to an analysis this week of station data by GasBuddy/OpenStore LLC, Speedway has rebranded 86 Hess sites since Oct. 15, 2014, when the fuel-price intelligence provider first started tracking the conversions.
"We're showing 1,550 outlets currently branded Speedway, while stations still flying the Hess flag have declined to 1,261 outlets," Patrick DeHaan, senior petroleum analyst at GasBuddy, Gaithersburg, Md., told CSP Daily News. "Since the conversion started in October, things have sped up, and Speedway has 16 active conversions right now based on our station data, up from single digits during October and early November."
The network of former Hess sites are spread across 16 states on the East Coast and Southeast. Under the terms of the deal, Speedway has three years to rebrand the sites, and it is investing $410 million between 2014 and 2017 in the process.
Tony Kenney, president of Speedway, Enon, Ohio, has described the chain's integration of the Hess sites as its "No. 1 focus." The acquisition has made Speedway the largest company-owned c-store chain in the United States by revenue, and pushed it from the sixth-biggest to third-biggest chain by site count, according to the 2015 CSP Convenience Top 101 ranking.
For Speedway, the rebranding is not merely a swapping of flags but also an opportunity to unlock the inside potential of the former Hess sites, with its Speedy Rewards loyalty program as an integral component.
"The opportunity really is upgrading or enhancing the performance inside the store," said Kenney during the company's last earnings call in late October. "Leveraging some opportunities in foodservice, general merchandise, in combination with partnerships with our key suppliers and consumer product companies, with our loyalty program is going to give us an opportunity to leverage the traditional Speedway performance into the Hess assets."
Woodbridge, N.J.-based Hess Corp. divested Hess Retail Holdings LLC to focus on the exploration and production of crude oil and natural gas.
Speedway is a wholly owned subsidiary of Findlay, Ohio-based Marathon Petroleum Corp.
Cover Story: Speedway-Hess, A Big Deal
Speedway Closes on Hess Retail Deal
Speedway's Acquisition of Hess C-Stores 'Fits Like a Glove'


The Trentino Cutlery collection from the professional line of Cozzini Cutlery is made from 100% high alloy carbon stainless steel mined in the Swiss Alps of Europe and is NSF certified. Our Trentino Cutlery line is celebrated by home and professional chefs alike for its comfort, ergonomic style, thus changing the way the chef handles the knife, minimizing stress on the body with a non-slip grip. Handles are sanitized antimicrobial protection prevents bacterial growth on the handles and is ideal for everyday kitchen use.


At the Hotel Saratz, our competent staff makes the difference. They are our "Marketing Delegates" and "Experience Providers".
Accommodation and cuisine merely serve as a platform – to create an experience it takes people. It is they who will generate added value and ensure a guest's comfort and well-being. As Johann-Wolfgang von Goethe wrote, "A noble person attracts noble people and knows how to hold on to them."
We thus regard the friendliness and cordiality of our staff as strategic success factors. It follows that our staff selection is all-important, as the right person in the right place will directly impact the implementation of our strategy and our business philosophy.
Are you ready to begin your career at Hotel Saratz? Vacancies are published here and updated daily. If you can't find what you are looking for, send us your unsolicited application so we can discuss the options with you. We look forward to hearing from you jobs@saratz.ch.
We also offer internships in our different departments. The minimum duration is 3 months.
We are a member of Private Selection Hotels and our employees benefit from their "Friends specials".
The Saratz features a gym and a spacious spa, which may be shared by employees at specific times.
Our kitchen team will not only spoil our guests - but also our employees. The all-day access to fruits, the daily salad buffet and a diverse meal planning allow for tasty and healthy eating.
Meals at the employee lounge Murütsch: There is also an alternative to Murütsch: Our employees have the opportunity to indulge with food and beverages in the Pitschna Scena.
Pontresina has a wide range of outdoor sports shops. Everything regarding sports can be found here!
The Coop supermarket is located near the end of the village in direction of the Bernina Pass (Mon-Fri 8 am to 12.15 pm / 1.30 pm to 6.30 pm and 8 am to 5 pm on Sat).
The pharmacy and perfumery Roseg is located in the village centre.
Train connection to Samedan, St. Moritz and Poschiavo. Departure every hour.
Discover innumerable other things in the Engadine St. Moritz holiday region. By the way, its proximity to Italy lends the valley a very special character. Milan and Innsbruck can be reached in just under 2.5 hours by car. The nearest major city is Chur (about 1.5 hours drive). In summer, Livigno can easily be reached in 30 minutes by car.


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Q: Add patch in git, all hunks matching regex in file Is there a way to add all hunks in a file matching a regex?
I know I can search for a given hunk with /, but that only finds the first. I want to add all matching.

A: Adding to the answer of @raphinesse and comments by @rubystallion, in addition to tips on how to format it from here, I made a git function to do this with the regex as the only input. Just enter this into your gitconfig (I used my global gitconfig) and use it with git regexadd <regex>.
[alias]
        regexadd = "!f() { git diff -U0 \
                | grepdiff -E $1 --output-matching=hunk \
                | git apply --cached --unidiff-zero; }; f"

A: The Solution
The following one-liner worked for me where <regex> is a POSIX Extended Regular Expression. This requires grepdiff but it was already included in my Ubuntu distribution.
git diff -U1 -G<regex> --pickaxe-regex \
| grepdiff -E <regex> --output-matching=hunk \
| git apply --cached

How it works
The first line shows all differences whose patch text contains added/removed lines that match <regex>. The problem is that this works on a file-level. That means that any changes in a file with at least one matching hunk will be included.
Thus we need line two. From all the hunks it receives grepdiff will only print those that include a match for <regex>. the problem here is, that it will not only match the lines that have actually been changed, but also the diff context. That's why the diff context is minimized¹ by passing -U1 to git diff.
Finally line three takes the filtered diff and applies it to the index, e.g. stages it.
Conclusion
All in all this means that there are edge cases where this will add more than desired. But those can probably avoided by tweaking the regex a bit.
Thanks to @VonC for providing the link to that other SO question. It had all the pieces, they just had to be put together appropriately.

¹: You can also pass -U0 to omit the context altogether. While that completely removes the need for the pre-filtering in line one, the resulting patch won't apply anymore in line three.

A: Sadly, the patch in July 2011 went nowhere for now.
It would have introduced a git add --hunks=magic option.
For now, you will have to do with:

*

*extracting a patch based on your regexp: see "Filtering a diff with a regular expression"

*git stash your changes

*apply your patch and git add

*reset your index and apply your stash (git stash pop)

Quite a cumbersome process.

redpajama_set_name: RedPajamaStackExchange


The Old Man thinks his problem is solved, but each time he picks out a cat, he sees a prettier and more charming cat that he can't bear to leave behind. Soon, he has chosen all of the cats to come home with him.
Along the way, the cats become hungry. He tells them to each take a mouthful of grass from the hillside, and soon the hill is bare dirt. When they then say they're thirsty, he tells them to each take a sip of water from a lake, and soon the lake is dry.
He reaches home, and the Old Woman is not pleased. "But we can never feed them all," she points out. So the Old Man tells the cats to choose the prettiest among them all – and predictably, they set to fighting. Soon they've battled each other to the death, all except a small, scrawny kitten. Since he's the only one left, the Old Man and the Old Woman must make do with him.
And a strange thing happens. Because they lavish all their attention on this one kitten, he soon grows plump and pretty.
And the Old Man and Old Woman, contented with their new pet, congratulate themselves on choosing the prettiest cat of all.
Whether you've ever read Millions of Cats (or had it read to you as a child), you may recognize the story from your experience in innovation portfolio management and new product development.
When ideas for new products and innovations are first proposed, they seem so pretty and promising that their champions insist that they must be added to the project portfolio. This often seems like a good policy on first glance, because everyone wants to give every idea a chance to prove out.
However, this leads to a portfolio cluttered with lots of projects, often with no clear way to compare their worth. Each project needs care, attention and feeding, and with so many, there's no way to ensure that they all get enough resources to reach their full potential.
So, quarterly or annually, management institutes a process to thin out the numbers of the projects in the portfolio. But if management doesn't have a way to make credible and comparable evaluations of the projects, the thinning-out process may devolve into a political beauty contest or an all-out brawl to claim scarce development resources. And the projects that survive the process may not in fact be the ones that had the greatest upside potential.
In fact, a project with huge upside potential may be nearly unrecognizable as such. Perhaps it has been starved of attention, resources, or vision, and so it looks small and scraggly. By taking a closer look to see what its upside potential really is, you may find that it has every chance of growing into a pretty, plump and profitable business if it gets the care and feeding it needs.
The key to having enough attention and food to give to your best projects — the ones with the best chances of growing into their upside — is to stop squandering attention and food on projects that will never make a big contribution to your business, not even if they reach their greatest potential. This requires you to have and use a tool for making credible and comparable evaluations of all of your projects. You can then identify which projects have strong upside potential and which ones will not significantly move the needle, even if they are completely successful.
And it requires that you act on those evaluations. You must decide what to do with your weakest projects. Do you abandon them outright and redirect the resources they were consuming to your more promising projects? Or do you look for hidden sources of significant upside within weaker projects and, if and when you find them, figure out ways to refocus or pivot those projects to strive for that upside?
Again, your best projects are often hard to recognize. It's easy to choose to fund and back projects that will clearly succeed and generate high growth. It's almost as easy to choose to kill off difficult projects that, even if successful, will never contribute much to the company's growth and profitability.
What's hard is deciding to back a project with uncertain prospects of success but that, if successful, will make a big contribution to the company's continued success. It's even harder to create a learning plan that guides you to finding out early on whether the project is viable and whether the perceived sources of upside are real. That learning plan will keep your focus on validating that upside, capturing it if it proves real and, if it isn't real, pivoting quickly to better uses of your resources.
Having the right tools and using them properly lets you distinguish clutter from value in your portfolio, and lets you realize that value through learning plans that bring hidden upside out into the open. You'll be left with the prettiest, plumpest opportunities in your portfolio.


text:
PHONE BOOK OF MICHIGAN
Area Code 313 and 586
Michigan Government
Farmington Hills, Michigan White Pages
Farmington Hills White Pages
Farmington Hills uses the Michigan area code. Other information about persons whose phone numbers start with this area code is also available on the Michigan white pages. Here, you can expect to find a person's name, address, and other publicly available information using their phone numbers.
County Oakland County
Area Code 231, 248, 269, 313, 517, 586, 616, 679, 734, 810, 906, 947, 989
Rank in State 12th
Rank in Country 453rd
Population 79,631
Land Area (mi²) 33,3 sq mi
Density (mi²) 2,393,70/sq mi
Zip codes 48167, 48322, 48331, 48334
Median Household Income $86,178
Estimated per capita income $45,125
Median Age 40,1 years
Males (50,6%)
Females (49,4%)
Website https://www.fhgov.com
© phonebookofmichigan.com | All rights reserved.

redpajama_set_name: RedPajamaCommonCrawl


Because you need a voice solution that works for your business.
Whether your business has two employees or 1,000, enTouch Business has a solution to help you stay connected. We offer a mix of traditional and Hosted Voice services suited for companies of all sizes and structured to scale as your business grows. We'll help you choose the service that will help your business run smoothly and back it with unparalled customer service. When you call for assistance, you'll always speak to a live customer service representative – never an automated attendant. But if you need one of those, we sell them too.


