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DISDads Hawaiian Shirt Pre-Orders Are Here!
by AJRitz | Jun 16, 2018 | DISDads Podcast, Disney Fan Community, Off-Topic, Site News
By popular demand, DISDads Hawaiian shirts are here. I LOVE the design, created by the supremely talented Kevin Quigley, co-host of The Thirty20Eight Podcast. Production of these shirts takes a bit more than a t-shirt. This design isn't printed on top of the shirt. Rather, the design is essentially "baked-in" to the 100% nylon fabric through a process called dye sublimation. Then the fabric is custom-cut and sewn to create the shirts. The result is a soft, slightly stretchy shirt in strong, vibrant color that doesn't crack, peel, or fade.
In order to raise the funds to kick start production of these shirts, the first 10 shirts are being sold as part of premium packages at a higher cost. Purchasers of one of the 5 Maui Packages will receive a DISDads coffee mug and a DISDads tumbler, in addition to their shirt. All three items together are a $121 retail value, being sold for a base price of $100 (plus any up charge for shirts larger than XL). The five Tamatoa Packages each include both a shirt and a tumbler, and all 10 of the premium packages will ship out two weeks before shirt-only purchases.
As soon as the 10 special package have been purchased, we'll set a deadline for shirt-only purchases. Because of the unit cost of these shirts, we can't really afford to carry inventory. So we can't send the shirts to production until the orders are in. Once we send to production, they get turned around pretty quickly. Packages will be sent first to Maui Package and Tamatoa Package purchasers, and then the rest of the shirt orders will be fulfilled approximately two weeks later. The sooner you all get your orders in, the sooner the shirts will go to production.
Bourboncast – Episode 532
by AJRitz | Oct 15, 2017 | Off-Topic, Podcast Episode, Pop Culture Reviews, WDW Dining
Host: Aaron Rittmaster
Panel: Chad King, David Juart, Joshua Shusterman
Sure, we ran a little long. But what did you expect? In Episode 532, the DISDads Podcast panel continues our irregularly scheduled alcoholic beverage series with BOURBONCAST. The panel discusses (and samples) a broad selection of bourbons and talks about some of the best places at Walt Disney World to find a bourbon-based cocktail. Near the end, David Juart gives us the recipe for the perfect bourbon old fashioned.
Are you enjoying our side trips to the bar? What beverage would you like us to report on next? Tell us about it!
Leave a comment here on the blog; Email us at podcast@disdads.com; tweet us at DisDadsPodcast on Twitter; and check out our DISDads Podcast Facebook page.
If you enjoyed the podcast, we'd also really appreciate it if you took a moment to leave us a review on iTunes (If you didn't enjoy the show, feel free to ignore that last link).
Our outro music is "Whiskey on the Mississippi" © Kevin MacLeod (incompetech.com)
The DISDads Podcast is sponsored by Fantastic Memories Travel, LLC, and its Mouse Master Travel service, a specialized Disney Destination vacation planning service.
http://media.blubrry.com/disdadspodcast/p/podcast.disdads.com/DisDadsPodcast532.mp3
Another kindred spirit died last night
by AJRitz | Dec 25, 2016 | Being a DIS Dad, DISDads Podcast, Disney Fan Community, Off-Topic
Last night, our extended community of Disney-fan dads lost another champion. Alan Mize was never an active member of the DISDads community, but he shared our interests and our spirit. I was lucky enough to get to know him just a little bit, through interactions on Twitter (where he went by the handle @NORMNB8S) and his role as a co-host of the Enchanted Tiki Talk Podcast (also on Facebook). I even had the privilege to interview him, along with his co-hosts, for a DAD Show in the early days of their podcast. Alan, along with Keith and Sean, built a wonderful community around their show. Now, it's time for the DISDads community to step up and help support the community that Alan leaves behind. A young man in his 30s, with a three-month old child, Alan suffered a massive heart attack at home and later died at the hospital on Christmas Eve.
Friends of Alan have set up a GoFundMe campaign, to help his family defray the costs of his sudden illness and death. Any remaining money raised will be used to help pay for his young daughter's college education. I encourage you, if you are able, to support this effort.
Rest in Peace Alan. My Festival of Lights will be dimmer for your light having left this mortal coil. Our thoughts and prayers – and hopefully some financial support – are with Alan's family. Just a few days ago, as he was reflecting on the coming holiday, Alan posted a series of tweets that make a beautiful, fitting epitaph, though he had no idea that's what he was composing at the time. We should all strive for this level of self-awareness and gratitude.
[View the story "Alan's Farewell" on Storify]
The DISDads Holiday Gift Guide Part 3: CyberWeek Tech Stuff
by AJRitz | Dec 1, 2015 | Off-Topic, Pop Culture Reviews
Part 3 of the DISDads Holiday Gift Guide focuses on gifts of a technological sort, most of which are part of Amazon's Cyber Monday Deals Week. Once again, the links in this article are affiliate links. Our recommendations unaffected by where you choose to make your purchase, we stand by these recommendations, even if you buy elsewhere or choose not to use our affiliate links.
1. GoPro Hero+ Starter Bundle ($179.78) – Sure, the Hero+ lacks some of the most advanced features of the latest and greatest GoPro models. But this version is more than sufficient to capture cool action video of your family's theme park adventures, and this Starter Bundle is less than half the price of the more heavily promoted (and excellent) GoPro HERO4 Silver Holiday Bundle ). The Hero+ is a bit less powerful than the Hero Silver, but the Hero+ can still capture full HD video (1080p at 60fps) and 5 burst mode photo frames per section at 8MP each. The biggest elements missing from the Hero+ as compared to the Silver are the lack of a built-in touch display and lack of support for GoPro's proprietary Protune production tool and time-lapse video. The Hero+ bundle comes with a 32 GB SDHC card, a head strap GoPro camera mount, and a floating hand grip (great for taking along on cruises and to waterparks and pools), in addition to the GoPro Hero+ camera.
2. I've written and talked about Anker external batteries for years now, and this model is the one I upgraded to just this past year. The Anker Astro E7 Ultra-High Capacity at a discounted price of $59.99 is the best value in an external battery that I've seen this holiday season. This is my third Anker external battery product, and I've had positive experiences with every one of them that I've used – I just keep upgrading to higher capacities to accommodate my family's growing appetite for power on the go. An external battery makes using the My Disney Experience app in the theme parks much less stressful (quick Jimmy, just switch that Fastpass to Ellen's Energy Adventure, my phone battery is down to 15%)! I can personally attest to Anker's claim that this model will recharge an iPhone 6 ten times or an iPad Air twice. And I've found that it does a great job holding its own charge between refills. The E7 is a bit larger and heavier than the external batteries I've used before. If you want to keep the size and weight down, consider the smaller and lighter Anker Astro E4 13000mAh Classic Portable Charger for $29.99 and which can still provide a full iPad Air recharge or five iPhone 6 recharges.
3. Blue Yeti USB Microphone – Silver ($90.99) – I've owned and used this microphone for 3 1/2 years now, and it's still probably the best podcasting technology investment I've made. If you are a regular podcast participant; would like to participate but feel like you need a decent microphone first; or you use your computer for other communications reasons (like recording singing children or Skype conversations with the grandparents), this microphone is a great choice. It offers multiple modes, allowing you to isolate just a single speaker, optimize recording sources on opposite side of the microphone, or capture it all equally. The build quality is excellent – the biggest problem with it is that it's remarkably heavy, especially when attached to the included base. And though I never thought I would make use of this feature, the 3.5mm no-latency headphone jack has proven to be amazingly convenient. It's MUCH less cumbersome than managing two wires coming out of the computer during a recording session, and gives you a lot better range of motion while connected. I actually use a pretty cheap pair of earbuds, with only one earbud in my ear, while recording, and combined with the Blue Yeti microphone, I don't have any issues with speaker sound "leaking" into my recording source.
4. TiVo Roamio HD DVR with Product Lifetime Service ($299.99 + tax). First things first, ignore the price that automatically generates with the link. It's only showing that price because Amazon itself is currently out of stock, and that's the best price offered by a partner merchant. You can place a back order at Amazon's price – even if you are in an area that has to pay sales tax on Amazon purchases, Amazon's price is nearly $100 cheaper. Some of you know that I "cut the cord" with pay TV almost two years ago now, and this is the device that allowed me to do it. Our sole source of broadcast television is OTA. The Roamio can record four OTA streams at the same time (technically, it can record five streams at once, but I use one port to feed a TiVo Mini in another room). In addition to being able to watch and record over-the-air network broadcasts, I can use the Roamio (and the Mini, for that matter) to access my Netflix account, Hulu+ Account, and Amazon Prime account. If you have a Vudu streaming video account, that can be linked to Disney Movies Anywhere and gives you access via the TiVo Roamio to all of your Disney Movies Anywhere content (it may be possible to access your DMA content via the included YouTube account as well, but I haven't tried it, so I can't guarantee that will work). But to me, the "killer app" feature for me as a cord-cutter has been the relatively-recent rollout of OnePass. OnePass allows me to do a single search for a show in the TiVo interface, and get access to the content I'm searching for, regardless of source. As an example, one of my favorite shows is Agents of Shield. An episode got pre-empted by weather reports. Thanks to OnePass, I could go to my show list and watch the same episode that was available on Hulu+ (the only difference was that I couldn't skip through Hulu's commercials like I could the broadcast commercials). It's possible that, once Apple rolls out it's long-awaited ala carte television subscription service that the TiVo Roamio will be eclipsed and a cord-cutters magic bullet. But for now, in my opinion, Roamio is even better than it's newer sibling the TiVo Bolt, because the Bolt maxxes out at four tuners, and would still lose one of those tuners to a Mini in another room. The only thing missing in all of this is a source for live televised sports. I don't have a great answer for that just yet, other than it being a good excuse to get out of the house and enjoy your local sports bar.
5. Disney Infinity 3.0: Rise Against the Empire Bundle – Amazon Exclusive ($78.71) – Only available through Amazon.com (at least for now), get five "classic" Star Wars characters (Chewbacca, Han Solo, Yoda, Obi-Wan Kenobi, and Darth Vader) AND the Rise Against the Empire playset, which includes Luke, Leia, and the Death Star-shaped playset piece. Note: these elements are only playable in the 3.0 version of Disney Infinity, and this bundle does NOT include the 3.0 game disk.
The DISDads Holiday Gift Guide Part 2: Gifts for the Home
by AJRitz | Nov 28, 2015 | Off-Topic, Pop Culture Reviews
Welcome to Part 2 of the DISDads Holiday Gift Guide, in which we tackle gifts for around the house. Once again, the links in this article are affiliate links or links to DISDads.com sponsors. Our recommendations are the same, even if you choose to make your purchase from another vendor or without using our links.
1. Capresso 465 CoffeeTeam TS ($209.95) – The version in the photo is the 464 GS ($159.95). The only real difference between the models is that the GS has a glass carafe and the TS has a stainless steel carafe. This one is a personal recommendation – the TS with the stainless steel carafe is worth the $50. I've gone through almost every major grind-and-brew coffee maker on the market over the last couple of decades, and this one is the best (by a fairly wide margin). It offers two major technical advantages over its competitors. First (and perhaps most importantly) it uses a burr grinder. Burr grinding means that the fresh coffee beans are ground more consistently and without being damaged by the friction heat generated by spinning-blade grinding. The advantage of the steel carafe is that the carafe does a good job of holding the coffee at temperature, without adding additional heat via a burner underneath the pot. That burner is the enemy of good-tasting coffee – it causes the first coffee through the brew basket to scald and can give the whole pot a burned taste. If you're not drip-brewing coffee into a steel carafe, you've likely gotten used to the slightly burned flavor such that you don't even realize its there — until the first time you brew a pot of coffee with good, fresh, beans that ISN'T burnt!
2. Disney 4 Mickey Waffle on a stick mini-waffle maker ($29.99). Sure, we've all seen other Mickey Waffle-makers, some of which were probably even less expensive. So why recommend this one? Because everything's better on a stick!
If being able to make Mickey Waffles at home is good, being able to make Mickey Waffles-on-a-stick at home is GREAT! They're not just cute to look at. The bargain waffle-makers make waffles that are too thin – they're all crust, and the ears in particular are prone to burning. This waffle-maker makes nice, thick, Belgian-style waffles with deep grooves (the better to hold syrup). And the non-stick plates generate sufficient heat to get a good, crisp, outer crust.(And it comes with 50 ready-to-use waffle sticks, just to get you started). It makes four mini-waffles at a time.
3. Masterbuilt 2-Door Propane Smoker ($194.92) – From the kitchen to the deck out back. This is a great option for someone interested in getting into smoking meat, without the challenges of managing a wood fire. The smoke is generated by wood chips (and there's no soaking necessary). Propane is generally cheaper than electricity as an alternative fuel to wood. And perhaps more importantly, propane smokers have a much better track record than electric for long term reliability. (Electric smokers tend to run into issues with their burner elements, especially when they're stored outdoors). Multiple racks, with easy and direct access to all of them, give you much more flexibility than bullet-type smokers. The 30-inch model in the photo and it's larger 44-inch cousin both offer a second, bottom, door that gives you access to the wood chip tray and water pan. This allows you to make adjustments to the smoke source without unduly affecting the temperature of the meat. For some reason, the middle-sized 40-inch unit lacks the separate door (so any time you need to make an adjustment to the wood chips or the water pan, you have to open the door for the entire smoker, allowing precious heat to escape).
4. Samsung UN55JS7000 55-Inch 4K Ultra HD Smart LED TV (2015 Model) ($997.99) – Let me start by saying that this is definitely a more expensive item than I was planning to put on this lis when I first started compiling it. But the deal on this set is simply too good not to mention. There are some even less expensive 4K TV deals this holiday season, and come of them are decent sets. The hard part is figuring out which ones and for how long. This is the first name-brand TV in this size class and resolution category that I've seen for under $1,000. List on this TV is nearly $2,000, so your not likely to find it any cheaper. If space is at a premium, Amazon has the 50-inch version of the pictured TV for $200 less. And if you've got room in both the location you want to use it and the budget, the 60-inch version is just $300 more. Oh – and this is a good example of why you need to be careful reading online reviews. There are some highly critical reviews listed with this TV on Amazon – but if you read carefully you'll see that they're almost all referring to much earlier TVs, from when 4K technology was in its infancy. They're really criticisms of the initial implementations of 4K TV technology than they are of the particular TV, and they're not reviewing a TV with the same technological specs as the ones currently being sold.
5. We'll wrap up this list by heading back to the kitchen for the VonShef Professional Fully Automatic Ice Cream Maker ($159.99). This is a great price on a true self-contained, automatic ice cream maker. It's more expensive than the Cuisinarts because it's got it's own built-in compressor. That means there's no need to pre-freeze the container for 12 hours or more, and no need to purchase a second container in order to make more than 1.5 quarts of ice cream in one night. Order some Dole Whip mix and you can feed your WDW addiction at will (and even make your own adult Dole Whip floats, with coconut rum and vanilla vodka).
a. None of these products were provided to the writer or DISDads.com for review purposes.
b. The opinions expressed about these gift options are solely our own and we have not been compensated to provide these opinions.
The DISDads Holiday Shopping Guide – Part 1: Gifts for Kids
As the holiday shopping season begins, here's our guide to great gifts for kids this year. In the interest of full disclosure, all of the links in this article are affiliate links or links to DISDads.com sponsors. Our recommendations are the same, even if you choose to make your purchase from another vendor or without using our links.
1. It probably comes as no surprise that the first recommendation from a website that started as a gathering place for fans of a travel destination is to book a vacation. But there's solid science behind our recommendation that the best gift is the gift of family vacation memories. More than a decade of research by Cornell University psychologist Dr. Thomas Gilovich suggests that, dollar-for-dollar, we derive more happiness from experiences (and particularly shared experiences) than we do from material things. Consider contacting site and podcast sponsor Fantastic Memories Travel / Mouse Master Travel to book your next family vacation as a holiday gift that keeps on giving, as the family participates in the planning process (and check our their "Choose your Disney Destination Bonus Offer for bookings made by the end of the year).
2. Bounce-Off Party! ($99.99) is a great (but more expensive) update to Bounce-Off ($17.25). The game is sort of halfway between beer pong (no beer is involved) and Connect Four. A challenge card is chosen, which sets the pattern required to win that round of the game. Players bounce balls onto the game grid, attempting to match the pattern for the game. The different patterns on the challenge cards ensure that each game is just a bit different, requiring a different ball arrangement to win the game. The Party! version expands the original, opening up play for more players. With both head-to-head and relay options, the six included game grids mean up to six players OR six TEAMS of players can play the Party version. (For Ages 7 and up. 2-4 players for the original game. The Party version can accommodate up to 6 – or more playing in teams).
3. I know that there's a lot of Lego love out there, but for my money there's no better building toy than Playmags ($89.99 for the linked 100-piece set). These are actually the slighly-less-expensive version of this sort of toy, but still well-made and playable. Geometric-shaped plastic tiles have magnets along the edges of the tiles. Careful placement of the magnets allows any two tiles to attract to one another along the edge. Resulting three dimensional structures are remarkably sturdy and stable. The originals from Magna-Tile are a bit stronger, so if your kids are hard on toys, they may be worth the price premium to you ($74.50 for the linked 48-piece set). These magnetic building toys are appropriate for ages 3 and up (and I've watched teens have fun building with them too).
4. Yes, I prefer iPads. Yes, I know that this is an older model. But the $34.99 "Black Friday Weekend" pricing for a 7″ Amazon Fire WiFi tablet that can take a microSD card for storage expansion is pretty darned hard to beat as a child's tech gift.
Its specs aren't top-of-the-line, but an IPS screen puts its image quality at the top of the line for a tablet in this price range. Frankly, it's biggest hardware weakness is the mono speaker and lack of Dolby Audio. Out of the box, the 7″ Fire tablet only connects to Amazon's app store. If that's not enough for you, relatively straightforward instructions for just adding the Google Play store or fully rooting the device are pretty readily available.
5. Thames & Kosmos Electricity and Magnetism ($40.99). Though this kit carries an ages 8-15 recommendation, it's probably more accurately good for ages 6-11. Those on the younger end can learn from and enjoy the experience of working with an adult on the 62 different experiments that are available. Those at the older end of the spectrum can safely explore the experiments on their own, without being bored by them. The well-organized kit divides the experiments into Electricity, Magetism, and Electromagnetism sections and includes 60 pieces to complete the various experiments. It should be noted that two rather important pieces are NOT included – those would be the two AA batteries that are necessary for any of the electricity-related experiments.
What are some of your favorite children's gifts that you're recommending this holiday season? Tell us about them in the comments!
a. None of the Amazon affiliate linked products were provided to the writer or DISDads.com for review purposes.
c. Mouse Master Travel is a site sponsor and podcast sponsor, but travel is a great gift regardless of which travel agency you choose or whether you choose to work with a travel agency at all.