Heard the one about the KTM 690 Duke, that walked into the Bike Shed for an oat milk flat white? No? You have now: welcome to Husqvarna's Svartpilen 701: Swedish name, styled by a Frenchman, built by Austrians. And if that isn't cosmopolitan enough, BikeSocial is at the launch in Portugal to find out if the Svartpilen 701 really is just KTM's 690 Duke in a frock, or if it offers a genuinely alternative riding experience.
Swedish motorcycle manufacturers since 1903, Husqvarna competed in Grand Prix and at the TT in the 1930s. In the 60s and 70s the company was focused on motocross and enduro, winning multiple world titles; in 1987 the business was sold to Cagiva and the factory moved to Varese in Italy. A group of employees remained in Sweden and founded Husaberg, continuing to build off-road bikes.
In 2007 BMW Motorrad bought Husqvarna with the intention of running it as a separate brand, but although the tuned, F800-based Nuda road bike was well-received, in 2013 BMW sold Husqvarna to rivals KTM – allowing Husaberg and Husqvarna to be reunited at KTM's factory in Mattighofen. Since then, Husqvarna has continued as a side-brand to KTM, growing from just over 16,000 sales in from 2014 to in excess of 48,000 bikes in 2018.
What's is Husqvarna's Svartpilen 701?
In 2015 Husqvarna released the Svartpilen (Black Arrow) and Vitpilen (White Arrow) 401 – KTM 390 Duke platform-based street bikes – and in 2018 they debuted the clip-on, café racer-style Vitpilen 701, built on a KTM 690 Duke platform.
Now it's the turn of the Svartpilen 701, similarly styled around the single-cylinder 690 Duke engine, chassis and electronics, but swapping the Vitpilen's clip-ons for handlebars, and adding flat-track form with a side number-board, 18in front wheel size, Pirelli MT60 flat track tyres, and extra wheel travel and suspension changes.
The Svartpilen 701's single-cylinder motor is a direct, platform-style lift from KTM's 690 Duke – same cases, crank, rod, piston, cams, cylinder head, bore and stroke (turns out KTM's 690cc is rounded down from an actual 692.7cc). A claimed 75bhp at 8500rpm makes it the most powerful single in production; 2bhp more than the Duke's 73bhp at 8000rpm as a result of minor revisions to intake, exhaust and engine mapping. The slight retune shifts peak torque 250rpm higher, but lowers the peak figure slightly – 53 lb.ft at 6750rpm instead of 54.6 lb.ft at 6500rpm.
The changes in performance are very small and probably more likely for differentiating between the KTM and Husqvarna brands on paper as much as providing a meaningful improvement; either way, think about those numbers for a moment, whichever badge is on the tank: over 70bhp from a single cylinder four-stroke with 6000-mile service intervals.
Just wow. That's such a big, fat, ton of go-motion from an engine weighing less than 50kg. We can all probably remember when an inline four sports 600 didn't make much more than 75bhp (or didn't rev all that much higher than 8500rpm either, really). The 690 (or 701, ahem) is a remarkable engine and if you're at all cynical about the point of a single cylinder in street bikes over 125cc, then five minutes in the company of the Austro-Swedish one-lung will forcefully show you the error of your ways.
And all that zap gives the Husky a super-aggressive edge to its explosive, instant grunt, shoveled out with snappy intensity from a light, quick-action throttle, barking through an aftermarket Akrapovic – or, to be fair, the equally loud standard exhaust. Peeling apart mid-morning downtown Lisbon traffic like a jet-black tin-opener on steroids, the Svartpilen is an urban weapon of considerable effectiveness, and the way it batters through its up/down quickshifter gearbox in a sonorous yelp of enthusiasm is as much time-saving as life-affirming. Why don't more people commute like this? They're mad, the losers.
Single cylinder motors offer just about the highest level of throttle-to-tarmac intimacy you can get from an engine, because their traction physics are easy for a human brain to decode and intuitive to control. We just seem to know, by magic, how much grip is available – and, in the Svartpilen 701's case on its Pirelli MT60 flat-track knobbly tyres, it's just as well because rear grip is at a premium. Give Husky a handful on rondabouts and corner exits and the back end will start to come round in a lovely mini-slip before it's captured by the 701's KTM-borrowed traction control. It's not a liability, it's a delight; the way the Svartpilen's throttle squares off turns is a whip-cracking delight. But you'd be well advised not to switch the traction control off (especially, I would imagine, in the wet) – which you can, while the bike is moving, from a button on the circular dash, for wheelies. But better road tyres would give more confidence and grip.
Traditional single-cylinder mechanical limitations don't seem to apply to the Husky 701/690 Duke motor – it doesn't vibrate particularly harshly and it revs to high heaven, made possible with a short stroke and wide bore to keep mean piston speed down. In the cylinder head, a single overhead cam carries a central lobe operating a forked rocker arm for the exhaust valves with lobes either side acting via rockers on each intake valve. This compact arrangement effectively frees-up space where the second camshaft would normally sit, and allows KTM to fit a second balancer shaft driven by the camchain (and in addition the conventional crank balancer), to help tame single cylinder vibes.
It works – at 80mph in top the Svartpilen is revving at 5500rpm and high-pitched vibes are present, but not insufferable (at that speed the Husky's piston is travelling up and down at an average of 32.8mph). This in turn gives the Svartpilen a great deal more flexibility than a conventional single – this is a motor that can turn its hand to longer rides without requiring a visit to a dentist (or optician) afterwards and will happily sit at illegal speeds on a motorway all day. Although to do that would be a near criminal waste of the Husky's true talent.
The Svartpilen 701's tank is 12 litres, of which 2.5 litres is reserve – at the start of the launch ride and with a topped-up tank, the on-board fuel range trip took 15 minutes' of riding to warm up and register an estimated 150 miles. But after some 50 miles or so, the gauge had dropped to half and the estimated range now showed 70 miles to empty – and then after another 30 miles or so the display gave up and showed empty with a refuel warning. I wouldn't expect much more than 100 miles before searching for a friendly fuel pump.
Like its motor, the Svartpilen borrows almost all its chassis from the KTM 690 Duke – same chrome moly steel tube trellis frame, same aluminium swingarm, same fully-adjustable WP suspension – with internals from the 690 Duke R taking wheel travel to 150mm front and rear, 15mm more than the Vitpilen. This is partly to accommodate the new 18in front wheel size (see below) and partly to give the Svartpilen a slightly more refined ride than the 'sportier' Vitpilen. Dry weight is a claimed 159kg, which works out at around 175kg with fuel and fluids.
And boy is it light and agile. Narrowly waisted, knees almost touching beneath the tank, the Svartpilen is keener and leaner than a hungry whippet. It steers with breath-taking spontaneity; just think it and it's already there, in and out of any gap and any corner in milliseconds. It's the kind of handling that encourages the taking of ever-larger liberties – diving past cars, squeezing through improbable gaps, bouncing off the odd mirror here and there if needs be. The Svartpilen has the capacity to be a very, very effective, and naughty, town tool.
Out on the open road the steering isn't nervous or hyperactive; at most speeds the Svartpilen tracks with steady determination. Crank it up to full volume and the lack of wind protection makes the scene go a bit blurry – hitting an apex starts to become guesswork – but Husqvarna/KTM have spent the money in the right place; the WP springs are quality suspension, and what a super-light chassis really needs to remain stable and have any chance of ride quality. Easy-adjust clickers on the fork tops make adjustment simple; compression on the left, rebound on the right.
The same 320mm Galfer disc and Brembo four-pot radial caliper at the front as the KTM Duke and the same Bosch ABS system generate plenty of anchorage – with only 175kg (plus rider, ahem) to haul up, twin discs would over-brake the front end.
The Svartpilen's cast aluminium wheels are, as pointed out by a fellow journalist on the launch, the same style as the old 2008-2013 690 Duke's cast items, only with a Husqvarna badge cast into the base of the spokes; that and the fact the front is an 18in rim as opposed to the KTM and Husky Vitpilen's 17in. Tyres are Pirelli MT60 flat track-style knobblies.
Wire spoked wheels are available from May as an optional extra; no price at the moment.
This is the most important part of the Husqvarna Svartpilen story; after all, it's what truly differentiates the Husky from the KTM. And there's a surprising amount to get into; it's not all skin-deep cosmetics (although much of it is).
First off, the Svartpilen has a very different priding position to the Duke – it's a lower, more-canted forward stance, much like a small naked bike, while the Duke still has more than a hint of upright, pulled-forward, off-road position.
The Husky's bars are lower and pulled further back than the upright KTM, but the 701's tank is much longer and its seat, although the same 835mm height, is further away. On the 690, the seat position is heavily sculpted and sits deep behind the tank; its lowest point is just set back from the frame's swingarm pivot. On the 701, the seat is much thinner and much flatter, putting the rider's hips a few inches still further back, and with less of a tank 'hump' to grip with knees or crotch.
The 701's sportier stance is supported by the Husky's footpegs, set back on rearsets and mounted a good six or seven inches behind the line of the swingarm pivot; the KTM 690 Duke's pegs are directly in line with it.
The overall result is actually fairly comfy; hard seats don't compress as much as deeply padded seats, meaning the initial impression is, "Cor, this is hard,", but which doesn't change much however many miles you ride. Deep seats feel more comfy initially, but soon flatten out and end up causing 'hot-spots' of bum-ache.
The pillion gets less of a deal; the 701 has the same rubberised-style pillion pad material as KTM's RC 390 – it'll carry a passenger at a push, but no-one will want to be on it for long.
The Husqvarna's clocks are a bit of a failure; the dial looks like a cheap, oversized tax disc holder. It's hard to read its information partly because the crammed numbers are small and partly because the dash is perma-lit by reflected sunlight, so it's hard to read. In the absence of a stronger contrast LCD, some sort of angle adjustment would be useful.
But surely everyone would rather Husky spend their money on quality suspension than clocks – and there are plenty of other details for build-quality nerds to get off on. The mag-style filler cap is cool, the fuel tank's chamfered sides are cool, the slash-cut styling line from seat/tank, through side panel, pillion footpeg hanger, exhaust hanger and even into the swingarm casting, is a coherent motif.
And then there's the Svartpilen's styling – designed by Frenchman Maxime Thouvenin, the flat-track theme is evident but not overplayed; at no point to you feel particularly obliged to go round a corner wearing a steel shoe, poking your foot out or, indeed, away from tarmac. Personally, I find the asymmetric number-board naff, but love the Diavel/GS-style swingarm mounted rear plate holder. More fashionable chaps than I on the launch found the reverse to hold true. Horses for courses, etc.
No surprises; the Svartpilen 701 comes with the same level of kit as the KTM 690 Duke – ABS, basic and switchable traction control, excellent up-down quickshifter, slipper clutch, LED lights all round; but no rider modes, cruise control or nail-polishing service.
KTM 690 in a frock? Well, the Svartpilen is a bit more than just that – there's enough difference primarily in the riding position to differentiate between the two with your eyes closed; depending on your point of view, the Husky has kept all the good bits (that incredible engine, the light weight, the intensely agile chassis, the cracking suspension and brakes) and simply altered the look, style and feel of riding it. Empirically and dynamically they're much the same – and the Svartpilen would certainly be on my list of about-town weapons if I actually lived in a city, loved carving up traffic with extreme prejudice, and didn't particularly have a thing for orange.
But the Svartpilen's big sticking point is its price. £8899; nearly nine grand for a single, however rampant, is a lot to ask. The 690 Duke, at £8299, is already a bit rich. You can get a 790 Duke for £8799, with over 100bhp, one extra cylinder, more gadgets and ton more versatility. You'd have to seriously not like orange to ignore that buying decision.


Due to the wet August weather a few of the mainstem salmon fry index sites were still outstanding. Dropping river levels recently provided an opportunity to complete all outstanding surveys as well as allowing us to repeat some surveys to assess the impact of the mega August spate and the mainstem mitigation stocking.
As usual we videod the surveys. The 3 minute survey at the Craigellachie site just upstream of the Telford Bridge can be viewed here. At the site we caught 100 wild salmon fry and 6 parr, better than the average from that site over the last three years. We also caught 12 fin-clipped (stocked fry) from the mitigation stocking carried out in some beats in the lower river following the August spate.
Watch the video and you will see that we caught fry on almost all the sweeps of the electrofisher -encouraging results after the spate.


text:
Fort Wayne Volunteer Organizations: Volunteer in Fort Wayne, IN 46802, USA | VolunteerMatch
Find the best places to volunteer in Fort Wayne. Learn more about Fort Wayne volunteer organizations in cause areas like education, crisis support, homeless & housing, and more.
There are 139 organizations in the Fort Wayne area with volunteer opportunities waiting to be filled.
Find an Organization
Organization Search Results
Organizations within 20.0 miles of Fort Wayne, IN 46802, USA
Sorted by Date Updated
31. MADINA VILLAGE SCHOOL INC
Everybody has dreams of helping someone they love. Some people have exceptional dreams of creating something that will help many loved ones far into the future. The proposal to build a school in... More >
FORT WAYNE, IN | ( No Reviews)
Interest Areas: Children & Youth, Education & Literacy, Faith-Based, International
Updated: 3/21/17 , Distance: 6.81 miles away
32. The League for the Blind & Disabled
The League engages in community change activities through systemic advocacy, public education, collaboration, and technical assistance. The agency advocates for change at local, state and national... (41.03107,-85.11177) More >
Interest Areas: Advocacy & Human Rights, Children & Youth, People with Disabilities
33. DOVES NEST INC
The first MCC Thrift Shop was started in 1972 by a group of 4 women in Altona, Manitoba as a way to raise money for Mennonite Central Committee. The women expected their thrift project to only last... (41.12109,-85.05692) More >
Interest Areas: Disaster Relief, Hunger, International, Justice & Legal
Updated: 2/2/17 , Distance: 7.24 miles away
34. Matthew 25 Clinic
Matthew 25 Health and Dental Clinic in partnership with the community will assist people in need to secure comprehensive healthcare and wellness education. (41.0772,-85.1337) More >
Interest Areas: Health & Medicine
Updated: 11/14/16 , Distance: 2.27 miles away
35. ALLEN COUNTY EDUCATION PARTNERSHIP
Project READS looks to pair a trained community volunteer with a K-3 student at one of 19 sites throughout Allen County, IN. (41.05001,-85.15095) More >
Interest Areas: Children & Youth, Community, Education & Literacy
36. National Military History Center
The National Military History Center (NMHC) in Auburn, IN is dedicated to honoring all branches of the military from all eras. It features the service and sacrifice of America's military through a... (41.32482,-85.09178) More >
Auburn, IN | ( No Reviews)
Interest Areas: Arts & Culture, Community, Education & Literacy, Seniors, Veterans & Military Families
Updated: 9/12/16 , Distance: 18.49 miles away
37. The Center for Whitley County Youth
You've found The Center for Whitley County Youth, a non-profit, community-based, youth development organization benefitting Whitley County since 2005. Watch our video to learn more. Healthy young... (41.1558,-85.490) More >
Columbia City, IN | ( No Reviews)
Interest Areas: Arts & Culture, Children & Youth, Community, Faith-Based, Sports & Recreation
Updated: 9/7/16 , Distance: 17.63 miles away
38. Turnstone
Turnstone serves people of all ages with orthopaedic and neurological disabilities and their families. We provide a wide array of therapeutic services and programs which include physical,... More >
Interest Areas: Children & Youth, People with Disabilities, Seniors, Sports & Recreation, Veterans & Military Families
39. SCAN INC
Since 1974, SCAN's focus on eliminating the abuse and neglect of children through education, family services and community partnerships has remained constant. SCAN's roots run deep. The... (41.07954,-85.1467) More >
Interest Areas: Advocacy & Human Rights, Children & Youth, Community, Crisis Support, Education & Literacy
40. Great Lakes Caring - Fort Wayne
Caring is at the heart of everything we do, especially when walking you through life's final chapters. When family, friends, loved ones and referring physicians entrust us with your care when... (41.07397,-85.23428) More >
Interest Areas: Community, Health & Medicine, Seniors


text:
Q: Eclipse GUI shows newly styled button incorrectly I'm starting with a fully working app, with all the buttons in the right locations and the right sizes... but now I wanted to try out using styles for the first time. In particular I wanted to have the text colour in my buttons a dark blue and the background white. So I wrote the following in styles.xml in res/values ->
<?xml version="1.0" encoding="utf-8"?>
<resources>

   <style name="mybut"  parent="@android:style/Widget.Button">

        <item name="android:textColor">@color/dblue</item>
         <item name="android:background">#ffffffff</item>
    </style>

</resources>

I modified my button code as follows:
    <Button
        android:id="@+id/spec"
        style="@style/mybut"  <-- I added this line here
        android:layout_width="match_parent"
        android:layout_height="wrap_content"
        android:layout_weight="0.12"
        android:text="@string/spec" />

In eclipse's XML viewer, the new button looked right in every way. But then at run time, on my android device, the button's height had shrunk by about a third! Any ideas?
EDIT: I'm not very confident about the parent="@android:style/Widget.Button" bit. I'm suspicious that perhaps I'm somehow already using some other style?/theme? and perhaps the line should look something akin to parent="@android:otherstyle/Widget.Button" or parent="@android:style/other.Widget.Button"... or similar.
EDIT: FYI... I'm trying this out on a kind of "home screen" activity which just contains two big buttons. I added the style="@style/mybut" to just one of the two buttons. They are now clearly very different sizes.
EDIT: I noticed that in the manifest I have android:theme="@android:style/Theme.NoTitleBar.Fullscreen" ... does that mean I need to make my button's parent android:theme="@android:style/Theme.NoTitleBar.Fullscreen.Widget.Button" ?? or something like that?

A: By default, a Button has a 9-patch background, not a simple color. This image has padding and a content area which alters the actual size of the button.
When you change the background, you're stripping that padding, and it appears smaller. The correct way to do this is to create a new 9-patch, based on the old, but with the colors changed.

A: The problem must be here:
  android:layout_weight="0.12"

If you have an action_bar or something like that when you run your app the button is going to shrink. In your preview the action_bar doesn't appear (I'm just guessing).
EDIT:
Here is a sample of a custom button that I use in my app, I put a min height and width.
Instead of using: parent="@android:style/Widget.Button" I use: parent="android:Widget.Button"
<style name="ButtonCustom" parent="android:Widget.Button">
    <item name="android:background">@drawable/btn_default_holo_light</item>
    <item name="android:minHeight">48dip</item>
    <item name="android:minWidth">64dip</item>
    <item name="android:textColor">#fff</item>
</style>


Law 661, the Law of Climate Change and Sustainable Development, will focus on ways in which the law is changing in view of likely climate change impacts and to promote sustainable development. It will focus on climate change adaptation (adjusting to the unavoidable impacts of climate change), not climate change mitigation (reducing greenhouse gasses). Thus, this course will not duplicate topics covered in environmental law and energy law courses.
Topics to be covered in Law 661 will include emerging developments concerning climate change in business law, such as corporations and securities law, disclosure requirements, fiduciary responsibilities of officers and directors, and insurance law. Legal regimes promoting adaptation to climate change and fostering sustainable development will also be covered, including land use planning (especially in coastal and vulnerable areas), regional transportation planning, stormwater management, and the promotion of "green" buildings through building codes and building efficiency requirements and incentives.
The class will meet alternating weeks, with attendance and active participation in all class sessions expected of all students. Class members will write six short papers over the semester and class grades will be based on these papers and on class participation. There will be no final exam or final paper.
Session 01: Introduction and the national U.S. framework for climate change adaptation and sustainable development planning.
Session 02: State adaptation planning, including vulnerability assessments and disclosure requirements.
Session 03: Sustainable cities and the built environment.
Session 04: Sustainable water resources management.
Session 05: Managing vulnerable areas, including insurance and disaster response.
Session 06: Managing working landscapes (farms and forests) and natural areas.
Session 07: Building a sustainable world; the international dimension.
Please send questions about Law 661 to mvanputt@umich.edu.


text:
\section{Introduction}

We consider the Maxwell-Dirac system
\begin{equation}\label{MD}
\left\{
\begin{aligned}
  &(-i\gamma^\mu \partial_\mu + M) \psi = A_\mu \gamma^\mu \psi,
  \\
  &\square A_\mu = \overline{\psi} \gamma_\mu \psi,
\end{aligned}
\right.
\end{equation}
on the Minkowski space-time $\mathbb{R}^{1+d}$ for space dimensions $d \le 3$. This fundamental model from relativistic field theory describes the interaction of an electron with its self-induced electromagnetic field. Our interest here is in the Cauchy problem with prescribed initial data at time $t=0$,
\begin{equation}\label{Data}
  \psi(0,x) = \psi_0(x), \quad A_\mu(0,x) = a_\mu(x), \quad \partial_t A_\mu(0,x) = b_\mu(x),
\end{equation}
and the question of local or global solvability, which has received some attention in recent years; see \cite{Chadam1973, Delgado1978, Bachelot2006, Huh2010, Okamoto2013, Zhang2014} for the case of one space dimension and \cite{Gross1966, Georgiev1991, Bournaveas1996, Esteban1996, Flato1997, Psarelli2005, Selberg2010b, Selberg2011, Gavrus2017} for higher dimensions, and the references therein.

The unknowns are the spinor field $\psi=\psi(t,x)$, taking values in $\mathbb{C}^N$ ($N=2$ for $d=1,2$; $N=4$ for $d=3$), and the real-valued potentials $A_\mu=A_\mu(t,x)$, $\mu=0,1,\dots,d$. $M \ge 0$ is the mass.

The equations are written in covariant form on $\mathbb{R}^{1+d} = \mathbb{R}_t \times \mathbb{R}_x^d$ with the Minkowski metric $(g^{\mu\nu}) = \mathrm{diag}(1,-1,\dots,-1)$ and coordinates $(x_\mu)$, where $x_0=t$ is the time and $x=(x_1,\dots,x_d)$ is the spatial position. Greek indices range over $0,1,\dots,d$, latin indices over $1,\dots,d$, and repeated upper and lower indices are implicitly summed over these ranges. We write $\partial_\mu = \frac{\partial}{\partial x_\mu}$, so $\partial_0=\partial_t$ is the time derivative, $\nabla = (\partial_1,\dots,\partial_d)$ is the spatial gradient, and $\square = \partial^\mu \partial_\mu = \partial_t^2-\Delta$ is the D'Alembertian. The $N \times N$ Dirac matrices $\gamma^\mu$ are required to satisfy
\begin{equation}\label{Matrices}
  \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 g^{\mu\nu} I,
  \qquad
  (\gamma^0)^* = \gamma^0, \qquad (\gamma^j)^* = - \gamma^j.
\end{equation}
We denote by $\psi^*$ the complex conjugate transpose, and write $\overline \psi = \psi^*\gamma^0$.

Key features of the Maxwell-Dirac system are the gauge invariance, the scaling invariance and the conservation laws, which we now recall.

Firstly, there is a $\mathrm{U}(1)$ gauge invariance
\[
  \psi \longrightarrow e^{i\chi}\psi, \qquad A_\mu \longrightarrow A_\mu + \partial_\mu \chi,
\]
for any real valued $\chi(t,x)$. This implies gauge freedom, allowing to specify a gauge condition on the potentials. The particular form \eqref{MD} of the Maxwell-Dirac system appears when the Lorenz gauge condition $\partial^\mu A_\mu = 0$ is imposed, that is,
\begin{equation}\label{Lorenz}
  \partial_t A_0 = \nabla \cdot \mathbf A,
\end{equation}
where $\mathbf A = (A_1,\dots,A_d)$. Since this gauge condition reduces to certain constraints on the data \eqref{Data}, we did not include it in \eqref{MD}. In addition to the obvious constraint, there is a constraint coming from the Gauss law (implied by \eqref{Lorenz} and the second equation in \eqref{MD})
\[
  \nabla \cdot \mathbf E = \abs{\psi}^2,
\]
where $\mathbf E = \nabla A_0 - \partial_t \mathbf A$ is the electric field. Thus, the data constraints are
\begin{equation}\label{Constraints}
  b_0 = \partial^j a_j, \qquad \partial^j (\partial_j a_0 - b_j) = \abs{\psi_0}^2.
\end{equation}
If these are satisfied (in some ball), then a solution of \eqref{MD}, \eqref{Data} will also satisfy the Lorenz gauge condition \eqref{Lorenz} (in the cone of dependence over the ball).

Secondly, the system is invariant under the rescaling, in the case $M=0$,
\[
  \psi(t,x) \longrightarrow \lambda^{3/2} \psi(\lambda t, \lambda x), \qquad A_\mu(t,x) \longrightarrow \lambda A_\mu(\lambda t, \lambda x) \qquad (\lambda > 0).
\]
For Sobolev data $(\psi_0,a_\mu,b_\mu) \in H^s(\mathbb{R}^d) \times H^r(\mathbb{R}^d) \times H^{r-1}(\mathbb{R}^d)$, the scale-invariant regularity (for the homogeneous Sobolev norms, to be precise) is $s=s_c(d)=\frac{d-3}{2}$ and $r=r_c(d)=\frac{d-2}{2}$. By the usual heuristic one does not expect well-posedness below this regularity.

Thirdly, we consider conservation laws. While the Maxwell-Dirac system does have a conserved energy, which is roughly speaking at the level of $H^{1/2}$ for the spinor, this energy does not have a definite sign, so it is difficult to see how to make use of it to prove global existence. On the other hand, one has the conservation of charge
\begin{equation}\label{Charge}
  \int_{\mathbb{R}^d} \abs{\psi(t,x)}^2 \, dx = \int_{\mathbb{R}^d} \abs{\psi(0,x)}^2 \, dx,
\end{equation}
which plays a key role in all the known global existence results for large data. We will refer to solutions at this regularity, that is, with $t \mapsto \psi(t,\cdot)$ a continuous map into $L^2(\mathbb{R}^d)$, as \emph{charge class solutions}. It should be noted that the charge regularity $s=0$ coincides with the scaling-critical regularity $s_c(d) = \frac{d-3}{2}$ when $d=3$. Thus, the Maxwell-Dirac system is charge-critical in three space dimensions, and charge-subcritical in dimensions $d=1,2$.

The first global result for \eqref{MD}, \eqref{Data} was obtained by Chadam \cite{Chadam1973}, in one space dimension, for data $(\psi_0,a_\mu,b_\mu) \in H^1(\mathbb{R}) \times H^1(\mathbb{R}) \times L^2(\mathbb{R})$. Chadam first proved local existence and uniqueness, and was able to extend the solution globally by proving an a priori bound on the $H^1(\mathbb{R}) \times H^1(\mathbb{R}) \times L^2(\mathbb{R})$ norm of the solution via a clever boot-strap argument making use of the conservation of charge \eqref{Charge}. But to be able to prove global existence with a more direct use of the conservation of charge, in any dimension, a natural strategy is to try to prove local existence of charge class solutions.\footnote{However, it should be noted that such a result does not immediately imply global existence via the conservation of charge, since one also needs a priori estimates for the potentials.}

We proceed to recall what is known about local and global well-posedness in the charge class.

Starting with one space dimension, we note that Bournaveas \cite{Bournaveas2000} proved global charge-class existence for the related Dirac-Klein-Gordon system, but the argument relies on a null structure in Dirac-Klein-Gordon which is not present in Maxwell-Dirac. Bachelot \cite{Bachelot2006} gave another proof that does not rely on null structure and applies also to Maxwell-Dirac; similar results have been obtained in \cite{Huh2010, Zhang2014, Selberg2018}.

In the charge-critical three space-dimensional case, local well-posedness remains an open question in the charge class, but has been proved almost down to that regularity by D'Ancona, Foschi and Selberg \cite{Selberg2010b}; see also \cite{Bournaveas1996,Gross1966} for earlier local results at higher regularity, and \cite{Georgiev1991,Flato1997,Psarelli2005} for small-data global results. The existence of stationary solutions was proved in \cite{Esteban1996}.