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We are now going to look at the Match Processor in more detail.
Click "Matching" and then "Match Processor" in the applicant's action menu. An empty grey grid appears. Please don't think that you have done anything wrong, or that you haven't matched anything. An empty grid simply indicates that you haven't created a new match yet.
If your applicant just wants one property, go to the match processor. Right click and select 'Add Property'. In order to send the match out right click and select 'Send E-mail'.
Go to match processor from your applicant. Right click and select the 'New Match' option.
If you include properties already sent you may end up with a duplication, although it may be beneficial to send applicants a reminder of any previously matched properties. Using the 'Clear Filters' option ensures any filters that have been previously applied to your match grid are removed, ensuring no results are hidden.
A prompt will display stating how many suitable properties this applicant has been matched to.
At this point it is good practice to choose and order your columns in the grid so that the Property and Applicant column are together. Run down the property column and read the advert. If not suitable, remove the include/exclude tick on the left. If you have decided to exclude any properties, right click and choose 'Reject excluded' and all the matches that you un-ticked will disappear.
If you need to track through to the ext page of matches you will find the page markers at the bottom of the screen.
Not Matching? - This option opens the Match Debugger to determine why an applicant may not be matching to a property.
In this example we are going to send a match e-mail.
Enter a match email subject.
Change the text of the e-mail if required.
Tick/untick the boxes depending on how you follow up the match afterwards. If you don't mark the matches as complete they will then be placed in "Outstanding Matches" under the Applicant drop down list in the Top Menu for you to chase up at a later stage.
As soon as you click the save option, the match e-mail will send. If the applicant has an e-mail address set and has "Send E-mail" ticked, the e-mail will get sent straight away.


China Huaneng Group, one of the largest electric power groups in China, has announced that it will transfer its Shantou power plant's coal shipping operations from Shantou Port, a major seaport in Guangdong, to a new port developed by the group itself. Huaneng Shantou Power Plant is the largest customer of Shantou Port Group's Mashan Port and the plant's coal transport volumes account for nearly 70% of the port's business. When contacted by Splash, an official at Shantou Port Group said the company is still working on strategy to deal with the situation at Mashan Port, and refused to offer more information. Local media reports suggest that it is very likely Shantou Port Gorup would close operations at Mashan Port. According to Huaneng Group, it has developed a new terminal at the nearby Haimen Power Plant, and it will use two 5,000dwt bulk carriers to transport coal between the two plants.
Nineteen people were rescued and three people remained missing following a shipwreck off the coast of south China's Guangdong Province Wednesday night, local marine authorities said. The municipal marine rescue center of Shantou received a report at 9:27 p.m. that a cargo ship capsized in the South China Sea near the Nan'ao Cross-sea Bridge, which is currently under construction, throwing its eight crew members into the water. A container vessel then hit the wrecked ship and lost power, stranding 14 people on board and endangering the nearby bridge construction, the center said. As of noon on Thursday, 19 people had been rescued from the water and from the stranded vessel, while rescuers continued to search for the three missing people.


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VOX ATL Commentary: Gentrification In Atlanta's West End — Bringing Positive Changes
Voxatl.com • Jun 21, 2018
The Atlanta BeltLine, a 22-mile trail designed to wrap around the metro Atlanta area, has brought attention to the West End area, specifically the neighborhoods of Adair Park, Pittsburgh and Westview.
Credit Screenshot from VOX ATL video
Add to My List In My List
By VOX Media Cafe reporters Aryanna Brown, Caleb Cherry, Jordan Owens
In recent years, Atlanta has been facing an increase in development across the city.
One major project is the Atlanta BeltLine, which is a 22-mile trail that is designed to wrap around the metro Atlanta area. This trail has brought attention to the West End area, specifically the neighborhoods of Adair Park, Pittsburgh and Westview.
VOX Media Café reporters looked at how changes in the West End are affecting residents and the community. (Courtesy of VOX ATL)
With the rise of construction, it begs the question of the effects this gentrification is having on the community.
According to the U.S. Census Bureau, from 1990 to 2016, the African-American population in the Atlanta area has decreased 13 percent, while the white population has increased 7 percent.
With the demographics in the area changing, neighborhoods are becoming more diverse. Some people see this as problematic, but contrary to popular belief, gentrification can have positive effects in communities.
We asked people in the West End area how these developments have impacted them and their community.
MAP: West End in 2007
Dedra Ridges, who says she's been living in the West End area since the age of 1, spoke with VOX Media Cafe reporters concerning the changes in the area.
She says it has had a positive impact.
"We have our own local farms, more grocery stores, less liquor stores," said Ridges. "We actually have more trees, see more green. We have more trails; it's been amazing so far."
The changes in the community have surprisingly been positive, as previous projects such as these haven't always had the best impact on a community.
This doesn't seem to be the case for the entire West End community, though.
Organizations such as the TransFormation Alliance help alleviate the potential problems gentrification can hold. This organization helps to create more equitable housing and transportation for residents in the Atlanta area.
Michael A. Bryan, a program intern with the TransFormation Alliance, told VOX, "Gentrification isn't necessarily a bad thing. If it's not controlled and maintained, it can be detrimental to communities."
With these positive changes, there are still the many negatives to gentrification, one of them being housing prices.
Ridges stated, "Housing prices have increased drastically, and it's making it harder for our seniors to afford housing."
Housing is a main concern for these new developments, but this is being helped by the involvement of the community.
"People really want to be involved in these processes," Bryan stated.
This seems to be what's making the West End community stick together, everyone being involved in what's occurring in their community.
Ridges added, "The engagement in the community has overall improved."
Ridges talked about the Facebook group chat some West End residents use to communicate with each other. In the Facebook group, they discuss the changes occurring and the building sites under construction.
The only negative about this form of communication is: "It's leaving out a lot of seniors, because a lot of them over here who are homeowners are seniors who've been here for 30 or more years," Ridges explained.
The community has tried distributing flyers to these homeowners to make them aware of changes that are occurring. But Ridges said the best way to get the news across is to talk to people face to face.
Some people who have lived in the area for a while aren't as receptive to the changes.
"Everything is happening so fast for her," Ridges says in regard to her mother, who is a senior homeowner in the area and has lived there for more than 50 years.
This is bittersweet for some neighbors.
As the community improves, it loses some of the sense of history that was once held. With this loss, there's also the gain of a greater sense of community, as residents must band together to represent their community.
Watch the video below to hear from West End community members.
This package was written and produced by VOX Media Café 2018 participants Caleb Cherry, Aryanna Brown and Jordan Owens.
This story was published at VOXAtl.com, Atlanta's home for uncensored teen publishing and self-expression. For more about the nonprofit VOX, visit www.voxatl.org.
Speakers Honoring Martin Luther King Jr. At Capitol Tribute Focus On Past And Present
Closer Look: Fulton County Chair Talks Transfer Of Arts Centers to South Fulton; Airbnb Shares Strategy For Working With Local Governments; Gwinnett Election Leaders Talk Preparations for 2020
Singer Deb Bowman Makes It To Atlanta To 'Make It': An Audio Postcard

redpajama_set_name: RedPajamaCommonCrawl


text:
Comscore Launches Comscore Everywhere
Wells Fargo Resumes Gray TV Coverage
Wheeler: Stay of Title II Rules Is High Hurdle
By John Eggerton published 27 February 15
FCC Chairman Tom Wheeler says he thinks the Title II order should stand up in court, and that he doesn't see a court staying the rules in advance of ruling on the underlying court challenges that have already been threatened by ISPs unhappy with being classified as common carriers.
In a press conference following the Feb. 26 vote, Wheeler said he thought it would be hard for carriers to get a stay to "put off" the implementation of the rules.
He pointed to the orders three, bright-line rules--no blocking, throttling, or prioritized fast lanes. "They all said 'oh, we never intend to do that,' so, they are going to go into court and say, 'no, court, you need to stay this because we intend to block, we intend to throttle, we intend to have fast lanes.' So, I think a stay is a high hurdle.
One of the key tests for a stay is to prevent immediate harms from implementation of a rule, and Wheeler is suggesting that since ISPs have said they are not going to do any of things, a rule preventing them poses no immediate harm.
As to whether Title II would hold up in an underlying challenge when it is adjudicated, Wheeler pointed out that in remanding the FCC's 2010 Open Internet order, the D.C. federal appeals court had signaled that Title II would be one way of repairing the legal justification of the rules.
"The D.C. Circuit sent the previous Internet order back to us and basically said that [we] were trying to impose common carrier-like regulation without saying that tese are common carriers. We have addressed that issue."
The court said that the FCC's absolute ban on unreasonable discrimination was too much like a common carrier reg. Initially Wheeler was going to address that by turning that ban into allowing only commercially reasonable discrimination, which was in essence a change in language and emphasis to get at the same thing. But he has since said he thought that approach could be interpreted as what was commercially reasonable for ISPs rather than consumers.
Instead, he decided to treat ISPs as common carriers under Title II to justify bans on throttling and blocking and paid prioritization. He said that approach "gives him great confidence going forward."
CNN Plus Taps Poppy Harlow to Host Biz Interview Show 'Boss Files'
Sen. Roger Wicker Seeks Better NTIA-FCC Cooperation
Hearst Media Production Group is Litton Entertainment's New Name
HBO Max Sets 20% Discount Across Tiers
'Welcome to Flatch' Premieres on Fox March 17


United Kingdom-based petroleum exploration and production company Enquest said it had completed the acquisition of the remaining 75% stake in Magnus, an additional 9.0% stake in the Sullom Voe Oil terminal and supply facility and additional interests in associated infrastructure from BP.
The Magnus transaction, as previously announced, is expected to add approximately 60 MMboe of 2P reserves (equating to approximately 30 percent of the Group's reserve base) as at 1 January 2018 and approximately 10 MMboe of 2C resources (equating to approximately 6 per cent of the Group's resource base) as at 1 January 2018.
The oil and gas production and development company listed on the London and Stockholm stock exchanges said in a press release that it will add $500 million of net present value to the Group at $70/bbl long-term oil price, as outlined in the Competent Person's Report on Magnus included in the Prospectus dated 7 September 2018.
Following the transaction, EnQuest now had a 100% equity stake in Magnus, 15.1% in Sullom Voe Oil terminal and supply facility, 18.0% in the Ninian Pipeline System and 41.9% in the Northern Leg Gas Pipeline, the company added.
EnQuest CEO, Amjad Bseisu, said: "We have completed the acquisition of Magnus, the Sullom Voe Terminal and associated infrastructure from BP. These assets are a strong strategic fit for EnQuest to which we can apply our life extension expertise and deliver value for all our stakeholders.
"The addition of Magnus is expected to add material production and cash flow from the addition of significant low-cost 2P reserves. These cash flows will help facilitate the planned reductions in the Group's debt," he said.


A fairly settled and mostly dry working week is expected over Gauteng, Mpumalanga, Limpopo, N parts of KZN and E parts of the North West until the weekend. Over the weekend thunderstorm activity is expected to become more widespread, particularly over the Limpopo and E parts of the North West.
With settled conditions expected over N parts of South Africa until the weekend, unsettled and wetter conditions are likely over E parts of the Northern Cape, E parts of the Eastern Cape, S parts of KZN, W parts of the North West and the whole of the Free State. Over these areas afternoon thunderstorms are likely to be quite widespread at times and total accumulated rainfall totals could reach 80-100mm – particularly over Lesotho, S parts of KZN and SE parts of the Free State. The majority of this rain is likely to fall between today (Monday) and Wednesday.
In the far W of the country, thunderstorms are also expected over W parts of the Northern Cape and the majority of the Western Cape until Thursday, unfortunately although there could be some thunder and lightning at times, no significant rainfall is expected at this stage over the drought stricken SW Cape.
Over the weekend a weak cold front is likely to pass through the SW Cape, as this moves up the coast it should cause any thunderstorm activity to ease over central parts with thunderstorms becoming confined to NE parts of the country. As the cooler air collides with humid air over KZN, there is the risk of some energetic thunderstorms, particularly over W and central areas of the province.
Temperatures are expected to be around normal for the time of the year although it will be hot along the West Coast today and tomorrow and also over the SW Cape tomorrow and over the Mpumalanga lowveld on Thursday and Friday.
Winds are expected to be mostly moderate although a strong SE is expected in Cape Town today, spreading to the Garden Route and Eastern Cape coastline tomorrow. Strong winds are then expected to re-develop along the KZN coastline over the weekend.