In two space dimensions, global well-posedness in the charge class was proved by D'Ancona and Selberg \cite{Selberg2011}.

To summarise, in the charge-subcritical dimensions $d=1,2$, there is global well-posedness in the charge class, and in the charge-critical dimension $d=3$, local well-posedness holds almost down to the charge regularity. Our aim here is to show that these results are sharp (or almost sharp, for $d=3$), by proving ill-posedness below the charge regularity. This result is somewhat surprising in the subcritical cases, and in particular for $d=1$. Indeed, it should be noted that in dimensions $d=2,3$, the proof of local existence at or near the charge regularity is quite involved and requires a subtle null structure that was uncovered in \cite{Selberg2010b}. By contrast, the proof in the case $d=1$ is elementary (see section \ref{WPproof}) and does not require this null structure. It was therefore expected that, by exploiting the latter, one should be able to go below the charge. But our result shows that this is not possible, and this means that the null structure is not helpful in the case $d=1$.

We remark that our proof of ill-posedness works also in dimensions $d \ge 4$, but then the critical regularity is above the charge, so this is not really of much interest. In dimensions $d \ge 4$, global existence and modified scattering for data with small scaling-critical norm has been proved in \cite{Gavrus2017}.

We now state our main results.

\section{Main results}

The following notation is used. For $1 \le p \le \infty$, $L^p(\mathbb{R}^d)$ denotes the standard Lebesgue space. For $s \in \mathbb{R}$, $H^s(\mathbb{R}^d)$ is the Sobolev space $(1-\Delta)^{s/2} L^2(\mathbb{R}^d)$. For an open set $U$ in $\mathbb{R}^d$ or $\mathbb{R}_t \times \mathbb{R}_x^d$, $\mathcal D'(U)$ is the space of distributions on $U$. We write
\begin{align*}
  X_0 &= \text{$H^s(\mathbb{R}^d)$ for some $s < 0$, or $L^p(\mathbb{R}^d)$ for some $1 \le p < 2$}.
  \\
  B &= \text{the open unit ball in $\mathbb{R}^d$, centred at the origin}.
  \\
  K &= \text{the cone of dependence over $B$}.
  \\
  K_T &= \text{$K \cap \left([0,T] \times \mathbb{R}^d \right)$, for $T > 0$}.
\end{align*}
Thus, $K = \{ (t,x) \in \mathbb{R} \times \mathbb{R}^d \colon 0 \le t < 1,\; \abs{x} < 1-t \}$. The interior of the truncated cone $K_T$ will be denoted $\mathrm{Int}(K_T)$.

We will use the following facts concerning $C^\infty$ solutions of \eqref{MD}, which follow from the general theory for semilinear wave equations. Assume we are given data \eqref{Data} belonging to $C^\infty(\mathbb{R}^d)$. Then there exists a corresponding $C^\infty$ solution $(\psi,A_\mu)$ of \eqref{MD} on an open subset $U$ of $[0,\infty) \times \mathbb{R}^d$ containing the Cauchy hypersurface $\{0\} \times \mathbb{R}^d$. Moreover, we may assume that $U$ is \emph{causal}, in the sense that for every point $(t,x)$ in $U$, the cone of dependence $K^{(t,x)}$, with vertex $(t,x)$ and base in $\{0\} \times \mathbb{R}^d$, is contained in $U$. The solution in the cone $K^{(t,x)}$ is uniquely determined by the data in the base of the cone. By the uniqueness, and since the union of two causal sets is again causal, there exists a maximal solution of the type described above, and we call this the \emph{maximal $C^\infty$ forward evolution} of the given data.

In the first version of our ill-posedness result, we take vanishing data for the potentials.

\begin{theorem}[Ill-posedness I]\label{MainThm1}
In space dimensions $d \le 3$, the Cauchy problem \eqref{MD}, \eqref{Data} is ill posed for data
\[
  \psi_0 \in X_0, \qquad a_\mu = b_\mu = 0 \qquad (\mu=0,\dots,d).
\]
More precisely, there exists $\psi_0^{\mathrm{bad}} \in X_0 \setminus L^2(\mathbb{R}^d)$ such that for any $T > 0$ and any neighbourhood $\Omega_0$ of $\psi_0^{\mathrm{bad}}$ in $X_0$, there fails to exist a continuous map
\[
  S \colon \Omega_0 \longrightarrow \mathcal D'\left( \mathrm{Int}(K_T) \right),
  \qquad \psi_0 \longmapsto S[\psi_0] = (\psi,A_\mu),
\]
with the property that if $\psi_0 \in \Omega_0 \cap C_c^\infty(\mathbb{R}^d)$, then $S[\psi_0]$ is $C^\infty$ in $K_T$ and solves \eqref{MD} there, with intial data $\psi_0$ and $a_\mu=b_\mu=0$ in $B$.
\end{theorem}

This result applies to \eqref{MD}, \eqref{Data} without regard to the data constraints \eqref{Constraints}, which of course are not compatible with the assumption $a_\mu=b_\mu=0$. We next state an alternative version of the result, which allows to take into account the constraints. In fact, Theorem \ref{MainThm1} is an immediate consequence of the following more precise result.

\begin{theorem}[Ill-posedness II]\label{MainThm2}
Let $d \le 3$. There exist $\psi_0^{\mathrm{bad}} \in X_0 \setminus L^2(\mathbb{R}^d)$ and $\psi_{0,\varepsilon}, a_{\mu,\varepsilon}, b_{\mu,\varepsilon} \in C_c^\infty(\mathbb{R}^d)$ for each $\varepsilon > 0$, such that
\begin{itemize}
\item[(i)] $\psi_{0,\varepsilon} \to \psi_0^{\mathrm{bad}}$ in $X_0$ as $\varepsilon \to 0$.
\item[(ii)] The maximal $C^\infty$ forward evolution $(\psi_\varepsilon,A_{\mu,\varepsilon})$ of the data $(\psi_{0,\varepsilon}, a_{\mu,\varepsilon}, b_{\mu,\varepsilon})$ exists throughout the cone $K$.
\item[(iii)] There exists $T > 0$ such that, as $\varepsilon \to 0$, $A_{0,\varepsilon}(t,x) \to \infty$ uniformly in any compact subset of $K_T \cap \{ (t,x) \colon \abs{x} < t \}$.
\end{itemize}
Moreover, we can choose the $a_{\mu,\varepsilon}, b_{\mu,\varepsilon}$ so that either
\begin{equation}\label{ZeroData}
  a_{\mu,\varepsilon} = b_{\mu,\varepsilon} = 0 \qquad \text{for $\mu = 0,\dots,d$},
\end{equation}
or
\begin{equation}\label{ConstrainedData}
  b_{0,\varepsilon} = \sum_{j=1}^d \partial_j a_{j,\varepsilon},
  \qquad
  \sum_{j=1}^d \partial_j \left( \partial_j a_{0,\varepsilon} - b_{j,\varepsilon} \right) = \abs{\psi_{0,\varepsilon}}^2 \qquad \text{in $B$}.
\end{equation}
\end{theorem}

Here, if we choose the alternative \eqref{ConstrainedData}, then $a_{\mu,\varepsilon}, b_{\mu,\varepsilon}$ do not have limits in the sense of distributions on $B$ as $\varepsilon \to 0$. This is not a deficiency of our construction, but is necessarily so, as our next result shows. The following theorem essentially says that the Gauss law for the initial data is ill posed when we are below the charge regularity.

\begin{theorem}[Ill-posedness of constraints]\label{MainThm3}
There exists $\psi_0^{\mathrm{bad}} \in X_0 \setminus L^2(\mathbb{R}^d)$ such that for any neighbourhood $\Omega_0$ of $\psi_0^{\mathrm{bad}}$ in $X_0$, there do not exist continuous maps
\[
  I_\mu, J_\mu \colon \Omega_0 \longrightarrow \mathcal D'(B)
\]
with the property that if $\psi_0 \in \Omega_0 \cap C_c^\infty(\mathbb{R}^d)$, then
\[
  a_\mu := I_\mu[\psi_0], \qquad b_\mu := J_\mu[\psi_0] \qquad (\mu=0,\dots,d)
\]
satisfy the constraint equations \eqref{Constraints} in $B$.
\end{theorem}

We conclude this section with a brief outline of the key steps in the proof of Theorem \ref{MainThm2}.

\medskip
\emph{Step 1.} We prove global well-posedness in the charge class for \eqref{MD}, \eqref{Data} in the case where the data only depend on a single coordinate, say $x_1$.

\medskip
\emph{Step 2.} To define the data $\psi_{0,\varepsilon}, a_{\mu,\varepsilon}, b_{\mu,\varepsilon} \in C_c^\infty(\mathbb{R}^d)$, we start with functions of $x_1$ and cut off smoothly outside the unit ball $B$. The corresponding maximal $C^\infty$ forward evolution $(\psi_\varepsilon,A_{\mu,\varepsilon})$ exists in the entire cone $K$, by Step 1, and depends only on $t$ and $x_1$ there.

\medskip
\emph{Step 3.} Using a null form estimate and a boot-strap argument we prove that there exists $T > 0$ such that $A_{j,\varepsilon}$, $j=2,3$, are uniformly bounded in $K_T$. A further boot-strap argument then yields a lower bound on $\abs{\psi_\varepsilon}$ in
$
  K_T \cap \{ (t,x) \colon 0 < t < x_1 \}.
$

\medskip
\emph{Step 4.}
Letting $\varepsilon \to 0$, we show that $A_{0,\varepsilon}(t,x) \to \infty$ uniformly in any compact subset of
$
  K_T \cap \{ (t,x) \colon \abs{x} < t \},
$
completing the proof of Theorem \ref{MainThm2}.
In fact, we prove this in the larger set $K_T \cap \{ (t,x) \colon \abs{x_1} < t \}$.
\medskip

The remainder of this paper is organised as follows. In section \ref{WPsection} we state the well-posedness result (Step 1), whose elementary proof is deferred until section \ref{WPproof}. In section \ref{MatricesSection}, we choose a particular representation of the Dirac matrices in dimensions $d \le 3$, write out the Maxwell-Dirac system in terms of the components of the spinor, and prove a null form estimate in one space dimension. In section \ref{DataSection} we specify the data (Step 2), and section \ref{IPsection} contains the proof of ill-posedness (Steps 3 and 4).

\section{Well-posedness for one-dimensional data}\label{WPsection}

We start by stating the result described in Step 1, the well-posedness in the case where the data only depend on the single coordinate $x_1$:
\begin{equation}\label{1dData}
  \psi(0,x) = \psi_0(x_1), \quad A_\mu(0,x) = a_\mu(x_1), \quad \partial_t A_\mu(0,x) = b_\mu(x_1).
\end{equation}
Then the solution of \eqref{MD} will depend only on $t$ and $x_1$. Indeed, if $(\psi,A_\mu)$ does not depend on $x_2,\dots,x_d$, then \eqref{MD} is equivalent to
\begin{equation}\label{MD'}
\left\{
\begin{aligned}
  (- i\gamma^0 \partial_t - i\gamma^1 \partial_1 + M) \psi &= \left( A_0 \gamma^0 + A_1 \gamma^1 + \dots + A_d \gamma^d \right) \psi,
  \\
  (\partial_t^2 - \partial_1^2) A_0 &= \psi^*  \psi,
  \\
  (\partial_t^2 - \partial_1^2) A_1 &= - \psi^* \gamma^0 \gamma^1 \psi,
  \\
  &\ \,\vdots
  \\
  (\partial_t^2 - \partial_1^2) A_d &= - \psi^* \gamma^0 \gamma^d \psi,
\end{aligned}
\right.
\end{equation}
and this is the system we will solve, with the initial condition \eqref{1dData}.

There is conservation of charge, for sufficiently regular solutions:
\begin{equation}\label{1dCharge}
  \int_{\mathbb{R}} \abs{\psi(t,x_1)}^2 \, dx_1 = \int_{\mathbb{R}^d} \abs{\psi(0,x_1)}^2 \, dx_1.
\end{equation}
Indeed, premultiplying the Dirac equation in \eqref{MD'} by $i\overline\psi=i\psi^*\gamma^0$, taking real parts, and using the fact that $M$ and the $A_\mu$ are real, and that $\gamma^0$ and $\gamma^0\gamma^j$ are hermitian, one obtains the conservation law $\partial_t \rho + \partial_1 j = 0$, where $\rho = \psi^*\psi = \abs{\psi}^2$ and $j = \psi^*\gamma^0\gamma^1\psi$. Integration then gives \eqref{1dCharge}.

We now state the global well-posedness result in the charge class. The $a_\mu$, $\mu=0,\dots,d$, will be taken in the space $AC(\mathbb{R})$ with norm
\[
  \norm{f}_{AC(\mathbb{R})} = \norm{f}_{L^\infty(\mathbb{R})} + \norm{f'}_{L^1(\mathbb{R})}.
\]
Thus, $AC(\mathbb{R})$ is the space of absolutely continuous functions $f \colon \mathbb{R} \to \mathbb{C}$ with bounded variation (cf.\ Corollary 3.33 in \cite{Folland1999}), and $AC_{\mathrm{loc}}(\mathbb{R})$ is the space of locally absolutely continuous functions.

\begin{theorem}\label{WPThm}
In any space dimension $d$, the Maxwell-Dirac system \eqref{MD} is globally well-posed for one-dimensional data \eqref{1dData} with the regularity
\[
  (\psi_0,a,b) \in \mathfrak X_0 := L^2(\mathbb{R};\mathbb{C}^N) \times AC(\mathbb{R};\mathbb{R}^{d+1}) \times L^1(\mathbb{R};\mathbb{R}^{d+1}),
\]
where $a = (a_0,\dots,a_d)$ and $b=(b_0,\dots,b_d)$. That is, for any $T > 0$, there is a unique solution
\[
  (\psi,A,\partial_t A) \in C([0,T];\mathfrak X_0), \qquad A=(A_0,\dots,A_d),
\]
depending only on $t$ and $x_1$. The solution has the following properties:
\begin{itemize}
\item[(i)] The data-to-solution map is continuous from $\mathfrak X_0$ to $C([0,T];\mathfrak X_0)$.
\item[(ii)] Higher regularity persists. That is, if $J \in \mathbb N$ and $\partial_1^j (\psi_0,a_\mu,b_\mu) \in \mathfrak X_0$ for $j \le J$, then $\partial_t^j \partial_1^k (\psi,A_\mu,\partial_t A_\mu) \in C([0,T];\mathfrak X_0)$ for $j+k \le J$.
\item[(iii)] If the data are $C^\infty$, then so is the solution.
\item[(iv)] The conservation of charge \eqref{1dCharge} holds.
\item[(v)] If the data constraints \eqref{Constraints} are satisfied for $x_1$ in an interval $I$, then the Lorenz gauge condition $\partial_t A_0 = \partial_1 A_1$ is satisfied in the cone of dependence over $I$.
\end{itemize}
\end{theorem}

In particular, taking $d=1$, this result provides an alternative to the charge-class results from \cite{Bachelot2006,Zhang2014}, with a stronger form of well-posedness and at the same time a much simpler proof. The elementary proof is given in section \ref{WPproof}. We use iteration to prove local existence, and to close the estimates we only rely on the energy inequality for the Dirac equation and an estimate for the wave equation deduced from the D'Alembert representation.

\section{The Dirac matrices and a null form estimate}\label{MatricesSection}

In this section we specify our choice of the Dirac matrices, in dimensions $d \le 3$. We do this in such a way that the Dirac equation in \eqref{MD'}, when written in terms of the spinor components, has a form which makes it easy to work with. Recall that that the spinor has $N=2$ components in space dimensions $d=1,2$, and $N=4$ components when $d=3$. We write
\[
  \psi = \begin{pmatrix} u \\ v \end{pmatrix},
\]
where $u,v$ are $\mathbb{C}$-valued for $d=1,2$ and $\mathbb{C}^2$-valued for $d=3$.

\subsection{Space dimension $d=1$}

We choose
\[
  \gamma^0 =
  \left( \begin{matrix}
    0 & 1  \\
    1 &0
  \end{matrix} \right),
  \qquad
  \gamma^1=
  \left( \begin{matrix}
    0 & -1  \\
    1 & 0
  \end{matrix} \right).
\]
Then \eqref{Matrices} is satisfied, and \eqref{MD'} becomes
\begin{equation}\label{MD1d}
\left\{
\begin{aligned}
  (\partial_t + \partial_x)u &= i (A_0 + A_1) u -iMv,
  \\
  (\partial_t - \partial_x)v &= i (A_0 - A_1) v -iMu,
  \\
  (\partial_t^2-\partial_x^2) A_0 &= \abs{u}^2 + \abs{v}^2,
  \\
  (\partial_t^2-\partial_x^2) A_1 &= - \abs{u}^2 + \abs{v}^2.
\end{aligned}
\right.
\end{equation}
Since $A_0$, $A_1$ are real valued, the first two equations imply
\begin{equation}\label{MD1dSquared}
\left\{
\begin{aligned}
  (\partial_t + \partial_x)\abs{u}^2 &= -2M \im\left( \overline v u \right),
  \\
  (\partial_t - \partial_x)\abs{v}^2 &= 2M \im\left( \overline v u \right).
\end{aligned}
\right.
\end{equation}

\subsection{Dimension $d=2$}

We choose
\[
  \gamma^0 =
  \left( \begin{matrix}
    0 & 1  \\
    1 &0
  \end{matrix} \right),
  \qquad
  \gamma^1=
  \left( \begin{matrix}
    0 & -1  \\
    1 & 0
  \end{matrix} \right),
  \\
  \qquad
  \gamma^2=
  \left( \begin{matrix}
    i & 0  \\
    0 & -i
  \end{matrix} \right)
\]
Then \eqref{Matrices} is satisfied, and \eqref{MD'} becomes, writing $x=x_1$ for simplicity,
\begin{equation}\label{MD2d}
\left\{
\begin{aligned}
  (\partial_t + \partial_x)u &= i (A_0 + A_1) u + A_2 v -iMv,
  \\
  (\partial_t - \partial_x)v &= i (A_0 - A_1) v - A_2 u -iMu,
  \\
  (\partial_t^2-\partial_x^2) A_0 &= \abs{u}^2 + \abs{v}^2,
  \\
  (\partial_t^2-\partial_x^2) A_1 &= - \abs{u}^2 + \abs{v}^2,
  \\
  (\partial_t^2-\partial_x^2) A_2 &= - 2 \im (u \overline v).
\end{aligned}
\right.
\end{equation}
Then we also have
\begin{equation}\label{MD2dSquared}
\left\{
\begin{aligned}
  (\partial_t + \partial_x)\abs{u}^2 &=  2A_2 \re \left( \overline v u \right)
  -2M \im\left( \overline v u \right),
  \\
  (\partial_t - \partial_x)\abs{v}^2 &= - 2A_2 \re \left( \overline v u \right) + 2M \im \left( \overline v u \right).
\end{aligned}
\right.
\end{equation}

\subsection{Dimension $d=3$}

The $4 \times 4$ Dirac matrices are, in $2 \times 2$ block form,
\[
  \gamma^0 =
  \left( \begin{matrix}
    0 & I  \\
    I &0
  \end{matrix} \right),
  \qquad
  \gamma^1=
  \left( \begin{matrix}
    0 & -I  \\
    I & 0
  \end{matrix} \right),
  \\
  \qquad
  \gamma^2=
  \left( \begin{matrix}
    \rho & 0  \\
    0 & -\rho
  \end{matrix} \right),
  \\
  \\
  \qquad
  \gamma^3=
  \left( \begin{matrix}
    \kappa & 0  \\
    0 & -\kappa
  \end{matrix} \right),
\]
where $I$ is the $2 \times 2$ identity matrix and $\rho$, $\kappa$ must satisfy
\[
  \rho^* = - \rho, \qquad \rho^2 = -I, \qquad \kappa^* = -\kappa, \qquad \kappa^2 = -I, \qquad \rho\kappa + \kappa\rho = 0.
\]
Then \eqref{Matrices} is satisfied. For example, we can choose
\[
  \rho =
  \left( \begin{matrix}
    0 & -1  \\
    1 & 0
  \end{matrix} \right),
  \\
  \qquad
  \kappa =
  \left( \begin{matrix}
    i & 0  \\
    0 & -i
  \end{matrix} \right).
\]
Then \eqref{MD'} reads (with $x=x_1$)
\begin{equation}\label{MD3d}
\left\{
\begin{aligned}
  (\partial_t + \partial_x)u &= i (A_0 + A_1) u - i A_2 \rho v - i A_3 \kappa v -iMv,
  \\
  (\partial_t - \partial_x)v &= i (A_0 - A_1) v + i A_2 \rho u + i A_3 \kappa u -iMu,
  \\
  (\partial_t^2-\partial_x^2) A_0 &= \abs{u}^2 + \abs{v}^2,
  \\
  (\partial_t^2-\partial_x^2) A_1 &= - \abs{u}^2 + \abs{v}^2,
  \\
  (\partial_t^2-\partial_x^2) A_2 &= - 2 \re (v^* \rho u),
  \\
  (\partial_t^2-\partial_x^2) A_3 &= - 2 \re (v^* \kappa u),
\end{aligned}
\right.
\end{equation}
where $u, v$ are now $\mathbb{C}^2$-valued. Then also
\begin{equation}\label{MD3dSquared}
\left\{
\begin{aligned}
  (\partial_t + \partial_x)\abs{u}^2 &=  2A_2 \im \left( v^* \rho u \right)
  + 2A_3 \im \left( v^* \kappa u \right)
  - 2M \im\left( v^* u \right),
  \\
  (\partial_t - \partial_x)\abs{v}^2 &= - 2A_2 \im \left( v^* \rho u \right)
  - 2A_3 \im \left( v^* \kappa u \right)
  + 2M \im\left( v^* u \right).
\end{aligned}
\right.
\end{equation}

\subsection{A null form estimate}

When we move from $d=1$ to $d=2$ or $d=3$, the decisive difference is that we pick up the additional fields $A_2, A_3$. These fields will be better behaved than $A_0, A_1$, since the right hand sides of the corresponding equations in \eqref{MD2d} and \eqref{MD3d} are null forms: They contain a product of  $v^*$ and $u$, which propagate in transverse directions. This fact will be exploited through the following crucial estimate (which fails for $uu$ and $u \overline u$).

We use the following notation. For $x \in \mathbb{R}$ and $t > 0$, let $K^{(t,x)}$ denote the backward cone with vertex at $(t,x)$, that is,
\begin{equation}\label{1dCone}
  K^{(t,x)} = \left\{ (s,y) \in \mathbb{R}^2 \colon 0 < s < t, \;\; x-t+s < y < x+t-s \right\}.
\end{equation}

\begin{lemma}[Null form estimate]\label{NullLemma}
Consider a system of the form
\[
\begin{alignedat}{2}
  (\partial_t + \partial_x)u &= F(t,x), & \qquad u(0,x) &= f(x),
  \\
  (\partial_t - \partial_x)v &= G(t,x), & v(0,x) &= g(x),
\end{alignedat}
\]
where $x \in \mathbb{R}$, $t > 0$, and the functions are $\mathbb{C}$-valued. For the solution $(u,v)$ we have the estimate, for all $X \in \mathbb{R}$ and $T > 0$,
\[
  \iint_{K^{(T,X)}} \abs{uv} \, dx \, dt
  \\
  \le
  \left( \norm{f}_{L^1} + \int_0^T \norm{F(t)}_{L^1} \, dt \right)
  \left( \norm{g}_{L^1} + \int_0^T \norm{G(t)}_{L^1} \, dt \right).
\]
\end{lemma}

\begin{proof}
Integrating, we have
\begin{align*}
  u(t,x) &= f(x-t) + \int_0^t F(s,x-t+s) \, ds,
  \\
  v(t,x) &= g(x+t) + \int_0^t G(s,x+t-s) \, ds.
\end{align*}
Taking absolute values, we see that for $0 \le t \le T$,
\begin{align*}
  \abs{u(t,x)} &\le \mu(x-t) := \abs{f(x-t)} + \int_0^T \abs{F(s,x-t+s)} \, ds,
  \\
  \abs{v(t,x)} &\le \nu(x+t) := \abs{g(x+t)} + \int_0^T \abs{G(s,x+t-s)} \, ds.
\end{align*}
By Fubini's theorem it is then obvious that
\[
  \iint_{K^{(T,X)}} \abs{uv} \, dx \, dt
  \le
  \norm{\mu}_{L^1(\mathbb{R})} \norm{\nu}_{L^1(\mathbb{R})}.
\]
But
\[
  \norm{\mu}_{L^1(\mathbb{R})}
  \le \norm{f}_{L^1(\mathbb{R})} + \int_0^T \norm{F(t)}_{L^1(\mathbb{R})} \, dt,
\]
and similarly for $\nu$, so we get the desired estimate.
\end{proof}

\section{Data for ill-posedness}\label{DataSection}

In this section we specify the data that are used to prove Theorem \ref{MainThm2}.