In HTML, this tag indicates that text should be strong. In most browsers, this logical style maps the the physical style of bold. To use it, put <strong> before the start of text you want to be "strong" and put </strong> at the end of what you want to be "strong".
Note that this tag is not case-sensitive, so you can use strong or STRONG.
This tag replaces the deprecated <b> tag.
%%%%      "%%"   %%%%%%"""  %%%%        %%%"
A combination of the radical for insect (at lower right) and an old Chinese character that means big/strong, as represented by a the radical meaning bow (as in bow and arrow) and the radical above insect.
The bow acts phonetically to express the concept of piercing, and also lends to the idea of strength. Thus all together, big, strong insect that pierces, which itself is actually a reference to the horsefly. This came to represent strength, possibly via an intermediate meaning of persistent.
KYOU, GOU: strength, might; strong person.
tsuyo(i): strong, powerful, mihgty, robust, vigorous, healthy; brave, courageous; severe, intense; durable, solid.
tsuyo(meru): strengthen, intensify, invigorate; confirm; emphasize, increase, redouble.
tsyo(gari): bluff, show of strength.
shi(iru): force, coerce, contrain, compel.
shi(ite): forcibly, against one's will.
-kyou: a litle over, and a fraction.
I like it! 1 C!
Having active physical power, or great physical power to act; having a power of exerting great bodily force; vigorous.
That our oxen may be strong to labor. Ps. cxliv. 14.
Orses the strong to greater strength must yield. Dryden.
Having passive physical power; having ability to bear or endure; firm; hale; sound; robust; as, a strong constitution; strong health.
Solid; tough; not easily broken or injured; able to withstand violence; able to sustain attacks; not easily subdued or taken; as, a strong beam; a strong rock; a strong fortress or town.
Having great military or naval force; powerful; as, a strong army or fleet; a nation strong at sea.
Having great wealth, means, or resources; as, a strong house, or company of merchants.
Reaching a certain degree or limit in respect to strength or numbers; as, an army ten thousand strong.
Moving with rapidity or force; violent; forcible; impetuous; as, a strong current of water or wind; the wind was strong from the northeast; a strong tide.
Adapted to make a deep or effectual impression on the mind or imagination; striking or superior of the kind; powerful; forcible; cogent; as, a strong argument; strong reasons; strong evidence; a strong example; strong language.
Ardent; eager; zealous; earnestly engaged; as, a strong partisan; a strong Whig or Tory.
Her mother, ever strong against that match. Shak.
Having virtues of great efficacy; or, having a particular quality in a great degree; as, a strong powder or tincture; a strong decoction; strong tea or coffee.
Full of spirit; containing a large proportion of alcohol; intoxicating; as, strong liquors.
Affecting any sense powerfully; as, strong light, colors, etc.; a strong flavor of onions; a strong scent.
Solid; nourishing; as, strong meat.
Well established; firm; not easily overthrown or altered; as, a strong custom; a strong belief.
He had offered up prayers and supplications with strong crying and tears. Heb. v. 7.
Having great force, vigor, power, or the like, as the mind, intellect, or any faculty; as, a man of a strong mind, memory, judgment, or imagination.
I was stronger in prophecy than in criticism. Dryden.
Like her sweet voice is thy harmonious song, As high, as sweet, as easy, and as strong. E. Smith.
Tending to higher prices; rising; as, a strong market.
Pertaining to, or designating, a verb which forms its preterit (imperfect) by a variation in the root vowel, and the past participle (usually) by the addition of -en (with or without a change of the root vowel); as in the verbs strive, strove, striven; break, broke, broken; drink, drank, drunk. Opposed to weak, or regular. See Weak.
Applied to forms in Anglo-Saxon, etc., which retain the old declensional endings. In the Teutonic languages the vowel stems have held the original endings most firmly, and are called strong; the stems in -n are called weak other constant stems conform, or are irregular.
Strong conjugation Gram., the conjugation of a strong verb; -- called also old, ∨ irregular, conjugation, and distinguished from the weak, ∨ regular, conjugation.
⇒ Strong is often used in the formation of self-explaining compounds; as, strong-backed, strong-based, strong-bodied, strong-colored, strong-fisted, strong-handed, strong-ribbed, strong-smelling, strong-voiced, etc.
Syn. -- Vigorous; powerful; stout; solid; firm; hardy; muscular; forcible; cogent; valid. See Robust.


text:
Knee Exercise Quiz
by Erik Meira · March 10, 2020
Take this quiz to advance to the next lesson.
According to the 2012 meta-analysis by Naugle et al, which type of exercise had the greatest hypoalgesic effects?
Dynamic resistance
Besides intensity what is the other key factor for loading tendinopathies?
Time under tension
Eccentric contraction
Concentric contraction
Isometric contraction
As new scientific information becomes available through basic and clinical research, recommended treatments undergo changes. The authors have done everything possible to make this content accurate, up to date, and in accord with accepted standards at the time of publication. The authors are not responsible for errors or omissions or for consequences from the application of the file, and make no warranty, expressed or implied, in regard to the contents of the file. Any practice described in this file should be applied by the reader in accordance with professional standards of care used in regard to the unique circumstances that may apply in each situation.
© 2023 The Science PT. All Rights Reserved.


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Bullying and Harassment in the Workplace – What Can You Do?
Personal Injury Accredited Specialist
Bullying or harassment are rightly considered unacceptable practices in any part of society but what are your options for dealing with this behaviour if you experience it at your place of work?
In contemporary society, many – though certainly not all – employers and organisations will have detailed anti-bullying policies in place to deal with any instances of this behaviour. They do so because to ignore or fail to deal with bullying or harassing behaviour can leave them liable to later legal action for not providing a safe place to work.
There are a number of options you can take if you're experiencing bullying or harassment at work, ranging from bringing it to the attention of the national Fair Work Commission, SafeWork NSW, or consulting specialist compensation lawyers about seeking workers' compensation because you have sustained a psychological injury as a result of the behaviour. This article deals with the last circumstance.
What is bullying behaviour?
Bullying and harassment can take different forms and could constitute overt or less obvious behaviour. Essentially it involves verbal, physical, psychological or social abuse by a manager, colleague or another group of people at work.
Non-exhaustive examples of bullying behaviour include:
abusive or offensive language or comments directed at you;
aggressive and intimidating behaviour towards you;
belittling or humiliating comments directed at you;
sexual harassment, including unwanted touching;
practical jokes or initiation;
excluding you from workplace activities;
unjustified criticism or complaints.
The effects of bullying on a person are well documented. They can affect a person's confidence and self-esteem, causing anxiety, depression and sleeplessness. Importantly for employers, these effects can also lead to a significant loss of productivity and wellbeing in the employee.
What should you do if you experience bullying and harassment at work?
The first thing to do is to check whether your company or organisation has an anti-bullying policy and, if so, what is the procedure for reporting this type of behaviour.
Secondly, if you're confident and able to, you should directly inform the person responsible that you find their behaviour unacceptable and would like them to stop. You should also report the behaviour to an immediate supervisor, manager, health and safety or human resources officer, or a union representative.
Remember that employers have a duty to provide a workplace that protects the health and safety of their employees, including the psychological health of workers.
At this stage you may also want to consult a lawyer experienced in workplace and employment law issues to see what options you may have to stop the behaviour or seek redress for the damage it has caused to your life both at and outside of work.
Making a workers' compensation claim as a result of bullying
If you believe bullying in your workplace has resulted in you having a psychological or psychiatric injury, you may be able to apply for workers' compensation through the NSW scheme.
In order for a claim of this type to succeed you will need to provide medical evidence that supports your claim that you've suffered psychological or psychiatric injury as a result of workplace bullying. This evidence must distinguish your claim from those where an employee's distress is caused by what's called "reasonable management action" – such as having your work responsibilities changed, being transferred, demoted, disciplined, retrenched, dismissed or otherwise treated in a way that management can argue is reasonable.
Permanent impairment
In certain cases, a person may find they are unable to continue working due to the effects of serious workplace bullying. This may be said to cause permanent impairment, for which they may be eligible for a lump sum compensation payment. In order to prove this level of impairment, a specialist medical assessor – trained in the exact method used to decide how much your psychological injury has affected your life and ability to function – needs to confirm that you have a permanent primary psychological injury that has produced at least 15 per cent "whole person impairment".
It's important to note that if you succeed in obtaining a lump sum payment for permanent impairment, any weekly workers' compensation payments you may have been receiving will end.
If you believe your psychological or psychiatric condition has been caused by the negligence of your employer, work injury damages may also be available to cover past and future economic loss. Again, the 15 per cent whole person impairment threshold is used to determine this type of claim.
How can legal advice help?
Expert compensation lawyers such as the professionals at BPC will tell you that proving psychological injury as a result of workplace stress caused by something such as bullying is considerably more difficult than cases involving physical injury at work. But that's not to say they are unheard of and the advice of legal professionals with expertise in this area is usually the difference between successful and unsuccessful claims.
In short, it can be a confusing and complicated landscape to navigate, particularly if you're already experiencing anxiety and stress as a result of the bullying behaviour.
For this reason it's highly advisable to seek out the guidance and advice of a law firm with expertise in workplace compensation claims help restore your quality of life and move on from the unreasonable behaviour of other people at your place of work. Contact award-winning personal injury lawyers in Sydney today on 1800 431 572 for a free legal consultation.
Work Injury Claims by Self-Employed People
Unless you are in full-time regular employment, you may be unclear about whether you can make a workers' compensation claim. In Australia, if you are an employee, your employer is required to cover you for workers' compensation so that you can make a claim for loss of income, medical and rehabilitation expenses and depending upon… Read More »Work Injury Claims by Self-Employed People
Read More »Work Injury Claims by Self-Employed People
The Role of Social Media in Personal Injury Claims
There is little doubt that social media has changed the way we live and interact. Whether it's Facebook, Twitter, Instagram, TikTok or any other social media app, many people now interact with friends and family via these platforms more often than they do face-to-face. And while there are many benefits to the growth of social… Read More »The Role of Social Media in Personal Injury Claims
Read More »The Role of Social Media in Personal Injury Claims
Are You Unrepresented in a CTP Claim?
Claiming compensation after a motor vehicle accident can be a very complex process. The introduction of the Motor Accident Injuries Act 2017 ('MAI Act') in December 2017 (which replaced the former Motor Accident Compensation Act 1999) introduced a no-fault statutory compensation scheme covering medical treatment expenses and lost wages for a period of up to… Read More »Are You Unrepresented in a CTP Claim?
Read More »Are You Unrepresented in a CTP Claim?
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Category Archives: Jason Pominville Jersey
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Incorporating Pub Watch, Hotel Watch, Shop Watch & Business Watch Schemes.
Warwickshire Police and West Mercia Police send alerts, information and crime prevention messages relating to pubs, hotels, shops and businesses across the county. The aim of these messages is to allow business owners and staff to review their security measures, and be aware of current incidents and trends, so that they can better protect themselves, their staff and their property from harm.
Being a member of a watch scheme is a two way process. We will provide you with crime messages about incidents, which have occurred in your area and, in exchange, we would expect you to tell us about suspicious persons, vehicles or activity. Before almost every crime there is usually some suspicious activity. Therefore it is vital that you help us, to help you.
For more information regarding any of the above Watch Schemes, message categories, or local initiatives in your area, please contact your Safer Neighbourhood Team (SNT).