Choose a cut-off $\chi \in C_c^\infty(\mathbb{R})$ such that $\chi=1$ on $[-1,1]$. Let $\varepsilon > 0$. For the spinor datum and its approximations, which are $\mathbb{C}^N$-valued, we then take
\begin{equation}\label{SpinorData}
  \psi_0^{\mathrm{bad}}(x) = \chi(x_1) \cdots \chi(x_d) \begin{pmatrix} f(x_1) \\ 0 \\ \vdots \\ 0 \end{pmatrix},
  \quad
  \psi_{0,\varepsilon}(x) = \chi(x_1) \cdots \chi(x_d) \begin{pmatrix} f_\varepsilon(x_1) \\ 0 \\ \vdots \\ 0 \end{pmatrix},
\end{equation}
where
\begin{equation}\label{SpinorData2}
  f(x_1) =  \frac{1}{\abs{x_1}^{1/2}},
  \qquad
  f_\varepsilon(x_1) =  \frac{1}{(\varepsilon^2+x_1^2)^{1/4}}.
\end{equation}
Thus, $\chi f_\varepsilon \in C_c^\infty(\mathbb{R})$, and for $1 \le p < 2$ we have $\chi f \in L^p(\mathbb{R}) \setminus L^2(\mathbb{R})$ and
\[
  \lim_{\varepsilon \to 0} \norm{ \chi f_\varepsilon - \chi f }_{L^p(\mathbb{R})} = 0.
\]
By the Hardy-Littlewood-Sobolev inequality,\footnote{We use the inequality $\norm{g}_{H^s(\mathbb{R})} \le C \norm{g}_{L^p(\mathbb{R})}$, valid for $s = 1/2-1/p$, $1 \le p < 2$.} we then conclude that $\chi f \in H^s(\mathbb{R})$ for $s < 0$, and that
\[
  \lim_{\varepsilon \to 0} \norm{ \chi f_\varepsilon - \chi f }_{H^s(\mathbb{R})} = 0.
\]
It follows that $\psi_{0,\varepsilon} \in C_c^\infty(\mathbb{R}^d)$, $\psi_0^{\mathrm{bad}} \in X_0 \setminus L^2(\mathbb{R}^d)$, and
\[
  \lim_{\varepsilon \to 0} \norm{ \psi_{0,\varepsilon} - \psi_0^{\mathrm{bad}} }_{X_0} = 0,
\]
where as before $X_0$ denotes either $H^s(\mathbb{R}^d)$, $s < 0$, or $L^p(\mathbb{R}^d)$, $1 \le p < 2$.

Next, we choose the data $a_{\mu,\varepsilon}, b_{\mu,\varepsilon} \in C_c^\infty(\mathbb{R}^d)$. The first alternative is to take vanishing data
\begin{equation}\label{abZero}
  a_{0,\varepsilon} = \dots = a_{d,\varepsilon} = 0,
  \qquad
  b_{0,\varepsilon} = \dots = b_{d,\varepsilon} = 0,
\end{equation}
as in \eqref{ZeroData}. The second alternative is to ensure that the constraints in \eqref{ConstrainedData} are satisfied. For this, we take all the data to vanish except $b_{1,\varepsilon}$, so the constraints reduce to
\[
  - \partial_1 b_{1,\varepsilon} = \abs{\psi_{0,\varepsilon}}^2 = \frac{1}{\sqrt{\varepsilon^2+x_1^2}} \quad \text{in $B$}.
\]
Integrating this, we obtain
\begin{equation}\label{abConstrained}
\left\{
\begin{aligned}
  &a_{0,\varepsilon} = \dots = a_{d,\varepsilon} = 0,
  \qquad
  b_{0,\varepsilon} = b_{2,\varepsilon} = \dots = b_{d,\varepsilon} = 0,
  \\
  &b_{1,\varepsilon}(x) = - \chi(x_1) \cdots \chi(x_d) \log\left(x_1 + \sqrt{\varepsilon^2 + x_1^2} \right),
\end{aligned}
\right.
\end{equation}
which satisfies \eqref{ConstrainedData}.

\section{Proof of ill-posedness}\label{IPsection}

We start by proving Theorem \ref{MainThm2}, which implies Theorem \ref{MainThm1}. Theorem \ref{MainThm3} is proved at the end of this section.

Let $d \le 3$, choose the Dirac matrices as in section \ref{MatricesSection}, and define the data
$
  (\psi_{0,\varepsilon},a_{\mu,\varepsilon},b_{\mu,\varepsilon}) \in C_c^\infty(\mathbb{R}^d)
$
by \eqref{SpinorData}, \eqref{SpinorData2}, and either \eqref{abZero} or \eqref{abConstrained}. Since the data depend only on $x_1$ in $B$, it follows from Theorem \ref{WPThm} that their maximal $C^\infty$ forward evolution
$
  (\psi_\varepsilon,A_{\mu,\varepsilon})
$
exists throughout the cone $K$ over $B$, and depends only on $t$ and $x_1$ there. Indeed, we can apply Theorem \ref{WPThm} with the data restricted to $x_2=\dots=x_d=0$.

We now claim that for $T > 0$ sufficiently small, the following holds for $\varepsilon > 0$:
\begin{equation}\label{Claim1}
  \abs{A_{j,\varepsilon}(t,x)} \le 1 \quad \text{in $K_T$, for $2 \le j \le d$},
\end{equation}
and
\begin{equation}\label{Claim2}
  \abs{\psi_\varepsilon(t,x)}^2 \ge \frac12 \abs{f_\varepsilon(x_1-t)}^2 \quad \text{in $K_T \cap \{ (t,x) \colon 0 < t < x_1 \}$}.
\end{equation}
Moreover,
\begin{equation}\label{Claim3}
  A_{0,\varepsilon}(t,x) \ge c(Q) \abs{\log\varepsilon}
  \quad \text{in any compact $Q \subset K_T \cap \{ (t,x) \colon \abs{x_1} < t \}$},
\end{equation}
for all sufficiently small $\varepsilon > 0$, and some constant $c(Q) > 0$ depending only on $Q$.

Once we have obtained \eqref{Claim3}, then Theorem \ref{MainThm2} is proved. The plan is now as follows: First, we prove that \eqref{Claim2} implies \eqref{Claim3}, then we prove \eqref{Claim1}, and finally we prove \eqref{Claim2}.

Since \eqref{Claim1}--\eqref{Claim3} are restricted to the cone $K$, where the solution depends only on $t$ and $x_1$, it suffices to prove them for $x_2=\dots=x_d=0$. For the remainder of this section we therefore restrict to $x_2=\dots=x_d=0$. The solution then exists for all $t \ge 0$ and $x_1 \in \mathbb{R}$, by Theorem \ref{WPThm}. To simplify the notation we also write $x=x_1$.

\subsection{Proof that $\eqref{Claim2} \implies \eqref{Claim3}$}

Since $(\partial_t^2 - \partial_x^2) A_{0,\varepsilon} = \abs{\psi_\varepsilon}^2$ with vanishing initial data, we have by d'Alembert's formula
\[
  A_{0,\varepsilon}(t,x) = \frac12 \iint_{K^{(t,x)}} \abs{\psi_\varepsilon}^2 \, dy \, ds
  =
  \frac12 \int_0^t \int_{x-t+s}^{x+t-s} \abs{\psi_\varepsilon(s,y)}^2 \, dy \, ds,
\]
with notation as in \eqref{1dCone}.
Take $\abs{x} < t < T \ll 1$ and restrict the integration to the cone $K^{(t,x)} \cap \{ (s,y) \colon s < y \}$. Assuming \eqref{Claim2} holds, we thus obtain
\begin{align*}
  A_{0,\varepsilon}(t,x) &\ge \frac12 \int_0^{\frac{x+t}{2}} \int_{s}^{x+t-s} \abs{\psi_\varepsilon(s,y)}^2 \, dy \, ds
  \\
  &\ge
  \frac14 \int_0^{\frac{x+t}{2}} \int_{s}^{x+t-s} \frac{1}{\sqrt{\varepsilon^2+(y-s)^2}} \, dy \, ds
  \\
  &\ge
  \frac14 \int_0^{\frac{x+t}{2}} \int_{s}^{x+t-s} \frac{1}{\varepsilon+y-s} \, dy \, ds
  \\
  &= \frac{x+t}{8} (-\log\varepsilon)
  + \frac18 (\varepsilon+x+t) \left( \log(\varepsilon+x+t) - 1 \right)
  - \frac12 \varepsilon (\log\varepsilon-1),
\end{align*}
and \eqref{Claim3} follows.

\subsection{Proof of \eqref{Claim1}} This is only relevant in dimensions $d = 2,3$. We show the proof for $d=2$, and comment on $d=3$ at the end.

Assuming now $d=2$, then the system is as in \eqref{MD2d}:
\begin{equation}\label{MD2dEps}
\left\{
\begin{aligned}
  (\partial_t + \partial_x)u_\varepsilon &= i (A_{0,\varepsilon} + A_{1,\varepsilon}) u_\varepsilon + A_{2,\varepsilon} v_\varepsilon -iMv_\varepsilon,
  \\
  (\partial_t - \partial_x)v_\varepsilon &= i (A_{0,\varepsilon} - A_{1,\varepsilon}) v_\varepsilon - A_{2,\varepsilon} u_\varepsilon -iMu_\varepsilon,
  \\
  (\partial_t^2-\partial_x^2) A_{0,\varepsilon} &= \abs{u_\varepsilon}^2 + \abs{v_\varepsilon}^2,
  \\
  (\partial_t^2-\partial_x^2) A_{1,\varepsilon} &= - \abs{u_\varepsilon}^2 + \abs{v_\varepsilon}^2,
  \\
  (\partial_t^2-\partial_x^2) A_{2,\varepsilon} &= - 2 \im (u_\varepsilon \overline{v_\varepsilon}),
\end{aligned}
\right.
\end{equation}
with data
\[
  u_\varepsilon(0,x) = \chi f_\varepsilon(x) = \frac{\chi(x)}{(\varepsilon^2+x^2)^{1/4}},
  \\
  \qquad v_\varepsilon(0,x) = 0,
\]
and either \eqref{abZero} or \eqref{abConstrained} (with $x=x_1$ and $x_2=0$). The solution exists globally and is $C^\infty$, by Theorem \ref{WPThm}.

We want to prove \eqref{Claim1}. This will follow if we can prove that for $T > 0$ sufficiently small,
\begin{equation}\label{Claim1'}
  \sup_{(t,x) \in [0,T] \times \mathbb{R}} \abs{A_{2,\varepsilon}(t,x)} \le 1
\end{equation}
for all $\varepsilon > 0$.

By d'Alembert's formula, since $a_{2,\varepsilon} = b_{2,\varepsilon} = 0$,
\begin{equation}\label{A2}
  A_{2,\varepsilon}(t,x)  =
  \iint_{K^{(t,x)}} \im (u_\varepsilon \overline{v_\varepsilon})(s,y) \, dy \, ds,
\end{equation}
were $K^{(t,x)}$ denotes the backward cone \eqref{1dCone}.

The idea is now to apply Lemma \ref{NullLemma} to the first two equations in \eqref{MD2dEps}. But first we need to integrate out the terms involving $(A_{0,\varepsilon} \pm A_{1,\varepsilon})$. Define $\phi_{+,\varepsilon}$, $\phi_{-,\varepsilon}$ by
\[
\begin{alignedat}{2}
  (\partial_t + \partial_x) \phi_{+,\varepsilon} &= A_{0,\varepsilon}+A_{1,\varepsilon}, & \qquad \phi_{+,\varepsilon}(0,x) &= 0,
  \\
  (\partial_t - \partial_x) \phi_{-,\varepsilon} &= A_{0,\varepsilon}-A_{1,\varepsilon}, & \qquad \phi_{-,\varepsilon}(0,x) &= 0,
\end{alignedat}
\]
that is,
\begin{align*}
  \phi_{+,\varepsilon}(t,x) &= \int_0^t (A_{0,\varepsilon}+A_{1,\varepsilon})(s,x-t+s) \, ds,
  \\
  \phi_{-,\varepsilon}(t,x) &= \int_0^t (A_{0,\varepsilon}-A_{1,\varepsilon})(s,x+t-s) \, ds.
\end{align*}
Then from the first two equations in \eqref{MD2dEps} we get
\begin{equation}\label{uvMod}
\begin{aligned}
  (\partial_t + \partial_x)(e^{-i\phi_{+,\varepsilon}} u_\varepsilon) &= e^{-i\phi_{+,\varepsilon}}[ A_{2,\varepsilon} v_\varepsilon -iMv_\varepsilon],
  \\
  (\partial_t - \partial_x)(e^{-i\phi_{-,\varepsilon}} v_\varepsilon) &= e^{-i\phi_{-,\varepsilon}}[- A_{2,\varepsilon} u_\varepsilon -iMu_\varepsilon],
\end{aligned}
\end{equation}
so by \eqref{A2} and Lemma \ref{NullLemma},
\begin{equation}\label{A2Est0}
\begin{aligned}
  \norm{A_{2,\varepsilon}(t)}_{L^\infty}
  &\le
  \sup_{x \in \mathbb{R}} \iint_{K^{(t,x)}} \abs{u_\varepsilon} \abs{v_\varepsilon} \, dy \, ds
  \\
  &=
  \sup_{x \in \mathbb{R}} \iint_{K^{(t,x)}} \abs{e^{-i\phi_{+,\varepsilon}} u_\varepsilon} \abs{e^{-i\phi_{-,\varepsilon}} v_\varepsilon} \, dy \, ds
  \\
  &\le
  \left( \norm{\chi f_\varepsilon}_{L^1} + \int_0^t (M + \norm{A_{2,\varepsilon}(s)}_{L^\infty})\norm{v_\varepsilon(s)}_{L^1} \, ds \right)
  \\
  &\quad \times \left( \int_0^t (M + \norm{A_{2,\varepsilon}(s)}_{L^\infty})\norm{u_\varepsilon(s)}_{L^1} \, ds \right).
\end{aligned}
\end{equation}
To control the $L^1$ norms of $u_\varepsilon(t)$ and $v_\varepsilon(t)$, we use again \eqref{uvMod}, which implies
\begin{align*}
  (e^{-i\phi_{+,\varepsilon}} u_\varepsilon)(t,x) &= \chi f_\varepsilon(x-t) + \int_0^t \left(e^{-i\phi_{+,\varepsilon}}[ A_{2,\varepsilon} v_\varepsilon -iMv_\varepsilon]\right)(s,x-t+s) \, ds,
  \\
  (e^{-i\phi_{-,\varepsilon}} v_\varepsilon)(t,x) &= \int_0^t \left( e^{-i\phi_{-,\varepsilon}}[- A_{2,\varepsilon} u_\varepsilon -iMu_\varepsilon] \right)(s,x+t-s) \, ds.
\end{align*}
Take $L^1$ norms in $x$ to get
\begin{align*}
  \norm{u_\varepsilon(t)}_{L^1} &\le \norm{\chi f_\varepsilon}_{L^1} + \int_0^t (M + \norm{A_{2,\varepsilon}(s)}_{L^\infty})\norm{v_\varepsilon(s)}_{L^1} \, ds,
  \\
  \norm{v_\varepsilon(t)}_{L^1} &\le \int_0^t (M + \norm{A_{2,\varepsilon}(s)}_{L^\infty})\norm{u_\varepsilon(s)}_{L^1} \, ds.
\end{align*}
Adding these and applying Gr\"onwall's inequality yields
\begin{equation}\label{uvBound}
  \norm{u_\varepsilon(t)}_{L^1} + \norm{v_\varepsilon(t)}_{L^1} \le \norm{\chi f_\varepsilon}_{L^1} e^{\int_0^t (M + \norm{A_{2,\varepsilon}(s)}_{L^\infty}) \, ds}.
\end{equation}
Observing that
\[
  \norm{\chi f_\varepsilon}_{L^1} \le C := \int_{\mathbb{R}} \frac{\abs{\chi(x)}}{\abs{x}^{1/2}} \, dx < \infty,
\]
and defining the continuous function $g_\varepsilon \colon [0,\infty) \to [0,\infty)$ by
\[
  g_\varepsilon(t) = \sup_{0 \le s \le t} \norm{A_{2,\varepsilon}(s)}_{L^\infty},
\]
we conclude from \eqref{A2Est0} and \eqref{uvBound} that
\begin{equation}\label{gEst}
  g_\varepsilon(t) \le C^2\left(1 + t (M + g_\varepsilon(t)) e^{t (M + g_\varepsilon(t))}\right)
  \left(t (M + g_\varepsilon(t)) e^{t (M + g_\varepsilon(t))}\right).
\end{equation}