text:
\section{Introduction}

Many reaction processes in nature and engineering take place in a
moving fluid.  The creation of ozone in the atmosphere, the
nuclear explosion of a supernova, pattern formation in
morphogenesis, wild fires, and the gasoline transformation in an
internal combustion engine are just a few examples. The effect of
fluid flow on a reaction process can be profound, especially if
the flow is strong. The problem has been studied for many years by
engineers, physicists, and mathematicians alike.
One of the most extensively studied mathematical models of the
premixed reaction process is the reaction-diffusion equation and
systems. The advection-reaction-diffusion equation is given by
\begin{equation}\label{rd1}
T_t + Au\cdot\nabla T = \kappa\Delta T +  M f(T).
\end{equation}
Here $u(x)$ is the fluid flow, which we will assume is
time-independent and passive. The coefficient $A$ is the flow
strength parameter, $\kappa$ is the diffusivity, $M$ is the
reaction strength, and $f(T)$ is the reaction term. The function
$T(x,t)$ is normalized so that $0 \leq T(x,t) \leq 1,$ and can
stand for normalized temperature, the mass fraction of a reactant,
or a share of a population with a certain feature, depending on the problem.
Classical works of Fisher and Kolmogorov-Petrovski-Piskunov (KPP)
\cite{Fisher,KPP} first considered equation \eqref{rd1} in the dimension
$d=1$ and with $u=0$. These works modeled the propagation of an advantageous
gene in the population, and established existence and, in a certain sense,
stability of the traveling fronts. The extensions have occupied some of
the best efforts of mathematicians since. Equation \eqref{rd1} in several
spacial dimensions in the absence of advection ($u=0$) is by now fairly well
understood: the existence of traveling waves, stability, and asymptotic propagation properties have been
extensively studied (see, e.g. \cite{AW,freidlin-4,HN,VVV} where
many more references can be found). The effects of advection,
however, are very important in many situations in combustion and
more general chemical reactions \cite{P1,Wil,ZBLM} as well as in
biology and ecology \cite{Levin,Murray}.

One important phenomena that is observed in many situations is the
enhancement of the reaction rate by fluid motion. The physical
reason for this observed speed-up is believed to be that fluid
advection tends to increase the area available for reaction.
Recently, there has been significant progress in the mathematical
understanding of the reaction rate enhancement for several different
classes of flows
\cite{ACVV1,ACVV2,ABP,CKOR,CKR,HPS,KB,KBM,KS,KRS,KR,MS,NX,VCKRR}.
The papers \cite{Berrev,Xin2} provide excellent reviews of some of
these results and further references.

In this paper, our goal is to study reaction enhancement within a
framework of a model given by a coupled system of one-dimensional
advection-reaction-diffusion equations:
\begin{equation}\label{mainmodel}
(T_{j})_t + A_j (T_j)_x - (T_j)_{xx} =  T_j(1-T_j) +
\alpha(T_{j-1}-T_j) + \alpha(T_{j+1}-T_j),
\end{equation}
$j =1, \dots, N.$ We assume that the model is periodic in $j;$
that is, in the equation for $T_1(x),$ we replace $T_0(x)$ with
$T_N(x),$ and in the equation for $T_N(x),$ we put $T_1(x)$
instead of $T_{N+1}(x).$ The initial data always satisfies $0
\leq T_j(x,0) \leq 1,$ and it is a simple consequence of the
maximum principle that $T_j(x,t)$ remains within these bounds for
all times (see Section~\ref{aux}). Our reaction
term is the classical KPP \cite{KPP}, and
 for simplicity we set $\kappa =
M=1$ (these parameters are normalized by a simple rescaling of
time and space). We assume that $\sum_{j=1}^N A_j =0,$ which
corresponds to the mean zero flow. Any nonzero mean is taken into
account by switching to a moving system of coordinates; we are
interested in a non trivial effect of front stretching rather than
simple transfer with a constant speed. One can regard
\eqref{mainmodel} as a model for reaction in a layered fluid,
where different layers move with different speeds. The role of the
parameter $\alpha>0$ is to provide diffusive coupling between
different layers, and its meaning will be further discussed in
Section~\ref{main}. Our main interest is in the case of the large
coefficients $A_j$, where the influence of the drift is most
pronounced. We consider the front-like initial data where
$T_j(x,0) \rightarrow 0$ as $x \rightarrow \infty$ and $T_j(x,0)
\rightarrow 1$ as $x \rightarrow -\infty$ exponentially fast (we
will state the exact conditions in Section~\ref{aux}). We adopt
the following natural quantity (called the bulk burning rate in
\cite{CKOR}) as the main measure of the reaction rate:
\begin{equation}\label{bbrdef}
V(t) = \frac{1}{N}\sum\limits_{j=1}^N
\int\limits_{-\infty}^\infty (T_j)_t(x) \,dx.
\end{equation} One of
the results that we prove is as follows.

\begin{thm}\label{main1}
Assume that the initial data $T_j(x,0)$ are front like
\eqref{id1}.
Then there
exists a universal constant $C$ such that for any $\tau>0,$ we
have
\begin{equation}\label{lb1}
\frac{1}{\tau}\int\limits^{\tau}_{0}V(t) dt \geq
\frac{C}{N}\sum\limits_{j=1}^N|A_{j-1}-A_j|
\left[\frac{1}{\alpha\tau^2}+ \frac{1}{\alpha\tau}+\frac{1}{\tau}+
\alpha+1\right]^{-1}.
\end{equation}
\end{thm}

There are several noteworthy properties the lower bound
\eqref{lb1} possesses. Firstly, the bound is linear in the flow
strength. Secondly, the estimate clearly shows the need to wait
for a certain time $\tau$ before the front propagation (and thus
the reaction rate) stabilizes and the long-time lower bound is
achieved. If the coupling constant $\alpha$ is large, we have the
lower bound $CA \alpha^{-1}$ starting from times $\tau \approx 1.$
If $\alpha$ is small, the lower bound is $CA,$ starting from
times $\tau \approx \alpha^{-1}.$ We interpret these results
further in Sections~\ref{main} and \ref{num}. The numerical
experiments discussed in Section~\ref{num} show very good
qualitative agreement with \eqref{lb1}.

We remark that the result of Theorem~\ref{main1} is reminiscent of
the bounds proved in \cite{CKOR} for the two dimensional equation
\eqref{rd1} with a shear flow $u = (u(y),0).$ This is not
surprising since our system \eqref{mainmodel} can be regarded as a
discrete model of a shear flow. However, the proofs in the case of
model \eqref{mainmodel} are conceptually much more transparent,
the estimates are more precise, and the dependence on the
averaging time is more explicit.

The paper is organized as follows. In Section~\ref{aux} we collect
some background estimates which are needed in the proof of main
results. In particular, we prove a sharp version of a functional
inequality which was first discovered in \cite{CKOR}. The
inequality is somewhat reminiscent of the indeterminacy principle
of quantum mechanics. Although finding a sharp constant is not
particularly important for the key results of this paper, we feel
that the result is elegant (even if elementary) and may be of
independent interest. In Section~\ref{two} we discuss a simpler
two-layer case. In Section~\ref{main} we prove
Theorem~\ref{main1}, and some related results for a slightly more
general model. In Section~\ref{num} we describe the numerical
simulations.

\section{The Auxiliary Tools and The Optimal Constant}\label{aux}

We make our model slightly more general by associating a width
$h_j$ with each layer. By dimensionality of the discrete gradient
terms providing coupling of the neighboring layers in
\eqref{mainmodel}, the parameter $\alpha$ is made to depend on $j,$ with $\alpha_j =\kappa /h_j:$
\begin{equation}\label{mainmodel1}
(T_{j})_t + A_j (T_j)_x - (T_j)_{xx} =  T_j(1-T_j) +
\alpha_j(T_{j-1}-T_j) + \alpha_j(T_{j+1}-T_j).
\end{equation}
The parameter $\kappa$ now plays the role of diffusivity between the
layers. As opposed to the diffusivity in $x$ direction, this
diffusion coefficient cannot be set to one by a simple rescaling.
Our mean zero flow condition now reads $\sum_{j=1}^N h_jA_j =0.$ Set
$H=\sum_{j=1}^N h_j.$

We first prove a lemma on the preservation of spacial decay of the
solutions of \eqref{mainmodel1} that will allow us to manipulate
the equations, in particular integrate by parts.

\begin{lem}\label{decay}
Assume that $T_j(x,t),$ $j=1,\dots,N$ satisfy \eqref{mainmodel1}
and that the initial data $T_j(x,0)$ satisfy
\begin{equation}\label{id1}
T_j(x,0) < C_0 e^{-\lambda x}, \,\,\,\,1-T_j(x,0) < C_0 e^{\lambda
x},\,\,\,\,|(T_j(x,0))_{x}| < C_0 e^{-\lambda |x|}
\end{equation}
for all $j$ and some $\lambda>0.$ Assume that $c_1$ is such that
\begin{equation}\label{upperspeed}
c_1 \geq {\rm max}_j |A_j| +\lambda+ \lambda^{-1}(1+ 2{\rm max}_j
\alpha_j).
\end{equation}
Then for all $x,$ $t > 0$ we have
\begin{equation}\label{expdecay}
T_j(x,t) \leq C_0 e^{-\lambda(x-c_1t)}, \,\,\,1-T_j(x,t) < C_0
e^{\lambda (x+c_1t)},\,\,\,|(T_j(x,t))_{x}| < C_0 e^{\lambda
(|x|+c_1t)}.
\end{equation}
\end{lem}
\begin{proof}
Let us start by establishing the first bound in \eqref{expdecay};
the second bound can be proved by an identical argument. Set
$\phi(x,t) = e^{-\lambda (x-c_1t)},$ and $G_j(x,t) = C_0 \phi(x,t)
- T_j(x,t).$ A direct computation using \eqref{upperspeed} shows
that
\begin{equation}\label{phi} \phi_t + max_{j}|A_j| \phi_x -\phi_{xx}
-(1+2{\rm max}_j \alpha_j)\phi \geq 0 \end{equation} for any $j.$
Note that $G_j(x,0)>0$ in view of \eqref{id1}. On the contrary, assume
that the first bound in \eqref{expdecay} is not satisfied
and that $t_0>0$ is the first time such that there exists $j$ and
$x_0$ so that $G_j(x_0,t_0)=0.$ Combining \eqref{phi} with
\eqref{mainmodel1}, we find that
\begin{equation}\label{subsol}
(G_j)_t +A_j (G_j)_x -(G_j)_{xx} \geq 0
\end{equation} for all $x,$ $t \leq t_0.$ Now, in \eqref{subsol},
$G_j(x,0)>0$ for all $x$ and $G_j(x_0,t_0)=0$ contradict the
well-known maximum principle for parabolic PDE (see e.g.
\cite{Evans}). The proof of the second inequality in
\eqref{expdecay} is similar and is omitted. To prove the
bound for $(T_j)_x,$ differentiate \eqref{mainmodel1}; denoting
$H_j = (T_j)_x,$ we have
\[ (H_j)_t + A_j (H_j)_x - (H_j)_{xx} =H_j (1-2T_j) +
\alpha_j(H_{j-1}-H_j) + \alpha_{j+1}(H_{j+1}-H_j). \] Given that
$0 \leq T_j(x)\leq 1$ by a simple application of maximum
principle, the bounds for $(T_j)_x$ are now obtained in the same
manner as above.
\end{proof}

The bulk burning rate for the model \eqref{mainmodel1} is now
defined accordingly via
\begin{equation}\label{bbr1}
V(t) = \frac{1}{H}\sum\limits_{j=1}^N
h_j\int\limits_{-\infty}^\infty (T_j)_t(x) \,dx.
\end{equation}
Assuming the initial data satisfies \eqref{id1}, the indefinite
integrals are well-defined (integrating over $[-B,B]$ and taking
$B$ to infinity). As a consequence of \eqref{mainmodel1},
Lemma~\ref{decay}, and the condition $\sum_{j=1}^N h_j A_j =0,$ we also
have
\begin{equation}\label{bbr2}
V(t) = \frac1H\sum\limits_{j=1}^N h_j\int\limits_{-\infty}^\infty
T_j(x)(1-T_j(x)) \,dx.
\end{equation}
We continue by computing:
\begin{eqnarray}
V' (t) = \partial_t \frac1H\sum\limits_{j=1}^N
h_j\int\limits_{-\infty}^\infty T_j(x)(1-T_j(x)) \,dx =
\frac1H\sum\limits_{j=1}^N h_j\int\limits_{-\infty}^\infty
(T_j)_t(x)(1-2T_j(x)) \,dx \nonumber \\
= \frac1H\sum\limits_{j=1}^N h_j\int\limits_{-\infty}^\infty
\left(2((T_j)_{x})^2 + T_j(1-T_j)(1-2T_j(x)) +
\frac{2\kappa}{h_j}(T_j(x) - T_{j-1}(x))^2\right) \,dx \nonumber \\
\geq \frac1H\sum\limits_{j=1}^N h_j\int\limits_{-\infty}^\infty
\left(2((T_j)_{x})^2  + \frac{2\kappa}{h_j}(T_j(x) -
T_{j-1}(x))^2\right) \,dx - V(t) \label{keycontrol}.
\end{eqnarray}
In the second step above, we substitute $(T_j)_t(x)$ from
\eqref{mainmodel1}, integrate by parts, and rearrange the terms.
These manipulations are justified by Lemma~\ref{decay}. In the last
step we use the expression \eqref{bbr2} and the fact that $0 \leq
T_j(x) \leq 1.$

As a warm up before our main results, we prove the following
estimate.
\begin{thm}\label{general}
For any choice of $A_j$ such that $\sum_j h_j A_j =0,$ we have
\begin{equation}\label{genbound}
V(t)^2 \geq \frac{\pi^2}{16} + e^{-2t}\left( V(0)^2 -
\frac{\pi^2}{16} \right).
\end{equation}
\end{thm}
A key element in the proof is a general functional inequality.
\begin{prop}\label{fineq}
For any function $T(x) \in C^1(\mathbb R),$ $0 \leq T(x) \leq 1,$ such
that $T(x) \to 1$ as $x\to -\infty$ and $T(x) \to 0$ as $x\to
\infty,$ we have
\begin{equation}\label{funin}
\int\limits_{-\infty}^\infty
\left(T_x\right)^2dx\int\limits_{-\infty}^\infty T(1-T)dx \geq
\left(\frac{\pi}{8}\right)^2.
\end{equation}
The constant $\left(\frac{\pi}{8}\right)^2$ is sharp.
\end{prop}
\begin{proof}
We note that the inequality \eqref{funin} has been proved in
\cite{CKOR} with a weaker universal constant on the right hand
side. Using the Cauchy-Schwartz inequality we find that
\begin{eqnarray*}
\int\limits_{-\infty}^\infty\left(T_x\right)^2dx\int\limits_{-\infty}^\infty
T(1-T)dx &\geq& \left(\,\int\limits_{-\infty}^\infty
T_x\left(T(1-T)\right)^{1/2}dx\right)^2.
\end{eqnarray*}
Changing the variable and taking into account the asymptotic
behavior of $T(x)$, we have
\[\int\limits^{\infty}_{-\infty}T_x(T(x)(1-T(x)))^{1/2}dx =
\int\limits^{1}_{0}(T(1-T))^{1/2}dT.\] The latter integral is
computed explicitly by setting $T = \cos \theta$, and evaluates to
$\frac{\pi}{8}$. This proves \eqref{funin}. To show that this
constant is sharp, note that if $T_x = (T(1-T))^{1/2}$, then
\[\int\limits^{\infty}_{-\infty}\left(T_x\right)^2dx\int\limits^{\infty}_{-\infty}T(1-T)dx
= \left(\int\limits^{\infty}_{-\infty}T_x(T(1-T))^{1/2}dx\right)^2
= \left(\frac{\pi}{8}\right)^2. \qquad  \] We can solve for $T(x)$
explicitly and use the solution to construct an explicit function
satisfying the necessary asymptotic behavior for which the
equality in \eqref{funin} is achieved. In particular,

\[ T(x) = \left\{ \begin{array}{ll}
           1 &{\rm if} \,\,\, x \leq -\frac{\pi}{2} \\
\frac{1-\sin x}{2}
         & {\rm if} \,\,\, -\frac{\pi}{2}\leq x \leq \frac{\pi}{2} \\
           0         &{\rm if} \,\,\, \frac{\pi}{2} \leq
           x. \end{array} \right. \]
is one such function.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{general}]
From \eqref{keycontrol}, we see that
\[ V' (t)+V(t) \geq \frac2H\sum\limits_{j=1}^N h_j\int\limits_{-\infty}^\infty
((T_j)_{x})^2dx. \] From \eqref{id1} and Proposition~\ref{fineq}, we
derive
\begin{eqnarray*}
 V' (t)+V(t) \geq  \frac{\pi^2}{32H}
\sum\limits_{j=1}^N \frac{h_j}{\int\limits_{-\infty}^\infty
T_j(x)(1-T_j(x))\,dx} \\ \geq \frac{\pi^2H}{32} \left(
\sum\limits_{j=1}^N h_j\int\limits_{-\infty}^\infty
T_j(x)(1-T_j(x))\,dx \right)^{-1}  = \frac{\pi^2}{32V(t)}.
\end{eqnarray*}
On the second step we use Cauchy-Schwartz inequality.
Therefore, we have
\[ \frac{d}{dt} (V(t)^2) +2V(t)^2 \geq \frac{\pi^2}{16}. \]
The inequality \eqref{genbound} follows from the standard
application of the Gronwall lemma.
\end{proof}
Theorem~\ref{general} gives a universal lower bound on the
reaction rate, independent of the flow. One consequence is that in
the framework of our model, a mean zero flow cannot quench the
reaction. However, we are interested in the bounds that show
enhancement of the reaction. We will prove such bounds in the next
two sections.