We now use a boot-strap argument to show that there exists a $\delta > 0$, depending only on $C$ and $M$, such that for $0 \le t \le \delta$,
\begin{equation}\label{gBound}
  g_\varepsilon(t) \le 1.
\end{equation}
Assuming this holds for some $t > 0$, then by \eqref{gEst} we have
\begin{equation}\label{gBound2}
  g_\varepsilon(t) \le C^2 \alpha(t),
\end{equation}
where the increasing function $\alpha \colon [0,\infty) \to [0,\infty)$ is defined by
\[
  \alpha(t) = \left(1 + t (M + 1) e^{t (M + 1)}\right)
  \left(t (M + 1) e^{t (M + 1)}\right).
\]
Since $\alpha(0) = 0$, there exists $\delta > 0$, depending only on $M$ and $C$, such that
\begin{equation}\label{deltaDef}
  C^2 \alpha(\delta) \le \frac12.
\end{equation}

By a continuity argument it now follows that \eqref{gBound} holds for all $t \in [0,\delta]$. Indeed, since $g_\varepsilon(0) = 0$, then \eqref{gBound} certainly holds for sufficiently small $t > 0$. And if \eqref{gBound} holds on some interval $[0,T] \subset [0,\delta]$, then by \eqref{gBound2} and \eqref{deltaDef} we have in fact $g_\varepsilon(t) \le 1/2$ on that interval, so \eqref{gBound} holds on a slightly larger interval.

This concludes the proof of \eqref{Claim1} for $d=2$. For $d=3$ the same proof goes through with some obvious changes. Indeed, the system \eqref{MD3d} has essentially the same structure as \eqref{MD2d}, and in particular the equations for $A_2, A_3$ have null forms in the right hand side. Thus, we obtain
\begin{equation}\label{Claim1'3d}
  \sup_{(t,x) \in [0,T] \times \mathbb{R}} \left( \abs{A_{2,\varepsilon}(t,x)} + \abs{A_{3,\varepsilon}(t,x)} \right) \le 1
\end{equation}
for $T > 0$ sufficiently small.

\subsection{Proof of \eqref{Claim2}}

Since we restrict to $x_2=\dots=x_d=0$ and write $x=x_1$, then \eqref{Claim2} reduces to proving that for $T > 0$ sufficiently small,
\begin{equation}\label{Claim2'}
  \abs{u_\varepsilon(t,x)}^2 \ge \frac12 \abs{f_\varepsilon(x-t)}^2
  \quad \text{for $0 < t < x < 1-t$ and $t < T$}.
\end{equation}
We do the proof for $d=2$, and comment on $d=1$ and $d=3$ at the end.

Assuming $d=2$, we use \eqref{MD2dSquared}. Thus,
\begin{align*}
  (\partial_t + \partial_x) \abs{u_\varepsilon}^2 &= F_\varepsilon := 2A_{2,\varepsilon} \re (u_\varepsilon \overline {v_\varepsilon}) - 2M \im (u_\varepsilon \overline {v_\varepsilon}),
  \\
  (\partial_t - \partial_x) \abs{v_\varepsilon}^2 &= G_\varepsilon:= - 2A_{2,\varepsilon} \re (u_\varepsilon \overline {v_\varepsilon}) + 2M \im (u_\varepsilon \overline {v_\varepsilon}),
\end{align*}
and therefore
\begin{align}
  \label{u}
  \abs{u_\varepsilon(t,x)}^2 &= \abs{\chi f_\varepsilon(x-t)}^2 + \int_0^t F_\varepsilon(s,x-t+s) \, ds,
  \\
  \label{v}
  \abs{v_\varepsilon(t,x)}^2 &= \int_0^t G_\varepsilon(s,x+t-s) \, ds.
\end{align}
By \eqref{Claim1'}, for $T > 0$ sufficiently small we have
\begin{equation}\label{FGbounds}
  \abs{F_\varepsilon},\abs{G_\varepsilon} \le (M+1)(\abs{u_\varepsilon}^2+\abs{v_\varepsilon}^2) \quad \text{in $[0,T] \times \mathbb{R}$},
\end{equation}
hence
\begin{align}
  \label{u2}
  \abs{u_\varepsilon(t,x)}^2 &\le \abs{\chi f_\varepsilon(x-t)}^2 + (M+1)\int_0^t (\abs{u_\varepsilon}^2+\abs{v_\varepsilon}^2)(s,x-t+s) \, ds,
  \\
  \label{v2}
  \abs{v_\varepsilon(t,x)}^2 &\le (M+1) \int_0^t (\abs{u_\varepsilon}^2+\abs{v_\varepsilon}^2)(s,x+t-s) \, ds
\end{align}
for $t \in [0,T]$, $x \in \mathbb{R}$.

The idea is now to apply a boot-strap argument. For $\rho \in (0,1-2T)$, define
\[
  B_{\rho,\varepsilon}(s) = \sup_{\rho + s \le y \le 1-s} \left( \abs{u_\varepsilon(s,y}^2+\abs{v_\varepsilon(s,y)}^2 \right)
  \qquad (0 \le s \le T).
\]
If $\rho + t \le x \le 1-t$,
the integrands in \eqref{u2}, \eqref{v2} are bounded by $B_{\rho,\varepsilon}(s)$, so
\[
  \abs{u_\varepsilon(t,x)}^2 + \abs{v_\varepsilon(t,x)}^2
  \le \frac{1}{\sqrt{\varepsilon^2 + (x-t)^2}} + 2(M+1)\int_0^t B_{\rho,\varepsilon}(s) \, ds
\]
Taking the supremum over $x \in [\rho+t,1-t]$ gives
\[
  B_{\rho,\varepsilon}(t)
  \le \frac{1}{\sqrt{\varepsilon^2 + \rho^2}} + 2(M+1)\int_0^t B_{\rho,\varepsilon}(s) \, ds.
\]
By Gr\"onwall's inequality we conclude that
\begin{equation}\label{Bbound}
  B_{\rho,\varepsilon}(t)
  \le \frac{1}{\sqrt{\varepsilon^2 + \rho^2}} e^{2(M+1)t} \le \frac{3}{\sqrt{\varepsilon^2 + \rho^2}} \quad \text{for $t \in [0,T]$},
\end{equation}
assuming $T > 0$ is so small that $2(M+1)T < 1$.

Combining \eqref{Bbound}, \eqref{u} and \eqref{FGbounds}, we obtain, for $\rho > 0$, $x \in [\rho+t,1-t]$ and $t \le T$,
\begin{align*}
  \abs{u_\varepsilon(t,x)}^2
  &\ge \abs{\chi f_\varepsilon(x-t)}^2 - (M+1)\int_0^t (\abs{u_\varepsilon}^2+\abs{v_\varepsilon}^2)(s,x-t+s) \, ds
  \\
  &\ge \frac{1}{\sqrt{\varepsilon^2 + (x-t)^2}} - (M+1) \int_0^t B_{\rho,\varepsilon}(s) \, ds
  \\
  &\ge \frac{1}{\sqrt{\varepsilon^2 + (x-t)^2}} -  \frac{3(M+1)t}{\sqrt{\varepsilon^2 + \rho^2}},
\end{align*}
where we also used the fact that $\chi=1$ on $[-1,1]$.
Choosing $\rho = x-t$ and assuming $T > 0$ so small that $6(M+1)T < 1$, we obtain the claimed inequality \eqref{Claim2'}.

This completes the proof of \eqref{Claim2} for $d=2$. The proof for $d=1,3$ works out the same way, but instead of \eqref{MD2dSquared} we use either \eqref{MD1dSquared} or \eqref{MD3dSquared}, and in the case $d=3$ we use \eqref{Claim1'3d} instead of \eqref{Claim1'}.

\subsection{Proof of Theorem \ref{MainThm3}}

Define $\psi_0^{\mathrm{bad}} \in X_0 \setminus L^2(\mathbb{R}^d)$ as in section \ref{DataSection}. Assume there exist (i) a neighbourhood $\Omega_0$ of $\psi_0^{\mathrm{bad}}$ in $X_0$, and (ii) continuous maps
\[
  I_\mu, J_\mu \colon \Omega_0 \longrightarrow \mathcal D'(B),
\]
such that if $\psi_0 \in \Omega_0 \cap C_c^\infty(\mathbb{R}^d)$, then defining
\[
  a_\mu = I_\mu[\psi_0], \qquad b_\mu = J_\mu[\psi_0] \qquad (\mu=0,\dots,d),
\]
the constraint equations \eqref{Constraints} are satisfied in $B$.

We will show that these assumptions lead to a contradiction. Define $\psi_{0,\varepsilon} \in C_c^\infty(\mathbb{R}^d)$ as in section \ref{DataSection}. Then $\psi_{0,\varepsilon} \to \psi_0^{\mathrm{bad}}$ in $X_0$ as $\varepsilon \to 0$, so $\psi_{0,\varepsilon}$ belongs to $\Omega_0$ for all $\varepsilon > 0$ small enough, and we may define
\[
  a_{\mu,\varepsilon} = I_\mu[\psi_{0,\varepsilon}], \qquad b_{\mu,\varepsilon} = J_\mu[\psi_{0,\varepsilon}] \qquad (\mu=0,\dots,d).
\]
By assumption, these fields satisfy the constraints \eqref{Constraints} in $B$, so in particular
\[
  \sum_{j=1}^d \partial_j (\partial_j a_{0,\varepsilon} - b_{j,\varepsilon}) = \abs{\psi_{0,\varepsilon}}^2 \quad \text{in $B$}.
\]
By the assumed continuity of the maps $I_\mu, J_\mu$, the left hand side must converge in $\mathcal D'(B)$ as $\varepsilon \to 0$. But the right hand side equals
\[
  \abs{\psi_{0,\varepsilon}(x)}^2 = \frac{1}{\sqrt{\varepsilon^2 + x_1^2}}
  \quad \text{for $x \in B$},
\]
and this function does not have a limit in the sense distributions in $B$, as $\varepsilon \to 0$.

\section{Proof of well-posedness}\label{WPproof}

In this section we prove Theorem \ref{WPThm}. To ease the notation we write $x=x_1$ throughout. To prove local existence we use an iteration and rely only on the following elementary estimates.

\subsection{Linear estimates}

Firstly, for the Dirac equation
\[
  (- i\gamma^0 \partial_t - i\gamma^1 \partial_x + M) \psi = F(t,x), \qquad \psi(0,x) = \psi_0(x),
\]
we shall use the energy inequality, for $t > 0$,
\begin{equation}\label{EnergyInequality}
  \norm{\psi(t)}_{L^2(\mathbb{R})} \le \norm{\psi_0}_{L^2(\mathbb{R})} + \int_0^t \norm{F(s)}_{L^2(\mathbb{R})} \, ds.
\end{equation}
This is proved as follows. By approximation, we may assume that $\psi_0$ and $F$ are smooth and compactly supported in $x$. Premultiplying the equation by $i\overline\psi=i\psi^*\gamma^0$ and taking real parts yields $\partial_t \rho + \partial_x j = \re(i\psi^*\gamma^0 F)$, where $\rho=\psi^*\psi$ and $j=\psi^*\gamma^0\gamma^1\psi$. Integration in $x$ gives
\[
  \frac{d}{dt} \int_{\mathbb{R}} \abs{\psi}^2 \, dx = 2\re \int_{\mathbb{R}} i\psi^* \gamma^0 F \, dx \le 2 \norm{\psi(t)}_{L^2} \norm{F(t)}_{L^2},
\]
which implies \eqref{EnergyInequality}.

Secondly, for the wave equation
\[
  \square u = G(t,x), \qquad u(0,x) = f(x), \qquad \partial_t u(0,x) = g(x),
\]
we shall use the estimates, for $t > 0$,
\begin{align}
  \label{WaveEstimateA}
  \norm{u(t)}_{L^\infty(\mathbb{R})} &\le \norm{f}_{L^\infty(\mathbb{R})} + \norm{g}_{L^1(\mathbb{R})}
  + \int_0^t \norm{G(s)}_{L^1(\mathbb{R})} \, ds,
  \\
  \label{WaveEstimateB}
  \norm{\partial_x u(t)}_{L^1(\mathbb{R})} &\le \norm{f'}_{L^1(\mathbb{R})} + \norm{g}_{L^1(\mathbb{R})}
  + \int_0^t \norm{G(s)}_{L^1(\mathbb{R})} \, ds,
  \\
  \label{WaveEstimateC}
  \norm{\partial_t u(t)}_{L^1(\mathbb{R})} &\le \norm{f'}_{L^1(\mathbb{R})} + \norm{g}_{L^1(\mathbb{R})}
  + \int_0^t \norm{G(s)}_{L^1(\mathbb{R})} \, ds,
\end{align}
which are immediate from D'Alembert's formula,
\[
  u(t,x) = \frac{f(x+t) + f(x-t)}{2} + \frac12 \int_{x-t}^{x+t} g(y) \, dy
  + \frac12 \int_0^t \int_{x-(t-s)}^{x+t-s} G(s,y) \, dy \, ds.
\]
Adding \eqref{WaveEstimateA}--\eqref{WaveEstimateC} gives
\begin{equation}\label{WaveEstimate}
  \norm{u(t)}_{AC} + \norm{\partial_t u(t)}_{L^1}
  \le 3 \left( \norm{f}_{AC(\mathbb{R})} + \norm{g}_{L^1} + \int_0^t \norm{G(s)}_{L^1} \, ds \right).
\end{equation}

\subsection{The local result}\label{Local}

With the above linear estimates, it is now an easy matter to prove the local well-posedness of \eqref{MD'} by iteration, for data with the regularity $\psi_0 \in L^2(\mathbb{R})$, $a_\mu \in AC(\mathbb{R})$ and $b_\mu \in L^1(\mathbb{R})$. Indeed, applying the energy inequality \eqref{EnergyInequality} to the Dirac equation in \eqref{MD'}, we use the trivial bilinear estimate
\[
  \int_0^T \norm{A_\mu \gamma^\mu \psi(s)}_{L^2} \, ds
  \le CT \norm{A}_{C([0,T];L^\infty)} \norm{\psi}_{C([0,T];L^2)},
\]
where $C([0,T];L^p)$ is equipped with the sup norm. Moreover, applying \eqref{WaveEstimate} to the wave equations in \eqref{MD'} we use the equally trivial bilinear bound
\begin{equation}\label{BilinearB}
  \int_0^T \norm{\psi^* \gamma^0 \gamma^\mu \psi(s)}_{L^1} \, ds
  \le CT \norm{\psi}_{C([0,T];L^2)}^2.
\end{equation}
By a standard contraction argument, which we do not repeat here, one now obtains local well-posedness with a time of existence $T > 0$ determined by
\[
  CT\left( \norm{\psi_0}_{L^2} + \sum_{\mu=0}^d \norm{a_\mu}_{AC} + \sum_{\mu=0}^d \norm{b_\mu}_{L^1} \right) \le 1,
\]
where $C$ is a universal constant. This proves Theorem \ref{WPThm} for such $T$. Next, we show that the results extend globally.

\subsection{The global result}\label{Global} To extend the local result globally in time, it suffices to obtain an a priori bound on the solution $(\psi,A,\partial_t A)(t)$ in $L^2(\mathbb{R}) \times AC(\mathbb{R}) \times L^1(\mathbb{R})$. For $\psi$, this bound is directly provided by the conservation of charge, \eqref{1dCharge}. The latter also provides the necessary bound for $(A,\partial_t A)$, via the linear estimate \eqref{WaveEstimate} and the bilinear estimate \eqref{BilinearB}. This concludes the proof of Theorem \ref{WPThm}.

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


text:
Peter in Current Affairs, Gender, Politics April 16, 2016
Wisdom and Justice are Not Always Popular and Easy
via Andrew Bolt:
Antigone's brave stance for Justice and Nature against unjust human laws is what electrified me. We in the Western world live in days of mad, accelerating, and bewildering change. A same-sex couple is the same as a man and a woman; male and female is an illusion; an unborn baby has a right to life only when her mother grants her that right; children may be taken from their natural parents for the gratification of adults; and anyone who opposes these schemes may, for their intolerance and bigotry, be ridiculed, sacked, fined, imprisoned, and, most sinister by far, re-educated.
At this time when the laws of nature and basic justice are conscientiously unravelled and suppressed, Antigone's defiance makes the heart swell. "I was not like, who feared no mortal's frown, To disobey these laws, and so provoke the wrath of Heaven."
As the storm-clouds of social change gather and loom, here is Sophocles' ancient and potent inspiration for all those who stand, whether in the Christian tradition or not, for Nature and Justice. Seeing that we must die, we had best do so, in this awesome battle, faithful and brave.
From a longer article by Campbell Markham at MercatorNet.
The Reverend Mr Markham refers to the bright and courageous Sophie Scholl. If you are not familiar with her story, this is a good place to start. With all respect to Anne Frank, Sophie Scholl is more inspirational by far:
https://www.youtube.com/watch?v=nXtC08tWxqA


PACE does small business budget consulting. Specifically, they help businesses level out their cash flow and find a steady pace. I was responsible for naming, branding, identity, collateral, and a responsive micro-site.


text:
Home » Uncategorized » United States Changing its Vote from Abstain to "No" on the Annual UN Resolution on the Golan Heights
United States Changing its Vote from Abstain to "No" on the Annual UN Resolution on the Golan Heights
November 18, 2018 November 16, 2018 by Coalition for the Israeli Golan
For the first time in history, the United States is changing its vote from 'Abstain' to 'No' on the annual UN resolution on the Golan Heights.
The results of the UN vote, 16.11.2018. In the red circle: The United States votes 'Against' the condemnation of Israeli control over the Golan Heights.
US Ambassador Haley said that in previous years, the United States has abstained from voting on this resolution. However, given the resolution's anti-Israel bias, as well as the militarization of the Syrian Golan border, and a worsening humanitarian crisis, this year the United States has decided to vote 'No' on the resolution.
The United States will no longer abstain when the United Nations engages in its useless annual vote on the Golan Heights. If this resolution ever made sense, it surely does not today. The resolution is plainly biased against Israel. Further, the atrocities the Syrian regime continues to commit prove its lack of fitness to govern anyone. The destructive influence of the Iranian regime inside Syria presents major threats to international security. ISIS and other terrorist groups remain in Syria. And this resolution does nothing to bring any parties closer to a peace agreement. The United States will vote 'No'.
The vote aroused many reactions. Israeli Prime Minister Benjamin Netanyahu referred to the change in the American position at the beginning of the cabinet meeting.
US Mission to the UN, Twitter account.
Israel's ambassador to the United Nations, Danny Danon, tanked Ambassador Haley in his Twitter account:
Israel's Ambassador to the United Nations, Twitter account.
Israel's Minister of Homeland Security, Gilad Erdan, also referred to the decision in his Twitter account:
See the Press Release >>
Memorandum to Congress


Via WGN-TV, the fourth-largest city in the U.S. has received more rain in the last week than Chicago receives in an average year.
Chicago's average annual precipitation is 910 mm. Since last Friday, Houston has gotten 1,070 mm. The wettest year in Houston history (1900) dumped 1,851 mm on it. So far this year, with 4 months left to go, Houston has gotten 1,798 mm. Of course, the odds are pretty good that the city will get another 53 mm of rain before December 31st.
We have no idea how bad the damage is yet. The entire Houston Chronicle website is about the flood. At least the rain has stopped for now—but officials worry about additional reservoir overflows and levee breaks.
We're just beginning to understand the magnitude of this disaster. And with key Federal posts, including FEMA Director, yet to be filled, President Trump is so out of his depth one can only hope that state and local governments can help.
Shameless plug: Inner Drive Technology can help!
The U.S. Army Corps of Engineers confirmed Tuesday morning that water was spilling from around the dam gates of the Addicks Reservoir, which has been overwhelmed by extreme rainfall from Hurricane Harvey. Officials said they expect the Barker Reservoir, to the south of Addicks, to begin overflowing similarly at some point Tuesday.
The reservoirs, which flank Interstate 10 on the west side of Houston, flow into the Buffalo Bayou and are surrounded by parks and residential areas. Water levels in the two reservoirs had already reached record levels Monday evening, measuring 32 m at Addicks and 30 m at Barker.
In response, City Lab asks, why can't the U.S. manage flooding?
For New Orleans, whose below-sea-level position makes it particularly imperiled, the August floods were a reminder of something we should take much more seriously than we have. We ought to apply more aggressively the lessons we claimed to be learning from the Dutch after Katrina. It's a course of action that would amount to a sea change in how we approach the wet threat that surrounds us on every side.
We need to get as smart and wily about water as Rotterdam. New Orleans's continued viability as a population center and commercial hub depends on it. We must learn to live with water, to absorb rainfall and storm surge in massive retention facilities, to designate greenspaces that double as parks. We need to stop paving our yardsto make nifty little pads for the family car. We need to build absorbent rooftop gardens on as many buildings as can be put to that purpose.
Speaking of, I'm planning to visit New Orleans this weekend. Harvey is expected to pass northwest of the city tomorrow and land in the Tennessee Valley by Friday afternoon. I'm still bringing water shoes and an umbrella.


Our client had been pulled over for driving without headlights, which led to a DUI investigation. When the officer approached the vehicle he had observed a strong odor of alcohol, however, his speech was not slurred. Our client participated in the field sobriety tests and performed poorly. He blew a .17 on the breath test. Client was arrested and charged with DUI. He was later arrested on an outstanding warrant for failure to appear in court. He contacted our office to resolve the case as it was causing issues with his Security Clearance with the United States Military.
RESULTS: During negotiations the State offered the minimum sanctions, but were convinced that they could apply the sanctions from the current date, instead of the earlier date, where the fines were lower and did not require probation. The attorney in this case informed the Court of his intent to file a motion to stay, as the client was stationed overseas. Instead of staying procedures, the court decided that DUI sanctions were close to impossible to complete while stationed abroad during war time. The State accepted these terms. NO DUI CONVICTION!


Registration is now open for Anchor Cross Cancer Foundation's 2nd annual Bass Fishing Tournament scheduled for Saturday, Oct. 15, 2016, at Live Oak Landing on the Tensaw River in Stockton, Alabama.
Tournament entry fee is $80 per boat and boats will be launched in the order they are registered. Cash prizes will be awarded for the top five places, with 1st place taking home $2,000. Participants will also enjoy free food, door prizes and multiple raffle drawings. All proceeds will benefit the Anchor Cross Cancer Foundation to help cancer patients in need along the Gulf Coast.
Registration forms may be found online at www.anchorcrossfoundation.org, or may be picked up at any Southern Cancer Center location in Mobile or Baldwin County. This is a catch and release tournament requiring boat inspections, life jackets and live wells, as well as polygraph testing for the winners.
ACCF's inaugural tournament last year hosted 150 fishermen and raised $8,000, funds used to provide local cancer patients transportation to and from appointments and provide financial assistance for costly medications and COBRA benefits.
The Anchor Cross Cancer Foundation, 501c3, is a community-based foundation dedicated to patient assistance, community education and advancing local cancer research while improving the health and wellness of the counties in the Gulf Coast region. All funds raised by the Anchor Cross Cancer Foundation stay in our local communities.
For tournament information, call 251.607.5081, email info@anchorcrossfoundation.org or visit www.anchorcrossfoundation.org.