\section{The Two-Layer Model}\label{two}

For the sake of clarity, we start with a simplified version of our
general model \eqref{mainmodel1} where only two layers of unit
width are involved. Namely, we look at a system
\begin{eqnarray}\label{twolayer}
(T_1)_t + A(T_1)_t - (T_1)_{xx} = T_1(1-T_1) + \kappa (T_2 - T_1)\\
(T_2)_t - A(T_2)_t - (T_2)_{xx} = T_2(1-T_2) + \kappa (T_1 - T_2)
\nonumber
\end{eqnarray}
with the initial data $T_{1,2}(x,0)$ satisfying \eqref{id1}. We
set $h_1 =h_2 =1$ in the definition \eqref{bbr1} of the bulk
burning rate. Throughout the rest of the paper we denote by $C$
different universal constants which enter the estimates.

\begin{thm}\label{2layer}
There exists a universal positive constant $C$ such that for any
$\tau>0,$ we have
\begin{equation}\label{lb2l}
\frac{1}{\tau}\int\limits^{\tau}_{0}V(t) dt \geq CA
\left[\frac{1}{\kappa\tau^2}+ \frac{1}{\kappa\tau}+\frac{1}{\tau}+
\kappa+1\right]^{-1}.
\end{equation}
\end{thm}
\noindent \it Remark. \rm Note that Theorem~\ref{2layer} is a
particular case of Theorem~\ref{main1} for $N=2.$ \\

\begin{proof}
We first integrate \eqref{twolayer} over all real axis. Given the
asymptotic behavior of $T_1$ and $T_2,$ \eqref{id1} and
Lemma~\ref{decay}, we obtain
\begin{eqnarray}\label{t1}
& \int_\mathbb{R} (T_1)_t dx -A = \int_\mathbb{R}
T_1(1-T_1)dx + \kappa \int_\mathbb{R}(T_2 - T_1)dx\\
\label{t2} & \int_\mathbb{R}(T_2)_t dx +A = \int_\mathbb{R}
T_2(1-T_2)dx + \kappa \int_\mathbb{R}(T_1 - T_2)dx.
\end{eqnarray}
\noindent Subtracting \eqref{t2} from \eqref{t1} and integrating
the result over $t \in [\tau_1,\tau_2]$, we find that
\begin{equation}
\begin{split}\label{2l3}
&\int_\mathbb{R} (T_1(\tau_2)-T_2(\tau_2))dx -
\int_\mathbb{R}(T_1(\tau_1)-T_2(\tau_1))dx \\+
&\int\limits^{\tau_2}_{\tau_1} \int_\mathbb{R}
(T_2(1-T_2)-T_1(1-T_1))dxdt \\
&+ 2\kappa \int\limits^{\tau_2}_{\tau_1}
\int_\mathbb{R}(T_1-T_2)dx dt = 2 A (\tau_2-\tau_1).
\end{split}
\end{equation}
We now claim the following Lemma.
\begin{lem}\label{keylem}
For any $t>0,$ we have
\begin{eqnarray}\label{keyest0}
\int_\mathbb{R} | T_1(x,t)-T_2(x,t) | dx  \leq  3
\left[\int_\mathbb{R}(T_1(x,t)-T_2(x,t))^2 dx + \right. \\ \left.
\int_\mathbb{R}
\left(T_1(x,t)(1-T_1(x,t))+T_2(x,t)(1-T_2(x,t))\right)dx \right]
\leq \frac{3}{\kappa} \left( V(t)+V^\prime(t)\right) + 3 V(t).
\nonumber
\end{eqnarray}
\end{lem}
\begin{proof}
\noindent We begin by proving the first inequality, which is valid
for any functions $T_1(x),$ $T_2(x) \in L^1(\mathbb{R}) \cap
L^2(\mathbb{R}):$
\begin{equation}\label{keyest}
\int_\mathbb{R} | T_1-T_2 | dx \leq 3
\left[\int_\mathbb{R}(T_1-T_2)^2 dx + \int_\mathbb{R}
(T_1(1-T_1)+T_2(1-T_2))dx\right].
\end{equation}
Consider two different cases. Denote $S$ the set of all $x$ such
that $|T_1(x) - T_2(x)| \geq 1/3.$ Then $|T_1(x)-T_2(x)| \leq
3|T_1(x)-T_2(x)|^2$ and \eqref{keyest} holds with $C=3$ if we
restrict the integration to set $S.$ Next, assume that $|T_1(x)
-T_2(x)| \leq 1/3.$ Note that if at least one of $T_1(x),$ $T_2(x)$
belongs to the interval $[1/3,2/3],$ then
$T_1(x)(1-T_1(x))+T_2(x)(1-T_2(x)) \geq 2/9.$ Consider the
alternative where both $T_1(x)$ and $T_2(x)$ lie in either $[0,1/3)$
or $(2/3,1].$ Note that the absolute value of the derivative of
function $x(1-x)$ satisfies $|(1-2x)| \geq 1/3$ in these intervals.
Then by the mean value theorem,
\[ T_1(x)(1-T_1(x))+T_2(x)(1-T_2(x)) \geq |T_1(x)(1-T_1(x))-
T_2(x)(1-T_2(x))| \geq \frac13
|T_1(x) -T_2(x)|. \] Therefore, if $x \in \mathbb{R} \setminus S,$
then
\[ |T_1(x)-T_2(x)| \leq 3 \left(
T_1(x)(1-T_1(x))+T_2(x)(1-T_2(x))\right). \] Combining the two
cases, we obtain \eqref{keyest}.

Now the second inequality in \eqref{keyest0} follows from the
definition \eqref{bbr1} of bulk burning rate, $V(t),$ and the
estimate \eqref{keycontrol} (recall that we set $h_{1,2}=1$).
\end{proof}

Given Lemma~\ref{keylem}, \eqref{2l3} implies that
\begin{equation}\label{bbrred}
\frac{3}{\kappa} [V^\prime(\tau_2)+ V^\prime(\tau_1)]
+(\frac{3}{\kappa}+9)[V(\tau_2)+V(\tau_1)] + (6\kappa + 7)
\int\limits^{\tau 2}_{\tau 1} V(t)dt \geq 2A (\tau_2-\tau_1).
\end{equation}

Let us now
apply the following averaging over $t\in[0,\tau]$ to the difference
of equations \eqref{t1} and \eqref{t2}:

\begin{equation}\label{aver1}
\frac{1}{\tau^3} \int\limits^{\frac{\tau}{4}}_{0}
\int\limits^{\frac{\tau}{4}+s}_{\frac{\tau}{4}-s}
\int\limits^{\frac{\tau}{2}+k}_{\frac{\tau}{2}-k} dt dk ds =
\frac{1}{\tau^3} \int\limits^{\tau}_{0}
G(\frac{\tau}{2},t-\frac{\tau}{2})dt
\end{equation}
\noindent where
\begin{equation}\label{averker}
G(h,\xi) = \left\{ \begin{array}{ll} \frac{1}{2}(h-|\xi|)^2 -
(\frac{h}{2}-|\xi|)^2 & {\rm if} \,\,
 0 \leq |\xi|\leq \frac{h}{2}\\
 \frac{1}{2}(h-|\xi|)^2 & {\rm if} \,\, \frac{h}{2}\leq
|\xi| \leq h.
\end{array} \right.
\end{equation}

\noindent
After the first integration, we obtain \eqref{2l3} and thus
\eqref{bbrred} with $\tau_1 = \tau-k,$ $\tau_2=\tau+k.$
\noindent Integrating twice more
leads to the following estimate:
\begin{eqnarray}\nonumber
\frac{3}{\kappa}\int\limits^{\tau}_{0}V(t)dt+\left(\frac{3}{\kappa}+9\right)
\int\limits^{\tau}_{0}V(t)h(t,\tau) dt  + \\  \label{2lf} (6\kappa
+7) \int\limits^{\tau}_{0} V(t)G(\frac{\tau}{2},t-\frac{\tau}{2})
dt \geq \frac{A\tau^3}{16}
\end{eqnarray}

\noindent where
\[ h(t,\tau) = \left\{ \begin{array}{ll} t
         & {\rm if} \,\,\, 0\leq t \leq \frac{\tau}{4} \\
           \frac{\tau}{4} &{\rm if} \,\,\, \frac{\tau}{4}\leq t \leq \frac{3\tau}{4} \\
           \tau-t         &{\rm if} \,\,\, \frac{3\tau}{4}\leq t \leq
           \tau. \end{array} \right. \]

\noindent Observe that $G(\frac{\tau}{2},t-\frac{\tau}{2}) \leq
\frac{\tau^2}{8}$ and that $h(t,\tau) \leq\frac{\tau}{4}$. Hence
\eqref{2lf} can be expressed as follows:
\[ \frac{1}{\tau}\int\limits^{\tau}_{0}V(t) dt \geq C
A\left[\frac{1}{\kappa\tau^2}+ \frac{1}{\kappa \tau}+
\frac{1}{\tau} + \kappa +1 \right]^{-1}. \]
This coincides with \eqref{lb2l}, thus proving the theorem. It is also
easy to estimate that the constant $C$ can be taken equal to
$1/48.$
\end{proof}

\section{The Multi-Layer Model}\label{main}

Here we prove a more general version of Theorem~\ref{main1}. Let
us introduce the notation
\[ R_1 =  1+ {\rm max}_j h_j^{-1} , \,\,\,R_2=  1+ {\rm
max}_j (h_j h_{j-1}^{-1}). \]

\begin{thm}\label{main2}
Assume that the initial data $T_j(x,0)$ for \eqref{mainmodel1} are
front like \eqref{id1}.
Then there exists a universal constant $C$ such that for any
$\tau>0,$ we have
\begin{equation}\label{lb2}
\frac{1}{\tau}\int\limits^{\tau}_{0}V(t) dt \geq
\frac{C}{HR_2}\sum\limits_{j=1}^N|A_{j-1}-A_j|h_j
\left[\frac{1}{\kappa\tau^2}+
\frac{1}{\kappa\tau}+\frac{R_1}{\tau}+ \kappa R_1+1\right]^{-1}.
\end{equation}
\end{thm}
\noindent \it Remark. \rm Theorem~\ref{main1} is a particular case
of Theorem~\ref{main2} when $h_j \equiv 1$ (with $\alpha$
substituted by $\kappa$). \\
\begin{proof}
Consider two neighboring layers, $j$ and $j-1.$ Assume without
loss of generality that $A_j \geq A_{j-1}.$ Subtracting the equation
for $T_j$ from the equation for $T_{j-1}$ and integrating, we find:
\begin{eqnarray}\label{diff1}
\int_\mathbb{R}((T_{j-1})_t-(T_j)_t)\, dx + A_j -A_{j-1} =
\int_\mathbb{R} \left( T_{j-1}(1-T_{j-1}) - T_j(1-T_j) \right)\,
dx + \\
\left(\frac{\kappa}{h_j}+\frac{\kappa}{h_{j-1}}\right)
\int_\mathbb{R}(T_{j}-T_{j-1}) \,dx +
\frac{\kappa}{h_j}\int_\mathbb{R}(T_{j}-T_{j+1}) \,dx +
\frac{\kappa}{h_{j-1}}\int_\mathbb{R}(T_{j-2}-T_{j-1}) \,dx.
\nonumber
\end{eqnarray}
Multiplying \eqref{diff1} by $h_j,$ and summing over $j=1, \dots, N$, we obtain:
\begin{eqnarray}
\begin{split}\label{diff2}
&\sum\limits_{j=1}^N (-1)^{\sigma_j}
h_j\int_\mathbb{R}(T_{j-1}-T_{j})_t
 dx + \sum\limits_{j=1}^N h_j |A_{j-1}-A_j| =\\
&-\sum\limits_{j=1}^N (-1)^{\sigma_j} h_j
\int_\mathbb{R}T_j(1-T_j) dx + \sum\limits_{j=1}^N (-1)^{\sigma_j}
h_{j-1}\int_\mathbb{R}T_{j-1}(1-T_{j-1})dx \frac{h_j}{h_{j-1}}\\
&+ \sum\limits_{j=1}^N (-1)^{\sigma_j} \kappa
\left[\left(1+\frac{h_j}{h_{j-1}}\right)\int_\mathbb{R}(T_{j}-T_{j-1})
dx + \int_\mathbb{R}(T_{j}-T_{j+1}) dx \right.
\\  &+ \left. \int_\mathbb{R}(T_{j-2}-T_{j-1}) dx
\frac{h_j}{h_{j-1}} \right], \\
\end{split}
\end{eqnarray}
where $\sigma_j =0$ if $A_j \geq A_{j-1}$ and $\sigma_j=1$
otherwise. Denote the last sum in \eqref{diff2} by $I_1.$ Using
Lemma~\ref{keylem} and \eqref{keycontrol}, we can estimate:
\begin{eqnarray}
\begin{split}\label{cruin}
& |I_1| \leq C  \kappa \sum\limits_{j=1}^N \left[
\int_\mathbb{R}(T_j-T_{j-1})^2 dx + \int_\mathbb{R}T_j(1-T_j)
\right]\left(1+\underset{j}{max}\left(\frac{h_j}{h_{j-1}}\right)\right)\\
& \leq CH \left( R_2  V^\prime(t) + \kappa V(t) R_1R_2 \right).
\end{split}
\end{eqnarray}
As in the proof of Theorem~\ref{2layer}, we average in $t \in [0,\tau]$ according to \eqref{aver1} and apply
Lemma~\ref{keylem} to estimate the expressions arising from the
time derivatives in \eqref{diff2} in terms of $V$ and $V'$. Taking
into account \eqref{cruin}, after the first integration we obtain,
with $\tau_1 = \frac\tau2-k$ and $\tau_2 = \frac\tau2+k,$
\begin{eqnarray*}
\sum\limits_{j=1}^N (-1)^{\sigma_j} h_j
\left[\int_\mathbb{R}(T_{j-1}(\tau_2)-T_{j}(\tau_2))
 dx-\int_\mathbb{R}(T_{j-1}(\tau_1)-T_{j}(\tau_1))
 dx \right] + \\   (\tau_2-\tau_1)\sum\limits_{j=1}^N h_j |A_{j-1}-A_j| \leq
CHR_2 \left(   V(\tau_2)-V(\tau_1) + (\kappa R_1+1)
\int\limits_{\tau_1}^{\tau_2} V(t)\,dt \right).
\end{eqnarray*}
Applying Lemma~\ref{keylem} once again to the expression on the
left hand side, and integrating twice more, we obtain
\begin{eqnarray*}
\frac{\tau^3}{H}\sum\limits_{j=1}^N h_j |A_{j-1}-A_j| \leq CR_2
\left[ (\kappa R_1+1) \int\limits_0^\tau V(t)G(\frac\tau2, t-
\frac\tau2)\,dt + \right. \\ \left. (\kappa^{-1}+R_1)
\int\limits_0^\tau V(t) h(t,\tau)\,dt + \kappa^{-1}
\int\limits_0^\tau V(t)\,dt \right].
\end{eqnarray*}
Taking into account the bounds on functions $G$ and $h,$ we arrive
at
\begin{equation}\label{fin1}
\frac1\tau\int\limits_0^\tau V(t)\,dt \geq
\frac{C}{HR_2}\sum\limits_{j=1}^N h_j |A_{j-1}-A_j| \left( 1+
\kappa R_1 + \frac{1}{\kappa \tau^2} +\frac{1}{\kappa \tau} +
\frac{R_1}{\tau} \right)^{-1}.
\end{equation}
This completes the proof of Theorem~\ref{main2}. In the situation
where all $h_j \equiv 1,$ and hence $R_1=R_2=2,$ we obtain Theorem~\ref{main1} exactly.
In general, we see that small $h_j$ (which translate to the large coupling constants $\alpha_j$),
or strongly non-uniform size distributions of the neighboring
$h_j,$ weaken the lower bound.

\end{proof}

\section{Numerical Simulations}\label{num}

We carried out numerical simulations for the two layer system
\eqref{twolayer}. Our main interest was in finding the
reaction propagation rate in the established regime, depending on
$A$ and $\kappa.$ Our results are obtained using the
Crank-Nicholson implicit scheme.  The implementation features two
noteworthy modifications. Firstly, the transient wave that occurs
between the initial condition and the eventual steady wavefront
shape was ignored when calculating the front speed.  This was
achieved by using a "$t_{transient}$" parameter, and then ignoring
the wave speed for times $0$ to $t_{transient}$.  Secondly, we
employed a computational trick to expedite simulation.  In a naive
implementation, we would apply the Crank-Nicholson scheme to the
entire field of points that the wave travels through.  Since
the matrices involved in the computation are $n \times n$ where
$n$ is the number of points being considered, considering the
entire field of points that the wave passes through can be
computationally expensive. We avoided this by only applying
Crank-Nicholson to points in the immediate neighborhood of the
front itself.  Then after every iteration we adjusted our
calculation frame to stay centered on the wavefront.

\begin{figure}
\begin{center}
\scalebox{0.75}{\includegraphics{SpeedVsAdv.eps}} \caption{The
bulk burning rate versus flow strength $A$ for different values of
the coupling $\kappa$} \label{lin}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\scalebox{0.75}{\includegraphics{Slopes.eps}} \caption{The slopes
corresponding to different values of $\kappa$} \label{slopes}
\end{center}
\end{figure}

We found clear linear dependence on the flow strength parameter
$A.$ The Figure~\ref{lin} shows this linear dependence for
different values of the coupling constant $\kappa.$ The constant
$\kappa$ varies from $0.1$ to $1$ with $0.1$ step. The slope
of the graph is monotone decreasing in $\kappa,$ so that the
steeper slopes on Figure~\ref{lin} correspond to smaller values of
$\kappa.$ The intuitive explanation for this effect is that
stronger diffusion between layers mollifies the front stretching
produced by the flow, thus reducing the reaction zone. The graph
on Figure~\ref{slopes} shows the dependence of slope on $\kappa.$
The shape of the graph is in good qualitative agreement with the
bound of Theorem~\ref{2layer}. We note that the results of our
simulations are similar to the results of \cite{VCKRR}, where
combustion in a shear flow was studied in a full PDE setting by
means of a more sophisticated numerical scheme.

{\bf Acknowledgments} This work has been a part of Collaborative
Undergraduate Research Lab (CURL) at the University of Wisconsin
during the 2004-2005 academic year. Support of the NSF VIGRE grant
is gratefully acknowledged.  We thank Carl Edquist, Morgan
Franklin, Jeremy Jacobson, Paul Heideman, Julie Mitchell and John
Vano for interesting discussions and help with this project. We
are especially indebted to Paul Milewski since without his
assistance with the code, the numerical simulations would be still
running. The work of LRM, DA and EB was supported in part by the
NSF VIGRE grant. The work of AK has been partially supported by
the Alfred P. Sloan Research Fellowship and NSF-DMS grant 0314129.

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Once again this proves how out of touch some of you guys are!
Obama is pretty much promising the illegals everything they demand -- but he simply cannot carry through with their demands for unlimited immigration and do anything about jobs for Americans -- no man can serve two masters.
By putting illegals first and foremost, he's putting Americans behind them. He's promised far too much to the illegals and they're expecting he come through on everything for them now.
So being "in touch" means erasing our immigration laws and rewarding those who break them? Is this how far down the road of anarchy that this country has gone? Is this a proud moment for our country? I don't think so!
A poll recently revealed that 69% of Hispanics think that illegal immigrants should be able to work in this country. Why is that with 23 million Americans out of work? Why have any immigration laws at all then? Or is that their desired result? When is Mexico going to erase its southern border so that illegal immigrants can flood their border?
Hispanic, and Hispanic citizens did vote in a block for Obama. But illegals have a different agenda then citizens do.
The first "amnesty" law (IRCA) brought in 2.8 million illegals. And was signed by Ronald Reagan. As long as the Democratic party considers unions its supporters, it would be difficult for Obama to push through Congress and sign another amnesty.
Break it down by ethnicity by who wants amnesty and who don't. There's your answer IMHO. Tho there are some greedy anglo whites who like their cheap labor.
The majority of Hispanic citizrens have the same agenda as illegal immigrants do and that's just a fact.
Americans are nice people. But you're completely out of touch if you really think we want to hand our citizenship and access to our welfare system to poorly educated foreign nationals who don't speak English and don't pay taxes. When Americans think illegals they tend to think nice guy who works hard, speaks English and does not demand special privileges based on race. They are ignorant that the reality is nothing of the sort.
I wouldn't use Arizona as a barometer of anything. We have too many nuts in this state to get a good reading on politics nationally.
One of the biggest union supporters of the Democratic Party is SEIU, and they have no problem supporting illegal aliens. In fact, they admit to having huge numbers of illegal alien members among their majority "immigrant" membership.
Listen from :55. . . . .
The same union leader pushing for amnesty. . .


text:
Ekesjön kan syfta på ett antal insjöar i Sverige:

Västra Ekesjön, sjö i Olofströms kommun,
Östra Ekesjön, sjö i Olofströms kommun,
Ekesjön, Halland, sjö i Falkenbergs kommun,  (3,8 ha)
Ekesjön (Herrljunga socken, Västergötland), sjö i Herrljunga kommun,
Ekesjön (Redslareds socken, Västergötland), sjö i Svenljunga kommun,  (6,86 ha)

Listor över Sveriges insjöar baserat på namn

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